ContentsIntroduction
Introduction: Sample Problem Solving Lesson
Mental Math
Listing of Worksheet Titles
Patterns & Algebra Teacher’s Guide Workbook 3:1
BLACKLINE MASTERS Workbook 3 - Patterns & Algebra, Part 1
Number Sense Teacher’s Guide Workbook 3:1
BLACKLINE MASTERS Workbook 3 - Number Sense, Part 1
Measurement Teacher’s Guide Workbook 3:1
BLACKLINE MASTERS Workbook 3 - Measurement, Part 1
Probability & Data Management Teacher’s Guide Workbook 3:1
BLACKLINE MASTERS Workbook 3 - Probability & Data Management, Part 1
Geometry Teacher’s Guide Workbook 3:1
BLACKLINE MASTERS Workbook 3 - Geometry, Part 1
Patterns & Algebra Teacher’s Guide Workbook 3:2
BLACKLINE MASTERS Workbook 3 - Patterns & Algebra, Part 2
Number Sense Teacher’s Guide Workbook 3:2
BLACKLINE MASTERS Workbook 3 - Number Sense, Part 2
Measurement Teacher’s Guide Workbook 3:2
BLACKLINE MASTERS Workbook 3 - Measurement, Part 2
Probability & Data Management Teacher’s Guide Workbook 3:2
BLACKLINE MASTERS Workbook 3 - Probability & Data Management, Part 2
Geometry Teacher’s Guide Workbook 3:2
BLACKLINE MASTERS Workbook 3 - Geometry, Part 2
JUMPMathTeacher’s Guide: Workbook 3
Copyright © 2008, 2010 JUMP Math
All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without written permission from the publisher, or expressly indicated on the page with the inclusion of a copyright notice.
JUMP MathToronto, Ontariowww.jumpmath.org
Writers: Dr. John Mighton, Dr. Sindi Sabourin, Dr. Anna KlebanovConsultant: Jennifer WyattCover Design: BlakeleySpecial thanks to the design and layout team.Cover Photograph: © iStockphoto.com/Michael Kemter
ISBN: 978-1-897120-70-5
Printed and bound in Canada
1Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
1. Key Components of the JUMP Math Resources
Student Workbooks Parts 1 and 2, Grades 1–8 - Solid foundation for each of the strands in the curriculum at grade level - Extensive review going back up to two grades - All strands complete the curriculum at grade level
Teacher’s Guides, Grades 1–8Overview of JUMP Math Mental Math Unit Detailed Table of Contents (Parts 1 and 2) Lesson plans provide clear explanations and explicit guidance on how to - Introduce one concept at a time - Explore concepts and make connections in a variety of ways - Assess students quickly - Extend learning with extra bonus questions and activities - Develop problem solving skills - Support material for each strand
BLMs (extra worksheets, games, manipulatives, etc.) Answer Keys (for Workbooks and Unit Tests), Grades 3–8 Unit Tests, Grades 3–8 Curriculum Correlations (WNCP, ON)
2. The Myth of AbilityThere is a prevalent myth in our society that some people are born with mathematical talent—and others simply do not have the ability to succeed.
Recent discoveries in cognitive science are challenging this myth of ability. The brain is not hard-wired; it continues to change and develop throughout life. Steady, incremental learning can result in the emergence of new abilities. The brain, even when damaged, is able to rewire itself and learn new functions through rigorous instruction. As Philip E. Ross points out in his 2006 Scientific American article “The Expert Mind,” this fact has profound implications for education:
The preponderance of psychological evidence indicates that experts are made not born. What is more, the demonstrated ability to turn a child quickly into an expert—in chess, music, and a host of other subjects—sets a clear challenge before the schools. Can educators find ways to encourage the kind of effortful study that will improve their reading and math skills? Instead of perpetually pondering the question, “Why can’t Johnny read?” perhaps educators should ask, “Why should there be anything in the world that he can’t learn to do?”1
JUMP Math builds on the belief that every child can be successful at mathematics by • promoting positive learning environments and building confidence through praise and encouragement; • maintaining a balanced approach to mathematics by concurrently addressing conceptual and procedural learning, explicit and inquiry-based learning; • achieving understanding and mastery by breaking mathematics down into sequential, scaffolded steps, while still allowing students to make discoveries;
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• keeping all students engaged and attentive by “raising the bar” incrementally; • guiding students strategically to explore and discover the beauty of mathematics as a symbolic language connected to the real world; and, • using continuous assessment to ensure all students are engaged and none are left behind.
JUMP Math is an approach to teaching mathematics that has been developed by John Mighton and a team of mathematicians and educators who are dedicated to excellence in mathematics teaching and learning.
3. About JUMP MathNine years ago I was looking for a way to give something back to my local community. It occurred to me that I should try to help kids who needed help with math. Mathematicians don’t always make the best teachers because mathematics has become obvious to them; they can have trouble seeing why their students are having trouble. But because I had struggled with math myself, I wasn’t inclined to blame my students if they couldn’t move forward. — John Mighton2
John Mighton is a mathematician, bestselling author, award-winning playwright, and the founder of JUMP Math, a national charity dedicated to improving mathematical literacy.
JUMP Math grew out of John’s work with a core group of volunteers in a “tutoring club” held in his apartment to meet the needs of the most challenged students from local schools. Over three years, John developed the early material—simple handouts for the tutors to use. This period was one of experimentation in developing the JUMP Math method through countless hours of one-on-one tutoring. Eventually, John began to work in local inner-city schools, placing tutors in classrooms. This led to the next period of innovation—working through the JUMP Math method in classrooms.
Teachers responded enthusiastically to their students’ success and wanted to adapt the method for classroom use. John and a group of volunteers and teachers developed workbooks to meet teachers’ needs for curriculum-based resources. These started out as a series of three remedial books with limited accompanying teacher materials, released in fall of 2003. The effectiveness of these workbooks led quickly to the development of grade-specific, curriculum-based workbooks and teacher’s guides, first released in 2004.
John documented his experience in two national bestselling books, The Myth of Ability (2003) and The End of Ignorance (2007). As a playwright, he has won several national awards, including the Governor General’s Literary Award for Drama, the Dora Award, the Chalmers Award, and the Siminovitch Prize. John was granted a prestigious Ashoka Fellowship as a social entrepreneur for his work in fostering mathematical literacy by building students’ self-confidence and competence through JUMP Math. In 2010 John was appointed an Officer of the Order of Canada.
In only ten years, JUMP Math has grown from a gathering around John’s kitchen table to a thriving organization reaching more than 50,000 students with high-quality learning resources and training for 3,000 teachers. JUMP Math is working in hundreds of schools across Canada and internation-ally, and has established a network of dedicated teachers who are mentoring and training teachers new to the program. As well, JUMP Math supports community organizations in reaching struggling students through homework clubs and after-school programs. Through the generous support of our sponsors, JUMP Math donates learning resources to classrooms and homework clubs across Canada. JUMP Math has inspired thousands of community volunteers and teachers to reach out to struggling students by donating their time as tutors, mentors, and trainers.
Introduction
3Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
4. JUMP Math WorksJUMP Math is a learning organization committed to evaluation and evidence-based practice. JUMP Math is a leader in encouraging and supporting third-party research to study the efficacy of the JUMP Math approach to mathematics teaching and learning.
Hospital for Sick Children 2008–2010Cognitive scientists from The Hospital for Sick Children in Toronto conducted a randomized-control-led study of the effectiveness of the JUMP Math program. Studies of such scientific rigour remain relatively rare in mathematics education research in North America. The results showed that, on well-established measures of math achievement, students who received JUMP instruction outper-formed students who received the methods of instruction their teachers would normally use.
Ontario Institute for Studies in Education (OISE) 2007–2008Researchers from OISE at the University of Toronto, led by Dr. Joan Moss, completed a one-year study on the efficacy of JUMP Math. Preliminary data indicate that • JUMP Math’s Grade 5 resources for the curriculum in multiplication provide a greater variety of representations of concepts and more practice than provincially recommended programs; and • JUMP Math significantly improves the conceptual understanding of math for struggling students.
Vancouver School Board 2006–2007Over the school year, 68 JUMP Math teachers were surveyed. The teachers indicated that • the JUMP Math methodology enhances student retention and transfer, promotes independent thinking, and creates excitement and curiosity; and • JUMP Math develops teacher confidence and self-efficacy.
Borough of Lambeth (London, UK) 2006–2009During the summer of 2006, 24 public schools in Lambeth participated in a pilot study on JUMP Math. As a result of the pilot’s impact on student behaviour, confidence, and achievement—as well as teachers’ strong reaction to JUMP Math as an effective teaching tool—a total of 35 local schools adopted the program for the 2006–2007 school year. In this second implementation, 69% of the students who initially performed two years below age-related expectations moved up multiple levels after using JUMP Math, and either reached or surpassed their desired level by the end of the school year. In 2008–2009, it was shown that 57% of students who were initially at grade level progressed three years ahead of national expectations for their age.
JUMP Math Evaluation Pilot 2007–2008After using JUMP Math as their exclusive math program during a 5-month pilot study, a class of Grade 3 students in British Columbia showed • an increase in math achievement equivalent to 9 months of instruction (80% more than the expected achievement in 5 months); and • a statistically significant decrease in math anxiety and a statistically significant increase in positive attitude toward math.
In 2010, JUMP Math was recommended by the Canadian Language and Literacy Research Network and the Canadian Child Care Federation in a report entitled “Foundations for Numeracy: An Evidence-Based Toolkit for the Effective Mathematics Teacher.”3
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In 2009, JUMP Math was recognized by the UK Government in the document “What Works for Children with Mathematical Difficulties?”4
See the JUMP Math website, www.jumpmath.org, for more details and updates on research.
5. What Is the JUMP Math Approach?JUMP Math is a balanced approach to teaching mathematics that supports differentiated instruc-tion. JUMP Math covers the full curriculum for both Ontario and Western Canada through student workbooks, teacher’s guides, and a range of support materials.
The JUMP Math student workbooks are not intended to be used without instruction. Teachers should use the workbooks and accompanying lesson plans in the teacher’s guides for dynamic lessons in which students are allowed to discover and explore ideas on their own. The careful scaffolding of the mathematics in the student workbooks make them an excellent tool for teachers to use for guided practice and continuous assessment.
The JUMP Math approach to teaching mathematics emphasizes: • Confidence-building • Guided practice • Guided discovery • Continuous assessment • Rigorously scaffolded instruction • Mental math • Deep conceptual understanding
Confidence-building JUMP Math recognizes that math anxiety is a significant barrier to learning for many students. The JUMP approach has been shown to reduce math anxiety by building on success in small steps. Raising the bar incrementally—a key component of the JUMP Math materials—encourages engagement and confidence. The research in cognition that shows the brain is plastic also shows that the brain can’t register the effects of training if it is not attentive. However, a child’s brain can’t be truly attentive unless the child is confident and excited and believes that there is a point in being engaged in the work. When students who are struggling become convinced that they cannot keep up with the rest of the class, their brains begin to work less efficiently, as they are never attentive enough to consolidate new skills or develop new neural pathways. That is why it is so important to give students the skills they need to take part in lessons and to give them opportunities to show off by answering questions in front of their classmates. To do this, try to constantly assess what strug-gling students know.
Guided practiceResearch in psychology has shown that our brains are extremely fallible: our working memories are poor, we are easily overwhelmed by too much new information, and we require a good deal of practice to consolidate skills and concepts. Repetition and practice are essential. Even mathemati-cians need constant practice to consolidate and remember skills and concepts. New research in cognition shows how important it is to practise and build component skills before students can understand the big picture. The workbooks are designed to provide guided practice when used with the lesson plans. The multiple representations of mathematics in JUMP Math combine with guided practice in small steps to promote mastery and understanding of key concepts.
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5Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Guided discoveryIn “Students Need Challenge, Not Easy Success,”5 Margaret Clifford makes the case that students benefit when they are given prompt, specific feedback on their work and when they are allowed to take “moderate risks.” JUMP Math provides moderate risks by providing tasks that are within students’ grasp. As students’ confidence grows, their risk tolerance grows as well, and they are ready to take more and more steps by themselves.
The lesson plans in the JUMP Math teacher’s guides show you how to build lessons around the material in the workbooks by creating tasks and questions similar to the ones on the worksheets. As much as possible, when students are ready, allow them to think about and work on these questions and tasks independently rather than teaching them explicitly. When you feel your students have sufficient confidence and the necessary basic skills, let them explore more challeng-ing or open-ended questions.
Students are more likely to become flexible and independent thinkers in math if you guide discovery through well-designed lessons. Hence, in creating discovery-based lessons it is important to balance independent work with practice and explicit hints and instruction. According to Philip E. Ross,6 research in cognition shows that to become an expert in a game like chess it is not enough to play without guidance or instruction. The kind of training in which chess experts engage, which includes playing small sets of moves over and over, memorizing positions, and studying the techniques of master players, appears to play a greater role in the development of ability than the actual playing of the game.
Continuous assessment JUMP Math workbooks are designed to allow for continuous formative assessment—the books show teachers how to break material into steps and assess component skills and concepts in every area of the curriculum.
The point of constantly assigning tasks and quizzes is not to rank students or to encourage them to work harder by making them feel inadequate. Quizzes should instead be treated as opportuni-ties for students to show off what they know, to become more engaged in their work by meeting incremental challenges, and to experience the collective excitement that can sweep through a class when students experience success together. Continuous assessment allows the teacher to differentiate instruction with small individual interventions.
Rigorously scaffolded instruction Consistent with emerging brain research, JUMP Math provides materials and methods that minimize differences among students, allowing teachers to more effectively improve student performance in mathematics. In “Why Minimal Guidance During Instruction Does Not Work,” Paul Kirschner, John Sweller, and Richard Clark argue that evidence from controlled studies almost uniformly supports direct, strong instructional guidance.7 Even for students with considerable prior knowledge, strong guidance while learning helps take into account the limitations of a student’s working memory: the mind can only retain so much of new information or so many component steps at one time.
Even in discovery-based lessons, in which there is little direct instruction, it is important to introduce new ideas through a series of well-designed tasks and explorations in which each new concept follows from the last; students are more likely to make discoveries if the progression of ideas makes sense to them and does not overwhelm them.
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6 Teacher’s Guide for Workbook 3COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Mental mathMental math is the foundation for all further study in mathematics. Students who cannot see number patterns often become frustrated and disillusioned with their work. Consistent practice in mental math allows students to become familiar with the way numbers interact, enabling them to make simple calculations quickly and effectively without always having to recall their number facts.
To solve problems, students must be able to see patterns in numbers and make estimates and predictions about numbers. It is a serious mistake to think that students who don’t know number facts can get by in mathematics using a calculator or other aids. Students can certainly perform operations and produce numbers on a calculator, but unless they have number sense, they will not be able to tell if their answers are correct, nor be able to develop a talent for solving mathematical problems.
Deep conceptual understanding JUMP Math scaffolds mathematical concepts rigorously and completely. JUMP Math materials were designed by a team of mathematicians and educators who have a deep understanding of, and a love for, mathematics. JUMP Math teaches symbolic and concrete understanding simultaneously, using a variety of approaches. JUMP Math materials offer multiple symbolic and concrete repre-sentations for all key mathematical concepts, and provide guided practice for mastery, allowing students to master and understand each representation completely before moving on.
JUMP Math shows teachers how to see the big ideas of mathematics in even the smallest steps, how to make sense of the individual steps in a mathematical procedure or problem, and how to relate them to the wider concept. JUMP Math teaches fundamental rules, algorithms, and procedures of mathematics for mastery, but students are enabled to discover those procedures themselves (as well as being encouraged to develop their own approaches) and are guided to understand the concepts underlying the procedures fully.
6. Building Confidence with the Introductory UnitIn the twenty years that I have been teaching mathematics to children, I have never met an educator who would say that students who lack confidence in their intellectual or academic abilities are likely to do well in school. Our introductory unit has been carefully designed and tested with thousands of students to boost confidence. It has proven to be an extremely effective tool for convincing even the most chal-lenged student that they can do well in mathematics. —John Mighton, in conversation
In recent years, research has shown that students are more likely to do well in subjects when they believe they are capable of doing well. It seems obvious, then, that any math program that aims to harness the potential of every student would start with an exercise that builds the confidence of every student. Getting Ready for JUMP Math: Introductory Unit Using Fractions, which can be downloaded from www.jumpmath.org, was designed for just this purpose. The Introductory Unit does not teach fractions in depth: you will find a more comprehensive approach to teaching fractions in the relevant JUMP Math workbooks and teacher’s guides. We recommend that teachers only use the unit for several weeks, preferably at the beginning of the school year.
The individual steps that teachers will follow in teaching the unit are extremely small, so even students who struggle most needn’t be left behind. Throughout the unit, students are expected to • discover or extend patterns or rules on their own, • see what changes and what stays the same in sequences of mathematical expressions, and • apply what they have learned to new situations.
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Students become very excited at making these discoveries and meeting these challenges as they learn the material. For many, it is the first time they have ever been motivated to pay attention to mathematical rules and patterns or to try to extend their knowledge in new cases.
The Introductory Unit Using Fractions, which consists of students worksheets and a short teacher’s guide, has been specifically designed to build confidence by:
• Requiring that students possess only a few simple skills — These skills can be taught to even the most challenged students in a very short amount of time. To achieve a perfect score on the final test in the unit, students need only possess three skills: they must be able to skip count on their fingers, add one-digit numbers, and subtract one-digit numbers.
• Eliminating heavy use of language — Mathematics functions as its own symbolic language. Since the vast majority of children are able to perform the most basic operations (counting and grouping objects into sets) long before they become expert readers, mathematics is the lone subject in which the vast majority of children are naturally equipped to excel at an early age. By removing language as a barrier, students can realize their full potential in mathematics.
• Allowing you to continually provide feedback — In the Introductory Unit, the mathematics are broken down into small steps so that you can quickly identify difficulties and help as soon as they arise.
• Keeping students engaged through the excitement of small victories — Children respond more quickly to praise and success than to criticism and threats. If students are encouraged, they feel an incentive to learn. Students enjoy exercising their minds and showing off to a caring adult.
Since the Introductory Unit is about building confidence, work with your students to ensure that they are successful. Celebrate every correct answer. Take your time. Encourage your students. Point out that fractions are considered to be one of the most difficult topics in mathematics. Have fun!
7. Using JUMP Math in the ClassroomJUMP Math supports a balanced approach to teaching mathematics. In the teacher’s guides you will find lesson plans which include everything from group work to explorations. Below are some recommendations for using JUMP Math.
Teach at regular intervals.Build a lesson around the material on a particular worksheet by creating questions or exercises that are similar to the ones on the worksheet. Discuss one or two skills or concepts at a time with the whole class, allowing students to develop ideas by themselves, but giving hints and guidance where necessary. Ask questions in several different ways and allow students time to think before you solicit an answer, so that every student can put their hand up and so that students can discover the ideas for themselves. After presenting a particular concept, do not go on until all of the students are assessed and show a readiness to move ahead.
Give mini-quizzes.Each time you cover a concept or skill, assign a mini-quiz consisting of several questions or a straightforward task to see exactly what students have understood or misunderstood. Write questions or instructions on the board and let students work independently in a separate notebook (or with concrete materials when indicated by the teacher’s guide). Depending upon the topic you are working on, assign questions from the workbooks only after going through several cycles of explanations (or explorations) followed by mini-quizzes. Check the work of students who might
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need extra help first, then take up the answers to the quiz at the board with the entire class. If any of your students finish a quiz early, assign extra questions.
Assess continuously.The secret to bringing an entire class along at the same pace is to use “continuous assessment.” When students are not able to keep up in a lesson it is usually because they are lacking one or two basic skills that are needed for that lesson, or because they are being held back by a simple misconception that is not difficult to correct. Make an effort to spot mistakes or misunderstandings right away. If you wait too long to correct an error, mistakes pile upon mistakes so that it becomes impossible to know exactly where a student is going wrong. To spot mistakes, it helps to break material into small steps or separate concepts, so the worksheets are an ideal tool for assessing mistakes and misunderstandings.
Prepare bonus questions.Be ready to write bonus questions on the board from time to time during the lesson for students who finish their quizzes or tasks early. Bonus questions and extensions are included in most of the lesson plans. While students who finish quickly are occupied with these questions, circulate around the class doing spot checks on the work of students who are struggling. The bonus questions you create should generally be simple extensions of the material (see How to Create Bonus Questions below).
If a student doesn’t understand your explanation, assume there is something lacking in your explanation, not in your student. Rephrase or reword explanations if a student doesn’t understand. Sometimes lessons go too fast for a student or steps are inadvertently skipped. Taking time to reflect on what worked and didn’t work in a lesson can help you reach even the most challenged students. When students are strug-gling always ask, “How could I have improved the lesson?”
In mathematics, it is always possible to make a step easier.The exercises in the JUMP Math workbooks break concepts and skills into small steps and in a coherent order that students will find easy to master. The lesson plans in the teacher’s guides provide many examples of extra questions that can be used to fill in a missing step in the develop-ment of an idea if a problem occurs.
Introduce one piece of information at a time. Teachers often inadvertently introduce too many new pieces of information at the same time. In trying to comprehend the final item, students can lose all memory and understanding of the material that came before, even though they may have appeared to understand this material completely as it was being explained.
According to Herb Simon, who won the Nobel Prize for his work on the brain, research in cognition shows that, “… a learner who is having difficulty with components can easily be overwhelmed by the processing demands of a complex task. Further, to the extent that many components are well mastered, the student wastes much less time repeating these mastered operations to get an opportunity to practice the few components that need additional effort.”8
John Mighton once observed an intern from teachers’ college who was trying to teach a boy in a Grade 7 remedial class how to draw mixed fractions. The boy was getting very frustrated as the intern kept asking him to carry out several steps at the same time. Here is how John separated the steps to facilitate understanding and success:
Introduction
9Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Introduction
I asked the boy to simply draw a picture showing the number of whole pies in the fraction 2 1/2. He drew and shaded two whole pies cut into halves. I then asked him to draw the number of whole pies in 3 1/2, 4 1/2 and 5 1/2 pies. He was very excited when he completed the work I had assigned him, and I could see that he was making more of an effort to concentrate. I asked him to draw the whole number of pies in 2 1/4, 2 3/4, 3 1/4, 4 1/4 pies, then in 2 1/3, 2 2/3, 3 1/3 pies, and so on. (I started with quarters rather than thirds because they are easier to draw.) When the boy could draw the whole number of pies in any mixed fraction, I showed him how to draw the fractional part. Within a few minutes he was able to draw any mixed fraction. If I hadn’t broken the skill into two steps (i.e., drawing the number of whole pies then drawing the fractional part) and allowed him to practise each step separately, he might never have learned the concept.
Before you assign workbook pages, verify that all students have the skills they need to complete the work.Before assigning a question from one of the JUMP workbooks, it is important to verify that all of your students are prepared to answer the question without your help (or with minimal help).
Never allow students to work ahead in the workbook on material you haven’t covered with the class. Students who finish a worksheet early should be assigned bonus questions similar to the questions on the worksheet or extension questions from the teacher’s guide. Write the bonus questions on the board or have extra worksheets prepared and ask students to answer the questions in their notebooks or on the worksheets. While students are working independently on the bonus questions, you can spend extra time with anyone who needs help.
Raise the bar incrementally.When a student has mastered a skill or concept, raise the bar slightly by challenging them to answer a question that is only incrementally more difficult or complex than the questions previously assigned.
Praise students’ efforts.We’ve found the JUMP program works best when teachers give their students a great deal of encouragement. Because the lessons are laid out in steps that any student can master, you’ll find that you won’t be giving false encouragement. One of the reasons that kids love the program so much is that it’s a thrill to be doing well at math!
Teach the number facts.It is a serious mistake to think that students who don’t know their number facts can always get by in mathematics using a calculator or other aids. Trying to do mathematics without knowing basic number facts is like trying to play the piano without knowing where the notes are. (See the Mental Math section of this guide for strategies to help students learn their number facts.)
Create excitement about math.Engaging the entire class in lessons is not simply a matter of fairness; it is also a matter of ef-ficiency. While the idea may seem counterintuitive, teachers will enable students who learn more quickly to go further if they take care of the students who struggle. Teachers can create a real sense of excitement about math in the classroom simply by convincing struggling students that they can do well in the subject. The class will cover far more material in the year and students who excel will no longer have to hide their love of math for fear of appearing strange or different.
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Introduction
8. How to Create Bonus QuestionsStudents love to show off to their teachers by solving difficult-looking puzzles and surmounting challenges, and they also love to succeed in front of their peers. You can make math lessons more exciting (and also make time to check the work of students who need extra time) if you know how to create engaging bonus questions. Bonus questions generally shouldn’t be based on new concepts and they don’t have to be extremely difficult to capture the attention of students. Here are some strategies you can use to create questions that will look hard enough to interest students who work quickly, but that all of your students can aspire to answer.
1. Make the numbers in a problem larger or introduce several new terms or elements without introducing any new skills or concepts.This is the simplest way to create bonus questions. Kids of all ages love showing off with larger numbers or with more-challenging looking rules and procedures. If your students know how to add a pair of three digit numbers without regrouping, let them impress you by adding a pair of five or six digit numbers. If they can add two fractions let them add three or four. You will be amazed at how excited students become when they can apply their skills with larger numbers or more difficult-looking calculations. You can use this strategy in almost any lesson.
Some things to bear in mind: First, bonus questions shouldn’t look tedious. You don’t want to give students an endless series of calculations that appear to have no purpose. It helps if you are excited when you assign bonus questions and if you assign only a few questions at a time. Students should feel they are involved in a quest, faced with a series of increasingly more difficult challenges that they believe they can meet.
Students will not necessarily gain a deeper conceptual understanding of a particular mathematical idea when they work on bonus questions that involve larger numbers or that have more terms or elements. But they will still make important conceptual gains. In addition to generalizing from smaller to larger numbers, they will, for instance, develop the ability to hold more material in their working memory, to follow a series of steps in a procedure, to stay on task, and to see patterns and apply rules in increasingly complex situations. They will also consolidate their understanding and commit the material memory. Their behaviour, confidence, and level of engagement will also likely improve.
2. Make a mistake and ask your students to correct it. Students love correcting a teacher’s mistakes—and you can find a way to make mistakes in any lesson! For instance, if you are teaching T-tables, you might draw the following T-table on the board:
INPUT OUTPUT
1 7
2 10
3 14
4 16
Tell your students you created the table by adding the same number repeatedly to the initial number, but you think you made a mistake. Ask them to find the mistake and explain where you went wrong.
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Introduction
3. Leave out several terms or elements in a sequence and ask your students to say what is missing.For instance, you might ask students to say what numbers are missing in the following T-tables, assuming the tables were made by adding the same number repeatedly. NOTE: The problem on the right is harder than the one on the left. Students might solve the problem on the right by guessing and checking or using a number line.
INPUT OUTPUT INPUT OUTPUT
1 8 1 7
2 2
3 14 3 15
4 17
When you create bonus questions, use number facts that your students are likely to know and give clear and concise instructions. Rather than giving lengthy explanations to struggling students, try to assign tasks that students can immediately see how to do. For instance, if the numbers in the charts above are too difficult for your students, use different numbers:
INPUT OUTPUT INPUT OUTPUT
1 2 1 5
2 4 2
3 3
4 8 4 20
5 10 5 25
Most students know how to count by twos or fives, so every student should see which numbers are missing from these charts. Once struggling students have succeeded with easier tasks, they will be more willing to take risks and to guess and check to solve more difficult problems.
4. Vary the task or problem slightly.You could use more challenging patterns in the output column, such as decreasing patterns (as shown on the left) or patterns with a gap that changes (as shown on the right).
INPUT OUTPUT INPUT OUTPUT
1 23 1 5
2 20 2 7
3 17 3 10
4 14 4 14
You could also add an extra column to the table, as you might do if you were following a recipe with three quantities that vary with each other. You could ask students to say how much of one ingredient they would need if they had a certain amount of another ingredient.
Number of pies Number of cups of flour
Number of cups of cherries
1 2 3
2 4 6
3 6 9
4 8 12
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5. Look for applications of the concepts.You might tell students that the numbers in the output column represent a student’s savings and ask them to predict how long it will take to save a certain amount of money. Or you could give them a T-table and ask them to draw a picture or describe a pattern that might be represented by the table.
6. Look for patterns in any work you assign, and ask students to describe the patterns.For example, if you ask students to find and state the rule for the T-table below, you can ask students who finish early to describe the pattern in the ones and the tens digits of the numbers in the right-hand column.
INPUT OUTPUT
1 7
2 12
3 17
4 22
5 27
(Answer: the ones digit repeats every second term (7, 2, 7, 2, 7,…) and the tens digit increases by 1 every second term (0, 1, 1, 2, 2,…). This happens because the numbers increase by 5 each time, so after two steps they have increased by 10.)
This kind of exercise can keep students who finish their work early occupied while you do a spot check on other students’ work.
7. Use extension questions from the teacher’s guides.As your students become more confident you will want to create questions that challenge them more and that extend the ideas in the lesson. Our lesson plans contain many suggestions for creating extension questions for your students.
9. Features of the JUMP Math Materials
Diversity of representationWhat makes sense to one student does not always make sense to another. Students learn in different ways. Multiple representations help to reach a broader number of students. Students who see multiple representations of a concept also develop a deeper understanding of that concept and are better able to explain it. They can then begin to make connections between the various representations. The more ways you as a teacher know how to teach a concept, the better you will understand it yourself and the more able you will be to teach it in ways that meet the varying needs of your students.
Differentiated InstructionBecause the math is broken into steps in the workbooks, the workbooks allow you to teach to the whole class while still addressing the needs of individual students (not leaving anyone behind and challenging those who are ready to move ahead). Part of the philosophy of JUMP Math is to teach to the collective—to ensure that the class meets success as a whole, thereby increasing excitement and momentum. On the JUMP Math worksheets, concepts and skills are introduced one step at a
Introduction
13Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Introduction
time, with lots of opportunities for practice. Struggling students can complete all of the questions on a worksheet while students who excel can skip some questions and do extra work provided in the teacher’s guides.
Text and layoutIn the workbooks, we tried to reduce the number of words per page and to use clear, simple language. This ensures that all students have equal access to the materials, regardless of their reading level. (However, in the teacher’s guides you will find many exercises that combine mathematics with language learning and communication.) Layout is simple and uncluttered to avoid distraction, which is particularly helpful for children with learning disabilities. Visual elements such as boxes, figures, background shading, and bold text emphasize changes in content to help students learn new steps or new ideas.
Review in the workbooks At the beginning of the school year, teachers often find that many of their students have forgotten material taught in the previous year. That’s why the workbooks for each grade level (after Grade 1) provide review that goes back one or two years in the curriculum. Research in education has shown that if teachers assess their students and start at the right level, they cover more curriculum in the year.
10. Hints for Helping Students Who Have Fallen Behind
Teach the number facts.Struggling students often have a weak grasp of number facts. They have trouble solving problems or doing calculations because their working memories are overwhelmed trying to recall number facts. Research shows that automatic recall of number facts helps enormously in the learning of math.
Give cumulative reviews.Even mathematicians constantly forget new material, including material they once understood completely. Giving reviews doesn’t have to create a lot of extra work for a teacher. John Mighton recommends that, once a month, you copy a selection of questions from the workbook for units already covered onto a single sheet and make copies for the class. Students rarely complain about doing questions they already did a month or more ago (and quite often they won’t even remember that they did those particular questions).
Make mathematical terms part of your spelling lessons.In some areas of math (e.g., geometry) the greatest difficulty that students face is in learning the terminology. If you include mathematical terms in your spelling lessons, students will find it easier to remember the terms and to communicate about their work.
Set aside five minutes every few days to give extra review to struggling students. Spending five minutes with a small group of students who need extra help can make the teaching component of lessons more productive. Ensuring that struggling students have the prerequisite skills and knowledge to participate fully in the lesson will enhance their learning and contribute to a positive learning environment for all your students.
14 Teacher’s Guide for Workbook 3COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Allow wait time.Wait time is the time between asking the question and soliciting a response. Wait time gives students a chance to think about their answer and leads to longer and clearer explanations. It is particularly helpful for more timid students, students who are slower to process information, and students who are learning English as a second language. Studies about the benefits of increasing wait time to three seconds or longer confirm that there are increases in student participation, better quality of responses, better overall classroom performance, more questions asked by students, and more frequent and unsolicited contributions.
11. Other Features of the JUMP Math Program
1. Professional Learning
TeachersTraining sessions offered across Canada give teachers opportunities to: • Learn about JUMP Math’s philosophy and guiding principles • Consider their own comfort levels with mathematics and math instruction • Watch a video of a lesson and discuss specific instructional techniques • Learn how to use all of the JUMP Math resources • Take home practical ideas on how to support all students’ learning in the classroom
Community VolunteersA new JUMP Math Essentials tutor program (Grades 3–8) is now available. There are three resource packages: Grades 3 & 4, Grades 5 & 6, and Grades 7 & 8. Each package includes a detailed tutor’s guide and student worksheets. Detailed information is available at www.jumpmath.org. Training sessions for tutors include instruction on how to: • Plan for 25 weeks of lessons • Assess student requirements • Engage students and get them excited about math • Use the support materials
2. National Book FundJUMP Math is committed to supporting vulnerable communities in schools. We provide free resources and supports through the National Book Fund.
Introduction
15Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
References
1 Phillip E. Ross, “The Expert Mind,” Scientific American, July 2006: 44.
2 John Mighton, The End of Ignorance (Toronto: Alfred A. Knopf/Random House, 2007), 6.
3 Canadian Child Care Federation and Canadian Language and Literacy Research Network (CLLRNet), “Foundations for Numeracy: An Evidence-Based Toolkit for the Effective Mathematics Teacher” (London, ON: CLLRNet, 2010), 44.
4 Ann Dowker, “What Works for Children with Mathematical Difficulties? The Effectiveness of Intervention Schemes,” Ref: 00086-2009BKT-EN (London, England: UK Government Department for Children, Schools and Families, 2009), 35–36.
5 Margaret M. Clifford, “Students Need Challenge, Not Easy Success,” Educational Leadership 48, no. 1 (1990): 22–26.
6 Phillip E. Ross, “The Expert Mind,” Scientific American, July 2006.
7 Paul A. Kirschner, John Sweller, and Richard E. Clark, “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential and Inquiry-Based Teaching,” Educational Psychologist 41, no. 2:75–86.
8 John R. Anderson, Lynne M. Reder, and Herbert A. Simon, “Applications and Misapplications of Cognitive Psychology to Mathematics Education,” Texas Educational Review (Summer 2008): 208.
Introduction
16 Teacher’s Guide for Workbook 3COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Sample Problem Solving Lesson
� TEACHER’S GUIDE
As much as possible, allow your students to extend and discover ideas on their own (without pushing them so far that they become discouraged). It is not hard to develop problem solving lessons (where your students can make discoveries in steps) using the material on the worksheets. Here is a sample problem solving lesson you can try with your students.
1. Warm-upReview the notion of perimeter from the worksheets. Draw the following diagram on a grid on the board and ask your students how they would determine the perimeter. Tell students that each edge on the shape represents 1 unit (each edge might, for instance, represent a centimeter).
Allow your students to demonstrate their method (ExAmplE: counting the line segments, or adding the lengths of each side).
�. Develop the IdeaDraw some additional shapes and ask your students to copy them onto grid paper and to determine the perimeter of each.
Check Bonus Try Again?
The perimeters of the shapes above are 10 cm, 10 cm and 12 cm respectively.
Have your students make a picture of a letter from their name on graph paper by colouring in squares. Then ask them to find the perimeter and record their answer in words. Ensure students only use vertical and horizontal edges.
Students may need to use some kind of system to keep them from missing sides. Suggest that your students write the length of the sides on the shape.
3. Go FurtherDraw a simple rectangle on the board and ask students to again find the perimeter.
Add a square to the shape and ask students how the perimeter changes.
Draw the following polygons on the board and ask students to copy the four polygons on their grid paper.
Sample problem Solving lesson
Isolate the problem
Immediate assessment
Introduce one concept at a time
Raise the bar incrementally
Review and test prior knowledge
Sometimes the big idea will only emerge at the end of the lesson, after you have developed the idea out of smaller skills and concepts. Always teach with the big idea in mind, but avoid overwhelming students with too much information. TIP
1
As much as possible, allow your students to extend and discover ideas on their own (without pushing them so far that they become discouraged). It is not hard to develop problem solving lessons (where your students can make discoveries in steps) using the material on the worksheets. The following lesson plan shows how you can create a rigorously scaffolded lesson that allows students to work independently and discover mathematical concepts on their own. The material is taken from several pages in the grade 6 workbooks.
Introduction
centimetre).
17Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Part 1 Number Sense �WORKBOOK 4
Sample Problem Solving Lesson (continued)
Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to add an additional square to each polygon and calculate the perimeter again.
4. Another StepDraw the following shape on the board and ask your students, “How can you add a square to the following shape so the perimeter decreases?”
Check Bonus Try Again?
Have your students demonstrate where they added the squares and how they found the perimeter.
Ask your students to discuss why they think the perimeter remains constant when the square is added in the corner (as in the fourth polygon above).
• Ask your students to calculate the greatest amount the perimeter can increase by when you add a single square.
• Ask them to add 2 (or 3) squares to the shape below and examine how the perimeter changes.
• Ask them to create a T-table where the two columns are labelled “Number of Squares” in the polygon and “Perimeter” of the polygon (see the Patterns section for an introduction to T-tables). Have them add more squares and record how the perimeter continues to change.
Ask students to draw a single square on their grid paper and find the perimeter (4 cm). Then have them add a square and find the perimeter of the resulting rectangle. Have them repeat this exercise a few times and then follow the same procedure with the original (or bonus) questions.
Check Bonus Try Again?
Discuss with your students why perimeter decreases when the square is added in the middle of the second row. You may want to ask them what kinds of shapes have larger perimeters and which have smaller perimeters.
Ask your students to add two squares to the polygons below and see if they can reduce the perimeter.
Have your students try the exercise above again with six square-shaped pattern blocks. Have them create the polygon as drawn above and find where they need to place the sixth square by guessing and checking (placing the square and finding the perimeter of the resulting polygon).NO
YES
Encourage students to communicate their understanding
Guide students in small steps to discover ideas for themselves
Scaffold when necessary
Check: Do they understand?
Bonus appears harder but requires no new explanation; allows teacher to attend to struggling students
When teaching a skill or concept to the whole class, give lots of hints and guidance. Ask each question in several different ways, and allow students time to think before soliciting an answer, so that every student can put their hand up and so that students can discover ideas for themselves.T
IP 2
Ask Your students how they would calculate the perimeter of the first polygon. Then instruct them to add an additional square to each polygon and calculate the perimeter again. Is there a place on the shape where they can add a square so the perimeter stays the same?
. Ask them to predict whether they could create a shape with odd perimeter by adding squares.
Repetition that is not tedious, subtle variations keep the task interesting
Introduction
18 Teacher’s Guide for Workbook 3COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Introduction
� TEACHER’S GUIDE
5. Develop the IdeaHold up a photograph that you’ve selected and ask your students how you would go about selecting a frame for it. What kinds of measurements would you need to know about the photograph in order to get the right sized frame? You might also want to show your students a CD case and ask them how they would measure the paper to create an insert for a CD/CD-ROM.
Show your students how the perimeter of a rectangle can be solved with an addition statement (ExAmplE: Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of equal lines and that, because of this, we only need two numbers to find the perimeter of a given rectangle.
3 cm ? Perimeter = cm
?
4 cm
Show your students that there are two ways to find this:
a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm
Ask your students to find the perimeters of the following rectangles (not drawn to scale).
1 cm ?
?
3 cm
3 cm ?
?
5 cm
4 cm ?
?
7 cm
6. Go FurtherDemonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.
Sample problem Solving lesson (continued)
Check Bonus Try Again?
Take up the questions (the perimeters of the rectangles above, from left to right, are 8 cm, 16 cm and 22 cm).
Continue creating questions in this format for your students and gradually increase the size of the numbers.
Have students draw a copy of the rectangle in a notebook and copy the measurements onto all four sides. Have them create an addition statement by copying one number at a time and then crossing out the measurement:
1 cm 1 cm
4 cm
4 cm
4 cm 4 cm 1 cm
+1 cm
Assign small sets of questions or problems that test students’ understanding of each step before the next one is introduced
In mathematics, it is always possible to make a step easier
Using larger numbers makes the problem appear harder and builds excitement
If your students are confident and engaged, try skipping steps when teaching new material, and challenge your students to figure out the steps themselves. But if students struggle, go back to teaching in small steps. T
IP 3
Use Extensions from the lesson plans for extra challenges. Avoid singling out students who work on extensions as geniuses and allow all of your students to try these questions (with hints if necessary). TIP
4
Make connec-tions explicit (Example: between math and the real world, between strands or between math and other subjects)
19Teacher’s Guide for Workbook 3 COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
Part 1 Number Sense �WORKBOOK 4
Ask your students to draw all the rectangles they can with a perimeter of 12 cm.
7. Raise the BarDraw the following rectangle and measurements on paper:
1 cm ? Perimeter = 6 cm
?
?
Ask students how they would calculate the length of the missing sides. After they have given some input, explain to them how the side opposite the one measured will always have the same measurement. Demonstrate how the given length can be subtracted twice (or multiplied by two and then subtracted) from the perimeter. The remainder, divided by two, will be the length of each of the two remaining sides.
Draw a second rectangle and ask students to find the lengths of the missing sides using the methods just discussed.
2 cm ? Perimeter = 14 cm
?
?
Sample Problem Solving Lesson (continued)
Check Bonus Try Again?
After your students have finished, ask them whether they were able to find one rectangle, then two rectangles, then three rectangles.
Ask students to find (and draw) all the rectangles with a perimeter of 18 cm. After they have completed this, they can repeat the same exercise for rectangles of 24 cm or 36 cm.
If students find only one (or zero) rectangles, they should be shown a systematic method of finding the answer and then given the chance to practise the original question.
On grid paper, have students draw a pair of lines with lengths of 1 and 2 cm each.
Ask them to draw the other three sides of each rectangle so that the final perimeter will be 12 cm for each rectangle, guessing and checking the lengths of the other sides. Let them try this method on one of the bonus questions once they accomplish this.
1 cm 2 cm
Keeping your students excited will help them focus on harder problems
Bonus questions do not require much background knowledge or extra explanations, so the stronger students can work independ-ently while you help weaker students
Continuously allow students to show off what they learned
Scaffold problem-solving strategies (Example: systematic search) when necessary
Only assign questions from the workbook after going through several cycles of explanations followed by mini-quizzes. Do not allow any student to work ahead of others in the workbooks. T
IP 5
Ask students to present their answers in a way that allows you to spot mistakes easily. You might say: “I can’t always keep up with you, so I need you to show me your steps occasionally” or “Because you’re so clever, you may want to help a friend with math one day, so you’ll need to know how to explain the steps.”
TIP
6
Introduction
20 Teacher’s Guide for Workbook 3COPYRIGHT © 2010 JUMP MATH: Not to be copied without permission
� TEACHER’S GUIDE
8. AssessmentDraw the following diagrams of rectangles and perimeter statements, and ask students to complete the missing measurements on each rectangle.
a) b) c)
2 cm ?
4 cm
?
Perimeter = 12 cm
? 4 cm
?
?
Perimeter = 18 cm
? ?
6 cm
?
Perimeter = 18 cm
Sample Problem Solving Lesson (continued)
Check Bonus Try Again?
Check that students can calculate the length of the sides (2 cm, 2 cm, 5 cm and 5 cm).
Give students more problems like above. For example:
Side = 5 cm; Perimeter = 20 cm Side = 10 cm; Perimeter = 50 cm Side = 20 cm; Perimeter = 100 cm Side = 65 cm; Perimeter = 250 cm
Be sure to raise the numbers incrementally on bonus questions.
Give students a simple problem to try (similar to the first demonstration question).
1 cm
Perimeter = 8 cm
Provide them with eight toothpicks (or a similar object) and have them create the rectangle and then measure the length of each side. Have them repeat this with more questions.
Check Bonus
Answers for the above questions (going clockwise from the sides given):
a) 2 cm, 4 cm b) 3 cm, 6 cm, 3 cm c) 5 cm, 4 cm, 5 cm
Draw a square and inform your students that the perimeter is 20 cm. What is the length of each side? (Answer: 5 cm.) Repeat with other multiples of four for the perimeter.
Observe the excitement that can sweep through a class when students experience success together
Continuous assessment is the secret to bringing an entire class along at the same pace
Use hands-on activities with concrete materials to consolidate learning
A full size copy of this lesson can be downloaded from our website.
NO
TE
If a student is failing to perform an operation correctly, isolate the problematic step. Give them a number of questions that have been worked out to the point where they have trouble and have them practise doing just that one step until they master it. T
IP 7
Introduction
Sample Problem Solving Lesson
page 1
Teacher’s Guide - Introduction
As much as possible, allow your students to extend and discover ideas on their own (without pushing
them so far that they become discouraged). It is not hard to develop problem solving lessons (where your
students can make discoveries in steps) using the material on the worksheets. Here is a sample problem
solving lesson you can try with your students.
1. Warm-up
Review the notion of perimeter from the worksheets. Draw the following diagram on a grid on the
board and ask your students how they would determine the perimeter. Tell students that each edge
on the shape represents 1 unit (each edge might, for instance, represent a centimeter).
Allow your students to demonstrate their method (e.g., counting the line segments, or adding the
lengths of each side).
2. Develop the Idea
Draw some additional shapes and ask your students to copy them onto grid paper and to determine
the perimeter of each.
Check Bonus Try Again?
The perimeters of the shapes above
are 10 cm, 10 cm and 12 cm
respectively.
Have your students make a picture of a
letter from their name on graph paper
by colouring in squares. Then ask them
to find the perimeter and record their
answer in words. Ensure students only
use vertical and horizontal edges.
Students may need to use some kind
of system to keep them from missing
sides. Suggest that your students write
the length of the sides on the shape.
3. Go Further
Draw a simple rectangle on the board and ask students to again find the perimeter.
Add a square to the shape and ask students how the perimeter changes.
Draw the following polygons on the board and ask students to copy the four polygons on their
grid paper.
Sample Problem Solving Lesson (continued)
page 2
Teacher’s Guide - Introduction
Ask your students how they would calculate the perimeter of the first polygon. Then instruct them to
add an additional square to each polygon and calculate the perimeter again.
Check Bonus Try Again?
Have your students demonstrate where
they added the squares and how they
found the perimeter.
Ask your students to discuss why they
think the perimeter remains constant
when the square is added in the corner
(as in the fourth polygon above).
� Ask your students to calculate the
greatest amount the perimeter can
increase by when you add a single
square.
� Ask them to add 2 (or 3) squares to
the shape below and examine how
the perimeter changes.
� Ask them to create a T-table where
the two columns are labelled
“Number of Squares” in the polygon
and “Perimeter” of the polygon (see
the Patterns section for an
introduction to T-tables). Have them
add more squares and record how
the perimeter continues to change.
Ask students to draw a single square
on their grid paper and find the
perimeter (4 cm). Then have them add
a square and find the perimeter of the
resulting rectangle. Have them repeat
this exercise a few times and then
follow the same procedure with the
original (or bonus) questions.
4. Another Step
Draw the following shape on the board and ask your students, “How can you add a square to the
following shape so the perimeter decreases?”
Check Bonus Try Again?
Discuss with your students why
perimeter decreases when the square
is added in the middle of the second
row. You may want to ask them what
kinds of shapes have larger perimeters
and which have smaller perimeters.
Ask your students to add two squares
to the polygons below and see if they
can reduce the perimeter.
Have your students try the exercise
above again with six square-shaped
pattern blocks. Have them create the
polygon as drawn above and find
where they need to place the sixth
square by guessing and checking
(placing the square and finding the
perimeter of the resulting polygon).
Sample Problem Solving Lesson (continued)
page 3
Teacher’s Guide - Introduction
5. Develop the Idea
Hold up a photograph that you’ve selected and ask your students how you would go about selecting
a frame for it. What kinds of measurements would you need to know about the photograph in order to
get the right sized frame? You might also want to show your students a CD case and ask them how
they would measure the paper to create an insert for a CD/CD-ROM.
Show your students how the perimeter of a rectangle can be solved with an addition statement (e.g.,
Perimeter = 14 cm is the sum of 3 + 3 + 4 + 4). Explain that the rectangle is made up of two pairs of
equal lines and that, because of this, we only need two numbers to find the perimeter of a given
rectangle.
Perimeter = ___ cm
Show your students that there are two ways to find this:
a) Create an addition statement by writing each number twice: 3 cm + 3 cm + 4 cm + 4 cm = 14 cm
b) Add the numbers together and multiply the sum by 2: 3 cm + 4 cm = 7 cm; 7 cm × 2 = 14 cm
Ask your students to find the perimeters of the following rectangles (not drawn to scale).
Check Bonus Try Again?
Take up the questions (the perimeters
of the rectangles above, from left to
right, are 8 cm, 16 cm and 22 cm).
Continue creating questions in this
format for your students and gradually
increase the size of the numbers.
Have students draw a copy of the
rectangle in a notebook and copy the
measurements onto all four sides.
Have them create an addition
statement by copying one number at a
time and then crossing out the
measurement:
6. Go Further
Demonstrate on grid paper that two different rectangles can both have a perimeter of 10 cm.
?
3 cm ?
4 cm
?
1 cm ?
3 cm
?
3 cm ?
5 cm
?
4 cm ?
7 cm
4 cm
4 cm
1 cm 1 cm
4 cm
4 cm
1 cm
+ 1 cm
Sample Problem Solving Lesson (continued)
page 4
Teacher’s Guide - Introduction
Ask your students to draw all the rectangles they can with a perimeter of 12 cm.
Check Bonus Try Again?
After your students have finished, ask
them whether they were able to find
one rectangle, then two rectangles,
then three rectangles.
Ask students to find (and draw) all the
rectangles with a perimeter of 18 cm.
After they have completed this, they
can repeat the same exercise for
rectangles of 24 cm or 36 cm.
If students find only one (or zero)
rectangles, they should be shown a
systematic method of finding the
answer and then given the chance to
practise the original question.
On grid paper, have students draw a
pair of lines with lengths of 1 and 2 cm
each.
Ask them to draw the other three sides
of each rectangle so that the final
perimeter will be 12 cm for each
rectangle, guessing and checking the
lengths of the other sides. Let them try
this method on one of the bonus
questions once they accomplish this.
7. Raise the Bar
Draw the following rectangle and measurements on paper:
Perimeter = 6 cm
Ask students how they would calculate the length of the missing sides. After they have given some
input, explain to them how the side opposite the one measured will always have the same
measurement. Demonstrate how the given length can be subtracted twice (or multiplied by two and
then subtracted) from the perimeter. The remainder, divided by two, will be the length of each of the
two remaining sides.
Draw a second rectangle and ask students to find the lengths of the missing sides using the methods
just discussed.
Perimeter = 14 cm
?
1 cm ?
?
?
2 cm ?
?
1 cm 2 cm
Sample Problem Solving Lesson (continued)
page 5
Teacher’s Guide - Introduction
Check Bonus Try Again?
Check that students can calculate the
length of the sides (2 cm, 2 cm, 5 cm
and 5 cm).
Give students more problems like
above.
For example:
� Side = 5 cm; Perimeter = 20 cm
� Side = 10 cm; Perimeter = 50 cm
� Side = 20 cm; Perimeter = 100 cm
� Side = 65 cm; Perimeter = 250 cm
Be sure to raise the numbers
incrementally on bonus questions.
Give students a simple problem to try
(similar to the first demonstration
question).
1 cm Perimeter = 8 cm
Provide them with eight toothpicks
(or a similar object) and have them
create the rectangle and then measure
the length of each side. Have them
repeat this with more questions.
8. Assessment
Draw the following diagrams of rectangles and perimeter statements, and ask students to complete
the missing measurements on each rectangle.
a)
Perimeter = 12 cm
b)
Perimeter = 18 cm
c)
Perimeter = 18 cm
Check Bonus
Answers for the above questions (going clockwise from the
sides given):
a) 2 cm, 4 cm
b) 3 cm, 6 cm, 3 cm
c) 5 cm, 4 cm, 5 cm
Draw a square and inform your students that the perimeter
is 20 cm. What is the length of each side? (Answer: 5 cm.)
Repeat with other multiples of four for the perimeter.
4 cm
2 cm ?
?
6 cm
? ?
?
?
? 4 cm
?
Mental Math 1WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
Sample use only - not for sale
Teacher
If any of your students don’t know their addition and subtraction facts, teach them to add and
subtract using their fi ngers by the methods taught below. You should also reinforce basic facts
using drills, games and fl ash cards. There are mental math strategies that make addition and
subtraction easier: some effective strategies are taught in the next section. (Until your students
know all their facts, allow them to add and subtract on their fi ngers when necessary.)
1. Add:
a) 5 + 2 = b) 3 + 2 = c) 6 + 2 = d) 9 + 2 =
e) 2 + 4 = f) 2 + 7 = g) 5 + 3 = h) 6 + 3 =
i) 11 + 4 = j) 3 + 9 = k) 7 + 3 = l) 14 + 4 =
m) 21 + 5 = n) 32 + 3 = o) 4 + 56 = p) 39 + 4 =
2. Subtract:
a) 7 – 5 = b) 8 – 6 = c) 5 – 3 = d) 5 – 2 =
e) 9 – 6 = f) 10 – 5 = g) 11 – 7 = h) 17 – 14 =
i) 33 – 31 = j) 27 – 24 = k) 43 – 39 = l) 62 – 58 =
Teacher
To prepare for the next section, teach your students to add 1 to any number mentally (by counting
forward by 1 in their head) and to subtract 1 from any number (by counting backward by 1).
Mental MathAddition and Subtraction
To ADD 4 + 8, Grace says the greater number (8) with her fi st closed. She counts up from 8, raising one
fi nger at a time. She stops when she has raised the number of fi ngers equal to the lesser number (4):
She said “12” when she raised her 4th fi nger, so: 4 + 8 = 12
To SUBTRACT 9 – 5, Grace says the lesser number (5) with her fi st closed. She counts up from 5 raising
one fi nger at a time. She stops when she says the greater number (9):
She has raised 4 fi ngers when she stopped, so: 9 – 5 = 4
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Teacher
Students who don’t know how to add, subtract or estimate readily are at a great disadvantage
in mathematics. Students who have trouble memorizing addition and subtraction facts can still
learn to mentally add and subtract numbers in a short time if they are given daily practice in
a few basic skills.
SKILL 1: Adding 2 to an Even Number
This skill has been broken down into a number of sub-skills. After teaching each sub-skill, you
should give your students a short diagnostic quiz to verify that they have learned the skill. I have
included sample quizzes for Skills 1 to 4.
i) Naming the next one-digit even number:
Numbers that have ones digit 0, 2, 4, 6 or 8 are called the even numbers. Using drills or
games, teach your students to say the sequence of one-digit even numbers without hesitation.
Ask students to imagine the sequence going on in a circle so that the next number after 8 is 0
(0, 2, 4, 6, 8, 0, 2, 4, 6, 8… ) Then play the following game: name a number in the sequence
and ask your students to give the next number. Don’t move on until all of your students have
mastered the game.
ii) Naming the next greatest two-digit even number:
CASE 1: Numbers that end in 0, 2, 4 or 6
Write an even two-digit number that ends in 0, 2, 4 or 6 on the board. Ask your students to name
the next greatest even number. Students should recognize that if a number ends in 0, then the
next even number ends in 2; if it ends in 4 then the next even number ends in 6, etc. For instance,
the number 54 has ones digit 4: so the next greatest even number will have ones digit 6.
CASE 2: Numbers that end in 8
Write the number 58 on the board. Ask students to name the next greatest even number.
Remind your students that even numbers must end in 0, 2, 4, 6, or 8. But 50, 52, 54 and 56
are all less than 58 so the next greatest even number is 60. Your students should see that an
even number ending in 8 is always followed by an even number ending in 0 (with a tens digit
that is one higher).
iii) Adding 2 to an even number:
Point out to your students that adding 2 to any even number is equivalent to fi nding the
next even number: EXAMPLE: 46 + 2 = 48, 48 + 2 = 50, etc. Knowing this, your students
can easily add 2 to any even number.
QU
IZ
Name the next greatest even number:
a) 52 : b) 64 : c) 36 : d) 22 : e) 80 :
QU
IZ
Name the next greatest even number:
a) 58 : b) 68 : c) 38 : d) 48 : e) 78 :
Addition and Subtraction
Mental Math 3WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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SKILL 2: Subtracting 2 from an Even Number
i) Finding the preceding one-digit even number:
Name a one-digit even number and ask your students to give the preceding number in the
sequence. For instance, the number that comes before 4 is 2 and the number that comes
before 0 is 8. (REMEMBER: the sequence is circular.)
ii) Finding the preceding two-digit number:
CASE 1: Numbers that end in 2, 4, 6 or 8
Write a two-digit number that ends in 2, 4, 6 or 8 on the board. Ask students to name the
preceding even number. Students should recognize that if a number ends in 2, then the
preceding even number ends in 0; if it ends in 4 then the preceding even number ends
in 2, etc. For instance, the number 78 has ones digit 8 so the preceding even number has
ones digit 6.
CASE 2: Numbers that end in 0
Write the number 80 on the board and ask your students to name the preceding even number.
Students should recognize that if an even number ends in 0 then the preceding even number
ends in 8 (but the ones digit is one less). So the even number that comes before 80 is 78.
ii) Subtracting 2 from an even number:
Point out to your students that subtracting 2 from an even number is equivalent to fi nding the
preceding even number: EXAMPLE: 48 – 2 = 46, 46 – 2 = 44, etc.
QU
IZ
Subtract:
a) 58 – 2 = b) 24 – 2 = c) 36 – 2 = d) 42 – 2 = e) 60 – 2 =
QU
IZ
Name the preceding even number:
a) 40 : b) 60 : c) 80 : d) 50 : e) 30 :
QU
IZ
Add:
a) 26 + 2 = b) 82 + 2 = c) 40 + 2 = d) 58 + 2 = e) 34 + 2 =
QU
IZ
Name the preceding even number:
a) 48 : b) 26 : c) 34 : d) 62 : e) 78 :
Addition and Subtraction
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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SKILL 3: Adding 2 to an Odd Number
i) Naming the next one-digit odd number:
Numbers that have ones digit 1, 3, 5, 7, and 9 are called the odd numbers. Using drills or
games, teach your students to say the sequence of one-digit odd numbers without hesitation.
Ask students to imagine the sequence going on in a circle so that the next number after 9 is 1
(1, 3, 5, 7, 9, 1, 3, 5, 7, 9…). Then play the following game: name a number in the sequence
and ask you students to give the next number. Don’t move on until all of your students have
mastered the game.
ii) Naming the next greatest two-digit odd number:
CASE 1: Numbers that end in 1, 3, 5 or 7
Write an odd two-digit number that ends in 1, 3, 5, or 7 on the board. Ask you students to
name the next greatest odd number. Students should recognize that if a number ends in 1,
then the next even number ends in 3; if it ends in 3 then the next even number ends in 5, etc.
For instance, the number 35 has ones digit 5: so the next greatest even number will have
ones digit 7.
CASE 2: Numbers that end in 9
Write the number 59 on the board. Ask students to name the next greatest number. Remind
your students that odd numbers must end in 1, 3, 5, 7, or 9. But 51, 53, 55, and 57 are all
less than 59. The next greatest odd number is 61. Your students should see that an odd
number ending in 9 is always followed by an odd number ending in 1 (with a tens digit that
is one higher).
iii) Adding 2 to an odd number:
Point out to your students that adding 2 to any odd number is equivalent to fi nding the next odd
number: EXAMPLE: 47 + 2 = 49, 49 + 2 = 51, etc. Knowing this, your students can easily add
2 to any odd number.
QU
IZ
Name the next greatest odd number:
a) 59 : b) 69 : c) 39 : d) 49 : e) 79 :
QU
IZ
Add:
a) 27 + 2 = b) 83 + 2 = c) 41 + 2 = d) 59 + 2 = e) 35 + 2 =
QU
IZ
Name the next greatest odd number:
a) 51 : b) 65 : c) 37 : d) 23 : e) 87 :
Addition and Subtraction
Mental Math 5WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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SKILL 4: Subtracting 2 from an Odd Number
i) Finding the preceding one-digit odd number:
Name a one-digit even number and ask your students to give the preceding number in the
sequence. For instance, the number that comes before 3 is 1 and the number that comes
before 1 is 9. (REMEMBER: the sequence is circular.)
ii) Finding the preceding odd two-digit number:
CASE 1: Numbers that end in 3, 5, 7 or 9
Write a two-digit number that ends in 3, 5, 7 or 9 on the board. Ask students to name the
preceding even number. Students should recognize that if a number ends in 3, then the
preceding odd number ends in 1; if it ends in 5 then the preceding odd number ends in 3,
etc. For instance, the number 79 has ones digit 9, so the preceding even number has
ones digit 7.
CASE 2: Numbers that end in 1
Write the number 81 on the board and ask your students to name the preceding odd number.
Students should recognize that if an odd number ends in 1 then the preceding odd number
ends in 9 (but the ones digit is one less). So the odd number that comes before 81 is 79.
iii) Subtracting 2 from an odd number:
Point out to your students that subtracting 2 from an odd number is equivalent to fi nding the
preceding even number: EXAMPLE: 49 – 2 = 47, 47 – 2 = 45, etc.
SKILLS 5 and 6
Once your students can add and subtract the numbers 1 and 2, then they can easily
add and subtract the number 3: Add 3 to a number by fi rst adding 2, then 1 (EXAMPLE:
35 + 3 = 35 + 2 + 1). Subtract 3 from a number by subtracting 2, then subtracting 1
(EXAMPLE: 35 – 3 = 35 – 2 – 1).
QU
IZ
Subtract:
a) 59 – 2 = b) 25 – 2 = c) 37 – 2 = d) 43 – 2 = e) 61 – 2 =
QU
IZ
Name the preceding odd number:
a) 41 : b) 61 : c) 81 : d) 51 : e) 31 :
QU
IZ
Name the preceding odd number:
a) 49 : b) 27 : c) 35 : d) 63 : e) 79 :
Addition and Subtraction
6 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Addition and Subtraction
NOTE: All of the addition and subtraction tricks you teach your students should be reinforced
with drills, fl ashcards and tests. Eventually students should memorize their addition and subtraction
facts and shouldn’t have to rely on the mental math tricks. One of the greatest gifts you can give
your students is to teach them their number facts.
SKILLS 7 and 8
Add 4 to a number by adding 2 twice (EXAMPLE: 51 + 4 = 51 + 2 + 2). Subtract 4 from a number
by subtracting 2 twice (EXAMPLE: 51 – 4 = 51 – 2 – 2).
SKILLS 9 and 10
Add 5 to a number by adding 4 then 1. Subtract 5 by subtracting 4 then 1.
SKILL 11
Students can add pairs of identical numbers by doubling (EXAMPLE: 6 + 6 = 2 × 6). Students
should either memorize the 2 times table or they should double numbers by counting on
their fi ngers by 2s.
Add a pair of numbers that differ by 1 by rewriting the larger number as 1 plus the smaller
number (then use doubling to fi nd the sum): EXAMPLE: 6 + 7 = 6 + 6 + 1 = 12 + 1 = 13;
7 + 8 = 7 + 7 + 1 = 14 + 1 = 15, etc.
SKILLS 12, 13 and 14
Add a one-digit number to 10 by simply replacing the zero in 10 by the one-digit number:
EXAMPLE: 10 + 7 = 17.
Add 10 to any two-digit number by simply increasing the tens digit of the two-digit number by 1:
EXAMPLE: 53 + 10 = 63.
Add a pair of two-digit numbers (with no carrying) by adding the ones digits of the numbers
and then the tens digits: EXAMPLE: 23 + 64 = 87.
SKILLS 15 and 16
To add 9 to a one-digit number, subtract 1 from the number and then add 10: EXAMPLE:
9 + 6 = 10 + 5 = 15; 9 + 7 = 10 + 6 = 16, etc. (Essentially, the student simply has to subtract 1
from the number and then stick a 1 in front of the result.)
To add 8 to a one-digit number, subtract 2 from the number and add 10: EXAMPLE:
8 + 6 = 10 + 4 = 14; 8 + 7 = 10 + 5 = 15, etc.
SKILLS 17 and 18
To subtract a pair of multiples of ten, simply subtract the tens digits and add a zero for the
ones digit: EXAMPLE: 70 – 50 = 20.
To subtract a pair of two-digit numbers (without carrying or regrouping), subtract the ones digit
from the ones digit and the tens digit from the tens digit: EXAMPLE: 57 – 34 = 23.
Mental Math 7WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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Mental MathFurther Strategies
Further Mental Math Strategies
1. Your students should be able to explain how to use the strategies of “rounding the subtrahend
(EXAMPLE: the number you are subtracting) up to the nearest multiple of ten.”
EXAMPLES:
a) 37 – 19 = 37 – 20 + 1
b) 64 – 28 = 64 – 30 + 2
c) 65 – 46 = 65 – 50 + 4
PRACTICE QUESTIONS:
a) 27 – 17 = 27 – + d) 84 – 57 = 84 – +
b) 52 – 36 = 52 – + e) 61 – 29 = 61 – +
c) 76 – 49 = 76 – + f) 42 – 18 = 42 – +
NOTE: This strategy works well with numbers that end in 6, 7, 8 or 9.
2. Your students should be able to explain how to subtract by thinking of adding.
EXAMPLES:
a) 62 – 45 = 5 + 12 = 17
b) 46 – 23 = 3 + 20 = 23
c) 73 – 17 = 6 + 50 = 56
PRACTICE QUESTIONS:
a) 88 – 36 = + = d) 74 – 28 = + =
b) 58 – 21 = + = e) 93 – 64 = + =
c) 43 – 17 = + = f) 82 – 71 = + =
3. Your students should be able to explain how to “use doubles.”
EXAMPLES:
a) 12 – 6 = 6 6 + 6 = 12
b) 8 – 4 = 4
PRACTICE QUESTIONS:
a) 6 – 3 = d) 18 – 9 =
b) 10 – 5 = e) 16 – 8 =
c) 14 – 7 = f) 20 – 10 =
Subtrahend
Subtrahend rounded to the nearest tens
You must add 1 because 20 is 1 greater than 19
You must add 2 because 30 is 2 greater than 28
Count by ones from 45 to the nearest tens (50)
Count from 50 until you reach the fi rst number (62)
The sum of counting up to the nearest ten
and the original number is the difference
What method did we use here?
Minuend
If you add the subtrahend to itself and
the sum is equal to the minuend then the
subtrahend is the same as the difference
Same value as
minuend
Minuend plus itself
8 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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NOTE TO TEACHER: Teaching the material on these worksheets may take several lessons.
Students will need more practice than is provided on these pages. These pages are intended
as a test to be given when you are certain your students have learned the materials fully.
Teacher
Teach SKILLS 1, 2, 3 AND 4 as outlined on pages 2 through 5 before you allow your students to
answer Questions 1 through 12:
1. Name the even number that comes after the number. Answer in the blank provided:
a) 32 b) 46 c) 14 d) 92 e) 56
f) 30 g) 84 h) 60 i) 72 j) 24
2. Name the even number that comes after the number:
a) 28 b) 18 c) 78 d) 38 e) 68
3. Add. REMEMBER: adding 2 to an even number is the same as fi nding the next even number:
a) 42 + 2 = b) 76 + 2 = c) 28 + 2 = d) 16 + 2 =
e) 68 + 2 = f) 12 + 2 = g) 36 + 2 = h) 90 + 2 =
i) 70 + 2 = j) 24 + 2 = k) 66 + 2 = l) 52 + 2 =
4. Name the even number that comes before the number:
a) 38 b) 42 c) 56 d) 72 e) 98
f) 48 g) 16 h) 22 i) 66 j) 14
5. Name the even number that comes before the number:
a) 30 b) 70 c) 60 d) 10 e) 80
6. Subtract. REMEMBER: subtracting 2 from an even number is the same as fi nding the preceding
even number:
a) 46 – 2 = b) 86 – 2 = c) 90 – 2 = d) 14 – 2 =
e) 54 – 2 = f) 72 – 2 = g) 12 – 2 = h) 56 – 2 =
i) 32 – 2 = j) 40 – 2 = k) 60 – 2 = l) 26 – 2 =
7. Name the odd number that comes after the number:
a) 37 b) 51 c) 63 d) 75 e) 17
f) 61 g) 43 h) 81 i) 23 j) 95
8. Name the odd number that comes after the number:
a) 69 b) 29 c) 9 d) 79 e) 59
Mental MathExercises
Mental Math 9WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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9. Add. REMEMBER: Adding 2 to an odd number is the same as fi nding the next odd number:
a) 25 + 2 = b) 31 + 2 = c) 47 + 2 = d) 33 + 2 =
e) 39 + 2 = f) 91 + 2 = g) 5 + 2 = h) 89 + 2 =
i) 11 + 2 = j) 65 + 2 = k) 29 + 2 = l) 17 + 2 =
10. Name the odd number that comes before the number:
a) 39 b) 43 c) 57 d) 17 e) 99
f) 13 g) 85 h) 79 i) 65 j) 77
11. Name the odd number that comes before the number:
a) 21 b) 41 c) 11 d) 91 e) 51
12. Subtract. REMEMBER: Subtracting 2 from an odd number is the same as fi nding the preceding
odd number.
a) 47 – 2 = b) 85 – 2 = c) 91 – 2 = d) 15 – 2 =
e) 51 – 2 = f) 73 – 2 = g) 11 – 2 = h) 59 – 2 =
i) 31 – 2 = j) 43 – 2 = k) 7 – 2 = l) 25 – 2 =
Teacher
Teach SKILLS 5 AND 6 as outlined on pages 5 and 6 before you allow your students to answer
Questions 13 and 14:
13. Add 3 to the number by adding 2, then adding 1 (EXAMPLE: 35 + 3 = 35 + 2 + 1):
a) 23 + 3 = b) 36 + 3 = c) 29 + 3 = d) 16 + 3 =
e) 67 + 3 = f) 12 + 3 = g) 35 + 3 = h) 90 + 3 =
i) 78 + 3 = j) 24 + 3 = k) 6 + 3 = l) 59 + 3 =
14. Subtract 3 from the number by subtracting 2, then subtracting 1 (EXAMPLE: 35 – 3 = 35 – 2 – 1):
a) 46 – 3 = b) 87 – 3 = c) 99 – 3 = d) 14 – 3 =
e) 8 – 3 = f) 72 – 3 = g) 12 – 3 = h) 57 – 3 =
i) 32 – 3 = j) 40 – 3 = k) 60 – 3 = l) 28 – 3 =
15. Fred has 49 stamps. He gives 2 stamps away. How many stamps does he have left?
16. There are 25 minnows in a tank. Alice adds 3 more to the tank. How many minnows are
now in the tank?
Exercises
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Exercises
Teacher
Teach SKILLS 7 AND 8 as outlined on page 6.
17. Add 4 to the number by adding 2 twice (EXAMPLE: 51 + 4 = 51 + 2 + 2):
a) 42 + 4 = b) 76 + 4 = c) 27 + 4 = d) 17 + 4 =
e) 68 + 4 = f) 11 + 4 = g) 35 + 4 = h) 8 + 4 =
i) 72 + 4 = j) 23 + 4 = k) 60 + 4 = l) 59 + 4 =
18. Subtract 4 from the number by subtracting 2 twice (EXAMPLE: 26 – 4 = 26 – 2 – 2):
a) 46 – 4 = b) 86 – 4 = c) 91 – 4 = d) 15 – 4 =
e) 53 – 4 = f) 9 – 4 = g) 13 – 4 = h) 57 – 4 =
i) 40 – 4 = j) 88 – 4 = k) 69 – 4 = l) 31 – 4 =
Teacher
Teach SKILLS 9 AND 10 as outlined on page 6.
19. Add 5 to the number by adding 4, then adding 1 (or add 2 twice, then add 1):
a) 84 + 5 = b) 27 + 5 = c) 31 + 5 = d) 44 + 5 =
e) 63 + 5 = f) 92 + 5 = g) 14 + 5 = h) 16 + 5 =
i) 9 + 5 = j) 81 + 5 = k) 51 + 5 = l) 28 + 5 =
20. Subtract 5 from the number by subtracting 4, then subtracting 1 (or subtract 2 twice,
then subtract 1):
a) 48 – 5 = b) 86 – 5 = c) 55 – 5 = d) 69 – 5 =
e) 30 – 5 = f) 13 – 5 = g) 92 – 5 = h) 77 – 5 =
i) 45 – 5 = j) 24 – 5 = k) 91 – 5 = l) 8 – 5 =
Teacher
Teach SKILLS 11 as outlined on page 6.
21. Add:
a) 6 + 6 = b) 7 + 7 = c) 8 + 8 =
d) 5 + 5 = e) 4 + 4 = f) 9 + 9 =
22. Add by thinking of the larger number as a sum of two smaller numbers:
a) 6 + 7 = 6 + 6 + 1 b) 7 + 8 = c) 6 + 8 =
d) 4 + 5 = e) 5 + 7 = f) 8 + 9 =
Mental Math 11WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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Teacher
Teach SKILLS 12, 13 AND 14 as outlined on page 6.
23. a) 10 + 3 = b) 10 + 7 = c) 5 + 10 = d) 10 + 1 =
e) 9 + 10 = f) 10 + 4 = g) 10 + 8 = h) 10 + 2 =
24. a) 10 + 20 = b) 40 + 10 = c) 10 + 80 = d) 10 + 50 =
e) 30 + 10 = f) 10 + 60 = g) 10 + 10 = h) 70 + 10 =
25. a) 10 + 25 = b) 10 + 67 = c) 10 + 31 = d) 10 + 82 =
e) 10 + 43 = f) 10 + 51 = g) 10 + 68 = h) 10 + 21 =
i) 10 + 11 = j) 10 + 19 = k) 10 + 44 = l) 10 + 88 =
26. a) 20 + 30 = b) 40 + 20 = c) 30 + 30 = d) 50 + 30 =
e) 20 + 50 = f) 40 + 40 = g) 50 + 40 = h) 40 + 30 =
i) 60 + 30 = j) 20 + 60 = k) 20 + 70 = l) 60 + 40 =
27. a) 20 + 23 = b) 32 + 24 = c) 51 + 12 = d) 12 + 67 =
e) 83 + 14 = f) 65 + 24 = g) 41 + 43 = h) 70 + 27 =
i) 31 + 61 = j) 54 + 33 = k) 28 + 31 = l) 42 + 55 =
Teacher
Teach SKILLS 15 AND 16 as outlined on page 6.
28. a) 9 + 3 = b) 9 + 7 = c) 6 + 9 = d) 4 + 9 =
e) 9 + 9 = f) 5 + 9 = g) 9 + 2 = h) 9 + 8 =
29. a) 8 + 2 = b) 8 + 6 = c) 8 + 7 = d) 4 + 8 =
e) 5 + 8 = f) 8 + 3 = g) 9 + 8 = h) 8 + 8 =
Teacher
Teach SKILLS 17 AND 18 as outlined on page 6.
30. a) 40 – 10 = b) 50 – 10 = c) 70 – 10 = d) 20 – 10 =
e) 40 – 20 = f) 60 – 30 = g) 40 – 30 = h) 60 – 50 =
31. a) 57 – 34 = b) 43 – 12 = c) 62 – 21 = d) 59 – 36 =
e) 87 – 63 = f) 95 – 62 = g) 35 – 10 = h) 17 – 8 =
Mental MathPractice Sheet
12 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Multiples of Ten
STUDENT: In the exercises below, you will learn several ways to use multiples of ten in mental
addition or subtraction.
1. Warm up:
a) 536 + 100 = b) 816 + 10 = c) 124 + 5 = d) 540 + 200 =
e) 234 + 30 = f) 345 + 300 = g) 236 – 30 = h) 442 – 20 =
i) 970 – 70 = j) 542 – 400 = k) 160 + 50 = l) 756 + 40 =
2. Write the second number in expanded form and add or subtract one digit at a time.
The fi rst one is done for you:
a) 564 + 215 = =
b) 445 + 343 = =
c) 234 + 214 = =
3. Add or subtract mentally (one digit at a time):
a) 547 + 312 = b) 578 – 314 = c) 845 – 454 =
4. Use the tricks you’ve just learned:
a) 845 + 91 = b) 456 + 298 = c) 100 – 84 = d) 1000 – 846 =
Mental MathAdvanced
EX
AM
PL
E 3
Sometimes in subtraction, it helps to think of a multiple of ten as a sum of 1 and a number consisting
entirely of 9s (EXAMPLE: 100 = 1 + 99; 1000 = 1 + 999). You never have to borrow or exchange
when you are subtracting from a number consisting entirely of 9s.
100 – 43 = 1 + 99 – 43 = 1 + 56 = 57
1000 – 543 = 1 + 999 – 543 = 1 + 456 = 457
Do the subtraction, using 99 instead of 100,
and then add 1 to your answer.
EX
AM
PL
E 2 If one of the numbers you are adding or subtracting is close to a number with a multiple of ten, add
the multiple of ten and then add or subtract an adjustment factor:
645 + 99 = 645 + 100 – 1 = 745 – 1 = 744
856 + 42 = 856 + 40 + 2 = 896 + 2 = 898
EX
AM
PL
E 1 542 + 214 = 542 + 200 + 10 + 4 = 742 + 10 + 4 = 752 + 4 = 756
827 – 314 = 827 – 300 – 10 – 4 = 527 – 10 – 4 = 517 – 4 = 713
Sometimes you will need to carry:
545 + 172 = 545 + 100 + 70 + 2 = 645 + 70 + 2 = 715 + 2 = 717
564 + 200 + 10 + 5 779
Mental Math 13WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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Purpose
If students know the pairs of one-digit numbers that add up to particular target numbers,
they will be able to mentally break sums into easier sums.
EXAMPLE: As it is easy to add any one-digit number to 10, you can add a sum more readily
if you can decompose numbers in the sum into pairs that add to ten.
7 + 5 = 7 + 3 + 2 = 10 + 2 = 12
To help students remember pairs of numbers that add up to a given target number I developed
a variation of “Go Fish” that I have found very effective.
The Game
Pick any target number and remove all the cards with value greater than or equal to the target
number out of the deck. In what follows, I will assume that the target number is 10, so you would
take all the tens and face cards out of the deck (Aces count as one).
The dealer gives each player 6 cards. If a player has any pairs of cards that add to 10 they are
allowed to place these pairs on the table before play begins.
Player 1 selects one of the cards in his or her hand and asks the Player 2 for a card that adds to
10 with the chosen card. For instance, if Player 1’s card is a 3, they may ask the Player 2 for a 7.
If Player 2 has the requested card, the fi rst player takes it and lays it down along with the card from
their hand. The fi rst player may then ask for another card. If the Player 2 doesn’t have the requested
card they say: “Go fi sh,” and the Player 1 must pick up a card from the top of the deck. (If this card
adds to 10 with a card in the player’s hand they may lay down the pair right away). It is then Player
2’s turn to ask for a card.
Play ends when one player lays down all of their cards. Players receive 4 points for laying down all
of their cards fi rst and 1 point for each pair they have laid down.
NOTE: With weaker students I would recommend you start with pairs of numbers that add to 5.
Take all cards with value greater than 4 out of the deck. Each player should be dealt only 4 cards
to start with.
I have worked with several students who have had a great deal of trouble sorting their cards and
fi nding pairs that add to a target number. I’ve found the following exercise helps:
Give your student only three cards; two of which add to the target number. Ask the student
to fi nd the pair that add to the target number. After the student has mastered this step with
3 cards repeat the exercise with 4 cards, then 5 cards, and so on.
NOTE: You can also give your student a list of pairs that add to the target number. As the
student gets used to the game, gradually remove pairs from the list so that the student learns
the pairs by memory.
Mental MathGame: Modifi ed Go Fish
These numbers add to 10.
14 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Student Name
Can Add
1 to
Any Number
Can Subtract
1 from
Any Number
Can Add
2 to
Any Number
Can Subtract
2 from
Any Number
Knows
All Pairs that
Add to 5
Can Double
1-Digit
Numbers
Mental MathChecklist #1
Mental Math 15WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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Mental MathChecklist #2
Student Name
Can Add
Near Doubles.
EXAMPLE:
6 + 7 =
6 + 6 + 1
Can Add a
1-Digit Number
to Any
Multiple of 10.
EXAMPLE:
30 + 6 = 36
Can Add Any
1-Digit Number
to a Number
Ending in 9.
EXAMPLE:
29 + 7 =
30 + 6 = 36
Can Add 1-Digit
Numbers by
“Breaking” them
Apart into Pairs
that Add to 10.
EXAMPLE:
7 + 5 =
7 + 3 + 2 =
10 + 2
Can Subtract
Any Multiple of
10 from 100.
EXAMPLE:
100 – 40 = 60
16 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Student Name
Can Mentally
Make Change
from a Dollar.
SEE: Workbook
Sheets on Money.
Can Mentally
Add Any
Pair of 1-Digit
Numbers.
Can Mentally
Subtract Any
Pair of 1-Digit
Numbers.
Student Can Multiply
and Count by:
Mental MathChecklist #3
2 3 4 5 6 7 8 9
Mental Math 17WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
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Teacher
Trying to do math without knowing your times tables is like trying to play the piano without knowing
the location of the notes on the keyboard. Your students will have diffi culty seeing patterns in
sequences and charts, solving proportions, fi nding equivalent fractions, decimals and percents,
solving problems etc. if they don’t know their tables.
Using the method below, you can teach your students their tables in a week or so. (If you set aside
fi ve or ten minutes a day to work with students who need extra help, the pay-off will be enormous.)
There is really no reason for your students not to know their tables!
DAY 1: Counting by 2s, 3s, 4s and 5s
If you have completed the Fractions Unit you should already know how to count and multiply by
2s, 3s, 4s and 5s. If you don’t know how to count by these numbers you should memorize the
hands below:
If you know how to count by 2s, 3s, 4s and 5s, then you can multiply by any combination of these
numbers. For instance, to fi nd the product 3 × 2, count by 2s until you have raised 3 fi ngers.
2 4 6
DAY 2: The Nine Times Table
The numbers you say when you count by 9s are called the MULTIPLES of 9 (zero is also a multiple
of 9). The fi rst ten multiples of 9 (after zero) are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. What happens
when you add the digits of any of these multiples of 9 (EXAMPLE: 1 + 8 or 6 + 3)? The sum
is always 9!
Here is another useful fact about the nine times table: Multiply 9 by any number between 1 and
10 and look at the tens digit of the product. The tens digit is always one less than the number
you multiplied by:
9 × 4 = 36 9 × 8 = 72 9 × 2 = 18
You can fi nd the product of 9 and any number by using the two facts given above. For instance, to
fi nd 9 × 7, follow these steps:
STEP 1: 9 × 7 = 9 × 7 =
Mental MathHow to Learn Your Times Tables in a Week
3 is one less than 4 7 is one less than 8 1 is one less than 2
Subtract 1 from the number
you are multiplying by 9: 7 – 1 = 6
Now you know the
tens digit of the product.
3 × 2 = 6
5
15 2010255
4
12 16820
43
9 12615
32
6 8410
2
18 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Teacher
1. Make sure your students know how to subtract (by counting on their fi ngers if necessary)
before you teach them the trick for the nine times table.
2. Give a test on STEP 1 (above) before you move on.
STEP 2: 9 × 7 = 9 × 7 =
Practise these two steps for all of the products of 9: 9 × 2, 9 × 3, 9 × 4, etc.
DAY 3: The Eight Times Table
There are two patterns in the digits of the 8 times table. Knowing these patterns will help you
remember how to count by 8s.
STEP 1: You can fi nd the ones digit of the fi rst fi ve multiples of 8, by starting at 8 and counting
backwards by 2s.
8
6
4
2
0
STEP 2: You can fi nd the tens digit of the fi rst fi ve multiples of 8, by starting at 0 and count up by 1s.
08
16
24
32
40
(Of course you don’t need to write the 0 in front of the 8 for the product 1 × 8.)
STEP 3: You can fi nd the ones digit of the next fi ve multiples of 8 by repeating step 1:
8
6
4
2
0
STEP 4: You can fi nd the remaining tens digits by starting at 4 and count up by 1s.
48
56
64
72
80
Practise writing the multiples of 8 (up to 80) until you have memorized the complete list. Knowing
the patterns in the digits of the multiples of 8 will help you memorize the list very quickly. Then you
will know how to multiply by 8:
8 × 6 = 48
How to Learn Your Times Tables in a Week
Count by eight until you have 6 fi ngers up: 8, 16, 24, 32, 40, 48.
So the missing digit is 9 – 6 = 3
(You can do the subtraction on your fi ngers if necessary).
These two digits add to 9.
Mental Math 19WORKBOOKS 3, 4, 5 & 6 Copyright © 2007, JUMP Math
Sample use only - not for sale
DAY 4: The Six Times Table
If you have learned the eight and nine times tables, then you already know 6 × 9 and 6 × 8.
And if you know how to multiply by 5 up to 5 × 5, then you also know how to multiply by 6
up to 6 × 5! That’s because you can always calculate 6 times a number by calculating 5 times
the number and then adding the number itself to the result. The pictures below show why this
works for 6 × 4:
6 × 4 = 4 + 4 + 4 + 4 + 4 + 4
6 × 4 = 5 × 4 + 4 = 20 + 4 = 24
Similarly:
6 × 2 = 5 × 2 + 2; 6 × 3 = 5 × 3 + 3; 6 × 5 = 5 × 5 + 5.
Knowing this, you only need to memorize 2 facts:
ONE: 6 × 6 = 36 TWO: 6 × 7 = 42
Or, if you know 6 × 5, you can fi nd 6 × 6 by calculating 6 × 5 + 6.
DAY 5: The Seven Times Table
If you have learned the six, eight and nine times tables, then you already know:
6 × 7, 8 × 7 and 9 × 7.
And since you also already know 1 × 7 = 7, you only need to memorize 5 facts:
1. 2 × 7 = 14 2. 3 × 7 = 21 3. 4 × 7 = 28 4. 5 × 7 = 35 5. 7 × 7 = 49
If you are able to memorize your own phone number, then you can easily memorize these 5 facts!
NOTE: You can use doubling to help you learn the facts above. 4 is double 2, so 4 × 7 (= 28)
is double 2 × 7 (= 14). 6 is double 3, so 6 × 7 (= 42) is double 3 × 7 (= 21).
Try this test every day until you have learned your times tables:
1. 3 × 5 = 2. 8 × 4 = 3. 9 × 3 = 4. 4 × 5 =
5. 2 × 3 = 6. 4 × 2 = 7. 8 × 1 = 8. 6 × 6 =
9. 9 × 7 = 10. 7 × 7 = 11. 5 × 8 = 12. 2 × 6 =
13. 6 × 4 = 14. 7 × 3 = 15. 4 × 9 = 16. 2 × 9 =
17. 9 × 9 = 18. 3 × 4 = 19. 6 × 8 = 20. 7 × 5 =
21. 9 × 5 = 22. 5 × 6 = 23. 6 × 3 = 24. 7 × 1 =
25. 8 × 3 = 26. 9 × 6 = 27. 4 × 7 = 28. 3 × 3 =
29. 8 × 7 = 30. 1 × 5 = 31. 7 × 6 = 32. 2 × 8 =
How to Learn Your Times Tables in a Week
Plus one more 4.
Plus one more 4.
PART 1Patterns & Algebra PA3-1 Counting 1
PA3-2 PreparationforIncreasingSequences 4
PA3-3 IncreasingSequences 5
PA3-4 CountingBackwards 7
PA3-5 PreparationforDecreasingSequences 10
PA3-6 DecreasingSequences 11
PA3-7 IncreasingandDecreasingSequences 12
PA3-8 Attributes 14
PA3-9 PatternsWhereTwoAttributesChange 16
PA3-10 RepeatingPatterns 17
PA3-11 ExtendingRepeatingPatterns 20
PA3-12 FindingCoresinPatterns 21
PA3-13 MakingPatternswithSquares 22
PA3-14 MakingPatternswithSquares(Advanced) 24
PA3-15 ExtendingaPatternUsingaRule 25
PA3-16 IdentifyingPatternRules 27
PA3-17 IntroductiontoT-tables 28
PA3-18 T-tables 31
PA3-19 ProblemsandPuzzles 32
Number Sense NS3-1 PlaceValue–Ones,Tens,andHundreds 33
NS3-2 PlaceValue 34
NS3-3 WritingandReadingNumberWords 35
NS3-4 WritingNumbers 36
NS3-5 RepresentationwithBaseTenMaterials 38
NS3-6 RepresentationinExpandedForm 40
NS3-7 RepresentingNumbers – Review 42–Review 42Review 42
NS3-8 ComparingNumbers 43
NS3-9 ComparingandOrderingNumbers 45
NS3-10 Differencesof10and100 46
NS3-11 ComparingNumbers(Advanced) 48
NS3-12 Countingby2s 49
NS3-13 Countingby5sand25s 50
NS3-14 Countingby2s,3sand5s 51
NS3-15 CountingBackwardsby2sand5s 52
NS3-16 Countingby10s 53
NS3-17 Countingby2s,3s,4s,5sand10s 54
Contents
NS3-18 Countingby100s 55
NS3-19 Regrouping 56
NS3-20 Regrouping(Advanced) 58
NS3-21 Adding2-DigitNumbers 59
NS3-22 AddingwithRegrouping(orCarrying) 60
NS3-23 AddingwithMoney 62
NS3-24 Adding3-DigitNumbers 63
NS3-25 Subtracting2-and3-DigitNumbers 65
NS3-26 SubtractingbyRegrouping 67
NS3-27 SubtractingbyRegroupingHundreds 69
NS3-28 MentalMath 70
NS3-29 PartsandTotals 72
NS3-30 PartsandTotals(Advanced) 74
NS3-31 SumsandDifferences 76
NS3-32 LargerNumbers 77
NS3-33 ConceptsinNumberSense 78
NS3-34 Arrays 79
NS3-35 AddingSequencesofNumbers 81
NS3-36 MultiplicationandRepeatedAddition 82
NS3-37 MultiplyingbySkipCounting 83
NS3-38 MultiplyingbyAddingOn 85
NS3-39 Doubles 87
NS3-40 TopicsinMultiplication 88
NS3-41 ConceptsinMultiplication 90
NS3-42 Pennies,NickelsandDimes 91
NS3-43 Quarters 93
NS3-44 CountingbyTwoorMoreCoinValues 94
NS3-45 CountingbyDifferentDenominations 96
NS3-46 LeastNumberofCoins 98
NS3-47 DimesandPennies 100
NS3-48 MakingChangeUsingMentalMath 101
NS3-49 Lists 103
NS3-50 OrganizedLists 105
Measurement ME3-1 EstimatingLengthsinCentimetres 107
ME3-2 MeasuringinCentimetres 108
ME3-3 Rulers 109
ME3-4 MeasuringCentimetreswithRulers 110
ME3-5 DrawingtoCentimetreMeasurements 111
ME3-6 EstimatinginCentimetres 112
ME3-7 EstimatinginMetres 113
ME3-8 EstimatinginMetres(Advanced) 114
ME3-9 Kilometres 116
ME3-10 OrderingandAssigningAppropriateUnits 118
ME3-11 MeasuringPerimeter 121
ME3-12 Perimeter 123
ME3-13 ExploringPerimeter 124
ME3-14 Investigations 125
ME3-15 MeasuringMass 127
ME3-16 MeasuringCapacity 128
ME3-17 MeasuringTemperature 129
Probability & Data Management PDM3-1 IntroductiontoClassifyingData 130
PDM3-2 VennDiagrams 132
PDM3-3 IntroductiontoTallyingData 135
PDM3-4 ReadingDatafromaTallyChart 136
PDM3-5 IntroductiontoPictographs 137
PDM3-6 PictographScale 138
PDM3-7 DisplayingDataonaPictograph 140
PDM3-8 IntroductiontoBarGraphs 141
PDM3-9 BarGraphs 143
PDM3-10 BarGraphs(Advanced) 145
PDM3-11 CollectingData 146
PDM3-12 PracticewithSurveys 147
PDM3-13 BlankTallyChartandBarGraph 149
PDM3-14 CollectingandInterpretingData 150
Geometry G3-1 SidesandVertices 151
G3-2 IntroductiontoAngles 153
G3-3 EquilateralShapes 155
G3-4 QuadrilateralsandOtherPolygons 156
G3-5 Tangrams 158
G3-6 Congruency 160
G3-7 Congruency(Advanced) 161
G3-8 RecognizingandDrawingCongruentShapes 162
G3-9 ExploringCongruencywithGeoboards 163
G3-10 ExploringCongruencywithGrids 164
G3-11 Symmetry 165
G3-12 LinesofSymmetry 166
G3-13 CompletingSymmetricShapes 167
G3-14 ComparingShapes 168
G3-15 SortingShapesbyProperty 170
G3-16 FindingPolygons 172
G3-17 PuzzlesandProblems 173
PART 2Patterns & Algebra PA3-20 PatternsInvolvingTime 176
PA3-21 Calendars 178
PA3-22 NumberLines 179
PA3-23 MixedPatterns 182
PA3-24 DescribingandCreatingPatterns 185
PA3-25 2-DimensionalPatterns 188
PA3-26 PatternsintheTwoTimesTables 191
PA3-27 PatternsintheFiveTimesTables 192
PA3-28 PatternsintheEightandNineTimesTables 193
PA3-29 PatternsinTimesTables(Advanced) 195
PA3-30 PatternswithIncreasingGaps 196
PA3-31 PatternswithLargerNumbers 197
PA3-32 ExtendingandPredictingPositions 198
PA3-33 Equations 199
PA3-34 AddingandSubtractingMachines 200
PA3-35 Equations(Advanced) 201
PA3-36 ProblemsandPuzzles 202
Number Sense NS3-51 OrdinalNumbers 204
NS3-52 RoundtotheNearestTens 206
NS3-53 RoundtotheNearestHundreds 208
NS3-54 Rounding 210
NS3-55 EstimatingSumsandDifferences 211
NS3-56 Estimating 212
NS3-57 MentalMathandEstimation 214
NS3-58 Sharing– Knowing the Number of Sets 216–KnowingtheNumberofSets 216 216
NS3-59 Sharing–KnowingHowManyinEachSet 218
NS3-60 Sets 220
NS3-61 TwoWaysofSharing 222
NS3-62 Division 225
NS3-63 DividingbySkipCounting 226
NS3-64 DivisionandMultiplication 230
NS3-65 KnowingWhentoMultiplyorDivide 231
NS3-66 Remainders 233
NS3-67 MultiplicationandDivision 235
NS3-68 MultiplicationandDivision(Review) 236
NS3-69 PatternsMadewithRepeatedAddition 238
NS3-70 CountingbyDollarsandCoins 239
NS3-71 DollarandCentNotation 240
NS3-72 CountingandChangingUnits 242
NS3-73 ConvertingBetweenDollarandCentNotation 244
NS3-74 CanadianBillsandCoins 245
NS3-75 AddingMoney 246
NS3-76 SubtractingMoney 248
NS3-77 Estimating 249
NS3-78 EqualParts 251
NS3-79 ModelsofFractions 252
NS3-80 FractionsofaRegionoraLength 254
NS3-81 EqualPartsofaSet 255
NS3-82 PartsandWholes 257
NS3-83 SharingandFractions 260
NS3-84 ComparingFractions 263
NS3-85 FractionsGreaterthanOne 265
NS3-86 PuzzlesandProblems 266
NS3-87 DecimalTenths 267
NS3-88 WordProblems(WarmUp) 269
NS3-89 WordProblems 270
NS3-90 PlanningaParty 271
NS3-91 AdditionalProblems 272
NS3-92 Charts 273
NS3-93 ArrangementsandCombinations 274
NS3-94 ArrangementsandCombinations(Advanced) 275
NS3-95 GuessandCheck 276
NS3-96 Puzzles 277
Measurement ME3-18 AnalogueClockFaces 278
ME3-19 HandsonanAnalogueClock 279
ME3-20 TellingTime→ TheHourHand 280
ME3-21 TellingTime→ Five-MinuteIntervals 282
ME3-22 TellingTime→ PuttingItTogether! 284
ME3-23 DigitalClockFaces 286
ME3-24 Timelines 287
ME3-25 IntervalsofTime 290
ME3-26 EstimatingTimeIntervals 292
ME3-27 CumulativeReviews 293
ME3-28 Area 294
ME3-29 AreainSquareCentimetres 297
ME3-30 HalfSquares 298
ME3-31 PuzzlesandProblems 300
ME3-32 InvestigatingUnitsofArea 302
Probability & Data Management PDM3-15 Outcomes 303
PDM3-16 EvenChances 304
PDM3-17 Even,LikelyandUnlikely 307
PDM3-18 Describing Probability 308DescribingProbability 308 308
PDM3-19 DescribingProbability(Advanced) 309
PDM3-20 FairGames 310
PDM3-21 ExperimentsandExpectation 311
PDM3-22 CumulativeReview 312
Geometry G3-18 IntroductiontoCoordinateSystems 314
G3-19 CoordinateSystems 316
G3-20 IntroductiontoSlides(orTranslations) 318
G3-21 Slides 319
G3-22 Slides(Advanced) 320
G3-23 SlidesonaGrid 321
G3-24 CoordinateSystemsandMaps 322
G3-25 MappingExercise 325
G3-26 Flips 326
G3-27 Reflections 327
G3-28 FlipsandSlides 328
G3-29 Turns 329
G3-30 Rotations 330
G3-31 Rotations(Advanced) 331
G3-32 Flips,SlidesandTurns 332
G3-33 BuildingPyramids 334
G3-34 BuildingPrisms 335
G3-35 Edges,VerticesandFaces 336
G3-36 PyramidNets 339
G3-37 PrismNets 340
G3-38 Prisms and Pyramids 341G3-38 PrismsandPyramids 341
G3-39 DrawingPyramidsandPrisms 342
G3-40 PropertiesofPyramidsandPrisms 343
G3-41 Sorting3-DShapes 345
G3-42 ClassifyingShapesandMakingPatterns 346
G3-43 GeometryintheWorld 348
G3-44 ProblemsandPuzzles 350
Patterns & Algebra Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
A Note for Patterns Before students can create or recognize a pattern in a sequence of numbers, they must be able to tell how
far apart the successive terms in a particular sequence are. There is no point in introducing students to
sequences if they don’t know how to find the gap between a given pair of numbers, either by applying their
knowledge of basic addition and subtraction, or by counting on their fingers as described below.
For weaker students, use the following method for recognizing gaps:
How far apart are 8 and 11?
STEP 1:
Say the lower number (“8”) with your fist closed.
8
STEP 2: Count up by ones, raising your thumb first, then one finger at a time until you have reached
the higher number (11).
9 10 11
STEP 3: The number of fingers you have up when you reach the final number is the answer
(in this case you have three fingers up, so three is the difference between 8 and 11.)
Using the method above, you can teach even the weakest student to find the difference between two
numbers in one lesson. (You may have to initially hold your student’s fist closed when they say the first
number—some students will want to put their thumb up to start—but otherwise students find this method
easy.)
Eventually, you should wean your student off using their fingers to find the gap between a pair of
numbers. The exercises in the MENTAL MATH section of this manual will help with this.
Patterns & Algebra Teacher’s Guide Workbook 3:1 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-1 Counting
Goal: Students will find differences between numbers mentally, by using fingers or by using a number line.
Prior Knowledge Required: Count to 100
Vocabulary: difference, number line.
To introduce the topic of patterns, tell your students that they will often encounter patterns in sequences of
numbers in their daily lives. For instance, the sequence 25, 20, 15, 10... might represent the amount of
money in someone’s savings account (each week) or the amount of water remaining in a leaky fish tank
(each hour).
Ask your students to give other interpretations of the pattern. You might also ask them to say what the
numbers in the pattern mean for a particular interpretation. For instance, if the pattern represents the
money in a savings account, the person started with $25 and withdrew $5 each week.
Tell your students that you would like to prepare them to work with patterns and you will start by making
sure they are able to find the difference between two numbers. There is nothing wrong with counting on
their fingers until they have learned their number facts—they are our built-in calculator. If there are students
who know their subtraction facts, tell them they are in perfect shape and can do the calculations mentally.
Show students how to find the difference between two numbers using the method described in the note at
the beginning of this unit. (Say the smaller number with your fist closed. Count up to the larger number
raising one finger at a time. The number of fingers you have raised when you say the larger number is the
difference).
NOTE: Before you allow students to try any of the questions on a particular worksheet you should assign
sample questions (like the ones on the worksheet) for the whole class to discuss and solve. You should
also give
a mini quiz or assessment consisting of several questions that students can work on independently (either
in a notebook or individual pieces of paper). The quiz or assessment will tell you whether students are
ready to do the work on the worksheet.
The bonus questions provided in a lesson plan may either be assigned during the lesson, (to build
excitement, by allowing students to show off with harder looking questions) or during the assessments (for
students who have finished their work early, so you have time to help slower students).
Sample Questions:
Find the difference:
a) 3 7 b) 23 25 c) 39 42 d) 87 92
Bonus:
a) 273 277 b) 1768 1777
c) 1997 2003 d) 32 496 32 502
Patterns & Algebra Teacher’s Guide Workbook 3:1 3 Copyright © 2007, JUMP Math For sample use only – not for sale.
Assessment:
Find the difference:
a) 4 7 b) 21 25 c) 37 42 d) 70 77
To introduce the topic on the second worksheet for this unit draw a number line on the board.
Show your students how to find the difference between 6 and 9 as shown below:
What number added to 6 gives 9? 6 + ? = 9
Count 3 spaces between 6 and 9 on a number line, checking with each hop:
Add 1, get 7, 7 is 1 more than 6.
Add 2, get 8, 8 is 2 more than 6.
Add 3, get 9, 9 is 3 more than 6,
SO: 6 + 3 = 9 AND: 3 is called the difference between 9 and 6
Ask several volunteers to come to the board and find the difference between 4 and 10, 2 and 7, 0 and 8.
Draw another number line and ask students to find the difference (or gap) between the numbers below.
a) 10 12 b) 15 25 c) 15 20 d) 14 19 e) 12 18 f) 18 24
Tell your students that the expression 33 36 (where they are expected to fill in the gap) could also be
written 33 + ____ = 36.
Ask a couple of volunteers to write the old exercises in the new way and to fill in the missing number.
4 10 4 + ____ = 10
Assessment:
0 1 2 3 4 5 6 7 8 9 10
1 2 3
15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 29
10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 24
?
Patterns & Algebra Teacher’s Guide Workbook 3:1 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
Find the missing number by counting up from the smaller to the larger number.
a) 20 + ____ = 23 b) 23 + ____ = 28 c) 18 + ____ = 25
d) 24 + ____ = 29 e) 22 + ____ = 31
Point at a pair of numbers on the number line, say 16 and 13. Ask your students to find the difference
between them. Which number is larger? How much larger than 13 is 16? Give an example—Jack has 13
stickers, Jill has 16. Who has more? How many more does Jill have?
Sample Questions:
Fill in the missing numbers:
a) 28 is ____ more than 22 b) 29 is ____ more than 25 c) 36 is ____ more than 25
Bonus:
Fill in the missing numbers:
a) 67 is ____ more than 63 b) 78 is ____ more than 71 c) 83 is ____ more than 78
d) 107 + ____ = 111 e) 456 + ____ = 459 f) 987 + ____ = 992.
NOTE: It is extremely important that your students learn their number facts and eventually move beyond
using their fingers. We recommend that you use the exercises in the MENTAL MATH section of this Guide
to help students learn their number facts.
Activity: Let your students play the Modified Go Fish Game in pairs (see the MENTAL MATH section for
instructions). Start with the number 5 as the target number and proceed gradually to 10. This will help them
find larger differences between numbers.
Patterns & Algebra Teacher’s Guide Workbook 3:1 5 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-2 Preparation for Increasing Sequences
Goals: Students will find a number that is more than another number using a given difference.
Prior Knowledge Required: Count to 100. Find difference between two numbers.
Vocabulary: difference
This worksheet is an essential step before starting increasing sequences.
Warm-up:
Add the number in the circle to the number beside it. Write your answer in the blank:
a) 6 _____ b) 7 _____ c) 9 _____
d) 19 _____
e) 17 _____ f) 26 _____ g) 14 _____ h) 23 _____
Then show an example: What is 4 more than 11? Use finger counting. Practice questions like:
a) _____ is 5 more than 7 b) _____ is 8 more than 3 c) _____ is 6 more than 9
Point out to your students that questions like the ones they just solved are the reverse of those they solved
in the last lesson—they now know the difference, but do not know one of the numbers.
Assessment:
Fill in the missing numbers:
a) _____ is 3 more than 13 b) _____ is 6 more than 15 c) _____ is 10 more than
26
d) _____ is 4 more than 32 e) _____ is 7 more than 21 f) _____ is 8 more than 17
Bonus:
Fill in the missing numbers:
a) _____ is 4 more than 59 b) _____ is 6 more than 75
2 3 2 4
5 3 6 8
Patterns & Algebra Teacher’s Guide Workbook 3:1 6 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-3 Increasing Sequences
Goal: Students will extend increasing sequences and solve simple problems using increasing sequences.
Prior Knowledge Required: Count to 100. Find difference between two numbers.
Vocabulary: difference, increasing sequence, pattern, number line.
In this lesson, weaker students can use their fingers to extend a pattern as follows:
EXAMPLE: Extend the pattern 3, 6, 9… up to six terms.
STEP 1: Identify the gap between successive pairs of numbers in the sequence (You may count on
your fingers, if necessary). The gap, in this example, is three. Check that the gap between
successive terms in the sequence is always the same, otherwise you cannot continue the pattern
by adding a fixed number. (Your student should write the gap between each pair of successive
terms above the pairs.)
3 , 6 , 9 , _____ , _____,
STEP 2: Say the last number in the sequence with your fist closed. Count by ones until you have
raised three fingers (the gap between the numbers). The number you say when you have raised
your third finger is the next number in the sequence.
3 , 6 , 9 _____ , _____ ,
STEP 3: Continue adding terms to extend the sequence.
3 , 6 , 9 , _____ , _____ , _____
To start the lesson write the sequence: 1, 2, 3, 4, 5, … on the board. Ask your students what the next
number will be. Then write the sequence 2, 4, 6, 8, … Ask what the next number is this time. Ask students
how they guessed that the next number is 10 (skip counting by 2). Ask what they did to the previous
number to get the next one. Ask what the difference between the numbers is.
Tell your students that the next question is going to be harder, but that you still expect them to find the
answer. Write the sequence 2, 5, 8, 11… with room to write circles between the numbers:
2 , 5 , 8 , 11 ,
Ask students if they can see any pattern in the numbers. If your students need help, fill in the difference
between the first two terms in the circle.
2 , 5 , 8 , 11
Add another circle and ask students to find the difference between 5 and 8.
2 , 5 , 8 , 11 .
3
3
3 3
3 3
12
12 18 15
Patterns & Algebra Teacher’s Guide Workbook 3:1 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
After students have found the difference between 8 and 11, ask them if they can predict the next term in the
sequence.
Explain that sequences like the one you just showed them are common in daily life. Suppose George wants
to buy a skateboard. He saves $7 during the first month and, in each of the next months, he saves $4 more.
How much money does he have after two months? Write 7 _____ , and ask a volunteer to fill in the
next number.
Ask volunteers to continue George’s savings’ sequence. Mention that patterns of this kind are called
“increasing sequences” and ask if the students can explain why they are called increasing. Point out that in
all of the exercises so far the difference between terms in sequence has been the same. Later your
students will study more advanced sequences in which the difference changes.
Extend the patterns:
a) 6 , 9 , 12 , 15 , _____ , _____ b) 5 , 11 , 17 , 23 , _____
c) 2 , 10 , 18 , 26 , _____
Do not progress to word problems before all your students can extend a given pattern.
Bonus:
a) 99, 101, 103, … b) 654, 657, 660, …
Now ask your students to solve the following problem:
A young wizard learned three new incantations every day during his studies at the wizards’ school. One
sunny Monday morning he told his friend that he already knows 12 incantations. How many incantations
will he know by Wednesday?
Draw a table on the board:
Number of
incantations 12
Days Monday Tuesday Wednesday
Students should see that to solve the problem they need to extend an increasing sequence (the “gap” here
is 3, as the wizard learns 3 spells per day).
Bonus:
A baby elephant grows 2 cm a day. Today it is 140 cm tall. What will its height be tomorrow?
When will it reach a height of 150 cm?
Sample Word Problems:
1. A newborn Saltwater Crocodile is about 25 cm long. It grows 5 cm in a month during the
first 4 months. How long is a 2-month-old Saltie? 4-mth old one? If its length is 40 cm,
how old is it?
2. An apple tree sapling grows 3 cm in a month. On May 1, it is 10 cm tall. What will its height be on
August 31?
4
Patterns & Algebra Teacher’s Guide Workbook 3:1 8 Copyright © 2007, JUMP Math For sample use only – not for sale.
Extensions:
1. For each sequence, create a word problem that goes with the sequence.
a) 7 , 9 , 11 , 13 , ___ , ___ b) 5 , 9 , 13 , 17 , ___ , ___
2. ANOTHER GAME: Each group needs two dice (say blue and red) and a token for each player. Use the
“Number Line Chasing Game” from the BLM. The blue die shows the difference in the pattern. The red
die shows the number of terms in the sequence to calculate. The players start at 1. If he throws 3 with
the blue die and 2 with the red die, he will have to calculate the next
2 terms of the sequence: 1 , _____ , _____ (4 and 7). 7 is a “smiley face”. When the player lands
on a “smiley face”, he jumps to the next “smiley face”, in this case 17. His next sequence will start at 17.
The first player to reach 100 is the winner.
Activities:
1. A Game for Pairs
One student writes a number, the other writes the difference in the increasing sequence. Then the
players take turns writing sequence terms. They have to continue the sequence up to 6 numbers.
2. Advanced
The first player writes a number and the second player writes the second number in the sequence. The
first player has to find the difference and then continue the sequence.
3 3
Patterns & Algebra Teacher’s Guide Workbook 3:1 9 Copyright © 2007, JUMP Math For sample use only – not for sale.
0 1 2 3 4 5 6 7 8 9 10
3 2 1
PA3-4 Counting Backwards
Goal: Students will find the difference between two numbers counting backwards using fingers and
a number line.
Prior Knowledge Required: Count backwards from 100 to 1.
Vocabulary: difference
ASK A RIDDLE: I am a number between 1 and 10. Subtract me from 8, and get 5. Who am I?
Explain that you can use a number line to find the difference. Draw the number line on the board and show how to find the difference by counting backwards. WRITE: “5 is 3 less than 8.” Show a couple more examples on the board, this time using volunteers.
Sample problems:
I am a number between 1 and 10. Who am I?
a) If you subtract me from 26, you get 21.
b) If you subtract me from 32, you get 27.
c) If you subtract me from 34, you get 26.
Ask your students how they could solve the problem when they do not have a number line. Remind them
how to find the difference counting backwards on their fingers.
You might need to spend more time with students who need more practice finding the gap between pairs of
numbers by subtracting or counting backwards. Your students will not be able to extend or describe
patterns if they cannot find the gap between pairs of numbers. You also might use flashcards similar to
those you used with the worksheet PA3-1.
Draw two number lines on the board:
20 21 22 23 24 25 26 27 28 29 30
20 21 22 23 24 25 26 27 28 29 30
20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 34
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What number added to 24 is 27? Solve by counting forward on the upper line.
What number is subtracted from 27 to get 24? Solve by counting backward on the lower line. (Draw arrows
to indicate the “hops” between the numbers on the number line).
What are the similarities between the methods? What are the differences?
Students should notice that both methods give the difference between 24 and 27 and are merely different
ways to subtract. ASK: How many times did you hop? What did you measure? (distance, difference, gap)
What is this called in mathematics? What other word for the gap did we use? (difference).
When you measured the distance (or difference) between two points on the number line, the first time you
went from left to right, and the second time you went from right to left.
What operation are you performing when you go from the smaller number to the larger? (adding) From the
larger number to the smaller one? (subtracting)
You might even draw arrows: from right to left and write “–”, and from left to right and write “+” below and
above the respective number lines.
Congratulate your students on discovering an important mathematical fact—addition and subtraction are
opposite movements on a number line, and both counting backwards and forwards are used to find the
difference.
Assessment:
MORE RIDDLES: I am a number between 1 and 10. Who am I?
a) If you subtract me from 27, you get 21.
b) If you subtract me from 32, you get 24.
c) If you subtract me from 34, you get 27.
Bonus: Find the gap between the numbers:
a) 108 105 b) 279 274 c) 241 238 d) 764 759
20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 34
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PA3-5 Preparation for Decreasing Sequences
Goal: Students will find the number that is less than the other number using a given difference.
Prior Knowledge Required: Count to 100. Find the difference between two numbers.
Vocabulary: difference
Repeat the lesson PA3-2: Preparation for Increasing Sequences, using decreasing sequences and
counting backwards.
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PA3-6 Decreasing Sequences
Goal: Students will find the differences between two numbers by subtraction, and extend decreasing
sequences.
Prior Knowledge Required: Count to 100. Find difference between two numbers.
Vocabulary: difference, decreasing sequence, pattern
Remind your students what an increasing sequence is. Ask them where they encounter increasing
sequences in life. POSSIBLE EXAMPLES: height, length, weight of plants and animals, growing buildings,
etc. Are all sequences that they will encounter in life increasing? Ask students to discuss this point and give
examples of sequences that are not increasing.
Give an example, such as: Jenny was given $20 as a monthly allowance. After the first week she still had
16 dollars, after the second week she had 12 dollars, after the third week she had 8 dollars. If this pattern
continues, how much money will she have after the fourth week? The amount of money that Jenny has
decreases: 20, 16, 12, 8, so this type of sequences is called a “decreasing sequence”. Write the term on the
board and later include it in spelling tests together with “increasing sequence”.
Have your students do some warm-up exercises, such as:
1. Subtract the number in the circle from the number beside it. The minus reminds you that you are
subtracting. Write your answer in the blank:
a) 13 – 2
_____ b) 12 – 6
_____ c) 11 – 8
_____ d) 9 – 5
_____
2. Fill in the missing numbers:
a) _____ is 4 less than 37 b) _____ is 5 less than 32
Then write two numbers on the board:
14 , 11
Ask what the difference between the numbers is. Write the difference in the circle with a minus sign then
ask students to extend the sequence.
14 , 11 , 8 , _____ , _____
Assessment:
Extend the decreasing sequences:
a) 21 , 19 , 17 , 15 , _____ , _____ c) 48 , 43 , 38 , 33 , _____ , _____
b) 34 , 31 , 28 , _____ , _____ , _____
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Bonus:
Extend the decreasing sequences:
a) 141 , 139 , 137 , 135 , _____
b) 548 , 541 , 534 , 527 , _____
c) 234 , 224 , 214 , _____ , _____
Activity:
A Game for Two
The students will need two dice and a hundreds chart (to help with counting backwards). The first player
throws both dice (blue and red). The number on the red die determines how many numbers in the sequence
the player has to calculate and the number on the red die indicates the difference of the decreasing
sequence. The first number in this sequence is 100. The next player starts at the number the first player
finished at. The player who ends at 0 or below is the winner.
EXAMPLE: The first throw yields red 4, blue 3. The first player has to calculate four numbers in the
sequence whose difference is 3 and whose first number is 100. The player calculates: 97, 94, 91, 88. The
next player starts with 88 and rolls the dice.
ADVANCED VERSION: Give 1 point for each correct term in the sequence and 5 points to the first player
to reach 0. The player with the most points is the winner.
ALTERNATIVE: Use the “Number Line Chasing Game” from the BLM. Each player uses a moving token,
similar to the game in section PA3-3.
Extension: If you used the “Number Line Chasing Game” from the BLM or the Activity above, your
students might explore the following question:
Two players start at the same place. John throws the dice and receives red 3 and blue 4.
Mike gets red 4 and blue 3. Do they end at the same place? Does this always happen? Why?
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PA3-7 Increasing and Decreasing Sequences
Goal: Students will distinguish between increasing and decreasing sequences, and extend sequences.
Prior Knowledge Required: Count to 100. Find difference between two numbers
Increasing sequences Decreasing sequences
Vocabulary: difference, increasing sequence, decreasing sequence, pattern
Remind your students about how they can extend a sequence using the gap provided. Give examples for
both increasing and decreasing sequences. For increasing sequences, you might draw an arrow pointing
upwards beside the sequence, and for decreasing sequences, an arrow pointing downwards.
Write several sequences on the board and ASK: Does the sequence increase or decrease? What sign
should you use for the gap? Plus or minus? What is the difference between the numbers?
Let your students practice finding the next term in a sequence using the gap provided:
a) 4 , 9 , … b) 25 , 22 , … c) 23 , 26 , … d) 49 , 44 , …
Ask students what they would do if they have the beginning of the sequence but they do not know the gap:
4 , 7 , 10 , …
First ask if the sequence goes up or down. Is the sign ‘+’ or ‘–‘? Then find the difference. When the gaps
are filled in, extend the sequence.
Assessment:
Extend the sequences. First find the difference (gap). (GRADING TIP: Check to see if the students have
the correct “gap”):
a) 21 , 18 , 15 , ____ , ____ , ____ d) 51 , 49 , 47 , ____ , ____ , ____
b) 34 , 36 , 38 , ____ , ____ , ____ e) 84 , 74 , 64 , ____ , ____ , ____
c) 47 , 50 , 53 , ____ , ____ , ____ f) 60 , 65 , 70 , ____ , ____ , ____
+5 –3 +3 –5
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Bonus:
a) 128 , 134 , 140 , ____ , ____ , ____
b) 435 , 438 , 441 , ____ , ____ , ____
c) 961 , 861 , 761 , ____ , ____ , ____
Activity:
A Game
Calculating three terms of a sequence each time, one point for each correct term. Students will need two
dice of differentcolours (say red and blue) and a cardboard coin with the signs “+” and “–” written on its
sides. The students throw the dice and the coin.
The number on the die is the difference between terms in sequence. The coin determines whether the
sequence is increasing (“+”) or decreasing (“-“). For example, if the die shows 4 and the coin shows “-“,
the student has to write: 30, 26, 22, 18. Give a point for each correct term.
Students could also play the game using the Number Line Chasing Game from the BLM.
Extension:
1. You could make up some fanciful word problems for your students, possibly based on things they are
reading:
A newborn dragon (called Herbert by its proud owner) grows 50 cm every day. At hatching, it is 30 cm
long. How long will Herbert be in 3 days? In a week?
2. A GAME FOR PAIRS: The first player throws a die so that the second player does not see the result.
The die indicates the difference between terms in a sequence. The first player chooses the first term of
the sequence. The first player then gives the second player the first and the third terms in the
sequence.
For instance, if the die gives 3, the first player might select 13 as the first term and write: 13, _____ , 19.
The second player then has to guess the second term and the difference. In this case, the second term
is 16.
(The trick is to find the number that is exactly in the middle of the two given terms) After students have
tried the game, you might ask them to make the T-table of differences:
Students will be soon able to see the pattern.
The die The difference between the first
and the second terms
The difference between the first
and the third terms
1
2
…
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PA3-8 Attributes
Goal: Students will distinguish the attributes that change in a sequence, and make sequences with the
given number of changing attributes.
Prior Knowledge Required: Geometric shapes: triangle, square, circle, pentagon
Vocabulary: attribute,colour, shape, size
Draw two similar triangles of different sizes on the board. Ask your students to describe the shapes. At this
stage, the description can be simple: large triangle, small triangle. Start by making a table, adding columns
and rows when you need them.
Shape triangle triangle
Size large small
Add another triangle of a different colour. Add the “colour” row. Then add a circle.
Shape triangle triangle triangle circle
Size large small small small
Colour white white red white
Add more shapes and ask volunteers to fill in the table.
You might also introduce attributes like orientation, visual pattern (dots, stripes, etc) and others.
FOR EXAMPLE:
1. Large red triangle, new attribute—orientation.
2. Large circle, new attribute—pattern.
Explain that the properties are called “attributes”, and write that word in the top-left cell of your table.
Ask students to practice in pairs—the first player says a shape and lists two attributes. The second player
then draws two versions of that shape: both shapes should have the first attribute, but only one shape
should have the second attribute. For example, the first player says “triangle, colour, size” and the second
player then draws two orange triangles, one large and one small.
For practice, give your students several patterns and ask them to write the attribute that changes. If
students have problems deciding which attribute changes you might give them a limited list to choose from.
W W
W W R W
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Sample Patterns:
a) b)
c)
Assessment:
Write the attribute that changes in each pattern:
a)
b)
Activity:
A Game for Pairs or Groups
Give your students a set of pattern blocks or beads. The beads may differ not only in size, colour and
shape, but also in texture (rough, smooth), visual pattern (striped, solid), etc. One player lays out three
shapes that share one attribute. The other has to add a shape that shares the same attribute.
If the second player succeeds, they receive a point. If not, the first player has to add another object with the
same attribute to help his peer. For a more advanced version ask students to say which attributes remain
the same and which ones change in their sequence of shapes. How many attributes change, how many
remain the same? It is a good exercise to ask students to write down all the possible attributes for their set
of shapes before they start playing. The second player only has two chances to guess the common attribute
before the game starts again, with the players roles reversed.
Extensions:
1. When your students have learned some terms from geometry, you can play the following game: Set out
three shapes that all share a common attribute and a fourth shape that lacks that attribute. Ask your
students to tell you which shape doesn’t belong in the set (and why).
EXAMPLES:
Three of the shapes have exactly one right angle. Three of the shapes are 4-sided. The parallelogram doesn’t belong. The triangle doesn’t belong.
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2. Circle the word that tells you which attribute of a figure or figures changes in the pattern:
a) b)
shape position size size orientation number
c) d)
number size shape shape position size
e) f)
size orientation number number size shape
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PA3-9 Patterns Where Two Attributes Change
Goal: Students will distinguish the attributes that changed in a sequence, and make sequences where two
attributes change.
Prior Knowledge Required: Geometric shapes: triangle, square, circle,
pentagon, cube, cylinder, cone, ball
Vocabulary: attribute, shape, colour, size.
Remind your students of what an attribute is are and ask them to tell you which attribute changed in the
following pattern:
Ask your students to name some attributes in a collection of objects, and to draw a sequence where at least
one of the attributes changes. If any of the following attributes are missing from the student’s list you should
add them to the list: shape, size,colour, visual pattern, number and position.
Give your students several patterns where two attributes change and ask them to name the attributes that
change.
a) b) c)
For an activity students could play the advanced version of the game in the Activity from the previous
section.
Assessment:
Write the attributes that changed in each pattern. How many attributes changed?
Activity:
SET® is an excellent game for developing pattern recognition skills. For an easy introduction, use the
SET® game with only two attributes—for example, take only red filled shapes.
If you do not have the game, go to http://www.setgame.com/ for rules. You may also find the online
version there.
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Extensions:
1. Show your students a necklace or bracelet and ask them to describe how the pattern changes:
Students might mention colour, size, shape, texture (rough, smooth), visual pattern (striped, solid), etc.
2. Give each student a set of pattern blocks or beads and ask them to pick out a pair of shapes that differ
by one attribute, two attributes etc. Ask them to find a pair of shapes that differ by the greatest number
of attributes. (For instance, they might pick a pair of beads that differ in shape, size, colour, texture
(rough, smooth), visual pattern (striped, solid), etc.)
3. Circle the two words that tell you which attributes of a figure or figures changes in the pattern:
a)
shape orientation size number colour
b)
orientation size number shape colour
c)
orientation size number shape colour
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PA3-10 Repeating Patterns
Goal: Students will find the core of the pattern and continue the repeating pattern.
Prior Knowledge Required: Ability to count. Attributes.
Vocabulary: attribute, length of core, core
Explain that a pattern is repeating if it consists of a “core” of terms or figures repeated over and over.
Show an EXAMPLE:
core
The core of the pattern is the part that repeats. Circle it. Write several more examples, such as:
a) b)
c) R A T R A T R A d) S T O P S T O P S e) u 2 u 2 u 2 u 2 u
Show the pattern M O M M O M and extend it as follows: M O M M O M O M M O M. Ask students if they
agree with the way you extended the sequence. You may pretend not to see your mistake or even to ask
the students to grade your extension. Give more difficult examples of sequences where the core starts and
ends with the same symbol, such as:
a) b) A Y A A Y A A Y c) 1 2 3 1 1 2 3 1 1
Warn your students that these patterns are trickier.
Explain that the “length” of the core is the number of terms in the core.
For instance, the pattern 3, 2, 7, 3, 2, 7… has a core of length 3. It’s core is “3, 2, 7”. Ask volunteers to find
the length of the cores for the examples you’ve drawn.
Assessment:
Circle the core of the pattern, and then continue the pattern:
a) C A T D O G C A T D O G C A T ___________________________________ Core length: _____
b) A M A N A A M A N A A M A ______________________________________ Core length: _____
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Circle the core of the pattern, and then continue the pattern.
Activity: Give your students a set of beads or pattern blocks and ask them to make a repeating pattern in
which one or more attributes change. Ask them to describe precisely how their pattern changes: EXAMPLES:
a)
In this example only the size of the shape changes.
b)
In this example both the size and the colour of the shape changes.
Extensions:
1. Can you always find the core of a pattern that repeats in some way? Do the following sequences have a
core? No. Can you still continue the pattern?
a) b)
2. What are the similarities and the differences of the two patterns of coloured counters below?
Pattern 1
Pattern 2
Explain to your students that these patterns were made by two different students according to the same
rule. Both students had counters of two colours only, and the rule described the way the attributes
changed: change colour, change colour, stay the same, change colour, change colour, stay the same,
repeat. Ask your students to describe the following patterns in terms of the change of attributes:
a)
b)
R
R
R
R
B R R B B B
B R R B B B
R
R
R
R
R W R R R W
B Y B B B Y
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3. Make a pattern of your own with blocks or beads, changing at least two attributes. Explain which
attributes you used in making your pattern, and how the attributes changed. Try to make both
sequences that have cores and sequences that do not. To make a pattern, you can change the colour,
shape, size, position or orientation of a figure or you can change the number of times a figure
occurs. Try to describe the change in the attribute as a rule.
EXAMPLES:
The shape changes, the size, colour and position stay the same.
The core consists of two squares and two circles.
The rule for the change of attributes is: Start with a small square. Stay the same, change shape, stay the
same, change shape, repeat.
The size changes, the shape, colour and position stay the same.
The core consists of one large square and two small squares.
The rule for the change of attributes is: Start with a large square. Change size, stay the same, change
size, change size, stay the same, change size, repeat.
The size and position change, the shape and the colour stay the same.
The core consists of one large square and two small squares.
The rule for the change of attributes is: Start with a large square. Change size and turn the square, turn
the square, change size, change size and turn the square, turn the square, change size, repeat.
Draw your pattern.
3. Ask your students to name some attributes of articles of clothing. For instance, shirts may have:
long sleeves short sleeves collars no collars stripes checks
• Make a list of the attributes on the board.
• Think of a simple repeating pattern that involves those attributes.
• Arrange students who are wearing articles of clothing that fit your pattern in a row.
• The rest of the class should try to guess the rule for your pattern.
Make your students aware that patterns result from repeating an action (say in music), operation,
transformation or in making some other change (colour, shape orientation, etc).
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PA3-11 Extending Repeating Patterns
Goal: Students will continue a repeating pattern given its core.
Prior Knowledge Required: Ability to count. Attributes.
Vocabulary: attribute, core, length of core, repeating pattern
Draw several groups of figures that are the cores of repeating patterns. Ask your students to continue the
patterns. You might also use the activity below for practice.
Draw several cores and extend some of the patterns in a wrong way. Ask the students to grade your work,
and then to correct your mistakes. Sample sequences:
R W W R W W R R W W R W W R R W W R W W R W R W R W R
Assessment:
Check if the following sequences were extended correctly and if not, write the correct extensions.
R W W R W W R R W W R W R R R W R R R W R W R W R W R
A A B B A A B B A C A C A A C A C A G Y G G Y G G Y
Activity: A game in pairs: The students will need a pair of dice and a set of coloured beads of different
shapes, patterns and sizes. If beads are not available, students might use pattern blocks or draw the
patterns. Make a list of at least 6 attributes of the shapes. The first player rolls the dice. He has to build the
core of the pattern, where the larger number on the dice is the length of the core, the other number is the
number of attributes that should change in the core. The second player has to continue the pattern and to
name the attributes that changed in the core.
For example, if the roll is 5 and 3; the attributes that change might be pattern, size and orientation:
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PA3-12 Finding Cores in Patterns
Goal: Students will find cores of repeating patterns.
Prior Knowledge Required: Ability to count. Attributes. Ordinal numbers.
Vocabulary: attribute, core, length of core, repeating pattern
Show your students how to recognize the core of a pattern. Draw a sequence of blocks with a simple
repeating pattern like the shown one below (where B stands for blue and Y for yellow).
Then draw a rectangle around a set of blocks in the sequence and follow the steps below:
STEP 1:
Ask students to say how many blocks you have enclosed in the rectangle (in this case, three).
STEP 2:
Ask students to check if the pattern (BYY) inside the rectangle recurs exactly in the next three blocks of the
sequence, and in each subsequent block of three, until they have reached the end of the sequence (if you
had enclosed four blocks in the rectangle, then students would check sets of four, and so on). If the pattern
in the rectangle recurs exactly in each set of three boxes, as in the diagram above, then it is the core of the
sequence. Otherwise students should erase your rectangle, guess another core and repeat steps 1 and 2.
(Students should start by looking for a shorter core than the one you selected.)
You could also have your students do this exercise with coloured blocks, separating the blocks into sets,
rather than drawing rectangles around them. Draw several sequences on the board and circle the cores in
some of them in the wrong way. Ask your students to grade your work. If they think you have not got it right,
ask them to correct the mistake.
EXAMPLES:
A A L L A A L L A A A S A S A A S A S A G E G G E G G E
Find the core and the length of the core:
R E D D R E D D D A D D A D D A G R E G G R E G
B Y Y B Y Y B Y Y
B Y Y B Y Y B Y Y
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Draw a pattern on the board:
Ask volunteers to circle the core of the pattern, find its length and continue the pattern. Ask which figures are
circles. (1st, 3
rd, 5
th, etc) What will the 20
th figure be – a circle or a diamond? Write down the sequence of the
places for circles (1, 3, 5, 7, …) and diamonds (2, 4, 6, 8, …), and ask students to say which sequence the
number 20 belongs to. Challenge them to predict the shapes of 36th, 55
th, 99
th figures.
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PA3-13 Making Patterns with Squares and PA3-14 Making Patterns with Squares (Advanced) 14
Goal: Students will make patterns with blocks and squares.
Prior Knowledge Required: Pattern extension
Vocabulary: attribute, core, length of core, repeating pattern
Lay out a couple of blocks and ask volunteers to add blocks in the positions you indicate. After that draw
several simple shapes on the board and ask more volunteers to draw an extra block in each of the places
marked by an arrow.
Sample problems:
After that draw several sequences of blocks, first adding only one block at a time, then two and or more
blocks, and ask your students to identify which blocks were added each time.
Assessment:
1. Shade two squares that were added to the figure.
2. Shade the squares added to each previous figure to get the next one in the pattern. Then draw the next
figure in the pattern on grid paper.
Activities:
1. Ask your students to build the sequence of QUESTION 5 a) of the worksheets using blocks. Then ask
them to build the next figure in the sequence. Repeat for QUESTIONS 5 b), c), and d).
2. Teach your students the terms “horizontal”, “vertical”, “column”, “row”, “right” and “left”. Ask them to
describe as precisely as they can (using mathematical terms) how the changes in QUESTION 5 a)
occur. For instance, they might say “the horizontal arm grows by one block each time” or even more
precisely: “You start with a column of 2 blocks. Place one block beside the top block on the right hand
side. Then place another block beside this one…” and so on.
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3. Have two students sit across from each other at a table. Place a barrier between the students to that
they can see each other but neither student can see the part of the table directly in front of the other
student. The first student builds a sequence of figures that grows in a regular way (as in QUESTION 5 a)
and describes their sequence as they build it. The second student has to try to build the same sequence.
The students then raise the barrier to see if their sequences match.
Extension: Give each student a set of 40 blocks of the same shape (either squares or triangles) and ask
them to build a sequence of 3 shapes in which the number of blocks in each shape grows by a fixed amount
(as in QUESTION 1). Have them count the number of blocks they have left after making 3 shapes and
predict using a T-table or calculator whether they have enough blocks to build the next step (or the next two
steps) in the pattern.
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PA3-15 Extending a Pattern Using a Rule
Goal: Students will extend patterns using a verbal rule.
Prior Knowledge Required: Addition. Subtraction. Extension of patterns. Sequence.
Vocabulary: pattern rule
This section provides a basic introduction to pattern rules. For more advanced work including word
problems, applications and communication, see sections PA3-20 TO 32.
Write a number sequence on the board: 10, 13, 16, 19, …
Draw the circles for differences.
10 , 13 , 16 , 19 , _____
Then ask them if they could describe the sequence without simply listing the individual terms. For a
challenge you might ask students to work in pairs. One student writes an increasing sequence without
showing their partner. The student must get their partner to reproduce their sequence but they are not
allowed to give more than one term of the sequence. Students should see that the best way to describe a
sequence is to give the first term and the difference between terms. The sequence above could be
described by the rule “Start at 10 and add 3 to get the next term.”
For practice, give your students questions such as:
Continue the patterns:
a) (add 3) 14, 17, ____ , ____ , ____ , ____ b) (subtract 2) 35, 33, ____ , ____ , ____ , ____
c) (add 5) 12, 17, ____ , ____ , ____ , ____ d) (subtract 10) 77, 67, ____ , ____ , ____ , ____
e) (add 4) 15, 19, ____ , ____ , ____ , ____ f) (subtract 5) 97, 92, ____ , ____ , ____ , ____
Create a pattern of your own: ____ , ____ , ____ , ____
My rule: ___________________________________________________________________________
For practice you might assign the game in the activity below.
Tell your students the following story and ask them to help the friends:
Bonnie and Leonie are two Grade 3 friends who are doing their homework together. The sequence in
their homework is: 43, 36, 29, 22, … Bonnie said: “If you count backwards from 43 to 36 you say eight
numbers:
1 2 3 4 5 6 7 8
43, 42, 41, 40, 39, 38, 37, 36
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So the difference between 43 and 36 is 8.”
Leonie said “If you count the number of “hops” on a number line between 43 and 36, you see there are
7 hops so the difference is 7.”
Who is right?
Assessment:
Continue the patterns:
a) (add 4) 32 , 36 , ____ , ____ , ____ , ____ b) (subtract 6) 77 , 71 , ____ , ____ , ____ , ____
Create a pattern of your own: ____ , ____ , ____ , ____
My rule: ___________________________________________________________________________
Activity:
A Game
The students need a die and a coin. The coin has the signs “+” and “–” on the sides. The player throws the
coin and the die and uses the results to write a sequence. The die gives the difference. The coin indicates
whether the difference is added or subtracted (according to the sign that faces up). The player chooses the
first number of the sequence. The game might also be played in pairs, where one of the players checks the
results of the other and supplies the initial term of the sequence. Give one point for each correct term.
Extension: Extend the sequences according to the rules:
a) (add 15) 12 , 27 , ____ , ____ , ____ , ____
b) (subtract 12) 177 , 165 , ____ , ____ , ____ , ____
c) (add 102) 12 , 114 , ____ , ____ , ____ , ____
d) (subtract 21) 276 , 255 , ____ , ____ , ____ , ____
e) (multiply by 2) 3 , 6 , ____ , ____ , ____ , ____ , ____
f) (add two previous numbers) 1, 2 , 3 , 5 , 8 , ____ , ____ , ____ , ____
36 37 38 39 40 41 42 43
Patterns & Algebra Teacher’s Guide Workbook 3:1 31 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-16 Identifying Pattern Rules
Goal: Students will identify simple pattern rules.
Prior Knowledge Required: Addition. Subtraction. Ordinal Numbers
Vocabulary: increasing sequence, decreasing sequence, term
Tell your students that today you will make the task from the previous lesson harder—you will give them a
sequence, but you will not tell them what the rule is; they have to find it themselves. Give them an
example—write a sequence on the board: 2, 6, 10, 14… Is it an increasing or a decreasing sequence?
What was added? Let students practice with questions like:
Write the rule for each pattern:
a) 69, 64, 59, 54 subtract ____
b) 98, 96, 94 subtract ____
c) 25, 28, 31, 34 add ____
d) 35, 39, 43, 47 add ____
e) 43, 54, 65, 76 add ____
f) 119, 116, 113 subtract ____
Tell another story about pattern fans Bonnie and Leonie: The teacher asked them to write a sequence that
is given by the rule “add 2”. Bonnie has written the sequence: “3, 5, 7, 9…”. Leonie has written the
sequence: “2, 4, 6, 8…”. They quarrel—whose sequence is the right one?
Ask your students: how can you ensure that the sequence given by a rule is the one that you want? What
do you have to add to the rule? Students should see that you need a single number as a starting point.
Write several sequences on the board and ask your students to make rules for them.
Explain to your students that mathematicians need a word for the members of the sequences. If your
sequence is made of numbers, you might say “number”. But what if the members of the sequence are
figures? Show an example of a pattern made of blocks or other figures.
Tell the students that the general word for a member of a sequence is “term”.
Form a line of volunteers. Ask them to be a sequence. Ask each student in the sequence to say in order:
“I am the first/second/third… term.” Ask each term in the sequence to do some task, such as: “Term 5, hop
3 times”, or “All even terms, hold up your right arm!” Ask your sequence to decide what simple task each of
them will do, and ask the rest of the class to identify which term has done what. Thank and release your
sequence.
MORE PRACTICE:
a) What is the third term of the sequence 2, 4, 6, 8?
b) What is the fourth term of the sequence 17, 14, 11, 8?
c) Extend each sequence and find the sixth term:
(i) 5, 10, 15, 20 (ii) 8, 12, 16, 20 (iii) 131, 126, 121, 116
Patterns & Algebra Teacher’s Guide Workbook 3:1 32 Copyright © 2007, JUMP Math For sample use only – not for sale.
Bonus:
What operation was performed on each term in the sequence to make the next term:
2, 4, 8, 16… (HINT: neither “add”, nor “subtract”. ANSWER: Multiply the term by 2.)
Activity: Divide your students into groups or pairs and ask them to play a game: One player or group
writes a sequence, the other has to guess what the rule is. Each correctly solved problem gives 5 points.
Warn them that the rules have to be simple—otherwise they will have to give clues. Each clue lowers the
score by 1 point for both players or groups.
Extensions:
1. Find the mistake in each pattern and correct it.
a) 2, 5, 7, 11 add 3
b) 7, 12, 17, 21 add 5
c) 6, 8, 14, 18 add 4
d) 29, 27, 26, 23 subtract 2
e) 40, 34, 30, 22 subtract 6
2. Divide the students into groups or pairs. One player writes a rule, the other has to write a sequence
according to the rule, but that sequence must have one mistake in it. The first player has to find the
mistake in the sequence. Warn the students that making mistakes on purpose is harder than simply
writing a correct answer, because you have to solve the problem correctly first! The easiest version is to
make the last term wrong. More challenging is to create a mistake in any other term.
3. Find the missing number in each pattern. Explain the strategy you used to find the number.
a) 2, 4, ____, 8
b) 9, 7, ____, 3
c) 7, 10, ____, 16
d) 16, ____, 8, 4
e) 3, ____, 11, 15
f) 15, 18, ____, 24, ____, 30
g) 14, ____, ____, 20
h) 57, ____, ____, 45
4. One of these sequences was not made by a rule. Find the sequence and state the rules for the other
two sequences. (Identify the starting number and the number added or subtracted.)
a) 25, 20, 15, 10 b) 6, 8, 10, 11 c) 9, 12, 15, 18
Patterns & Algebra Teacher’s Guide Workbook 3:1 33 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-17 Introduction to T-tables
Goal: The students will be able to create a T-table for growing block patterns and to identify rules of
number patterns using T-tables.
Prior Knowledge Required: Addition
Skip counting
Subtraction. Number patterns.
Vocabulary: T-table, growing pattern, chart, core, term
Draw the following sequence of figures on the board and tell your students that the pictures show several
stages in the construction of a castle made of blocks:
Ask your students to imagine that they want to keep track of the number of blocks used in each stage of the
construction of the castle (perhaps because they will soon run out of blocks and will have to buy some
more). A simple way to to keep track of how many blocks are needed for each stage of the construction is
to make a T-table (the central part of the chart resembles a T- hence the name). Draw the following table on
the board and ask students to help you fill in the number of blocks used in each stage of construction.
Figure Number of
Blocks
1 4
2 6
3 8
Ask students to describe how the numbers in the table change—they should notice that the number of
blocks in each successive figure increases by 2, or that the difference between successive terms in the right
hand column is 2. Write the number 2 (the “gap” between terms) in a circle between each pair of terms.
Figure Number of
Blocks
1 4
2 6
3 8
2
2
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Ask students if they can state a rule for the pattern in the table (Start at 4 and add 2) and predict how many
blocks will be used for the fifth figure. Students should see that they can continue the pattern in the chart
(by adding the gap to each new term) to find the answer. Point out that the T-table allows them to calculate
the number of blocks needed for a particular structure even before they have built it.
Draw the following sequence of figures on the board and ask students to help you make a T-table and to
continue the table up to five terms.
Before you fill in any numbers, ask your students if they can predict the gap between terms in the T-table.
They should see—even without subtracting terms in the table—that the gap between terms is 3 because
the castle has three towers and you add one block to each tower at every stage of the construction.
Draw the following T-table on the board:
Figure Number of
Blocks
1 13
2 18
3 23
Tell students that the T-table gives the number of blocks used at each stage in the construction of a castle.
The castle was built in the same way as the others (towers are separated by a gate with a triangular roof,
and one block is added to each of the towers at each stage), but this particular castle has more towers. Ask
students if they can guess, from the pattern in the number of blocks, how many towers the castle has. From
the fact that the gap is 5, students should see that the castle must have five towers. Ask a student to come
to the board to draw a picture of the first stage in the building of the castle. Then ask students to help you
extend the T-table to five terms by adding the gap to successive terms.
After completing these exercises, give students a quiz with several questions like QUESTIONS 1 a) and
2 a) on the worksheet, or have them work through the actual questions on the worksheet.
Activities:
1. Ask your students to construct a sequence of shapes (for instance, castles or letters of the alphabet)
that grow in a fixed way. You might also use pattern blocks for this activity.
Patterns & Algebra Teacher’s Guide Workbook 3:1 35 Copyright © 2007, JUMP Math For sample use only – not for sale.
Ask students to describe how their pattern grows and to predict how many blocks they would need to
make the 6th figure. (If students have trouble finding the answer in a systematic way, suggest that they
use a T-table to organize their calculation.) This activity is also good for assessment.
2. Give each student a set of blocks and ask them to build a sequence of figures that grows in a regular
way (according to some pattern rule) and that could be a model for a given T-table. Here are some
sample T-tables you can use for this exercise.
Extensions:
1. A castle was made by adding one block at a time to each of four towers. Towers are separated by a
gate with a triangular roof. Altogether 22 blocks were used. How high are the towers? How many
blocks are not in the towers?
2. Claude used one kind of block to build a structure. He added the same number of blocks to his structure
at each stage of its construction. He made a mistake though in copying down the number of blocks at
each stage. Can you find his error and correct it?
3. You want to construct a block castle following the steps shown below. You would like each tower to be
5 blocks high. Each block costs five cents and you have 80 cents altogether. Do you have enough
money to buy all the blocks you need? (HINT: Make a T-table with three columns: Figure, Number of
Blocks and Cost).
STEP 1 STEP 2 STEP 3
Figure Number of
Blocks
1 4
2 6
3 8
Figure Number of
Blocks
1 3
2 7
3 11
Figure Number of
Blocks
1 1
2 5
3 9
Figure Number of
Blocks
1 5
2 7
3 11
4 14
Patterns & Algebra Teacher’s Guide Workbook 3:1 36 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-18 T-tables
Goal: Students will extend patterns using T-tables.
Prior Knowledge Required: Addition. Subtraction. Skip counting. Number patterns.
Ordinal numbers.
Vocabulary: T-table, chart, term
Draw several figure patterns on the board (or build them using one type of pattern block for each pattern).
Ask your students to predict how many blocks you will need for the next figure in each pattern. Ask them how
they made their prediction.
Invite volunteers to draw T-tables for the patterns you built. Use more volunteers to extend the tables and to
check the results by building the next figure in the pattern. For one of the simpler figures you might say that
each block costs two cents and ask the students to find the cost of each figure in the pattern. Ask students
to make a new table for the cost of the figures. For the figures in the picture, you may also ask: Each block
costs 3 cents. I have 40 cents; will that be enough to build the sixth figure?
Assessment:
Shade the blocks added to each figure to make the next one.
How many blocks will the 6th figure in the pattern have?
Extension: In many sequences that students will encounter in life—for instance in a recipe—two
quantities will vary in a regular way.
If a recipe for muffins calls for 3 cups of flour for every 2 cups of blueberries, you can keep track of how
many cups of each ingredient you need by making a double chart. Give students several questions about
recipies that they can solve with a double chart.
Number of
Muffins Trays
Number of Cups
of Flowers
Number of Cups
of Blueberries
1 3 2
2 6 4
3 9 6
Patterns & Algebra Teacher’s Guide Workbook 3:1 37 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-19 Problems and Puzzles
This worksheet is the final review and may be used for practice.
Workbook 3 - Patterns & Algebra, Part 1 1BLACKLINE MASTERS
Hundreds Charts _______________________________________________________2
Number Line Chasing Game ______________________________________________3
PA3 Part 1: BLM List
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Hundreds Charts
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Workbook 3 - Patterns & Algebra, Part 1 3BLACKLINE MASTERS
Number Line Chasing GameS
tart
Fin
ish
10
80
50
40
30
70
1
10
0
2 8
116 9
14
12
16
1922
24
26
29
32
34
36
39
44
38
42
46
41
59
48
49
51
52
54
56
61
58
62
69
64
66
68
71
74
79
76
72
78
81
84
86
89
82
96
92
91 90
18
= M
ISS
A T
UR
N
= T
AK
E A
NO
TH
ER
TU
RN
= G
O T
O T
HE
NE
XT
= G
O T
O T
HE
PR
EV
IOU
S10
20
4
21
312
8
60
88
94
98 99
Number Sense Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-1 Place Value – Ones, Tens and Hundreds
Goals: Students will identify the place value of digits in 2- and 3-digit numbers.
Prior Knowledge Required: Number Words — one, ten, hundred— and their corresponding
numerals
Vocabulary: the numbers from 1–10, both the sounds and the numerals
Photocopy the BLM “Place Value Cards” and cut out the three cards. Write the number 321 on the board,
leaving extra space between all the digits, and hold the “ones” card under the 3.
ASK: Did I put the card in the right place? Is 3 the ones digit? Have a volunteer put the card below the
correct digit. Invite volunteers to position the other cards correctly. Cards can be affixed to the board
temporarily using tape or sticky tack.
Now erase the 3 and take away the hundreds card. ASK: Are these cards still in the right place?
Write the 3 back in, put the hundreds card back beneath the 3, erase the 1, and remove the ones card.
ASK: Are these cards still in the right place? Have a volunteer reposition the cards correctly. Repeat this
process with 3 1 (erase the 2).
Write 989 on the board and ask students to identify the place value of the underlined digit. (NOTE: If you
give each student a copy of the BLM “Place Value Cards,” individuals can hold up their answers. Have
students cut out the cards before you begin.) Repeat with several 2- and 3-digit numbers that have an
underlined digit.
Vary the question slightly by asking students to find the place value of a particular digit without underlining it.
(EXAMPLE: Find the place value of the digit 4 in the numbers: 401, 124, 847.) Continue until students can
identify place value correctly and confidently. Include examples where you ask for the place value of the digit 0.
Then introduce the place value chart and have students write the digits from the number 231 in the correct
column:
Do more examples together. Include numbers with 1, 2, and 3 digits and have volunteers come to the board
to write the numbers in the correct columns.
Extensions:
1. Teach students the Egyptian system for writing numerals, to help them appreciate the utility of
place value.
1 = (stroke) 10 = (arch) 100 = (coiled rope)
Hundreds Tens Ones
a) 231 2 3 1
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Write the following numbers using both the Egyptian and our Arabic systems:
234
848
423
Invite students to study the numbers for a moment, then ASK: What is different about the Egyptian system
for writing numbers? (It uses symbols instead of digits. You have to show the number of ones, tens, and so
on individually—if you have 7 ones, you have to draw 7 strokes. In our system, a single digit (7) tells you how
many ones there are.) Review the ancient Egyptian symbols for 1, 10, and 100, and ask students to write a
few numbers the Egyptian way and to translate those Egyptian numbers into regular numbers (using Arabic
numerals). Emphasize that the order in which you write the symbols doesn’t matter:
234 = =
ASK: Does the order in which you write regular digits matter? Is 234 the same as 423? In the Egyptian way,
does the value of a symbol depend on its place? In our way, does the value of a digit depend on its place?
Are the ones, tens and so on always in the same place in our system? In the Egyptian system? Why is our
way called a place value system?
Have students write a number that is really long to write the Egyptian way (EXAMPLE: 798). ASK: How is our
system more convenient? Why is it helpful to have a place value system (i.e. the ones, tens, and so on are
always in the same place)? Having a place value system allows you to use the same symbol to mean many
different values. The digit 7, for example, can mean 7 ones, 7 tens or 7 hundreds depending on where it is
in the number.
Students might want to invent their own number system using the Egyptian system as a model.
2. Have students identify and write numbers given specific criteria and constraints.
a) Write a number between 30 and 40.
b) Write an even number with a 6 in the tens place.
c) Write a number that ends with a zero.
d) Write a 2-digit number.
e) Write an odd number greater than 70.
f) Write a number with a tens digit one more than its ones digit.
Harder
g) Which number has both digits the same: 34, 47, 88, 90?
h) Write a number between 50 and 60 with both digits the same.
i) Find the sum of the digits in each of these numbers: 37, 48, 531, 225, 444, 372.
j) Write a 2-digit number where the sum of the digits is 11.
k) Write a 2-digit number where the digits are the same and the sum of the digits is 14.
l) Write a 3-digit number where the digits are the same and the sum of the digits is 15.
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Bonus:
Is there a 2-digit number satisfying the same conditions?
m) Which number has a tens digit one less than its ones digit: 34, 47, 88, 90?
n) Write a 2-digit number with a tens digit eight less than its ones digit.
o) Write a 3-digit number where all three digits are odd.
p) Write a 3-digit number where the ones digit is equal to the sum of the hundreds digit
and the tens digit.
Make up more such questions, or have students make up their own.
Number Sense Teacher’s Guide Workbook 3:1 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-2 Place Value
Goals: Students will understand the value of digit in 2-, and 3-digit numbers.
Prior Knowledge Required: Place Value: Ones, Tens, Hundreds
Vocabulary: ones, tens, and hundreds digit, value
Write 836 on the board. SAY: The number 836 is a 3-digit number. What is the place value of the digit 8?
(If necessary, point to each digit as you count aloud from the right: ones, tens, hundreds). SAY: The 8 is in
the hundreds place, so it stands for 800. What does the digit 3 stand for? (30) The 6? (6)
Explain that 836 is just a short way of writing 800 + 30 + 6. The 8 actually has a value of 800, the 3 has a
value of 30, and the 6 has a value of 6. Another way to say this is that the 8 stands for 800, and so on.
ASK: What is 537 short for? 480? 35? 601? Write out the corresponding addition statements for each
number (also known as the expanded form).
ASK: What is the value of the 6 in 608? In 306? In 762? In 506?
ASK: In the number 831, what does the digit 3 stand for? The 1? The 8?
ASK: What is the value of the 0 in 340? In 403? In 809? Emphasize that 0 always has a value of 0, no matter
what position it is in.
ASK: In the number 856, what is the tens digit? Ones? Hundreds? Repeat for 350, 503, 455, 770, 820.
Write the following numbers on the board: 350, 503, 435, 537, 325, 753. Ask students to identify which digit,
the 5 or the 3, is worth more in each number. Students should be using the phrases introduced in the
lesson—stands for, has a value of, is short for. (EXAMPLE: In 350, the 5 stands for 50 and the 3 stands for
300, so the digit 3 is worth more.)
Extension: If your students are familiar with the concept “how many times more”, ASK: What is the value
of the first 1 in the number 11? What is the value of the second 1? How many times more is the first 1 worth
than the second 1? Repeat with more numbers in which the digit 1 is repeated (EXAMPLES: 131, 110, 101,
211, 171).
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NS3-3 Writing and Reading Number Words
Goals: Students will read and write number words to twenty and multiples of ten up to ninety.
Prior Knowledge Required: Reading and writing number words to ten
Place value (ones and tens)
Saying the alphabet
Vocabulary: numeral, number word, ones and tens digits
Write the following words on the board, all in a row:
eighteen thirteen seventeen sixteen nineteen fifteen
Ask the class to read the words out loud together and then ask volunteers to write the corresponding
numerals under the words.
ASK: What number does the word “teen” remind you of? Guide them by asking them to look at the letters—is
it spelled almost the same as a number they know? Tell them that eighteen is 8 + 10 = 18. ASK: Where can
you see “eight” in eighteen? Where can you see a word that looks like “ten” in the
word eighteen?
Have volunteers fill in the blanks with the correct number words:
a) fourteen = ________ + ten b) seventeen = ________ + seven
c) eighteen = ________ + _______ d) nineteen = ________ + __________
e) thirteen = ________ + _______ f) fifteen = ________ + __________
g) _______ = six + ten h) twelve = ________ + __________
i) eleven = ________ + _______
Have individual students write the missing words in their notebooks:
a) sixteen = ________ + ten b) seventeen = ________ + ten
c) nineteen = nine + _______ d) thirteen = ________ + ten
e) fourteen = ________ + four f) fifteen = ________ + ten
Have student volunteers circle the beginning letters that are the same.
a) six sixteen b) five fifteen c) nine nineteen
d) four fourteen e) three thirteen f) two twelve
Then, for each pair above, have students write the correct numerals in their notebooks and to circle the digits
that are in common.
Repeat the above exercise with ending letters instead of beginning letters for the following pairs.
a) thirteen fourteen b) seventeen eighteen c) nineteen fifteen
Then write on the board: twenty = 20 two = 2
ASK: What two beginning letters do those words have in common? (tw) What digit is in both numbers? (2)
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Write on the board: thirty. ASK: Can anyone think of a word for a 1-digit number that starts with the same two
letters? (three) Then write: thirty = 0 three = 3
Have a volunteer fill in the blank.
Write: forty = 0 fifty = 0 thirty = sixty =
Have volunteers fill in the blanks by looking carefully at the beginning letters and asking themselves what
one-digit number those letters remind them of.
ASK: What ones digit do these numbers all have? What letter do the words all end with? Tell them that any
number word ending with “y” will always mean a number having ones digit 0.
Ask volunteers to guess how the following number words are written as numbers:
eighty ninety seventy
Challenge them to find a 2-digit number having ones digit 0 whose number word doesn’t end with “y”. (10)
Have students write the numerals for the following number words individually:
a) thirty thirteen three b) twenty two twelve
c) four fourteen forty d) eighteen eighty eight
e) seven ninety thirteen eighty nine fourteen
f) nineteen sixty forty fifteen twelve eight
Have students write individually the number word ending for these words:
a) 30 = thir____ b) 20 = twen ______ c) 13 = thir_____
d) 17 = seven____ e) 40 = for____ f) 80 = eigh___
g) 18 = eigh____ h) 19 = nine_____ i) 90 = nine____
Finally, have students write the full number words:
a) 20 = _______ b) 19 = _______ c) 90 = ________ d) 17 = __________
e) 13 = _______ f) 80 = _______ g) 50 = ________ h) 15 = __________
Activity: On the web-site: http://www.funbrain.com/numwords/index.html students can use Method 1
to write the number word in the correct place on the cheque or use Method 2 to read the number word and
write the correct numeral. You may choose between numbers from 0 to 10, 0 to 100, 0 to 1000 or 0 to
10 000, depending on the level of your students.
Extensions:
1. Provide the BLM “Number Word Search.” Encourage students to use the message they find after
finishing the puzzle as a way to check that they did the puzzle correctly.
2. Write the alphabet on the board with enough spacing between the letters to circle some of them.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Write the word “act” on the board and ask a volunteer to circle, in the list, the letters that appear in the
word “act”. ASK: Are the letters in the same order in the word “act” as they are in the alphabet?
Number Sense Teacher’s Guide Workbook 3:1 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
Have another student, using a different colour of chalk, circle the letters from the word “sun”.
ASK: Are the letters in the same order in the word “sun” as they are in the alphabet? What order do they
appear in the alphabet? (n-s-u). Have students decide whether or not each of the following words are
written alphabetically: bat, box, cat, mom, snow, most, now, win, lose, knot, knots, stone, ghost.
Challenge students to find the longest alphabetical word that they can.
ASK: Is “dog” alphabetical? Is “doghouse” alphabetical? Fun? Funny? On? One? Pony? Phone? Bone?
Top? Stop? Tops?
Tell your students that you know that since “on” is not alphabetical, you know that the following words
cannot be alphabetical either: pony, money, gone, only. Ask them to explain your thinking.
Then make the connection to number words: Are any of the number words from one to ten written
alphabetically? Eleven to twenty? Did they need to check all the number words from eleven to twenty? Is
there a sequence of letters common to many of the number words? (teen is in many of them and is not
alphabetical, so we don’t even need to check thirteen to nineteen)
Which of the following multiples of ten is written alphabetically?
a) ten b) twenty c) thirty d) forty e) fifty f) sixty
3. Make a chart on the board with headings as follows:
3 letters 4 letters 5 letters 6 letters 7 letters 8 letters 9 letters
Have student volunteers write number words that fit in each column. Students should use number words
from zero to twenty as well as multiples of ten up to ninety (thirty, forty, and so on to ninety). When most
words are on the list, draw the following puzzle on the board:
Tell students that we want to solve this puzzle using number words. Point to the vertical group of 3
squares and ask students if the word FIVE will fit. Why not? How many letters does the word need to
have to fit? Refer your students to the list of 3-letter words they made and ask if there are any they
missed.
THEN SAY: How many letters should the other word have? Repeat the chart for words with 4 letters
(zero, four, five, nine).
Then tell students that one of the letters from the 3-letter word has to be the same as one of the letters
from the 4-letter word. Ask if they can tell which letter from each word needs to overlap the other word.
Have a volunteer circle the second letter from each 3-letter word and have another volunteer circle the
first letter from each 4-letter word. Tell them that the 2nd
letter from the 3-letter word is either n, w, i or e
and that the 1st letter from the 4-letter word is either f, f or n. Tell them that if there are going to be words
that fit in the puzzle, there had better be a letter in both lists. What letter is in both lists? (n) Which 3-letter
word has n as its second letter? (one) Which 4-letter word starts with n? (nine) Write the words into the
puzzle for them. Below are more puzzles (with the answers in brackets) your students can practice with.
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(zero, one) (four, one) (one, zero (five, three) (zero, four
or two, four) or nine, three) or five, zero
or nine, zero)
(four, five) (five, nine) (fourteen, one)
(six, sixteen, ten) (seventy thirty, four) (eight, fifty, five, ten)
4. Give students the BLM “Number Words Crossword Puzzle”.
5. Give students the BLM “Crossword Without Clues”.
6. Hand out the BLM “Recognizing Number Words”. The sheet asks students to circle the number words
and to cross out the words that only sound like number words. Have a copy of the BLM on the board or
overhead projector. Read the page out loud and point to the words as you say them. Give lots of hints.
For example, “Eight children ate pie”. What were the people in this sentence doing? Were they sleeping,
playing, eating or working? What were they eating? How many children ate pie?” Repeat the sentence
several times so that all students can see that “eight” is the number word and “ate” only sounds like a
number word. Remind the students that they should circle the number words and cross out the words
that only sound like number words. When a word sounds like a number word other than the one in the
sentence, students will benefit from hearing you read the sentence out loud and then saying some of the
number words from one to ten and then repeating the sentence out loud as often as necessary. When all
students have correctly done this sheet, hand out the BLM “Spelling Number Words” and have students
look at their completed sheet to answer the questions. This sheet will give students a taste of how they
can use the context of words to figure out the correct spelling. It will also show them that some words
that sound the same can be spelled differently.
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NS3-4 Writing Numbers
Goal: Students will read and write number words up to nine hundred ninety-nine. Prior Knowledge Required: Reading and writing number words to twenty and
multiples of ten to ninety
Vocabulary: numeral, number word, digit
Write “twenty” on the board and ask a volunteer to write the corresponding numeral. Ask them what number
they think the number word “twenty-three” means. Can they think of an addition sentence from this word?
(20 + 3 = 23) Repeat for twenty-seven and twenty-one. Have students individually write the numbers for the
following words:
twenty-two twenty-five twenty-nine twenty-six twenty-eight twenty-four
Then write: thirty-six. SAY: if thirty means 30 and six means 6, what number do you think thirty-six means?
What addition sentence can you write from that? (30 + 6 = 36) To help them find 30 + 6, provide a number
line or use a metre stick as a number line. Show them where 30 is on the number line so that they just have
to move ahead six places.
Have a volunteer write the number for thirty-five with the addition sentence (35 = 30 + 5), then have students
write the numbers with addition sentences for each number word below:
thirty-three thirty-two thirty-eight thirty-four
Provide them with a number line so that they can see how to add the numbers.
Show them where to find 10, 20 and 30 on the number line and then challenge them to find 40 on the
number line. Have a volunteer write the 2-digit number ” forty-seven” on the board by looking at a number
line and adding the two parts of the number they see. Summarize to the class how the volunteer is finding
the number 40 and then adding 7 to find 47. Repeat: thirty-six, twenty-seven, forty-two, thirty-one, forty-five,
fifty-four.
Write the number sentences on the board:
73 = 70 + 3
seventy-three
15 = 5 + 10
fifteen
32 = 30 + 2
thirty-two
18 = 8 + 10
eighteen
54 = 50 + 4
fifty-four
13 = 3 + 10
thirteen
61 = 60 + 1
sixty-one
16 = 6 + 10
sixteen
If available, use an overhead projector and write the parts in bold in a different colour. Point to each question
and ASK: Where do you see the first digit of the number in the number word – at the beginning or at the
end? Which numbers have the first digit at the beginning? (twenty and higher) Which numbers have the first
digit at the end? (thirteen to nineteen).
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When you write twenty-seven, where do you see the first digit in the number word? Where do you see
the last digit? Have them compare this with the number word seventeen. Tell them that number words
for numbers twenty and higher are a bit different from what they’ve seen so far because the first digit
is read first and the last digit is read last. Have students individually write the numbers for the following
number words:
thirty-eight forty-five twenty-six thirty-four fifty-one fifty-four
sixty-seven eighty-nine seventy-four ninety-one eighty-eight forty-two
Then have students write numerals for number words between zero and ninety-nine:
twenty-eight eighteen sixteen four forty
forty-three zero fifty fifty-eight thirteen
twelve nineteen twenty-nine fifty-nine forty-eight
thirty-four thirty-one eleven six fifteen
Have students write number words for numerals between 0 and 99:
a) 41 b) 32 c) 90 d) 9 e) 89 f) 74 g) 99 h) 0 i) 50 j) 25 k) 17 l) 11
Invite students to find any mistakes in the way the following number words are written and to correct them
(some are correct):
forty-zero forty-three twenty-eight thirty nine eight-five seventy-six
Summarize the process for writing numbers between 20 and 99: You can write the 2-digit number by writing
the word for the first digit times ten, a hyphen, and then the word for the second digit, as long as it isn’t zero.
If the second digit is zero, you write only the word for the first digit times ten.
EXAMPLE: 35 = 3 x 10 + 5 and is written as thirty-five, but 30 is written as thirty, not thirty-zero.
ASK: How is writing the number words for 11 to 19 different? (They don’t follow the same pattern.)
Write the number words for 11 through 19 on the board and invite students to look for patterns and
exceptions (eleven and twelve are unique; the other numbers have the ending “teen”).
Once students have mastered writing numbers up to 99, tell them that writing hundreds is even easier.
There’s no special word for three hundreds like there is for three tens:
30 = 10 + 10 + 10 = thirty but
300 = 100 + 100 + 100 = three hundred (not three hundreds)
SAY: You just write what you see: three hundred. There’s no special word to remember.
Have students write the number words for the 3-digit multiples of 100: 200, 300, 400, and so on.
Remind them not to include a final “s” even when there is more than one hundred.
Tell students that they can write out 3-digit numbers like 532 by breaking them down. Say the number out
loud and invite students to help you write what they hear: five hundred thirty-two. Point out that there is no
dash between “five” and “hundred.” Have students practice writing number words for many 3-digit numbers.
EXAMPLES: 134, 761, 898, 903, 740, 500, 601. Emphasize that the word “and” should not appear: 301 is
written as “three hundred one” not as “three hundred and one.”
Write some typical text from signs and banners and have students replace any number words with numerals
and vice versa.
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EXAMPLES:
a) Montreal 181 km b) Speed Limit – 110 km/h
c) Max. Height 3 m d) Seventy-Four Queen Street
e) Saskatoon next four exits f) Bulk Sale! Buy ten for the price of five!
g) Highway 61 h) Bus Stop: Route 18
i) Montreal Canadiens – j) Top Racing Broom for Witches and Wizards –
24 Stanley Cup Titles! only $599!
Then have students individually write the correct number words in the following sentences:
a) There are ______ months in a year.
b) There are ______ days in a week.
c) There are (52) ______ weeks in a year.
d) February normally has _______ days.
e) A year normally has ______ days.
f) A leap year has _______ days.
Then have students write number words that make sense:
a) There are _______ girls and _____ boys in grade ______ at my school.
b) My house is about _______ city blocks from my school.
c) I can run ______ km in _______ minutes
d) My teacher is about _______ years old.
e) There are about ______ days in summer vacation.
f) My birthday is in about ______ days from now.
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NS3-5 Representation with Base Ten Materials
Goal: Students will practice representing numbers with base ten materials.
Prior Knowledge Required: Place value
Base ten materials
Vocabulary: digit, ones digit, tens digit, hundreds digit, ones block, tens block, hundreds block
Photocopy the BLM “Hundreds Chart and Base Ten Materials” onto a transparency if available. Demonstrate
how to find 3 + 4 by taking 3 ones blocks and then another 4 ones blocks and placing them on the chart in
order, so that the last block is on square 7. ASK: How can I find 13 + 5 by using ones blocks and the
hundreds chart?
ASK: How is the counting already done for them when they put the ones blocks on in order?
Emphasize that they can see the answer by looking under the last ones block.
Tell your students that instead of using ten ones blocks to cover a row, you find it easier just to use one
bigger block. Show them a tens block and ask if anyone remembers what the block is called.
Provide your students with the BLM “Hundreds Charts” as well as 10 tens blocks and 9 ones blocks each.
Have students use 3 tens blocks and 5 ones blocks and cover the squares in order. The hundreds charts
were drawn to be 10 cm by 10 cm so that a ones block will cover a grid square exactly. ASK: How many
squares are covered? How do you know? (They should look under the last ones block to see the number
35.) Repeat for several examples. (41, 23, 59, 74, 99) Then ask your students what number they get if they
use two tens blocks and no ones blocks (20). 5 tens blocks? 7 tens blocks? 10 tens blocks?
Tell your students that we used a tens block instead of ten separate ones blocks. ASK: What can we use
instead of 10 tens blocks? (a hundreds block) Give your students 2 hundreds blocks to add to their 10 tens blocks and 9 ones blocks. ASK: What number do you get if you place a hundreds block on the first hundreds chart and then 3 tens blocks and 7 ones blocks in order on the next hundreds chart? Repeat with:
a) 1 hundreds blocks, 5 tens blocks, 4 ones blocks
b) 1 hundreds block, 6 tens blocks, 2 ones blocks
c) 1 hundreds blocks, 7 tens blocks, 5 ones blocks
d) 1 hundreds blocks, 3 ones blocks
e) 1 hundreds blocks, 2 tens block, 2 ones block
f) 1 hundreds blocks, 1 tens block
g) 1 hundreds block, 3 tens blocks
h) 2 hundreds blocks.
Then show models of base ten blocks without using the hundreds chart and have students tell you what
number is represented. EXAMPLES: 3 hundreds blocks, 4 tens blocks and 2 ones blocks; 5 hundreds blocks
and 8 ones blocks.
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Now write only the expanded form and have students tell you what number is represented:
a) 7 hundreds + 5 tens + 3 ones
b) 9 hundreds + 0 tens + 6 ones
c) 8 hundreds + 1 ten + 1 one
d) 4 hundreds + 7 tens + 0 ones
Have your students write out the expanded form from the numerals.
EXAMPLE: 790 = 7 hundreds + 9 tens + 0 ones.
Demonstrate drawing a base ten model for 145 on grid paper:
Shade the blocks and ASK: How many little squares are shaded altogether? (145) Have students draw base
ten models for other 2- and 3-digit numbers: 45, 60, 74, 104, 251, 300, 260.
Activities:
1. Give your students ones, tens, and hundreds blocks. Students might work in teams (with each team
scoring a point for each right answer). Students might also sketch their answers (so you can verify that
they have successfully completed the work):
Hundreds block Tens block Ones block
Instruction:
a) Show 17, 31, 252, etc. with base ten blocks.
b) Show 22 using exactly 13 blocks.
c) Show 31 using 13 blocks.
HINT: for b and c: Start with a standard model and trade for blocks of equal value.
Harder
d) Show 315 using exactly 36 blocks.
Extension: Change the order of the words hundreds, tens and ones and have students fill in the blanks.
EXAMPLE: 793 = ____ tens + ____ hundreds + ___ ones
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NS3-6 Representation in Expanded Form
Goal: Students will replace a number with its expanded numeral form and vice versa.
Prior Knowledge Required: Place value (ones, tens, hundreds, thousands)
Vocabulary: digit, numeral
Show the base ten model for 145 again:
ASK: How many little squares are coloured? (145) Point to the hundreds block, then the 4 tens blocks and
finally the ones blocks and ask in turn, how many little squares are coloured from each type of block. Then
write on the board: 145 = 100 + 40 + 5. Ask a volunteer to change this to expanded form using the words:
hundreds, tens and ones (1 hundred + 4 tens + 5 ones).
Have students draw base ten models on grid paper for several numbers and to record the expanded form in
two different ways (using numerals and words or numerals only).
EXAMPLES: 135, 241, 129, 302.
Have students expand several numbers using numerals instead of words.
EXAMPLES: 348 = 300 + 40 + 8, 640 = 600 + 40, 301 = 300 + 1.
Ensure students understand that when we have a 0 digit, we do not include 0 in the expanded sum;
70 is just 70, not 70 + 0.
Have students write the numeral for several sums written in this expanded form.
EXAMPLES: 200 + 70 + 6 = ___, 300 + 50 = ____, 300 + 5 = _____
Bonus:
3000 + 400 + 20 + 7 = _______ 5000 + 40 + 9 = ______
Have students write in the missing numbers:
a) 500 + 30 + ____ = 534 b) 641 = 600 + ____ + 1 c) 812 = ____ + 10 + 2
d) 700 + ___ + 2 = 742 e) 400 + ___ = 420 f) 400 + ____ = 402
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Bonus:
54 327 = 50 000 + 4 000 + ____ + 20 + 7
Teach your students to draw rough sketches of base ten models: a large square for a hundreds block, a strip
for a tens block and a small square for a ones block:
Then have students write various numbers in expanded form and then draw a rough sketch of a base
ten model. EXAMPLES: 732, 456, 57, 507, 570.
Tell your students that you read one book with 300 pages, and another book with 70 pages. ASK: How many
pages did I read altogether? Have a volunteer write the corresponding addition statement (300 + 70 = 370)
Repeat with several similar word problems:
a) A store has 100 red bikes, 40 blue bikes and 6 green bikes. How many bikes does the store have
altogether? (100 + 40 + 6 = ______ )
b) On a class field trip, there were 200 children, 10 parent volunteers and 7 teachers. How many people
went on the field trip? (200 + 10 + 7 = ______ )
c) Bonus: In a school in Toronto with 498 children, 400 children were from Canada and 90 children were
from the United States. How many were not from Canada or the United States? (498 = 400 + 90 +
______ )
Activity:
I have ---, who has ----?
Using the BLM “Make Up Your Own Cards,” make enough cards so that everyone in the class can have one
(or for everyone in small groups to have one). Use the expanded sum and base ten materials to make the
cards. For example, if the student has the card,
I have
300 + 40 + 8
----------
Who has
?
they say, “I have 348, who has 213?” and the person with 200 + 10 + 3 then says “I have 213, who has ---?”
depending on which base ten model is on the bottom of their card. Play continues until everyone gets a turn.
Ensure that the bottom of the last card matches with the top of the first card, so that students know when
they get back to the first card.
hundreds block
tens block
ones block
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Extensions:
1. Fill in the blanks.
a) 200 + 6 + 90 = ____ b) 30 + 800 + 5 = _____ c) 9 + 300 = _____
d) 854 = 50 + 4 + ____ e) 743 = 3 + ____ + 40 f) 912 = ____ + 900 + 2
2. Tell your students that when we write numbers, we write them as sums of 100s, 10s and 1s.
For example,
723 = 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1. Have your students write
various numbers as a sum of 100s, 10s and 1s. EXAMPLES: 341, 213, 411, 315. Then tell your
students that when we talk about money, we write the money as a sum of coin values. ASK: What are
the values of the coins we have that are less than a dollar? (25¢, 10¢, 5¢, 1¢) Show how 13 can be
written in various ways:
13 = 10 + 1 + 1 + 1 or 5 + 5 + 1 + 1 + 1 or 5 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 or a sum of 13 ones.
Show how each of these corresponds to different ways of making 13 cents. ASK: If we were to make a
standard way of writing money, how would we write 13? Why? Emphasize that the standard way should
use the fewest number of coins just as the standard base ten model uses the fewest number of blocks.
Then have students attempt to find the standard way of making:
a) 21¢ b) 16¢ c) 24¢ d) 29¢ e) 56¢ f) 62
3. Have your students demonstrate expanded form in other ways than base 10.
a) Ask your students to show different ways to make 100 as sums of 25, 10, and 5
(i.e. 100 = 25 + 25 + 10 + 10 + 10 + 10 + 5 + 5 or 100 = 2 twenty fives + 4 tens + 2 fives).
b) Show different ways to make 200 as sums of 100, 50, 10.
c) Show different ways to make 1000 as sums of 500, 250, 100.
4. Which numbers are representations of 352?
a) 300 + 50 + 2 or
b) 1 hundred + 20 tens + 2 ones or
c) 2 hundreds + 15 tens + 2 ones or
d) 34 tens + 12 ones
Make up more problems of this sort.
5. Decompose a number in as many different ways as you can.
EXAMPLE:
312 = 3 hundreds + 1 ten + 2 ones or
2 hundreds + 9 tens + 22 ones, etc…
Students could use base ten blocks if it helps them: they can exchange blocks for smaller denominations
to find different representations.
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NS3-7 Representation Numbers - Review
Goal: Students will consolidate the learning in number sense done so far.
Prior Knowledge Required: Expanded form of 2- and 3-digit numbers using words and numerals
(e.g. 2 hundreds + 3 tens + 7 ones)
Expanded form of 2- and 3-digit numbers using numerals alone
(e.g. 200 + 30 + 7)
Reading and writing number words for 2- and 3-digit numbers
Base ten models of 2- and 3-digit numbers, including drawing a rough
sketch of hundreds, tens and ones blocks
Give your students ones and tens blocks. ASK: Which numbers have standard base ten models that
can be arranged as rectangles of width at least 2? That is, if tens blocks are arranged horizontally,
there are at least 2 rows.
EXAMPLES:
60 =
30 + any multiple of 3
up to 39 (33, 36, 39)
22 55
Definitely all numbers with identical digits
(A standard base ten model uses the minimum number of blocks. For example, 35 = 3 tens and 5 ones is
standard, 35 = 2 tens + 15 ones is not.) Note that any number can be modelled as a rectangle of width 1,
hence the restriction!
EXAMPLES:
11 =
35 =
NOTE: This is an open-ended activity: there are many possible answers.
Numbers in the 60s are especially interesting to consider because 2, 3, and 6 divide evenly into the tens digit
(6); if the ones digit is any multiple of these factors, the number can be modelled by a rectangle with width at
least 2:
A good hint, then, is to first make just the tens blocks into a rectangle with at least 2 rows. This is necessary
since the (at most 9) ones blocks cannot rest on top of the tens blocks. The numbers 60, 62, 63, 64, 66, 68,
and 69 can all be modelled this way but 61, 65, and 67 cannot.
Allow students time to make these discoveries on their own.
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Extensions:
1. Ask students to explain and show with base ten blocks the meaning of each digit in a number with all digits the same (EXAMPLE: 3333).
2. Have students solve these puzzles using base ten blocks:
a) I am greater than 20 and less than 30. My ones digit is one more than my tens digit.
b) I am a 3-digit number. My digits are all the same. Use 9 blocks to make me.
c) I am a 2-digit number. My tens digit is 5 more than my ones digit. Use 7 blocks to make me.
d) I am a 3-digit number. My tens digit is one more than my hundreds digit and my ones digit is one
more than my tens digit. Use 6 blocks to make me.
3. Have students solve these puzzles by only imagining the base ten blocks. QUESTIONS (a) through (d)
have more than one answer—emphasize this by asking students to share their answers.
a) I have more tens than ones. What number could I be?
b) I have the same number of ones and tens blocks. What number could I be?
c) I have twice as many tens blocks as ones blocks. What 2-digit number could I be?
d) I have six more ones than tens. What number could I be?
e) You have one set of blocks that make the number 13 and one set of blocks that make the
number 22. Can you have the same number of blocks in both sets?
f) You have one set of blocks that make the number 23 and one set of blocks that make the
number 16. Can you have the same number of blocks in both sets?
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NS3-8 Comparing Numbers
Goals: Students will use base ten materials to determine which is larger.
Prior Knowledge Required: Naming numbers from base ten materials
Modeling numbers with base ten materials
Vocabulary: hundreds, tens, ones, base ten blocks, greater than, less than
Introduce the phrases “greater than” and “less than”. Emphasize that to say that one number is greater than
another means the first number represents more objects than the second, so 4 is greater than 3 since a
collector of objects 4 objects contains more objects than a collection of 3 objects (4 dollars is more money
than 3 dollars, 4 metres is longer than 3 metres, 4 goals is more than 3 goals, 4 minutes is more time than 3
minutes). It is crucial that students understand that 4 of anything is more than 3 of the same thing and so it
makes sense to compare the numbers 3 and 4 by saying that 4 is “more than” 3. The correct mathematical
expression is 4 is greater than 3, and students should get used to using the expression. Then write on the
board: 4 ____ 5 and have student volunteers say either “greater than” or “less than” as appropriate (in this
case, “less than” is correct). Repeat with several pairs of single-digit numbers to ensure that students are
comfortable with the words “greater than” and “less than.”
If your students are comfortable with base ten blocks and trading a tens block for 10 ones blocks, you may
use base ten materials for this lesson. Otherwise, you might find more convenient to use link-it cubes built
into stacks of 10. This way, instead of trading, students can simply pull apart one stack of 10 if necessary.
Some students may find this more natural than trading.
Make the numbers 25 and 35 using base ten blocks:
Have students name the numbers. ASK: Which number is greater? How can we show which number is
greater using base ten blocks? Explain that 3 tens blocks is more than 2 tens blocks and 5 ones blocks is the
same as 5 ones blocks, so 35 is greater than 25.
Have students use base ten blocks to determine which number is greater:
a) 26 or 28 b) 42 or 32 c) 67 or 57 d) 23 or 83 e) 74 or 78
ASK: Do you need to use base ten blocks to determine which number is greater? (no, we can just look at the
digits) How does looking at the digits tell us which number is greater? (if one of the digits is the same, just
look at the other digit to see which number is greater)
Have students use base ten blocks to determine which number is greater:
a) 35 or 47 b) 26 or 15 c) 48 or 32 d) 57 or 68 e) 3 or 14
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ASK: How does looking at the digits tell us which number is greater? (if the tens and ones digit of one
number are both greater than the tens and ones digit of the other number, then that number must be greater)
Make the numbers 43 and 26 using base ten materials.
SAY: Which has more tens blocks? Which has more ones blocks? Hmmm, 43 has more tens blocks,
but 26 has more ones blocks—how can we know which one is bigger?
Show students how trading a tens block for more ones blocks can help them compare the two numbers (or
break apart a stack of 10 link-it cubes):
43 = 4 tens + 3 ones = 3 tens + 13 ones
26 = 2 tens + 6 ones
Now, it is clear that 43 is more than 26—43 has more tens and more ones.
Give students base ten blocks or link-it cubes and ask students make models for each pair of numbers and
then to compare the numbers by trading blocks if necessary (or splitting apart stacks of 10 link-it cubes) so
that the tens and ones in one number are both the same or greater than the tens and ones in the other
number.
EXAMPLES: Which number is greater:
a) 56 or 38 b) 39 or 45 c) 17 or 46 d) 38 or 55 e) 63 or 24
Have students compare several pairs of numbers in their notebook where the tens digit is greater in one
number and the ones digit is greater in the other number. They should draw rough sketches of the base ten
blocks to help them and then trade ten ones blocks for a tens block.
Invite volunteers to show their work on the board. ASK: If two numbers have different tens and ones digits,
which number is greater—the number with greater tens digit or the number with greater ones digit?
Draw pairs of base ten models and have students individually write the numbers modeled and circle the
greater number in each pair. Students should see that the number with more tens is always greater.
EXAMPLE:
34 27
Repeat the exercise, but this time have students write the names for the numbers in words. Make the words
below a regular part of your spelling lessons and have them visible to all students during math lessons: one,
two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, twenty, thirty, forty, fifty, sixty, seventy,
eighty, ninety, one hundred.
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When providing pairs of numbers for the students to compare, include examples where both numbers have
the same tens digit and ASK: If both numbers have the same tens digit, how can you tell which number is
greater? (the number with the greater ones digit will be greater) If the numbers have different tens digits, how
can you tell which number is greater? (the number with a greater tens digit will be greater).
Then ask students to make (or draw rough sketches of) base ten models for the numbers 238 and 153.
ASK: Which number has more hundreds blocks? Tens blocks? Ones blocks? Which number do you think is
greater? How can you trade some blocks so that one number has at least as many ones, tens and hundreds
blocks as the other number? Repeat with other pairs of numbers.
a) 345, 169 b) 541, 355 c) 102, 45 d) 600, 497 e) 436, 429 f) 810, 801
ASK: Which number is greater—the number with more hundreds, more tens or more ones? Is a 3-digit
number always greater than a 2-digit number? Is a 2-digit number always greater than a 1-digit number?
How do you know?
Then draw base ten models of several pairs of 2- and 3-digit numbers. Have students write the number word
and the numeral for each number and then to circle the greater number.
Extensions:
1. Create base ten models of a pair of two-digit numbers. Ask students to say how they know which number
is greater. You might make one of the numbers in non-standard form, as shown for the first number
below.
EXAMPLE:
To compare the numbers students could remodel the first number in standard form by regrouping ones
blocks as tens blocks.
2. Ask students to create base ten models of two numbers where one of the numbers…
a) is 30 more than the other
b) is 50 less than the other
c) has hundreds digit equal to 6 and is 310 more than the other
3. Ask students where they tend to see many numbers in increasing order (page numbers, houses,
mailboxes, apartment numbers, line-ups when people need to take a number tag).
First Number Second Number
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NS3-9 Comparing and Ordering Numbers
Prior Knowledge Required: One-to-one correspondence
Pairing things up
More and less
Recognizing the number of fingers held up
Write 52 and 42 on the board as follows:
5 2 4 2
Have volunteers write the value of each digit in the appropriate box and then to tell you which number is
greater. Write several similar questions on the board comparing either 2- or 3-digit numbers and have
students determine the greater number in each pair individually in their notebooks.
Repeat the exercises by writing the numbers in expanded form:
52 = 50 + 2
42 = 40 + 2
Since 50 is larger than 40, and 2 is the same as 2, 52 is larger than 42.
Show your students a shortcut for comparing two numbers without having to write the expanded form.
475 = 400 + 70 + 5
465 = 400 + 60 + 5
Since 70 is greater than 60, they know from the expanded form that 400 + 70 + 5 is greater than 400 + 60 + 5.
Without doing the expanded form, we can see this from the digits themselves.
475
465
Since the 7 means 70 and the 6 means 60, we can see the same thing just from the digits in the numbers.
Have students compare two numbers by circling the digit that is different and writing the greater number in
the space below:
a) 475 b) 356 c) 297 d) 493 e) 527
465 358 497 490 507
475
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Write 43 and 26 in expanded form and compare them:
43 = 40 + 3 Since 10 is always greater than any 1-digit number, we can split
26 = 20 + 6 43 (the one with more tens) into 30 + 10 + 3:
43 = 30 + 10 + 3
26 = 20 + 6
This makes it clear that 43 is greater than 26, since it wins on all counts. Compare more 2-digit numbers in
this way until it becomes clear that the number with more tens is always greater.
Then move on to 3-digit numbers. Tell students you want to compare 342 and 257. ASK: Which number has
more hundreds? More tens? More ones? Which number do you think is greater, the one with the most
hundreds, the most tens or the most ones? Why? Then write:
342 = 200 + 100 + 42
257 = 200 + 57
Since 100 is greater than any 2-digit number (it has 10 tens and any 2-digit number has at most 9 tens), it
doesn’t matter what the tens and ones are of 257. Ask students to compare more 3-digit numbers with
different hundreds digits in this way (e.g., 731 and 550, 403 and 329). Then look at a pair in which the
hundreds digit is the same (e.g., 542 and 537). This is just like comparing 42 and 37: 542 = 500 + 40 + 2 =
500 + 30 + 12, which is greater than 500 + 30 + 7. ASK: If two 3-digit numbers have different hundreds digit,
how can you tell which one is greater? If they have the same number of hundreds, how can you tell which
one is greater?
Have students compare numbers that do not have the same number of digits (e.g., 350 and 93). Can they
explain why any 3-digit number is always greater than any 2-digit number?
Write the following pairs of numbers on the board:
a) 743 b) 583 c) 392 d) 278
693 591 267 249
Tell your students that 743 has more hundreds and 693 has more tens. ASK: Which number is greater?
Remind your students that the number of hundreds is more important than the number of tens. Circle the
hundreds and tell them that 743 is greater because it has more hundreds. Have volunteers circle the first
digit from the left that is different in each number and then determine the greater number.
Then have students compare numbers given in context. These questions do not have to be given in written
form if some students are uncomfortable reading at this stage. The workbook questions could be read aloud
together before being assigned.
a) Rita’s mother is 43 years old. Anna’s mother is 51 years old. Whose mother is older?
b) Rita has $540. Anna has $259. Who has more money?
c) Montreal is 539 km from Toronto and Ottawa is 399 km from Toronto.
Which city is closer to Toronto?
d) Maurice Richard scored 544 career goals in the NHL. Wayne Gretzky scored 894 career goals. Who
scored more career goals?
Number Sense Teacher’s Guide Workbook 3:1 24 Copyright © 2007, JUMP Math For sample use only – not for sale.
Extensions:
1. Use the digits 5, 6, and 7 to create as many 3-digit numbers as you can (only use each digit once when
you create a number). Then write your answers in descending order.
2. List 4 numbers that come between 263 and 527.
3. Name 2 places you might see more than 1000 people.
4. Say whether you think there are more or fewer than 1000…
a) hairs on a dog b) fingers and toes in a class c) students in the school
d) grains of sand on a beach e) left handed students in school
5. Two of the numbers are out of order in each increasing or decreasing sequence. Circle each pair.
a) 28, 36, 47, 42, 95, 101
b) 286, 297, 310, 307, 482
c) 87, 101, 99, 107, 142, 163
6. Introduce students to the > and < notation for “greater than” and “less than”. Have them discover the
notation by using the following trick. Draw a face:
Draw two piles of apples on the board, one with 3 apples and one with 4 apples and have students point
the face towards the pile with more apples because this person is hungry. Repeat with several examples,
always drawing the picture of the apples. Then, instead of the pictures of apples, draw only the number
of apples. ASK: Which number of apples should the hungry person eat? (EXAMPLE: 5 2)
Then erase the face and leave only the mouth. Tell students to just imagine what the mouth will look like
for several pairs of numbers. (EXAMPLE: 7 < 9). Then, for several pairs of numbers, have students write
both the mouth in between the numbers and the words “is less than” or “is more than”. (EXAMPLE: 8 > 7,
8 is more than 7). When students have done this several times, tell them that mathematicians have
invented a symbol to mean “is more than” and another symbol to mean “is less than” and that they just
discovered it. Can they see which symbol (< or >) means more than and which means less than?
Discuss with students why mathematicians may have chosen to invent symbols for these words just like
they did for plus and minus. (Comparing numbers is an important part of mathematics and is done often
enough that they wanted a short form). Ask students if they think that mathematicians could have defined
> to mean “less than” and < to mean “greater than” instead. Emphasize that people often have to make
arbitrary decisions and it is just important to be consistent so that there is no confusion. Brainstorm other
situations where arbitrary decisions like this are necessary (which hand on the clock is longer, which side
of the road do people drive on, which sound does the symbol “b” represent, and so on).
7. For extra practice recognizing differences of 1 and 10, provide the BLM “Hundreds Chart Pieces.” After
students finish, they (or a partner) should check their answers by using an actual hundreds chart.
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NS3-10 Differences of 10 and 100
Goals: Students will recognize when numbers are different by 10 or 100.
Prior Knowledge Required: Expanded form
Place Value
3-digit numbers
Vocabulary: difference, expanded form, sum
Have students write “1 more” or “1 less” in the blanks.
a) 8 is _________ than 7 b) 4 is __________ than 5 c) 2 is _________ than 3
d) 10 is _________than 9 e) 11 is _________ than 10 f) 39 is _________ than 40
Have students write “10 more” or “10 less” in the blanks.
a) 50 is _________ than 40 b) 30 is __________ than 40 c) 20 is _________ than 10
d) 60 is _________than 70 e) 40 is _________ than 30 f) 70 is _________ than 80
Have students write “100 more” or “100 less” in the blanks.
a) 600 is _______than 500 b) 300 is _______ than 400 c) 300 is ______ than 200
d) 600 is _______than 700 e) 500 is _______ than 400 f) 700 is ______ than 600
Have students write each number in expanded form and then tell you how much more or less the first
number is than the second number.
a) 345 = 300 + 40 + 5 b) 259 c) 567 d) 431
335 = 300 + 30 + 5 269 467 432
40 is 10 more than 30, so 345 is 10 more than 335.
ASK: How can you tell how much more or less the number is by looking at which digit is different? Have
students practise this skill with several pairs of numbers which differ in only 1 digit. Students should not use
the expanded form, but should only pay attention to which digit is different.
(EXAMPLES: 756, 746; 430, 431; 542, 442; 542, 552)
Bonus: Have students compare pairs of numbers that differ in only 1 digit, but by more than 1.
(EXAMPLES: 563, 263; 412, 432; 743, 703; 703, 709)
Have students fill in the blanks:
a) ___ is 100 more than 352 b) ____ is 10 more than 352 c) ____ is 1 more than 352.
d) ___ is 100 less than 352 e) ____ is 10 less than 352 f) ____ is 1 less than 352
Provide several problems of this sort with different numbers.
EXAMPLES: ____ is 10 less than 890; ___ is 100 more than 743, ___ is 1 less than 502
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Bonus: ____ is 30 less than 572; ____ is 40 more than 641, ___ is 200 more than 738
Have students add or subtract 1, 10 or 100.
a) 354 – 1 b) 653 + 10 c) 987 – 100 d) 432 + 100 e) 581 + 10 f) 436 + 1
Ensure that students can add and subtract 1 comfortably when crossing over a multiple of 10.
EXAMPLES: 59 + 1; 90 – 1; 39 + 1; 20 – 1; 50 – 1; 79 + 1.
Then write on the board: 304 – 10. ASK: How many tens are in 304? (30) If students do not see this
immediately, you may need to ask a series of questions to guide them: How many tens are in 20? 30? 80?
90? 100? 110? 120? 150? 190? 200? 210? How many tens are in 27? 37? 84? 96? 105? 115? 122? 151?
196? 201? 213? Which digit should I cover up to see how many tens are in 304? (cover up the ones digit)
Demonstrate covering up the ones digit. Then tell them that there are 30 tens and ASK: If I take away 1 ten,
how many tens are left? What is 1 less ten than 30 tens? (29 tens) Do I change the ones digit by subtracting
10? (no) What number has 29 tens and 4 ones? (294) Write on the board: 304 – 10 = 294. Similarly,
students can add 496 + 10 by realizing that 49 tens plus one more ten is 50 tens, so 496 + 10 is 506.
Give students practice with crossing over multiples of a hundred when adding and subtracting 10.
(EXAMPLES: 598 + 10; 703 – 10; 392 + 10; 909 – 10; 800 – 10)
Have students state the pattern rules and then extend the patterns:
a) 432, 442, 452, ____, ____, _____ b) 201, 301, 401, ____, ____, ____
c) 947, 847, 747, ____, ____, _____ d) 759, 758, 757, ____, ____, ____
Bonus:
a) 531, 521, 511, ____, ____, _____ b) 703, 723, 743, ____, ____, ____
Activities:
1. Give your students 5 loonies and ask them to show you how much money they would have left if they
took away a dime. Students will see that they have to regroup one of the loonies as ten dimes, and that
there will only be 9 dimes left once one is taken away. (Ask students to translate the result into pennies:
500 pennies becomes 490 pennies.)
2. Ask students to show you in loonies how much money they would have altogether if they had $2.90 and
they were given a dime.
3. Ask students to make a base ten model of 207 and to show you how the model would change if you took
away 10. (Students will see that they have to regroup one of the hundreds as 10 tens and that there
would be 9 tens left)
Extensions:
1. Circle the greater number in each pair: 2. What number is 110 less than…
a) 3 125 or 3 225 b) 4 508 or 4 608 a) 273 b) 860 c) 922 d) 508
2. For more practice adding and subtracting 10, provide the BLM “Hundreds Chart Pieces.”
Number Sense Teacher’s Guide Workbook 3:1 27 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-11 Comparing Numbers (Advanced)
Goals: Students will order sets of two or more numbers and will find the greatest and smallest number possible when
given particular digits.
Prior Knowledge Required: Comparing pairs of numbers
Vocabulary: greater than, less than
Review reading number words and comparing 2- and 3-digit numbers.
Have students list all the 2-digit numbers they can make using only digits chosen from:
a) 4 b) 4, 5 c) 3, 0 d) 7, 1 e) 8, 5, 2
ANSWERS:
a) 44 b) 44, 45, 54, 55 c) 30, 33 d) 71, 17, 11, 77
e) 88, 55, 22 85, 58, 52, 25, 28, 82
ASK: Which is larger: 54 or 45? 76 or 67? 83 or 38? 91 or 19?
Tell your students that you want to make the largest possible 2-digit number using the digits 3 and 6. Which
number should you make? (63) What if you want the smallest number using the digits 4 and 7? (47)
Have students list the 3-digit numbers that use each digit exactly once:
a) 4, 5, 8 b) 3, 1, 9 c) 5, 3, 2
Which number is greatest? Least? Have students reflect: Could they have found the largest possible number
using the digits 4, 5 and 8 without listing all the possible numbers? How do they know which order to write
the digits in? Should they put the largest digit they have in the ones place, the tens place or the hundreds
place? How do they know? Is any 3-digit number with 4 hundreds more or less than any number with 8
hundreds? Since any 3-digit number with 4 hundreds is less than any 3-digit number with 5 hundreds which
in turn is less than any 3-digit number with 8 hundreds, we should put 8 in the hundreds place.
Have students find the least and greatest 3-digit numbers that use each digit exactly once, this time without
listing all the possible numbers they can make:
a) 3, 4, 1 b) 7, 5, 8 c) 6, 9, 1
Have students first compare pairs of numbers and then order lists of 3 numbers. ASK: Which number is
larger: 74 or 76? 74 or 54? Have students order the 3 numbers starting with the least: 74, 76, 54 (ANSWER:
54, 74, 76). Repeat with other numbers:
a) 64, 91, and 64, 25 b) 543, 523, 543, 743 c) 908, 926 908, 876
Have students order from least to greatest other lists of 3 or 4 numbers by comparing two at a time:
a) 79, 82, 75 b) 872, 864, 587 c) 993, 939, 399 d) 765, 781, 762, 709
Bonus: Provide longer lists of numbers for students to order.
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Write on the board the heights of some towers:
Eiffel Tower: 324 m, Tower of Pisa: 56 m, CN Tower: 558 m. Have students list the towers in order from
tallest to shortest.
Bonus: Include other towers in the list as well, such as: Calgary Tower, 191 m; Eureka Tower, 297 m.
Activity: Have students roll a pair of dice 5 times, create a 2-digit number from each roll and try to
achieve the lowest possible sum of their 5 numbers. They win if their total is less than 100.
Variations:
• They win if their sum is at least 200
• Roll 3 dice, create 3-digit numbers and win if their sum is less than 1000
• Roll 3 dice, create 3-digit numbers and win if their sum is at least 2000
NOTE: students can check their sums by using a calculator; they do not need to be able to add 2- or 3-digit
numbers at this time; rather, they should simply be choosing either the least possible or greatest possible
numbers from each roll.
ASK: Which is easier to win: playing with 2 dice or with 3 dice? Can you explain why? (they have more
freedom to make the leading digit as small as possible if there are 3 numbers to choose from than if there
are 2 numbers to choose from, so playing with 3 dice should be easier)
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NS3-12 Counting by 2s
Goals: Students will use patterns to skip count by 2s.
Prior Knowledge Required: Counting forwards
Counting backwards
Number lines
Hundreds charts
Repeating patterns
Vocabulary: skip counting, odd, even
Draw a number line on the board. Tell them that to skip count, you have to skip numbers. If you only
count every 2nd
number, you are skip counting by 2. Emphasize the connection between the ordinal number
2nd
and the ordinary number 2. Demonstrate this:
0 1 2 3 4 5 6
Then draw a number line from 0 to 10 and ask if anyone wants to show skip counting by 2 on the
number line.
0 1 2 3 4 5 6 7 8 9 10
Ask them how the arrows show skip counting by 2. Be sure that all students understand that you say the
numbers that the arrows touch and the arrow always points to the number you say next. It tells you to start at
0 and then to say every second number in order.
Then show the first row of a hundreds chart and tell them that we can show skip counting by 2 by colouring
the numbers we say and not colouring (or skipping) the numbers we don’t say. Have a volunteer colour the
right squares to show skip counting by 2 starting from 2:
1 2 3 4 5 6 7 8 9 10
Ask how this is the same as using a number line to skip count and how it is different. Then have 3 rows of a
hundreds chart on the board and ask students how they would show skip counting by 2 on the hundreds
chart. Have a volunteer colour the right numbers in the first row, another volunteer do the second row and
another volunteer do the third row. Have the class read the skip counting out loud as a group.
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Ask them if they see anything the same about the numbers in the first row and the numbers from the second
row. If they could remember the numbers they say when skip counting by 2 up to 10, how would this help
them skip count up to 20? Up to 30? Up to 40? When all students understand that the numbers they say
when counting by 2s starting from 2 have ones digit 0, 2, 4, 6 or 8, write a random number on the board
between 0 and 100 and ask students to raise their hand if they think you say that number when skip counting
by 2. Repeat several times. Then, instead of writing the number on the board, say the number out loud.
Have volunteers skip count by 2s from 20 to 30, from 60 to 70, and so on.
Skip count by 2s starting from:
a) 36, ___, ___, ___ b) 48, ___, ___, ___ c) 90, ___, ___, ___ d) 68, ___, ___, ____
Bonus: Skip count by 2s starting from: a) 138 b) 846 c) 896
Tell your students that when you count by 2s starting from 0, the numbers you say have a special name.
ASK: Does anyone know what that special name is? (even numbers) What happens when you count by 2s
starting at an even number—are the numbers you say still even?
ASK: Does anyone know the name for numbers that are not even? (odd) What happens when you count by
2s starting at an odd number—are the numbers you say still odd? What are the ones digits of the even
numbers? Of the odd numbers?
Skip count by 2s starting from:
a) 43, ___, ___, ___ b) 51, ___, ___, ___ a) 67, ___, ___, ___ a) 39, ___, ___, ___
Bonus: Skip count by 2s starting from a) 255 b) 849 c) 897
Ask them why skip counting by 2 might be useful – is there anything they can think of that comes in 2s?
(Feet, hands, shoes, gloves, mittens, etc.) Have students count the number of shoes in the room. Ask them
why skip counting by 2 is a natural way to do so.
Once students learn to count by 2s and to recognize even and odd numbers, they can use these skills as a
foundation for doing mental addition and subtraction: see the Mental Math section of this manual for details.
Activity:
House Numbers
Before doing this activity, have students do the extension below. Have students walk around a residential
neighborhood with an adult and look at house numbers along one side of the street. What are the ones digits
of the numbers they see. Do the numbers seem to be skip counting by any specific number? Are any
numbers missed? Which numbers are missed? Why might numbers be missed? (sometimes, there used to
be more houses in between two houses and then they got torn down). ASK: Why might some houses get
torn down? Why wouldn’t they change all the addresses to make the numbers go up by 2s again? What
problems would this cause?
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Extension: Have students find the missing number in each pattern:
35, 37, 39, ___, 43, 45,
62, 64, ___, 68, 70
78, 80, ___, 84, 86
432, 434, ____, 438, 440
Now don’t tell them where the missing term is:
706, 708, 712, 714, 716
56, 58, 62, 64, 66, 68
32, 34, 36, 40, 42, 44
55, 57, 59, 63, 65, 67
317, 321, 323, 325, 327
836, 838, 840, 842, 846
Literacy Connection:
“Two of Everything”, L.T. Hong
(A Chinese folktale where everything gets counted by 2s.)
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NS3-13 Counting by 5s and 25s
Goals: Students will use patterns to skip count by 5s and 25s.
Prior Knowledge Required: Number lines Repeating patterns Counting forwards and backwards
Vocabulary: skip counting
Draw a number line on the board:
0 10 20 30 40
ASK: What number is the at? (4) If I skip count by 5 starting at 4, what will be the next number I say?
Have a volunteer draw an at that point. Have another volunteer continue marking all the numbers you say
when skip counting by 5 starting at 4. Underline the ones digits for them and ASK: Do you see a pattern in
the ones digits? Have students describe the pattern rule: 4, 9, then repeat. Have students predict the next
three numbers you say when counting by 5: 4, 9, 14, 19, 24, 29, 34, 39, __, __, __.
Bonus: Is there a pattern in the tens digits? Challenge them to describe it. (start at 0, then increase by 1,
stay the same and repeat). Ask them to be clear about what is repeating (the increase by 1 and the stay the
same).
Show them other sequences formed by counting by 5s and have them find the pattern in the ones digits:
a) 23, 28, 33, 38, 43, 38 b) 79, 84, 89, 94, 99, 104 c) 281, 286, 291, 296, 301, 306
Bonus: Describe the pattern in the number of tens (EXAMPLE: c) start at 28, stay the same, increase by 1,
then repeat).
Have students extend each sequence above. Discuss the patterns in the ones digits in more detail. How are
the patterns the same and how are they different? (Similarities include: the core always has length 2, the first
two terms always differ by 5, differences include: the two starting numbers can be different in each case)
Draw a number line on the board:
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Have a volunteer start at 0 and circle the numbers they would say when counting by 25.
Then ASK: If I know how to count by 25s from 0 to 100, how can I count by 25s up to 200?
Number Sense Teacher’s Guide Workbook 3:1 33 Copyright © 2007, JUMP Math For sample use only – not for sale.
0 25 50 75 100
125 150 175 200
Discuss the patterns in a) the number of hundreds and b) the remaining digits (the tens and ones together).
a) 0, 0, 0, 0, 1, 1, 1, 1, 2, ____, ____, ____
b) 0, 25, 50, 75, 0, 25, 50, 75, 0, ____, ____, _____
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
Have students complete the number sequences by skip counting by 25:
50, 75, 100, ____, ____, ____.
125, 150, 175, ____, ____, ____,
275, 300, 325, ____, ____, ____,
800, 825, ____, ____, ____.
650, 675, ____, ____, ____.
Activities:
1. Skip Counting Machines
Tell students that they are going to make a machine to help them skip count by 5 from any number on a
hundreds chart. Give them the BLM “Counting by 5s” and ensure that students understand all
instructions before allowing them to start. Emphasize that students must not cut out the grid; they should
only cut along the lines suggested on the BLM; otherwise they will be left with pieces. When they are
finished counting by 5, provide students with the BLM “Hundreds Charts up to 200”. Ask students to
place their new skip counting machine so that they can start counting at various numbers. Then have
them say the counting sequences out loud with you. EXAMPLE: start at 5, start at 3, start at 9, start at
14, start at 20, start at 102, start at 134, start at 109, start at 145. When students are comfortable with
this, have them make a machine to count by 2s or by 10s.
Students may wish to investigate whether flipping their sheet over either vertically or horizontally will
result in a machine that also works (as long as the mirror line is vertical, the flipped sheet will still work).
Bonus:
Make a machine to count by 25s. Does their tool work for starting at any number? Suppose that they
make their holes arranged as shown by the shaded squares:
Number Sense Teacher’s Guide Workbook 3:1 34 Copyright © 2007, JUMP Math For sample use only – not for sale.
This works for counting by 25s starting at 13 (13, 38, 63, 88) by placing the top left shaded square over
the 13. However, it does not work for the sequence starting at 18 (18, 43, 68, 93); not even rotating or
flipping the sheet will work. In order to make it work, the shaded squares should appear in 5 rows, so that
the sequence can start at either the top or the second top hole.
2. Quarters and Nickels
Give students play money (quarters only or nickels only) and have students count the money they have
(in cents, not dollars) by skip counting by 5 or by 25. Then demonstrate counting a combination of
quarters and nickels, first in random order and then in organized order. Have volunteers count the
money first in random order and then in organized order. ASK: Which way is easier? Why do you think
that is? Students may wish to investigate any differences between counting first by 25 and then by 5 and
counting first by 5 and then by 25. Does it matter whether they count the nickels or the quarters first?
(no, as long as they count all the quarters first or all the nickels first)
Tell your students that when they skip count just by 25 and then just by 5 (or vice versa), they can use
the patterns discussed in class, but if they count in random order, there is no pattern they can use, so
each time they have to think separately. That’s why it takes longer to count in random order than in an
organized order.
Extensions:
1. Repeat the exercises above, starting at 10 instead of 0, first having a volunteer circle the correct numbers
and then finding the same patterns:
a) 0, 0, 0, 0, 1, 1, 1, 1, 2, ____, ____, ____.
b) 10, 35, 60, 85, 10, 35, 60, 85, 10, ____, ____, ____.
ASK: What is similar about the patterns for starting at 0 and the patterns for starting at 10? What is the
length of the core in each case?
Have students complete the number sequences by skip counting by 25:
a) 20, 45, 70, 95, ____, ____, ____, ____.
b) 310, 335, ____, _____, ____, ____.
c) 560, 585, _____, _____, _____, ____.
Bonus: 312, 337, ____, ____, ____, ____.
Number Sense Teacher’s Guide Workbook 3:1 35 Copyright © 2007, JUMP Math For sample use only – not for sale.
2. Show your students how they can group objects into groups of 2 or 5 to make it easier to count them.
Draw the following picture on the board:
1 2 3 4 5 6 7 8
2 4 6 8
Tell your students that you are counting the dots twice, once by counting normally and once by counting
by 2. Ask your students to compare the two ways of counting. ASK: Did I get the same answer both
ways? How many dots are there? Which way do you find easier? Why? (Some may find counting by 1s
easier because they know the next number to say more easily and others may find counting by 2s easier
because they have to say less numbers – both answers are good answers). Which way is faster? If you
had a lot of things to count, which way would you be done sooner?
3. Have students count the number of letters in the alphabet by counting by 5s. Then provide the BLM
“Foreign Alphabets.” Students will need to count the number of letters in foreign alphabets by counting
by 5s and then counting on by 1s. Before assigning this BLM, it is a good idea to hand out the BLM and
allow students time to familiarize themselves with the different alphabets. Discuss various differences
and similarities between foreign alphabets and our own. Discuss how, in Russian, letters that are
different actually look very similar. Have students write pairs of very similar-looking Russian letters on the
board. Are there any pair of letters like that in English? (e.g. O and Q, C and G, P and R) Students may
wish to discuss symmetry in letters as well, or how b and d are backwards from each other. Are there
any foreign alphabets on the BLM that are like that?
Ask students to look carefully at the Russian alphabet – are there any symbols where they aren’t sure if
the symbol represents a single letter or more than one letter? Correct any such misconceptions before
they start grouping by 5s.
Draw your students’ attention to the Cherokee alphabet. Are there any letters that look like our letters?
Are there any letters that look like our numbers? Tell them that the Cherokee alphabet was made up by
someone who saw English writing, was amazed by the way people could communicate through symbols,
and decided to assign sounds in their language to these symbols. They had no idea that 4 didn’t mean a
sound or even which sound D or R meant, but they just assigned sounds from their own language to
these symbols arbitrarily. As soon as one person invented the writing system, everyone in the
community got excited about it and virtually everyone learned to read very quickly.
If you have a diverse class, ask if anyone’s first language is on the sheet. Ask if anyone’s first language
is missing. Have volunteers write their own alphabet on the board so that other volunteers can group by
5 and count the letters in those alphabets.
Notice that students may group fairly arbitrarily and might have more than 5 single letters left over at the
end if they grouped randomly. That is okay. They might, for example, count the Korean letters as 5, 10,
15, 20, 25, 30, 31, 32, 33, 34, 35 instead of 5, 10, 15, 20, 25, 30, 35. Both are faster than counting by 1s.
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4. Ask students to estimate how many dots are on a pre-made page. For example,
Hold a card with dots arranged randomly like this up and ask students to estimate how many they think
there are. Ask how many groups of 5 they think there are. Then circle a group of 5 and ask if anyone
wants to change their guess. Then circle another 5 and again ask if anyone wants to change their guess.
Continue in this way. Ask them if showing a group of 5 made it easier to estimate how many there are.
Then finish putting all groups of 5 and ask them how grouping by 5 made it easier to count them all. One
strategy could be to tally the groups of 5 as you cross them out.
5. Ask the following 2 questions to your class:
a) To make $1.75 in quarters, how many quarters would you need?
b) Which term (1st, 2
nd, 3
rd, etc.) in the sequence 25, 50, 75, …is 175?
Discuss the similarities and the differences between these two questions.
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NS3-14 Counting by 2s, 3s and 5s
Goals: Students will count by 3s. Students will decide whether skip counting by 2, 3 or 5 should be done
given endpoints on a number line with a specified number of places.
Prior Knowledge Required: Skip counting by 2 and 5
Number lines
Vocabulary: skip counting
Review skip counting by 2 and by 5 and then introduce skip counting by 3. Draw two rows of skip counting
as follows:
0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51,
Have students continue the pattern for 2 more rows by skip counting and then discuss patterns in the rows
and the columns. Challenge them to find the next 2 rows by extending the pattern rather than by skip
counting.
Have students describe the pattern in the ones digits: 0, 3, 6, 9, 2, 5, 8, 1, 4, 7 and then repeat
Have students practice skip counting by 3, starting at various numbers. (EXAMPLES: start at 13, 27, 34, 55,
201, 836) ASK: When you skip count by 2s starting at 13, what can be the ones digits of the numbers you
say? (only 1, 3, 5, 7 and 9). When you skip count by 5s starting at 13, what can be the ones digits of the
numbers you say? (only 3 and 8). When you skip count by 3s starting at 13, what can be the ones digits of
the numbers you say? (any digit)
Write the following number line on the board:
5 7
Ask a volunteer to write what goes in the missing place. Ask how they can check their answer.
Make sure students understand that by finding what comes after their guess, they can make sure it’s 7.
Continually leave more and more spaces in the number line.
Then show them a number line like this:
10 14
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Tell your students that you want to skip count from 10 to 14 and only say one number in between. If you
count by 1s, you know that 11 comes right after 10. Does 14 come right after 11?. What should I skip count
by if I want the same number to come right after 10 and right before 14?
Have students choose between what to skip count by in the following order:
1. Skip counting by 1 or skip counting by 2.
2. Skip counting by 2 or by 5.
3. Skip counting by 2 or by 3.
4. Skip counting by 3 or by 5.
5. Skip counting by 2, by 3 or by 5.
Always begin by leaving only one space between the numbers and then progress to leaving more spaces.
When choosing between counting by 2, 3 or 5, a good strategy is to start counting by 3 and then either guess
a higher or lower number based on checking their first guess. Does counting by 3 get them too far or not far
enough? Should their next guess be higher or lower? Do they want to go further or less far with their next
guess?
Extension: For the sequence of skip counting by 3 starting at 0, have students describe the pattern in
the number of tens. ANSWER: The number of repetitions is 4, 3, 3 and then repeat. The number being
repeated starts at 0 and increases by 1 each time. (0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, …)
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NS3-15 Counting Backward by 2s and 5s
Goals: Students will count backwards by 2s from any number and by 5s from any multiple of 5.
Prior Knowledge Required: Counting forwards
Counting backwards
Skip counting forwards by 2 and 5
Vocabulary: skip counting, counting forwards, counting backwards
Say number sequences and have students tell you what you’re counting forwards by when you say the
numbers – students might hold up 2 or 5 fingers to show their answer. Then repeat by writing number
sequences on the board instead of saying them. ASK: How can you tell what I’m counting by – what do the
ones digits tell you?
Then make it a bit harder. They not only have to tell you what you’re counting by, they have to find the
missing number; the first step is to find what they’re counting by.
Demonstrate: 10 12 14 16 18 20 22 _____ 26 28
Ask them what they are counting by and how they know. Ask what comes next after 22 when they count by
2. Repeat with several examples of counting by 2 or 5, always asking first what they are counting by and
then what comes next. Do not always start with multiples of 2 or 5.
Then tell them that you are going to make it even harder for them by counting backwards instead of
forwards. Write on the board:
20 18 16 14 12 10 8
Ask them if they can tell what number you are counting back by. To help them, tell them to read the numbers
in backwards order – what number are they counting forwards by? Ask them how you could tell what to say
next after saying 8. To help them, ASK: What would you say before 8 when counting forwards by 2? What
operation do you use to get from 6 to 8? (add 2) To get from 8 to 6? (subtract 2) Students should see that
they always need to subtract 2 to find the next number. Repeat with several sequences of counting back by 2
and 5; have students extend the sequences to find the next 3 terms.
Now put a blank in the middle of the sequence: 20 15 _____ 5.
Ask what you’re counting back by and then what goes after the 15 – ask what they would say before 15
when counting forwards by 5. ASK: What is 15 – 5? Then ask them to check their answer by asking
themselves what comes after their answer – What comes right after 10 when counting back by 5? If their
answer of 10 is correct, then 5 should be the next number. Is it? So do they think their answer of 10 is right?
Do several examples of this, counting back by 2 or 5.
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Teach students to count back by 10s by skipping every second number they would say when counting back
by 5s. A number line is a good visual for this:
25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Activities:
Countdown by 2s
The teacher counts back from 20 to 0 by 2s. Tell the students that they have to stand up before you get to 6.
The last person to stand up before you get to 6 wins. Then they have to focus on which number comes
before 6 when you count backwards by 2s. They won’t want to stand up when you say 10 because someone
else might stand up when you say 8. They also can’t wait until you say 6 because then they won’t win. Vary
this at first only by changing what number you count down from. E.g. Count down from 16 to 6, then from 14
to 6, then from 18 to 6, etc. When everyone in the class understands that 8 comes right before 6 when
counting back, then you can vary the number you count down to as well as from. Eventually, you can ask for
volunteers to count back while the rest of the class plays the game.
Variation: Count back by 5s from 100 or by 10s from 200.
Zero
Have students start at 10 and take turns saying the next number when counting back by 2s and the one to
say 0 wins. Does the person who starts win or lose? Start at higher numbers as students become ready.
Which numbers could I start at if I want to win?
A Strategy Game for Counting Back
Tell your students that they have to say the numbers counting back by 2s from 20, taking turns as in the
game “Zero”, but they have to decide whether to say one or two numbers. The winner is again the person
who says 0. Challenge your students to find the strategy for this game.
EXAMPLE:
Player 1 Player 2 P1 P2 P1 P2 P1
20 18,16 14,12 10 8,6 4 2,0 Player 1 wins
The main strategy is to try to say 12, but not 10, since this will also allow you to say 6 regardless of whether
your partner says both 10 and 8 or just 10. Whatever the case, do not say 4. Make your partner say either 4
or both 4 and 2. Then you can say 2 and 0 or just 0.
Then change the rules so that the person who says 0 loses.
Extension: Give students a sequence counting back by 2, 5, or 10 with a missing number, but don’t tell
them where the missing number is – challenge them to find it. For example:
28 26 24 20 18 16 14 12 10
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NS3-16 Counting by 10s
Goals: Students will extend patterns that count by 10.
Prior Knowledge Required: Adding 10
Subtracting 10
The relation between adding and counting on
The relation between subtracting and counting back
Review adding 10 from section NS3-10: Differences of 10 and 100. Have students practice adding 10 to
extend the patterns in the following order. Patterns that consist of:
• 2-digit numbers with ones digit 0 (EXAMPLE: 40, 50, 60, …)
• 3-digit numbers with ones digit 0 and do not cross multiples of 100
(EXAMPLE: 230, 240, 250, …)
• 3-digit numbers with ones digit 0 and do cross multiples of 100 (EXAMPLE: 270, 280, 290, …)
• 2-digit numbers with non-zero ones digit (EXAMPLE: 26, 36, 46, …)
• 3-digit numbers with non-zero ones digit and do not cross multiples of 100
(EXAMPLE: 217, 227, 237, …)
• 3-digit numbers with non-zero ones digit and do cross multiples of 100
(EXAMPLE: 479, 489, 499, …)
Repeat the sequencing for counting back by 10.
Extension: Complete the sequences
785, 795, _____, 815, _____
675, 685, _____, _____, 715
365, 375, 385, _____, _____
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NS3-17 Counting by 2s, 3s, 4s, 5s and 10s
Goals: Students will count by 4s and then will choose between counting by 2s, 3s, 4s, 5s or 10s.
Prior Knowledge Required: Counting by 2s, 3s, 5s and 10s
Write the following pattern on the board: 0 4 8 12 16
Have volunteers finish the second, third and fourth rows. Discuss any patterns they see in the rows and
columns. Have your students individually predict the next two rows in their notebooks. ASK: What is the
pattern in the ones digit? (0, 4, 8, 2, 6, then repeat) What is the pattern in the number of tens? (The number
of times each term is repeated is 3, 2, then repeat. The terms start at 0 and then increase by 1)
Have students practice skip counting by 4, starting at various numbers.
(EXAMPLES: start at 13, 27, 34, 55, 201, 836)
Review choosing between skip counting by 2, 3 or 5 (See NS3-14).
Then write the following number line on the board:
30 26
Tell your students that you want to skip count back from 30 to 26 and only say one number in between. If you
count by 1s, you know that 29 comes right after 30. Does 26 come right after 29?. What should I skip count
by if I want the same number to come right after 30 and right before 26?
Have students choose between what to skip count backwards by in the following order:
1. By 1 or 2
2. By 2 or 5
3. By 2 or 3
4. By 3 or 5
5. By 2, 3 or 5
6. By 3 or 4
7. By 2, 3 or 4
8. By 2, 3, 4 or 5
9. By 5 or 10
10. By 3, 5 or 10
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11. By 2, 3, 5 or 10
12. By 2, 3, 4, 5 or 10
Always begin by leaving only one space between the numbers and then progress to leaving more spaces.
When choosing between counting by 2, 3 or 5, a good strategy is to start counting by 3 and then either guess
a higher or lower number based on checking their first guess. Does counting by 3 get them too far or not far
enough? Should their next guess be higher or lower? Do they want to go further or less far with their next
guess?
Provide examples of numbers where missing numbers are filled in incorrectly, so that students need to find
the error.
EXAMPLE:
57 54 52 48
Students will need to first determine what number to skip count back by.
Literacy Connection:
What Comes in 2s, 3s and 4s? S. Aker
(Describes everyday situations where items naturally come in 2s, 3s and 4s.) Have students think of other
things that come in 2s, 3s and 4s. (EXAMPLES: wheels on a bike, eyes on a face, legs on a person, wheels
on a tricycle, sides on a triangle, tennis balls in a can, legs on a dog, legs on a chair, wheels on a car)
Discuss which is easiest to find: things that come in 2s, 3s or 4s and which is hardest to find.
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NS3-18 Counting by 100s
Goals: Students will extend patterns that count by 100.
Prior Knowledge Required: Adding 100
Subtracting 100
The relation between adding and counting on
Review adding 100 from section NS3-10: Differences of 10 and 100. Have students practice adding 100 to
extend the patterns in the following order. Patterns that consist of:
• 3-digit numbers with ones and tens digit 0 (EXAMPLE: 400, 500, 600, …)
• 4-digit numbers with ones and tens digit 0 and do not cross multiples of 100
(EXAMPLE: 2300, 2400, 2500, …)
• 4-digit numbers with ones and tens digit 0 and do cross multiples of 100
(EXAMPLE: 2700, 2800, 2900, …)
• 3-digit numbers with non-zero ones and tens digits (EXAMPLE: 267, 367, 467, …)
• 4-digit numbers with non-zero ones and tens digits and do not cross multiples of 100
(EXAMPLE: 3217, 3227, 3237, …)
• 4-digit numbers with non-zero ones digit and do cross multiples of 100
(EXAMPLE: 5479, 5489, 5499, …)
Repeat the sequencing for counting back by 100.
Activities:
The two activities below are from “A Guide to Effective Instruction in Mathematics, Kindergarten to
Grade 3.”
1. This is a game for pairs. Player A thinks of a starting number. Player B may add either 10 or 100 to the
starting number. Player A, in turn, adds either 10 or 100 to the number given by player B. Continue
taking turns in this fashion. The winner is the player who gets to 500 or closest to 500 without going over.
Variation: Use 1000 as the target number. More advanced students may choose between skip counting
by 10 and skip counting by 25. Weaker students may need to begin the counting at 0. Another variation
is for students to skip count backwards starting from 500 or 1000 with the winner being the player who
gets to 0 or closest to 0.
2. Students stand in a circle. The teacher chooses a volunteer to start the game. The first student begins by
saying 25, the next student says 50, then 75, and so on. Whenever a multiple of 100 (100, 200, 300,
etc.) is said, the student must sit down. The counting continues with only students who remain standing.
The game ends when only one student is standing.
Extension: Complete the sequences
723, 823, 923, _____, 1123 5833, 5933, _____, _____, 6233
1780, 1880, 1980, _____, 2180 7981, _____, 8181, 8281, 8381
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NS3-19 Regrouping
Goals: Students will extend patterns that count by 10.
Prior Knowledge Required: Link-it cubes
Ones blocks, tens blocks
A tens block is exactly like ten ones blocks, only easier to handle
because there is only one of them instead of ten of them
A 2-digit number is written according to the number of tens and ones
Vocabulary: tens blocks, ones blocks, trading
Show students a pile of 12 link-it cubes. Have a volunteer count them. Then group ten of them in a stack so
that you have 2 left over. ASK: How many are in this stack? How do you know? Tell them that you have one
stack of 10 and 2 more ones, so you have 12 link-it cubes altogether. ASK: How many link-it cubes would I
need to build 3 stacks of 10? 7 stacks of 10? Tell them that you have 4 stacks of 10 and 6 more blocks and
draw a model of this on the board:
ASK: How many link-it cubes did I use? (46)
Show students a pile of 27 link-it cubes. Tell them that you find them too hard to handle because there are
too many loose link-it cubes. Ask for strategies to deal with this. Suggest grouping some of them together in
ways that make it easier to count. Demonstrate grouping by 2s, 5s, 10s and 25s and use skip counting to
find out how many you have. ASK: Which way was easiest? When you group into 10s, did you need to skip
count at all to find the answer or is there another way? Remind them that the tens digit of a 2-digit number is
the number of tens and the ones digit tells how many more ones, so 2 stacks of ten with 7 more ones can be
easily read as 27. Draw several stacks of 10 on the board with ones left over and ask students to identify the
number of blocks used.
Bring out the base ten materials and show students that these blocks are just like link-it cubes except that we
don’t have to stack the blocks ourselves; the tens blocks are already together. Instead of grouping together
ten link-it cubes, they can trade 10 ones blocks for a tens block. Demonstrate with piles of 13 link-it cubes
and 13 ones blocks. Group 10 of the 13 link-it cubes and exchange 10 of the 13 ones blocks for a tens block.
Emphasize that you end up with a stack of 10 and 3 leftover blocks in each case, so trading ten ones blocks
for a tens block is just like stacking ten link-it cubes together.
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Then show how to model this process on the board.
Tell your students that this process of grouping ten single blocks into one stack of ten is called regrouping
because you are rearranging the blocks into groups.
ASK: How many ones blocks are left after you trade the ten ones for a tens block? How many ones blocks
were there before trading? How does breaking it up into tens and ones make it easier to count them?
Emphasize that 1 ten + 6 ones is the same thing as 16 ones and we use the blocks to show the numbers.
Then do examples where the number of blocks is more than 20, so that 2 tens blocks are required.
Draw base ten models with more than ten ones and have students practice trading ten ones blocks for a tens
block. They should draw models to record their trades in their notebooks. EXAMPLE:
4 tens + 19 ones = 5 tens + 9 ones
Students should also be comfortable translating these pictures into number sentences:
40 + 19 = 40 + 10 + 9 = 50 + 9 = 59
ASK: What number is 6 tens + 25 ones? How can we regroup the 25 ones to solve this question?
25 = 2 tens + 5 ones = 10 + 10 + 5, so 6 tens + 25 ones = 60 + 20 + 5 = 80 + 5 = 85
Tens Ones
6 25
6 + 2 = 8 25 – 20 = 5
Students can practice using such a chart to regroup numbers. Then have them regroup numbers without
using the chart:
3 tens + 42 ones = _________ tens + __________ ones
Draw several arrays with 10 rows of dots in each row:
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Bonus:
Have students count by tens to find out how many dots there are and then to say how many tens there are.
(EXAMPLE: 50 ones = 5 tens)
Progress to arrays that have only partial rows of 10, so that students need to count by 10s and then continue
to count by 1s to find out how many dots there are. Have them write how many tens and ones there are.
EXAMPLE:
____ ones = ____ tens + ____ ones
Bonus: Include examples with more than 10 rows.
Then have students fill in the blanks without using arrays:
a) 43 = ___ tens + ___ ones b) 75 = ___ tens + ___ ones
c) 80 = ___ tens + ___ ones d) 19 = ___ tens + ___ ones
e) 27 = ___ tens + ___ ones f) 98 = ___ tens + ___ ones
g) 63 = ___ tens + ___ ones h) 36 = ___ tens + ___ ones
Bonus: a) 126 = ___ tens + ___ ones b) 874 = ___ tens + ___ ones
Activity:
Pick-up Straws
You will need straws and elastics for this activity.
Cut up several straws into thirds. Make sure there are enough pieces so that everyone can have more than
10, preferably an average of about 20. Hide them around the room. Have everyone pick up as many as they
can. Have them count how many straws they have. Then have them pair up with a partner and find how
many straws they got together. Tell them they might have to group their leftover single straws together. Then
have each pair group with another pair. At the end, ask them how did grouping them in tens make it easier to
count the total number they had with a partner?
Extension: Regroup: 5 ten thousands + 7 thousands + 12 hundreds + 15 tens + 6 ones NOTE: You will have to regroup twice.
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NS3-20 Regrouping (Advanced)
Goals: Students will regroup base ten materials and coins to make standard models of numbers.
Prior Knowledge Required: Trading ten ones for a tens block
Show your students a hundreds block and ASK: How many tens blocks would I need to glue together to
make this block? How many ones blocks would I need? Tell them that they can trade a hundreds block for 10
tens blocks or for 100 ones blocks. Tell your students that you have 7 tens blocks ASK: How many more
ones blocks would you need to be able to trade for a hundreds block? Tell your students that you could trade
7 tens blocks and 30 ones blocks for a hundreds block.
Introduce the hundreds, tens and ones chart and continue with problems that require:
1. regrouping ones to tens (EXAMPLE: 4 hundreds + 5 tens + 22 ones)
2. regrouping tens to hundreds (EXAMPLE: 5 hundreds + 34 tens + 2 ones)
3. regrouping ones to tens and tens to hundreds (EXAMPLE: 3 hundreds + 14 tens + 25 ones)
4. regrouping ones to tens and tens to hundreds, but you don’t realize you need to regroup the tens to
hundreds until you have regrouped the ones to tens (EXAMPLE: 2 hundreds + 9 tens + 14 ones)
Give students play coins (pennies and dimes) so that each student has more than 10 pennies. Have
students record the number of each coin they have in a “dimes and pennies” chart:
Then have students trade coins so that they have the fewest number of coins but still the same amount of
money. They should give you 10 pennies in exchange for a dime. They should then record the number of
dimes and pennies they have after regrouping.
Repeat this exercise several times. ASK: How is this similar to regrouping tens and ones blocks? Which coin
is like the ones block? Which coin is like the tens block?
Repeat the exercise, but using loonies, dimes and pennies and including a hundreds column in your chart.
Ensure that each student has either more than 10 dimes or more than 10 pennies.
dimes pennies
3 16
dimes pennies
3 16
4 6
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Extensions:
1. Ask students to show the regrouping in QUESTIONS 1 to 3 with base ten blocks and with play money.
2. If you taught your students Egyptian writing (see Extension for NS3-1: Place Value – Ones, Tens, and
Hundreds) you could ask them to show regrouping using Egyptian writing.
EXAMPLE:
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NS3-21 Adding 2-Digit Numbers
Goals: Students will add 2-digit numbers without regrouping.
Prior Knowledge Required: Base ten materials
Adding 1-digit numbers
Vocabulary: sum
Have volunteers draw base ten models of the numbers 15 and 43 on the board. Students can draw sticks for
tens and dots for ones to make it easier. If students have trouble modelling 15 (or 43), ASK: Which digit is
the ones digit, the 1 or the 5? How many ones do we have? How many tens?
Tell students you want to add these two numbers. Write the following sum on the board:
15
+ 43
ASK: If we add 15 and 43, how much do we have in total? Prompt students to break the problem down into
smaller steps and to refer to the base ten models: How many tens are there altogether? How many ones
altogether? What number has 5 tens and 8 ones? What is 15 + 43?
Now draw a tens block and five ones:
Ask your students to count all the little squares, or ones, including those in the tens block. Tell students that
we use tens blocks because it’s easier to count many objects when we put them in groups of 10. ASK: When
would we use hundreds blocks?
ASK: How does using tens and ones blocks make it easier to add 2-digit numbers? Would using fives blocks
be just as easy? (No.) Why not? (Because we don’t have a fives digit, we have a tens digit. If we make and
count groups of ten, we can immediately write the tens digit in a number. If we make and count groups of
five, we have to do more work to translate the fives into a tens digit and maybe a ones digit. For example,
3 fives = 5 + 5 + 5 = 10 + 5 = 1 ten and 5 ones = 15.)
In their notebooks, have students draw base ten models to add more 1- and 2-digit numbers where
regrouping is not required (EXAMPLES: 32 + 7, 41 + 50, 38 + 21, 54 + 34, 73 + 2).
When students have mastered this, write on the board:
57
+ 21
ASK: How many tens are in 57? How many tens in 21? How many are there altogether? Did we need base
ten blocks to find out how many there are altogether? (no, there are 5 + 2 = 7 tens altogether) There are 5
tens and how many more ones in 57? There are 2 tens and how many more ones in 21? Do we need base
ten models to find out how many extra ones there are in total? (No, there are 8 in total. We can just add the 7
and the 1.) There are 7 tens and 8 more ones altogether—what number is that?
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As students answer your questions, write the digits in the correct position beneath the line, to demonstrate
the standard algorithm for addition:
57
+ 21
78
ASK: Why did I write the 8 under the 7 and the 1? Why did I write the 7 under the 5 and the 2? Ensure that
students understand that the ones digit of the answer is written under the ones digits of the addends, and
similarly for the tens digit. Have students add several pairs of 2-digit numbers that requires no regrouping.
(EXAMPLES: 24 + 32, 14 + 73, 25 + 41, 34 + 44, 26 + 13)
Bonus:
Have faster students add three 2-digit numbers (EXAMPLES: 41 + 23 + 15, 30 + 44 + 23, 21 + 21 + 21).
Extension:
Have students use base ten materials to add 2- and 3-digit numbers. Include only questions that
do not involve regrouping.
EXAMPLE: Find the sum:
132
+ 45
STEP 1: Create base ten models for 132 and 45.
132 = 45 =
STEP 2: Count the base ten materials you used to make both models:
132 and 45 = 1 hundred, 7 tens, 7 ones
STEP 3: Now that you know the total number of base ten materials in both numbers, you
have the answer to the sum:
132 + 45 = 177 (since 1 hundred + 7 tens + 7 ones = 177)
STEP 4: Check your base ten answer by solving the question using the standard algorithm
for addition (EXAMPLE: line up the two numbers and add one pair of digits at a time).
132
+ 45
177
Remind students that when they use the standard algorithm for addition, they are simply combining the ones,
tens, and hundreds as they did when they added up the base ten materials above.
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NS3-22 Adding with Regrouping (or Carrying)
Goals: Students will add 2-digit numbers with regrouping.
Prior Knowledge Required: Adding 2-digit numbers without regrouping
Vocabulary: regrouping, carrying
Tell students you want to add 27 and 15. Begin by drawing base ten models of 27 and 15 on the board:
27 = 15 =
Then write the addition statement and combine the two models to represent the sum:
27
+ 15 =
ASK: How many ones blocks do we have in the total? How many tens? Replace 10 ones with 1
tens block. ASK: Now how many ones do we have? How many tens? How many do we have altogether?
27
+ 15 =
Use a tens and ones chart to summarize how you regrouped the ones:
After combining the base ten materials
After regrouping 10 ones blocks for 1 tens block
Have students draw the base ten materials and the “tens and ones” charts for:
36
+ 45
28
+ 37
46
+ 36
19
+ 28
Bonus:
32
46
+ 13
29
11
+ 34
Tens Ones
2 7
1 5
3 12
4 2
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Show the tens and ones chart for the first question you gave the students (36 + 45).
Show your students the first step of the standard algorithm alongside a tens and ones chart.
ASK: Which step or steps from the addition is being shown on the right? (adding the ones digit and
regrouping the 16 ones as 1 ten and 6 ones). ASK: How does this step show the regrouping? Tell your
students that when we regroup 10 ones for a ten, we put the 1 on top of the tens column. Mathematicians
call this process “carrying the 1.” Ask students for reasons why this name is appropriate for the notation.
ASK: How are the tens and ones shown separately in the chart? How are the tens and ones shown
separately in the algorithm? Emphasize that the ones digit of the sum is always lined up with the ones digit of
the addends and the tens digit of the sum is lined up with the tens digit of the addends.
Have volunteers do the first step of the standard algorithm for several problems.
1
3 5
+ 2 8
3
4 4
+ 1 9
3 7
+ 4 7
4 8
+ 3 9
5 6
+ 2 5
3 8
+ 5 8
Explain to your students that even though we no longer have a tens column and a ones column in a chart
format, we still keep the ones in the same column and the tens in another column. This is why the tens digit
of the total number of ones is put on top of the tens column instead of on top of the ones column. Show them
how it is easier to line up the digits if they are using grid paper. Then have students practice lining up the
digits properly and doing the first step of the algorithm in their notebooks for several more problems.
Add each digit separately
Regroup 10 ones with 1 ten: 70 + 11
= 70 + 10 + 1
= 80 + 1 = 81
Tens Ones
3 6
4 5
7 11
8 1
Tens Ones
2 7
1 9
3 16
4 6
1
27
+ 19
6
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Tell them to leave enough space below their work so that they can finish the addition later. When they are
done, show the second step of the algorithm for the initial problem you showed them:
1
2 7
+ 1 9
4 6
Discuss the differences and similarities of these two methods of adding.
Similarities: you are still adding the ones and tens separately; you still get the same answer, you are still
trading the 10 ones from the 16 to get 1 ten and 6 ones. Differences: you don’t have to write the 3 at all; you
put the 1 still in the tens column, but this time it’s on top of the other tens; there is less writing. Emphasize
that you do not need to re-write the ones digit from the number of ones as you do using the chart, and that
you are just doing 2 steps at the same time. When using the standard algorithm, instead of finding 2 + 1 = 3
and then 1 + 3 = 4, you are doing both at the same time by finding: 1 + 2 + 1 = 4.
ASK: What is wrong with the following addition:
2 3
+ 2 5
2 5 5
Have a volunteer do the problem correctly. Discuss the importance of lining up the digits properly when adding.
Tell them that it is not just to make it look better; it actually makes it easier to add the numbers correctly.
ASK: How does grid paper make it easy to line up the digits?
Give students many problems to practice with, using grid paper. Include problems where the numbers add
up to more than 100 (EXAMPLES: 85 + 29, 99 + 15).
Tell your students that you had a student once who always added the tens before the ones. Show them the
student’s work and challenge them to find the answers that the student got wrong:
1 1
+ 5 8
6 9
1
1 7
+ 2 7
3 4
1
2 6
+ 2 6
4 2
4 3
+ 2 5
6 8
Lead a discussion on why it is important to add the ones first – if they add the tens first, they will forget to add
the extra 1 that was traded for 10 ones. Tell them that it is a bit tricky because they have to add from right to
left instead of from left to right. Tell them that even many grade 4 students will sometimes have trouble
remembering to add from right to left because it is so different from what they are used to, so that’s why it’s
important to practice a lot.
Tens Ones
2 7
1 9
3 16
4 6
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Give students more problems to practice with.
NOTE: Here is another exercise that will help your students visualize regrouping two digit numbers.
For this exercise you will need real or play money (dimes and pennies), and cards that have been divided
into ten squares, using the layout of tens frames.
Add: 38 + 25
STEP 1: Make a model of the two numbers by placing dimes on the tens cards and pennies on
the ones cards.
38 =
25 =
STEP 2: Move as many pennies from the lower ones-card as you need to fill the upper ones-card.
38
+ 25
STEP 3: Exchange the ten pennies on the upper ones card for a dime and place the dime in the
upper tens card. (This is the equivalent to the carrying step.) Notice that there are only 3
pennies left on the lower ones car:
tens ones
10¢ 1¢ 10¢ 10¢
1¢
1¢
1¢
1¢
1¢
1¢ 1¢
10¢ 10¢ 1¢ 1¢ 1¢ 1¢ 1¢
tens ones
10¢ 10¢ 10¢ 1¢
1¢
1¢
1¢
1¢
1¢
1¢ 1¢
1¢ 1¢
10¢ 10¢ 1¢ 1¢ 1¢
tens ones
1
38 + 25 3
1¢ 1¢ 1¢
10¢ 10¢ 10¢
10¢ 10¢
10¢
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STEP 4: Move all the coins to the upper cards and count the total (63).
Addition Rummy
This activity was adapted from: “A Companion Resource for Grade Two Mathematics,” by Saskatchewan
Learning. This activity will consolidate the students’ understanding of 2-digit addition and will prepare them
for 3-digit addition.
Give your students the BLMs “Addition Rummy Preparation” and “Addition Rummy Blank Cards”.
Explain to your students three different ways of adding and how they all correspond to the same problem.
For example:
1
23 2 tens + 3 ones +
+ 39 + 3 tens + 9 ones
62 5 tens + 12 ones
Always make sure to draw the model on the board as well as using the actual base ten materials so that
students can see how the symbol looks on “paper”.
Then have them do the BLM “Addition Rummy Preparation”. They have to create the two matching cards
themselves. They could then make up their own addition questions, solve them using the standard algorithm
and show them in the two different ways on the BLM “Addition Rummy Blank Cards”. Suggest to the
students that they fill the left-hand column with the standard algorithm, the second column with __ tens + ___
ones and the third column with how they would show it with base ten materials. Actually give the students the
base ten materials if it helps them.
Once all students are done the worksheet, they can cut out their cards and they are ready to play Addition
Rummy. They should play with a partner; together they will have made a total of 36 playing cards (12 new
cards each and 12 shared that they can only use one copy of). They should deal out 8 cards to each player
and the remaining cards will be in a pile, face down, between them.
tens ones
1 38 + 25
63
10¢ 10¢ 10¢ 10¢
10¢
10¢ 1¢ 1¢ 1¢
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The top card should be placed face up in a separate discard pile. The first player decides to either pick
up the top card from the face up discard pile or to pick up the top card from the face down pile. He or she
then discards a card face up for the next player. It is then the next player’s turn. Once a player has 3
complete sets of 3 matching cards, they win. Note that, on their last turn, the student will not need to discard
any cards.
Estimating Game
You will need dice for this game. Draw on the board:
If you do not have dice, you can have students make their own dice (see below).
Either roll one die 4 times, 2 dice twice, or 4 dice once. Write the digits you rolled on the board for the
students to see. Tell them that you want to place your roll in the top 4 squares so that the sum of the 2-digit
numbers is as close to 100 as possible. Let several volunteers guess and add their 2-digit numbers on the
board. Ask which one is the closest.
You do not need to make a big deal out of who was the closest, but make sure all students agree on which
answer is the closest. Then have them try to make those same numbers as close to 70 as possible and
finally as close to 40 as possible.
Then have a student roll 4 times and write their numbers on the board. Repeat the exercise with those 4
numbers.
Provide students with the BLM “Estimating Game”. Students can work in groups if there is not enough dice
for everyone. They could each have their own sheet, and take turns recording their numbers before anyone
begins the actual page.
As an extension, you could make it harder by having them record the number in a square after each roll and
not allowing them to change their minds based on their next roll.
How to make your own dice:
Students could use the nets from the BLM “Cubes” to make dice. Another more fun (but time-consuming)
way to make dice is to have your students bring in 6-pack egg cartons, but bring in a few extra in case some
students forget. It is a good idea to start collecting them several weeks before actually doing the activity. Let
them know that a 12-pack cut in half will not work for the activity, they actually need a closed 6-pack that you
could shake a coin in and the coin won’t fall out.
To make the dice, have them write different numbers in each egg-hole in the carton. You could have them
write the numbers on paper first and tape or glue them to the carton. Then they put four counters into the
carton and shake. This is their “roll”.
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I have ---, who has ----?
Using the BLM “Make Up Your Own Cards,” (see NS3-6: Representation in Expanded Form) make cards,
depending on the number of students in the class. Example,
I have
28 ----------
Who has
17 + 25
The student must read the answer from the bottom (the student says, “I have 28, who has 42?” and the
person with 42 then says “I have 42, who has ---?” depending on what question they have on the bottom of
their card. Play continues until everyone gets a turn.
Dominoes
This is a variation of the “I have --- who has ---?” game. Have one student go up to the board or and tape
their card. Then the person whose top matches the bottom of the other goes to play theirs in domino fashion.
Or have two teams, randomly distributing the cards to two teams and the team that can make the longest
chain wins.
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NS3-23 Adding with Money
Goals: Students will add dimes and pennies.
Prior Knowledge Required: Adding 2-digit numbers with regrouping
Using cents notation properly
Vocabulary: fewer, fewest, ¢
Write on the board: 51 = tens + ones. Have volunteers fill in the blanks with all the possible
combinations (EXAMPLE: 51 = 4 tens + 11 ones).
ASK: Which combination uses the most tens? Which combination uses the most ones? Which is the
standard way of writing 51 as tens and ones? (5 tens + 1 one) Why do you think that is? (It uses the fewest
number of tens and ones, i.e. the fewest number of blocks in a base ten model.)
SAY: “I have 2 quarters and 1 penny, so 51¢ in total. I want to trade my 51¢ for dimes and pennies. How
many dimes and pennies could I trade for? Have volunteers provide different answers to the question:
51¢ = dimes + pennies
ASK: Which combination uses the most dimes? Which combination uses the most pennies? Which
combination uses the least number of coins? If you were to invent a standard way of writing amounts of
money as sums of dimes and pennies, how would you do it? Would you use the most dimes or the most
pennies? Why?
Remind your students that you asked how they could write 51 as a sum of tens and ones and also how they
could write 51¢ as a sum of dimes and pennies. ASK: What is the same about these two questions?
(Answers include: the same numbers work in both; a ten is worth te n ones and a dime is worth ten pennies;
the way that uses the fewest blocks also uses the fewest coins.) Have students answer several questions
using both tens and ones, and dimes and pennies. Students should use standard form.
EXAMPLE: 36 = tens + ones 36¢ = dimes + pennies
Then take away the context of tens and ones and have students do examples using only dimes
and pennies.
Draw a dimes and pennies chart and ask students if
they’ve seen another chart that looks similar.
ASK: What do you think the second row is for? What would you do if the headings were “tens” and “ones”?
Tell students that they can regroup 10 pennies as a dime just like they can regroup 10 ones as
a ten. Complete the second row in the chart.
Have students complete more such charts independently. Ensure that every student can regroup 10 pennies
for 1 dime, 20 pennies for 2 dimes, and so on, before moving on. Then have students use charts to add
dimes and pennies the same way they added 2-digit numbers. Finally, have them add dimes and pennies
using regrouping and carrying. Encourage them to write the cents sign (¢) when adding money.
Dimes Pennies
2 12
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NS3-24 Adding 3-Digit Numbers
Goals: Students will add 3-digit numbers with and without regrouping.
Prior Knowledge Required: Adding 2-digit numbers
Place value Base ten materials
Vocabulary: algorithm
Have volunteers draw base ten models for 152 and 273 on the board and tell students that you want to add
these numbers.
ASK: How many hundreds, tens, and ones are there altogether? Do we need to regroup? How do you know?
How can we regroup? (Since there are 12 tens, we can trade 10 of them for 1 hundred.) After regrouping,
how many hundreds, tens, and ones are there? What number is that?
Write out and complete the following statements as you work through the EXAMPLE:
152 hundred + tens + ones
+ 273 hundred + tens + ones
hundred + tens + ones
After regrouping: hundred + tens + ones
Have students add more pairs of 3-digit numbers. Provide examples in the following sequence:
• the ones need to be regrouped (EXAMPLES: 238 + 147, 426 + 165)
• the tens need to be regrouped (EXAMPLES: 456 + 381, 277 + 392)
• either the ones or the tens need to be regrouped (EXAMPLES: 349 + 229, 191 + 440).
• both the ones and the tens need to be regrouped (EXAMPLES: 195 + 246, 186 + 593).
• you have to regroup the tens, but you don’t realize it until you regroup the ones (EXAMPLES: 159 +
242, 869 + 237) Use this to emphasize the importance of regrouping the ones first.
Now show students the standard algorithm alongside a hundreds, tens, and ones chart for the first example
you did together (152 + 273):
Hundreds Tens Ones
1 5 2
2 7 3
3 12 5
3 + 1 = 4 12 – 10 = 2 5
1
1 5 2
+ 2 7 3
4 2 5
Point out that after regrouping the tens, you add the 1 hundred that you carried over from the tens at the same time as the hundreds from the two numbers, so you get 1 + 1 + 2 hundreds.
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Demonstrate both the chart and the algorithm for adding a 3-digit number to a 2-digit number and emphasize
the importance of lining up the digits – ones with ones, tens with tens and hundreds with hundreds.
Activities:
1. Play addition rummy as in NS3-21, except use 3-digit numbers in place of 2-digit numbers.
2. Play “I have ---, who has ---?” From NS3-6, but this time use pairs of 3-digit numbers that sum to at most
999. These cards can also be used to play the dominoes game introduced in NS3-22.
Extensions:
1. Game for two players:
Each player makes a copy of the grid shown.
Players take turns rolling a die and writing the number rolled on one of their grid boxes.
The winner is the player who creates two 3 digit numbers with the greatest sum.
2. If you have already taught your students Egyptian writing (see Extension for NS3-1: Place Value –
Ones, Tens, and Hundreds) you could ask them to show adding with regrouping using Egyptian writing.
EXAMPLE:
3. Have students add more than 2 numbers at a time:
a) 427 + 382 + 975 + 211
+
+
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NS3-25 Subtracting 2- and 3-Digit Numbers
Goals: Students will subtract without regrouping.
Prior Knowledge Required: Subtraction as taking away
Base ten materials Place value Separating the tens and the ones
Vocabulary: standard algorithm
Tell your students that you want to subtract 48 – 32 using base ten materials. Have a volunteer draw a base
ten model of 48 on the board. SAY: I want to take away 32. How many tens blocks should I remove? (3)
Demonstrate crossing them out. ASK: How many ones blocks should I remove? (2) Cross those out, too.
ASK: What do I have left? How many tens? How many ones? What is 48 – 32?
Have student volunteers do other problems with no regrouping on the board (EXAMPLES: 97 – 46, 83 – 21,
75 – 34). Have classmates explain the steps the volunteers are taking. Then have students do similar
problems in their notebooks.
Give students examples of base ten models with the subtraction shown and have them complete the tens
and ones charts:
48 – 32: Tens Ones
4 8 – 3 2
1 6
Have students subtract more 2-digit numbers (no regrouping) using both the chart and base ten models.
When students have mastered this, have them subtract by writing out the tens and ones (as in question 2 on
the worksheet):
46 = 4 tens + 6 ones
– 13 = 1 tens + 3 ones
= 3 tens + 3 ones
= 33
Then have students separate the tens and ones using only numerals (as in QUESTION 3 on the worksheet):
36 = 30 + 6
– 24 = 20 + 4
= 10 + 2
= 12
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Now ask students to subtract:
54
– 23
ASK: Which strategy did you use? Is there a quick way to subtract without using base ten materials, or tens
and ones charts, or separating the tens and ones? (Yes—subtract each digit from the one above.) What are
you really doing in each case? (Subtracting the ones from the ones and the tens from the tens, and putting
the resulting digits in the right places.)
Have students draw a base ten model of 624 and show how to subtract 310. Have them subtract using the
standard algorithm (i.e. by lining up the digits) and check to see if they got the same answer both ways.
Repeat with several 3-digit numbers that do not require regrouping.
Extensions:
1. Subtract using the base ten blocks:
a) 7 2 9 b) 8 9 5 c) 5 2 4 d) 3 9 8 e) 5 9 2
– 3 1 6 – 2 5 4 – 4 0 1 – 1 6 3 – 1 7 0
Bonus: Have students subtract from many-digit numbers:
9964387 678439841
– 2541263 – 138210731
Students may subtract from right to left or left to right. If you teach borrowing from right to
left as many teachers do, you may wish to develop the habit of subtracting from right to left
at this stage.
2. Write on the board: 100 – 36. ASK: How is this problem different from problems they have seen so far?
Challenge them to change it to a problem they already know how to do. After letting them work for a few
minutes, suggest that they think of a number close to 100 that does have enough tens and ones to
subtract directly and then adjust their answer. (Students can use any of 96, 97, 98 or 99; for example,
98 – 36 = 62, so: 100 – 36 = 64.)
3. Teach students to subtract numbers like 100 – 30 by counting the number of tens in each number:
10 tens – 3 tens = 7 tens = 70. Give several practice problems of this type and then ask:
What is 100 – 40? What is 40 – 36? (students can count up to find this answer) How does this help to
find 100 – 36? (show a number line to help them see the addition they need to do)
4. Have students subtract by changing to a problem they already know how to do and then adjusting their
answers:
a) 61 – 28 b) 34 – 15 c) 68 – 39
Example solution: 58 – 28 = 30, so: 61 – 28 = 30 + 3 = 33.
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NS3-26 Subtracting by Regrouping
Goals: Students will subtract with regrouping using base ten materials and using the standard algorithm.
Prior Knowledge Required: Subtraction without regrouping
Base ten materials Counting by 10s from any number
Vocabulary: standard algorithm, regrouping, “borrowing”
Ask students how they learned to subtract 46 – 21. Have a volunteer demonstrate on the board. Ask the
class if you can use the same method to subtract 42 – 28. What goes wrong? Should you be able to subtract
28 from 42 If you have 42 things, does it make sense to take away 28 of them? (Sure it does). Challenge
students to think of a way to change the problem to one that looks like a problem they did last time. ASK:
What is a number close to 42 that has a larger ones digit as 28? (39) What is 39 – 28? (11) How is this
problem more like the ones from last class? How can we use 39 - 28 to find 42 – 28? Show a number line on
the board to help them.
28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Show students how to subtract by adding: 11 + 3
28 39 42
The difference between 42 and 28 is 11 + 3 = 14. ASK: Is there another way to change the problem to make
it more like the problems we did last time? (finding 38 – 28 and then adding 4 would also work) Have
students do more problems of this type by finding a number that is close to the number they are subtracting
from but has the same or larger ones digit as the number they are taking away.
a) + b) + c) +
37 81 38 63 56 72
EXAMPLE: Students could find 81 – 37 by finding 79 – 37 = 42 and then counting up from 79 to 81 to
find 81 – 79 = 2, so 81 – 37 = 44. Students could also find 78 – 37 or 77 – 37 and then count up and add
the differences.
Now change the rules: tell students they are not allowed to use adding anymore; they need to think of a way
to change 46 – 28 into a problem just like last time where 46 has more tens and more ones than 28. ASK:
How can you trade tens and ones so that 46 has more ones than 28? Have pairs work on 46 – 28 using
these new rules. Give them a few minutes to think about the problem, and then have volunteers share their
strategies with the group.
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ASK: Which number required trading—the larger number or the smaller number? Why does it help to have
more ones blocks in the number 46? Use base ten materials to illustrate—make a standard model of 46 and
then regroup 1 ten as 10 ones:
46
– 28
SAY: Now there are 16 ones, so we can take away 8 of them. Since we didn’t change the value of 46—we
just traded blocks—we will get the right answer. Removing the shaded blocks (2 tens and 8 ones), we are
left with 1 ten and 8 ones. So: 46 – 28 = 18.
Tell your students that mathematicians have come up with a standard notation to say that you are borrowing
a ten from the tens digit and trading it for ten ones:
4 14
54 = 5 tens + 4 ones 4 tens + 14 ones
Emphasizing that you are taking 1 ten away and replacing it with 10 ones. Have volunteers come to the
board to show this standard notation for various examples alongside the trading of base ten materials. (First,
have students show each number using base ten materials and then have another volunteer show the
trading and the notation for trading at the same time).
When students have mastered this, move on to subtracting using the standard algorithm.
Draw on the board:
7 5 – 4 8
ASK: Can you take away 48 from 75? (yes, 75 is bigger than 48, so if I have 75 objects, I can just remove 48
of them and count how many are left) Can you take away 8 ones from the 5 ones: Do they need to borrow?
How can they change the 5 ones so that they have enough ones? What can they borrow from so that the 75
still has the same value? Have a volunteer show using the standard notation how they would borrow a ten
from the tens digit and replace it with ten ones – emphasize that the 48 stays the same; only the 75 is
changing. When they are done the borrowing should look like:
Tens Ones 4 6
5 4
7 5 6 0 8 9 5 3 3 0
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6 15
7 5 – 4 8
Students can check their borrowing by adding 60 + 15 = 75. Then ask if we can take 8 ones from 15 ones?
Can we take 4 tens away from 6 tens? Have a volunteer show this and say the answer to 75 – 48.
Do several examples of this where borrowing is always required and then give them lots of practice where
borrowing is always required.
Then tell them that sometimes borrowing is not required and they have to decide when to use borrowing.
Show them what would happen if you borrowed when you didn’t need to.
6 15
7 5
– 5 2
1 13
They could still do it this way if they remember to trade the ten ones back for a ten to end up with 2 tens and
3 ones, but it is a lot faster not to borrow in the first place:
Give several examples where students only need to decide whether to borrow or not. Have several
subtraction questions on the board and have them raise their hand if they need to borrow and not raise their
hand if they don’t need to borrow as you point to each question. Encourage students to explain how they
know. Emphasize that if there are more ones in the first number, they don’t need to borrow, but if there are
more ones in the second number, they do need to borrow.
Then give them several examples where they need to decide whether or not to borrow and then do the
subtraction. Have student volunteers explain at each step what they are doing.
Activity: Ask students to show the regrouping in QUESTIONS 1 a) to d) with base ten blocks.
7 5 – 5 2 2 3
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Extensions:
1. Have students subtract by adding and use the standard algorithm for several problems.
EXAMPLE: 54 – 26 = + + =
26 30 50 54
When is the standard algorithm easiest? (when there is no borrowing necessary) When is subtraction by
adding easiest? (when the adding does not require carrying) Notice that the addition will require carrying
precisely when the subtraction does not require borrowing. For example, 54 – 26 = 4 + 20 + 4 = 20 + 8 =
28 and no carrying is required, whereas 54 – 23 = 4 + 20 + 7 = 20 + 11 = 31 and there is carrying
required.
2. Make a 2-digit number using consecutive digits (EXAMPLE: 23). Reverse the digits of your number to
create a different number, and subtract the smaller number from the larger one (EXAMPLE: 32 – 23).
Repeat this several times. What do you notice? Some students may wish to investigate what happens
when we don’t use consecutive digits (EXAMPLE: 42 – 24 = 18, 63 – 36 = 27, 82 – 28 = 54; the result in
this case is always from the 9 times tables).
Journal:
Have students explain which way of subtracting (subtracting by adding or the standard algorithm) they like
better and why.
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NS3-27 Subtracting by Regrouping Hundreds
Goals: Students will subtract 3-digit numbers by borrowing when necessary.
Prior Knowledge Required: subtracting by using base ten materials and regrouping
subtracting 2-digit numbers by borrowing (using the standard algorithm) standard notation for regrouping
Vocabulary: subtraction, borrowing, regrouping, standard notation, standard algorithm.
Have students subtract 3-digit numbers using base ten materials and transfer their results to
standard notation:
For example, to subtract 543 – 365, they would trade a hundreds block for ten tens blocks and a tens block
for ten ones blocks and the borrowing, in standard notation would look like:
13
4 14 4 14 13
5 4 3 5 4 3
3 6 5 3 6 5
Students have to keep trading until all numbers in the top row are larger than all numbers in the bottom row.
They could also trade the tens and ones first instead:
13
3 13 4 3 13
5 4 3 5 4 3
3 6 5 3 6 5
1 7 8
In any case, the answer is 178. Students should be encouraged to check that their borrowing is done
correctly by adding: 4 hundreds + 13 tens + 13 ones = 400 + 130 + 13 = 543. Students should also check
that their subtraction is done correctly by adding 178 + 365 = 543.
Notice that if you teach students to subtract each digit as it becomes available so that they are borrowing
then subtracting then borrowing then subtracting and so on, then you must teach them to borrow from right to
left. If students borrowed the ten tens from the hundreds digit first, their subtraction would start with hundreds
digit 1 and tens digit 8. When they then borrow from the tens, they would find that the 8 is incorrect.
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If, however, you teach students to set up all the borrowing first so that the problem looks just like one without
borrowing, then students may borrow in any order. The advantage of this approach is that it reinforces the
idea of turning the problem into one they already know how to do; the disadvantage is that it is less
“standard” in that later teachers might expect your students to borrow from right to left. Even if you teach the
standard method, it will deepen the students’ understanding if you show them that they will get the correct
answer by doing the subtraction after all regrouping is done.
To teach the standard method, provide problems in the following sequence:
• regrouping a ten as ten ones (EXAMPLE: 754 – 216)
• regrouping a hundred as ten tens (EXAMPLE: 754 – 281)
• regrouping a hundred as ten tens and regrouping a ten as ten ones (EXAMPLE: 754 – 467) Show
students how this will obtain the incorrect answer if they borrow from left to right and subtract the
digits as they borrow.
• regrouping a hundred as ten tens is required, but only after regrouping a ten as ten ones (EXAMPLE:
754 – 357) Use this to again emphasize the importance of borrowing from right to left.
Activities:
1. Ask students to make a base ten model for the number 81. Then have students take away any 3 blocks.
Have volunteers draw their model for the number they have left on the board. ASK: What number did the
student take away? What number is left?
Have other volunteers show different answers—how many answers are there?
Give students base ten materials and an individual number with sum of digits at least 5 (81, 63, 52, 23,
41, 87) Ask students to make base ten models for their number and to find as many numbers as they
can by taking away exactly 3 blocks. They should make a poster of their results by drawing models of
base ten materials and showing how they organized their answers. This can be done over 2 days, giving
2-digit numbers the first day and 3-digit numbers the second day.
2. Ask students to show the regrouping in QUESTIONS 1 a) to d) with base ten blocks.
Extensions:
1. Teach your students the following fast method for subtracting from 100, 1 000, 10 000 (it helps them
avoid regrouping).
You can subtract any 2-digit number from 100 by taking the number away from 99 and then adding 1 to
the result. EXAMPLE:
100 = 99 + 1 – 42 = – 42 = 57 + 1 = 58
You can subtract any 3-digit number from 1 000 by taking the number away from 999 and adding 1 to
the result. EXAMPLE:
1000 = 999 – 423 = – 423 = 576 + 1 = 577
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2. (See Extension 2 from Worksheet NS3-26: Subtracting by Regrouping) Create a 3-digit number using
consecutive digits (EXAMPLE: 456). Reverse the digits of your number to create a different number, and
subtract the smaller number from the larger one. Repeat this several times. What do you notice? (The
result is always 9 for 2-digit numbers and 198 for 3-digit numbers). Have them find the result for 4-digit
numbers and predict the result for 5-digit numbers.
3. Have students determine which season is the longest if the seasons start as follows (in the northern
hemisphere):
Spring: March 21
Summer: June 21
Fall: September 23
Winter: December 22
Assume non-leap years only.
Students will need to be organized and add several numbers together as well as use some subtraction.
For example, the number of days in spring is March: 31 – 20 (since the first 20 days are not part of
spring) April: 30 May: 31 June: 20.
Total: Students may either add: 11 + 30 + 31 + 20 or notice that they are adding and subtracting 20 and
simply add 31 + 30 + 31 = 92 days. Summer has 10 + 31 + 31 + 22 = 94 days, fall has 8 + 31 + 30 + 21
= 90 days and winter has 10 + 31 + 28 + 20 = 89 days, so in order from longest to shortest: summer,
spring, fall, winter.
Students should be encouraged to compare this ordering with temperature. It is premature to discuss
scientific reasons for this; just noticing the pattern is enough.
Students may wish to examine this question for their own region and the current year.
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NS3-28 Mental Math
Goals: Students will decompose numbers into sums in an organized way and will use pairs making 10 to
add single-digit numbers. Students will add multiples of 10 to 1-digit numbers.
Prior Knowledge Required: Decomposing numbers in different ways
Models for addition
Missing addend problems
Addition is commutative
Vocabulary: pair, making 10
Draw 6 rows of 7 circles, with a vertical line between them as shown.
Write 1 + 6 = 7 beside the first row and have different volunteers come up and write number sentences for
each row of circles. Ask your students what number is the same in each addition sentence and what
numbers change. Ask how the numbers change each time. What happens to the first number? (It increases
or grows by 1) What happens to the second number? (It decreases or shrinks by 1).
Ask a volunteer to show 7 + 0 or 0 + 7. Tell them that when finding all the number sentences that add to 7,
counting those two would almost be cheating, because they’re too easy.
Then write 8 = 1 + ____
8 = 2 + ____
8 = 3 + ____
8 = 4 + ____
8 = 5 + ____
8 = 6 + ____
8 = 7 + ____
Ask them for strategies to fill in the rest of the numbers. Possibilities include:
• Draw 8 circles and a vertical line between them at various points and then count the circles to the
right of the vertical line.
• Start at the bottom and write the numbers 1, 2, 3, 4, 5, 6 and 7 in order from the bottom up;
• Start at the top, and write the numbers in backwards order, starting at 7;
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Emphasize that from one addition sentence to the next, you are adding one to the first number and
subtracting one from the second number, so you are not changing the total number. That is just like moving
the line one more circle to the right.
Tell them that you don’t need all of the number sentences because some of them have the same numbers,
and you don’t care what order the addends are in. Show how you can erase the bottom three equations and
still have all the pairs that add to 8.
Ask your students how many number sentences they think they’ll need to write all the ways you can sum to
6. (Don’t include 0 + 6 or 6 + 0 because it’s too easy of a sum, so you don’t care about that one).
Now teach students how to find pairs that make 10. Hold up all your fingers on both hands. ASK: How many
fingers do I have up? Then hold up 7 fingers and say, “How many fingers do I have up? How many do I not
have up? What is 7 + 3? How do you know?” Repeat with several examples. Then say, “I want to know what
number makes 10 with 4,” and write on the board: 4 + ____ = 10. ASK: How could I use my 10 fingers? How
many fingers should I hold up? What does the number of fingers I’m not holding up tell me?
Write 3 numbers on the board: 4 5 6. Ask them if 4 makes 10 with either of the next two numbers. Yes it
does, so we can circle the 4 and the 6.
4 5 6
Then try: 2 3 7 and ask if 2 makes 10 with any of the other numbers. What number would have to be there
for 2 to make 10 with it? If some students aren’t sure, remind them that they can hold up 2 fingers and count
the number of fingers that they are not holding up. So we can cross out the 2 because we know it’s not one
of the numbers we have to circle:
2 3 7
Then look at the last 2 numbers – do they make 10? Yes they do, so circle them. Do several examples of
this, either circling the first number with one of the other 2, or crossing out the first number and circling the
other 2.
Bonus: Use longer lists of numbers. (EXAMPLES: 3 4 5 6, 2 3 4 7 9)
Write 9 + 4 on the board. ASK: How can we write 4 as a sum of two numbers? (4 = 1 + 3 or 2 + 2).
Show students how to use this to add 9 + 4: 9 + 4 = 9 + 1 + 3 OR 9 + 4 = 9 + 2 + 2.
ASK: Does one of these make it easier to add 9 + 4? Why? Is it easier to add 10 + 3 or 11 + 2? Why?
Emphasize that we don’t have just one number on either side of the equal sign. Both sides have more than
one number. But that’s okay – it’s still true that if I have 9 “anythings” and add 4 more “anythings”, I get the
same number as if I started with 9, added 1 and then added 3.
Some students might find it helpful to have 3 piles of counters (of 9, 1 and 3) and then put the last 2 piles
together to see that they now have 4 in the second pile and still 9 in the first. Stress that the counters neither
disappear nor appear out of the air, so their number does not change.
Or you could draw three groups on the board, one of 9 circles, one of 1 circle and one of 3 circles. Write the
number sentence on the board as review: 9 + 1 + 3 = 13. Then circle the two last groups to show that you
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are grouping them together and write: 9 + 4 = 13. Emphasize that you have not changed the total number of
circles just by grouping the last two piles.
Then ask students how to separate the second number to make 10 with the first number for several
problems, following the sequence below:
• Problems with first addend 9
(EXAMPLES: 9 + 5 = 9 + 1 + 4 = 10 + 4 = 14, 9 + 7, 9 + 3, 9 + 6, 9 + 8, 9 + 4)
• Problems with first addend 8
(EXAMPLES: 8 + 5, 8 + 6, 8 + 4, 8 + 7, 8 + 9, 8 + 8)
For each problem above, students should show what is happening with a model. For example, to show
9 + 5 = 9 + 1 + 4 = 10 + 4 = 14, draw a row of 9 beside a row of 5 separated as shown:
To the left of the line there are 10 dots and to the right, there are 4 dots, so this picture, before the line was
drawn, showed 9 + 5 and now shows 9 + 1 + 4 or 10 + 4. So 9 + 5 = 10 + 4 = 14.
Now provide many problems with first addend ranging from 6 to 9
(EXAMPLES: 6 + 7, 9 + 5, 7 + 7 , 8 + 5, 9 + 8, 7 + 4, 8 + 5, 8 + 7, 9 + 6) Have one volunteer decompose the
second addend using all the number sentences possible and another volunteer find the number that makes
10 with the first addend. Students can then fill in the blanks for problems of the form:
6 + 7 = 6 + ____ + ____
These make 10 Left over
Since 6 + 4 makes 10, students need to think: 7 = 4 + _____.
Then have students use this strategy to add pairs of single-digit numbers. Finally, have them add single-digit
numbers to 2-digit numbers using this strategy:
36 + 7 = 36 + _____ + _____
These make 40 Left over
Give students word problems which require them to use this new skill. (EXAMPLE: 57 students went on a
field trip. 4 teachers went with them. How many people went on the field trip?)
NOTE: The next lessons will begin to teach students how to solve word problems. If some students are
struggling with the word problems on this sheet, go back to these problems after they are more comfortable
with word problems. This will allow students to see their progress if they will later find easy what they
struggle with now.
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Activities:
Magic Number Dice
Make a die from a photocopy of one of the nets on the BLM “Cubes”. Copy the numbers from a die onto your
cube. Be sure that opposite sides of the die add to 7 as is true for a regular die, so that 1 is opposite 6, 2 is
opposite 5 and 3 is opposite 4. Tell them that you made a bigger die by copying a smaller one. Tell them
there is a magic number on every die that many people don’t even know about and their goal is to find that
magic number. Tell them that instead of rolling two dice and adding the total, you are just going to roll one
and add the top and the bottom. Roll the die, show the top face and then the bottom face. Ask the students
what the total you rolled is. Ask a volunteer to come write the addition sentence on the board. Then ask
another volunteer to roll the die and write their addition sentence on the board by looking at the top and
bottom numbers. Continue having volunteers do this. Ask if anyone sees a magic number. Ask them if they
think it would make sense, if they didn’t have two dice, to just roll one and add the top and bottom numbers.
Why wouldn’t it make sense?
Tell them that you would like to make a different pair of dice, but this time, you want the magic number to be
10. Ask them what pairs of numbers they could put on the top and bottom to make a total of 10. Ask
volunteers to write addition sentences on the board such as 1 + 9 = 10, 2 + 8 = 10, etc.
Then give each student a copy of the BLM “Cubes” and have them make their own die with a magic number
of 10. Demonstrate cutting out a cube first, being careful not to cut out the tabs. If your students use tape
instead of glue, this is not as important.
Once their cubes are made, they can practice finding the missing number that makes 10 with the top number
and checking the bottom to see if they’re right. They can switch dice with different partners as well to practice
different addends, since some might have chosen different addition sentences that add to 10. Have students
roll their dice onto a plate or a shoebox lid, so that they don’t throw them across the room.
Tens
This is a solitaire game. You will need a deck of cards – remove the 10, J, Q and K. The player shuffles the
cards and turns over the first 10, putting them in 2 rows of 5 cards. The goal of the game is to find cards that
make 10 with the top card of the remaining pile. Each card they turn over, they either place it on top of
another card it makes 10 with or they discard it. After they run through the pile, they can take the discard pile
and use them to make 10 with cards that are at the top of one of the 10 piles that are face up. They can go
through the discard pile as many times as they want. They then count the number of cards in the discard pile
after they cannot place any more in piles. This is the number of points they get. The fewer points they get,
the better.
Game: Modified Go Fish
This game requires students to hold 6 cards and find a pair that makes 10. Details can be found in the
Mental Math section of this guide.
The activities below are designed to give students practice adding single-digit numbers that add to more than
10. They are all adapted from “A Companion Resource for Grade Two Mathematics,” by Saskatchewan
Learning.
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Adding on a 9 × 9 grid
Give each student a copy of the BLM “Addition Table (Ordered)” and a small counter. Have the students toss
their counters as many times as they can in 2 minutes and write the answers to the addition: if their counter
lands on the column numbered 4 and the row numbered 9.
They write the answer to 4 + 9 in that square. In this way, students randomly generate questions for
themselves. Repeat this over several periods so that students can see how they are improving, then suggest
that they challenge their parents to a competition—can they add more pairs than their parents can?
Number Sentence Practice
You will need the BLM “Number Sentence Practice,” enough two-colour counters to give each pair of
students 30 of them. Photocopy the BLM enough times so that each pair of students can have one game
card between them. If you don’t have two-colour counters, you can use coins with heads and tails as the two
“colours”.
Group students in pairs. Give each pair of students their own game card and 20 counters. Photocopy a game
card onto a transparency so that you can demonstrate on the overhead projector. If you do not have an
overhead projector, copy a game card onto the board. Tell students that their goal is to get four in a row
before their partner does and that you’ll demonstrate playing against the class. To show them, you will go
first, but you want their help. Tell them that you are allowed to choose how many counters to use and then
you want to shake them to find a number sentence. Demonstrate this with 7 counters and tell students that
you will get a number sentence from the red counters and the yellow counters. Ask them what number
sentence you got. For example, if you roll 5 red and 2 yellow, 5 + 2 = 7 and 2 + 5 = 7 both work. Ask them if
your number sentence is on the card. Ask them if any number sentence on your card uses a total of 7
counters. How can they tell? Ask them if it was a good choice to use only 7 counters? Why not? Then
challenge them to use a better number of counters, one that they can find a lot of number sentences on the
game card for. Are there a lot of number sentences on the card that add to the same number? Which
number is a good number of counters to try? Does someone want to come up and shake that many counters
and see what number sentence they get? If they get a number sentence that is on the card, they get to put
their colour of counter on the board. So if we decide that the teacher is red and the class is yellow, then the
student puts a yellow counter on the game board. Repeat this several times so that students understand,
always demonstrating good strategy on your turn. Make sure they know that once a square is covered, it
cannot be used again. Then let them play against each other in pairs.
Also, since they only have 20 counters, they will need to fill the places that add to 17 and 18 quickly, or they
never will.
Extensions:
1. To practice other missing addend or subtraction problems, students can make dice with other magic
numbers, anywhere from 5 to 16.
2. For extra practice, provide the BLM “Ten-Dot Dominoes”.
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NS3-29 Parts and Totals
Goals: Students will use pictures and charts to solve word problems.
Prior Knowledge Required: Subtraction as take away
Subtraction as comparison (how much more than, how much shorter than, and so on)
Addition as how much altogether, or how many altogether, or how long altogether
Vocabulary: difference, total, altogether, how many more than
Write on the board:
Red apples
Green apples
Tell students that each square represents one apple and ASK: How many red apples are there? How many
green apples? Are there more red apples or green apples? How many more? (Another way to prompt
students to find the difference is to ASK: If we pair up red apples with green apples, how many apples are
left over?) How many apples are there altogether?
Label the total and the difference on the diagram, using words and numerals:
Difference: 4 apples
Red apples Total: 8 apples
Green apples
Do more examples using apples or other objects. You could also have students count and compare the
number of males and females in the class. (If yours is a single-gender class, you can divide the class by age
instead of by gender). ASK: How many girls are in the class? How many boys? Are there more boys or girls?
How many more? What subtraction sentence could you write to express the difference? How many children
are there altogether in the class? What addition sentence could you write to express the total?
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NS3-30 Parts and Totals (Advanced)
Goals: Students will recognize what they are given and what they need to find in a word problem.
Prior Knowledge Required: Subtraction as take away
Subtraction as comparison (how much more than, how much shorter than, and so on) Addition as how much altogether, or how many altogether, or how long altogether
Vocabulary: equation, fact family
Then draw the following chart and have volunteers fill in the blanks:
Red Apples Green Apples Total Number of
Apples
How many more of
one colour of apple
2 3
7 5
3 6 more green than red
5 3 more red than green
9 2 more green than red
7 1 more red than green
3 11
5 8
Then ask students to fill in the columns of the chart from this information:
a) 3 red apples, 5 green apples
b) 4 more red apples than green apples, 5 green apples
c) 4 more red apples than green apples, 5 red apples
d) 11 apples in total, 8 green apples
e) 12 apples in total, 5 red apples
Have students find the total number of apples and the difference between how many of each colour.
a) 2 red apples and 4 green apples b) 7 red and 3 green c) 3 red and 8 green
Have students find the number of green apples and the total number of apples.
a) 2 red and 6 more green than red b) 7 red and 3 more red than green.
Have students find the number of green apples and how many more of one colour:
a) 2 red and 8 altogether b) 6 red and 10 altogether c) 7 red and 12 altogether
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Draw on the board:
Ask your students to suggest number sentences (addition and subtraction) represented by this model.
Ensure that either you or the students state what the numbers in each number sentence represent. Continue
until all possible answers have been identified and explained: 3 + 4 = 7 because 3 white circles and 4 dark
circles make 7 altogether; 4 + 3 = 7 because 4 dark circles and 3 white circles make 7 altogether; 7 – 4 = 3
because removing the 4 dark circles leaves 3 white circles; 7 – 3 = 4 because removing the 3 white circles
leaves 4 dark circles. Tell them that these number sentences all belong to the same fact family because they
come from the same picture and involve the same numbers. Tell them that these number sentences are all
examples of equations because they express equality of different numbers, for example 3 + 4 and 7 or 7 – 4
and 3. Write on the board:
a) 3 + 5 = 8 b) 43 – 20 > 7
c) 9 – 4 < 20 – 8 d) 5 + 12 + 3 = 11 + 9 = 31 – 11
Ask which number sentences are equations (parts a and d only).
Then have students list all the possible equations in the fact family for different pictures. Include pictures in
which more than one attribute varies.
EXAMPLE: (dark and white, big and small)
NOTE: The fact families for 4 + 5 = 9 and 6 + 3 = 9 can both be obtained from this picture, but they are not
themselves in the same fact family.
When students are comfortable finding the fact family of equations that correspond to a picture, give them a
number sentence without the picture and have them write all the other number sentences in the same fact
family. Use progressively larger numbers, sometimes starting with an addition sentence and sometimes with
a subtraction sentence (EXAMPLES: 10 + 5 = 15, 24 – 3 = 21, 46 + 21 = 67, 103 – 11 = 92).
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Green
Grapes
Purple
Grapes
Total
Number of
Grapes
Fact Family for
the Total Number
of Grapes
How many more
of one type
of grapes?
7 2 9 7 + 2 = 9 2 + 7 = 9
9 – 7 = 2 9 – 2 = 7 5 more green than purple
5 3
7 3 more purple than green
7 3 more green than purple
5 12
6 10
Look at the completed first row together, and review what the numbers in the fact family represent.
Demonstrate replacing the numbers in each number sentence with words:
7 + 2 = 9 becomes: green grapes + purple grapes = total number of grapes
2 + 7 = 9 becomes: purple grapes + green grapes = total number of grapes
9 – 7 = 2 becomes: total number of grapes – green grapes = purple grapes
9 – 2 = 7 becomes: total number of grapes – purple grapes = green grapes
Have students complete the chart in their notebooks, and ask them to write out at least one of the fact
families using words.
Look at the last column together and write the addition sentence that corresponds to the first entry, using
both numbers and words:
5 + 2 = 7: how many more green than purple + purple grapes = green grapes
Have students write individually in their notebooks the other equations in the fact family, using both numbers
and words:
2 + 5 = 7: purple grapes + how many more green than purple = green grapes
7 – 5 = 2: green grapes – how many more green than purple = purple grapes
7 – 2 = 5: green grapes – purple grapes = how many more green than purple
Make cards labelled:
• # of green grapes
• # of purple grapes
• Total number of grapes
• How many more purple than green
• How many more green than purple
Stick these cards to the board and write “=” in various places.
EXAMPLE: # green grapes # purple grapes = how many more green than purple
Have students identify the missing operation and write in the correct symbol (+ or –).
For variety, you might put the equal sign on the left of the equation instead of on the right.
Be sure to create only valid equations.
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For example: How many more purple than green # purple grapes = # green grapes is invalid, since
neither addition nor subtraction can make this equation true.
Teach students to identify which information is given and what they need to find out. They should underline
the information they are given and circle the information they need to find out.
For example:
Cari has 15 marbles. 6 of them are red. How many are not red?
Teach students to write an equation for what they have to find out in terms of what they are given.
EXAMPLE:
not red = marbles red marbles
+ or −
not red = 15 − 6 = 9
They then only need to write the appropriate numbers and add or subtract.
Finally, do a simple word problem together. Prompt students to identify what is given, what they are being
asked to find, and which operation they have to use. Encourage students to draw a picture similar to the bars
drawn in exercise 1 of worksheet NS5-14: Parts and Totals.
EXAMPLE:
Sera has 11 pencils and Thomas has 3 pencils. How many more pencils does Sera have?
I know: # of pencils Sera has (11), # of pencils Thomas has (3)
I need to find out: How many more pencils Sera has than Thomas
How many more Sera has = # of pencils Sera has – # of pencils Thomas has
= 11 – 3
= 8
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NS3-31 Sums and Differences
Draw:
10 m
13 m
Tell your students that you asked two people to tell you how much longer the second stick is than the first
stick. One person answered 3 m and the other person answered 23 m. How did they get their answers?
Which person
is right? How do you know?
Emphasize that something measuring only 13 m long cannot be 23 m longer than something else.
Ask if anyone has a younger sister or brother. Ask the student their age, then ask how old their sister or
brother is. Ask them how much older they are than their sister or brother. How did they get their answer? Did
they add the ages or subtract? Why? Emphasize that when they’re asked how much older or how much
longer that they’re being asked for the difference between two numbers so they should subtract. Write:
Smaller Larger How much more?
10 m 13 m 3 m longer
5 years 9 years 4 years older
Give several word problems like this and continue to fill in the chart. Try to wait until most hands are raised
before allowing a volunteer to come to the board. EXAMPLE:
1. Katie has 13 jelly beans. Rani has 7 jelly beans. Who has more jelly beans? How many more? 2. Sara weighs 29 kg. Ron weighs 34 kg. Who weighs more? How much more? 3. Anna’s pet cat weighs 15 kg and her new kitten weighs 7 kg. How much more does the cat weigh than
the kitten?
Continue to use the chart, but this time have students insert a box ( ) when they aren’t given the quantity.
Ask questions such as: Are you given the larger number or the smaller number? How do you know?
EXAMPLES:
1. Sally’s yard is 21 m long. Tony’s yard is 13 m longer than Sally’s. How long is Tony’s yard?
2. Teresa read 14 books over the summer. That’s 3 more than Randi. How many books did
Randi read?
3. Mark ran 5 100 m on Tuesday. On Thursday, he ran 400 m less. How far did he run on Thursday?
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Write on the board: larger number – smaller number = how much more 1. – = 2. – = 3. – = Have students write the correct numbers and box under each heading: 1. – 21 = 13 2. 14 – = 3 3. 5100 – = 400
Remind students that there are 4 different equations in the fact family: 7 – 2 = 5 and have a volunteer write
the other three on the board. Then show your students how you can change the first equation to find all the
equations in its fact family.
– 21 = 3 – 3 = 21 3 + 21 = 21 + 3 =
ASK: Do you think that these four problems will have the same missing number? Emphasize that all three
numbers in the equation will be the same for any equation in the same fact family, so the missing number
should be the same. To help them see this, draw on the board: 7 – 2 = 5 7 – 5 = 2 5 + 2 = 7 2 + 5 = 7
ASK: How do you use the numbers 2 and 5 to find the 7? Do you add or subtract? How do you use the
numbers 21 and 3 to find the missing number? Do you add or subtract? How do you know? Emphasize that
they should find the equation where the missing number is by itself; in this case, they add 21 + 3 (or 3 + 21).
Have a volunteer write the fact family for the second equation and have another volunteer find one of the
equations so that the box is by itself. Do they need to add or subtract to find the missing number? Repeat for
the third equation.
Students should be aware that phrases like “in all” or “altogether” in a word problem usually mean that one
should combine quantities by adding.
Similarly, phrases like “how many more”, “how many are left” and “how many less” indicate that one should
find the difference between quantities by subtracting.
However the wording of a question doesn’t always tell you which operation you should use to solve the
question. The two questions below have identical grammatical structure, but in the first question you must
add to find the answer and in the second you must subtract:
1. Paul has 3 more marbles than Ted. Ted has 14 marbles. How many marbles does Paul have?
2. Paul has 3 more marbles than Ted. Paul has 14 marbles. How many marbles does Ted have?
In solving a problem, it helps to make a model, draw a simple diagram, or make a mental model of the
situation. Students should start by getting a sense of which quantity in a question is greater and which is
smaller. This will usually help them decide whether they should add or subtract in a question and whether
their answer makes sense. In the first question, they should recognize that the 14 is the smaller quantity,
so they have – 14 = 3, which they can change to 14 + 3 = . In the second question, they should
recognize that 14 is the larger number, so 14 – = 3, or 14 – 3 = .
When students have mastered this, combine the problems and use a chart similar to the following:
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First
number
Second
number
How many
altogether?
How much more
is larger than
smaller?
Have students put the given numbers and the box for the unknown in the right place, and then put an X in
the part that doesn’t apply for each question. Then have them solve the problem by expressing the as the
sum or difference, then adding or subtracting.
Extensions:
1. Here are some 2 step problems your students could try:
a) Alan had 352 stickers. He gave 140 to his brother and 190 to his sister. How many did he keep?
b) Ed is 32 cm shorter than Ryan. Ryan is 15 cm taller than Mark. If Ed is 153 cm high, then how
tall is Mark?
2. Ask your students to say what information is missing from each problem. Then ask them to make up an
amount for the missing information and solve the problem.
a) Howard bought a lamp for $17. He then sold it to a friend. How much money did he lose?
b) Michelle rode her bike 5 km to school. She then rode to the community center. How many km did
she travel?
c) Kelly borrowed 3 craft books and some novels from the library. How many books did she borrow
altogether?
Make up more problems of this sort.
3. Students should fill in the blanks and solve the problems.
a) Carl sold _____ apples. Rebecca sold _____ apples. How many more apples did Carl sell than
Rebecca.
b) Sami weighs _____ kg. His father weighs _____ kg. How much heavier is Sami’s father?
c) Ursla ran _____ km in gym class. Yannick ran _____ km. How much further did Ursla run?
d) Jordan read a book _____ pages long. Digby read a book _____ pages long. How many more pages
were in Jordan’s book?
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NS3-32 Larger Numbers
Goals: Students will apply the concepts they have learned so far to numbers with 4 or more digits.
Prior Knowledge Required: Place value for ones, tens, and hundred
Writing number words for numbers with up to 3 digits
Comparing and ordering numbers with up to 3 digits
Writing numbers in expanded form
Addition and subtraction
Review NS3-1 and NS3-2 by asking volunteers to identify the place value of each digit in these numbers:
a) 7 b) 37 c) 237
Write on the board: 4 156. ASK: How is this number different from most of the numbers we have worked with
in this unit? (It has 4 digits.) Underline the 4 and tell students that its place value is “thousands.” Then have a
volunteer identify the place value of each digit in d) 4 237.
Write all the place value words on the board for reference: ones, tens, hundreds, and thousands.
Then write many 4-digit numbers, underline one of the digits in each number, and have students identify and
write the place value of the underlined digits. Be sure to include the digit 0 in some of the numbers.
Have students write the numbers in a) to d) in expanded form using numerals. Then have them write more 4-
digit numbers in expanded form. Again, assign numbers with more digits as a bonus.
Read these numbers aloud with students and write the number words for each one:
4 000, 8 000, 1 000.
Then write on the board: 4 502 = 4 000 + 502. Tell students that to read the number 4 502, they read both
parts separately: four thousand five hundred two.
Have students read these numbers aloud with you: 8 430, 5 001, 3 500, 4 782. Then have them write
corresponding addition sentences for each number (EXAMPLE: 8 430 = 8 000 + 430) and the number words
(EXAMPLE: eight thousand four hundred thirty). Have students write the number words for more 4-digit
numbers. Include the digit 0 in some of the numbers.
Review comparing and ordering numbers with up to 3 digits, then write on the board:
a) 7 834 b) 987 c) 9 050
1 963 4 802 8 950
ASK: What is the largest place value in which these pairs of numbers differ? Why do we care only about the
largest place value that differs? SAY: 9 050 has 1 more in the thousands place, but 8 950 has 9 more in the
hundreds place. Which number is greater? How do you know? Do a few addition and subtraction questions
together, to remind students how to add and subtract using the standard algorithm (EXAMPLES: 354 + 42,
52 + 119, 401 – 259) Then teach students to extend the algorithm to larger numbers by adding and
subtracting 4-digit numbers.
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Activity: Give each student a pair of dice, one red and one blue. Have them copy the following diagram
into their notebooks:
<
Each student rolls their dice and places the number from the red die in a box to the left of the less-than
sign and the number from the blue die in a box to the right. If, after 4 rolls, the statement is true (the
number on the left is less than the number on the right) the student wins. Allow students to play for several
minutes, then stop to discuss strategies. Allow students to play again. Variation: Use 5 digit numbers and roll
the die 5 times:
<
Extensions:
1. What is the number halfway between …
a) 42 and 50? b) 42 and 60? c) 42 and 70?
d) 42 and 100? e) 420 and 1000? f) 9 420 and 10 000?
g) 9 870 and 10 000? h) 9 160 and 10 000?
Bonus: What is the number halfway between 99 330 and 100 000?
2. Write out the place value words for numbers with up to 12 digits:
ones tens hundreds
thousands ten thousands hundred thousands
millions ten millions hundred millions
billions ten billions hundred billions
Point out that after the thousands, there is a new word every 3 place values. This is why we put spaces
between every 3 digits in our numbers—so that we can see when a new word will be used. This helps us
to identify and read large numbers quickly. Demonstrate this using the number 3 456 720 603, which is
read as three billion four hundred fifty-six million seven hundred twenty thousand six hundred three.
Then write another large number on the board—42 783 089 320—and ASK: How many billions are in
this number? (42) How many more millions? (783) How many more thousands? (89) Then read the
whole number together.
Have students practise reading more large numbers, then write a large number without any spaces and
ASK: What makes this number hard to read? Emphasize that when the digits are not grouped in 3s, you
can’t see at a glance how many hundreds, thousands, millions, or billions there are. Instead, you have to
count the digits to identify the place value of the left-most digit. ASK: How can you figure out where to put
the spaces in this number? Should you start counting from the left or the right? (From the right, otherwise
you have the same problem—you don’t know what the left-most place value is, so you don’t know where
to put the spaces. You always know the right-most place value is the ones place, so start counting from
the right.)
Write more large numbers without any spaces and have students re-write them with the correct spacing and
then read them (EXAMPLE: 87301984387 becomes 87 301 984 387).
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NS3-33 Concepts in Number Sense
Goals: Students will review the number sense concepts learned so far.
Prior Knowledge Required: Ordering numbers
Addition and subtraction
Word problems
The standard algorithm for addition and subtraction
Worksheet NS3-33 is a review of number sense concepts. It can be used as an assessment.
Extensions:
1. Place the numbers 1, 2, 3, 4 in the top four boxes to make the least possible sum and the least possible difference.
+ –
ANSWER: least possible sum: 23 + 14 or 24 + 13; least possible difference; 31 – 24 = 7
2. Find a question where Bob’s method from question 4 on the worksheet NS3-33, would still (albeit
accidentally) give the correct answer (ANSWER: Any pair of numbers whose ones digits add to 11.
EXAMPLES: 47 + 24, 36 + 45, 28 + 53)
3. Ask students to explain…
a) Could 200 people fit in 2 school buses?
b) Could you fit 50 eggs into 4 cartons?
c) Could you take 5 friends to see a movie with $20?
4. A palindrome is a number that looks the same written forwards or backwards: i.e. 212; 37 873, etc.
a) Find as many palindromes as you can using at most 9 tens blocks and 12 ones blocks.
b) Find as many palindromes as you can where the sum of the digits is 10.
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NS3-34 Arrays
Goals: Students will model multiplication using arrays.
Prior Knowledge Required: Counting
The multiplication sign
Vocabulary: times, multiplication, product, array, row, column
Draw the following array on the board and circle each row:
ASK: How many rows are there? How many dots are in each row? If you wanted to know how many dots
there are altogether, would you have to count them one by one? How can you take advantage of there
being 5 in each row to find out how many there are altogether?
Then write on the board: 4 5 20 and ask if anyone knows what signs we can put between the numbers
to make a multiplication sentence. Prompt students by suggesting that you have 4 sets with 5 objects in
each set. If no one suggests it, write in the multiplication and equal signs: 4 × 5 = 20.
Draw several arrays on the board and ASK: How many rows? How many dots in each row? How many dots
altogether? What multiplication sentence could you write? Emphasize that the number of rows is written first
and the number in each row is written second. Include arrays that have only 1 row and arrays that have only
1 dot in each row.
In the arrays you’ve drawn, point out that each row contains the same number of dots. Show students that
there are 3 ways to arrange 4 dots so that each row contains the same number of dots:
Then give students 6 counters each and have them create as many different arrays as they can. They
should draw their arrays in their notebooks and write the corresponding multiplication sentences (or
“products”). As students finish with this first problem, give them 2 more counters for a total of 8, then 4 more
for a total of 12, then 3 more for 15, 1 more for 16, and finally 8 more for 24 counters. Some students will
work more quickly than others; not all will get to 24 counters. Allow enough time for every student to make
and record arrays using 6, 8, and 12 counters. Then ask students to share their answers. For example,
some students may have 12 as 4 rows of 3 (4 × 3) or 3 rows of 4 (3 × 4), others as 6 rows of 2 or 2 rows of
6, and some as 12 rows of 1 or 1 row of 12.
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Now give students multiplication sentences and have them draw the corresponding arrays (EXAMPLE: 2 ×
4 = 8; 5 × 2 = 10, 3 × 3 = 9). Remind your students that the first number is the number of rows and the
second number is the number in each row.
Finally, give students multiplication word problems that refer directly to rows and the number in each row.
Mix it up as to which is mentioned first:
EXAMPLES:
There are 3 rows of desks. There are 5 chairs in each row.
There are 4 desks in each row. There are 2 rows of chairs.
How many desks are there altogether? How many chairs are there altogether?
(3 × 4 = 12) (2 × 5 = 10)
Have students draw corresponding arrays in their notebooks (they can use actual counters first if it helps
them) to solve each problem. Ensure that students understand what the dots in each problem represent. (In
the examples above, dots represent desks and chairs.)
Demonstrate the commutativity of multiplication (the fact that order doesn’t matter—3 × 4 = 4 × 3). Revisit
the first array and multiplication sentence you looked at in this lesson (4 × 5 = 20):
Then have students count the number of columns and the number of dots in each column and write a
multiplication sentence according to those (5 × 4 = 20):
ASK: Does counting the columns instead of the rows change the number of dots in the array?
Are 4 × 5 and 5 × 4 the same number? What symbol do mathematicians use to show that two numbers or
statements are the same? Write on the board: 4 × 5 5 × 4 and have a volunteer write the correct sign in
between (=). Tell students this is a short way of saying they are both the same number without saying which
number that is (20). ANALOGY WITH LANGUAGE: Instead of saying “The table is brown” and “Her shirt is
brown” we can say “The table is the same colour as her shirt” or “The table matches her shirt.”
To emphasize the commutativity of multiplication (that order doesn’t matter), you could draw several arrays
on chart paper, record the multiplication statements your students observe on the board and then rotate the
large sheet of paper and ask what multiplication statements they see after the rotation. Then record these
new statements underneath the corresponding statement for each array.
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Have students write similar multiplication sentences for the arrays they drew in their notebooks earlier. Then
have them fill in the blanks in multiplication sentences that include large numbers (EXAMPLES:
4 × 7 = 7 × , 19 × 27 = × 19, × 5 = 5 × 102, 387 × , = 2 501 × 387)
Remind students about how they can change the order when they add and now when they multiply.
Ask them if they can do that when they subtract.
Activities:
1. Different perspectives. Put 10 counters on the table in 2 rows of 5. Have a pair of students come up
and look at the array from different directions, so that one student sees 2 rows of 5 and the other sees
5 rows of 2. Have pairs of students view different arrays from different directions at their own desks:
I see 2 rows of 5.
I see 5 rows of 2.
2. Tables in Microsoft Word. If your students have access to Microsoft Word, have them create tables.
Draw the tables on the board that you want them to draw in their computer file. They then have to know
which are rows and which are columns when they enter them into the Microsoft Word format. After they
create the tables, they can write addition and multiplication sentences to match. For example, to create
the following table:
They would have to:
• click on Table in the bar on top of the screen
• click on Insert Table
• Put 2 in the “Number of rows” section
• Put 5 in the “Number of columns” section
• Click OK
If they accidentally do 5 rows and 2 columns, their table will not match, so they will have to re-do
their table.
Once they have made this table correctly, they can type on the screen:
2 + 2 + 2 + 2 + 2 = 10 5 + 5 = 10
5 × 2 = 10 2 × 5 = 10
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3. Build a multiplying machine. Teach students to make a multiplying machine for up to 6 × 6.
STEP 1: Draw a 6 × 6 array of dots.
STEP 2: Cut out a square corner from a sheet of paper so that the corner is large enough to
cover the array. Then throw away that corner and use the remaining part of the sheet.
STEP 3: Place the remaining part of your sheet so that you see the first number of rows and the second
number of dots in each row.
2 × 5
STEP 4: Count the dots you see. (In the example, we find that 2 × 5 = 10)
Extensions:
1. Students can use counters to model the arrays in QUESTIONS 6 and 7.
2. Use tiles or counters to create arrays for:
a) 3 × 3
b) 2 × 3
c) 1 × 3
d) 0 × 3
3. Create a story problem that you could solve by making 3 rows of 4 counters.
4. Teach students how to draw arrays to represent statements that involve addition and multiplication. For
instance 3 × 5 and 2 × 5 may be represented by:
3 × 5
2 × 5
(3 + 2) × 5 = 5 × 5
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Students should see from the above array that: 3 × 5 + 2 × 5 = (3 + 2) × 5
(This is the distributivity law.)
To practice, have students draw arrays for:
a) 2 × 2 + 3 × 2
b) 3 × 4 + 2 × 4
c) 4 × 5 + 2 × 5
5. After doing Activity 3 from this section, teach your students how to make a multiplication table by
counting squares to the left and above each corner:
× 1 2 3
1 6 goes here (there are 2 rows of 3 squares and 6 in total, so 2 × 3 = 6)
2 6
3
Students can use this method to fill in a whole 3 × 3 times table chart. Provide the BLM “Arrays in the
Times Tables” for practice with this skill.
6. Using that order doesn’t matter, complete the multiplication tables:
a) × 1 2 3
1 1 2 3
2 4 6
3 9
b) × 1 2 3 4
1 1
2 2 4
3 3 6 9
4 4 8 12 16
c) × 1 2 3 4 5
1 1 2 5
2 4 8 10
3 3 6 9 15
4 4 12 16
5 20 25
Then have students shade the square that has the same number as the bolded square.
× 1 2 3
1
2
3
× 1 2 3
1
2
3
× 1 2 3 4
1
2
3
4
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Bonus: use a 7 × 7 multiplication table instead. If your students are familiar with symmetry, ASK: where
is the symmetry line.
Literature Connection:
One Hundred Hungry Ants, E. J. Pinczes
(Rhymes describes one hundred ants marching toward a picnic. Efficiency dictates that the ants divide into
two lines of fifty, then four lines of twenty-five, and finally ten lines of ten.)
Read the story up until the ant makes the suggestion of dividing up and give pairs of students 100
manipulatives and allow them to discover different combinations/arrays to make the walk quicker. They can
record their findings in their journals and encourage them to write the matching multiplication sentence.)
Spunky Monkeys on Parade, S.J. Murphy
(This book is a fun approach to introducing multiplication. Count by twos, threes, and fours as the monkeys
parade down the street. A list of suggested activities is included in the book.)
Journal:
Have students write about when they can change the order of numbers and when they cannot. Which
numbers can they change the order of? When adding or multiplying, we can change the order of the
addends or the factors and the sum or product will remain the same, as in 3 + 4 = 7 and 4 + 3 = 7 or as in
3 × 4 = 12 and 4 × 3 = 12. Some students might notice that, when subtracting, we can change the order of
the difference and the subtrahend, as in 6 – 1 = 5 and 6 – 5 = 1. This is very different from addition and
multiplication, since the numbers they are changing the order of are not on the same side of the equal sign.
They don’t get the same answer by switching numbers. Although 6 – 1 = 5, it doesn’t even make sense to
write 1 – 6 since if we have only one object, we cannot take away 6 of them.
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NS3-35 Adding Sequences of Numbers and NS3-36 Multiplication and Repeated Addition
Goals: Students will understand multiplication as determining how many altogether when objects are
divided into equal sets.
Prior Knowledge Required: The multiplication sign
Addition by counting on
Vocabulary: multiplication, product, sum, group, set, array, row
Write on the board: 2 + 5 + 3 + 4 + 6 + 3 + 3 + 2.
Tell your students that this is a lot of adding to do, so we’re going to break it down and do one step at a time.
ASK: What is 2 + 5? Draw a box over the 5 and write 7 in it:
7
2 + 5 + 3 + 4 + 6 + 3 + 3 + 2
ASK: What is 7 + 3? Count aloud as a class from 8 until you have 3 fingers up. Draw a box over the 7 and
write 10 in it. Continue in this way until the sum is found:
7 10 14 20 23 26
2 + 5 + 3 + 4 + 6 + 3 + 3 + 2 = 28
Then ask volunteers to do smaller problems:
a) 2 + 4 + 3 = b) 3 + 5 + 3 = c) 4 + 2 + 4 = d) 2 + 5 + 1 =
e) 3 + 6 + 3 + 4 = f) 4 + 2 + 2 + 3 = g) 3 + 3 + 3 + 3 =
Give students similar problems to do individually in their notebooks.
Bonus: 3 + 7 + 3 + 4 + 1 + 3 + 5 + 2 + 6
Draw a 2 × 3 array on the board.
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ASK: What multiplication sentence does the array show? If most students do not raise their hands, prompt
them by asking: How many rows are there? How many dots are in each row? Which number do we write
first? How many are there altogether? Then write 2 × 3 = 6 next to the array.
Tell your students that instead of arranging the dots in rows, we can arrange them in sets or groups. (Use
the words “group” and “set” interchangeably.) Rearrange the above array as follows:
ASK: How many sets are there? How many dots are in each set? Tell students that we can write a
multiplication sentence for any grouping of objects as long as there is an equal number in each group. Have
students write both addition and multiplication sentences for the following, or similar, pictures:
Then draw the following pictures on the board and ASK: Can you write multiplication sentences
for these pictures? Why not? (There are not an equal number in each group.)
Draw a few more pictures, with both equal and unequal numbers in each group, and have volunteers write
the corresponding addition and, where possible, multiplication sentences.
Then, without using pictures, have students rewrite addition sentences as multiplication sentences and vice
versa. (Given 4 + 4 + 4, they should be able to write 3 × 4) It is important that students understand that the
second number in a multiplication statement is the number that is repeated and the first number is the
number of times the second number is repeated.
First have students write the addition statement given the multiplication statement.
(EXAMPLES: 4 × 3 = 3 + 3 + 3 + 3, 2 × 5, 5 × 2, 1 × 7, 7 × 1, 5 × 4, 4 × 6)
Bonus: 3 × 117, 4 × 204, 2 × 80147.
Then have students write the multiplication statement given the addition statement.
(EXAMPLES: 1 + 1 + 1 + 1 + 1 = 5 × 1, 4 + 4 + 4, 2 + 2 + 2 + 2, 6 + 6, 3, 9 + 9)
Bonus: 217 + 217 + 217, 1047 + 10 47)
Show students how to use circles and dots to model different groups and the objects in them. For example,
to model 3 cars with 4 people in each car, ASK: What are the groups? (the cars) What are the objects in
each group? (people) How many groups are there? How many circles should we draw? How many objects
are in each group? How can we show this using dots? (Draw 3 dots in each circle.) Then have a volunteer
write the multiplication and addition sentences for the picture. Repeat with several examples.
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Then show students a pile of 15 counters on the overhead projector (or use circles taped to the board).
Arrange the counters into 5 piles ( 3, 4, 3, 3 and 2). Have a volunteer write the addition statement that the
picture shows: 3 + 4 + 3 + 3 + 2 = 15.
ASK: Can we write this addition statement as a multiplication statement? Can we move one of the counters
so that we can? Have a volunteer show how to do this. Have another volunteer write the new addition
statement and then a multiplication statement. Repeat with several examples and then move on to examples
where they need to move two counters from the same pile: 3 + 5 + 3 + 1 + 3.
When students are comfortable with this, have volunteers change two numbers so that each addition
statement can be turned into multiplication statements.
a) 4 + 3 + 4 + 4 + 4 + 5 b) 3 + 2 + 2 + 2 + 1 c) 6 + 8 + 6 + 6 + 4
Activities:
1. Give each pair of students 20 cards (2 of each card marked 1 through 10) and have them shuffle the
deck and deal out 10 cards to each student. The student with the highest total of cards wins. Students
should add their cards by using either method discussed in Extension 1 below.
2. In groups of 3, have students write an addition sentence and a multiplication sentence for the number of
shoes people in their group are wearing (2 + 2 + 2 = 6 and 3 × 2 = 6). Then have them write sentences
for the number of left and right shoes in the group (3 + 3 = 6 and 2 × 3 = 6). Have them repeat the
exercise for larger and larger groups of students.
3. Instead of left shoes and right shoes, students could group and count circles and triangles on paper:
They can count this as 2 + 2 + 2 = 6 (3 × 2 = 6) or they can count 3 circles + 3 triangles = 6 shapes
(2 × 3 = 6). Then provide students with the 2-page BLM “Multiplication and Order”.
Extensions:
1. Have students find each product by adding:
a) 6 × 1 = ______ Add 6 ones: 1 + 1 + 1 + 1 + 1 + 1
b) 5 × 1 = ______ Add 5 ones: 1 + 1 + 1 + 1 + 1
c) 4 × 1 = ______ Add 4 ones: 1 + 1 + 1 + 1
d) 3 × 1 = ______ Add 3 ones: 1 + 1 + 1
e) 2 × 1 = ______ Add 2 ones: 1 + 1
f) 1 × 1 = ______ Add 1 one: 1
Have students predict: 8 × 1, 11 × 1, 23 × 1, 89 × 1, 9063 × 1.
g) 3 × 3 = ______ Add 3 threes: 3 + 3 + 3
h) 2 × 3 = ______ Add 2 threes: 3 + 3
i) 1 × 3 = ______ Add 1 three: 3
j) 3 × 5 = ______ Add 3 fives: 5 + 5 + 5
k) 2 × 5 = ______ Add 2 fives: 5 + 5
l) 1 × 5 = ______ Add 1 five: 5
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Have students predict: 1 × 8, 1 × 17, 1 × 11, 1 × 20, 1 × 304, 1 × 2 987, 1 × 3 446 035
For extra practice, provide the BLM “Multiplying by 1”. Remind students about the commutativity of
multiplication (that order doesn’t matter) and ASK: Does this apply to 1 as well? Is 7 × 1 = 1 × 7? What
are they both equal to?
2. Have students find each product by adding:
a) 5 × 0 = ______ Add 5 zeroes: 0 + 0 + 0 + 0 + 0
b) 4 × 0 = ______ Add 4 zeroes: 0 + 0 + 0 + 0
c) 3 × 0 = ______ Add 3 zeroes: 0 + 0 + 0
d) 2 × 0 = ______ Add 2 zeroes: 0 + 0
e) 1 × 0 = ______ Add 1 zero: 0
f) 0 × 0 = ______ Add no zeroes at all:
Have students predict: 11 × 0, 8 × 0, 14 × 0, 23 × 0, 732 × 0, 79 846 × 0.
g) 5 × 3 = ______ Add 5 threes: 3 + 3 + 3 + 3 + 3
h) 4 × 3 = ______ Add 4 threes: 3 + 3 + 3 + 3
i) 3 × 3 = ______ Add 3 threes: 3 + 3 + 3
j) 2 × 3 = ______ Add 2 threes: 3 + 3
k) 1 × 3 = ______ Add 1 three: 3
l) 0 × 3 = ______ Add no threes at all:
Have students predict: 0 × 5, 0 × 9, 0 × 6, 0 × 32, 0 × 97, 0 × 436, 0 × 50 980
For extra practice, provide the BLM “Multiplying by 0”. ASK: Is 7 × 0 = 0 × 7? What are they both
equal to?
3. Have students find pairs that add to 10 to add longer sequences of numbers.
(EXAMPLES: 3 + 6 + 7 + 2 + 4 + 8 + 1 + 5 + 9)
Students might be given cards with the numbers from 1 to 10 on them, arranged randomly, and asked to
find the sum by recording each step of the process as in the lesson and then rearranging the order so
that pairs adding to 10 are together. Students can then record the new order of numbers and check that
they get the same answer using both methods.
4. Find the number of days in a (non-leap) year by adding the sequence of 12 numbers:
31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 =
J F M A M J J A S O N D
Start by writing the sequence: J F M, and so on, on the board. ASK: Which months have 30 days?
Have a volunteer write 30 above those months. ASK: Which months have 31 days? Have a volunteer
write 31 above those months. Then have another volunteer fill in the missing number. Then write “+”
between the numbers and have students add the sequence of numbers.
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5. Have students add strings of numbers that are “almost” arrays using multiplication and adjusting their
answers as needed. EXAMPLE: 3 + 3 + 3 + 2 is not quite 3 + 3 + 3 + 3, but is one less, so you can
solve it by multiplying 4 × 3 (12) and then subtracting 1 (11). Other examples include:
4 + 4 + 4 + 4 + 4 + 5 = 4 × 6 + 1 = 24 + 1 = 25
2 + 2 + 2 + 2 + 3 = 5 × 2 + 1 = 11
6 + 6 + 6 + 6 + 5 = 5 × 6 – 1 = 30 – 1 = 29
6. Students could rewrite the problems in QUESTIONS 4 on the worksheet as complete word problems.
For example, QUESTIONS 4 a) might become: Sally has 3 boxes of pencils. There are 2 pencils in
each box. How many pencils does she have altogether?
7. Teach students to multiply by 10 by using arrays or rows of a hundreds chart. Extend by having them
multiply really large numbers by 10; after they are comfortable with 3 × 10 = 30, 7 × 10 = 70, 2 × 10 = 20,
9 × 10 = 90, have them find 11 × 10, 12 × 10, 23 × 10, 47 × 10, 60 × 10, 64 × 10, 82 × 10, 90 × 10,
93 × 10, 100 × 10.
8. Teach multiplication of 3 numbers using 3-dimensional arrays. Give students blocks and ask them to
build a “box” that is 4 blocks wide, 2 blocks deep, and 3 blocks high.
When they look at their box head-on, what do they see? (3 rows, 4 blocks in each row, a second layer
behind the first) Write the math sentence that corresponds to their box when viewed this way:
3 × 4 × 2.
Have students carefully pick up their box and turn it around, or have them look at their box from a
different perspective: from above or from the side. Now what number sentence do they see? Take
various answers so that students see the different possibilities. ASK: When you multiply 3 numbers,
does it matter which number goes first? Would you get the same answer in every case? (Yes, because
the total number of blocks doesn’t change.)
9. You could also teach multiplication of 3 numbers by groups of groups:
This picture shows 2 groups of 4 groups of 3. ASK: How would we write 4 groups of 3 in math
language? (4 × 3). How could we write 2 groups of 4 groups of 3 in math language? (2 × 4 × 3). If I
want to write this as a product of just 2 numbers, how can I do so? Think about how many groups of 3
hearts I have in total. I have 2 × 4 = 8 groups of 3 hearts in total, so the total number of hearts is 8 × 3.
SO: 2 × 4 × 3 = 8 × 3.
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Invite students to look for a different way of grouping the hearts. ASK: How many hearts are in each of
the 2 groups? (There are 4 × 3 = 12 hearts in each of the 2 groups, so 2 × 4 × 3 = 2 × 12.) Do I get the
same answer no matter how I multiply the numbers? (Yes; 8 × 3 = 2 × 12).
Put up this math sentence: 2 × 4 × 3 = 2 × .
ASK: What number is missing? (12) How do you know? (because 4 × 3 = 12). Is 12 also equal to 3 × 4?
Can I write 2 × 4 × 3 = 2 × 3 × 4? Remind students that 2 × 4 × 3 is the same as both 8 × 3 and 2 × 12.
Have students explore different orderings of the factors. Tell them that just like we use the term addend
in sums, we use the term factor in products. Have them try this with 2 × 6 × 5. Ask if there is any way to
order the factors that make it particularly easy to find the product. What is the answer? (2 × 6 × 5 =
2 × 5 × 6 = 10 × 6 = 60). Then challenge them to find more products of 3 numbers: 2 × 9 × 5, 3 × 5 × 2,
5 × 7 × 2, 2 × 4 × 5.
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NS3-37 Multiplying by Skip Counting
Goals: Students will multiply by skip counting.
Prior Knowledge Required: Number lines
Multiplication
Skip counting
Vocabulary: product, sum, skip counting
Show a number line on the board with arrows, as follows:
0 1 2 3 4 5 6 7 8 9 10
Ask students if they see an addition sentence in your number line. ASK: What number is being added
repeatedly? How many times is that number added? Do you see a multiplication sentence in this number
line? Which number is written first—the number that’s repeated or the number of times it is repeated? (The
number of times it is repeated.) Then write “5 × 2 = ” and ASK: What does the 5 represent? What does
the 2 represent? What do I put after the equal sign? How do you know?
Have volunteers show several products on number lines.
(EXAMPLES: 6 × 2, 3 × 4, 2 × 5, 5 × 1, 1 × 5)
Then have a volunteer draw arrows on an extended number line to show skip counting by 3s:
Have students find 5 × 3, 2 × 3, 6 × 3, 4 × 3, and other multiples of 3 by counting arrows.
Then draw 2 hands on the board and skip count with students by 3 up to 30 (you can use your own hands
or the pictures):
3
6 12 9
15 18
21 27 24
30
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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Have students skip count on their hands to find 5 × 3, 2 × 3, 6 × 3, 4 × 3, and other multiples of 3.
ASK: Which is easier to use to multiply by 3, number lines or your hands? Why? (Since they probably don’t
need to count fingers to tell when 6 fingers are held up but they do need to count 6 arrows, students are
likely to find it easier to read answers on their hands.)
For extra practice, provide your students with the BLM “Multiplication Practice”.
Draw the following picture on the board:
5 pizzas.
2 pieces in each pizza.
Tell your students that you want to know how many pieces there are altogether. ASK: What is a multiplication
statement for this? An addition statement? What should we skip count by to find the answer?
(skip count by 2).
Repeat with similar questions, but have your students draw the pictures.
Extension: Use skip counting and that order doesn’t matter to fill in the unshaded parts of the
multiplication table.
× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7
1 5 1
2 10 2
3 15 3
4 4
5 5
6 6
7 7
× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7
1 1
2 2
3 3
4 4
5 5
6 6
7 7
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NS3-38 Multiplying by Adding On
Goals: Students will turn products into smaller products and sums.
Prior Knowledge Required: Arrays
Multiplication
Vocabulary: product, sum, skip counting
Draw the following three pictures on the board:
Have students write a multiplication sentence for each picture. Then have them draw their own pictures and
invite partners to write multiplication sentences for each other’s pictures. Demonstrate the various ways of
representing multiplication sentences by having volunteers share their pictures.
Then draw 2 rows of 4 dots on the board:
ASK: What multiplication sentence do you see? (2 × 4 = 8) (Prompt as needed: How many rows are there?
How many in each row? How many altogether?) What happens when I add a row? Which numbers change?
Which number stays the same? (The multiplication sentence becomes 3 × 4 = 12; we added a row, so there
are 3 rows.) How many dots did we add? Invite a volunteer to finish writing the math sentence that shows
how 2 × 4 becomes 3 × 4 when you add 4. (2 × 4 + ____ = 3 × 4).
Look back at the number line you drew above:
0 1 2 3 4 5 6 7 8 9 10
ASK: What multiplication sentence do you see? What addition sentence do you see? (Prompt as needed:
Which number is repeated? How many times is it repeated?) Ensure all students see 5 × 2 = 10 from this
picture. Then draw another arrow:
0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6 7 8 9 10
1 2
3
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ASK: Now what multiplication sentence do you see? What are we adding to 5 × 2 to get 6 × 2? Have a
volunteer show how to write this as a math sentence: 5 × 2 + 2 = 6 × 2.
Repeat using the 4 sets of 3 hearts and adding another set of 3 hearts. Have students use arrays to practise
representing products as smaller products and sums. Begin by providing an array with blanks (as in
QUESTION 1 on the worksheet) and have volunteers come up and fill in the blanks, as has been done in the
following EXAMPLE:
3 × 5
4 × 5
+ 5
In their notebooks, have students draw an array (or use counters) to show that:
a) 3 × 6 = 2 × 6 + 6 b) 5 × 3 = 4 × 3 + 3 c) 3 × 8 = 2 × 8 + 8
Have students do the following questions without using arrays:
a) If 10 × 2 = 20, what is 11 × 2? b) If 5 × 4 = 20, what is 6 × 4?
c) If 11 × 5 = 55, what is 12 × 5? d) If 8 × 4 = 32, what is 9 × 4?
e) If 6 × 3 = 18, what is 7 × 3? f) If 8 × 2 = 16, what is 9 × 2?
g) If 2 × 7 = 14, what is 3 × 7?
Finally have students turn products into a smaller product and a sum without using arrays.
Begin by giving students statements with blanks to fill in:
a) 5 × 8 = 4 × 8 + b) 9 × 4 = × 4 + c) 7 × 4 = × +
Extensions:
1. Ask students to circle the correct answer:
2 × 5 + 5 = 2 × 6 or 3 × 5
2 × 5 + 2 = 2 × 6 or 3 × 5
4 × 3 + 3 = 4 × 4 or 5 × 3
4 × 3 + 4 = 4 × 4 or 5 × 3
3 × 9 + 9 = 4 × 9 or 3 × 10
3 × 9 + 3 = 4 × 9 or 3 × 10
4 × 6 + 4 = 5 × 6 or 4 × 7
4 × 6 + 6 = 5 × 6 or 4 × 7
2. Use adding on to fill in the bolded squares of the multiplication tables.
× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7
1 1
2 8 2
3 3 9
4 12 4 20
5 30 5 10
6 12 6
7 7 28
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NS3-39 Doubles
Goals: Students will use doubles and doubling to multiply mentally.
Prior Knowledge Required: Counting by 2s
Doubling numbers from 1 to 5
Relationship between skip counting and multiplying
Arrays
Vocabulary: double
ASK: What does the word “double” mean? Allow several students to give definitions and explanations in their
own words. Make sure students understand that to “double” a number means to add the number to itself, or
to multiply the number by 2. Then ASK: How would you find the double of “3”? If students say they “just
know” it’s 6, ask them how they would explain it to someone in grade 2. Some may add 3 + 3, others may
skip count by 3s until they have 2 fingers up. Take several answers.
Then draw 2 rows of 6 dots on the board and pair them up as follows and demonstrate counting the number
in the first row and the total number:
1 2 3 4 5 6
2 4 6 8 10 12
Ask if they can tell from this picture what the double of 4 is. What is the double of 6? Of 3? Of 2? 5? 1?
Then ask if anyone notices a pattern in the second row. What are they counting by to get the numbers in the
second row? Then demonstrate counting by 2s to double a number. Show that that if you count by 2s on
your fingers, then to double 3, for example, they can count by 2s until they have 3 fingers up. Make the
connection to products and ask students to find each product by doubling:
2 × 3, 2 × 4, 2 × 2, 2 × 5, 2 ×1.
Ask students to find the double of 9 by skip counting. ASK: What is another way to find the double of 9? If no
one says it, suggest splitting 9 into 5 + 4 and doubling each number separately. Use the following array to
illustrate this solution:
ASK: What is the double of 5? What is the double of 4? What is the double of 9? Why is it convenient to use
5 to find the double of 9?
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Use the above method (splitting a number into 5 + something) to find the doubles of 6, 7, and 8. Be sure all
students know, or can quickly calculate, the doubles of 6–9 before proceeding. Make the connection to
multiplying by 2 and have students find the following products: 2 × 7, 2 × 6, 2 × 9, 2 × 8.
SAY: I want to find the double of 12, so I want to split it up into 5 and something the same way I did before.
Write: 12 = 5 + ____
12 + 12 = 10 + ____ = ____
Have a student volunteer fill in all the blanks. ASK: Is there another way that we could split the 12 to make it
easier to double? Is there a number that is even easier to double than 5? If anyone suggests 10, write out
the split (12 = 10 + 2) and ASK: What is the double of 10? The double of 2? The double of 12? If no one
suggests 10, ASK: What does the 1 in the number 12 represent? How much is it worth? What is 10 doubled?
What is left after I take 10 away from 12?
Then show:
12 + 12 = 20 + 4 = 24.
ASK: How does splitting the number 12 into 10 and 2 make it very easy to double? Did we get the same
answer as when we used 5 to double 12?
Then ask students to help you find the double of 14. Draw 2 rows of 14 dots on the board. Have a student
volunteer explain where you should draw a line to help you see the double of 14 as a sum of two numbers.
Then write:
14 = 10 + ____
14 + 14 = 20 + ____ = _____
Have a volunteer fill in the blanks. Repeat with the number 13, but this time don’t draw the dots. Just write
13 = 10 + ___, so 13 + 13 = 20 + ____ = ____. Finally, double some numbers between 16 and 19. Then
have students multiply several numbers by 2: 2 × 13, 2 × 17, 2 × 14, 2 × 18, 2 × 16. When all students are
able to double numbers less than 20, ask them how they would split 23 to find its double.
ASK: What does the 2 in 23 stand for? Is that number easy to double? Solve the problem together:
23 = 20 + 3 23 + 23 = 40 + 6 = 46
Have students double more 2-digit numbers by splitting the numbers into tens and ones, as in the above
examples. Give students numbers in the following sequence:
– both digits are less than 5 (EXAMPLES: 24, 32, 41)
– the tens digit is less than 5 and the ones digit is more than 5 (EXAMPLES: 39, 47, 28)
– the tens digit is more than 5 and the ones digit is less than 5 (EXAMPLES: 81, 74, 92)
– both digits are more than 5 (EXAMPLES: 97, 58, 65)
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Bonus: Students who master this quickly can double 3- and 4-digit numbers.
Then have students multiply several numbers by 2: 2 × 31, 2 × 43, 2 × 22, 2 × 61, 2 × 84, 2 × 49, 2 × 89.
When all students are able to double 2-digit numbers, show them how they can use doubling when they
multiply. Draw 1 row of 6 on the board. ASK: What number is this? How would I show this number doubled?
(Draw another row of 6.) Then draw 3 rows of 8 and ASK: How many dots do I have here?
What multiplication sentence does this show? (3 × 8 = 24) I have 3 rows of 8. If I want to double that, how
many rows of 8 should I make? Why? (I should make 6 rows of 8 because 6 is the double of 3.) Add the
additional rows and SAY: Now I have twice as many dots as I started with. How many dots did I start with?
(24) How many dots do I have now—what is the double of 24? Ask students to give you the multiplication
sentence for the new array: 6 × 8 = 48.
Ask students to look for another way to double 3 rows of 8. ASK: If we kept the same number of rows, what
would we have to double instead? (the number of dots in each row) So how many dots would I have to put in
each row? (16, because 16 is the double of 8) What is 3 × 16? (It is the double of 3 × 8, so it must be 48.)
How we can verify this by splitting the 16 into 10 and 6? Have someone show this on the board:
3 × 16 = 3 × 10 + 3 × 6 = 30 + 18 = 48.
Have students fill in the blanks:
a) 4 × 7 is double of 2 × 7 = 14 , so 4 × 7 = 28
b) 5 × 8 is double of , so 5 × 8 =
c) 6 × 9 is double of , so 6 × 9 =
d) 3 × 8 is double of , so 3 × 8 =
e) 4 × 8 is double of , so 4 × 8 =
Encourage students to find two possible answers for the first blank in part e) (2 × 8 or 4 × 4 are both possible).
Ask volunteers to solve in sequence: 2 × 3; 4 × 3; 8 × 3: 16 × 3; What is double of 16? What is 32 × 3? What
is double of 32? What is 64 × 3? Repeat with the sequences beginning as follows. Take each sequence as
far as your students are willing to go with it.
a) 3 × 5, 6 × 5 b) 2 × 6, 4 × 6 c) 3 × 4, 6 × 4
d) 2 × 7, 4 × 7 e) 2 × 9, 4 × 9 f) 3 × 9, 6 × 9
Activities: Toothpick Tic Tac Toe
1 2 3
4 6 8
9 12 16
1 2 3 4
1 2 4
5 8 10
16 20 25
1 2 4 5
1 2 3 4 5
6 7 8 9 10
12 14 15 16 18
20 21 24 25 28
30 35 36 42 49
1 2 3 4 5 6 7
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For each game board, you will need two toothpicks and enough tokens of different colours for each
player (5 each for the smaller game board and 12 each for the larger game board). In each case, the
goal for each player is to place 3 tokens of their colour all in a row horizontally, vertically or diagonally, as
in tic tac toe. There are two toothpicks, which Player 1 can begin by playing on any two of the numbers
below the grid (they may put both toothpicks on the same number if they wish). They can then place their
token on the product of the two numbers. For example, if they placed the toothpicks both on 3, they can
place their token on 9. Player 2 must then move only 1 toothpick. In this case, Player 2 is forced to move
their token to some multiple of 3, but can choose between 3, 6 and 12 (9 is already taken and cannot be
used again). Play continues until one player gets 3 in a row.
Variation A: Change the game boards by arranging the numbers in random order.
Variation B: Use the multiplication table as a game board. This time, if a student places their toothpick
on 2 and 3, they may choose which 6 to cover with their token. This adds an added element of strategy.
With larger game boards, students may aim for 4 in a row.
1 2 3
2 4 6
3 6 9
1 2 3
1 2 3 4
2 4 6 8
3 6 9 12
4 8 12 16
1 2 3 4
1 2 3 4 5
2 4 6 8 10
3 6 9 12 15
4 8 12 16 20
5 10 15 20 25
1 2 3 4 5
Students might also use 6 × 6 and 7 × 7 multiplication tables.
Extensions:
1. Can a double ever be an odd number?
2. Use doubling to fill in the blank squares in the multiplication tables:
× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7
1 1 2
2 2 4 6 8 10 12 14 2 4
3 3 6
4 4 8
5 5 10
6 6 12
7 7 14
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× 1 2 3 4 5 6 7
1 3
2 6
3 3 6 9 12 15 18 21
4 12
5 15
6 18
7 21
3. Show students how the doubling strategy in this lesson can be combined with adding on from the
previous lesson (NS3-38).
× 1 2 3 4 5 6 7 × 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 1 3
2 double 2 6
3 then add on 3 9
4 4 12
5 5 15
6 6 18
7 7 21
double
then add on
4. Review the strategies students have seen so far for finding multiplication facts up to 7 × 7 (skip counting,
adding on, doubling) Challenge students to fill out a blank 7 × 7 times table by combining all the
strategies they have learned so far.
5. Tell students that knowing their multiplication facts (or “times tables”) will help them solve problems
which would otherwise require a good deal of work adding up numbers. For instance, they can now
solve the following problems by multiplying.
a) There are 4 pencils in a box. How many pencils are there in 5 boxes?
b) A stool has 3 legs. How many legs are on 6 stools?
c) A boat can hold 2 people. How many people can 7 boats hold?
6. Introduce sports scores. Tell them that some professional leagues award 2 points for every game you
win. So, for example, if a team has 3 wins and 4 losses, they don’t get any points for the 4 games they
lost, but they get 2 points for each of the 3 games they win. Ask a volunteer to write an addition sentence
for the number of points they will have: 2 + 2 + 2 = 6.
Ask if anyone knows a multiplication sentence that means the same thing (3 × 2) and then encourage
them to think of a different multiplication sentence that will still get the same answer (2 × 3).
Then encourage them to write an addition sentence that means the same thing as the multiplication
sentence (3 + 3).
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Encourage students to think of a word for adding 3 to itself. ASK: What are you doing to the 3 – remind
them that there’s a special word for that (doubling it).
Then repeat with other examples until all students understand that the number of points is double the
number of wins. When students understand this, ask volunteers to find the total number of points each
team has:
Blue: 3 wins and 1 loss Blue: 7 wins and 3 losses
Red: 1 win and 3 losses Red: 6 wins and 4 losses
Yellow: 2 wins and 2 losses Yellow: 8 wins and 2 losses
Which team has the most points? Does the team with the most points have the most wins?
Encourage students to double higher numbers: 23 wins and 38 losses means you double
23 = 20 + 3, so you get 40 + 6 = 46.
To double numbers such as 28, students can either write:
28 = 20 + 8 40 + 16 = 56 or 28 = 20 + 5 + 3 40 + 10 + 6 = 56.
When students are comfortable with this, introduce ties. In many sports leagues, a team gets 2 points for
a win and 1 point for a tie (no points for a loss). Then have students use the doubling strategy and then
the standard algorithm for addition to find point totals for teams with the following records:
Team Wins Losses Ties Points
Red 23 38 19 46 + 19 = 65
Blue 41 30 9
Yellow 32 29 19
Green 38 25 17
Orange 28 40 12
Literature Connection:
Anno’s Magic Seeds, M. Anno
(Two seeds are given and one is planted to grow. Explores the concept of doubling while engaging readers –
links to patterns in skip counting as well.)
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NS3-40 Topics in Multiplication
Prior Knowledge Required: Multiplication
Introduce the phrase “times as many.” Tell students that Jenny has 3 stickers and Ryan has 12 stickers.
Draw a diagram to illustrate this situation, and group the larger number of items (12) by the smaller
number (3):
Jenny:
Ryan:
SAY: Since 12 is 4 times 3, we say that Ryan has 4 times as many stickers as Jenny.
Draw similar diagrams, but don’t group the larger number of items. Have volunteers group the larger number
by the smaller number to find out how many times as many items one person has than the other.
Then draw the following partial pictures and have students finish the pictures:
a)
There are three times as many triangles as stars.
b)
There are twice as many circles as stars.
c)
There are 4 times as many squares as stars.
Extension: I am a 2-digit number.
a) Use 6 blocks to make me. Use twice as many tens blocks as ones blocks.
b) Use 12 blocks to make me. Use twice as many ones blocks as tens blocks.
c) Use 12 blocks to make me. Use three times as many ones blocks as tens blocks.
d) Use 10 blocks to make me. Use four times as many tens blocks as ones blocks.
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NS3-41 Concepts in Multiplication
Goals: Students will consolidate their learning done on multiplication so far.
Prior Knowledge Required: Multiplying by 1 or 0
Multiplication strategies: repeated addition, skip counting, arrays,
doubling, adding on
Patterns
Vocabulary: sum, product
This worksheet can be used as review or as an assessment.
Activities:
1. Give your students tiles or counters and ask them to find the numbers by making models.
I am less than ten. You can show me with:
2 equal rows of tiles
3 equal rows of tiles
Solution: The number is 6.
I am between 15 and 25. You can show me with:
2 equal rows of tiles
5 equal rows of tiles
Solution: The number is 20
2. Ask your students to solve these questions by making base ten models (using only ones or tens blocks).
I am a multiple of 5. You can make me with 6 base ten blocks (there are two answers: 15, 60).
I am a multiple of 3. You can make me with 3 base ten blocks (there are three answers: 12, 21, 30).
I am a multiple of 4. You can make me with 5 base ten blocks (there is one answer: 32).
Extensions:
1. Write the same number in both boxes to make the multiplication statement true:
a) × = 1
b) × = 25
c) × = 9
d) × = 4
e) × = 16
f) × = 36
2. Look at the numbers from Extension 1 (1, 25, 9, 4, 16, 36). Where do these numbers appear in the times
table in QUESTION 9? Can you describe the position of these numbers?
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NS3-42 Pennies, Nickels and Dimes
Goals: Students will be able to count any combination of pennies, nickels and dimes (two coin types)
up to a dollar.
Prior Knowledge Required: Skip counting by 5 and 10
Adding
Vocabulary: penny, nickel, dime, skip counting
Begin by reviewing the names and values of pennies, nickels and dimes. Try asking the class, “What coin
has a maple leaf on it? What is its value?” etc.
Hold up each coin and list its name and value on the board. It’s also a good idea to have pictures of the
coins on the classroom walls at all times while studying money.
Review skip counting by 1, 5 and 10 with your students. Tell them that adding money is really easy when you
just have pennies because each penny is worth 1 cent. Demonstrate counting out a pile of pennies (use the
BLM “Cut Out Coins” if you don’t have actual coins). Say the word “cent” after every penny: 1 cent, 2 cents,
3 cents, etc. Then give each student a copy of the BLM “Cut Out Coins”. Have them cut out paper coins –
they can cut out the squares if it is easier. Have them separate the pennies from the other coins. Then put
different prices on the board and tell them to put that much money to the side of their desk (or an envelope)
as though they were going to buy something for that price.
Repeat the counting activity with nickels and dimes. When you ask your students to put some money aside
as if they were going to buy an item, ask them first to put aside enough coins of the same denomination.
For example, for an item that costs 69¢, the students should put aside 7 dimes. Explain that different coins
can add up to the same amount. Ask students how you could make 5¢ using different coins. (5 pennies, or
1 nickel.) Then ask how you could make 10¢. (10 pennies, 2 nickels, 1 dime and a bonus answer: 1 nickel
and 5 pennies.)
Pick one of the items that the students set money aside for (for example, a pencil for 37 cents).
ASK: can you make exactly 37 cents with dimes? With nickels? This means you need pennies to make the
exact amount. Ask your students to remove a dime from the pile and to replace it with the necessary
amount of pennies. Invite a volunteer to count the coins. Repeat with other items. Write several more prices
on the board and ask your students to put aside the exact amount of money using two types of coins.
45¢ (dimes and nickels) 78¢ (dimes and pennies) 23¢ (nickels and pennies) and so on.
Assessment:
Count the given coins and write the total amount:
a) 10¢, 5¢, 5¢, 5¢ Total amount = b) 5¢, 1¢, 1¢, 1¢, 1¢, 1¢ Total amount =
c) 10¢, 10¢, 10¢, 1¢, 1¢, 1¢ Total amount =
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NS3-43 Quarters
Goals: Students will be able to identify 25¢ in any combination of coins.
Prior Knowledge Required: Skip counting by 5, 10 and 25
Adding
Penny, nickel, dime
Vocabulary: penny, nickel, dime, quarter, skip counting
Introduce the quarter the same way you introduced dimes, nickels and pennies last lesson. Ask your
students to find several ways to make 25 cents using nickels, dimes and pennies. Remind them how they
made various amounts of money last lesson and suggest using the same technique.
ASK: Can you make a quarter with dimes only? Why? Can you make a quarter with nickels only? How would
you do that?
Review skip counting by 25s. Repeat the activity involving setting money aside with prices such as 35 cents
(quarters and dimes, quarters and nickels), 27 cents, 30 cents. Use larger amounts of money, like 55 cents,
85 cents, and even more than a dollar.
Activity: Ask students to estimate the total value of a particular denomination (EXAMPLE: quarters,
pennies, etc.) that would be needed to cover their hand or book. Students could use play money to test their
predictions. This activity is a good connection with the measurement section.
Extension: When skip counting by 25, leave out some numbers without telling them where you are
leaving out the numbers (example: 0 25 50 100 125 150) and see if students can figure
out where to insert the missing number and what the missing number is.
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NS3-44 Counting by 2 or More Coin Values
Goals: Students will be able to count any combination of coins up to a dollar.
Prior Knowledge Required: Skip counting by 5, 10 and 25
Adding
Penny, nickel, dime, quarter
Counting on by 1,5 and 10
Vocabulary: penny, nickel, dime, quarter, skip counting
Review the two previous lessons. Complete the first page of the worksheet together. Review using the
finger counting technique to keep track of your counting. It might help to point to a large number line when
skip counting with numbers over 100. It’s a good idea to keep this large number line on the wall while
studying money. Let your students continue with the worksheets independently. Stop them after
QUESTION 4. Give each student a handful of play money coins. Ask the students to sort them by
denomination—putting all the pennies together, all the nickels together, etc. Once they are sorted,
demonstrate how to find the value of all the coins by skip counting in different units starting by tens. Do
some examples on the board (EXAMPLE: if you had 3 dimes, 2 nickels and 3 pennies you would count 10,
20, 30, 35, 40, 41, 42, 43). Have the students practice counting up the play money they have. They can
also trade some coins with other classmates to make different amounts each time.
NOTE: Allow your students to practice the skill in QUESTIONS 5 AND 6 of the worksheet with play money.
Do not move on until they can add up any combination of coins up to a dollar. Another way to help
struggling students is to cross out the counted money on the worksheet.
Assessment: Count the given coins and write the total amount:
a) 25, 1, 10, 5, 5, 25 Total amount = b) 10, 1, 1, 25, 25, 5 Total amount =
Activities:
1. Place ten to fifteen play money coins on a table. Ask students to estimate the amount of money, and
then to count the value of the coins. (Students could play this game in pairs, taking turns placing the
money and counting the money.)
2. Adding Money Game – Adapted from “A Companion Resource for Grade One Mathematics” by
Saskatchewan Learning.
Each pair should have the BLM “Adding or Trading Game” as a game board and each player should
have a different token to use as their playing piece. They will also need a die to know how many pieces
to move forward. When they roll, they move forward the correct number of squares and receive the coin
shown on the board. When both players are at the end of the board (not necessarily by the exact amount
shown on the die), they count up their money – the player with the most amount of money wins.
Number Sense Teacher’s Guide Workbook 3:1 114 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-45 Counting by Different Denominations
Goals: Students will identify how many cons of any particular denomination would need to be added to an
amount to make a certain total.
Prior Knowledge Required: Skip counting by 5, 10 and 25
Adding
Penny, nickel, dime, quarter
Counting on by 1,5 and 10
Vocabulary: penny, nickel, dime, quarter, skip counting
Begin by reviewing how to skip count by 10s and 5s from numbers not divisible by 10 or 5. (i.e. count by 5s
from 3. Count by 10s from 18).
Have students practice writing out skip counting sequences in their notebooks. Assign start points and what
to count by. Some sample problems:
a) Count up by 10s from 23
b) Count up by 10s from 27
c) Count up by 5s from 7
d) Count up by 5s from 41
Have students notice the number patterns. When counting by 10s, all the numbers will end in the same
digit. When counting by 5s, all the numbers will end in one of two digits. If you used the “skip counting
machines” in section NS3-13, review the activity with the students.
Give your students some play coins (cents only), and let them play the following game—one player takes 3
or 4 coins of two denominations, counts the money he has and says the total to the partner. Then he hides
one coin and gives the rest to the partner. The partner has to guess which coin is hidden. Make the task
harder allowing more denominations.
Next, modify the above game. Give the students additional play coins. One player will now hide several
coins (at most three coins that must be of the same denomination). He then says the total value of all the
coins, as well as the denomination of the hidden coins. The other player has to guess the number of hidden
coins (of each denomination).
Assessment:
Draw the additional money you need to make the total. You may use only 2 coins (or less) for each
question.
a) 25¢, 10¢, 5¢, 1¢ Total amount = 61¢
b) 25¢, 25¢, 10¢, 1¢, 1¢ Total amount = 92¢
Number Sense Teacher’s Guide Workbook 3:1 115 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Bring in flyers for some local businesses that sell inexpensive items (i.e. the grocery store, the drug
store, a convenience store).
Give each student a handful of play money change including quarters, dimes, nickels and pennies.
Have students count up their change. How much money do they have? What item or items could they
afford to buy?
Ask the students consult the flyers and to select one item that they would like to ‘buy’. This item must
cost less than $1, but it must also cost more money than they currently have. Ask them to figure out
how many additional coins of any one denomination they would need to have enough money to buy
their item (EXAMPLE: ask how many nickels would you need in addition to what you have now to afford
this item? How many dimes would you need?) Students are allowed to go over the amount and get
change back.
2. Extend the activity above. Ask students if it would be possible to pay for their item using only four coins.
Have them determine which four coins they would need. Then ask if it would be possible to buy the item
with three or two coins. For some students, it will not be possible to make their total
with only four coins or fewer. Ask them to determine how many coins they would need to make
their target amount.
Extensions:
1. Quan says he can make 76¢ using only nickels, dimes and quarters. Is he correct?
Explain.
2. Lynne says she can make 80¢ using only quarters and dimes. Is she correct?
Explain.
3. Kirsten says she can make $4.18 using only quarters and dimes. Is she correct?
Explain.
4. Rennish says he can make $5 using only nickels. Is he correct?
Explain.
5. What coins am I counting if I say …
a) … 10, 20, 30, 35, 40, 45, 50?
b) … 25, 50, 75, 85, 95, 100?
c) … 100, 200, 300, 325, 350, 351, 352, 353?
d) … 200, 400, 500, 600, 700, 710, 715, 720?
Number Sense Teacher’s Guide Workbook 3:1 116 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-46 Least Number of Coins
Goals: Students will make specific amounts of money using the least number of coins.
Prior Knowledge Required: Creating amounts of money using a variety of coins.
Vocabulary: penny, nickel, dime, quarter, skip counting
Begin with a demonstration. Bring in $2 worth of each type of coin (EXAMPLE: 4 rolls of pennies, 2 rolls of
nickels, etc.). Let students pick up each $1 amount to compare how heavy they are. Explain that it is
important to figure out how to make amounts from the least number of coins, because it is too heavy to
carry extra coins around.
Tell the class that the easiest way to do this is to start with the largest possible denomination and then
move to smaller ones.
Ask students how they could make exactly 10¢ with the smallest number of coins. They will probably give
the right answer immediately. Next, ask them to make up a more difficult amount, such as 35¢.
A quarter could be used, so put one aside (you can use the large pictures of the denominations to
demonstrate). We have 25¢, so what else is needed? Check a dime. Now we have 25¢ + 10¢ = 35¢. That’s
just right. Then look for another combination. Ask the students how to make 35¢ using dimes, nickels and
pennies, but no quarters. Compare the arrangements and the number of coins used in each.
Ask students to compare how many dimes they need compared to how many nickels they need to make the
same amount of money, example: 12¢, 14¢, 13¢, 32¢, 51¢. Give students either dimes and pennies or
nickels and pennies and have them work in pairs to see who used the smaller number of coins, the person
with dimes and pennies or the person with nickels and pennies.
Demonstrate how they can keep track of how many of each coin they’ve used by using a chart similar to the
one on the worksheet:
Dimes Nickels Pennies
8¢ � ���
11¢ � �
14¢ � ����
20¢ ��
19¢ � � ����
Let your students practice making amounts with the least number of coins.
Try totals such as 15¢, 17¢, 30¢, 34¢, 50¢, 75¢, 80¢, 95¢.
Number Sense Teacher’s Guide Workbook 3:1 117 Copyright © 2007, JUMP Math For sample use only – not for sale.
Ask if they noticed what coin is best to start with (the largest possible without going over the total).
When giving larger sums, you might use more than one volunteer—ask the first to lay out only the
necessary quarters, the next one—only the dimes, etc.
Show a couple of examples with incorrect numbers of coins, like 30¢ with 3 dimes, 45¢ with 4 dimes and a
nickel, 35¢ with a quarter and two nickels. Ask if you laid the least number of coins in the right way. Let your
students correct you.
Assessment:
Make the following amounts with the least number of coins:
a) 9¢ b) 45¢ c) 53¢ d) 17¢ e) 3¢ f) 80¢
Activities:
1. Use play money to trade the following amounts for the least number of coins.
a) 7 quarters b) 9 quarters c) 7 nickels d) 12 dimes e) 7 dimes and 1 nickel
2. Money Memory
Use the first page of the BLM “Money Memory” to play concentration by matching equivalent sums of
money. When students excel at this game and are ready for a greater challenge, you can add the
second page of the BLM.
Extension: Have students look at their answers for QUESTIONS 9 and 10, and ask them if there are any
answers where they use more than one nickel. Ask them to explain why there should never be more than
one nickel in these kinds of problems.
Literature Connection:
Smart by Shel Silverstein, contained in “Where the Sidewalk Ends”.
Read this poem to your class. With each stanza, stop and ask the students how much money the boy in the
story has. Give students play money to use as manipulatives so they can each have the same amount of
coins as described in the poem. This is an excellent demonstration that having more coins does not
necessarily mean having more money.
Number Sense Teacher’s Guide Workbook 3:1 118 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-47 Dimes and Pennies
Goals: The students will represent amounts up to $1 in dimes and pennies.
Prior Knowledge Required: Dimes and pennies
Count by two denominations
Skip counting by 10
Vocabulary: dimes, pennies, tens, ones, tens digit, ones digit
Give your students some dimes and pennies and ask them to make various amounts of money using only
these two types of coins. Ask your students to write their answers in a T-table:
Dimes Pennies Amount in ¢
Ask your students if they notice a pattern. Is the number of dimes the same as the tens digit or the ones
digit? Which digit is the same as the number of pennies? Write several money amounts in the third column of
the table and ask your students to fill in the other two columns without using play money.
You may also use the activity that involved hiding coins from the lesson NS3-45: Counting by Different
Denominations.
Extensions:
1. I have 35¢ in dimes and pennies. How many coins do I have?
2. I have 3 coins (dimes and/or pennies), How much money can I have? Find all the possible answers.
Number Sense Teacher’s Guide Workbook 3:1 119 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-48 Making Change Using Mental Math
Goals: Students will make change for amounts less than $1 using mental math.
Prior Knowledge Required: Counting by 1s and 10s
Subtraction
Vocabulary: skip counting, penny, nickel, dime, quarter, loonie, cent, dollar, change
Pose a sample problem. You would like to buy a pen which costs 18¢, but you only have a quarter to pay
with. How much change should you get back?
Rather than starting with a complicated subtraction question that includes decimals and carrying, show
students how to ‘count up’ to make change. Demonstrate that you can first count up by 1s to the amount.
In this case you would start at 18 and count up: 19, 20, 21, 22, 23, 24, 25. You would need 7¢ as change.
Review the finger counting technique, keeping track of the number counted on their fingers.
Model more examples, getting increasingly more volunteer help from the class as you go along.
Have students practice this skill with play money, if helpful. They should complete several problems of this
kind before moving on. Some sample problems:
a) Price of an orange = 39¢ Amount paid = 50¢
b) Price of a hair band = 69¢ Amount paid = 75¢
c) Price of a sticker = 26¢ Amount Paid = 30¢
Pose another problem: You would like to buy a flower that costs $80¢, but you only have a loonie to pay
with. How much change should you get back? How many cents are in $1? Before the students start to
count by 1s to 100, tell them that you will show them a faster way to do this.
Explain that instead of counting up by 1s, you can count by 10s to 100. Starting at 80 you would count up:
90, 100 to $1.
Give students several problems to practice. Some sample problems:
a) Price of a newspaper = 50¢ Amount Paid = $1.00
b) Price of a milk carton = 80¢ Amount Paid = $1.00
c) Price of a paper clip = 10¢ Amount Paid = $1.00
Pose another problem: You would like to buy a postcard that costs 55 cents, but you only have a loonie to
pay with. How much change should you get back?
Explain that you can count up by 1s to the nearest 10, then you can count by 10s to $1 (or the amount of
money paid). In this case you would start at 55 and count up to 60: 56, 57, 58, 59, 60.
Number Sense Teacher’s Guide Workbook 3:1 120 Copyright © 2007, JUMP Math For sample use only – not for sale.
Then count up by 10s. Starting at 60 you would count up: 70, 80, 90, 100. At the end, you will have counted
five ones and four 10s, so the change is 45¢.
Demonstrate the above problem using the finger counting technique. Explain to the students that this can
be clumsy, because it is easy to forget how many 1s or how many 10s you counted by.
How could you use this method to solve the problem without the risk of forgetting?
Model another example. You would like to buy a candy bar costs 67¢, but you only have a loonie
to pay with.
STEP 1: Count up by 1s to the nearest multiple of 10 (Count 68, 69, 70 on fingers).
STEP 2: Write in the number that you have moved forward and also the number that you are now ‘at’ (Write
“counted 3, got to 70¢”).
STEP 3: Count up by 10s to $1.00. Write in the number that you have counted up.
(Count 80, 90, 100, write 30¢)
STEP 4: Add the differences to find out how much change is owing (3¢ + 30¢ = 33¢).
Let a couple of volunteers model this method with sample problems. Give the students another several
problems to complete in their notebooks, such as:
a) Price of a bowl of soup = 83¢ Amount Paid = $1.00
b) Price of a stamp = 52¢ Amount Paid = $1.00
c) Price of a pop drink = 87¢ Amount Paid = $1.00
d) Price of an eraser = 45¢ Amount Paid = $1.00
e) Price of a candy = 9¢ Amount Paid = $1.00
Assessment:
a) Price of a pencil = 80¢ Amount Paid = $1.00
b) Price of a chocolate bar = 62¢ Amount Paid = $1.00
c) Price of a candy = 17¢ Amount Paid = $1.00
Bonus:
a) Price of an action figure = $9.99 Amount Paid = $10.00
b) Price of a book = $4.50 Amount Paid = $5.00
Activity:
Play Shop Keeper
Set up the classroom like a store, with items set out and their prices clearly marked. The prices should be
under $1.00. Tell the students that they will all take turns being the cashiers and the shoppers. Explain that
making change (and checking you’ve got the right change!) is one of the most common uses of math that
they will encounter in life!
Number Sense Teacher’s Guide Workbook 3:1 121 Copyright © 2007, JUMP Math For sample use only – not for sale.
Allow the students to explore the store and select items to ‘buy’. Give the shoppers play money to ‘spend’
and give the cashiers play money to make change with. Ask the shoppers to calculate the change in their
heads at the same time as the cashiers when paying for the item. This way, students can double check and
help each other out. Reaffirm that everyone needs to work together, and should encourage the success of
all of their peers.
Allow the students a good amount of time in the store. Plan at least 30 minutes for this exploration.
Extensions:
1. Once students have calculated the change for any question, have them figure out the least number of
coins that could be used to make that change.
2. This would be a great lead up to an actual sale event. If your community or school has an initiative that
your class wanted to support, you could plan a bake sale, garage sale, etc.
that would raise funds for the cause. Students would find the pretend task even more
engaging if it was a ‘dress rehearsal’ for a real event.
3. How much change should you get back if…
…you have 4 dimes and a quarter and you want to buy something for 58¢?
…you have 3 quarters and 4 pennies and you want to buy something for 66¢?
…you have 2 quarters and 3 pennies and you want to buy something for 37¢?
4. Word Problems:
I have 2 quarters, 3 dimes and a nickel. How much more money do I need if I want to buy something
worth 93¢?
I have 1 quarter, a dime and 3 nickels. How much more money do I need if I want to buy something
worth a dollar? Worth 60¢? Worth 84¢?
Number Sense Teacher’s Guide Workbook 3:1 122 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-49 Lists
Goals: Students will use lists to find numbers which have two properties.
Prior Knowledge Required: Comparing Numbers
Multiplication and skip counting
Even and odd numbers
Vocabulary: even, odd, multiple, greater than, less than
Have a volunteer write all the numbers from 0 to 9 in order. ASK: Which numbers are greater than 7?
Where can you find them? (to the right of 7) Which numbers are less than 5? Where can you find them?
(to the left of 5)
Have students individually write the numbers that are:
a) less than 4
b) greater than 6
c) greater than 3
d) less than 7
ASK: How can you find the numbers that are less than 7 and greater than 3? (find the numbers that are in
both the “less than 7” list and the “greater than 3” list)
Have students find numbers that are:
a) less than 6 and greater than 2
b) less than 9 and greater than 7
c) less than 8 and greater than 5
d) less than 7 and greater than 1
Remind students that 0 is a multiple of any number. For example, 0 = 2 × 0, so 0 is a multiple of 2.
Have students individually write the numbers that are:
a) multiples of 2
b) multiples of 3
c) multiples of 4
d) multiples of 5
e) multiples of 2 and multiples of 3
f) multiples of 3 and less than 7
g) multiples of 4 and greater than 7
Ask if anyone remembers what the words “odd” and “even” mean and have a student explain.
Then have students individually write the numbers that are:
a) odd
b) even
Number Sense Teacher’s Guide Workbook 3:1 123 Copyright © 2007, JUMP Math For sample use only – not for sale.
c) odd and less than 6
d) even and less than 5
e) odd and more than 4
f) odd and a multiple of 3
g) even and a multiple of 3
Bonus:
a) List the odd numbers that are multiples of 3 and are less than 5
(odd numbers are: 1, 3, 5, 7, 9; multiples of 3 are: 0, 3, 6, 9; numbers less than 5 are: 0, 1, 2, 3, 4.
The only number that belongs to all 3 lists is 3.)
b) List the even numbers that are multiples of 3 and are greater than 5 (even numbers are: 0, 2, 4, 6, 8;
multiples of 3 are: 0, 3, 6, 9; numbers greater than 5 are: 6, 7, 8, 9. The only number that belongs to all
three lists is 6.)
Extensions:
Introduce combinations of properties where no number has both or all properties.
a) Which numbers are odd multiples of 2? (none)
b) Which numbers are greater than 6 and less than 4? (none)
c) Which numbers are odd multiples of 5 that are less than 4? (none)
Bonus: Find another combination of properties which gives a list of no numbers.
Number Sense Teacher’s Guide Workbook 3:1 124 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-50 Organized Lists
Goals: Students will make various amounts of money using specific coins by creating an organized list.
Prior Knowledge Required: Canadian coins
Vocabulary: penny, nickel, dime, quarter, loonie, cent, dollar, list
Present the following problem: You need to program a machine that will sell candy bars for 45¢. Your
machine accepts only dimes and nickels. It does not give change. But you have to teach your machine to
recognize when it’s been given the correct change. The simplest way to do this is to give your machine a list
of which coin combinations to accept and which to reject. So you need to list all combinations of dimes and
nickels that add up to 45¢.
Explain that there are two quantities—number of dimes and number of nickels. The best way to find the
possible combinations is to list one of them (usually the larger—dimes) in increasing order. You start with
no dimes, then with one dime, etc. Where do you stop? How many dimes will be too much? Then next to
each number of dimes in the list, you write down the number of nickels needed to make 45¢. Make a list
and use volunteers to fill it in. You can also use a table, as shown below.
Dimes Nickels
0
1
2
3
4
Let your students practice making lists of dimes and nickels (for totals such as 35¢, 75¢, 95¢), nickels and
pennies (for totals such as 15¢, 34¢, 21¢), quarters and nickels (for totals such as 60¢, 85¢, 75¢).
Bonus:
Use dimes and nickels to make 135¢, 175¢.
Assessment:
List all the combinations of dimes and nickels to make 55¢.
Now let your students solve a more complicated riddle:
Dragons come in two varieties: the One-Headed Fearsome Forest Dragons and the Three-Headed Horrible
Hill Dragons. A mighty and courageous knight is fighting these dragons, and he has slain 5 of them. There
are 9 heads in the pile after the battle. Which dragons did he slay?
Number Sense Teacher’s Guide Workbook 3:1 125 Copyright © 2007, JUMP Math For sample use only – not for sale.
Remind your students that it is convenient to solve such problems with a list and to start with the largest
number. How many 3-headed dragons could there be? Not more than 3—otherwise there are too many
heads. How many 3-headed dragons could there be? There are 5 dragons in total. Make a list and ask the
volunteers to fill in the numbers:
Ask another volunteer to pick out the right number from the table—there were two 3-headed dragons and
three 1-headed dragons slain.
Give your students additional practice, with more head numbers: 6 heads, 2 dragons; 10 heads, 4 dragons.
What is the least and the most number of heads for 10 dragons?
Extensions:
1. List all the combinations of quarters and dimes to make 60¢, 75¢.
2. Monsters come in two varieties: Three-Headed Danger-of-Dale Monsters and Ever-Quarrelling-Nine-
Headed-Terror-of-Tundra Monsters. A mighty and courageous knight fought a mob of these monsters
and cut off all of their heads. There were 24 heads in the pile after the battle (and none of them
belonged to the knight). How many of each type of monster did the knight slay? Find all possible
solutions (ANSWERS: two 9-headed and two 3-headed, one 9-headed and five 3-headed, no 9-headed
and eight 3-headed).
3. The monsters from the previous question all have 7 tails each. When the mighty knight next got into a
fight, he produced a pile with 35 tails and 27 heads. Which monsters were destroyed? (ANSWER: two
9-headed and three 3-headed monsters)
3-Headed Horrible
Hill Dragons
1-Headed Fearsome
Forest Dragons
Total Number
of Heads
0 5 5
1
2
3
Workbook 3 - Number Sense, Part 1 1BLACKLINE MASTERS
Adding or Trading Game _________________________________________________2
Addition Rummy Blank Cards _____________________________________________3
Addition Rummy Preparation _____________________________________________4
Addition Table (Ordered) _________________________________________________5
Arrays in the Times Tables ________________________________________________6
Counting by 5s _________________________________________________________7
Crossword Without Clues _________________________________________________8
Cubes ________________________________________________________________9
Cut Out Coins _________________________________________________________10
Estimating Game ______________________________________________________11
Foreign Alphabets _____________________________________________________12
Hundreds Chart and Base Ten Materials ____________________________________13
Hundreds Chart Pieces __________________________________________________14
Hundreds Charts_______________________________________________________16
Hundreds Charts up to 200 ______________________________________________17
Make Up Your Own Cards ________________________________________________18
Money Memory _______________________________________________________19
Multiplication and Order ________________________________________________21
Multiplication Practice __________________________________________________23
Multiplying by 0 _______________________________________________________24
Multiplying by 1 _______________________________________________________25
Number Sentence Practice ______________________________________________26
Number Word Search ___________________________________________________27
Number Words Crossword Puzzle _________________________________________28
Place Value Cards ______________________________________________________29
Recognizing Number Words _____________________________________________30
Spelling Number Words _________________________________________________31
Ten-Dot Dominoes _____________________________________________________32
NS3 Part 1: BLM List
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Adding or Trading Game
END 1¢ 5¢ 1¢
1¢ 5¢ 1¢ 5¢10¢ 5¢
1¢ 5¢ 1¢25¢ 10¢ 1¢
START 5¢ 10¢ 1¢ 10¢25¢ 1¢ 1¢
1¢
1¢
10¢5¢
1¢
1¢
10¢ 25¢
1¢
1¢
10¢1¢10¢25¢
Addition Rummy Blank Cards
Workbook 3 - Number Sense, Part 1 3BLACKLINE MASTERS
Addition Rummy Preparation
1
37
+ 17
54
1
23
+ 39
62
2 tens + 3 ones
+ 3 tens + 9 ones
5 tens + 12 ones
+
4 tens + 7 ones
+ 2 tens + 6 ones
6 tens + 13 ones
+
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
+ 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
Addition Table (Ordered)
Workbook 3 - Number Sense, Part 1 5BLACKLINE MASTERS
Arrays in the Times Tables
Count the squares in the rectangle.
Fill in the blanks.
2 × 3 = ____ 3 × 1 = _____ 4 × 2 = _____
Now draw the rectangle yourself and write the product in the
bottom right square of the rectangle.
3 × 2 = 6 1 × 3 = _____ 3 × 4 = _____
Now complete the whole chart.
BONUS:
× 1 2 3
1
2
3
× 1 2 3
1
2
3
× 1 2 3 4
1
2
3
4
× 1 2 3
1
2
3
× 1 2 3 4 5 6 7
1
2
3
× 1 2 3
1
2
3 6
× 1 2 3
1
2
3
× 1 2 3 4
1
2
3
4
6 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Counting by 5s
How to Make a Skip-Counting Machine:
1. Fold along line A.
2. Cut along lines B, C, D, and E.
3. Cut along lines F and G.
4. Unfold.
5. Place this sheet over a hundreds chart. What numbers do you see?
B F C D
G E
A
Workbook 3 - Number Sense, Part 1 7BLACKLINE MASTERS
Crossword Without Clues
eighty
fifteen
forty
nine
one
seventeen
seventy
six
ten
three
twenty
two
zero
1. Group the words according to the number of their letters.
2. Which word is by itself in a group? Where does it fit?
3. Solve the puzzle. HINT: Cross out the words as you use them.
3 letters 4 letters 5 letters 6 letters
one
six
ten
two
7 letters 8 letters 9 letters
8 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Cubes
Workbook 3 - Number Sense, Part 1 9BLACKLINE MASTERS
10 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Cut Out Coins
Estimating Game
I rolled , , , and .
+
100
+
40
+
70
I rolled , , , and .
+
100
+
40
+
70
Workbook 3 - Number Sense, Part 1 11BLACKLINE MASTERS
Foreign Alphabets
Count by 5s and then by 1s to # nd the number of letters or symbols in
each foreign alphabet.
Hawaiian:
A, E, I, O, U, H, K, L, M, N, P, W, ‘
5, 10, 11, 12, 13
Russian:
Korean:
Spanish:
a, b, c, ch, d, e, f, g, h, i, j, k, l, ll,
m, n, ñ, o, p, q, r, s, t, u, v, w, x, y, z
Greek:
Cherokee:
12 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Hu
nd
red
s C
har
t an
d B
ase
Ten
Mat
eria
ls
12
34
56
78
91
0
11
12
13
14
15
16
17
18
19
20
2 r
ow
s o
f a
hu
nd
red
ch
art
:
Ten
s b
lock
:
On
es
blo
cks:
Workbook 3 - Number Sense, Part 1 13BLACKLINE MASTERS
Hundreds Chart Pieces
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 204 + 10 =
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 5038 + 10 =
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 10081 + 10 =
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
7 + 10 =
18 + 10 =
23 – 10 =
30 + 10 =
47 – 10 =
Shade the next 10 numbers after the bolded square. Add 10.
Move down a row to add 10.
Move up a row to subtract 10.
14 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Hundreds Chart Pieces (continued)
The boxes below are pieces from a hundreds chart.
Add 1 or 10 to find the missing numbers.
3 24 1939 48 60
Subtract 1 or 10 to find the missing numbers.
19
36
34
70
47 70
Find the missing numbers.
32
32
32
32
29
24
1826
57 74
Workbook 3 - Number Sense, Part 1 15BLACKLINE MASTERS
Hundreds Charts
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
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Hundreds Charts Up to 200
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190
191 192 193 194 195 196 197 198 199 200
Workbook 3 - Number Sense, Part 1 17BLACKLINE MASTERS
Make Up Your Own Cards
I have I have
Who has Who has
I have I have
Who has Who has
18 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Workbook 3 - Number Sense, Part 1 19BLACKLINE MASTERS
Money Memory
20 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Money Memory (continued)
Add before multiplying.
3 + 3 + 3 + 3 =
4 × 3 =
4 + 4 + 4 =
3 × 4 =
2 + 2 + 2 + 2 + 2 =
5 × 2 =
5 + 5 =
2 × 5 =
3 + 3 + 3 + 3 + 3 =
5 × 3 =
5 + 5 + 5 =
3 × 5 =
4 + 4 + 4 + 4 + 4 + + +
× = ×
Make a prediction. Write the same number in each box.
Check your answer by adding.
Fill in the blanks.
+ + + + = 5 × =
+ = 2 × =
Multiplication and Order
Workbook 3 - Number Sense, Part 1 21BLACKLINE MASTERS
Fill in the blanks.
Which number sentences make sense?
5 + 6 = 6 + 5 5 – 6 = 6 – 5 5 × 6 = 6 × 5
4 × = 6 ×
3 × = 5 ×
2 × = 7 × =
Multiplication and Order (continued)
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Multiply.
5 10 15 20 25 30
35 40 45 50
Count by 3s.
3 × 5 = 8 × 5 = 6 × 5 = 7 × 5 = 10 × 5 =
Multiply.
4 × 3 = 1 × 3 = 2 × 3 = 9 × 3 = 5 × 3 =
Now use your own # ngers.
7 × 2 = 4 × 4 = 9 × 2 = 6 × 4 = 8 × 2 =
3 6
30
Multiplication Practice
Workbook 3 - Number Sense, Part 1 23BLACKLINE MASTERS
Multiplying by 0
0 + 0 + 0 =
3 × 0 =
0 + 0 + 0 + 0 + 0 =
5 × 0 =
12 × 0 =7 × 0 = 23 × 0 =
213 × 0 = 0 × 174 = 11 × = 0
15 × = 0 × 38 = 0 × 94 = 0
When you add 3 zero times, you are not adding it at all!
So, 0 × 3 = 0.
Fill in the blanks:
0 × 1 =
0 × 4 =
0 × 7 =
0 × 8 =
0 × 16 =
0 × 21 =
0
0 + 0 + 0 + 0 + 0 + 0 =
6 × 0 =
0 + 0 + 0 + 0 =
4 × 0 =
24 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Multiplying by 1
1 + 1 + 1 + 1 + 1 + 1 = so 6 × 1 =
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = so 9 × 1 =
1 + 1 + 1 + 1 = so 4 × 1 =
Make a prediction.
7 × 1 = 26 × 1 = 583 × 1 =
When you add 3 once, you just get 3. So 1 × 3 = 3.
Fill in the blanks.
1 × 2 = 1 × 7 = 1 × 98 =
1 × = 4 1 × = 8 1 × = 67
× 5 = 5 × 6 = 6 × 93 = 93
Isobel says: 1 × 71 = 71 × 1. Is she right?
Explain:
Fill in the blanks.
Workbook 3 - Number Sense, Part 1 25BLACKLINE MASTERS
8 + 4 9 + 5 7 + 6 5 + 8
4 + 7 8 + 8 6 + 9 7 + 7
9 + 7 6 + 6 9 + 8 9 + 9
8 + 7 8 + 6 9 + 3 4 + 9
9 + 8 8 + 7 6 + 5 6 + 6
8 + 6 8 + 5 6 + 7 9 + 7
8 + 4 3 + 9 7 + 4 5 + 9
6 + 6 9 + 4 5 + 7 7 + 7
5 + 9 7 + 4 8 + 9 9 + 9
9 + 6 7 + 8 7 + 7 8 + 4
9 + 2 8 + 8 6 + 5 8 + 3
8 + 6 9 + 3 5 + 7 6 + 6
Number Sentence Practice
26 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Number Word Search
Find:
one ten eleven two twenty
twelve three thirty four forty
fifty zero seventeen eight
Use the leftover letters to finish the message.
The four seasons are fall, ,
.
This puzzle was made using the Internet tool at http://www.superkids.com/aweb/tools/words/search
w t i t w e n t y
n w t e o e r t s
f o u r v n y w p
s e v e n t e e n
z t l e r r i l f
e e f i f t y v o
r n h g n g a e r
o t t h r e e n t
d s u t m m e r y
Workbook 3 - Number Sense, Part 1 27BLACKLINE MASTERS
3
4
1 2
6
5
7
8 9
10
11
Across
2. Four less than ten
4. Rhymes with fine
7. Ten + Seven
8. Fifty + Thirty
10. Twenty + Twenty
11. Nothing
Down
1. Eleven – Ten
2. Two more than sixty-eight
3. Twenty – Five
5. Two tens
6. Seven + Three
9. Seven – Four
Number Words Crossword Puzzle
28 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Place Value Cards
Ones
Hundreds
Tens
Workbook 3 - Number Sense, Part 1 29BLACKLINE MASTERS
Recognizing Number Words
1. Eight children ate pie.
2. Ravi ate eight cookies.
3. She won two games.
4. He only won one game.
5. Four friends played soccer for fun.
6. She had to fix six bikes.
Circle the number words.
Cross out the words that only sound like number words.
30 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Spelling Number Words
Circle the spelling of the number words.
HINT: Look at the words you circled.
one won wun
to too two
for four fore
six sicks siks
ate eight ait
1
2
4
6
8
Workbook 3 - Number Sense, Part 1 31BLACKLINE MASTERS
Ten-Dot Dominoes
Draw the missing dots on the blank side.
Finish the number sentence.
All of these dominoes have a total of 10 dots.
8 + = 10 9 + = 105 + = 10
10 = + 2 10 = + 710 = 6 +
3
+
10
1
+
10
+ 4
10
32 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Measurement Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-1 Estimating Lengths in Centimetres Goals: Students will estimate measurements in centimetres.
Prior Knowledge Required: The ability to measure
Units of measurements
Linear measurement
The ability to estimate
Skip counting
Vocabulary: measurement, centimetre, estimate, index finger
Point out the centimetre marks on a ruler, and explain to your students that a centimetre is a unit of
measurement. Write the word “centimetre” (circle the letters c and m) and the abbreviation “cm,” and explain
that these are the two ways to write centimetre.
Give your students a number of small items (tacks, buttons, blocks, etc.) that range in length from 1 mm to
10 cm—several of them should measure 1 cm long. Ask them to estimate if the item measures 1 cm exactly,
or longer or shorter than 1 cm. Remind students that “estimate” means close to or about, and explain that
they won’t be able to determine exact lengths without measuring. Then have each student present an item
with an explanation for his or her estimate.
Now give your students a centimetre measure that they can take anywhere and always have. Can they
guess what it is? Show them that their index finger is approximately 1 cm wide.
Have students compare the small items measuring 1 cm with the width of their index fingers. How close are
they to matching in length? Then have students measure the width of their index fingers with a ruler to
determine how close it is to a centimetre.
Collect all of the items given out at the beginning of the lesson and put the items measuring 1 cm in a box
that will be stored somewhere prominent.
As your students learn the many units of measurement, they will add respectively measured items to the box.
Then demonstrate to your students how they can use their index fingers, measuring 1 cm, to measure
something (a book, an eraser, etc.).
Have students select two objects from their desks or backpacks, measure the objects with their index fingers
and then complete the following sentence for both objects.
___________________ is approximately_________ cm long.
Show your students, by having a volunteer measure with his or her index finger, that the width of a penny is
about 2 cm. Then ask if they can measure their two objects with a penny instead of their index fingers.
Explain that it can be done by skip-counting by two. Distribute play-money pennies, have students measure
(in pennies) the length of their hands, notebooks, pencils, etc., and then convert their measurements into cm.
Measurement Teacher’s Guide Workbook 3:1 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activity: Create a box to collect benchmark items that correspond with each unit of measurement
introduced. Everyone can then refer to and compare these items as the lessons progress.
Extensions:
1. Students can use benchmarks, such as the width of their thumb or index finger (approximately 1 cm) or
the width of their hand with fingers spread (approximately 10 cm) to estimate lengths.
Ask students to estimate the length of various objects in the classroom using their hands or thumbs as
benchmarks before they measure the actual lengths with a ruler.
2. Students could use non-standard units, such as the width of a penny (which is about 2 cm) to estimate or
measure lengths. Give students play money pennies and have their measure the length of their hand, a
notebook, a pencil, etc…in pennies. Students should convert their measurements into cm. (This is a
good exercise in skip counting by 2s or in ‘doubling’.)
3. Estimate the height of a classmate in cm. Then measure their height using your hand. (Your hand with
fingers spread should be about 10 cm wide.) Finally, use a metre stick to check your result. How close
were you?
HINT: Measure them against a wall to get an accurate result.
Estimate ____ cm Hand Measurement ____ cm Actual Measurement ____ cm
Measurement Teacher’s Guide Workbook 3:1 3 Copyright © 2007, JUMP Math For sample use only – not for sale.
0 cm 1 2 4 3 5 6 8 7 10
9
1 2 3 4 5
1 2 3
0 cm 1 2 4 3 5 6 8 7 10
9
ME3-2 Measuring in Centimetres Goals: Students will take measurements in centimetres on a ruler.
Prior Knowledge Required: The ability to count using a number line
Units of measurements
The ability to estimate
Skip counting
Centimetres
Vocabulary: measurement, centimetre, estimate, index finger, number line
Ask your students what they always have with them that can be used as a rough measure of a centimetre.
Then ask them how they could measure a fairly large item (a poster, the blackboard, etc.). They might
suggest using their hand with fingers spread slightly (which is about 10 cm wide) or some other referent.
What if the measurement had to be exact, not approximate? They should know that a ruler will make exact
measurements.
Draw a number line and explain that counting on a number line is just like using a ruler. Demonstrate this by
asking a volunteer to count (by “hopping”) to five using a number line. Then ask another volunteer to
demonstrate how to measure 5 cm with the ruler.
Ask your students how they can find a measurement without starting at the zero mark of a ruler.
Demonstrate this by starting at the 2 cm mark and hopping to the 5 cm mark, measuring 3 cm.
Have your students practice this by assigning them the worksheet.
Assessment:
Record the distance between two points.
a) from 3 to 8 b) from 2 to 11 c) from 5 to 15 d) from 12 to 14
Measurement Teacher’s Guide Workbook 3:1 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
0 cm 1 2 4 3 5 6 8 7 10
9 0 cm 1 2 4 3 5 6 8 7 9
ME3-3 Rulers and ME3-4 Measuring in Centimetres with Rulers Goals: Students will take measurements in centimetres using a ruler.
Prior Knowledge Required: The ability to count using a number line.
Units of measurements
Linear measurement
The ability to estimate
Skip counting
Centimetres
Vocabulary: measurement, centimetre, estimate, index finger, number line, ruler
Present a variety of objects and ask your students to estimate their length. Remind your students that they
can make a better estimate if they measure the objects with their built-in benchmarks: for instance, they
might use their index fingers or their hand with their fingers spread slightly. Ask your students to record
their estimates.
Hold up an object about 10 cm long and explain that you need to know if it fits into a box that is 10 cm long.
Ask your students to explain why estimating may give them an incorrect answer. What other method could they
use to determine whether the object will fit into the box? Ask your students to explain why they need to know
how to measure things. In which situations in life they might need to take exact measurement?
Draw several objects on the board together with rulers so that some pictures show a correct way to measure
the lengths of the objects and some do not (as shown below):
Ask your students if measurements taken in the way shown would be correct. Review the method of
measuring from the previous lesson.
Ask your students to measure some objects whose lengths they have estimated.
Measurement Teacher’s Guide Workbook 3:1 5 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Have students use rulers to measure items in the classroom. Have them find an item that:
a) is 6 cm long b) is 10 cm long c) is between 12 and 15 cm long
d) is 1 cm long e) is greater than 15 cm long f) is about 35 centimetres long
2. Draw a triangle on grid paper and measure its sides to the nearest cm. Calculate the total distance
around the shape. This distance is called the perimeter of the shape. (see sections ME3-11 to ME3-14).
As a project, students could find examples of optical illusions (in books or on the Internet) where the objects
appear to have different lengths, but actually to have the same length when measured. Students might try to
find an explanation of how the brain is tricked by the illusion.
Measurement Teacher’s Guide Workbook 3:1 6 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-5 Drawing to Centimetre Measurements Goals: Students will measure and draw items to a specific length in centimetres.
Prior Knowledge Required: The ability to count using a number line
Using a ruler
Centimetres
Vocabulary: measurement, centimetre, ruler, grid paper
Give your students several objects of standard length (paper clips, unused pencils, etc.) and ask them to
measure the lengths.
Demonstrate how to draw a line 2 cm in length with a ruler. Have your students practice these steps—
separately, if necessary—in their notebooks.
STEP 1: Find the zero mark on a ruler/number line. Draw a vertical line to mark the zero.
STEP 2: Count forward from zero by two hops.
STEP 3: Draw a vertical line to mark the two.
STEP 4: Draw a line connecting the two vertical marks.
Have your students draw lines of several lengths with a ruler. If the class does not have a standard set of
rulers, have students trade rulers with their classmates so that each measurement is made using a different
ruler. This reinforces the idea that rulers have equal measurement markings, even when they look different.
Ask your students to draw another line 2 cm in length, but to start it at 3 on the number line. Following the
steps taught earlier in this lesson, but beginning at 3 and ending at 5 on the number line, have a volunteer
demonstrate how the line is drawn. Have students reproduce the problem in their workbooks and solve
several others using this method (EXAMPLE: a line 8 cm in length beginning at 3 on the number line,
a line 11 cm in length beginning at 2 on the number line, a line 14 cm in length beginning at 1 on the
number line, etc.).
Distribute centimetre grid paper to your students and have them estimate the width of the squares. Then
have them measure the width with their rulers. Demonstrate how to draw items of specific length by using the
centimetre grid as a guide. On your grid, draw a pencil (or some other item) that is 5 cm long. Explain that
you selected a starting point and then hopped forward by 5. Draw vertical marks at the start and end points,
then draw the item to fill the length between the marks.
Assessment:
Draw a line 4 cm long and a pencil 8 cm long.
Measurement Teacher’s Guide Workbook 3:1 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Draw each object to the given measure.
a) A shoe 6 cm long b) A tree 5 cm high c) A glass 3 cm deep
2. Draw a collection of long items.
a) Draw a collection of alligators, each one being 1 cm longer than the previous.
b) A pencil shrinks when it is sharpened. Draw a collection of pencils, each one being 1 cm shorter than
the previous.
c) Draw a sequence of toboggans where each one is 2 cm longer than the last.
d) A carrot shrinks when it is eaten. Draw a collection of carrots, each one being 2 cm shorter
than the previous.
3. Write a story about one of the growing or shrinking items in Activity 2 (or invent your own!). Tell the story
of how the item grows or shrinks. How does the carrot get eaten? Who wants to ride the toboggans?
Write the story to go with the pictures that you have drawn.
Measurement Teacher’s Guide Workbook 3:1 8 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-6 Estimating in Centimetres Goals: Students will measure and draw items to a specific length in centimetres
Prior Knowledge Required: The ability to count using a number line
Using a ruler
Centimetres
Vocabulary: measurement, centimetre, ruler, grid paper
Draw a line on the board which measures 20 cm. Have students volunteer a non-standard unit to measure it
with. The units (link cubes, unit cubes, paperclips, string, shoelace, link its, etc.) can be displayed where the
students can easily see them.
Show the unit to the students and have them compare its size to the line’s size. Ask how many of the unit
they think it would take to measure the line. Discuss, focusing on differentiating between “informed” guesses
and “wild” guesses. Write students responses on board. Test the predictions.
Choose another unit of measurement (preferably one that is distinctly larger or smaller than the first) and
have students estimate how many non-standard units it would take to measure the line. Test the predictions
again and record the new measurement. Assuming that the predictions are more accurate the second time
around, discuss why this is so. If it is not, then ask students how they think they might achieve closer
estimates.
Ask students to look at the two measurements.
NOTE: You may want to record estimates and real measurements on a chart to get students use to
organizing their data. Discuss why it takes more of one unit and less of the other to measure the same line.
Students should see the connection between size of unit and number of units required.
Suggest the students to measure one of the non-standard units of measurement they used with a ruler and
to obtain the estimate for the length of the line in cm.
Ask for three student volunteers. Assign each a task, two students to find objects which are longer than the
line, and the other, an object which is shorter than the line. Then have them order themselves beside the line
on the board, from shortest to longest object, including the line itself. Have each student explain how they
knew where to stand.
Measurement Teacher’s Guide Workbook 3:1 9 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-7 Estimating in Metres Goals: Students will estimate lengths in metres.
Prior Knowledge Required: Using a ruler
Centimetres
Vocabulary: measurement, centimetre, metre, metre stick, ruler
Identify the length and marking of a metre on a ruler or tape measure, and explain to your students that a
metre is another unit of measurement. Write the word “metre” and the abbreviation “m”, and explain that
these are the two ways to write metre. Explain that there are 100 centimetres in one metre. Explain to your
students that they have a natural benchmark in their body that is close to 1 m. Their arm span is close to
1 m, and so is a large step.
As a class, select an assortment of items to measure with a metre stick. First estimate if each item is closer
in length to 0 m, 0.5 m or 1 m, then measure each item to verify its length. Hold the item up to the zero mark
of a metre stick and note how far the end of the item is from the zero mark, the 50 cm mark and the 1 m
mark. The shortest distance between either of these marks and the end of the item is the correct estimate for
the item’s length.
Help your students to develop an estimate for the length of the blackboard. First, ask your students to guess
the length of the blackboard in metres. You might wish to record their estimates in a table. Place a metre
stick along the board. Ask your students to compare the length of the board to the stick. Draw a line that
extends from one end of the board to the other and mark of a length of 1 metre at one end of the line. Ask
your students if they wish to adjust estimates for the length of the board. Record any new estimates in the
table. Continue to mark off lengths of 1 metre on your line and allow your students to adjust their estimates.
When the last unmarked section of the line is clearly less than 1 metre, ask your students to guess whether
the section is more or less than half a metre (50 cm) or one quarter of a metre (25 cm) long.
Activities:
1. Ask students to measure and compare the lengths of various body parts using a string and a ruler.
EXAMPLE:
a) Is your height greater than your arm span?
b) Is the distance around your waist greater than your height?
c) Is your leg longer than your arm?
2. Encourage your students to predict the answers before they perform the measurements. Estimate the
width of the school corridor: First, try to compare the length to some familiar object. For example, a
minivan is about 5 m long. Will a minivan fit across the school corridor? Then, estimate and measure the
width with giant steps or let several students stand across the corridor with arms outstretched. Estimate
the remaining length in centimetres. Measure the width of the corridor with a metre stick or measuring
tape to check your estimate.
Measurement Teacher’s Guide Workbook 3:1 10 Copyright © 2007, JUMP Math For sample use only – not for sale.
3. Measure your shoe. Use this natural benchmark to measure long lengths: the classroom, the width of the
hallway, or the length of the whole school!
4. Your students should know how to make and record measurements in various forms; for instance they
might write a measurement as 1 metre and 25 centimetres, or 1 m and 25 cm, or 125 cm. When
students measure various objects in the classroom, ask them to record their measurements in these
three ways shown above.
Extension: Some natural benchmarks you can use to make estimates are the index finger (for 1 cm), the
hand with fingers slightly spread (for 10 cm), the arm span and a giant step (for 1 m). Will these benchmarks
represent approximately the same lengths in 5 years? Compare the length of your “benchmarks” with those
of some adult members of your family.
Measurement Teacher’s Guide Workbook 3:1 11 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-8 Estimating in Metres (Advanced) Goals: Students will estimate lengths in metres.
Prior Knowledge Required: Using a ruler
Metres
Addition, multiplication
Skip counting
Vocabulary: measurement, centimetre, metre, metre stick, ruler
Ask your students to think of objects that can act as benchmarks for estimating lengths, heights, and
distances larger than the ones they used in the previous lesson. Would you use paper clips to measure a
height of a tree? You may measure the actual length or height of some of these objects if they are
available. Here are some possible benchmarks:
• A (very tall) adult and a door are about 2 m tall.
• A level, or storey, in a building (viewed from outside the building) is about 2 doors tall,
so about 4 m tall.
• A school bus is about 10 m long.
• A typical car is about 3 m long.
Invite students to use these benchmarks to estimate greater lengths and heights, such as:
• A basketball field is about 9 cars long. How long is a basketball field in metres?
• Two minivans are as long as 1 school bus. How long is each minivan?
• A playground is about 10 cars long. How long is the playground in metres? How many minivans can
be parked along the playground?
• Daniel lives in an apartment building with 8 storeys. About how many metres tall is the building?
About how many school buses standing end-to-end, one on top of the other, are as tall as Daniel’s
building?
• Ileana wants to estimate the height of a tree that grows near the school building. The tree is
3 storeys tall. How tall is the tree?
• A storey is about 4 m tall. How tall is 5-storey-high building? How many storeys are in a building
that is about 100 m high?
Students can use either multiplication or skip counting to solve these problems. Ask your students to
explain their solutions and encourage them to try different ways of thinking. For example, the last problem
could be solved in different ways, apart from skip counting by 4s to 100:
1. Skip count by 4s: 4, 8, 12, 16, 20. The building is 20 metres tall. I can skip count by 20 to 100: 20,
40, 60, 80, 100. I said 5 numbers. Each 20 metres is 5 storeys, so I can skip count by 5s 5 times to
get the height in storeys: 5, 10, 15, 20, 25. The building is 25 storeys high.
Measurement Teacher’s Guide Workbook 3:1 12 Copyright © 2007, JUMP Math For sample use only – not for sale.
2. Five storeys are 20 metres high. Make a T-table:
Height in
storeys
Height in
metres
5 20
10 40
15 60
20 80
25 100
3. Each storey is 4 metres high. 5 storeys are 4 x 5 = 20 metres high. What should I multiply 20 by to
get 100? I should multiply by 5. The 100 m-tall building is 5 times higher than 20 m-tall building (of 5
storeys). This means I have to multiply 5 by 5 to get the height of 100 m-tall building in storeys.
4. If your students are familiar with division, they could use the following method:
Each storey is 4 m high. 5-storeys are 20 m tall. What do you do to get 20 from 4 and 5?
(Multiply them.) What do you do to get 5 from 20 and 4? (Divide 4 into 20) The building is 100 m
high. What should I multiply 4 by to get 100? I divide 4 into 100, and get 25. This is the height of
the building in storeys.
Extension: Use a number line to solve the next question: A swimming pool is 5 m deep. If you place a
school bus upright into this swimming pool, and place a car upright near this swimming pool, which will be
taller? Hint: Put the zero mark at the bottom of the swimming pool.
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ME3-9 Kilometres Goals: Students will estimate lengths in metres.
Prior Knowledge Required: Using a ruler
Centimetres
Vocabulary: measurement, kilometre, metre, metre stick, ruler
Explain to your students that a kilometre is another unit of measurement. Write the word “kilometre” (circle
the letters k and m) and the abbreviation “km”, and explain that these are the two ways to write kilometre.
Explain that there are one thousand metres in one kilometre.
A kilometre can then be represented in the measurement box (see ME3-1) with a spool of fishing line. [The
packaging always labels the length of fishing line on each spool in metres; one spool of line should suffice.]
A kilometre is about the same distance that we can walk in fifteen minutes, the length of about ten football
fields, or the length of about ten small city blocks. Ask your students to name a place that they think is about
1 km from the school.
As a class, skip-count by 100s to 1000, by 10s to 100, by 25s to 100. Ask your students: A soccer field is
about 100 m long. How many soccer fields make a kilometre? A school bus is 10 m long. How many school
buses can be parked along the soccer field? Along 1 km? A tennis court is about 25 m long. How many
tennis courts make 100 metres? How can you use this information to find out how many tennis courts will fit
in 1 km?
Tell your students a story: A university student planning a cheap trip to Saskatchewan can only afford
enough gas to travel a total of 500 km. View the map and determine the most interesting places to visit.
Display an overhead of the BLM “Map of Saskatchewan” and ask students to identify the distances between
sets of two adjacent points (EXAMPLE: Weyburn to Regina, and Regina to Saskatoon). Then ask them to
calculate the distance between three points (EXAMPLE: Saskatoon to Moose Jaw to Regina).
Assessment:
1 Jenny drives from Monkton to Yarmouth. How long is her trip?
2. A minivan is 5 m long. How many minivans parked in a line will it take to equal…
a) 10 m? b) 20 m? c) 100 m? d) 1 km?
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Activities:
1. Kilometre for a Cause
If your school is planning any charitable events, such as a food or toy drive, set a goal to raise a
kilometre of goods. Have students measure a variety of items that might be collected in the drive
(canned goods, bags of pasta, etc.), and calculate the average length. If the items are lined up end to
end, how many will need to be collected to create a line a kilometre long? Set that as a goal. When
the drive is done, try to snake a line of all the items in a large open space (maybe the gym). How long is
the line?
2. Have students use a web application like Google Maps or MSN Maps to see how far their house is from
the school.
Extensions:
1. An adult Chinese Alligator is about 2 m long. How many crocodiles lined end to end will equal kilometre?
What about a…
• Saltwater Crocodile, 5 m long
• Newborn Saltwater Crocodile, 20 cm long (How many equal 1 m? 10 m? 100 m?)
2. Create a BLM map of a place that the class has actually visited. Bring in photographs of attractions to
enrich the experience for your students. Alternately, if the class is planning a trip, bring in a map of the
destination and ask students to help plan a good route.
3. If you lined up the following objects would they be: (i) close to 1 km, (ii) less than 1 km, or (iii) more
than 1 km? Explain your answer on a separate piece of paper.
a) 1000 paper clips b) 1000 bikes c) 1000 JUMP books d) 1000 baseball bats
HINT: First decide if the individual object is close to a metre, less than a metre or more than a metre.
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ME3-10 Ordering and Assigning Appropriate Units
Goals: Students will estimate lengths in metres.
Prior Knowledge Required: Using a ruler
Centimetres
Metres
Kilometres
Vocabulary: measurement, centimetre, metre, kilometre, appropriate
Individually present the items from the measurement box (see ME3-1) and have your students express the
measurement that each item represents. Review the full name and abbreviation for each unit.
Ask your students to tell you how far it is from Halifax to Calgary in centimetres. Then ask them to calculate
the width of a chocolate bar in kilometres.
Explain that it is important to choose the appropriate unit of measurement for the length/distance being
measured. Ask your students to tell you which unit of measurement will best express the distance from
Halifax to Calgary. Which unit of measurement will best express the width of a chocolate bar?
Practice this by displaying a variety of objects (a book, a stapler, a coin, etc.) and asking your students to tell
you which unit of measurement will best express the length of the object. Is the length of a stapler expressed
best by a centimetre or a metre?
Have your students select five items in the classroom and guess which unit of measurement will best
express each item’s height, width or length.
Have a volunteer measure one of the practice items (the blackboard, for example), but do not specify a unit
of measurement. After the volunteer relates the measurement to the class, identify the unit of measurement
used and ask your students why the volunteer chose that unit of measurement. Students will likely respond
that metres were used because the board is about the right size. It would be many centimetres long, and it’s
much smaller than a kilometre.
Have your students measure their five objects. Which units of measurement do they automatically use?
Do these units of measurement lend themselves easily to the task? Would alternate units of measurement
offer simpler measurements?
Ask your students to list in their notebooks five things that could be measured and are not in the room
(a bicycle, a video game, a rocket ship, anything). Ask students to arrange the five things in order from
smallest to largest. Then ask them to indicate which unit of measurement will give the simplest measurement
for each item.
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Play a game with the class. State a length and have your students identify the best unit of measurement
for it. Allow your students to identify the best unit of measurement by displaying the appropriate benchmark
item from the measurement box (see ME3-1).
Round 1: Centimetres and Metres
Which unit of measurement will make these statements correct?
1. Your index finger is about 1 wide.
2. Blue whales are an average of 25 long.
3. A pencil is about 20 long.
4. A car is about 4 long.
5. A paper clip is about 3 long.
Round 2: Metres and Kilometres
Which unit of measurement will make these statements correct?
1. An average-sized adult is about 1 and .5 tall.
2. The distance between Edmonton and Calgary is about 300 .
3. An average alligator is about 4 and .5 long.
4. A marathon course is about 42 long.
5. A male African elephant may grow to be as tall as 4 .
Round 3: Bonus Round
Which unit of measurement will best express the length (or height) of each item?
1. Length of an ice cube
2. Length of an ice rink
3. Length of Antarctica
4. Length of a maple leaf
5. Height of a polar bear
Ask your students how many centimetres are in a metre. Write “100 centimetres in a metre, 100 cm = 1 m.”
Ask them to identify the bigger number, 35 or 2. Then ask them to identify the bigger distance, 35 cm or 2 m.
Have them explain their reasoning. Remind them that there are 100 cm in every metre. Then have your
students measure out both lengths with the centimetre and metre benchmarks from the measurement box
(see ME3-1).
Explain that the easiest way to compare measurements that are expressed in different units is to convert all
measurements to the small unit. For instance, to compare 3 m and 250 cm, convert 3 m to 300 cm, and it
becomes clear that 3 m is greater than 250 cm.
Remind students to multiply by 100 when converting metres into centimetres. Solve a couple example
problems (1 × 100, 2 × 100, etc.) together as a class. It may help them to skip-count by 100s when
converting from metres to centimetres.
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Let your students practice with questions like:
Circle the greater amount:
1 m or 80 cm 6 m or 79 cm 450 cm or 5 m 230 cm or 2 m
Ask students to order these lengths from shortest to longest and ask them to mark these lengths on a
number line:
0 cm 100 cm 200 cm 300 cm 400 cm 500 cm 600 cm
Assessment:
Thickness of a book: 3 Length of a chocolate bar: 20 _____
Height of the school: 8 Height of a tree: 12 ____
Height of a mountain: 4 or 4000 Distance from a window to a door: 4 ____
Distance from your school to your home: 700 Distance from your nose to your toes: 113 ____
Distance from the Earth to the Moon: 385 000 Distance between your ears: 20 _____
Convert the distances to cm and then order them from greatest to smallest.
a) 75 cm b) 85 m c) 230 cm d) 7 m e) 4 cm
Activity: Divide your class into three groups and assign one unit of measurement (cm, m or km) to each
group. Give each group a sheet of chart paper, and ask them to write the full name of their unit of
measurement and the abbreviation at the top of the page. Have them list as many things as they can that
could be measured with that unit of measurement. Set a target quantity (maybe twenty) and ask students to
try and list more than that quantity. Have each group share their ideas.
Extensions:
1. Give students extra practice exercises like the ones in QUESTIONS 9 to 11 on the worksheet. For more
advanced work, students could compare measurements in mm and cm, or in mm and m. For example,
which is larger:
a) a piece of string 150 mm long or one that is 17 cm long?
b) a piece of string 2 m long or one that is 1387 mm long?
2. Could you…? Challenge:
Here are some questions you could ask your students to help them practice estimating.
(Younger students could use a calculator to multiply with or you could teach them the standard method
for multiplying larger numbers.) You might also have students round all numbers to the leading digit and
then count by 10s, 100s, or 1000s to estimate.
QUESTION 1 (Warm Up): Could you…fit all the students in your school onto three school buses?
QUESTION 1 Solution:
ASK: How many students fit in one school bus?
ASK: How many students in the school?
ESTIMATE: Round the numbers.
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ANSWER: – Unlikely (unless it is a very small school) that you could fit all the students in your school
onto three school buses.
QUESTION 2 (Warm Up): Could you…walk up 100 steps from your classroom to the main office?
QUESTION 2 Solution:
Have the students try this. Estimate how many steps it takes to get to a certain location, then use this
information to decide if it is possible to walk 100 steps from your classroom to the office.
QUESTION 3 (Harder): Could you…reach as high as the CN Tower with all the rulers in your school?
QUESTION 3 Solution:
RESEARCH: What is the height of the CN Tower? The CN Tower is about 533 metres tall. This is equal
to 53 300 cm.
What is the length of the ruler most commonly used in the school? Usually 30 cm.
How many rulers are in your classroom? Equal to the number of students. Multiply by the number of
classrooms and round.
Multiply the estimate for the number of rulers by the height of one ruler.
EXAMPLE: For a school with 600 rulers, the height of all the rulers is 18 000 cm. Far shorter than
the CN Tower.
Here are some other questions your students could try.
QUESTION 4. Could you…stack 100 pennies to be as high as the school?
QUESTION 5. Could you…read 50 books in a month?
QUESTION 6. Could you…fit 100 boxes of macaroni and cheese into a suitcase?
QUESTION 7. Could you…place 100 oranges in a line as long as your classroom?
QUESTION 8. Could you…walk 10 kilometres in a day?
QUESTION 9. Could you…fill a fish tank with 100 bottles of juice?
QUESTION 10. Could you…find a rhinoceros as heavy as everyone in your class put together?
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ME3-11 Measuring Perimeter Goals: Students will estimate and measure perimeters using non-standard units.
Prior Knowledge Required: What measurement means
Practice and comfort with other forms of linear measurement and non-
standard unit use
How to count
Add strings of 1-digit numbers
Vocabulary: perimeter, around, area
Hold up a piece of artwork. Ask students what parts of the artwork they would need to measure in order to
frame the picture. Then, ask how many link cubes or pattern block triangles it might take those
measurements.
Draw a rectangle on the board (22 cm × 30 cm) and demonstrate how to line up the link cubes along the
border of the rectangle. Note the importance of only counting the units whose edges touch the edge of the
object being measured. Review the importance of ensuring that units are lined up in a straight line and that
they touch sides but do not overlap.
Next, ask students what they would do if they only had one link cube to measure the distance around the
rectangle. Show students how to use only one unit and make marks to show where the unit starts and
finishes so that they can keep track of what they have measured.
As you line up the link cube, create a number sentence which encompasses the length and width of each
side, e.g., ____ + ____ + ____ + ____ = ____
Ask students if they know what the mathematical term is for measuring distance around an object. If they do
not say perimeter, introduce the term and explain that when measuring perimeter, they are measuring the
“outside edge of any area.” As you explain, use the rectangle from before and with a marker/chalk, go over
the outside edge of the shape to reinforce the concept. You may also want to write this on a sentence strip
and post it somewhere in the classroom for easy student reference.
Challenge students to come up to the board to create two shapes, one which would have a smaller
perimeter, and one with a larger perimeter than the rectangle used to demonstrate the concept of perimeter.
Other students can predict what the perimeter of each is and then test and measure the shapes. Encourage
students to write corresponding number sentences to find each shape’s perimeter.
Be prepared to address concerns about half units and get students to think of solutions. Possible solutions
could be to write that the perimeter is about X units, while some students may realize that if they have two
halves, that makes a whole and they would add it to the total of units.
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Activities:
1. Have students work in partners to measure the distance around various objects in the classroom using
link and/or unit cubes and check each others measurements for discrepancies.
2. Give students link cubes and ask them to create various shapes with a perimeter of 12 cubes.
3. Have students create fences for fields by using 4 pieces from a tangram set (two small triangles, medium
triangle, and square). Challenge them to make different shaped fields with the same four pieces. Have
them measure the perimeter with a link cube. Does the P remain the same? Why or why not?
Literature/Cross Curricular Connection:
How big is a foot? R. Myller
(The King wants to order a bed for his Queen but beds have not yet been invented. Begin the story and stop
at the point of figuring out how big the bed should be. Have students brainstorm how to solve this dilemma.
They should be focusing on figuring what perimeter the bed should be. Encourage them to use actual size to
solve the problem, and then give them grid paper to record a solution with a partner. They then will write a
letter to the King’s apprentices to explain their work and thinking. Have a group discussion to compare pairs’
solutions and then finish reading the story to them to find out how the characters solved the problem.)
Extensions:
1. What unit of measurement would students use to measure the distance around the classroom? What
would be most effective and efficient? Find out the perimeter of the classroom. (Giant steps?)
2. Have students draw a square that has a perimeter of 12 cubes. Have them figure out the length of each
side. Next, tell them to draw a rectangle that has a perimeter of 12 cubes and tell what the length of each
of the sides will be. Finally, have them draw a triangle with a perimeter of 12 cubes and have them figure
out the length of each of the sides.
3. Paul bought 13 bushes to place around the perimeter of his yard (shown in the diagram below).
For each edge of the diagram, he planted one bush. He measured the perimeter before going to the
nursery but he thinks he made a mistake because he doesn’t have enough bushes. Can you help him?
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ME3-12 Perimeter Goals: Students will estimate and measure perimeters using the standard and non-standard units.
Prior Knowledge Required: What measurement means.
Practice and comfort with linear measurement and
non-standard unit use.
How to count.
Vocabulary: perimeter, around, area, cm
Write the word “perimeter” and explain to your students that perimeter is the measurement around the
outside of a shape. Illustrate the perimeters of some classroom items; run your hand along the perimeter
of a desk, the blackboard or a chalkboard eraser. Write the phrase “the measurement around the outside
of a shape.”
Draw this figure:
1 cm
Explain that each edge of the squares represents 1 cm, and that perimeter is calculated by totalling the
outside edges. Demonstrate a method for calculating the perimeter by marking or crossing out each edge
as it is counted. Demonstrate this several times.
Examine the figure again. Squares have four sides, so why is the perimeter only 10 cm and not 16 cm
(4 × 4)? Remind your students that perimeter is the measurement around the outside of a shape.
The squares on the ends each have three outside edges. The squares in the middle have two outside
edges. The inside edges—sides that touch—are not totalled in the perimeter. Refer your students to the
definition again.
Ask them why they might want to know the perimeter of an object. (To border a picture with ribbon? To wrap
a present? To fence in a garden?)
Demonstrate another method for calculating perimeter by counting the entire length of one side, instead
of counting one edge at a time, then adding the lengths. Remind your students that addition is a quick way
of counting.
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Activity: Distribute one piece of string, about 30 cm long, and a geoboard to each student. Have them tie
the ends of the string together to form a loop, and then create a variety of shapes on the geoboard with the
string. Explain that the shapes will all have the same perimeter because the length of the string, which forms
the outside edges, is fixed. How many different shapes can all have the same perimeter?
Extension: Distribute Pentamino pieces (a set of twelve shapes each made of five squares – see the
BLM) to your students and have them calculate the perimeter of each shape. Create a table and order the
perimeters from smallest to greatest. Have students also calculate the amount of square edges inside each
shape. Can they notice a pattern emerging in the table?
Shape Perimeter Number of Inside Edges
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ME3-13 Exploring Perimeter Goals: Students will measure perimeters of given and self-created shapes.
Prior Knowledge Required: Perimeter
Practice and comfort with linear measurement and
non-standard unit use
How to count
Add strings of 1-digit numbers
Vocabulary: perimeter, around, grid paper
Review the perimeter, its definition and how it is calculated by totalling the outside edges of a figure.
Demonstrate the method for calculating perimeter by counting the entire length of each side and creating an
addition statement. Write the length of each side on the picture. Draw several figures on a grid and ask your
students to find the perimeter of the shapes. Include some shapes with sides one square long, like the shape
in the assessment exercise, as students sometimes overlook these sides in calculating perimeter. Ask your
students to draw several shapes of their own design on grid paper and exchange the shapes with a partner.
For the last exercise, suggest that students draw a letter, or simple word (like CAT), or their own names.
Assessment:
Write the length of each edge beside each edge and count the perimeter of this shape.
Do not miss any sides—there are ten!
2 cm
Extensions:
1. Explain that different shapes can have the same perimeter. Have your students draw as many shapes as
they can with a given perimeter (say ten units). (From the Western Curriculum)
2. Can a rectangle be drawn with sides that measure a whole number of units and have a perimeter…
a) of seven units?
b) with an odd number of units? [Both are impossible.]
Students can use a geoboard rather than grid paper, if preferred.
3. Create three rectangles with perimeters of 12 cm. (Remember: A square is a rectangle.)
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ME3-14 Investigations Goals: Students will estimate and measure perimeters of given and self-created shapes in standard units.
They will also predict the perimeter of figures in growing patterns.
Prior Knowledge Required: Perimeter
Practice and comfort with linear measurement and
non-standard unit use
Patterning
Add strings of 1-digit numbers
Vocabulary: perimeter, around, grid paper
Draw this figure:
and have your students demonstrate the calculation of the perimeter by totalling the outside edges.
Then draw the same rectangle without the inside edges:
and ask them how they could calculate the perimeter again. Explain that it can be measured, and then have
volunteers measure each side with a metre stick.
Draw several shapes on the board, as shown below:
Ask your students to estimate the sides of the shapes using natural benchmarks (remind them that the
widths of their hands are about 10 cm). Then ask your students to measure the sides and to find the
perimeter of each shape.
Review appropriate units. Have students determine the best units of measurement for calculating the
perimeters of the schoolyard, a chalkboard eraser, a sugar cube, etc. Point out to your students that if you
had used 1 cm to represent a kilometre in the drawings above, their measurements would have to be
expressed in kilometres.
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Draw a growing pattern on the board:
As a class, find the perimeter of each shape in the sequence. Examine the number of squares and then ask
your students how the perimeters change with the addition of each square. [The perimeter increases by two.]
Why does the perimeter only change by two, even though each added shape has four sides? Reiterate that
perimeter is the measurement of outside edges only. Every time a new square is added to the sequence it
covers one of the edges that had previously been on the outside.
Invite volunteers to extend the sequence and have your students predict the perimeter of the three
subsequent shapes. Draw each shape to check the predictions.
Draw the following shape:
Ask a volunteer to find the perimeter. Ask another volunteer to add a square to the figure (so that at least
one edge of the square is coincident with an edge of the figure) and calculate the new perimeter. Ask
students to repeat this exercise by adding squares in different positions. Summarize the results in a table. (It
is also good to mark congruent shapes. Do they have the same perimeter?) Why does the perimeter change
the way it does?
How many edges that had previously been on the outside of the figure are now inside? Are there any
positions where you can add a square so that the perimeter does not change?
Present a word problem:
Sally wants to arrange eight square (1 m × 1 m) posters into a rectangle. How many different rectangles can
she create? She plans to border the posters with a trim. For which arrangement would the border be least
expensive? Explain how you know.
Ask your students to create different rectangles from 8 squares. They can use blocks, grid paper or
geoboards. What should Sally do to find how much material she needs for the border? (Find perimeter) Ask
your students to find the perimeter of each rectangle. Tell your students that material for the boarder costs 5
cents per metre.
How much does the border cost for each arrangement?
Assessment:
1. Add a square to this shape so the perimeter’s measurement…
a) …increases by two. b) …remains the same.
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2. Measure the sides of the shape and find its perimeter.
TEACHER: Draw the lengths of all sides in whole centimetres.
Extensions:
1. A hexagon has equal sides 5 cm long. What is its perimeter?
2. On grid paper, draw 3 different figures with perimeter 8. (The figures don’t have to be rectangles.)
3. The sides of a regular pentagon are all 7 m long. What is the pentagon’s perimeter?
4. Will 500 toothpicks, placed end to end be enough to cover perimeter of your school? First, estimate the
length and the width of the school with giant steps. Draw the shape of the school building and find the
perimeter. Estimate and measure the length of a toothpick. What is the length of 100 toothpicks? Of 500
toothpicks? As a challenge, approximate the perimeter of the school in toothpicks.
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ME3-15 Measuring Mass Goals: Students will estimate and measure perimeters of given and self-created shapes in standard units.
They will also predict the perimeter of figures in growing patterns.
Prior Knowledge Required: What is mass
Gram
kilogram
Ordering from greatest to least, from least to greatest
Vocabulary: gram, kilogram, mass
NOTE: Mass is a measure of how much substance, or matter, is in a thing. Mass is measured in grams and
kilograms. A more commonly used word for mass is weight: elevators list the maximum weight they can
carry, package list the weight of their contents, and scales measure your weight. The word weight however,
has another very different meaning. To a scientist, weight is a measure of the force of gravity on an object.
An object’s mass is the same everywhere—on Earth, on the Moon, in space—but it’s weight changes
according to the force of gravity. When we use the term weight in this and subsequent lessons, we use it as
synonym for mass.
Remind students that mass (which we often call weight) is measured in grams (g) and kilograms (kg). Give
several examples of things that weigh about 1 gram or about 1 kilogram:
1 g: a paper clip, a dime, a chocolate chip
1 kg: 1L bottle of water, a bag of 200 nickels, a squirrel.
List several objects on the board and ask students to say which unit of measurement is most appropriate for each one—grams or kilograms:
• A whale • A cup of tea
• A table • A workbook
• A napkin • A minivan
Ask students to match these masses to the objects above:
2000 kg 50 000 kg 10 g 150 g 400 g 10kg
Have students order these objects from heaviest to lightest.
Ask students think of three other objects that they would weigh in grams and three objects that would
demand kilograms.
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Activities:
1. Let students feel the weight of objects that are close to 1 g (EXAMPLE: paperclip) or 1 kg (EXAMPLE: a
1 L bottle of water). They can use these referents to estimate the weight of the objects in the classroom,
such as books, erasers, binders, games and calculators. (You could ask students to order the objects
from lightest to heaviest.) Students should use scales to weight the objects and check their estimates.
2. Weigh an empty container, then weigh the container again with some water in it. Subtract two masses to
find the mass of the water. Repeat this with a different container but the same amount of water. Point out
to students that the mass of a substance doesn’t change even if its shape does.
Extension: Jane wants to estimate the weight of one grain of rice. She weighs 100 grains of rice and
divides the total by 100. Try to weigh 1 grain of rice. Explain why Jane uses this method above. Use Jane’s
method to estimate the weight of a bean or a lentil.
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ME3-16 Measuring Capacity Goals: Students will determine the capacity of various containers.
Prior Knowledge Required: Litres
Millilitres
Half, quarter
Vocabulary: litre, millilitre, capacity
Explain that the capacity of a container is how much it can hold. Write the term on the board. Explain that
capacity is measured in litres (L) and millilitres (mL). Explain that there are 1 000 millilitres in 1 litre.
Write on the board: 1 litre = 1 000 millilitres 1 L = 1 000 mL
Put out several containers (EXAMPLES: milk and juice boxes, medicine bottles, measuring cups, cans of
paint, cans of pop) with capacities clearly marked on them. Invite students to help you separate the
containers into two groups: those that can hold 1 or more litres and those that can hold less than 1 litre. Then
ask students to help you order the containers by capacity, from least to greatest. The containers that hold 1L,
500 mL or 250 mL can also act as “capacity benchmarks” that you can keep in a class measurement box.
Write the following on the board and ask students whether they would measure the capacity of each
container in millilitres or litres:
• a glass of juice
• a bowl of soup
• a pail of water
• a pot of soup
• an aquarium
• a backyard pool
Ask students to think of three more quantities that are measured in litres and three that are measured in
millilitres.
Review the concepts of halves and quarters. ASK: How many halves make one whole? How many quarters
make one whole? Show your students an empty 1 L milk carton but don’t tell them that its capacity is 1 L.
Ask a volunteer to find the capacity written on the carton. Next show your students an empty 500 mL milk
carton with the capacity marking covered up. Tell students that you want to find the capacity of this carton
using the larger carton. Fill the smaller carton with water and empty it into the 1 L carton. Invite a volunteer to
look into the larger carton and indicate on the outside (using a finger or marker) how much water is inside.
ASK: About how much water is now in the carton? Refill the small carton and empty if into the larger carton
again. Now the carton is full. ASK: How many smaller cartons did we need to fill the 1 L carton? What is the
capacity of the smaller carton in terms of parts of a litre? (one half) Repeat with a 250 mL carton in place of
the 500 mL carton.
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Show students a small (200 mL) glass. ASK: How can we determine the capacity of the glass in millilitres
using the 1 L carton? You might ask a volunteer to check how many times they can fill the glass with water
from the carton. Then ASK: What is the capacity of the milk carton in millilitres? How many times did we fill
the glass? What should we do to the capacity of the large carton to get the capacity of the glass? What is the
capacity of the glass? (NOTE: You will need a large bowl or pot in which to empty out the glass each time it
is filled.)
After you have determined the capacity of the cartons and the glass, ask your students to order the
containers according to capacity, from smallest to largest.
Activities:
1. Measure the capacity of several glasses or containers in your classroom. Estimate the capacity of the
containers before you measure their capacity. Students should select and justify appropriate units to
measure the capacity of a container. NOTE: Students will need a measuring cup and several containers
for this activity.
2. Ask students to fill a measuring cup marked in millilitres and estimate how many cups they would need to
fill a 1 L container. Then, knowing the number of millilitres their cup holds, they should estimate how
many millilitres are in a litre. Students could then test their prediction by filling up the litre container and
keeping track of how many millilitres they need.
3. Give students several glasses of containers on which the capacities have been covered or removed. Ask
them to estimate the capacity of each container and then check their answers. Remind students to
include the appropriate units (mL or L) with their estimates.
4. Weigh a measuring cup. Pour 10 mL of water in the cup. Calculate the mass of the water by subtracting
the weight of the cup from the weight of the water with the cup. How much do 10 mL of water weigh?
(How much does 1 mL of water weigh?)
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ME3-17 Measuring Temperature Goals: Students will learn to read a thermometer and will identify given temperatures as hot or cold.
Prior Knowledge Required: Thermometer
Proficiency in using number lines
Degree Celsius
Vocabulary: thermometer, degree Celsius
Hold up a large thermometer and ask students to identify what it is. ASK: What do we measure with a
thermometer? (temperature) Where and when do people use thermometers? (to measure the temperature of
a home, a greenhouse, the outdoors, the fridge or freezer, the body, and so on)
Explain how a thermometer works. The liquid inside the bulb goes up and down depending on the
temperature. It goes up as the temperature gets warmer, and down as it gets colder. You can demonstrate
this using cups of water at different temperatures. Fill one cup with hot water, one with water at room
temperature, and another with cold water and ice. Invite a volunteer to touch the cups and to order them from
coldest to hottest. Then insert the thermometer into each cup and let students observe the movement of the
liquid in the thermometer.
Explain to your students that temperature is measured in units called degrees. In Canada, we use degrees
Celsius, which can also be written like this: °C. Ask your students to observe the numbers alongside the liquid
in thermometer. What do they notice? Explain that the zero on the thermometer indicates the freezing mark.
This is the temperature at which water freezes. When the liquid in the thermometer drops below the zero, we
say the temperature is “below freezing.” You might point out that water becomes ice below 0°C, but other
liquids—like the liquid inside the thermometer—stay unfrozen.
Draw several thermometers on the board and shade them in to show different temperatures. (Only the zero
mark needs to be indicated.) Ask students to tell whether the thermometers are showing temperatures above
or below freezing.
Draw a thermometer on the board and divide it into five sections as shown. Ask your
students to think of words that describe different air temperatures (words they may
suggest include: freezing, cool, warm). Record the words and have students order
them from coldest to hottest. ASK: When it’s so cold outside that water turns to ice,
how would you describe the temperature? (possible answers: very cold, freezing
cold) Ask a volunteer to identify the part of the thermometer where the temperatures
are below freezing. The temperatures in this section can be labelled “freezing cold.”
Ask student volunteers to label the temperatures in the other sections on the
thermometer as cold, cool, warm, and hot.
As the liquid in the thermometer rises above zero, the temperature get warmer, so we count up. The larger
the number above zero, the warmer it is. As the liquid in the thermometer drops below zero, the temperature
0°C
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gets colder, so we put a minus sign in front of the number and count up. The larger the number below zero,
with the minus sign, the colder it is.
Ask students to help you add a scale to the thermometer on the board. Ask them to count up by tens from
zero, and mark the degrees on the thermometer so that each mark sits on the boundary between two
sections. Write the range of temperatures beside each descriptive word: hot 30°C and up; warm 20°C to
30°C, cool 10°C to 20°C, cold 0°C to 10°C, and freezing cold 0° and below. Remind students that these
ranges and descriptions refer to the temperature of air.
Point to different temperatures on the thermometer (between the ten degree increments marked) and ASK: If
the liquid in the thermometer was here, how would you describe the temperature? Then ask students to think
about the temperature that is appropriate or necessary for various outdoor activities. For each activity, draw
two thermometers showing very different temperatures and ask students which one shows the temperature
that is more appropriate for the activity. EXAMPLES: water skiing: –10°C or 30°; berry picking: 0°C or 17°C;
ice hockey: 15°C or –3°C.
Draw part of a vertical number line on the board and mark every fifth increment: 0°C, 5°C, 10°C, 15°C.
Explain to your students that this is the enlarged scale of the thermometer. On a real thermometer there is no
room to write the other numbers, but you will teach them how to read the temperature even when these
numbers are missing.
Invite a volunteer to count up by fives from 0 to 10 and by ones from 10 to 15. Then point at the mark
showing 12°C and explain that, to read this temperature, people first count by fives and then continue by
ones until they reach the mark. Invite students to count up with you until you reach 12°C. Repeat with several
examples until all of your students are able to count first by fives then by ones to any mark you indicate. If
necessary, separate the tasks – ask students who have trouble counting by both fives and then ones to
count by fives only or to count by ones only from the nearest marked increment. Ask your students to tell
whether the weather outside is hot, cold, warm, or cool for each of the temperatures you show.
Explain that water feels differently than air at different temperatures. For instance, what feels colder – the air
on your face when it is 0°C outside or a piece of ice that you hold in your bare hand? Students can do the
activity below to compare the temperatures of water and air.
ASK: Is –5˚C hot or cold? Is water at this temperature a liquid or is it ice? Explain that if you have a piece of
ice at –5˚C, you have to heat it for some time to make it melt. How much do you have to heat it? (enough to
raise its temperature by 5˚C, to 0˚C)
Write several temperatures on the board. Ask students to describe each temperature and to tie it to familiar
events or activities, as well as states of water, such as:
–10˚C – freezing cold; skating, skiing
0˚C – cold, water freezes
5˚C – cold, water is nearly frozen, ice has melted
20˚C – air is warm, water still cold; bike riding
36˚C – air is very hot, near to normal human body temperature; swimming, sitting in the shade
50˚C – very hot, desert (You can bake eggs in the sand!)
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Add to this list the range of temperatures for each season where you live, to give students the information
they need to complete the worksheet.
Have students solve several problems in which they have to add or subtract to find the temperature.
SAMPLE PROBLEMS:
• The temperature today is 15˚C. Yesterday it was 5˚C higher. What was the temperature yesterday?
• The average temperature in winter is –10 ˚C. The average temperature in summer is 25˚C. What is the
difference between the average temperatures in winter and in summer? (Let students use a
thermometer to solve this problem.)
Assessment:
a) What was the temperature on Monday?
b) What was the temperature on Wednesday?
c) How much warmer was in on Monday than on Wednesday?
d) A jar of water was left outside. On which day did the water
turn to ice?
Activity: Students will need thermometers. Give students some hot tap water. (NOTE: The water
temperature should not be greater than 49°C for safety reasons.) Let your students measure its temperature.
Add some cold tap water, let students feel the water to estimate its temperature, and then have students
measure the temperature. Repeat so that students can feel and measure the temperature of water at
different temperatures. Student can describe the various temperatures of water as they did the temperatures
of air during the lesson, and then compare them. For example, how does at air at 36˚C compare to water at
or near that temperature?
Extension: Explain that the scale of the thermometer is based on the properties of water, so that
anyone can build his or her own thermometer. Explain that 0˚C is the temperature at which water freezes
and 100˚C is the temperature at which water boils. So to make a new thermometer, you need to freeze some
water to find the 0˚C mark and then boil some water to establish the 100˚C mark. Then you divide the scale
into 100 parts and your thermometer is ready. You can illustrate the process while drawing a thermometer on
the board.
– 25
0°C
10
20
30
5
15
25
– 15
– 5
– 20
– 10
– 25
0°C
10
20
30
5
15
25
– 15
– 5
– 20
– 10
Monday Wednesday
Workbook 3 - Measurement, Part 1 1BLACKLINE MASTERS
Map of Saskatchewan ___________________________________________________2
Pentamino Pieces _______________________________________________________3
ME3 Part 1: BLM List
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Map of Saskatchewan
Workbook 3 - Measurement, Part 1 3BLACKLINE MASTERS
Pentamino Pieces
Probability & Data Management Teacher’s Guide Workbook 3:1 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-1 Introduction to Classifying Data
Goals: Students will group data into categories. Students will identify attributes shared by all members
of a group.
Prior Knowledge Required: The difference between “and” and “or”
Vocabulary: data, classify, attribute, and, or
Have eight volunteers stand up. Ask the students to suggest ways in which to classify them (for example,
long or short hair, boy or girl, nine or ten years old, wearing jeans or not wearing jeans, wearing yellow or
not wearing yellow). Then have one student classify the eight volunteers into two groups without telling the
class how he or she chose to classify them. The student tells each of the eight volunteers which side of the
room to stand on. Each remaining student then guesses which group he or she belongs to. (If the student
guesses incorrectly, the student who classified the volunteers moves that student into the right group but
does not reveal the classification.) Stop when five consecutive students have guessed correctly. The last
student to guess correctly appoints each remaining student in the class to either of the two groups and is
told if they’re right or not.
Repeat this exercise several times, with different students doing the classifying. Note that the student
doing the classifying never reveals the classification. To make guessing the classification harder, students
may decide to combine attributes, such as grouping “boys not wearing yellow” and “boys wearing yellow
and girls.”
Write the following words on the board:
J.K. Rowling lion Alberta Ottawa Anna Klebanov
dog cat Canada mouse Rita Camacho
Ask students to put the words into the following categories:
People Places Animals
Have students add more words to each category. Then tell students that the category “People” could be
divided further. For example: adults and children, girls and boys, first language English or first language
other. Have students suggest other ways to categorize people and have volunteers put their own names in
the appropriate categories. Then do the same for Places and Animals.
Have students think of categories for the following data:
Weather (e.g., sunny, cloudy, rainy)
Time of day (e.g., morning, afternoon, evening, night)
Foods
Fruits
Vegetables
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Have students identify attributes shared by all members of the following groups:
a) grey, green, grow, group (EXAMPLE: starts with “gr,” one-syllable word)
b) pie, pizza, peas, pancakes (EXAMPLE: food, starts with p)
c) 39, 279, 9, 69, 889, 909 (EXAMPLE: odd number, ones digit 9, less than 1000)
d) 5412 9807 7631 9000 9081 (EXAMPLE: number, 4 digits, greater than 5000)
e) hat, cat, mat, fat, sat, rat (EXAMPLE: three-letter word, ends in “at”) Have students write 2 attributes for the following groups:
a) 42, 52, 32, 62, 72 b) lion, leopard, lynx
Activity: Show students cards from the game SetTM
and ask students to say what categories can be used
to sort the cards (e.g. shape, colour, number, shading) and then to sort the shapes using those categories.
Have groups of students play the card game SetTM
. Include only the solid shapes (no stripes or blanks) until
students are very comfortable finding sets.
Extensions:
1. Which 2 attributes could have been used to sort the items into the groups shown?
Group 1:
Group 2:
� at least 1 right angle � 4 sides or more � 2-D shapes � quadrilaterals
� no right angles � 3-D objects � 3 sides � hexagons
Which categories do all shapes in both groups belong to? Which categories do no shapes in either group
belong to? What categories do some shapes in group 1 and some shapes in group 2 belong to? What
category do all shapes in group 1 and no shapes in group 2 belong to? What category do no shapes in
group 1 but all shapes in group 2 belong to? How were these groups categorized?
2. Give pairs of students the BLM “Shapes” (it shows 8 shapes that are either big or small, triangles or
squares) or the Set™ card game. The BLM has 2 of each shape, so that students can work individually.
Ask students to separate the small triangles from the rest of the shapes, then challenge them to
describe the remaining group of shapes (big or square). Encourage them to use the words “or” or
“and” in their answers, but do not encourage use of the word “not” at this point since, in this case,
“not small” simply means “big.” If students need help, offer them choices: big or triangle, big and
square, big or square.
Challenge students to describe the remaining group of shapes when they separate out the:
a) small squares e) shapes that are big or triangular
b) big squares f) shapes that are square or big
c) big triangles g) shapes that are both square and big
d) shapes that are small or triangular h) shapes that are both triangular and small
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PDM3-2 Venn Diagrams
Goals: Students will sort data using Venn diagrams.
Prior Knowledge Required: Identifying attributes
Classifying according to attributes The words “and,” “or,” and “not”
Vocabulary: Venn diagram, or, and, not
Label a cardboard box with the words “toy box,” then ask individual students if various items (some items
toys and some items not toys) belong inside or outside the box. Then tell students that you want to classify
items as toys without having to put them in a box. Draw a circle, write “toys” in the circle, and tell your
students that you want all of the toy names written inside the circle and all of the other items’ names written
outside the circle. Ask your students to tell you what to write inside the circle. Make sure to crowd the words.
ASK: Is there a way we can save space and not write the whole word in the circle? If students suggest you
write just the first letter, ask them what will happen if two toys start with the same letter. Explain to them that
you will instead write one letter for each word. (EXAMPLE: A. blocks B. bowl C. toy car D. pen)
ASK: Which letters go inside the circle, and which letters go outside the circle?
Draw several shapes and label them with letters.
A. B. C. D. E. F.
Draw:
shapes
ASK: Do all the letters belong inside the box? Why? Which letters belong inside the circle? Which letters
belong outside the circle but still inside the box? Why is the circle inside the box? Are all of the triangles
shapes? Students should understand that everything inside the box is a shape, but in order to be inside the
circle the shape has to be a triangle. Change the word inside the circle and repeat the sorting exercise.
triangles
toys
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[Suggested words to use include: dark, light, quadrilaterals, polygons, circles, dark triangles, dark circles.]
Ask your students why the circle is empty when it’s labelled “dark circles.” Can they think of another property
that would empty the circle of all the shapes? Remind them that the word inside the circle should reflect the
fact that the entire box consists of shapes, so “rockets” probably isn’t a good word to use, even though the
circle would still be empty.
Lay two hula hoops on a table or on the floor. Clearly label one hula hoop “pens” and another “blue,” then
ask your students to assign several coloured pens (black and red) and pencils (blue, red, and yellow) to the
proper position—inside either of the hoops or outside both of them. Do not overlap the hoops at this point.
Then present your students with a blue pen and explain that it belongs in both hoops. ASK: How can we
move the hoops so that the blue pen is circled by both hoops at the same time? Ask a student volunteer to
show the others how it’s done. If the student moves the hoops closer together without overlapping them, and
positions the blue pen such that part of it is in the “blue” hoop and part of it is in the “pens” hoop, be sure to
ask them if it makes sense for part of the pen to be outside of the “pens” hoop. Shouldn’t the entire pen be
circled by the “pens” hoop?
Draw:
A. B. C. D. E. F.
And draw:
Dark Triangles
Explain to your students that this is called a Venn diagram. Ask them to explain why the two circles are
overlapping. Have a volunteer shade the overlapping area of the circles and ask the class which letters go in
that area. Have a second volunteer shade the area outside the circles and ask the class which letters go in
that area. Ask them which shape belongs in the “dark” circle and which shape belongs in the “triangle” circle.
Change the words representing both circles and repeat the exercise. [Suggested words to use include: light
and quadrilateral, light and dark, polygon and light, circles and light.]
This is a good opportunity to tie in concepts from other subjects. For example, students can categorize words
by the beginning or ending letters, by rhyme patterns, or by the number of syllables. (EXAMPLE: “rhymes
with tin” and “2 syllables”: A. begin; B. chin; C. mat; D. silly). Start with four words, then add four more words
that belong to the same categories. Encourage students to suggest words and their place in the diagram.
[Cities, provinces, and food groups are also good categories.]
Activity: Create a big circle on the floor with masking tape and label it “8 years old”. Have students stand
in the circle if they are 8 years old and have them stand outside the circle if not.
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Repeat with several other properties:
Wearing Girl Blue Takes the
yellow eyes bus to
school
Have your students stand inside the circle if the property applies to them. Have them suggest properties and
try to move appropriately (either to the circle or away from it) before you finish writing the words. Students
should learn to strategically pick properties that take a long time to write. Repeat the activity by overlapping
two circles.
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PDM3-3 Introduction to Tallying Data
Goals: Students will switch between numbers and their corresponding tallies.
Prior Knowledge Required: Ability to count
Number recognition
Counting by 5
Vocabulary: tally marks, tallies, diagonal, vertical
Explain to students that experts in mathematics and other subjects often use tally marks to keep track of
what they are counting. Tell students that you want to count the number of boys in the room. Draw a vertical
stroke each time you say a boy’s name:
IIIIIIIIII
Then repeat using diagonal lines for each fifth boy:
IIII IIII
ASK: Which set of marks is easier to read? Explain that in a tally, each vertical stroke represents one, but
every fifth stroke is drawn diagonally across the first four. This makes it easy to count by fives.
Then write the numbers 1–5 on the board, making sure they are well spaced out. Under the number 1, draw
the corresponding tally: one vertical stroke. Ask students what they think the tally for the number 2 will look
like. Have a volunteer write that on the board. Repeat with the numbers 3 and 4. For number 5, remind
students that tally marks are grouped in fives and we draw the fifth line diagonally across the other four—the
fifth line “bundles” the other four together.
Next, write the numbers 6–10 and show students the tally for the number 6. Ask volunteers to write the
tallies for the numbers 7–9. For the number 10, ask students to predict what the tally will look like and then
show them.
Repeat the process with numbers 11–15 and then 16–20.
Now show students the tally for the number 4 and ask them to identify it. Then do 5, 8, 12, 15, and others
until all students can quickly and easily read the tallies.
Next, draw 5 apples on the board and remind students that tallies are useful for tracking data. Cross out an
apple, and draw a tally mark. Ask a volunteer to continue the process. Repeat with an array of 12 circles.
Bonus Questions: Who can count quickly? Show 15 as a tally. Then show 20, 25, 35… and 100!
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Activity:
Paper Clip Search. Place enough paper clips around the room to average about 10 per student. Have
students find and collect as many paper clips they can. Show students how to use 1 paper clip to bundle
another 4:
Ask students how many paper clips they have in total. They should use the bundles of 5 to count by 5s. Then
have students pair up and find out how many the pair has in total; they may need to bundle once more.
Extension: Show students a penny and ask them to show what the penny is worth with tally marks.
Continue showing tallies for a nickel, a dime, a quarter, a loonie, and a toonie.
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PDM3-4 Reading Data from a Tally Chart
Goals: Students will compare, order, add, subtract or multiply data to find new information
Prior Knowledge Required: Tallies
Addition, subtraction, and multiplication by small numbers
Vocabulary: add, subtract, multiply, twice as many, three times as many
Take a survey of your students and ask what type of fruit is their favourite among: apples, bananas,
oranges, grapes, and peaches. Record the tallies yourself and have students complete the chart by writing
the appropriate number for each tally.
Fruit Apples Bananas Oranges Grapes Peaches
Tally
Number
Ensure that students can read the data directly from the tally chart. ASK: How many students like grapes
best? How many like apples best? Oranges? Bananas? Peaches?
Tell students that they can compare and order the data to find new information. ASK: Did more people
like oranges or bananas better? Which fruit was the most popular? Which fruit was the least popular?
Working independently, have students list the fruits in order from most popular to least popular. Ask them
to explain their thinking. (PROMPTS: How did you know which fruit to put first? How did you decide which
one came next?) Tell students that they can add the data values together. ASK: How many students
answered the survey? Was it everyone in the class? How many students are in the class? How many
students liked either apples or grapes best? Bananas or oranges? Peaches or oranges or apples?
Bonus: Combine comparing and ordering with addition. For example, ASK: Did more people like bananas
best than liked peaches and grapes best?
Tell students that sometimes they need to subtract data to find new information. ASK: How many more
students liked apples best than peaches best? (Or “peaches best than apples best” if appropriate). Make
more such comparisons. Then ASK: Of all the people who answered the survey, how many did not choose
oranges? What number do we need to subtract from what to figure this out? How many students did not
choose bananas? Can we answer these questions without using subtraction? (Yes, we can add the other
four data values.)
Bonus: How many chose students neither bananas nor oranges? Did you use addition or subtraction or both?
Tell students that sometimes they need to multiply data to find new information. ASK: Is any fruit twice as
popular as another fruit? Is any fruit three times as popular (or almost three times as popular) as another
fruit?
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PDM3-5 Introduction to Pictographs
Goals: Students will read and interpret pictographs that have a scale of 1.
Prior Knowledge Required: Ability to count
One-to-one correspondence
Understanding of symbols
Comparing and ordering numbers
Addition, subtraction, multiplication
Vocabulary: pictograph, key, symbol, more, less, least
ASK: What is a symbol? Allow students to discuss this in pairs and then debrief as a class. If students need
hints, remind them of maps—what pictures/symbols they have seen there? What are some of our country’s
symbols? (beaver, maple leaf, loon, etc.)
Tell students that they will learn how to show data in a pictograph today. Explain that pictographs use
symbols which represent the data in order to show how many of each set of data there are. Explain that the
symbol should match who is being asked the question or what is being asked about.
Display this data on the board:
What time do students go to bed during the week?
Before 8:30 x
8:30 x x x x
9:00 x x x x x x x
9:30 x x x x
After 9:30 x x
Ask students what symbols they could use to show the above data and choose one. (Suggestions include: a
stick person, a pillow.) Invite a volunteer to replace the Xs with the new symbol.
Next, introduce the word “key” to students. Draw the symbol chosen for the above example and an equal
sign next to it. Ask students what each symbol in the chart represents (one student). Write the number 1 next
to the equal sign. Explain that this is the key and it tells us what each symbol represents. In this case, each
symbol represents or is equal to one student. Tell students that all pictographs usually include a key.
Encourage students to talk about what the data tells us. Encourage them to make comparisons and to ask
questions. ASK: How many students go to bed at 8:30? How many go to bed after 9:30? What is the most
popular bedtime? How many students were surveyed? (Show students how to count the symbols to find out.)
How many more students go to bed at 9:00 than after 9:30? How many more students go to bed at 9:00 or
earlier than at 9:30 or later? How many students go to bed at 8:30 or later? To answer this last questions, did
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they add all the data values or did they subtract from the total number of students surveyed that they found
earlier?
Discuss good symbols to show various data. Have students first choose the best possible symbol from a list
and then move to having them choose their own.
a) Number of books Tanya read in each season:
b) Number of sunny days in each season: c) Number of rainy days in each season. d) Number of people whose favourite fruit is apples, bananas, oranges, grapes or peaches.
ERRATA NOTE: The chart in the workbook should say “Number of Rainy Days,” not “Number of Sunny Days.”
Activities:
1. Have students create concrete graphs by collecting small items (e.g., leaves, shapes, beads, buttons)
and sorting them. A partner can interpret the data and write a few sentences about the items collected.
(EXAMPLE: Most of the buttons are big. There are more red beads than blue beads.)
2. Place students into small groups and give them a package of Smarties, M&Ms, or jelly beans. Ask
students to sort the candies (you can choose categories or they can), count how many are in each
group, record the data, and display it. Encourage students to analyze the data—what does it tell them?
What are there more of and less of? Why do they think that happened?
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PDM3-6 Pictograph Scale
Goals: Students will read pictographs that have a scale larger than 1.
Prior Knowledge Required: Pictographs that have scale 1
Vocabulary: symbol, key, scale, pictograph
Tell your students that you have a garden but you are very secretive about how many flowers are in it. You
want to make a pictograph of how many of each kind of flower you have. ASK: What is a good symbol to use
for the key? Tell your students that instead of using one symbol to mean one flower, you will use one symbol
to mean many flowers—that way, your students won’t know exactly how many flowers you have without
knowing your key.
Draw on the board:
Daffodil F F F F
Buttercup F F F F F
Daisy F F F
Tell your students that each symbolic flower could mean any number of actual flowers, but it’s always the
same number for each symbol. If the first symbol means 2 flowers, then all the symbols mean 2 flowers.
ASK: If each symbol means only 1 flower, how many flowers do I have in my garden? What numbers did you
add together to find the answer? What if each symbol means 2 flowers—then how many flowers would I
have? What strategy did you use to find the answer? (Allow several students to explain how they found the
answer, to illustrate the diversity of strategies. ) What if each symbol means 3 flowers? 4 flowers? 5 flowers?
ASK: If each symbol means 3 flowers how many daisies do I have?
If each symbol means 100 flowers, how many buttercups do I have?
If each symbol means 5 flowers, how many daffodils do I have?
If each symbol means 4 flowers, how many daisies do I have?
Bonus: “Accidentally” tell your students that you have 12 daffodils. Can they figure out the key?
Ask students whether or not it would make a difference if the data in the pictograph was presented vertically
instead of horizontally. Re-create the graph so that the symbols are stacked vertically, starting from the
bottom and moving upwards. Then erase each flower and draw an “x” in its place.
Ask students if this reminds them of any type of graph they saw last year. It should remind them of line plots.
Remind your students that they learned pictographs and line plots last year, but they always used the symbol
to mean only one object. Now they are using a symbol to mean more than one object.
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Tell your students that sometimes people will pick a symbol just because it’s easier to draw, and circles are
often a good choice. Draw the following pictograph and tell students that each circle means 4 flowers:
Number of Flowers
Daffodil
Buttercup Key: 1 means 4 flowers
Daisy
ASK: How is the circle easier to draw than the flower? Use the following questions to address the half circle:
If means 4 people, how many people does mean? What if means 6 people—then how many people
does mean? Repeat with the circle meaning a variety of even numbers. Then return to the pictograph
above and ask how many flowers of each type there are.
Draw the following pictograph, and ask how many people like each colour:
Number of People with Each Favourite Colour
Red
Blue
Yellow
Orange Key: 1 means 10 people
Green
Purple
Other
Write the number of people who like each colour as students tell you the answers. Then have students draw
a pictograph of the same data using a different key: 1 means 5 people. Discuss the similarities and
differences between the two pictographs. How would you tell from each graph which colour was most often
picked as favourite?
Explain that the data sometimes makes it easy to choose a scale. Ask students to choose a scale of two, five, or ten for the following data:
a) 12, 4, 10 b) 5, 25, 35 c) 40, 20, 70 d) 35, 50, 20 e) 16, 8, 14
Choose a scale of two, three, or five for the following data:
a) 3, 9, 18 b) 20, 10, 25 c) 6, 12, 15 d) 8, 18, 6 e) 40, 25, 30
Bonus:
f) 18, 15, 9, 21, 27, 30 g) 40, 105, 35, 70, 60, 95, 35 h) 40, 20, 36, 18, 24, 16, 32
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PDM3-7 Displaying Data on a Pictograph
Goals: Students will select a symbol for a pictograph that is easy to draw based on the choice of scale.
Prior Knowledge Required: The symbol on a pictograph can mean more than one object.
Vocabulary: scale, key, pictograph, symbol
Take a survey of your students’ favourite colours among the choices: blue, red, or yellow. Tally the results on
the board and then write:
Favourite Primary Colour
Blue
Red
Yellow
1 means 2 students
Have a volunteer complete the pictograph, then demonstrate how difficult it would be if you chose the same
symbol to mean 5 students—it is hard to draw one fifth of the happy face!
Brainstorm symbols that are easy to draw when you need to draw half a symbol and symbols that are easy
to draw when you need to show a fifth of a symbol. (EXAMPLES: flower with 5 petals, star with 5 points)
As in the last lesson, ask your students which scale they would use (5 or 10) for each data set below:
a) 80, 90, 95 b) 20, 10, 25 c) 120, 45, 90 e) 40, 25, 30
To guide your students, ASK: For the scale you used, do you need to draw a half symbol? How many whole
symbols would you need to draw? Emphasize that students don’t want to have to draw too many symbols.
It’s okay to use half symbols! Students should choose the symbol carefully so that they can draw half of the
symbol easily. Have students practice drawing a pictograph using the data from part c) above, a scale of 10,
and the symbol of their choice.
Discuss with your students what is wrong with the following pictograph:
Tell your students that one happy face represents one
student who picked that sport as their favourite.
ASK: Which sport is the most popular? Which sport has
the longest row of faces? Why is it easier to read the
graph when all the faces are the same size?
Tell your students that it is easier to make all the happy
faces the same size if they draw on grid paper. That
way, they can draw each happy face in one grid square.
Students should use 2 cm grid paper to make drawing
the objects easier.
Favourite Sport of Students in Class A
Soccer
Hockey
Basketball
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PDM3-8 Introduction to Bar Graphs
Goals: Students will read and interpret bar graphs.
Prior Knowledge Required: Pictographs
Scale of a pictograph
Vocabulary: bar, graph, bar graph, common
Review pictographs. Tell your students that there are other ways, besides pictographs, to show data. Write
the words ‘bar graph’ and ‘pictograph’ on the board side by side. Underline the word ‘graph’ in each. A
pictograph uses pictures to display data. Ask students how they think a bar graph will display data.
Create a pictograph and a bar graph for the same data ahead of time, or use the examples below. Identify
the parts of the bar graph: the two axes (vertical and horizontal), the scale, the labels (including the title), and
the data (shown in the bars).
What time do students go to bed during the week?
Before 8:30 x
8:30 x x x x
9:00 x x x x x x x
9:30 x x x x
After 9:30 x x
1 x = 2 students
Student Bed Times During the Week
0
2
4
5
8
10
12
14
16
Before
8:30
8:30 9:00 9:30 After 9:30
Times
Nu
mb
er
of
Stu
de
nts
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Remind students that a pictograph can be drawn vertically as well and have a volunteer draw a vertical
pictograph that shows the same data as the horizontal pictograph.
Allow your students to compare the bar graph and the vertical pictograph, then ask them if they think the
graphs show the same data. Explain to them that the scale of the bar graph expresses how much each
mark on the grid represents. Ask them what scale was used for the bar graph. Was the same scale used in
their pictograph? Instead of using a symbol, how does the bar graph represent two students? How would
the bar graph represent one student? How does the pictograph represent one student? Emphasize that a
bar graph is like a pictograph that uses one grid square as a symbol, and that grid square can mean any
number of objects, just like the symbol on a pictograph. The nice thing about the grid square as a symbol, is
that it is easy to draw half a symbol, or a third of a symbol, or a quarter of a symbol—just use a ruler to
measure half the height or a third of the height or a quarter of the height.
Half a symbol A third of a symbol A quarter of a symbol
Just as there is usually space between the pictures on a pictograph, there is always a space between bars
on a bar graph.
Draw students’ attention to the height of the bar and the number where the bar stops. Explain that this shows
how many students answered in each of the categories. Ask students how many students responded in each
category. Change the data and repeat to ensure that students are able to read the vertical axis correctly.
Then move on to information that follows more indirectly from the bar graph, such as: How many students go
to bed at 9:00 or earlier? How many students do not go to bed before 8:30? How many more students go to
bed at 9:00 than at 9:30? Have students indicate the concepts and/or operations they used to answer the
questions—addition, subtraction, comparing, and/or ordering. Challenge students to think of a question that
would require multiplication.
Next, introduce ‘common’ by writing it on the board and ask students to define it. Then, ask them what the
most and least common answer was to the question “What time do you go to bed?” Prompt discussion by
asking why 9:00 is the most common bedtime? In other cases, students may use the word ‘popular’ to
describe data such as favourite foods or sports. Discuss this as a group and explain why it would not be
accurate to say that the most common bedtime is the same as the most popular bedtime.
Activity: Ask students to discuss, and then write everything they can about, the data displayed in the
following bar graph. There is no title on the graph. This is done on purpose, so that students may interpret
the data in different ways. Discuss the importance of a title on a graph and how it clarifies what the data
represents even further than the labels on the bars.
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0
1
2
3
4
5
6
7
8
dogs cats fish none
Literature Connection:
Tikki Tikki Tembo by A. Mosel
(This is a Chinese folktale with a moral which lends itself well to student participation through chanting.
A boy’s long name causes a dilemma.)
Survey the class about how many letters there are in each of the students’ names. Graph the data.
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PDM3-9 Bar Graphs and PDM3-10 Bar Graphs (Advanced)
Goals: Students will read, interpret, and create bar graphs.
Prior Knowledge Required: Bar graphs and pictographs
Scale
Vocabulary: data, bar graph
Using a scale of 10, draw a bar graph for the following data:
Skis sold by a sports store during each season
Fall: 120 Winter: 60 Spring: 45 Summer: 15
ASK: In which season did the store sell 3 times as many skis as in another season? In which season did
the store sell twice as many skis as in another season? In which season were there 15 more skis sold than
in Spring?
Then draw a partially completed bar graph, with only the “tennis” bar shown:
Favourite sports of people in a Grade 3 class
Tennis: 4 Hockey: 5 Basketball: 12 Soccer: 10
Tell your students that hockey was chosen one more time than tennis was and have a volunteer draw
the bar. Then tell them that basketball was chosen three times as often as tennis and have a volunteer add
that bar. Finally, tell them that soccer was chosen twice as often as hockey and have a volunteer add the last
bar. Ask students what other conclusions they can draw from the graph.
Activities:
1. You will need a large open space for this activity. Create a ‘human’ birthday graph. Place cards with the
names of the months of the year along a horizontal line. Students should line up in rows behind their
birth month. If a camera is available, take a photo of the graph. After determining which months have the
most and least birthdays, ask students why it is easy to tell this. (The ‘bars’ are the rows of students—
some rows are long and some are short.) Students can then transfer the data into a graph on paper.
2. Give students a paragraph of text and ask them to create a graph that shows the number of words on
each line.
3. Some students might enjoy the Internet game found at:
http://pbskids.org/cyberchase/games/bargraphs/bargraphs.html
Students need to be quick with their hands to catch bugs. A bar graph charts the number of each colour
caught. The game gives practice at changing scale.
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4. Encourage students to look at bar graphs in books, in magazines, on the Internet, or on television
(such as on The Weather Network). Have them record the number of markings on the scales and the
number of bars. About how many markings do most bar graphs use? About how many bars do most bar
graphs use?
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PDM3-11 Collecting Data
Goals: Students will ask good survey questions.
Prior Knowledge Required: Ability to distinguish between a statement and a question
Ability to distinguish between “and” and “or”
Vocabulary: survey, other, none
Explain to the class that the quality of a survey question determines the quality of the data collected. For
students to be successful at conducting surveys and collecting data, they must learn how to ask a good
question.
Conduct a survey with your students by asking them what their favourite flavour of ice cream is—do not limit
their choices at this point.
Tally the answers, then ask students how many bars will be needed to display the results on a bar graph.
How can the question be changed to reduce the number of bars needed to display the results? (Remind
them of what they learned from Activity 4 in PDM3-9,10). How can the choices be limited? Should choices
be limited to the most popular flavours? Why is it important to offer an “other” choice?
Explain to your students that the most popular choices to a survey question are predicted before a survey is
conducted. Why is it important to predict the most popular choices? Could the three most popular flavours
of ice cream have been predicted?
Have your students predict the most popular choices for the following survey questions:
• What is your favourite colour?
• What is your favourite vegetable?
• What is your favourite fruit?
• What is your natural hair colour?
• What is your favourite animal?
Students may disagree on the choices. Explain to them that a good way to predict the most popular choices
for a survey question is to ask the survey question to a few people before asking everyone.
Emphasize that the question has to be worded so that each person can give only one answer. Which of the
following questions are worded so as to receive only one answer?
a) What is your favourite ice cream flavour?
b) What flavours of ice cream do you like?
c) Who will you vote for in the election?
d) Which of the candidates do you like in the election?
e) What is your favourite colour?
f) Which colours do you like?
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Have your students think about whether or not an “other” category is needed for the following questions:
What is your favourite food group?
Vegetables and Fruits Meat and Alternatives
Milk and Alternatives Grain Products What is your favourite food?
Pizza Burgers Tacos Salad
Then ask students how they know when an “other” category is needed. Discuss which of the following questions would require an “other” category and why:
• What is your favourite day of the week? (List all seven days.)
• What is your favourite day of the week? (List only Friday, Saturday, and Sunday.)
• What is your favourite animal? (List horse, cow, dog, pig, cat.)
• How many siblings do you have? (List 0, 1, 2, 3, 4 or more)
• Who will you vote for in the election? (List all candidates.)
Then bring up the point that an “other” category may not be an option. For example, if the teacher wants to
bring 2 movies to show on the last day before Christmas holidays and she has only 5 movies at home, she
would give only those 5 movies as choices and would bring the 2 most popular ones to class.
Ask students what they think most people in the class would prefer to do for a party: go to a movie or go on
a skating trip. Ask students what kind of question you should ask to gather this data. Record all of their
suggestions on the board. Once this is done, review all the questions and determine all the possible answers
for each one.
Questions and answers might include the following:
1. Do you want to have a party? Yes/no
2. If you had a party, would you like to go to a movie? Yes/no
3. If you had a party, would you like to go on a skating trip? Yes/no
4. If you had a party, would you like to go to a movie or go on a skating trip? Movie/skating trip/neither/both
5. If you had a party, choose one of these things to do: go to a movie, or go skating? Movie/skating trip
Now, discuss with students which is the best questions to ask. Questions 1, 2, and 3 are limiting and will not
capture all data. Question 4 makes it difficult to make a decision (i.e., determine what is the most popular
choice) but Questions 5 makes it clear which activity is preferred. (NOTE: Make sure that you emphasize the
positive in each suggestion, showing how it’s a good start and demonstrating where to go from there to get to
the targeted question.)
Activity: Discuss the difference between ‘and’ and ‘or’ in a question and what they mean. For example,
the questions “Do you have a cat and a dog?” and “Do you have a cat or a dog?” will produce different
answers. For the first question, respondents can answer no (I don’t have a cat AND a dog) or yes (I have
both a cat AND a dog). For the second, respondents can answer yes and no but the meaning changes:
yes means I have a cat OR a dog; no means I don’t have either.
If possible, bring in pictures (perhaps cut from magazines) of people with cats and/or dogs: some with a cat
and a dog, some with only a cat, some with only a dog, and some with neither. Hold up the pictures one at a
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time and ask students how each person would answer these two questions: Do you have a cat or a dog? Do
you have a cat and a dog? Tally the results. For example, depending on the pictures you brought, your tallies
might look like:
YES NO
I have a cat or a dog
I have a cat and a dog Then have students summarize the answers in a separate bar graph for each question.
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PDM3-12 Practice with Surveys and PDM3-13 Blank Tally Chart and Bar Graph
Goals: Students will create a survey question, tally the data and present their data in a bar graph
Prior Knowledge Required: How to show data (graphs)
How to collect data
How to ask a question
What choices to give survey takers
How to analyze data
Vocabulary: survey, tallies, pictograph, bar graph
Tell students you want to find out what they will be doing on their summer holidays. Then discuss the
question and choices: What will you be doing over the summer holidays? Camp, family trip, summer school,
staying at home, other. (Modify the choices according to your students’ interests and activities.)
Ask each student to identify their summer activity by raising their hand when you call out the choice. Record
the data in a tally chart with the choices listed.
Count the tallies for each category to determine how much space you will need for the bar graph. Draw a grid
with a fixed number of markings, say 8. ASK: What scale should we use?
Remind students that there is a space between the bars in a bar graph and have students independently
create a bar graph for the data collected. Remind them to include a title and labels on their graphs.
Analyze the data together. Ask students what the data is telling them. Record those statements.
Activity: Ask students to name some of their favourite authors. (Have a selection of books which are
popular during read alouds and independent reading time on hand for students to refer to.) Then, select
the top 3 authors as well as “other” for categories. Ask students to identify their favourite author (explain that
they can only raise their hand once when ‘voting’ and record data using tally marks) and collect the data
from the class.
When you’re finished, ask how students can tell if everyone voted or not. Do they think anyone voted twice?
Does the number of votes equal the number of students in the class? What would happen if someone didn’t
vote? If someone voted twice?
ASK: If we want no more than 5 markings on our graph (have a graph with 5 markings on the vertical axis on
the board), what scale should we use? Would it make sense to have only 1 marking on the graph? How
would that make the graph hard to read? Decide on the number of markings and the scale for the graph.
Then complete the graph.
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Tell your students that they will be designing their own survey and then surveying their classmates.
Everyone can ask a different question, so suggest several topics if they have trouble getting started.
(EXAMPLES: How do you get to school? What is your favourite colour? How many people are there in your
family? What time do you wake up on weekdays? How long do you take to get ready for school? Does your
jacket have a hood? What pizza toppings do you like? What is your favourite meal? What is your favourite
cereal? What is your favourite season? Who is your favourite person? What type of home do you live in?
What is your favourite summer or winter activity?)
Extensions:
1. After completing their survey, students can transfer their graph data to KidPix to create a computer-
generated graph.
2. Ask student how they could find out the favourite colour of every teacher in the school. How would they
collect the data? How would they organize the data? Students can work on this in pairs or small groups
after they have a plan of action. After collecting and representing the data, students can report their
findings orally.
Literature Connection:
So you want to be president? by J. St. George
(A Caldecott winner. Anyone can be president, no matter what they look like or where they are from.)
After reading the book, have students work in small groups to collect data from their classmates. They can
ask the following question: What do you want to be when you grow up? Students should be encouraged to
organize the data in more than one way and be prepared to present their final work.
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PDM3-14 Collecting and Interpreting Data
Goals: Students will understand when surveys are appropriate to find out information and when other
methods are required.
Prior Knowledge Required: Asking a good survey question
Vocabulary: survey, measuring, researching, observing
Conduct a survey of your students to determine how many of them were born before noon and how many of
them were born after noon. It is likely that very few, if any, students will raise their hands for either option.
Explain that deciding when to conduct a survey, or when to use another method to collect data, depends on
whether or not the people being surveyed can answer the question. Let’s look at the example above:
Question: When were you born?
Choices: Before or at noon / After noon
The question is very clear and there is no “other” category because no other choice is possible. But if
people don’t know the answer, the survey is pointless.
Have your students think about which topics from the list would be good survey topics:
• What are people’s favourite colours? (good)
• Are people left-handed or right-handed? (good)
• Can people count to 100 in one minute or less? (not so good)
• How many sit-ups can people do in one minute? (not so good)
• What are people’s resting heart rates? (not so good)
• What are people’s favourite sports? (good)
• How fast can people run 100 m? (not so good)
• How do people get to school? (good)
• How many siblings do people have? (good)
Is a survey needed for the following topics or can the data be obtained by observation?
• Do students in the class wear eyeglasses?
• Do students in the class wear contact lenses?
• What are adults' hair colours?
• What are adults' natural hair colours?
Sometimes a measurement or calculation is needed to obtain data. Have your students decide if
observation, a survey or a measurement is needed to obtain the data for the following topics:
• What are people’s heart rates?
• What are people’s favourite sports?
• How do people get to school?
• How many sit-ups can people do in one minute?
• How long are people’s arm spans?
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• What colour of shirt are people wearing?
• How tall are people?
Sometimes students cannot find the information themselves but need to rely on measurements or
observations that other people have taken. Have students brainstorm examples of such situations (e.g.,
sports data from last year’s championships, world records, how fast different types of birds can fly, the time
of year that different animals hibernate, and so on).
Workbook 3 - Probability & Data Management, Part 1 1BLACKLINE MASTERS
Shapes ________________________________________________________________2
PDM3 Part 1: BLM List
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Shapes
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G3-1 Sides and Vertices Goals: Students will identify polygons, sides and vertices, and distinguish polygons according to the
number of sides.
Prior Knowledge Required: Count to 10
Distinguish straight line
Vocabulary: polygon, sides, vertex, vertices, triangle, quadrilateral, pentagon, hexagon
Draw and label a polygon with the words “polygon,” “sides” and “vertex/vertices”. Remind your students of
what a side and vertex are and explain that a side has to be straight. Show students how to count sides—
marking the sides as you count—then have them count the sides and the vertices of several polygons. Ask
them if they can see a pattern between the number of vertices and the number of sides. Be sure that all
students are marking sides properly and circling the vertices, so they don’t miss any sides or vertices.
Construct a large triangle, quadrilateral, pentagon and hexagon using construction paper or bristol board.
Label each figure with its name and stick them to the chalkboard. Explain that “gon” means angle or corner
(vertex), “lateral” means sides. You might want to leave these figures on a wall throughout the geometry unit.
Explain that “poly” means many, and then ask your students what the word polygon means (many angles or
vertices). Explain that a polygon is a shape that only has straight sides. Draw a shape with a curved side and
ask if it is a polygon. Label it as “not polygon”.
Draw several shapes on the board and ask students to count the sides and sort the shapes according to the
number of sides. Also ask them to draw a triangle, a pentagon, a figure with six sides, a figure with four
angles, and a figure that is not a polygon but has vertices.
Bonus:
Draw a figure that has:
a) Two curved sides and three straight sides
b) Two straight sides and three curved sides
Assessment:
1. Draw a polygon with seven sides.
2. Draw a quadrilateral. How many vertices does it have?
3. Draw a figure that is not a polygon and explain why it is not a polygon.
Possible answers: it has a curved side, a circle is not a polygon, a rectangle with rounded edges does
not have proper vertices.
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Activity:
Give each student a set of pattern blocks or several tangram pieces with the following instructions:
a) Group your pieces according to how many sides they have.
How many of each type do you have?
b) Can you make a shape with four sides using two triangles? Three triangles?
c) Can you make a large triangle using four triangles?
d) Can you make a triangle from two small tangram triangles and a square?
e) Can you make a pentagon with pattern blocks?
f) Can you make a seven-, eight- or nine-sided figure with pattern blocks or tangram pieces?
Extension:
1. How many sides does each group of shapes have?
a) 2 pentagons b) 3 pentagons c) 4 pentagons
Students should see the connection with multiplication: 4 pentagons have 4 × 5 = 20 sides.
2. How many sides would 2 pentagons and 3 hexagons have?
3. Count the sides of a paper polygon. Count the vertices. Cut off one of the vertices. Count the sides and vertices
again. Cut off another vertex. Repeat the count. Do you notice a pattern? (The number of sides will increase by
one and the number of vertices will increase by one.)
4. “Word Search Puzzle (Shapes)” in the BLM section.
S
S
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smaller larger
G3-2 Introduction to Angles Goals: Students will identify right angles in drawings and objects.
Vocabulary: right angle
Ask your students if they know what a right angle is—a right angle is the corner of a square, but there is no
need to define it in terms of degrees at this stage. Ask them where they can see right angles in real life
(EXAMPLE: of a sheet of paper corners, doors, windows, etc.). Draw a right angle and then show them how
to mark right angles with a small square. Explain to your students that not all angles are right angles; some
are sharper than a right angle, some are less sharp. Tell them to think of corners; the sharper the corner is,
the smaller the angle is. NOTE: You may want to perform Activity 1 here.
Draw two angles.
Ask your students which angle is smaller. Which corner is sharper? The diagram on the left is larger, but the
corner is sharper, and mathematicians say that this angle is smaller. The distance between the ends of the
arms is the same, but this does not matter. What matters is the “sharpness”. The sharper the angle is, the
narrower the space between the angle’s arms. Explain that the size of an angle is the amount of rotation
between the angle’s arms. The smallest angle is closed—with both arms together. Draw the following picture
to illustrate what you mean by smaller and larger angles.
With a piece of chalk you can exhibit how much an angle’s arm rotates. Draw a line on the chalkboard then
rest the chalk along the line’s length. Fix the chalk to one of the line’s endpoints and rotate the free end
around the endpoint to any desired position.
You might also illustrate what the size of an angle means by opening a book to different angles.
Draw some angles and ask your students to order them from smallest to largest. EXAMPLE:
a)
A B C
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b)
A B C
Students that have trouble comparing the angles could use a book and open its pages so that the cover
halves coincide with arms of the angles.
Discuss whether flipping the third angle changed its size.
Bonus: Provide longer lists of angles.
Draw a polygon and explain that the polygon’s angles are inside the figure, not outside the figure. Draw
several polygons and ask volunteers to mark the smallest angle in each figure.
Note that the smallest angle in the rightmost figure is A and not B—B is actually the largest angle, since the
angles are inside the polygon. (You might hold up a pattern block or a cut-out of a polygon so students can
see clearly which angles are inside the figure.)
Do some of the activities before proceeding to the worksheets.
Draw several angles and ask volunteers to identify and mark the right angles. For a short assessment, you
can also draw several shapes and ask your students to point out how many right angles there are. Do not
mark the right angles in the diagram.
Activities:
1. Make a key from an old postcard. Have students run their fingers over the corners to identify the
sharpest corner. The sharper the corner is, the smaller the angle is.
2. Ask students to use any object (a piece of paper, an index card) with a square corner to identify various
angles in the classroom that are “more than," “less than” or “equal to” a right angle.
• corner of a desk • angle made by an open door and the wall
• window corners • corners of base ten materials
A
B
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3. Use a geoboard with elastics to make …
a) a right angle
b) an angle less than a
right angle
c) an angle greater than
a right angle
4. Use a geoboard with elastics to make a figure with…
a) no right angles
b) 1 right angle
c) 2 right angles
5. Have your students compare the size of angles on pattern blocks by superimposing various pattern
blocks and arranging the angles in order according to size.
Students may notice that there are two angles on the trapezoid that are greater than the angles in the
square, and that there are two angles that are less than the angles in the square.
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G3-3 Equilateral Shapes Goals: Students will classify polygons according to the number and the lengths of sides and the number of
right angles. Students will also identify equilateral shapes.
Prior Knowledge Required: Sides, vertices
Right angles
Vocabulary: polygon, sides, vertex, vertices, triangle, quadrilateral, pentagon, hexagon
Write the word “equilateral” on the board. Ask the students what little words they see in it. What might the
parts mean? Where did they meet these parts? (“Equi” like in “equal” and “lateral” like in “quadrilateral”). Give
your students the set of shapes below. Students can either cut out the shapes and fold them to check if the
sides are equal or measure the sides with a ruler. Ask your students to extend the following chart to classify
the shapes.
Number of
Right Angles
Number of
Sides Equilateral
4 4 Yes
Include the word “equilateral” on your next spelling test.
Extensions:
1. Pick a property (i.e. same number of sides, vertices, right angles, etc.) and find three shapes in
QUESTION 3 that all have that property in common.
2. Find a group of four shapes where three share a common property and one doesn’t belong.
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G3-4 Quadrilaterals and Other Polygons Goals: Students will distinguish the quadrilaterals from other polygons.
Prior Knowledge Required: Count sides of polygons
What is a polygon
Measure straight lines with a ruler
Vocabulary: polygon, sides, vertex, vertices, triangle, quadrilateral, pentagon, hexagon
Draw a quadrilateral on the board. Ask how many sides it has. Write the word “quadrilateral” on the board
and explain what it means. Explain that “quad” means “four”, “lateral” means “sides” in Latin. Ask if students
have ever encountered any other word having either of these parts in it. (EXAMPLES: quadrangle,
quadruple). You might also mention that “tri” means “three” and ask if they know what the French words are
for 3 and 4.
Emphasize the similarities: “tri” and “trois”, “quad” and “quatre”. Draw several polygons on the board and
ask whether they are quadrilaterals. Write the number of sides for each and mark the answer on the board.
Make two columns (quadrilaterals and non-quadrilaterals) on the board.
Use the polygons from the shape game (see the BLM section) with tape on the back side. Invite volunteers
to come and affix the shapes at the right column on the board. You may ask them to explain their choice.
You may also ask the students to sort the pattern block pieces into quadrilaterals and non-quadrilaterals.
You may also ask your students to sort shapes according to different properties—those that have right
angles and those that do not, number of sides or angles or right angles, equilaterality, etc. Include the word
“quadrilateral” on your next spelling test.
Activities:
1. Give each of your students a set of 4 toothpicks and some model clay to hold them together at vertices.
Ask to create a shape that is not a quadrilateral. (This might be either a 3-dimensional shape or a self-
intersecting one). Ask also to make several different quadrilaterals.
2. Provide each of your students a set of 10 toothpicks. Ask them to check how many different triangles
they can make with these toothpicks. Each figure should use all the picks. How many different
quadrilaterals you can make using 10 toothpicks? The answer for the second problem is infinity—a slight
change in the angles will make a different shape.
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Extension: Explain that a kite is a quadrilateral with two pairs of equal adjacent sides. Draw a kite and
ask a volunteer to mark the equal pairs of sides. Point out that a kite has no indentation – illustrate the
meaning with a picture.
Draw several polygons on the board, some kites, some not, some resembling kites, and ask your students to
measure the sides of the shapes and to determine which of these shapes are kites.
Hold up a cut-out paper kite and ask your students how they could check whether this shape is a kite
without measuring the sides. Would folding help? Invite a volunteer to fold the shape and to check whether
the sides are equal. Ask: Are all the angles in a kite different? How do you know? (Folding the kite along the
diagonal that is also a line of symmetry will show that the opposite angles between the non-equal sides are
also equal.)
What about the other pair of opposite angles? Ask a volunteer to check whether these two angles are equal
by folding the kite along the other diagonal. (The Atlantic curriculum)
Indentation, so this is not a kite.
These angles are equal
Fold
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G3-5 Tangrams Goals: Students will become familiar with polygons and develop the use of the vocabulary.
Prior Knowledge Required: Quadrilateral, square, rectangle, pentagon, hexagon
Vocabulary: polygon, sides, vertex, vertices, triangle, quadrilateral, pentagon, hexagon
Give your students copies of the Tangram sheet in the BLM section of this manual. Have them cut out the
pieces and follow the instructions on the worksheet.
Activity:
Communication Game. Two players are separated by a barrier that prevents each player from seeing the
table directly in front of the other player. Player 1 makes a simple shape using a limited number of tangram
pieces. (For instance ). Player 1 then tells player 2 how to build the shape. For instance: “I used the
small triangle and the square. I put the triangle on the left side of the square.”
NOTE: There are several ways to carry out this instruction: or or or so Player 1
would have to give more precise instructions such as “Put one of the short sides of the triangle against the
side of the square so the right angle of the triangle is at the bottom”.
As students play this game (and as they see how difficult it is to describe their shapes) teach them the
terms they will need (“right”, “left”, “short side”, “long side”, “horizontally”, “vertically”, etc.) and show them
some situations in which more precise language would be needed. For instance, in the example above you
might say that the short side of the triangle is adjacent to the square and the long side goes from top left
corner right and down.
NOTE: Each player is allowed to ask questions but students should only use only geometric terms in the
game, rather than describing what a shape looks like (EXAMPLE: NOT “It looks like a house.”) Here are
some simple shapes students could start with.
S = small triangle M = medium triangle L = large triangle
S S
S S
L M
S S
M M
L
L
S S
S S
S S S
S
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Extensions:
1. How many rectangles can you make using the tangram pieces? (REMEMBER: squares are
rectangles.)
EXAMPLES:
2. How many ways can you find to construct a square as in QUESTION 5 on the worksheet? Which way
uses the smallest number of shapes? The greatest number of shapes? (The Ontario curriculum)
3. Try to make various polygons using all the tangram pieces. Sample shapes may be found at
www.tangrams.ca. SEE ALSO:
http://tangrams.ca/puzzles/asso-02.htm and
http://tangrams.ca/puzzles/asso-02s.htm
M S S S S
S
M
S S
M L
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G3-6 Congruency Goals: Students will become familiar with polygons and develop the use of the vocabulary.
Prior Knowledge Required: Count to 10
Vocabulary: congruent shapes
Explain that two shapes are congruent if they are the same size and shape. A pair of congruent two-
dimensional figures will coincide exactly when one is placed one on top of the other. Have students actually
do this with tangrams or pattern blocks. As it isn’t always possible to check for congruency by superimposing
figures, mathematicians have found other tests and criteria for congruency.
A pair of two-dimensional figures may be congruent even if they are oriented differently in space (see, for
instance, the figures below):
As a first test of congruency, your students should try to imagine whether a given pair of figures would
coincide exactly if one were placed on top of one another. Have them copy the shapes onto grid paper.
Trace over one of them using tracing paper and try to superpose it. Are the shapes congruent? Have your
students rotate their tracing paper and draw other congruent shapes. Let your students also flip the paper!
You might also mention the origin of the word: “congruere”—“agree” in Latin.
Assessment: Circle the pair of shapes that are congruent:
a) b)
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G3-7 Congruency (Advanced) Goals: Students will identify congruent shapes regardless of their position and colour.
Prior Knowledge Required: Count to 10
Vocabulary: congruent shapes
Review the previous lesson. Draw two congruent shapes of different colours and ASK: Are the shapes
congruent? Are they of the same size? Are they of the same shape? Remind your students that congruent
shapes are of the same size and shape, and they can have different colors. Give your students several
shapes of different colours and with different patterns (stripes, dots, etc.) and ask them to find congruent
pairs. Increase the number of congruent pairs in the same group of shapes gradually.
Make several shapes of blocks, such as the shapes below and ask your students to explain why these
shapes are not congruent. Repeat with pairs of polygons.
Ask your students to build shapes that are congruent to the shapes above.
Activities:
1. Give each student a set of pattern block shapes and ask them to group the congruent pieces. Make sure
students understand that they can always check congruency by superimposing two pieces to see if they
are the same size and shape.
2. Give students a set of square tiles and ask them to build all the non-congruent shapes they can find
using exactly 4 blocks. They might notice that this is like Tetris game. Guide them in being organized.
They should start with two blocks, and then proceed to three blocks. For each shape of 3 blocks, they
should add a block in all possible positions and to check whether the new shape is congruent to one of
the previous ones.
SOLUTION:
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G3-8 Recognizing and Drawing Congruent Shapes
Goals: Students will identify and draw congruent shapes.
Prior Knowledge Required: Count to10
Identify congruent shapes
Vocabulary: congruent shapes
Review the previous lesson. Draw a group of shapes as shown below and ask your students to find a pair of
congruent shapes among the shapes below. Ask your students also to explain why the other shapes are not
congruent to the two congruent shapes.
Activity: Review the names of the following polygons: triangle, rectangle, square, rhombus, pentagon,
hexagon. Each pair of students will need a spinner as shown. Player 1 spins the spinner and draws a
polygon according to the result of the spinner. Player 2 has to draw two polygons of the same type, so that
one is congruent to the polygon drawn by Player 1, and the other polygon is not.
Advanced: Review the concepts of attributes before starting the activity. Player 1 spins the spinner and
draws a polygon according to the result on the spinner. Player 1 decides on an attribute (such as striped
pattern) and then names the attribute. Player 2 draws a shape congruent to the shape drawn by her partner,
so that it differs in the given attribute (in this case, a different pattern). Then Player 2 draws a shape that is
not congruent to the given shape so that it shares the given attribute (it will still be striped). For example, the
spinner reads “Triangle”: Player 1 says: Pattern Different pattern Same pattern
Rectangle
SquareRhombus
Triangle Pentagon
Hexagon
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G3-9 Exploring Congruency with Geoboards Goals: Students will create congruent and non-congruent shapes on a geoboard.
Prior Knowledge Required: Count to10
Identify congruent shapes
Vocabulary: congruent shapes
Draw a shape on a grid on the board and ask your students to make a copy of it on their geoboards. (This
exercise could also be done on dot paper). Then ask your students to create a new shape, congruent to the
first one, but differently oriented. Repeat with several other shapes. Ask your students also to make shapes
that are not congruent to the given shape, and to explain why these shapes are not congruent.
Activities:
1. Make 2 shapes on a geoboard. Use the pins to help you say why they are not congruent.
EXAMPLE:
2. Repeat the activity from the previous lesson with geoboards.
The two shapes are not congruent: one has a larger base
(you need 4 pins to make the base).
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G3-10 Exploring Congruency with Grids Goals: Students will create congruent and non-congruent shapes on a grid.
Prior Knowledge Required: Congruent shapes
Vocabulary: congruent shapes
Ask your students to check how many different squares they can draw on a 4 × 4 grid. Ask your students:
How can you check if the squares are congruent? What do you have to check to make sure your shape is a
square? Suggest that your students measure the sides with rulers and angles with benchmarks to make sure
that the shapes they created are squares, and then to check congruency.
Include the word “congruent” into your next spelling test.
Bonus: Repeat with the 5 × 5 grid. Can you find eight non-congruent squares? (HINT: some of the squares
may be oriented so that their sides are diagonal to the grid).
Extension: The picture shows one way to cut a 3 × 4 grid into 2 congruent shapes. Show how many
ways you can cut a 3 by 4 grid into 2 congruent shapes.
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G3-11 Symmetry Goals: Students will find lines of symmetry using paper folding.
Prior Knowledge Required: Congruent shapes
Vocabulary: line of symmetry
Explain that a line of symmetry is a line that divides a figure into parts that have the same size and shape
(i.e. into congruent parts), and that are mirror images of each other in the line of symmetry. You can check
whether a line drawn through a figure is a line of symmetry by folding the figure along the line and verifying
that the two halves of the figure coincide.
Hold up a large paper parallelogram. Mark a line on it as shown below.
Ask your students: Is this line a line of symmetry? Are the halves of the figure on both sides of the line
congruent? Fold the shape along the line and ask: Do the halves of the figure coincide? Students should see
that the two halves do not coincide so they are not mirror images of each other in the line. Hence the line is
not a line of symmetry. Invite volunteers to check if other lines on the shape are lines of symmetry. (They can
try a line that connects the other pair of the diagonals or pairs of sides.) They will find that the shape has no
lines of symmetry.
Let your students cut out various polygons and check how many lines of symmetry the shapes have. You
might suggest that your students predict the number of lines of symmetry first and then check their prediction.
Extension: These shapes are called regular shapes. All their sides and angles are equal. Fill in the
T-table:
Figure
Number of Sides
Number of Symmetry
Lines
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G3-12 Lines of Symmetry Goals: Students will find lines of symmetry in pictures and draw mirror images.
Prior Knowledge Required: Congruent shapes
Horizontal
Vertical
Vocabulary: lines of symmetry horizontal, vertical
Review the definition of a line of symmetry. Give an example using the human body. You may wish to draw
a symbolic human figure on the board and mark the line of symmetry. Review the meaning of the words
“horizontal” and “vertical”. To help the students remember the word “horizontal”, remind them of how they
might draw the horizon line in art class.
Draw several pictures and ask students to find the horizontal and the vertical lines of symmetry.
Ask them to circle the pictures that have a vertical line of symmetry and to draw a square around the
pictures that have a horizontal line of symmetry. Does every picture have a line of symmetry? Are there
pictures with more than one line of symmetry?
EXAMPLES:
Challenge your students to find all the possible lines of symmetry for the following shapes:
Assessment:
Draw all the possible lines of symmetry:
a) b) c)
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Activities:
1. Find a picture in a magazine that has a line of symmetry and mark the line with a pencil. Is it a horizontal
or a vertical line? Try to find a picture with a slanted line of symmetry.
2. Cut out half an animal or human face from a magazine and glue it on a piece of paper, draw the missing
half to make a complete face.
Extension:
Cross-curriculum Connection: Check the flags of Canadian provinces for lines of symmetry.
Possible Source: http://www.flags.com/index.php?cPath=8759_3429.
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G3-13 Completing Symmetric Shapes Goals: Students will draw mirror images and find lines of symmetry in figures.
Prior Knowledge Required: Congruent shapes
Symmetry lines
Vocabulary: symmetry line, mirror line, mirror image
Draw several simple images such as the ones below and ask your students to draw the missing halves so
that the resulting pictures have a line of symmetry. Ask students to mark the line of symmetry. Is it a
vertical or a horizontal line?
Explain to your students that the halves of pictures that they have drawn are called mirror images of the
original pictures. Can your students explain why the pictures are called so? Let your students put a mirror or
a MIRA along the symmetry line and compare the images in the mirror with their pictures. Explain that the
symmetry line is also often called mirror line.
Activities:
1. Using each pattern block shape at least once, create a figure that has a line of symmetry. Choose one
line of symmetry and explain why it is a line of symmetry. Draw your shape in your notebook.
2. Using exactly 4 pattern blocks, build as many shapes as you can that have at least one line of symmetry.
Record your shapes in your notebook.
Extension: Take a photo of a family member’s face (such as an old passport photo) and put a mirror
along the line of symmetry. Look at the face that is half the photo and half the mirror image. Does it look the
same as the photo?
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G3-14 Completing Symmetric Shapes Goals: Students will compare shapes according to a given pattern.
Prior Knowledge Required: Congruent shapes
Lines of Symmetry
Sides and vertices
Polygons
Right angles
Equilateral
Vocabulary: line of symmetry, equilateral shapes, polygon, square, rectangle, triangle, pentagon,
hexagon, right angle, vertices, vertex, sides
Draw a regular hexagon on the blackboard. ASK: How many sides and how many vertices does it have?
What is it called? Does it have any right angles? Is it an equilateral shape? How many lines of symmetry
does it have? Have volunteers mark the lines of symmetry. Then draw a hexagon with two right angles and
make the comparison chart shown below. Ask volunteers to help you fill in the chart.
Property Same? Different?
Number of vertices 6 6 �
Number of edges
Number of right angles
Any lines of symmetry?
Number of lines of symmetry
Is the figure equilateral?
Ask students to summarize the information from the table in a short paragraph that describes any
similarities and differences of the shapes.
Use the worksheet for more practice.
Assessment: Write a comparison of the two shapes. Be sure to mention the following properties:
The number of vertices
The number of sides
The number of right angles
Number of lines of symmetry
Whether the figure is equilateral
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G3-15 Sorting Shapes by Property Goals: Students will compare shapes according to a given pattern.
Prior Knowledge Required: Congruent shapes
Symmetry lines
Sides and vertices
Polygons
Right angles
Equilateral
Vocabulary: symmetry line, equilateral shapes, polygon, square, rectangle, triangle, pentagon, hexagon,
right angle, vertices, vertex, sides
Give each student (or team of students) a deck of shape cards and a deck of property cards from “2-D
Shape Sorting Game” of the BLM section. Let them play the first game in Activity 1 below. The game is an
important preparation for Venn diagrams.
Draw a Venn diagram on the board. Show an example—you may do the first exercise of the worksheets
using volunteers. In process remind your students that any letters that cannot be placed in either circle
should be written outside the circles (but inside the box). FOR EXAMPLE, the answer to QUESTION 1 a)
of the worksheet should look like this:
Also remind students that figures that share both properties, in this case D and A, should be placed in the
overlap. Ask your students where they would put a figure that looks like the one shown below:
Let your students play the game in the second activity. Then draw the following set of figures on the board.
Ask your students to make a list of figures satisfying each of the properties in a given question below they
draw a Venn diagram to sort the figures.
D A
C
A B F G H
E I
Has at least 2 right angles
Quadrilateral
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Here are some properties students could use to sort the figures in a Venn diagram.
a) 1. At least two right angles
2. Equilateral
b) 1. Equilateral
2. Has exactly one line of symmetry
(In this case the centre part contains only one irregular pentagon—figure J)
c) 1. Quadrilateral
2. Has at most one right angle
d) 1. Pentagon
2. Equilateral
Assessment: Use the same list of figures to create a Venn diagram with properties:
1. Has at least one line of symmetry
2. Has at least two right angles
Activities:
Students will need a deck of shape cards and a deck of property cards from the “2-D Shape Sorting Game”
of the BLM section.
1. 2-D Shape Sorting Game
Each student flips over a property card and then sorts their shape cards onto two piles according to
whether a shape on a card has the property or not. Students get a point for each card that is on the
correct pile. (If you prefer, you could choose a property for the whole class and have everyone sort
their shapes using that property.) Once students have mastered this sorting game they can play the
next game.
2. 2-D Venn Diagram Game
Give each student a copy of the Venn diagram sheet in the BLM section (or have students create their
own Venn diagram on a sheet of construction paper or bristol board) in addition to the shape cards and
property cards. Ask students to choose two property cards and place one beside each circle of the
Venn diagram. Students should then sort their shape cards using the Venn diagrams. Give 1 point for
each shape that is placed in the correct region of the Venn diagram.
Extension:
A Game for Two: Player 1 draws a shape without showing it to the partner, then describes it in terms
of number of sides, vertices, right angles, lines of symmetry, etc. Player 2 has to draw the shape from
description.
A
B
C
D
E
F
G
H J
K
I
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G3-16 Finding Polygons Goals: Students will identify polygons in drawings.
Prior Knowledge Required: Congruent shapes
Symmetry lines
Sides and vertices
Polygons
Right angles
Equilateral
Vocabulary: symmetry line, equilateral shapes, polygon, square, rectangle, triangle, pentagon, hexagon,
right angle, vertices, vertex, sides
As a preparation for QUESTION 2 of the worksheet, draw two shapes on the board:
Invite volunteers to make a comparison chart and to write a comparison paragraph for these two shapes.
Properties you might mention:
• The number of vertices
• The number of sides
• The number of right angles
• Number of lines of symmetry
• Whether there are pairs of equal sides
• Whether the equal sides adjacent or opposite
• Whether the figure is equilateral
QUESTION 5 on the worksheet can be done with pattern blocks.
Activity: On a picture from a magazine or a newspaper, ask the students to mark as many polygons as
possible.
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G3-17 Problems and Puzzles
This worksheet is the final review and may be used for practice.
Activities:
1. Give students a set of circular tiles of two colours. Ask them to make as many 3-tile triangles as they can
inside a 6-tile triangle.
Solution:
Repeat this exercise with a larger triangle:
2. Take any 2 congruent pattern blocks. Predict the shapes you can make by putting the blocks edge to
edge. (Can you make a quadrilateral? a pentagon? a hexagon? or a shape with more sides?) Trace
around the pattern blocks to show how they combine to make your figure. Repeat this exercise with 3
different pairs of pattern blocks.
3. Using pattern block triangles, try to make the following shapes:
a) A quadrilateral. b) A hexagon. c) A bigger triangle. d) A pentagon.
Extension: In QUESTION 6 students can make a square as shown:
Workbook 3 - Geometry, Part 1 1BLACKLINE MASTERS
2-D Shape Sorting Game _________________________________________________2
Tangram ______________________________________________________________6
Venn Diagram __________________________________________________________7
Word Search Puzzle (Shapes) ______________________________________________8
G3 Part 1: BLM List
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2-D Shape Sorting Game
Workbook 3 - Geometry, Part 1 3BLACKLINE MASTERS
2-D Shape Sorting Game (continued)
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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2-D Shape Sorting Game (continued)
Three or
more sides
Four or
more vertices
Three or
more vertices
More than
one line
of symmetry
No lines
of symmetryHexagon
Workbook 3 - Geometry, Part 1 5BLACKLINE MASTERS
No right
angles
2-D Shape Sorting Game (continued)
Two or
more sides
One right
angle only
At least
fi ve sides
Equilateral
Quadrilateral
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Tangram
Workbook 3 - Geometry, Part 1 7BLACKLINE MASTERS
Venn Diagram
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Word search puzzle—can also be assigned as homework.
WORDS TO SEARCH:
triangle
vertices
quadrilateral
sides
pentagon
angle
vertex
polygon
l a o l g i o l d l r l
a e n e e a e g l g r o
r n i l i s e v c e t e
e l g n a i r t s n n n
t n e o v d v t p e e g
a c l g l e i a y a v p
l g a a r s o i i r e r
i e e t a p e e i i v x
r o e n i p n r t t r n
d x s e c i t r e v n x
a i o p o l y g o n t a
u a q q a e s n q s a e
q l n n x s r g q e s t
Word Search Puzzle (Shapes)
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PA3-20 Patterns Involving Time Goal: Students will create T-tables for growing and shrinking patterns and use them to identify the rules for
the patterns.
Prior Knowledge Required: Addition, Subtraction, Skip counting, Number patterns
Days of the week, Months of the year, T-tables
Vocabulary: T-table, chart, term, growing pattern, shrinking pattern
Explain that today you are going to use T-tables to solve problems involving money and time. Students
are sure to encounter similar problems in their day-to-day lives; it is important they know how to approach
such problems systematically. Make sure all of your students know the days of the week and the months of
the year.
Give an EXAMPLE: Jenny started babysitting in the middle of June and earned $10 in June. She continued
babysitting throughout the summer and earned $15 in July and August. She saved all the money. How much
money did she save through the whole summer? Make a T-table and label the columns Month and Savings
($). Fill in the starting amount and add circles at the side, as shown:
Month Savings ($)
June 10
July
August
ASK: How much money did Jenny add to her savings in July? Write “+ 15” in the upper circle. How much
money did Jenny have saved by the end of July? Ask volunteers to continue the pattern and to finish
the problem.
Let your students practice solving similar problems, like the two below. Help your students to create a T-table
for each problem.
1. Lucy’s family spent 9 days hiking the Bruce Trail. They started by hiking 15 km on Saturday. They hiked
20 km every day after that until the next Sunday. Make a chart to show their progress. How many
kilometers did they hike in total over the course of their trip?
2. Natalie started saving for her mom’s birthday present in March. She saved $8 in March and plans to
save $5 every month after that. Her mom’s birthday is July 1. Will she have enough money by the end of
June to buy a present that costs $22? Make a T-table to show her savings.
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Invite volunteers to help you solve the following problem:
Ronald received $50 as a birthday present on March 31. Every month he spends $4. How much money
will remain by the end of July?
Allow your students to practise solving more problems that involve decreasing patterns.
Assessment:
1. There are 15 fish in an aquarium on Monday. On Tuesday, Thomas the cat catches and eats two fish.
He eats two more fish every day after that. If Thomas’ owners don’t stop him, how many fish will they find
in the tank on Saturday night? Use a T-table to solve the problem.
2. The zoo is breeding a group of critically endangered Golden Lion tamarins. Every year 6 new monkeys
are born. At the end of 2006, the zoo had 13 tamarins. Make a T-table to show how many tamarins will
be at the zoo in the years 2007–2011.
Activity: Divide your students into pairs, give them play money and some beads or counters, and let them
play a selling and buying game: one player is a seller (starts with all the beads or counters and no money),
the other is a buyer (starts with all the money). The buyer can only buy and the seller can only sell (i.e., the
buyer can’t “sell” anything back to the seller). Have buyers and sellers record the money they have after
each transaction in a T-chart.
After a few transactions, ask the students to look at the patterns in their T-charts. Point out that each buyer’s
amount is a decreasing sequence, or shrinking pattern. ASK: What other patterns do you know? Is the
seller’s money shrinking, growing, or repeating?
Extension: Samia collects 4 stamps every month. She starts collecting at the beginning of November.
How many stamps does she have at the end of February? When will her collection reach 20 stamps? When
will she have 28 stamps?
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PA3-21 Calendars Goal: Students will identify patterns in calendars.
Prior Knowledge Required: Addition. Subtraction, Skip counting, Number patterns
Days of the week, Months of the year, T-tables
Vocabulary: T-table, chart, growing pattern, shrinking pattern, vertically, horizontally, diagonally
Remind your students which months have 30 days, 31 days, and 28 days. Mention the leap year.
Give your students blank BLM “Calendars” to work with and draw a large calendar on the board or use the
overhead projector. Solve the following problems as a class, but give students time to solve each problem
independently before calling on volunteers to share their answers on the board.
Fill in the calendar for August, so that August 1st is Tuesday.
a) Steven gets a pet snake on his birthday, August 5th. The snake should be fed every five days. Mark the
days when Steven has to feed his pet with a little snake. How are the snake-filled squares situated—
vertically, horizontally, or diagonally? Review the meaning of these terms with your students.
b) Steven plays the drums every Wednesday. Mark the days he drums on the calendar with a D or small
picture of a drum. Ask your students to describe the pattern of the drum days. Are there any days when
Steven has to feed his pet snake and play the drums? (yes, August 30th)
c) Steven bought his mother an orchid on August 3rd
. The orchid has to be watered every 6 days. Shade
the days when the orchid has to be watered. What pattern do the shaded squares make? (a diagonal
pattern) On which date does Steven play the drums and water the orchid? (August 9th) Are there any
dates when the snake needs feeding and the flower needs watering? (August 15th)
Activity: Ask students to create a calendar for the current month. They should label the days of the week
and mark the dates of personal events, such as lessons, chores, and family activities. Encourage students to
think of both special events, like birthdays or parties, and recurring events, such as feeding a pet, cleaning
their room, going to the library, taking out the garbage, or visiting a relative.
Extensions:
1. Fill in a blank calendar for any month, beginning on any day. Draw a square around any 4 numbers. Add
the pair of numbers on one diagonal. Then add the pair of numbers on the other diagonal. What do you
notice about the two sums? Will this always happen? Can you explain why?
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2. PROJECT: Calendars of the World. Explain to students that different calendars are used in other parts of
the world to mark time. Students can learn more about any one of these calendars. Questions to
consider: How many months are in the year in your calendar? How long is the year? How long are the
months? What defines the months and the year (movement of the Sun, the Moon, the Nile)? Is there a
leap year? What is a leap year (additional day or additional month)? How often does a leap year occur?
What patterns can be found in the calendar? (The Chinese calendar is particularly interesting in this
respect.)
POSSIBLE SOURCE: http://webexhibits.org/calendars/calendar.html
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PA3-22 Number Lines Goal: Students will use number lines to solve pattern problems.
Prior Knowledge Required: Addition, Subtraction, Skip counting, Number patterns
Days of the week, Months of the year
Vocabulary: number line
Write a sequence on the board: 36, 31, 26, 21. ASK: What is this kind of sequence called? (a decreasing
sequence) What is the difference between successive terms in the sequence? (5) Ask a volunteer to
continue the sequence and to explain how he or she figured out what the next term should be. Students
might say that they counted backwards or used their number facts to subtract. Ask students if they can think
of another way to solve the problem—if, for example, the numbers are so large that counting backwards or
subtracting is difficult. If no one suggests it, tell students they can use a number line and you will now teach
them how.
Present the following EXAMPLE: A snail crawls 4 cm in an hour. It is 24 cm away from the end of the branch.
How far from the end of the branch will it be after four hours of crawling?
Draw a number line and add arrows showing the snail’s progress every hour:
You could ask:
When will the snake reach the end of the branch? (after 6 hours)
A cherry hangs 11 cm from the end of the branch. Invite a volunteer to mark the place where the cherry
is on the number line. When would the snail reach the cherry? (in the 4th hour)
Let your students draw number lines in their notebooks. Here are some problems they could practice with:
A messenger pigeon flies 3 km in an hour. How long will it take this pigeon to carry a letter to a person
who lives 21 km away from the sender? The pigeon sets out at 1:00 p.m. When will it reach the
addressee?
Jonathan has $20. He spends $6 a week for snacks. How much money will he have after three weeks?
When will his money be completely spent?
3rd
hour 2nd
hour 1st hour 4
th hour
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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Activity: Draw a number line on the floor. It should have at least 12 intervals, but preferably a few more.
Ask a volunteer to demonstrate a hopping pattern: Four hops forward, one backward. Let the volunteer start
from 0 and repeat the core several times. Ask students to draw a number line on paper and record the
movement of their classmate with arrows. ASK: Where will the volunteer be after 6 hops? 7 hops? 9 hops?
12 hops?
As a challenge, repeat the exercise but have the volunteer hop from the other end of the line.
Now ask a volunteer to stand at the 7th mark, take 3 hops backward, then 6 hops forward. Where does the
volunteer end up? Ask your students to predict where the volunteer will end up if he or she hops forward 5
and backward 2. Make up more questions, beginning at different points on the number line. (EXAMPLE:
Start at 4, take 6 hops forward then 1 hop backward.)
Extension: A painter’s ladder has 12 steps. The painter spills red paint on every second step and
blue paint on every third step. Which steps have red and blue paint on them? Which steps will have no paint
on them?
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PA3-23 Mixed Patterns Goal: Students will extend patterns of various kinds
Prior Knowledge Required: Patterning, Core of the pattern
Vocabulary: growing pattern, shrinking pattern, repeating pattern
Review the concept of the core of repeating patterns with the students. Draw several repeating patterns on
the board and invite volunteers to circle the core of each pattern and to extend the patterns. Here are some
sample patterns:
A H H T A H H T R U N R U N R U N R U 1 4 6 7 2 2 1 4 6 7 2 2 1 4 6 7 2 2
Draw a pattern with a core that begins and ends with the same symbol and repeat the task. ASK: Was it
harder to find the core and continue the pattern? Why? Give students more such patterns to practice with.
Here are some samples:
A H H A A H H A R U R R U R R U R R U 2 1 4 6 2 2 2 1 4 6 2 2 2 1 4 6 2 2
Then allow your students to practise with a mixed assortment of repeating patterns.
Draw the following pattern on the board:
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ASK: Is this a repeating pattern? Does this pattern have a core? (no) Can you continue this pattern? How
many diamonds come next? Explain that this pattern is called a growing pattern and ask your students to
explain why they think this pattern is called that. (One part of the pattern grows, or increases.) Let your
students extend more growing patterns, such as:
S I S I I I S I I I I I S I I I I I I I K a a K K a a a K K K a a a a K K K K a a a a a
Ask your students what they think a shrinking pattern might look like. If they do not guess correctly, prompt:
In growing patterns, one of the parts was growing. What should some part of a shrinking pattern do? If we
start with M U U U U U U M, what might the next group of letters in the pattern be? (The number of Us
should shrink; the next term in the pattern could be M U U U U U M [1 fewer U] or M U U U U M [2 fewer U’s]
or even M U U U M [3 fewer U’s]. You could have students find the third term for each sequence, too.) Invite
students to draw examples of more shrinking patterns on the board. They can create shrinking patterns using
letters, numbers, or shapes. EXAMPLES:
5 5 5 5 5 4 4 4 4 3 3 3 2 2 1 9 9 9 9 8 8 8 7 7 6 A a a a a a a a a B b b b b b b C c c c c D
Invite a volunteer to write the letters of the alphabet on the board and ask your students to continue the
following patterns:
A Z B Y C X D Aa Bbb Cccc D F H J L
If some students have trouble seeing the rule for a pattern, suggest that they circle the letters in the alphabet
as they appear in the pattern. Invite students who see the rule to describe it to their classmates.
Finally, show your students a mixed assortment of patterns—repeating, growing, shrinking, letter—and ask
them to continue the patterns.
Activity: A game for pairs. Each pair of students will need a spinner as shown. Player 1 spins the spinner
so that his partner does not see the result. Player 1 draws a pattern according to the result on the spinner.
Player 2 has to write the next three terms of the pattern.
Repeating pattern
Growing pattern
Shrinking pattern
Letter pattern
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PA3-24 Describing and Creating Patterns Goal: Students will identify and describe increasing, decreasing, and repeating patterns. They will also
create increasing, decreasing, and repeating patterns.
Prior Knowledge Required: Patterning, Core of the pattern, Increasing sequences
Decreasing sequences, Repeating patterns, Addition
Subtraction, Difference between pattern terms
Vocabulary: increasing sequence, decreasing sequence, repeating pattern, difference
Write a sequence on the board: 3, 5, 7, 9, 11, 13, 15. Ask your students what kind of sequence this is. (The
numbers grow, or increase, so this is an “increasing sequence.”) Write another sequence: 95, 92, 89, 86, 83,
80, 77, and ask students what this sequence is called. Write “decreasing sequence” on the board.
ASK: Is every sequence an increasing or decreasing sequence, or can a sequence do both—can it increase
and decrease? Give an example: Jennifer saved $80. She receives a $20 allowance every two weeks. She
spends money occasionally:
Date Balance ($)
Sep 1 80
+20 Allowance
Sep 5 100
–11 Book
Sep 7 89
–10 Movie + Popcorn
Sep 12 79
+20 Allowance
Sep 19 99
Jennifer’s balance increases between the first two terms, decreases for the next two terms, and then
increases again.
Write several similar sequences on the board and ask volunteers to put a “+” sign in any circle where the
sequence increases, and a “–” sign where it decreases. Let your students practice this skill, and then ask
them to mark the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence
sometimes increases and sometimes decreases.
Sample Sequences
a) 7 , 8 , 7 , 10 b) 2 , 4 , 7 , 9
c) 10 , 7 , 4 , 2 d) 2 , 5 , 1 , 17
e) 15 , 23 , 29 , 28 f) 22 , 52 , 59 , 62
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Explain to your students that now their task will be more difficult. This time they will have to determine if the
sequence is increasing or decreasing and find the magnitude of the difference between successive terms.
Invite volunteers to find differences by counting forwards and backwards using their fingers or number lines.
Sample Sequences
a) 7 , 9 , 11 , 13 b) 2 , 5 , 9 , 12
c) 10 , 7 , 4 , 1 d) 32 , 36 , 40 , 44
e) 15 , 23 , 31 , 39 f) 72 , 52 , 32 , 12
Ask your students to find the differences between the terms in the previous sample sequences. Then list
several possible descriptions of sequences, such as:
A: Increases by different amounts
B: Decreases by different amounts
C: Increases by a constant amount
D: Decreases by a constant amount
Ask your students to match the descriptions with the sequences on the board. When the terms of the
sequence increase or decrease by the same amount, you might ask students to write a more precise
description. For example, the description of sequence (d) above could be “Increases by 4 each time” or,
more precisely, “Start at 32 and add 4 each time.” Encourage your students to give more precise
descriptions.
Next let your students do the activity below.
Assessment:
1. Put a “+” sign in any circle where the sequence increases, and a “–” sign where it decreases. Then mark
the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence sometimes
increases and sometimes decreases. Write the next term in each sequence.
a) 12 , 17 , 14 , 19 ___ b) 1 , 5 , 8 , 3 ___ c) 18 , 13 , 17 , 23 ___
d) 28 , 26 , 19 , 12 ___ e) 17 , 8 , 29 , 25 ___ f) 53 , 44 , 36 , 38 ___
2. Match the descriptions to the patterns. Each description may fit more than one pattern.
17, 16, 14, 12, 11 3, 5, 7, 3, 5, 7, 3 A: Increases by the same amount
10, 14, 18, 22, 26 4, 8, 11, 15, 18, B: Decreases by the same amount
54, 49, 44, 39, 34 4, 5, 3, 2, 6, 2, 4 C: Increases by different amounts
4, 8, 12, 8, 6, 4, 6 11, 19, 27, 35, 43 D: Decreases by different amounts
74, 70, 66, 62, 58 12, 15, 18, 23, 29 E: Repeating pattern
58, 55, 52, 49, 46 67, 71, 75, 79, 83 F: Increases and decreases
For patterns that increase or decrease by the same amount, write an exact rule. Don’t forget to include the
starting point!
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Bonus:
1. Put a “+” sign in any circle where the sequence increases, and a “–” sign where it decreases. Then mark
the sequences: “I” for increasing, “D” for decreasing, and “B” (for “both”) if the sequence sometimes
increases and sometimes decreases.
a) 257 , 258 , 257 , 260 b) 442 , 444 , 447 , 449 c) 310 , 307 , 304 , 298
d) 982 , 952 , 912 , 972 e) 815 , 823 , 829 , 827 f) 632 , 652 , 649 , 572
2. Match the descriptions to the patterns.
97, 96, 94, 92, 91 A: Increases by the same amount
3, 5, 8, 9, 3, 5, 8, 9 B: Decreases by the same amount
210, 214, 218, 222 C: Increases by different amounts
444, 448, 451, 456 D: Decreases by different amounts
654, 647, 640, 633 E: Repeating pattern
741, 751, 731, 721 F: Increases and decreases by different amounts
For patterns that increase or decrease by the same amount, write an exact rule.
Activity: A game for pairs. Your students will need the spinner shown. Player 1 spins the spinner so that
Player 2 does not see the result. Player 1 has to write a sequence of the type shown by the spinner. Player 2
has to write the rule for the sequence or describe the pattern. For example, if the spinner shows, “Decreases
by the same amount,” Player 1 can write “31, 29, 27, 25.” Player 2 has to write “Start at 31 and subtract 2
each time.”
Extension: Write the differences for the patterns. Identify the rule for the sequence of differences.
Extend first the sequence of differences, then the sequence itself.
a) 7 , 10 , 14 , 19 b) 12 , 15 , 20 , 27
c) 57 , 54 , 50 , 45 d) 32 , 30 , 26 , 20
e) 15 , 18 , 22 , 25 , 29 f) 77 , 72 , 69 , 64 , 61
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PA3-25 2-Dimensional Patterns Goal: Students will identify and describe patterns in 2-dimensional grids.
Prior Knowledge Required: Patterning, Core of the pattern, Increasing sequences
Decreasing sequences, Repeating patterns, Addition, Subtraction
Difference between pattern terms
Understanding that reading is done from left to right
and that text wraps onto the next line
Vocabulary: increasing sequence, decreasing sequence, repeating pattern, difference
Draw a 2-dimensional grid on the board. Remind your students that rows are vertical and columns are
horizontal. Point out the diagonals as well. Draw this diagram as a reference:
Explain that in 2-dimensional patterns, you generally count columns from left to right, but there are two
common ways of counting rows. In coordinate systems, which students will learn later, you count from the
bottom to the top. But in this lesson, students will count rows from the top to the bottom. Ask a volunteer
number the columns and rows on your grid.
Ask your students to draw a 4 × 4 grid in their notebooks and to fill it in as shown:
Ask them to shade the first column, the first row, and the third row, and to circle both diagonals. Let them
describe the patterns that they see in the shaded rows and columns and in the diagonals. Then invite
students to do the activities below for more practice.
Write this sentence on the board exactly as shown, i.e., over two lines:
“The big monkey ate the bananas
that were almost too ripe.”
13 17 21 25
10 14 18 22
7 11 15 19
4 8 12 16
D C I R O W A L G U O M N N A L
V
E
R
T
I
C
A
L
H O R I Z O N T A L
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Ask students where the sentence starts and where the sentence ends. Ask them to explain how they know.
Address any misconceptions students may have about sentences ending at the end of a line. Explain
that the “reading pattern” rule is that we read from left to right and that a sentence can continue on to the
next line.
Next, ask students to identify the word that comes right before the following words in the sentence: monkey,
ate, ripe, that. Then, ask students to identify the words that come right after these words: too, bananas, big,
almost, ate.
Draw a five-frame on the board and place only the numbers 1 and 5 in the row:
1 5
ASK: Which number comes right after the 1? Which number comes right before the 5? Can you predict
which number comes right after the 2 but right before the 4?
Add a row to the five-frame turning it into a ten-frame:
1 2 3 4 5
Challenge students to write the number that comes next on the next line, reminding them (if necessary) of
the reading pattern and of the number that comes after 5.
Next, draw the following charts on the board and challenge students to use the reading pattern to fill in the
missing numbers.
1 1 1
3 5 3
7 6
Emphasize that students should always start at the top row from the left, then go on to the next row, again
starting at the left. Then give students blank charts and ask them to fill in the charts starting at 1:
Students could also fill the above charts by skip counting by 2s, 3s, and 5s.
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Activities:
1. Give students a hundreds chart with the following instructions: Put your finger on the first number given
in each question below. Then move your finger to show the result of the addition. Then describe how
your finger moved.
EXAMPLES:
a) Start at 2. Add 20 (Answer: My finger moved 2 rows down.)
b) Start at 5. Add 21 (My finger moved 2 rows down and 1 column across.)
c) Start at 17. Add 42 (My finger moved 4 rows down and 2 column across.)
Make up more questions of this sort. With practice, students should be able to predict how their finger
will move before they carry out the addition, so they can add simply by moving their finger.
2. Students will need the “Hundreds Charts” BLM. Write the following patterns on the board and ask
students to identify in which rows and columns in the hundreds chart the patterns occur:
After students have had some practice finding patterns in the hundreds chart, ask them to fill in the
missing numbers in patterns. (Challenge them to do this without looking at the hundreds chart.)
EXAMPLES:
Make up more such problems for students who finish their work early.
Extensions:
1. Add the digits of some 2-digit numbers on the hundreds chart. Can you find all the numbers that have
digits that add to ten? Describe any pattern you see.
2. Sudoku is an increasingly popular mathematical game that is now a regular feature in many newspapers.
In the BLM section of this guide, you will find Sudoku suitable for children (“Sudoku – Warm Up” and
“Mini Sudoku”) with step-by-step instructions. Once students master this easier form of Sudoku, they can
try the BLM “Sudoku – The Real Thing.”
3
13
23
33
38
48
58
68
43
54
63
74
12
23
34
45
12
22
32
57
77
87
7
18
38
45
56
67
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PA3-26 Patterns in Two Times Tables Goal: Students will identify even and odd numbers.
Prior Knowledge Required: Skip counting by 2s, Rows, Columns, Diagonals
Vocabulary: row, column, diagonal, multiple, even, odd
Explain to your students that the numbers you say when skip counting by 2s, starting from 2, are called
multiples of 2. ASK: What other words similar to “multiple” do you know? (multiplication, multiply) What do
these other words mean? Ask your students to find the sequence of multiples of 2 on a multiplication chart.
What do you do to the numbers 1, 2, 3, 4, 5, … to get 2, 4, 6, 8, 10, …? (You multiply by 2.) What is a
“multiple”? (The multiple of a number is what you get when you multiply that number by any other number.)
Let your students complete the worksheet independently, or guide the class through the worksheet in stages.
You might wish to explain where the name “even” comes from: Give your students 20 small objects, like
markers or beads or base ten unit blocks, to act as counters. Ask them to take 12 blocks and to divide the
blocks into two equal groups. Does 12 divide into two groups evenly? Ask students to repeat the exercise
with 13, 14, 15, and any other number of blocks. When the number of blocks is even, they divide into two
groups evenly. If they do not divide evenly, the number is called “odd.” Write the on the board: “Even
numbers: 2, 4, 6, 8, 10, 12, 14… All numbers that end with .” Ask volunteers to finish the sentence.
After that write: “Odd numbers: 1, 3, 5, 7, 9, 11, 13… All numbers that end with ” and ask another
volunteer to finish this sentence.
Assessment: Circle the even numbers. Cross out the odd numbers.
23, 34, 45, 56, 789, 236, 98, 107, 3 211, 468 021.
Activity: Call up a group of volunteers and give each one a card with a whole number on it.
1. Ask the rest of the class to give the volunteers orders, such as “Even numbers, hop” or “Odd numbers,
raise your right hand.”
2. ADVANCED: Ask a student to skip count by 3s, starting with three. These are the multiples of three. Ask
your “numbers”: Are you a multiple of three? Repeat, skip counting by 5s or other numbers.
3. ADVANCED: Give the volunteers more complex orders, such as “All numbers that are multiples of five
or even numbers, clap” or “All numbers that are odd and multiples of five, stomp.”
Extension: The “Colouring Exercise” in the BLM section of this guide. (ANSWER: The Quebec flag.)
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PA3-27 Patterns in the Five Times Tables Goal: Students will identify multiples of five.
Prior Knowledge Required: Skip counting by 5s, Rows, Columns, Diagonals
Vocabulary: row, column, diagonal, multiple
Let your students complete the worksheet independently, or guide the class through the worksheet in stages.
Then let students play the game below (SEE: Activity).
After students have played the game, write several numbers with blanks for the missing digits on the board
and ask your students to fill in each blank so that the resulting number is divisible by 5. Ask students to list all
possible solutions for each number. EXAMPLES:
2___ 32___ 8___ 56___ 5___5 ___35 8___0
Assessment:
1. Circle the multiples of 5.
75 89 5 134 890 40 234 78 4 99 100 205 301 45675
2. Fill in the blanks so that each number become a multiple of 5. In two cases it is impossible to do so. Put
an “X” through these numbers.
3___ 3___5 3___9 ___0 70___ 5___0 ___3
Bonus: Fill in the blanks so that the numbers become multiples of 5. (List all possible solutions.) In two
cases it is impossible to do so. Put an “X” beside these numbers.
3 44___ 34 ___25 13 6___9 6 783 45___ 786 7___0
4 567 70___ 234 5___6 780 13 460 0___3 15 4___0 000
Activity: A game for pairs. One player will be the “pro-5” player, and the other will be the “anti-5” player.
The anti-5 player starts by adding either 3 or 5 to the number 4. The pro-5 player is then allowed to add
either 3 or 5 to the result. Each player then takes one more turn. If either player produces a multiple of 5, the
pro-5 player gets a point. Otherwise, the anti-5 player gets a point. The players exchange roles and start the
game again. Students should quickly see that there is always a winning strategy for the pro-5 player.
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Extension: “Who am I?”
1. I am a number between 31 and 39. I am a multiple of five.
2. I am an even multiple of five between 43 and 56.
HINT: Write down all even numbers between 43 and 56.
3. I am an odd multiple of five between 76 and 89.
4. I am the largest 2-digit multiple of five.
5. I am the smallest 3-digit multiple of five.
6. I am the smallest 2-digit odd multiple of 5.
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PA3-28 Patterns in the Eight and Nine Times Tables
Goal: Students will identify multiples of eight and nine.
Prior Knowledge Required: Skip counting is a quick way to add
Ones, Tens, Place value, Patterning
Addition, Rows, Columns, Diagonals
Vocabulary: column, diagonal, ones, tens, multiple
The worksheet for this lesson is similar to the worksheets for the previous two lessons—it guides students to
discover the properties of certain multiples. Because the patterns in the eight times table are more
complicated than those seen previously, you might want to guide students through the worksheet in stages
or teach the following lesson and have students complete the worksheet as a review or for reinforcement.
Ask students to skip count orally by 4s to 40. Then, write the corresponding number sentences on the board:
4
4 + 4 =
4 + 4 + 4 =
4 + 4 + 4 + 4 = and so on.
Have a volunteer find the sums. Now, ask students if they see a connection between skip counting by 4s and
repeated addition. Have students extend the pattern beyond the addition sentences written on the board.
Remind your students that the numbers that you say when skip counting by 4s, starting from 4, are called
“multiples of 4.” Review the meaning of the term “multiple” and its connection to multiplication.
Now ask your students to skip count by 8s. Suggest that they write a similar group of addition sentences for
the first five multiples of 8 and write the sums in a column:
8 08
8 + 8 = 16
8 + 8 + 8 = 24
8 + 8 + 8 + 8 = 32
8 + 8 + 8 + 8 + 8 = 40
Ask your students to circle the ones digits in the multiples. ASK: What pattern can you see in the ones digits
of the multiples of 8? What is the rule for this pattern? (Subtract 2.) What is the rule for the pattern in the tens
digits? (Add 1.)
Now write the addition sentences for the next five multiples of 8. List the sums in a column, as before. Look
at the patterns in the ones digits and the tens digit. Have the patterns changed? Have the rules changed?
Patterns & Algebra Teacher’s Guide Workbook 3:2 19 Copyright © 2007, JUMP Math For sample use only – not for sale.
(The ones digit went back to 8, and you again subtract 2 each time. The tens digit started at 4—the same
digit as the fifth multiple of 8—and you again add 1 each time.)
ASK: Can you use these patterns to write the next five multiples of 8? Invite a volunteer to write the ones
digits, then the tens digits, for the first two terms (88 and 96). ASK: What should the ones digit in the next
term be? (4) What should the tens digit in the next term be, according to the pattern? (9 + 1 = 10) Is 10 a
digit? (No; it’s a 2-digit number.) What should we do? Invite your students to find the next multiple of 8 by
adding 8 to the previous term (96), and point out that the pattern they spotted in the tens digits holds—there
are indeed 10 tens in the next term! (96 + 8 = 104 = 10 tens and 4 ones) ASK: How many tens are in 80? In
88? In 100? In 101? Let a volunteer find the next two multiples of 8. Then ask your students to write the next
five multiples of eight.
Ask your students to find the first five multiples of 9 by addition. Write them in a column (starting from 09).
ASK: Can you see a pattern in the ones digits? What’s the rule? Ask students to continue the pattern. Repeat
with the tens digits.Ask your students to add the digits in any of the multiples of 9. What do they notice?
Activity:
“Eight-Boom” Game Players stand in a circle and count up from one, each saying one number in turn.
When a player has to say a multiple of 8, he says “Boom!” instead: 1, 2, 3, 4, 5, 6, 7, “Boom!”, 9, … If a
player makes a mistake, he leaves the circle.
ADVANCED VERSION: When a number has “8” as one of its digits, the player says “Bang!” instead. If the
number is a multiple of 8 and has 8 as a digit, the player says both. EXAMPLE: “6, 7, “Boom, Bang!”, 9,” or
“15, “Boom!”, 17, “Bang!”, 19.”
Both games are a fun way to learn the eight times table, and can be used for multiples of any other number
you would like to reinforce.
Extensions:
1. Let your students use base ten materials to build multiples of nine in the standard way. Pick a multiple of
nine, say 27. How many blocks are in the standard model? (9: 2 tens and 7 ones) What do you do when
you add 9 to 27? You add 9 ones blocks, so now you have 16 ones blocks. Trade 10 ones blocks for 1
tens block. How many blocks do you have now? (9: 3 tens and 6 ones) When does this pattern break?
(When you add 9 to 90. You do not have 10 ones to trade.)
2. Continue the pattern:
9 × 1 = 09
9 × 2 = 18
9 × 3 = 27
9 × 4 = 36
What do you have to do to the second factor in the multiplication sentence to get the tens digit? Why?
(Nine is one less than ten. To find, say, 9 × 3, you add three tens and remove three ones – one for each
nine (30 - 3). You have to regroup one of the tens to do this. This means the number of tens in a multiple
of nine is one less than the second factor in the multiplication statement.)
Patterns & Algebra Teacher’s Guide Workbook 3:2 20 Copyright © 2007, JUMP Math For sample use only – not for sale.
3. Show your students a method to remember the multiples of 9 using fingers. Put your hands on the table
with fingers spread. You want to find, say, 9 × 3. Count three fingers from the left and fold the third
finger. The fingers to the left of the folded finger are the tens, and the fingers to the right of the folded
finger are the ones. The answer is 27. Ask your students to use Extensions 1 and 2 to explain why the
trick works.
4. “Who am I?”
a) I am a number between 31 and 39. I am a multiple of eight.
b) I am the largest 2-digit multiple of eight.
c) I am the smallest 3-digit multiple of eight.
d) I am a two digit number larger than 45. I am a multiple of eight and a multiple of five as well.
Twenty seven
Patterns & Algebra Teacher’s Guide Workbook 3:2 21 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-29 Patterns in Times Tables (Advanced) Goal: Students will use Venn diagrams to reinforce their knowledge of multiples of two, five, eight,
and nine.
Prior Knowledge Required: Multiples of two, five, eight and nine, Venn diagrams
Vocabulary: multiple
Students can use patterns in the times tables of various 1-digit numbers to help them learn their
multiplication facts. See “How to Learn Your Times Tables in a Week” in the Mental Math section of this
manual for some effective strategies for using patterns to learn times tables.
Review Venn diagrams (SEE: PDM3-2, empty Venn Diagrams are provided in the BLM section). Remind
your students that any numbers that cannot be placed in either circle should be written outside the circles
(but inside the box). Then draw a Venn diagram with the properties:
1. Multiples of five 2. Multiples of two
Ask your students to sort the numbers between 20 and 30 in the diagram.
Draw another Venn diagram, this time with multiples of five in one circle and multiples of eight in the other.
Ask your students to sort the numbers between 30 and 41, then between 76 and 87, into this diagram. Ask
volunteers to sort some 3-digit numbers in the diagram (for example, 125 and 140).
Draw another Venn diagram:
1. Multiplies of two 2. Multiplies of eight
Ask volunteers to sort into it the numbers: 5, 8, 14, 15, 27, 28, 56, 40, 25, 30, 99. ASK: Why is there an
empty part in the diagram? (All multiples of eight are also multiples of two.) Ask your students to add two
numbers to each part of the diagram that is not empty.
Assessment:
Make a Venn diagram with properties:
1. Multiplies of nine 2. Multiplies of five
Bonus: Add three 3-digit numbers to each part of the last diagram.
Activity: Give students a set of base ten blocks (ones and tens only). Ask them to build base ten models
of the following numbers.
a) You need 3 blocks to build me. I am a multiple of 5. (Solution: 30)
b) You need 6 blocks to build me. I am a multiple of 5. (There are 2 solutions: 15 and 60.)
c) You need 3 blocks to build me. I am a multiple of 4. (Solution: 12)
Challenging
d) You need 6 blocks to build me. I am a multiple of 3. (There are 6 solutions: 6, 15, 24, 33, 42, 51.)
Patterns & Algebra Teacher’s Guide Workbook 3:2 22 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-30 Patterns with Increasing Gaps Goal: Students will extend patterns with increasing gaps
Prior Knowledge Required: Increasing sequences, T-tables
Vocabulary: increasing sequence, difference, gap
Draw the following pattern on the board or build it with blocks:
Figure 1 Figure 2 Figure 3 Figure 4
Ask a volunteer to build the next term of the sequence. Ask your students to complete a T-table for the
number of blocks in each term. Then ASK: How many blocks are added each time? What are you adding to
each figure to make the next figure in the sequence? (A new column is being added on the right side—you
could shade or highlight the new column in each figure, to help students see it.) How many blocks are in the
column added to figure 1? How many in the column added to figure 2? This is the difference in the total
number of blocks at each stage. Ask a volunteer to write the difference in the circles beside the table.
Ask your students if they can see a pattern in the differences. Ask a volunteer to determine the next term in
the pattern of differences, i.e., how many blocks are added to figure 4 to make figure 5. Ask another
volunteer to fill in the next row of the table. Ask a third volunteer to build figure 5 to check the result.
For practice, ask students to find the differences between the terms of these sequences, extend the
sequence of the differences, and then extend the sequence itself.
a) 5 , 8 , 12 , 17 , _____ , _____ c) 11 , 14 , 20 , 29 , , _____
b) 3 , 5 , 9 , 15 , 23 , , _____ d) 6 , 8 , 13 , 21 , 32 , , _____
Figure Number
of blocks
1
2
3
4
Patterns & Algebra Teacher’s Guide Workbook 3:2 23 Copyright © 2007, JUMP Math For sample use only – not for sale.
Show the following geometrical pattern and ask how many triangles will be in the next design:
Draw a T-table for the pattern. How many triangles do you add each time? (Point out that a new row is being
added each time, and that it is always two triangles longer than the previous row.) The number of triangles in
the new row is the difference, or gap, between the total number of triangles in two successive figures. This
means that the pattern in the gaps follow the rule “Start at 3 and add 2 each time.” Where in the T-table do
you write the gaps? Ask volunteers to extend first the sequence of gaps in the circles beside the T-table,
then the sequence itself.
Assessment:
a) 15 , 18 , 22 , 29 , ______ , ______
b) 13 , 16 , 22 , 31 , 43 , ______ , ______
Bonus:
a) 7 , 17 , 37 , 67 , ______ , ______
b) 88 , 189 , 391 , 694 , 1098 , ______ , ______
Activity: Let your students build growing patterns using pattern blocks. For each sequence of patterns,
find out how many blocks you need for the next term using a T-table.
Extension: Janet is training for a marathon. On Monday she ran 5 km. Every day after that, she ran 1 km
more than on the previous day. How many kilometres did she run in total from Monday to Sunday?
Draw the T-table. The first 3 entries should appear as follows:
Day Km from the beginning
of the week (total)
1. Monday 5
2. Tuesday 11
3. Wednesday 18
6 Each day,
Janet runs
1 km more. 7
Patterns & Algebra Teacher’s Guide Workbook 3:2 24 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-31 Patterns with Larger Numbers Goal: Students will practise extending sequences of larger numbers using T-tables.
Prior Knowledge Required: Addition, Subtraction, Increasing and decreasing sequences, T-tables,
Canadian money
Vocabulary: T-table, chart, term
Tell your students that they have done so well with patterns that today you are going to give them patterns
with HUGE numbers. Present several problems and invite volunteers to solve them using T-tables.
EXAMPLES:
A normal heartbeat rate is 72 times in a minute. How many times will your heart beat in five minutes?
There are 60 minutes in an hour. George sleeps for six hours. How many minutes does he sleep?
A sprinting ostrich’s stride is 700 cm long. A publicity-loving ostrich spots a photographer 3 000 cm away and
runs towards him. How far from the camera will the ostrich be after three strides?
An extinct elephant bird weighed about 499 kg. Make a T-table to show how much five birds would weigh.
Do you see a pattern in the numbers? (HINT: Look at the ones, tens, and hundreds separately.) Can you
write the weights of six, seven, and eight birds without actually adding?
Extension: A regular year is 365 days long, a leap year (e.g., 2000, 2004, …) is 366 days long. Tom was
born on January 8, 2002. How many days old was he on January 8, 2003? On January 8, 2005?
Patterns & Algebra Teacher’s Guide Workbook 3:2 25 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-32 Extending and Predicting Positions Goal: Students will extend patterns using T-tables.
Prior Knowledge Required: Addition, Subtraction, Skip counting, Number patterns
Ordinal numbers
Vocabulary: T-table, chart, term
Ask the students if they remember what special word mathematicians use for the figures or numbers in
patterns or sequences. Write the word “term” on the board. Review with the students how they to find the
core of a pattern. Give students several patterns and ask them to identify the cores.
Draw a pattern on the board:
Ask volunteers to circle the core of the pattern, identify its length, and continue the pattern. ASK: Which
terms are circles? What will the 20th term be—a circle or a diamond? Write down the sequence of the term
numbers for the circles (1, 3, 5, 7, …) and diamonds (2, 4, 6, 8, …), and ask which sequence the number 20
belongs to.
For more advanced work students could try predicting which elements of a pattern will occur in particular
positions. EXAMPLE: If the pattern below were continued, what colour would the 23rd
block be?
The worksheet for this lesson illustrates a method for solving this sort of problem using a hundreds chart.
Students could also use number lines to solve the problem, as shown below.
STEP 1: Find the length of the core of the pattern
The core is 5 blocks long.
R R Y Y R R Y Y Y Y
R R Y Y R R Y Y Y Y
Patterns & Algebra Teacher’s Guide Workbook 3:2 26 Copyright © 2007, JUMP Math For sample use only – not for sale.
STEP 2: Mark off every fifth position on a number line and write the colour of the last block in the core above
the marked position.
The core ends on the 20th block and starts again on the 21
st block. Write the letters of the core in order on
the number line starting at 21. The 23rd
block is yellow.
Eventually you should encourage students to solve problems like the one above by skip counting. Students
might reason as follows:
“I know the core ends at every fifth block so I will skip count by 5s until I get close to 23. The core ends at 20,
which is close to 23, so I’ll stop there and write out the core above the numbers 21, 22, and 23.
Students can use the activity (below) to practise predicting terms in patterns and sequences.
Assessment:
What is the 20th term of the pattern: A N N A A N N A A N N A A? What is the 30
th term?
Bonus:
Find the core of the following pattern, then find terms 20, 30, 40, 50, …, 100 (i.e., all multiples of 10 to 100).
R W W R R W W R W W R R W W R W W…
Activity: Each pair of players will need a die and a set of coloured beads or blocks. Player 1 rolls the die
so that Player 2 does not see the result. Player 1 builds a pattern of blocks or beads with a core of the length
given by the die. Then Player 2 throws the die and multiplies the result by 10. Player 2 predicts the bead for
the term he or she got. For instance, if Player 2 rolled three on the die, he or she has to predict the 30th term
of the pattern.
I know the core ends here
19 20 21 22 23
×
R R Y
A new core starts again here
The 23rd
block is yellow
Each × shows where the core ends. The core starts again here.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
× × × ×
Y Y Y Y R R Y Y Y
24 25
Patterns & Algebra Teacher’s Guide Workbook 3:2 27 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-33 Equations Goal: Students will write and solve (by guessing and checking) algebraic equations of the type
a + x = b and x – a = b.
Prior Knowledge Required: Addition, Subtraction
Vocabulary: equation
Tell your students that today they will solve algebraic equations. Let them know that equations are like the
scales people use to weigh objects (such as apples). But in equations, there’s a problem: A black box
prevents you from seeing part of whatever object you are weighing. Solving an equation means figuring out
what is inside the black box. Write the word “equation” on the board. Ask your students if they know any
similar words (EXAMPLES: equal, equality, equivalence). So the word “equation” means “making the same,”
or balancing the scales.
Make a line down the middle of a table (e.g., with tape). Put 7 apples on one side of the line and put 3 apples
and a bag or box containing 4 more apples on the other side. Tell your students that there is the same
number of apples on both sides of the line. (You could represent the same problem with a scale and a
collection of objects of equal weight.) Ask your students to guess how many apples are in the box.
Tell your students that it is easy to represent the problem they just solved by guessing with a picture:
+ =
Invite a student to draw the missing apples in the box.
Ask your students how they could make sure that their guess about the number of apples is right without
looking into the box. Your students might suggest adding the number of apples inside the box and the
number of apples outside the box to check if it is the same as the number of apples on the other side of the
equal sign (or line on the table).
After students have had practice with this sort of problem, explain that it is inconvenient to draw the apples
all the time, so people use numbers to represent the visible quantities. This is called an equation. Let
students write the equation for the picture above:
+ 3 = 7
Patterns & Algebra Teacher’s Guide Workbook 3:2 28 Copyright © 2007, JUMP Math For sample use only – not for sale.
+ 3 = 7
Ask students to make models for the following equations, using circles for the known numbers of apples
(e.g., 7 and 11 in the first equation) and a square for the unknown number.
7 + = 11 6 + = 13 4 + = 10 + 9 = 12
Then ask your students to solve the equations they have written by guessing and checking. They should
refer to their models as required. As an alternative, your students might use a chart, as shown below for the
equation:
Guess Right side Left side Is my guess good? Why not?
2 2 + 3 = 5 7 no Too low
6 6 + 3 = 9 7 no Too high
Show students how they can solve such equations by removing the same number of items from each side of
the equation. Whatever is left on the right side is the amount on the left (in the box). Use one of the
equations students have already solved to model this process, to illustrate that it produces the same result.
Present this word problem: Sindi has a box of apples. She took two apples from the box and four were left.
How many apples were in the box before she removed the apples?
Draw the box. SAY: There are some apples inside, but we do not know how many. Draw two apples and
cross them to show that they are taken away. Four apples were left in the box, so draw them too. How many
were there from the beginning? (Six)
= =
Explain that when we draw an equation, we draw it differently. We draw the apples that we took out of the
box outside the box, with the “ – ” sign, to show that they were taken away:
– =
So to solve the equation we have to put all the apples into the box—the ones that we took out and the ones
that were left inside.
Remind your students that they also learned to write equations using numbers instead of pictures. Can they
guess what the equation for this problem will look like?
– 2 = 4
Patterns & Algebra Teacher’s Guide Workbook 3:2 29 Copyright © 2007, JUMP Math For sample use only – not for sale.
Draw several models like the ones on the worksheet, and ask your students to write the corresponding
equations. Ask volunteers to present the answers on the board. Students could use a chart, like the one
used above, to solve the equations by guessing and checking.
Then ask your students to draw models for these equations and to solve them by drawing the original
number of apples in the box:
– 6 = 9 – 7 = 12 – 5 = 3 – 3 = 10
Assessment:
Draw models for the equations. Solve the equations by guessing and checking.
3 + = 9 12 – = 7 – 4 = 11 + 7 = 14
Extensions:
The next two extensions satisfy the demands of the Western Curriculum.
1. What is the same and what is different in the following equations?
3 + = 9 3 + = 9 3 + = 9
Discuss with the students the similarities and the differences. Does the solution depend on the
symbol used in the equation?
2. a) Alina says that 3 and 4 solve the equation: 3 + + = 10. Jane says that 2 and 5 solve
this equation, too. Are both answers correct? Which other solutions can you find for this
equation?
b) Janet says that she can find different solutions for the equation 3 + = 9, too. Is she
correct? Explain.
Patterns & Algebra Teacher’s Guide Workbook 3:2 30 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-34 Adding and Subtracting Machines Goal: Students will solve (by counting forward and backward) algebraic equations of the types:
a + x = b, x + a = b, a – x = b, x – a = b.
Prior Knowledge Required: Addition, Subtraction, Counting forward and backward
Vocabulary: equation
Prepare an adding machine. First, make a strip with the numbers from 1 to 20:
For a machine that adds 3 to a number, make a rectangle of length 3 and make two slots with distance 2
between them:
Write “+ 3” in the space between the slots. Pull the strip through the slots so that two numbers are covered:
Fold the strip, so that you can show only the numbers adjacent to the slots. You will need several adding
machines like this one, for different numbers (e.g., + 4, + 6). A subtracting machine can be made the same
way; the only difference is the numbers on the strip are written in decreasing order (20, 19, … 1).
Show your students the adding machine, and explain that it adds the number 3 to any number between 1
and 17. Show them how the machine works with two or three examples, then ask them to predict the sum for
several more examples. Check their answers with the machine. Show them another adding machine, and
ask them to predict some answers.
Show your students the back of the machine, so that they see the strip of paper going through the slots.
ASK: How many numbers are covered by the rectangle? Suggest that the students count up using their
fingers to check how many numbers are covered. (The machine adds 3, and 4 + 3 = 7; I see 4 and 7, but 5
and 6 are covered, so two numbers are covered.)
SAY: I have a machine that adds 3 to a number. The answer is 8. (Show this example on the “+ 3” machine,
covering the 5 with your hand.) What number was fed into the machine? Suggest to students that they count
backward to find the number that was fed into the machine. (The answer is 8, the machine adds 3, two
1 2 3 …
+ 3 3 6 … …
Patterns & Algebra Teacher’s Guide Workbook 3:2 31 Copyright © 2007, JUMP Math For sample use only – not for sale.
numbers are covered, and the third is shown. I count backward: 7, 6 are covered, 5 is on the other side of
the machine.)
Show your students some adding machines without telling them what number the machine adds, and ask
them to find that out from the numbers that are put through the machine.
Repeat the exercises above with subtracting machines.
To conclude, show your students several machines with the numbers that go “in” and “out,” but do not show
the number that is added or subtracted, or the operation that is performed (i.e., cover the front of the
machine). Ask your students to find out what was added or subtracted.
Activity: A game for pairs. Give your students a deck of cards with signs and 1-digit numbers, such as
+ 3 or - 5 on each. Player 1 takes a card at random without showing it to Player 2. Player 2 gives Player 1 a
number that is more than 12. Player 1 performs the operation shown on the card with the number given by
Player 2 and tells Player 2 the result. Player 2 has to guess what card Player 1 has. For example: Player 1
has the card “- 5,” Player 2 says “13,” Player 1 then says “8.”
Extensions:
1. Robin has an adding machine. Robin enters the number 12 into the machine, takes the answer, and
enters it back into his machine. The answer is 16. What number does Robin’s machine add to the
numbers he enters?
2. Colin has an adding machine that adds 3 to a number. Colin enters a number into the machine, takes the
answer, and enters it into the machine again. He takes the second answer and enters it into the machine
a third time! The machine gives him the number 15. What was the first number that Colin fed into his
machine?
Patterns & Algebra Teacher’s Guide Workbook 3:2 32 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-35 Equations (Advanced) Goal: Students will solve (by guessing and checking) algebraic equations of the types:
a + x = b, x + a = b, a – x = b, x – a = b.
Prior Knowledge Required: Addition, Subtraction, Counting forward and backward
Vocabulary: equation
Review the previous lesson. Write the equation + 3 = 7 on the board and ASK: Which adding machine
does this equation remind you of? Which number is added? The result is 7, so how did you find out what
number was fed into the machine? Encourage your students to solve several equations of this type by
counting backward.
Continue with equations of the type 5 + = 8. Explain that this is also an adding machine, but you do not
know what number is added. Ask your students to use finger counting to find what number was added.
Repeat with equations of the types 12 – = 8 and – 3 = 6. Give your students a mix of equations to
solve. They can either use finger counting or guess and check. As a bonus, add several equations of the
type 3 × = 12.
Remind your students about the models they were using for equations in lesson PA3-34. Ask a volunteer to
draw a model for an addition equation. Ask your students to tell a story that fits the model. Invite your
students to draw several different models and tell different stories for the same equation. Repeat with
subtraction equations.
Tell your students that they can also create problems for multiplication equations. Explain that “2 ×” means
that some quantity is taken two times.
For example:
2 × =
Present the equation 2 × = 10 and ask your students to draw the appropriate number of circles
in the box:
2 × =
Patterns & Algebra Teacher’s Guide Workbook 3:2 33 Copyright © 2007, JUMP Math For sample use only – not for sale.
Explain that this model may represent several problems, such as:
a) Jane has some apples. George has twice as many apples as Jane. He has 10 apples. How many
apples does Jane have?
b) Tim and Tom ate the same number of cookies. They ate 10 cookies together. How many cookies did
each of them eat?
c) How many sets of two are in 10?
Present more equations and ask your students to draw models for these equations. Ask your students to
create short stories that fit their models.
Extension: Provide students with the BLM “Hanji Puzzles.” These 4 worksheets introduce the students
to Hanji Puzzles and then asks them to use what they have learned about finding the missing number in
equations.
Patterns & Algebra Teacher’s Guide Workbook 3:2 34 Copyright © 2007, JUMP Math For sample use only – not for sale.
PA3-36 Problems and Puzzles PA3-36 is a review worksheet, which can be used for practice.
Extensions:
1. An empty box or a letter can represent a number. Ask students if they can find the solutions to these
equations.
a) + 2 = 5 b) + 5 = 9 c) + 3 = 4
d) a + 4 = 10 e) a + 17 = 25 f) b + 12 = 19
g) – 3 = 2 h) – 5 = 4 i) + 7 = 3
j) a – 2 = 8 k) x – 6 = 12 l) b – 9 = 15
Advanced
m) 25 – 4 = 15 + n) 32 + 6 = – 9 o) 41 + 7 = 50 –
The equation + 2 = 6 could represent the problem: “Peter has six marbles. He has 2 more marbles
than Fran. How many marbles does Fran have?” (The box represents Fran’s amount.)
Ask students to make up a problem and then write an equation that represents the problem. Students
should recognize that it doesn’t matter what symbol they use to represent the unknown.
2. Fill in the missing numbers (from 1 to 25) so that the numbers increase by one:
HINT: The sequence can travel up, down, left, right, or in a combination of directions.
a) b) c)
1 10
4
7
13
17
4 1
7 9
15
7 9
1
5 3
17 13
Workbook 3 - Patterns & Algebra, Part 2 1BLACKLINE MASTERS
Calendars _____________________________________________________________2
Colouring Exercise ______________________________________________________3
Hanji Puzzles ___________________________________________________________4
Hundreds Charts________________________________________________________8
Mini Sudoku ___________________________________________________________9
Sudoku—The Real Thing ________________________________________________11
Sudoku—Warm Up ____________________________________________________12
Venn Diagram _________________________________________________________14
PA3 Part 2: BLM List
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Calendars
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
NAME OF MONTH:
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
NAME OF MONTH:
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
NAME OF MONTH:
Workbook 3 - Patterns & Algebra, Part 2 3BLACKLINE MASTERS
Colouring Exercise
Colour the even numbers blue, and leave the odd numbers white.
45
93 91 3 42
5
1038
9872
1670
12
66102
4
40
1845
68
36
24 113
1
34
10620
32
14
3
27
15
17
39
53
29
31
49
120
8
2 14
2626
18
6
42
4
10 917
3
15
7
38
345
19
74
16
76
18
8
44
86
45
58
48
79
37
51
75
278
3373
17
22
46
80 56
86
20
48
52
35
6
54
88
4524
84
1351
6917
4
57
531
528243
55
45
19
57
64
26
21
47
41
23
5925
71
54
68
60
96
94
30
92 2810
90
94
33
73 71
357
9
1
What do you get? _________________________________________________________________________
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Hanji PuzzlesCount the shaded squares in each row.
Write an addition sentence to fi nd out how many shaded squares altogether.
Write 2 addition sentences for the total number of shaded squares.
1
+ 0
+ 2
3
1
+ 0
+ 0
3
1
+ 0
+ 0
3
Now count by column.
3 + 1 = + + 1 =
2
1
+ 2
1 + 3 + 1 =
2
1
+ 2
1 + 3 + 1 =
+ + 1 =
Workbook 3 - Patterns & Algebra, Part 2 5BLACKLINE MASTERS
Hanji Puzzles (continued)
Circle the full rows and columns.
Shade the full rows and columns. Is the right number shaded in each row and column?
5 04 03 02 0
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1 3 4 23 12 1
2
1
1
2
1
1
1
2
1
2
2
1
6 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Hanji Puzzles (continued)
Finish solving these puzzles. Start by crossing out the squares you can’t shade.
1 2 3 3 3
4
5
3
4 2 3
3
2
3
1
5 3 2
1
3
1
3
2
BONUS:
4 2 4 2 3 2 2 4 2 3 2 4
12
6
12
4
Find the rows that have enough shaded. Cross out the white squares in those rows.
Then do the columns.
1 4 4 2
2
2
3
4
2 1 3
1
3
2
2 3 0 3 3
4
4
3
Workbook 3 - Patterns & Algebra, Part 2 7BLACKLINE MASTERS
Hanji Puzzles (continued)
This Hanji puzzle is not possible. Can you see why? HINT: Try solving it.
3 4 2 1
4
2
4
0
Solve the Hanji puzzles.
STEP 1: Shade the full rows and columns.
STEP 2: Cross out the squares you can’t shade.
STEP 3: Finish the puzzle. Check your answer.
4 2 5
3
2
1
3
2
1 5 1 4
4
2
2
2
1
2 3 0 3 3
4
4
3
BONUS:
4 3 2
2
3
1
3
3 3 4
4
2
4
1
4
3 + + 2 = 3 + 1 + 3 + 2
3 2
3
1
3
2
Find the missing number. Then solve the puzzle.
8 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Hundreds Charts
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 60
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Workbook 3 - Patterns & Algebra, Part 2 9BLACKLINE MASTERS
Mini Sudoku
In these mini Sudoku problems, the numbers 1, 2, 3 and 4 are used.
Each number must appear in each row, column and 2 × 2 box.
When solving Sukoku problems:
1. Start with a row or square that has more than one number.
2. Look along rows, columns and in the 2 × 2 boxes to solve.
3. Only put in numbers when you are sure the number belongs there (use a pencil with
an eraser in case you make a mistake).
EXAMPLE:
Here’s how you can fi nd the numbers in the second column:
The 2 and 4 are given so we have to decide where to place the 1 and the 3.
There is already a 3 in the third row of the puzzle so we must place a 3 in the fi rst row of the
second column and a 1 in the third row.
Continue in this way by placing the numbers 1, 2, 3 and 4 throughout the Sudoku. Before you
try the problems below, try the Sudoku warm-up on the following worksheet.
1. a) b) c)
2. a) b) c)
1
2
3
4
1 3
2
1 3
4
1 4
4 3 2
4 1 3
3 4
2 1
1 4 3
4 3 2
3 4
1 4
4 1
2 3
3 2
2 1
3
1 4
1
2
4 3
4
1
3
3 2
10 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Mini Sudoku (continued)
Try these Sudoku Challenges with numbers from 1 to 6. The same rules and strategies apply!
Bonus
3. 4.
5. 6.
1 4 3 5
6 4 5 1
2 3 6
4 1 6 2 3
5 1 2
2 5 3 4 6
5 4 2 1
1 5 6
3 6 4 5
6 4
2 3 1
2 3
4 3
6 4 5 2
6
5 1
1 5 2 3
2
2 3
4 1 2
6 5 1
4 3
6
3 5 1
Workbook 3 - Patterns & Algebra, Part 2 11BLACKLINE MASTERS
Sudoku—The Real Thing
Try these Sudoku puzzles in the original format 9 × 9.
You must fi ll in the numbers from 1 through 9 in each row, column and box. Good luck!
Bonus
Super Bonus
For more Sudoku puzzles, check the puzzle section of your local newspaper!
4 2 1 8 3 9
7 1 4 6
6 7 3 2
3 6 9 5
6 2 8 4 5 1 3
8 7 6
9 4 8
1 8 3 6 5 7
5 7 1 4 2
5 3 6 7 8
2 8 6
4 7 3 9
7 1 3 4
1 4 5 2
4 9 1
2 4 6 8 7
8 3 9 5 1
9 1 4
12 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Sudoku—Warm Up
1. Each row, column or box should contain the numbers 1, 2, 3, and 4. Find the missing
number in each set.
a) c) d) e) f)
b)
2. Circle the pairs of sets that are missing the same number.
a) b) c)
3. Find the number that should be in each shaded square below. REMEMBER: In sudoku puzzles
a number can only appear once in each row, column, or box.
a) b) c)
d) e) f)
4. Fill in the shaded number. Remember that each row, column, and box must have the numbers
1, 2, 3, and 4.
a) b) c)
4
1 31
3
2
1 3 4 1
4 2
2
4
34 1 2
3 4
2
3
2
1
2 3 4
3
4 2
4
3
1
1 3 2
2
3
2 4 3
4
4
3
1
1
4 2 1
1 2
3
1
3 4
3
4
4 2
1 2
2 1 3
3
2
4
2 4
1
4
3
4 2
Workbook 3 - Patterns & Algebra, Part 2 13BLACKLINE MASTERS
Sudoku—Warm Up (continued)
d) e) f)
Bonus
Can you fi nd the numbers for other empty squares (besides the shaded ones)?
6. Try to solve the following puzzles using the skills you’ve learned.
a) b) c)
d) e) f)
7. Find the missing numbers in these puzzles.
a) b) c)
d) e) f)
Now go back and solve the mini Sudoku puzzles!
1 3
2
1
2
4
1
3
4
3 2 4
4 2 1
2 4
4 1 2
4 2 1
1 2
1 4 2
3 4
3 1
2 4
1 3
4 2
1 3 4
4 3 1
1
2 4
3 2 4
2
1 2
4
4
3 1
3 2 4
1 2
2 3
1 2 4
4 2
1 4
2
4 2
3 1
2 1 4
3 1
3
4
14 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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Venn Diagram
Number Sense Teacher’s Guide Workbook 3:2 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-51 Ordinal Numbers Goal: Students will use ordinal numbers correctly.
Prior Knowledge Required: Number lines Counting
Vocabulary: ordinal numbers, position, ordinals from first to tenth.
Teach students that ordinal numbers are used to tell the position of objects.
Write the ordinal words from first to tenth on the board in order. Leave a lot of space between words. Ask
volunteers if they know what the words mean.
Hand out cards numbered 1 to 10 to ten volunteers. Ask the volunteers to line up in the order of their
number, underneath the appropriate word. The student with “1” should stand under the word “first” and so
on. Tell the class to pretend that these students are in line for tickets and ask for examples of what the
tickets might be for (a sporting event, a play, a movie theatre, and so on). Ask various volunteers to
demonstrate that they know where they are in line. EXAMPLES: Will the second person please clap their
hands? Will the 6th person please take one step forward and then one step back? Will the 8th person please
turn around in a complete circle?
Ask the class to tell you who is 3rd in line. Who is 4th? Who is 9th? Who is first? Who is last? How many
people are before the 4th person in line? How many are after the 8th person? Use your students’ names to
ask a question like “How many places before Mark is Sara?” Demonstrate counting back from Mark’s place
in line to Sara’s. SAY: “Tony is 1 place before Sara, Lisa is 2 places before Sara, Bilal is 3 places before
Sara” and so until you reach Mark.
ASK: How many are before the 8th person and after the 4th person? How many people are between the 2nd
and 4th people? Between the 2nd and 5th people? The 2nd and 7th? The 5th and 9th? Tell students that Sally
finds the number of people between the 2nd and 4th people by finding 4 – 2, so she says that there are 2
people between the 2nd and 4th people—is she right? How much is she off by? Then repeat for the other
examples: the 2nd and 5th people, the 2nd and 7th, the 5th and 9th. How much is Sally off by each time? How
can you change Sally’s answer to make it right? Then ask students to see if this strategy works for various
other pairs of small ordinal numbers. (EXAMPLES: 3rd and 8th, 2nd and 9th, 3rd and 6th, 6th and 9th. 7th and 9th,
8th and 9th.) Then challenge students to extend this pattern to larger ordinal numbers. For example, if there
are 30 people in line, have students find the number of people in between the 14th and the 28th by subtracting
and then subtracting 1, or between the 3rd and the 25th, and so on.
Review the words “vowel” and “consonant.” Then ASK: What is the 4th letter in the alphabet? What is the 4th
vowel in the alphabet? What is the 3rd letter in the word “Montreal”? What is the 3rd vowel in the word
“Montreal”? Which two letters are the same in the word “apple”? (Have students phrase their answer in terms
of ordinal numbers, i.e. 2nd and 3rd). Have students find two letters (and describe them by their ordinal
position) that are the same in each word:
a) moon b) sleep c) penny d) counting
Number Sense Teacher’s Guide Workbook 3:2 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
Bonus: Make up a word that has 2 letters the same and describe the position of those letters. As an extra
challenge, make up a word with 3 letters the same and describe the position of those letters.
Have students count by 5s, starting at 5. ASK: What is the 3rd number you say? What is the 7th number you
say? Teach students to keep track by using their fingers.
ASK: What is the 4th letter you say if:
a) You say the alphabet starting from H?
b) You say the alphabet starting from V?
c) You say the alphabet backwards starting from T?
d) You say the vowels starting from A?
What is the 7th number you say if:
a) You count by 2s starting at 8?
b) You count by 2s starting at 17?
c) You count by 5s starting at 35?
d) You count by 5s starting at 49?
e) You count by 100 starting at 433?
Then draw a number line from 0 to 30, labelled with only the multiples of 10. Show students how to use the
number line to find the 7th number they say when counting by 3, starting from 3. Students can mark 3 with an
X, mark every third number with an X and then count to find the 7th X.
Activities:
1. Working in Pairs
For a random way to pair up students, give half the class cards with numbers (say 1 to 10 and 21 to 25
if there are 30 students in the class) and give the other half cards with ordinal number endings put st,
nd, rd, and th in quotation marks “st”, “nd”, “rd”, and “th” (st (2 cards), nd (2 cards), rd (2 cards) and th
(9 cards)). Have students find a partner that matches, for example, 2 matches with “nd” because the
corresponding ordinal is 2nd or second, 4 through 10 all match with “th”. Students will see quickly that
“th” is the easiest to match with.
2. Decoding Messages
Teach students how to use skip counting by 5 to find the position of letters in the alphabet quickly:
A B C D E = 5th
F G H I J = 10th
K L M N O = 15th
P Q R S T = 20th
U V W X Y = 25th
Z
Number Sense Teacher’s Guide Workbook 3:2 3 Copyright © 2007, JUMP Math For sample use only – not for sale.
ASK: What is the 22nd letter? The 14th? The 11th? The 19th? The 26th? And then, to make sure they’re paying
attention, ask: The 27th?
Have students decode messages using ordinal numbers. For example, the code for “recess” is:
18th 5th 3rd 5th 19th 19th
Some examples of messages you could encode for your students to decode include:
• To make glue, mix water and flour.
• No math homework today.
• Ordinal numbers are used to tell position.
• Ordinal number words are used for fractions too.
Students could make up a message for a partner to decode.
3. Money
The following is a sample problem from an Ontario Ministry of Education Guide for Teachers. Students
could solve the problem using play money:
Remove the 3rd coin.
Move the last coin into the second place.
Remove the 4th coin.
Move the 5th coin into 1st place.
Remove the second coin.
How much money do you have left?
Extension: Ask students to write the third letter of the word “fast,” the second letter of the word “puppy,”
the last letter of the word “mop,” the most common letter in the word “green” and then the fourth letter in the
word “fourth.” Ask students what word they spelled? Encourage them to make up their own similar puzzles,
either by using their classmates’ names as words or making up their own names.
25¢ 5¢ 5¢ 5¢ 5¢ 10¢ 10¢
Number Sense Teacher’s Guide Workbook 3:2 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-52 Round to the Nearest Tens and NS3-53 Round to the Nearest Hundreds Goal: Students will round to the closest ten or hundred, except when the number is exactly half-way
between two multiples of ten or a hundred.
Prior Knowledge Required: Number lines
Concept of closer
Vocabulary: multiple
Show a number line from 0 to 10 on the board:
0 1 2 3 4 5 6 7 8 9 10
Circle the 2 and ask if the 2 is closer to the 0 or to the 10. When students answer 0, draw an arrow from the
2 to the 0 to show the distance. Repeat with several examples and then ask: Which numbers are closer to
0? Which numbers are closer to 10? Which number is a special case? Why is it a special case?
Then draw a number line from 10 to 30, with 10, 20 and 30 a different colour than the other numbers.
Circle various numbers (not 15 or 25) and ask volunteers to draw an arrow showing which number they
would round to if they had to round to the nearest ten.
Repeat with a number line from 50 to 70, again writing the multiples of 10 in a different colour. Then repeat
with number lines from 230 to 250 or 370 to 390, etc.
Ask students for a general rule to tell which ten a number is closest to. What digit should they look at? How
can they tell from the ones digit which multiple of ten a number is closest to?
Then give several examples where the number line is not given to them, but always giving them the two
choices. EXAMPLE: Is 24 closer to 20 or 30? Is 276 closer to 270 or 280?
Tell students that the multiples of 10 are the numbers they say when they start at 0 and skip counting by 10,
namely 0, 10, 20, 30, and so on. ASK: Is 70 a multiple of ten? Is 130 a multiple of 10? What about 37? How
can you tell whether or not a number is a multiple of 10?
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Number Sense Teacher’s Guide Workbook 3:2 5 Copyright © 2007, JUMP Math For sample use only – not for sale.
ASK: Which two multiples of 10 is 37 between? (30 and 40) How can you tell? How many tens are in 37? (3)
What is one more ten? (4) So 37 is between 3 tens and 4 tens. How many tens are in 97? (9) What number
has exactly 9 tens? (90) What is one more ten than 9 tens? (10 tens) What number has 10 tens? (100) What
two multiples of 10 is 97 between? (90 and 100) How many tens are in 794? (79) Show students how they
can cover up the ones digit to find the number of tens. What is one more ten than that? (79 + 1 = 80 tens).
Which two multiples of ten is 794 between? (790 and 800)
Ensure that all students can tell you which number is another way of saying:
a) 54 tens b) 3 tens c) 10 tens d) 99 tens e) 100 tens f) 1430 tens
Have students find which two multiples of 10 the following numbers are between:
a) 53 b) 153 c) 671 d) 809 e) 998 Bonus: f) 789 432 g) 12 349 087
Then have students round each number to the nearest ten. Explain that to round a number to the nearest ten
means to find the multiple of ten that the number is closest to.
STEP 1: Decide which two multiples of ten the number is between.
STEP 2: Look at the ones digit to decide which multiple of ten the number is closest to.
a) 327 b) 411 c) 32 d) 48 e) 196 Bonus: 53 098 006
Tell students that when you round a 3-digit number to the nearest ten, you usually get a 3-digit number.
Challenge your students to find an exception. (The exceptions are 995, 996, 997, 998 and 999—995 is not
closer to either 990 or 1000; this case will be discussed in the next lesson)
Repeat the lesson with a number line from 0 to 100, that shows only the multiples of 10.
0 10 20 30 40 50 60 70 80 90 100
At first, only ask students whether numbers that are multiples of 10 (30, 70, 60 and so on) are closer to 0
or 100. (EXAMPLE: Is 40 closer to 0 or 100? Draw an arrow to show this.) ASK: Which multiples of 10
are closer to 0 and which multiples of 10 are closer to 100? Which number is a special case? Why is it a
special case.
Then include numbers that are not multiples of 10. First ask your students where they would place the
number 33 on the number line. Have a volunteer show this. Then ask the rest of the class if 33 is closer to 0
or to 100. Repeat with several numbers. Then repeat with a number line from 100 to 200 and another
number line from 700 to 800.
Review the word “multiple” and ASK: If a multiple of 10 means “the numbers you say when skip counting by
10s starting from 0,” what do you think a multiple of 100 is? Say various numbers and have students tell you
whether each number is a multiple of 100.
(EXAMPLES: 320; 1 500; 78 000; 341; 12 341; 12 300; 890)
ASK: How can you tell whether or not a number is a multiple of 100?
Remind students that to find the number of tens, we can cover up the ones digit and read the number we
see. ASK: How can we find the number of hundreds in a number? What digits should we cover up? (Cover
up the ones and tens digits.)
Number Sense Teacher’s Guide Workbook 3:2 6 Copyright © 2007, JUMP Math For sample use only – not for sale.
Have students find the number of hundreds in various numbers.
(EXAMPLES: 349; 890; 1 954; 39 876 421)
ASK: Once we’ve found the number of hundreds in a number, how can we find the two multiples of a
hundred the number is between?
Have students decide which two multiples of 100 the above examples are between.
Ask your students for a general rule to tell which multiple of a hundred a number is closest to.
What digit should they look at? How can they tell from the tens digit which multiple of a hundred a number is
closest to? When is there a special case? Emphasize that the number is closer to the higher multiple of
100 if its tens digit is 6, 7, 8 or 9 and it’s closer to the lower multiple of 100 if its tens digit is 1, 2, 3 or 4. If
the tens digit is 5, then any ones digit except 0 will make it closer to the higher multiple. Only when the tens
digit is 5 and the ones digit is 0 do we have a special case where the number is not closer to either.
Activity: Attach 11 cards to a rope so that there are 10 cm of rope between each pair of cards. Write the
numbers from 30 to 40 on the cards so that you have a rope number line. Make the numbers 30 and 40 more
vivid than the rest. Ensure that the numbers are stuck to the rope so that they cannot move. Take a ring that
can slide freely on the rope and pull the rope through it.
Ask two volunteers to hold the number line taught. Ask a volunteer to find the middle number between 30
and 40. How do you know that this number is in the middle? What do you have to check? (the distance to the
ends of the rope—make a volunteer do that). Let a volunteer stand behind the line holding the middle.
Explain to your students that the three students with a number line make a rounding machine. The machine
will automatically round the number to the nearest ten. Explain that the machine finds the closest ten. Put the
ring on 32. Ask the volunteer who is holding the middle of the line to pull it up, so that the ring slides to 30.
Try more numbers. Ask your students to explain why the machine works.
Number Sense Teacher’s Guide Workbook 3:2 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-54 Rounding Goal: Students will round whole numbers to the nearest ten or hundred.
Prior Knowledge Required: Knowing which two multiples of ten or a hundred a number is between
Finding which multiple of ten or a hundred a given number is closest to
Vocabulary: rounding, multiple
Review rounding to the nearest ten when the ones digit is not 5.
Then tell your students that when the ones digit is 5, it is not closer to either the smaller or the larger ten,
but we always round up. Give them many examples to practice with: 25, 45, 95, 35, 15, 5, 85, 75, 55, 65. If
some students find this hard to remember, you could give the following analogy: “I am trying to cross the
street, but there is a big truck coming, so when I am part way across I have to decide whether to keep going
or to turn back. If I am less than half way across, it makes sense to turn back because I am less likely to get
hit. If I am more than half way across, it makes sense to keep going because I am again less likely to get
hit. But if I am exactly half way across, what should I do? Each choice gives me the same chance of getting
hit.” Have them discuss what they would do and why. Remind them that they are, after all, trying to cross
the street. So actually, it makes sense to keep going rather than to turn back. That will get them where they
want to be.
Another trick to help students remember the rounding rule is to look at all the numbers with tens digit 3 (i.e.
30-39) and have them write down all the numbers that we should round to 30 because they’re closer to 30
than to 40. Which numbers should we round to 40 because they’re closer to 40 than to 30? How many are
in each list? Where should we put 35 so that it’s fair?
Then move on to 3-digit numbers, still rounding to the nearest tens: 174, 895, 341, 936, etc.
Bonus: Include 4- and 5-digit numbers.
Then move on to rounding to the nearest hundreds. ASK: Which multiple of 100 is this number closest to?
What do we round to? Start with examples that are multiples of 10: 230, 640, 790, 60, 450 (it is not closest
to either, but we round up to 500). Then move on to examples that are not multiples of 10. (EXAMPLES:
236, 459, 871, 548)
SAY: When rounding to the nearest 100, what digit do we look at? (The tens digit). When do we round
down? When do we round up? Look at these numbers: 240, 241, 242, 243, 244, 245, 246, 247, 248, 249.
What do these numbers all have in common? (3 digits, hundreds digit 2, tens digit 4). Are they closer to 200
or 300? How can you tell without even looking at the ones digit?
Then tell your students to look at these numbers: 250, 251, 252, 253, 254, 255, 256, 257, 258, 259. ASK:
Which hundred are these numbers closest to? Are they all closest to 300 or is there one that’s different?
Why is that one a special case? If you saw that the tens digit was 5, but you didn’t know the ones digit, and
you had to guess if the number was closer to 200 or 300, what would your guess be?
Number Sense Teacher’s Guide Workbook 3:2 8 Copyright © 2007, JUMP Math For sample use only – not for sale.
Would the number ever be closer to 200? Tell your students that when you round a number to the nearest
hundred, mathematicians decided to make it easier and say that if the tens digit is a 5, you always round
up. You usually do anyway, and it doesn’t make any more sense to round 250 to 200 than to 300, so you
might as well round it up to 300 like you do all the other numbers that have tens digit 5.
Then ASK: When rounding a number to the nearest hundreds, what digit do we need to look at? (The tens
digit.) Then write on the board:
Round to the nearest hundred: 234 547 651 850 493
Have a volunteer underline the hundreds digit because that is what they are rounding to.
Have another volunteer write the two multiples of 100 the number is between, so the board
now looks like:
Round to the nearest hundred: 234 547 651 850 973
200 500 600 800 900
300 600 700 900 1000
Then ask another volunteer to point an arrow to the digit they need to look at to decide whether
to round up or round down. Ask where is that digit compared to the underlined digit? (It is the
next one). ASK: How do you know when to round down and when to round up? Have another volunteer
decide in each case whether to round up or down and circle the right answer.
Tell students that most 3-digit numbers, when rounded to the nearest hundred, will have 3 digits. Which
numbers will be exceptions? (any number from 950 to 999)
Extension: Ask students to round a 3-digit number to all possible places.
EXAMPLE: 1382
1000
thousands
1400
Hundreds
1380
tens
Number Sense Teacher’s Guide Workbook 3:2 9 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-55 Estimating Sums and Differences Goal: Students will estimate sums and differences by rounding each addend to the nearest ten or
hundred.
Prior Knowledge Required: Rounding to the nearest ten or hundred
Adding and subtracting
Vocabulary: the “approximately equal to” sign ( ≈ ), estimating
Show students how to estimate 52 + 34 by rounding each number to the nearest ten: 50 + 30 = 80.
SAY: Since 52 is close to 50 and 34 is close to 30, 52 + 34 will be close to, or approximately, 50 + 30.
Mathematicians have invented a sign to mean “approximately equal to.” It’s a squiggly “equal to” sign: ≈.
So we can write 52 + 34 ≈ 80. It would not be right to put 52 + 34 = 80 because they are not actually equal;
they are just close to, or approximately, equal.
Tell students that when they round up or down before adding, they aren’t finding the exact answer, they are
just estimating. They are finding an answer that is close to the exact answer. ASK: When do you think it
might be useful to estimate answers?
Have students estimate the sums of 2-digit numbers by rounding each to the nearest ten. Remind them to
use the ≈ sign.
EXAMPLES:
41 + 38 52 + 11 73 + 19 84 + 13 92 + 37 83 + 24
Then ASK: How would you estimate 93 – 21? Write the estimated difference on the board with students:
93 – 21 ≈ 90 – 20
= 70
Ensure that students can add and subtract multiples of 10 (EXAMPLES: 30 + 20, 70 – 40, 130 – 50). Have
students estimate the differences of 2-digit numbers by again rounding each to the nearest ten.
EXAMPLES:
53 – 21 72 – 29 68 – 53 48 – 17 63 – 12 74 – 37
Then have students practise estimating the sums and differences of:
• 3-digit numbers by rounding to the nearest ten (EXAMPLES: 421 + 159, 904 – 219).
• 3- and 4-digit numbers by rounding to the nearest hundred (EXAMPLES: 498 + 123, 4 501 – 1 511).
Ensure that students can add and subtract multiples of 100 (EXAMPLES: 300 – 100, 600 + 300,
800 – 200)
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Bonus: Students who finish quickly may add and subtract larger numbers, rounding to tens, hundreds, or
even thousands.
Teach students how they can use rounding to check if sums and differences are reasonable.
EXAMPLE:
Daniel added 273 and 385, and got the answer 958. Does this answer seem reasonable?
Students should see that even rounding both numbers up gives a sum less than 900, so the answer can’t
be correct. Make up several examples where students can see by estimating that the answer cannot be
correct.
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NS3-56 Estimating Goal: Students will solve word problems by estimating rather than calculating.
Prior Knowledge Required: Estimating sums and differences by rounding
Reading word problems and knowing when to add or subtract
Vocabulary: estimating
Tell students that you want to estimate how many apples were sold altogether if 58 red apples were sold
and 21 green apples are sold. Tell students that it would be easier to add these numbers if they were
multiples of 10 and ASK: Are these numbers close to multiples of 10? What is the closest multiple of 10 to
58? (60) To 21? (20) What is 60 + 20? (80) Do you think that 80 is a good estimate for 58 + 21? Will
58 + 21 be close to 60 + 20? Why? What is the actual answer to 58 + 21? (79) Was 80 a good estimate?
Have students estimate the total number of apples sold in these situations:
a) 27 red apples were sold and 42 green apples were sold;
b) 46 red apples were sold and 78 green apples were sold;
c) Jenn sold 52 apples and Rita sold 31 apples;
d) Jenn sold 42 apples and Rita sold 29 apples.
Write out some of the answers on the board, using the “approximately equal to” sign
(EXAMPLE: 27 + 32 ≈ 30 + 30 = 60).
Write the following question on the board:
About how many more green apples than red apples were sold in part a)?
ASK: What word in that question tells you I only want an estimate? Does the question ask for the sum of the
numbers of green and red apples or the difference between them? How do you know? What operation
should I use to find the difference—addition or subtraction? Tell students that you would find it easier to
subtract if the numbers were multiples of 10. ASK: What multiples of 10 are closest to the number of red
and green apples? If there are about 30 red apples and about 40 green apples, about how many more
green apples were sold than red apples?
Have students estimate how many more green apples were sold than red apples in b), and then how many
more apples Jenn sold than Rita in c) and d).
Tell students (and write on the board): Greg collected 37 stamps, Ron collected 72 stamps, and Sara
collected 49 stamps. ASK: How many more stamps did Ron collect than Sara? How many more did Sara
collect than Greg? How many more stamps does Ron have than Greg?
Give students similar problems with 3-digit numbers, asking them to round to the nearest ten to estimate the
answer. Then give problems with 3- and 4-digit numbers and ask students to round to the nearest hundred.
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Ask students to estimate this sum: 27 + 31. Then ask them to find the actual answer. What is the difference
between the estimate and the actual answer? Which number is larger, the estimate or the actual answer?
How much larger? What operation do they use to find how much more one number is than another? (They
should subtract the smaller number from the larger number.) Repeat with sums and differences of:
2-digit numbers, rounding to the nearest ten
(EXAMPLES: 39 + 41, 76 – 48).
3-digit numbers, rounding to nearest ten
(EXAMPLES: 987 – 321, 802 + 372).
3- and 4-digit numbers, rounding to nearest hundred
(EXAMPLES: 3 401 + 9 888, 459 – 121).
Draw on the board:
ASK: How many balls are in each box? If I want to know how many are in 4 boxes, what is an easier number
to multiply 4 by that is close to 12? (10) What makes that number easy to multiply by? About how many balls
are in 4 boxes? (40).
Repeat with the following pictures, having students estimate how many balls are in 7 boxes:
a) b) c)
NOTE: Encourage your students to use estimating to judge the reasonableness of their answers. Give them
the following questions and ask them to tell you what they would estimate the answer will be before they
perform the operation.
a) 382 + 217 b) 427 + 604 c) 923 – 422 d) 875 – 215
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Activities:
1. Count the number of slices in an orange and estimate the number of slices in 6 oranges.
2. Take a handful of counters and drop them onto a sheet of paper folded into 4 equal regions. Count the
number of counters in one of the four regions and estimate the number of counters you dropped.
3. Fill up a clear plastic container with beans or small blocks. Ask students to estimate the number of beans
or blocks in the container. Ask them how they arrived at their estimate. Did they count out 10 blocks to
use 10 as the known quantity? Or 100 blocks? 10 beans or 100 beans?
Extensions:
1. Estimate the number of pages in your JUMP Math Part 2 workbook. Note that the page number on the
last page of the workbook shows the number of pages in both Part 1 and Part 2, so students will need to
round both this number and the number of pages in their Part 1 workbook and then subtract. To guide
students, ASK: What page does Part 2 end at? What page does it start at? So what page does Part 1
end at? How many pages are in both Part 1 and Part 2 together? How many pages are in just Part 1?
How can we find the number of pages in just Part 2? What operation should we use? To make the
numbers easier to subtract, what numbers close to these numbers are easier to work with?
2. Estimate the number of pages in all the JUMP Math Part 2 workbooks in the class. Hint: Round the
number of pages to the nearest hundred and the number of workbooks to the nearest ten.
The following three extensions are adapted from the Atlantic Curriculum Guide (A2):
3. Which estimate is closer to find 46 + 25? 50 + 20 OR 50 + 30 How do you know?
4. Is the following estimate for 82 – 47 too high or too low: 80 – 50? How do you know?
5. If you have a loonie, do you have enough money to buy:
• A pencil for 12¢
• An eraser for 25¢
• A notebook for 29¢
• A pen for 19¢
Explain your strategy.
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NS3-57 Mental Math and Estimation Goal: Students will use doubles to add mentally.
Prior Knowledge Required: Counting by 2s
Doubling 2-digit numbers
Relationship between skip counting and multiplying
Arrays
Vocabulary: Double, double plus one, double minus one
Review doubling (see NS3-38) with your students.
Prepare 20 paper circles to use as counters and to tape to the board. Draw 2 circles and put 5 counters in
each of the two circles. ASK: What double have I shown? (Double 5)
Draw two new circles and put 3 counters in one circle and 5 in another. ASK: What addition statement does
this show? (3 + 5 = 8) Challenge students to think of a way to move one of the counters so that there is the
same number in each circle. What double have you shown? (4 + 4 = 8)
Repeat with larger numbers. EXAMPLES: 8 + 6, 7 + 9, 5 + 7, 11 + 9. Bonus: Move two counters to change
7 + 11 into a double.
Teach students to change addition statements into doubles without using counters to help them. Emphasize
that when you move a counter from one pile to the other, you are adding 1 counter to one pile and
subtracting 1 counter from the other pile, so 8 + 6 becomes 8 – 1 + 6 + 1 = 7 + 7.
Have students change 7 + 5 to a double, then 6 + 8, then 10 + 12, then 33 + 31, then 62 + 60.
Draw the following picture on the board:
Tell your students that the four counters are in front of a mirror. Ask a volunteer to draw what they would see
in a mirror on the other side of the dotted line. Have another volunteer write an addition sentence based on
the number of circles on one side of the mirror, and the total number of circles they see altogether. Do they
see a double anywhere?
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Do more examples of this and then ask what happens if we put a circle over here outside the range of
the mirror:
Have a volunteer show where to draw the circles we would see in the mirror. Ask how many circles there
appear to be altogether. Suggest two different number sentences: 4 + 4 + 1 = 9 and 5 + 4 = 9 and ask your
students to tell you what you are thinking by each addition sentence. Ask what number is being doubled and
what they think you mean by a double plus one. Repeat with several other examples of doubles plus one.
Then write the sum: 3 + 4 and ask if that can be written as a double plus one. What number would be
doubled? Write on the board:
3 + 4 = + + 1 and tell students that you want to put the same number in each box. Ask them
how writing the sum this way can help them add 3 + 4. Show students how they can make a doubles chart if
they don’t have the doubles memorized:
0 1 2 3 4 5 6 7 8 9 10
0 2 4 20
Have students guide you in finishing this double’s chart. Then demonstrate how to use the chart to find
doubles:
Have students use the method above to find the following sums: 8 + 7; 4 + 5; 9 + 10; 9 + 8.
Bonus: 23 + 24; 35 + 36; 57 + 56;
NOTE: Students might also try finding the sums above by doubling the larger number and subtracting one:
7 + 6 = 7 + 7 – 1 = 14 – 1 = 13.
Teach students to subtract numbers from 100 using the following mental math strategy. Ensure that
students can …
1) Subtract single-digit numbers easily from 10 (EXAMPLE: 10 – 4 = 6) See the Modified Go Fish game in
the Mental Math section of this guide.
2) Subtract 2-digit multiples of 10 from 100 (EXAMPLE: 100 – 40 = 60)
3) Subtract single-digit numbers from multiples of 10 (EXAMPLE: 80 – 4 = 76)
4) Subtract two-digit numbers from 100 (EXAMPLE: 100 – 74 = 100 – 70 – 4 = 30 – 4 = 26)
Bonus: Subtract 2-digit numbers from multiples of 100 (EXAMPLE: 800 – 74 = 726)
Teach students to find sums by adding the digits separately. Ensure that students can …
1) Add 2 single-digit numbers (EXAMPLE: 7 + 8 = 15)
2) Add 3 single-digit numbers (EXAMPLE: 6 + 7 + 9 = 22)
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3) Separate the digits (EXAMPLES: 48 = 40 + 8; 132 = 100 + 30 + 2)
4) Add a 2-digit number and a 1-digit number by separating the tens and ones
(EXAMPLE: 48 + 6 = 40 + 8 + 6 = 40 + 14 = 54)
5) Add three 2-digit numbers by separating the tens and ones
(EXAMPLE: 48 + 16 + 23 = 40 + 10 + 20 + 8 + 6 + 3 = 70 + 17 = 87)
6) Add 3 numbers that include a 3-digit number
(EXAMPLE: 532 + 54 + 7 = 500 + 30 + 50 + 2 + 4 + 7 = 500 + 80 + 13 = 593)
Activities:
1. If you have miras available, students can show different examples of doubles plus one, such as 3 + 3 + 1
using counters.
2. Compete (teacher against the class) to see who can come up with the most strategies to find 78 – 29.
Some strategies include:
• 78 – 28 = 50, so 78 – 29 = 49.
• 78 – 29 = 79 – 30 (because if I have two piles of counters, one with 78 counters and one with 29
counters and I add a counter to each pile, the difference stays the same) = 49.
• 78 – 29 = 80 – 31 (adding two counters to each pile instead) = 49.
• 29, 30, 70, 78 has differences 1, 40 and 8, so the total difference is 1 + 40 + 8 = 49.
• Separating the tens and ones and then regrouping:
70 + 8 60 + 18
– 20 + 9 – 20 + 9
40 + 9 = 49.
3. Repeat Activity 2, but with adding 27 + 49. Sample strategies include:
• 27 + 50 = 77, so 27 + 49 = 76
• 20 + 49 = 69, so 27 + 49 = 69 + 7 = 76
• 27 + 49 = 20 + 40 + 7 + 9 = 60 + 16 = 76
• 25 + 50 = 75, so 27 + 49 = 75 + 2 – 1 = 76
Extensions:
1. Teach students to subtract 2-digit numbers from 100 by adding: 100 – 73 = 100 – 80 + 80 – 73. = 20 + 7.
This can be represented as:
7 20 27
73 80 100
2. Find an easy way to add: 299 + 198 + 399.
3. Tell students that the numbers 40 and 60 are partners because they add to 100. Have students find a
number which is its own partner. (50). Then have students make a chart of various multiples of ten and
find their partners. What is an easy way to find the partner of a multiple of ten? Then move on to
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examples that are not multiples of ten. Have students make a chart with headings: “Number” and
“Partner that adds to 100.”
Give students various numbers to put in the first column (24, 33, 19, 47, 85, 76, 93, 4, 11). Then have
students use various numbers of their choice to add more pairs to their chart. Challenge students to look
for a pattern so that they can find an easy way to find a number’s partner without using the method
taught in class. In particular, guide students’ attention to the ones digits and the tens digits. Look at the
ones digits of the pairs of numbers. What do they notice about them? (The ones digits add to ten.) Look
at the tens digits of the pairs of numbers. What do they notice about them? (The tens digits add to 9).
Why does this make sense? If the sum is 9 tens and 10 ones, what number is that? (90 + 10 = 100).
Have students use this method to find the partner of various numbers. Start by filling in the tens digit for
them and having the students only find the ones digit, then fill in the ones digit and have them only find
the tens digit. The mix up which digit you give them and which digit they need to find. Finally, have them
find both digits.
The following four extensions were taken from Atlantic Curriculum B6:
4. Why might someone find it easier to subtract 123 – 99 than 123 – 87?
5. Which sum is closest to 500? Explain how you know.
329 + 189 329 + 217 329 + 207
6. Which difference is closest to 50? Explain how you know.
125 – 30 168 – 115 103 – 82
7. You subtracted a number in the 3 hundreds from a number in the 5 hundreds. The answer was about
100. What might the numbers have been?
8. Teach students to estimate by clustering. For example: 23 + 24 + 34 is estimated, by rounding, as
20 + 20 + 30 = 70. But if students notice that 3 + 4 + 4 is about 10, a better estimate is 70 + 10 = 80.
Similarly, 232 + 244 + 322 is about 200 + 200 + 300 + 100.
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NS3-58 Sharing – Knowing the Number of Sets ERRATA NOTE: This 2-sheet spread in the workbook is accidentally titled NS3-58: Multiplication and
Division (Review) and NS3-59: Knowing the Number of Sets
Goal: Given the total number of objects divided into a given number of sets, students will identify the
number of objects in each set.
Prior Knowledge Required: Dividing equally
Word problems
Vocabulary: set, divide, equally
Divide 12 volunteers into 4 teams, numbered 1-4, by assigning each volunteer a number in the following
order: 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 1. Separate the teams by their numbers, and then ask your students what
they thought of the way you divided the teams. Was it fair? How can you reassign each of the volunteers a
number and ensure that an equal amount of volunteers are on each team? An organized way of doing this
is to assign the numbers in order: 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. Demonstrate this method and then
separate the teams by their numbers again. Does each team have the same amount of volunteers? Are the
teams fair now?
Present your students with 4 see-through containers and ask them to pretend that the containers represent
the 4 teams. Label the containers 1, 2, 3 and 4. To evenly divide 12 players (represented by a counter of
some sort) into 4 teams, one counter is placed into one container until all 12 counters are placed into the 4
containers. This is like assigning each player a number. Should we randomly distribute the 12 counters and
hope that each container is assigned an equal amount? Students should see that it makes more sense to
place the students’ counters (or name tags) into the container one at a time.
Now, suppose you want to share 12 cookies between yourself and 3 friends. How many people are sharing
the cookies? [4.] How many containers are needed? [4.] How many counters are needed? [12.] What do the
counters represent? [Cookies.] What do the containers represent? [People.] Instruct your students to draw
circles for the containers and dots inside the circles for the counters. How many circles will you need to
draw? [4.] How many dots will you need to draw inside the circles? [12.]
Draw 4 circles.
Counting the dots out loud as you place them in the circles, have your students yell “Stop” when you reach
12. Ask them how many dots are in each circle? If 4 people share 12 cookies, how many cookies does each
person get? If 12 people are divided among 4 teams, how many people are on each team? Now what do
the circles represent? Now what do the dots represent?
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If 12 people ride in 4 cars, how many people are in each car? What do the circles and dots represent? Have
students suggest additional representations for circles and dots.
Assign your students practice exercises, drawing circles and the correct numbers of dots in each circle.
If it helps, allow students to first use counters and count them ahead of time so they know automatically
when to stop.
a) 12 cookies, 3 people b) 15 cookies, 5 people c) 10 cookies, 2 people
When students have mastered this, write the following word problem.
5 friends shared 20 strawberries. How many strawberries does each friend get?
Have a volunteer read the word problem out loud and then ask how many dots are needed. What is to be
divided into groups? [Strawberries.] How many circles are needed? What will the circles represent?
[Friends.] What will the amount of dots in each circle illustrate? [The number of strawberries that each friend
will receive.] Have another volunteer solve the problem for the rest of the class. Then assign your students
several word problems. Read all the word problems out loud, and remind students that they can use a
dictionary if they don’t understand a word.
EXAMPLES:
a) 3 friends picked 15 cherries. How many cherries did each friend pick?
b) Joanne shared 15 marbles among 5 people (4 friends and herself). How many marbles
did each person receive?
c) There are 18 plums on 6 trees. How many plums are on each tree?
d) There are 16 apples on 2 trees. How many apples are on each tree?
e) 20 children sit in 4 rows. How many children sit in each row?
f) Lauren’s weekly allowance is $21. What is Lauren’s daily allowance?
g) An egg carton has 12 eggs divided into 2 rows. How many eggs are in each row?
Bonus:
Have students use base ten materials for the following questions.
a) 3 friends picked 69 cherries. How many cherries did each friend pick?
b) Joanne shared 84 marbles among 4 people (3 friends and herself). How many marbles
did each person receive?
c) There are 63 plums on 3 trees. How many plums are on each tree?
d) There are 68 apples on 2 trees. How many apples are on each tree?
Activity: Students might act out their solutions to questions 5 and 6.
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NS3-59 Sharing – Knowing How Many in Each Set
Goal: Given a number of objects to be divided equally and the number in each set, students will determine
the number of sets.
Prior Knowledge Required: Dividing objects into equal sets
Word problems
Vocabulary: sets, groups, divide, row, column
Distribute 12 counters to each student and have them divide the counters evenly into 3 piles, then 2 piles,
then 6 piles, and then 4 piles. Then have them divide the 12 counters first into piles of 2, then into piles of 3,
then 6, then 4. Tell your students that, last class, the question always told them how many piles (or sets) to
make. This class, the question will tell them how many to put in each pile (or in each set) and they will need
to determine the number of sets.
Tell your students that Saud has 30 apples. Count out 30 counters and set them aside. Saud wants to
share his apples so that each friend gets 5 apples. He wants to know how many people can get apples.
What can be used to represent Saud’s friends? [Containers.] Do you know how many containers we need?
[No, because we are not told how many friends Saud has.] How many counters are to be placed in each
container? [5.]
Put 5 counters in one container. Are more containers needed for the counters? [Yes.] How many counters
are to be placed in a second container? (5) How do they know? (Each container gets 5 counters because
each friend gets 5 apples.) Will another container be needed? (Yes, because there are at least 5 apples, or
counters, left) Continue until all the counters are evenly distributed. The 30 counters have been distributed
and each friend received 5 apples. How many friends does Saud have? How do they know? [6, because 6
containers were needed to evenly distribute the counters.]
Now draw dots and circles like in the last lesson. What will the circles represent? [Friends.] How many
friends does Saud have? How many dots are to be placed in each circle? [5.] Place 5 dots in each circle
and keep track of the amount used.
5 10 15 20 25 30
6 circles had to be drawn to evenly distribute 30 dots, meaning 6 friends can share the 30 apples. What is
the difference between this problem and the problems in the previous lesson? [The previous lesson
identified the number of sets, but not the number of objects in each set.]
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Saud has 15 apples and wants to give each friend 3 apples. How many friends can he give apples to? How
can we model this problem? What will the dots represent? What will the circles represent? Does the
problem tell us how many circles to draw? (no) Or how many dots to draw in each circle? (yes, 3) Do we
know how many dots to draw altogether? (yes, 15) Draw 15 dots on the board and demonstrate distributing
3 dots in each circle:
There are 5 sets, so he can give apples to 5 people.
Have your students distribute 3 dots into each set, and then ask them to count the number of sets.
a)
b)
c)
Repeat with 2 dots into each set:
a)
b)
Have students draw the dots to determine the correct amount of sets.
a) 15 dots, 5 dots in each set b) 12 dots, 4 dots in each set c) 16 dots, 2 dots in each set
Bonus:
24 dots and
a) 2 dots in each set b) 3 dots in each set c) 4 dots in each set
d) 6 dots in each set e) 8 dots in each set f) 12 dots in each set
ASK: As the number of dots in each set gets bigger, what happens to the number of sets?
Activity: Students might act out their solutions to questions 5 and 6.
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NS3-60 Sets Goal: Students will identify the objects to be divided into sets, the number of sets and the number of
objects in each set.
Prior Knowledge Required: Sharing equally
Vocabulary: groups or sets, divided or shared
Draw 12 children divided equally into 4 canoes, as illustrated in the worksheet. Ask your students to identify
the objects to be shared or divided into sets, the number of sets and the number of objects in each set.
Repeat with 15 apples divided into 3 baskets and 12 plates placed on 3 tables. Have students complete the
following examples in their notebook.
a) Draw 9 people divided into 3 teams (Team A, Team B, Team C)
b) Draw 15 flowers divided into 5 flowerpots.
c) Draw 10 fish divided into 2 fishbowls.
Note that, in division problems, the word that tells you what is being divided or shared will almost always
come right before the word “each” (“in each”, “on each”, “to each”, “for each”, “at each”). The word coming
after “each” is usually the set.
For instance, in the sentence: “There are 4 kids in each boat,” the word ‘”kids” comes right before the phrase:
“in each boat”. Boats are the sets and kids are being divided into sets.
Students should also think of the set as a kind of container that holds the things that are being divided
or shared.
On the board, write several phrases or sentences with the word “each” in them and ask your students to say
what is being divided or shared, and what are the sets.
a) 5 boxes, 4 pencils in each box (pencils are being divided, boxes are sets)
b) 3 classrooms, 20 students in each classroom (students are being divided, classrooms are sets)
c) 4 teams, 5 people on each team (people are being divided, teams are the sets)
d) 5 trees, 30 apples on each tree (apples are being divided, trees are the sets)
e) 3 friends, 6 stickers for each friend (stickers are being divided, friends are the sets)
f) There are 3 sides on each triangle.
g) There are 6 houses on each block.
h) There are 30 kids on each school bus.
i) There are 3 school buses for each school.
j) There are 6 schools in each neighbourhood.
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To ensure contextual understanding, you might ask your students to draw some of the above situations.
Then have students use circles to represent sets and dots to represent objects to be divided into sets.
a) 5 sets, 2 dots in each set
b) 7 groups, 2 dots in each group
c) 3 sets, 4 dots in each set
d) 4 groups, 6 dots in each group
e) 5 children, 2 toys for each child
f) 6 friends, 3 pencils for each friend
g) 3 fishbowls, 4 fish in each bowl, 12 fish altogether
h) 20 oranges, 5 boxes, 4 oranges in each box
i) 4 boxes, 12 pens, 3 pens in each box
j) 10 dollars for each hour of work, 4 hours of work, 40 dollars
Bonus:
k) 12 objects altogether, 4 sets
l) 8 objects altogether, 2 objects in each set
m) 5 fish in each fishbowl, 3 fishbowls
n) 6 legs on each spider, 4 spiders
o) 3 sides on each triangle, 6 triangles
p) 3 sides on each triangle, 6 sides
q) 6 boxes, 2 oranges in each box
r) 6 oranges, 2 oranges in each box
s) 6 fish in each bowl, 2 fishbowls
t) 6 fish, 2 fishbowls
u) 8 boxes, 4 pencils in each box
v) 8 pencils, 4 pencils in each box
Activities:
1. The teacher starts by saying a sentence that describes some objects that are divided into groups.
EXAMPLE: There are 4 kids in each boat. The first student then says a sentence where the sets become
the objects being divided. For example, the first student might say: “There are 7 boats on each river.” Or
“There are 4 boats on each dock.” Or “There are 5 boats in each boathouse.” Students continue in this
way. For example, “there are 7 boathouses in each river” and then “there are 5 rivers in each province.”
2. A student makes up a division-related phrase, for instance, “There are 4 kids in each boat.” They then try
to make up a sentence where things that were previously divided are now the sets and the sets are the
things divided: ie, “Each kid has 4 boats.” (In some cases this will be impossible to do.)
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NS3-61 Two Ways of Sharing Goal: Students will recognize which information is provided and which is absent among: how many sets,
how many in each set and how many altogether.
Prior Knowledge Required: Solving problems where either the number of sets or amount of
objects in each set is given
Samuel has 15 cookies. There are two ways that he can share or divide his cookies equally.
1. He can decide how many sets (or groups) of cookies he wants.
If he wants to share his cookies with two friends, he will have to divide
the cookies into 3 sets. He will draw 3 circles and place one cookie
into each circle until all 15 cookies are placed in the circles.
2. He can decide that he wants each person to receive 5 cookies.
He counts out sets of 5 cookies until he has counted all 15 cookies.
Show students how to divide 3 rows of 8 dots into 4 circles.
And so on. Or, instead of crossing out the dots, students might count the total number of dots (or multiply to
find the total number of dots) and then count as they place the dots in the circles.
Assign your students several similar problems.
a) 2 rows of 6 squares into 3 circles
b) 3 rows of 8 hearts into 6 circles
c) 1 row of 18 dots into 2 circles
d) 1 row of 12 vertical lines into 6 circles
NOTE: Some students may find it easier to draw dots instead of triangles, squares or hearts.
Then ask your students if they remember how to group the dots so that there are 4 dots in
each set, and to explain how this is different from the previous problem.
a)
b)
Have students draw 12 dots and group them so that there are…
a) 4 dots in each set
b) 2 dots in each set
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c) 6 dots in each set
d) 3 dots in each set
Sometimes the problem provides the number of sets, and sometimes the problem provides the number of
objects in each set. Have your students tell you if they are given the number of sets or the number of
objects in each set for the following problems.
a) There are 15 children. There are 5 children in each canoe.
b) There are 15 children in 5 canoes.
c) Aza has 40 stickers. She gives 8 stickers to each of her friends.
Have your students draw and complete the following table for PROBLEMS a)–i). Insert a box for
information that is not provided in the problem.
What has been shared
or divided into sets?
How many sets? How many in each set?
EXAMPLE 18 6
a) 24 4
EXAMPLE: There are 6 strings on each guitar. There are 18 strings.
a) There are 24 strings on 4 guitars.
b) There are 3 hands on each clock. There are 15 hands altogether.
c) There are 18 holes in 6 sheets of paper.
Make sure your students understand that words such as “book shelves” or “tables” or “containers” or
“vehicles” might refer to the sets or the objects being divided into sets.
Ask students to decide in each statement below whether the words “book shelves” represent sets or objects
being divided into sets.
a) There are 5 books on each bookshelf.
b) There are 6 bookshelves in each room.
c) Each bookshelf has several books on it.
d) Each library has several bookshelves.
NOTE: In question 8 all of the examples tell you the number of containers (or sets). For variety assign your
students several questions which give the number of items in each set. For example:
1. Paul has 15 stamps. He put 5 on each page.
2. 12 kids sit down to dinner. There are 3 kids at each table.
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NS3-62 Division and NS3-63 Dividing by Skip Counting
Goal: Students will learn the division symbol through repeated addition and skip counting.
Prior Knowledge Required: Skip counting using fingers or a number line
Sharing or dividing into sets or groups
Vocabulary: division (and its symbol ÷), divided by, dividend, divisor
Ensure that students can tell you, for various pictures: a) how many objects altogether, b) how many sets
and c) how many objects in each set.
Then write two division statements for each picture: 8 ÷ 4 = 2 and 8 ÷ 2 = 4 for the first picture and
15 ÷ 5 = 3 and 15 ÷ 3 = 5 for the second picture.
Explain that 15 objects divided into sets of 5 equals 3 sets. This is written as 15 ÷ 5 = 3 or as 15 ÷ 3 = 5 and
read as “fifteen divided by 5 equals 3” or “fifteen divided by 3 equals 5.”
Distribute 12 counters to each of your students and then ask them to divide the counters into sets of 3. How
many sets do they have? What two division statements can they write?
The following website provides good worksheets for those having trouble with this basic definition of
division:
http://math.about.com/library/divisiongroups.pdf.
Using various symbols, have your students find 12 ÷ 2, 12 ÷ 3, 12 ÷ 4 and 12 ÷ 6. For example:
So 12 ÷ 2 = 6.
Have your students illustrate each of the following division statements with two pictures.
6 ÷ 3 = 2
12 ÷ 3 = 4 9 ÷ 3 = 3 8 ÷ 2 = 4 8 ÷ 4 = 2
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Then have your students write two division statements for each of the following illustrations.
A B C
D E F G H
I J K
Challenge students to find another way to group the objects in each illustration that will show the same
division statement. For example, instead of 3 groups of 4 in diagram A, students can draw 4 groups of 3;
instead of 4 groups of 1 in diagram I, students can draw 1 group of 4. You might also draw several
pictures and have students pair up the pictures that show the same division statements. (In the pictures
above, G and I both show 4 ÷ 1 = 4 and 4 ÷ 4 = 1; B and C both show 12 ÷ 2 = 6 and 12 ÷ 6 = 2).
WRITE: 15 ÷ 3 = 5 (15 divided into sets of size 3 equals 5 sets).
3 + 3 + 3 + 3 + 3 = 15
Explain that every division statement implies an addition statement. Ask your students to write the addition
statements implied by each of the following division statements. Allow them to illustrate the statement first,
if it helps.
15 ÷ 5 = 3 12 ÷ 2 = 6 12 ÷ 6 = 2 10 ÷ 5 = 2
10 ÷ 2 = 5 6 ÷ 3 = 2 6 ÷ 2 = 3
Add this number
WRITE: 15 ÷ 3 = 5 This many times.
Ask your students to write the following division statements as addition statements, without
illustrating the statement this time.
12 ÷ 4 = 3 12 ÷ 3 = 4 18 ÷ 6 = 3 18 ÷ 3 = 6
18 ÷ 2 = 9 18 ÷ 9 = 2 25 ÷ 5 = 5
Bonus:
132 ÷ 43 = 3 1700 ÷ 425 = 4 90 ÷ 30 = 3 1325 ÷ 265 = 5
Then have your students illustrate and write a division statement for each of the following
addition statements.
4 + 4 + 4 = 12 2 + 2 + 2 + 2 + 2 = 10 6 + 6 + 6 + 6 = 24
3 + 3 + 3 = 9 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21 5 + 5 + 5 + 5 + 5 + 5 = 30
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Then have your students write division statements for each of the following addition statements, without
illustrating the statement.
17 + 17 + 17 + 17 + 17 = 85
21 + 21 + 21 = 63
101 + 101 + 101 + 101 = 404
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 36
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 21
Draw:
What division statement does this illustrate? What addition statement does this illustrate? Is the addition
statement similar to skip counting? Which number could be used to skip count the statement?
Explain that the division statement 18 ÷ 3 = ? can be solved by skip counting on a number line.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
How many skips of 3 does it take to reach 18? (6.) So what does 18 ÷ 3 equal? How does the number line
illustrate this? (Count the number of arrows.) NOTE: Some students might find it helpful if you start with 18
counters and count out 3 at a time, drawing an arrow for each set of 3 counters that you set aside. You
know to stop when all 18 counters are used up, or when you reach 18 on the number line.
Explain that the division statement expresses a solution to 18 ÷ 3 by skip counting by 3 to 18 and then
counting the arrows.
Ask volunteers to find, using the number line: 12 ÷ 2; 12 ÷ 3; 12 ÷ 4; 12 ÷ 6;
0 1 2 3 4 5 6 7 8 9 10 11 12
ASK: If I want to find 10 ÷ 2, how could I use this number line?
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
What do I skip count by? How do I know when to stop? Have a volunteer demonstrate the solution.
Ask your students to solve the following division statements with a number line to 18.
8 ÷ 2, 12 ÷ 3, 15 ÷ 3, 15÷ 5, 14 ÷ 2, 16 ÷ 4, 16 ÷ 2, 18 ÷ 3, 18 ÷ 2
Then have your students solve the following division problems with number lines to 20 (see the BLM
“Number Lines to Twenty”). Have them use the top and bottom of each number line so that each number
line can be used to solve two problems. For example, the solutions for 6 ÷ 2 and 8 ÷ 4 might look like this:
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
6 ÷ 2 8 ÷ 4 6 ÷ 3 14 ÷ 2 9 ÷ 3 15 ÷ 3
10 ÷ 5 20 ÷ 4 20 ÷ 5 18 ÷ 2 20 ÷ 2 16 ÷ 4
Provide various solutions of division problems and have your students express the corresponding division
and addition statements. For example, students should give the statements “20 ÷ 4 = 5”
and “4 + 4 + 4 + 4 + 4 = 20” for the following number line.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Draw number lines for 16 ÷ 8, 10 ÷ 2, 12 ÷ 4, and 12 ÷ 3.
Explain that skip counting can be performed on the fingers, as well as on a number line. If your students
need practice skip counting without a number line, see the MENTAL MATH section of this teacher’s guide.
They might also enjoy the following interactive website:
http://members.learningplanet.com/act/count/free.asp
Draw 12 dots on the board. Ask your students if they can figure out how to divide a set of 12 dots into 3
equal sets by skip counting by 3s. As students count up by 3s ask them to imagine putting one dot into each
set (thus placing 3 dots altogether) every time they say a multiple of 3. The skip counting helps them keep
track of the number of dots placed altogether and the number of fingers raised helps them keep track of the
number of dots placed in each set.
“3” I’ve placed one dot in each
set (3 altogether).
“6” I’ve placed 2 dots in each set
(6 altogether).
“9” I’ve placed 3 dots in each set
(9 altogether).
“12” I’ve placed 4 dots in each set
(12 altogether).
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To solve 12 ÷ 3, skip count by 3 to 12. The number of fingers it requires to count to 12 is the answer. [4.]
Perform several examples of this together as a class, ensuring that students know when to stop counting
(and in turn, what the answer is) for any given division problem. Have your students complete the following
problems by skip counting with their fingers.
a) 12 ÷ 3 b) 6 ÷ 2 c) 10 ÷ 2 d) 10 ÷ 5 e) 12 ÷ 4 f) 9 ÷ 3
g) 16 ÷ 4 h) 50 ÷ 10 i) 25 ÷ 5 j) 15 ÷ 3 k) 15 ÷ 5 l) 30 ÷ 10
Bonus:
a) 200 ÷ 50 b) 125 ÷ 25
Two hands will be needed to keep track of the count for the following questions.
a) 12 ÷ 2 b) 30 ÷ 5 c) 18 ÷ 2 d) 28 ÷ 4 e) 30 ÷ 3 f) 40 ÷ 5
Bonus:
a) 450 ÷ 50 b) 175 ÷ 25
As an extra challenge, provide problems that require counting beyond the fingers on two hands.
Bonus:
a) 22 ÷ 2 b) 48 ÷ 4 c) 65 ÷ 5 d) 120 ÷ 10 e) 26 ÷ 2
Then have your students express the division statement for each of the following word problems and
determine the answers by skip counting. Ask questions like: What is to be divided into sets? How many sets
are there and what are they?
a) 5 friends share 30 tickets to a sports game. How many tickets does each friend receive?
b) 20 friends sit in 2 rows at the movie theatre. How many friends sit in each row?
c) $50 is divided among 10 friends. How much money does each friend receive?
Have your students illustrate each of the following division statements and skip count to
determine the answers.
3 ÷ 3 5 ÷ 5 8 ÷ 8 11 ÷ 11
Without illustrating or skip counting, have your students predict the answers for the following division
statements.
23 ÷ 23 180 ÷ 180 244 ÷ 244 1 896 ÷ 1 896
Then have your students illustrate each of the following division statements and skip count to determine the
answers.
1 ÷ 1 2 ÷ 1 5 ÷ 1 12 ÷ 1
Then without illustrating, have your students predict the answers for the following
division statements.
18 ÷ 1 27 ÷ 1 803 ÷ 1 6 692 ÷ 1
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Extension: Teach students that division is similar to repeated subtraction. Start with 20 counters and
take away 4 each time until there is none left. Symbolize this on the board by writing:
20 – 4 – 4 – 4 – 4 – 4 = 0
ASK: How many times did you subtract 4? What is 20 ÷ 4?
Compare this method to dividing on a number line or by skip counting backwards by 4s:
20 16 12 8 4 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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NS3-64 Division and Multiplication Goal: Students will understand the relationship between division and multiplication
Prior Knowledge Required: Relationship between multiplication and skip counting
Relationship between division and skip counting
Vocabulary: divided by
Write “10 divided into sets of 2 results in 5 sets.” Have one volunteer read it out loud, another illustrate it,
and another write its addition statement. Does the addition statement remind your students of
multiplication? What is the multiplication statement?
10 ÷ 2 = 5
2 + 2 + 2 + 2 + 2 = 10
5 × 2 = 10
Another way to express “10 divided into sets of 2 results in 5 sets” is to write “5 sets of 2 equals 10.”
Have volunteers illustrate the following division statements, write the division statements, and then rewrite
them as multiplication statements.
a) 12 divided into sets of 4 results in 3 sets. (12 ÷ 4 = 3 and 3 × 4 = 12)
b) 10 divided into sets of 5 results in 2 sets.
c) 9 divided into sets of 3 results in 3 sets.
Assign the remaining questions to all students.
d) 15 divided into 5 sets results in sets of 3.
e) 18 divided into 9 sets results in sets of 2.
f) 6 people divided into teams of 3 results in 2 teams.
g) 8 fish divided so that each fishbowl has 4 fish results in 2 fishbowls.
h) 12 people divided into 4 teams results in 3 people on each team.
i) 6 fish divided into 3 fishbowls results in 2 fish in each fishbowl.
Then ask students if there is another multiplication statement that can be obtained from the same picture as
3 × 4 = 12? How can the dots be grouped to express that 3 sets of 4 is equivalent to 4 sets of 3?
3 sets of 4 is equivalent to 4 sets of 3.
ANSWER: The second array of dots should be circled in columns of 3.
ASK: If 12 ÷ 4 = 3 and 3 × 4 = 12 are obtained from the same picture, what division statement comes from
the same picture as 4 × 3 = 12? (12 ÷ 3 = 4)
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Draw similar illustrations and have your students write two multiplication statements and
two division statements for each. Have a volunteer demonstrate the exercise for the class
with the first illustration.
Demonstrate how multiplication can be used to help with division. For example, the division statement
20 ÷ 4 = _____ can be written as the multiplication statement 4 × _____ = 20. To solve the problem, skip
count by 4 to 20 and count the number of fingers it requires. Demonstrate this solution on a number line,
as well.
20 divided into skips of 4 results in 5 skips. 20 ÷ 4 = 5
5 skips of 4 results in 20. 5 × 4 = 20, SO: 4 × 5 = 20
Assign students the following problems.
a) 9 × 3 = 27, SO: 27 ÷ 9 = _____
b) 2 × 6 = 12, SO: 12 ÷ 2 = _____
c) 8 × _____ = 40, SO: 40 ÷ 8 = _____
d) 10 × _____ = 30, SO: 30 ÷ 10 = _____
e) 5 × _____ = 30, SO: 30 ÷ 5 = _____
f) 4 × _____ = 28, SO: 28 ÷ 4 = _____
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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NS3-65 Knowing When to Multiply or Divide Goal: Students will understand when to use multiplication or division to solve a word problem.
Prior Knowledge Required: Relationship between multiplication and division
Vocabulary: each, per, altogether, in total
Have your students fill in the blanks.
sets 2 sets
objects per set 5 objects per set
objects altogether 10 objects altogether
When you are confident that your students are completely familiar with the terms “set,” “group,” “for every,”
and “in each,” and you’re certain that they understand the difference between the phrases “objects in each
set” and “objects” (or “objects altogether,” or “objects in total”), have them write descriptions of the
diagrams.
2 groups
4 objects in each group
8 objects
Explain that a set or group expresses three pieces of information: the number of sets, the number of objects
in each set, and the number of objects altogether. For the problems, have your students explain which piece
of information isn’t expressed and what the values are for the information that is expressed.
a) There are 8 pencils in each box. There are 5 boxes. How many pencils are there altogether? (5 groups,
8 objects in each group, how many altogether?)
b) Each dog has 4 legs. There are 3 dogs. How many legs are there altogether?
c) Each cat has 2 eyes. There are 10 eyes. How many cats are there?
d) Each boat can fit 4 people. There are 20 people. How many
boats are needed?
e) 30 people fit into 10 cars. How many people fit into each car?
f) Each apple costs 20¢. How many apples can be bought for 80¢?
g) There are 8 triangles divided into 2 sets. How many triangles are
there in each set?
h) 4 polygons have a total of 12 sides. How many sides are on
each polygon?
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I I I I I I I I I I I I I I I I
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Introduce the following three problem types:
Type 1: You know the number of sets and the number of objects in each set.
Example: You have 4 sets of objects and 2 objects in each set. How many objects do you have in total?
STEP 1: Draw 4 boxes to represent the 4 sets:
STEP 2: Fill each box with 2 objects:
STEP 3: Count the number of objects: 8 objects (or “8 objects altogether” or
“8 objects in total”)
Your student can then write a multiplication statement to represent the solution: 4 � 2 = 8
ASK: If you know the number of objects in each set and the number of sets, how can you
find the total number of objects? What operation should you use—multiplication or division?
Write on the board:
Number of sets × Number of objects in each set = Total number of objects
Other examples you could use:
a) 3 sets b) 3 groups
5 objects in each set 7 objects in each group
How many objects? How many objects in total?
Type 2: You know how many objects there are altogether and how many objects there are in each set.
Example: You have 6 objects altogether and 3 objects in each set. How many sets do you have?
STEP 1: Draw the total number of objects:
STEP 2: Draw a box around three objects
at a time until you’ve put all the
objects in boxes:
STEP 3: Count the number of boxes: 2 boxes (or "sets")
Your student should then write a division statement to represent the solution: 6 ÷ 3 = 2
Again use the same multiplication statement as before.
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Number of sets × Number of objects in each set = Total number of objects
If there are 4 objects in each set and 12 objects in total, how many sets are there?
_____ × 4 = 12, SO: 12 ÷ 4 = 3. There are 3 sets.Other examples you could use:
a) 12 objects altogether b) 16 objects
4 objects in each group 2 objects per set
How many groups? How many sets?
Type 3: You know how many objects there are and how many sets there are.
Example: You have 10 objects and 5 sets. How many objects are there in each set?
STEP 1: Draw the total number of sets:
STEP 2: First, put one object in each set.
STEP 3: Check to see if you have placed all
the objects. If not, put one more object
in each set and continue until you have
placed all the objects:
STEP 4: Count the number of objects in each set: 2 objects in each set
Your student should then write a division statement to represent the solution: 10 ÷ 5 = 2
Number of sets × Number of objects in each set = Total number of objects
If there are 6 sets and 12 objects in total, how many objects are in each set?
6 × _____ = 12, SO: 12 ÷ 6 = 2. There are 2 objects in each set.
Other examples you could use:
a) 15 objects altogether b) 3 sets
5 sets 12 objects altogether
How many objects in each set? How many objects in each set?
Have students write multiplication statements for the following problems with the blank in the correct place.
a) 2 objects in each set. b) 2 objects in each set. c) 2 sets.
6 objects in total. 6 sets. 6 objects in total.
How many sets? How many objects in total? How many objects in
each set?
[ _____ × 2 = 6 ] [ 6 × 2 = _____ ] [ 2 × _____ = 6 ]
Which of these problems are division problems? [Multiplication is used to find the total number
of objects, and division is used if the total number of objects is known.]
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Assign several types of problems.
a) 5 sets, 4 objects in each set. How many objects altogether?
b) 8 objects in total, 2 sets. How many objects in each set?
c) 3 sets, 6 objects in total. How many objects in each set?
d) 3 sets, 6 objects in each set. How many objects in total?
e) 3 objects in each set, 9 objects altogether. How many sets?
When your students are comfortable with this lesson’s goal, introduce alternative contexts for objects and
sets. Start with point-form problems (EXAMPLE: 5 tennis courts, 3 tennis balls on each court. How many
tennis balls altogether?), and then move to complete sentence problems (EXAMPLE: If there are 5 tennis
courts, and 3 balls on each court, how many tennis balls are on the tennis courts altogether?).
Have your students solve these problems:
a) 20 apples; 4 baskets.
How many apples in each
basket?
b) 3 birds; 5 cages.
How many birds?
c) 18 bottles of water, 3 cases.
How many bottles in each
case?
When your student is able to distinguish between (and solve) problems of Type 1, 2, and 3 readily, you can
teach them how to solve more general word problems involving multiplication and division. Tell them to think
of a container (like a box or pot) or a carrier (like a car or a boat) as a set, and the things contained or carried
as objects in the set.
EXAMPLE: Ten people need to cross a river. A boat can hold two people. How many boats are needed to
take everyone across?
Hint: Think of the boats as sets (or boxes) and the people as objects placed in the sets. This is a problem of
Type 2, as discussed in above, ie. you know the total number of objects and the number of objects in
each set.
Draw 10 lines to represent 10 people:
Put boxes around every 2 lines (each
box represents a boat):
Count the number of boats: 5 boats
This approach also works for things that have parts (think of the things that have parts as sets, and the parts
as objects in the set).
Example: A cat has 2 eyes. How many eyes are there on 5 cats?
Hint: Note that the first statement is really saying that “Each cat has 2 eyes,” so cats are the sets and eyes
are being divided into sets. Use boxes to represent each cat and lines in each box to represent the eyes.
This is a problem of Type 1, as discussed above, i.e. you know the number of sets and the number of
objects in each set.
I I I I I I I I I I
I I I I I I I I I I
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Draw 5 boxes to represent the 5 cats:
Draw 2 lines in each box
(representing the eyes):
The number of lines gives you the answer: 10 eyes
This approach also works for things that have a value or a price (think of the thing with value as a set or box,
and the price or value as the objects in the set, i.e. you can think of dollars or cents as lines which you can
place inside the box representing the thing you are buying).
Example: A piece of gum costs 5 cents. You have 15 cents. How many pieces of gum can you buy?
Hint: Again, this is a problem of Type 2.
Draw 15 lines to represent 15 cents:
Put boxes around every 5 lines (each
box represents a piece of gum):
Count the number of boxes: 3 boxes (or pieces of gum)
Once you give your students enough practice with this type of problem, they should eventually see that they
simply have to divide 15 by 5 to find the answer.
Activity: Students could model their solutions to questions 5 a) and b) with counters. It is important,
however, that students also be able to solve the problems by drawing a sketch with dots or lines.
Extension: Tell your students that you met someone from Mars last weekend, and they told you that
there are 3 dulgs on each flut. If you count 15 dulgs, how many fluts are there? Explain the problem-solving
strategy of replacing unknown words with words that are commonly used. For example, replace the object in
the problem (dulgs) with students, and replace the set in the problem (flut) with bench. So, if there are 3
students on each bench and you count 15 students, how many benches are there? It wouldn’t make sense to
replace the object (dulgs) with benches and the set (flut) with students, would it? [If there are 3 benches on
each student and you count 15 benches, how many students are there?] A good strategy for replacing words
is to replace the object and the set and then invert the replacement with the same two words. Only one of the
two versions of the problem with the replacement words should make sense.
Students may wish to create their own science fiction word problems for their classmates. Encourage them
to use words from another language, if they speak another language.
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NS3-66 Remainders Goal: Students will divide with remainders using pictures.
Prior Knowledge Required: Relationships between division and multiplication, addition, skip
counting, number lines
Vocabulary: remainder R, quotient, divisor
Draw:
6 ÷ 3 = 2
7 ÷ 3 = 2 Remainder 1
8 ÷ 3 = 2 Remainder 2
9 ÷ 3 = 3
10 ÷ 3 = 3 Remainder 1
Ask your students if they know what the word “remainder” means. Instead of responding with a definition,
encourage them to only say the answers for the following problems. This will allow those students who don’t
immediately see the pattern a chance to detect it.
7 ÷ 2 = 3 Remainder _____
11 ÷ 3 = 3 Remainder _____
12 ÷ 5 = 2 Remainder _____
14 ÷ 5 = 2 Remainder _____
Challenge volunteers to find the remainder by drawing a picture on the board. This way, students who do
not yet see the pattern can see more and more examples of the rule being applied.
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SAMPLE PROBLEMS:
9 ÷ 2 7 ÷ 3 11 ÷ 3 15 ÷ 4
15 ÷ 6 12 ÷ 4 11 ÷ 2 18 ÷ 5
What does “remainder” mean? Why are some dots left over? Why aren’t they included in the circles? What
rule is being followed in the illustrations? [The same number of dots is placed in each circle, the remaining
dots are left uncircled]. If there are fewer uncircled dots than circles then we can’t put one more in each
circle and still have the same number in each circle, so we have to leave them uncircled. If there are no
dots left over, what does the remainder equal? [Zero.]
Introduce your students to the word “quotient”: Remind your students that when subtracting two numbers,
the answer is called the difference. ASK: When you add two numbers, what is the answer called? In
7 + 4 = 11, what is 11 called? (The sum). When you multiply two numbers, what is the answer called? In
2 × 5 = 10, what is 10 called? (The product). When you divide two numbers, does anyone know what the
answer is called? There is a special word for it. If no-one suggests it, tell them that when you write
10 ÷ 2 = 5, the 5 is called the quotient.
Have your students determine the quotient and the remainder for the following statements.
a) 17 ÷ 3 = _____ Remainder _____ b) 23 ÷ 4 = _____ Remainder _____
c) 11 ÷ 3 = _____ Remainder _____
Write “2 friends want to share 7 apples.” What are the sets? [Friends.] What are the objects being divided?
[Apples.] How many circles need to be drawn to model this problem? How many dots need to be drawn?
Draw 2 circles and 7 dots.
To divide 7 apples between 2 friends,
place 1 dot (apple) in each circle.
Can another dot be placed in each
circle? Are there at least 2 dots
left over? So is there enough to put
one more in each circle? Repeat this
line of instruction until the diagram
looks like this:
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How many apples will each friend receive? Explain. [There are 3 dots in each circle.] How many apples will
be left over? Explain. [Placing 1 more dot in either of the circles will make the compared amount of dots in
both circles unequal.]
Repeat this exercise with “5 friends want to share 18 apples.” Emphasize that the process of division and
placing apples (dots) into sets (circles) continues as long as there are at least 5 apples left to share. Count
the number of apples remaining after each round of division to ensure that at least 5 apples remain.
Have your students illustrate each of the following division statements with a picture, and then determine
the quotients and remainders.
Number in each circle
a) 11 ÷ 5 = _____ Remainder _____ Number left over
b) 18 ÷ 4 = _____ Remainder _____
c) 20 ÷ 3 = _____ Remainder
d) 22 ÷ 5 = _____ Remainder
e) 11 ÷ 2 = _____ Remainder
f) 8 ÷ 5 = _____ Remainder
g) 19 ÷ 4 = _____ Remainder
Explain to students that the word “Remainder” is sometimes written just as “R.” For example,
11 ÷ 5 = 2 R 1.
Teach students that there are many ways to think about division. To find the answer to 14 ÷ 3, students might use
any of the following methods.
a) Forming equal groups (of size 3) using a picture of 14 objects: b) Sharing 14 things (candies for instance), three apiece, among friends: c) Adding threes repeatedly. Stop before you reach 14:
There are 4 groups of 3 with 2 objects left over
3 6 9 12
You stop here, one more three would take you beyond 14 to 15. (You could show this with a number line).
You can share with 4 friends. Two are left over.
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So 3 goes into 12 (and 14!) four times.
The remainder of 14 ÷ 3 is the difference between 14 and the number you reached when you were
counting up (12). So the remainder is 2.
d) Guessing and multiplying by 3 until you get close to 14:
e) Subtracting 3 from 14 repeatedly until you get close to 0 (this is not the most practical method). You can
subtract 3 four times from 14 before you reach 2 (the remainder).
Activity: Students could model their solutions to all of the questions on this worksheet with counters.
Extensions:
1. 22 kids go on a picnic. Hot dogs come in packs of 8. Buns come in packs of 12. How many packs of
buns and hot dogs should the children take if each kid wants one hot dog and bun? Will there be any
buns or hotdogs left over?
2. Which number is greater, the divisor (the number by which another is to be divided) or the
remainder? Will this always be true? Have your students examine their illustrations to help explain.
Emphasize that the divisor is equal to the number of circles (sets), and the remainder is equal to the
number of dots left over. We stop putting dots in circles only when the number left over is smaller
than the number of circles; otherwise, we would continue putting the dots in the circles. See the
journal section below.
Which of the following division statements is correctly illustrated? Can one more dot be placed
into each circle or not? Correct the two wrong statements.
15 ÷ 3 = 4 Remainder 3 17 ÷ 4 = 3 Remainder 5 19 ÷ 4 = 4 Remainder 3
Without illustration, identify the incorrect division statements and correct them.
a) 16 ÷ 5 = 2 Remainder 6 b) 11 ÷ 2 = 4 Remainder 3 c) 19 ÷ 6 = 3 Remainder 1
3. Explain how a diagram can illustrate a division statement with a remainder and a multiplication
statement with addition.
14 ÷ 3 = 4 Remainder 2
3 × 4 + 2 = 14
4 × 3 = 12 5 × 3 = 15
This is too high, so 3 divides into 14 four times (with 2 remainder).
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Ask students to write a division statement with a remainder and a multiplication statement
with addition for each of the following illustrations.
4. Compare mathematical division to normal sharing. Often if we share 5 things (say, marbles) among 2
people as equally as possible, we give 3 to one person and 2 to the other person. But in mathematics, if
we divide 5 objects between 2 sets, 2 objects are placed in each set and the leftover object is
designated as a remainder. Teach them that we can still use division to solve this type of real-life
problem; we just have to be careful in how we interpret the remainder. Have students compare the
answers to the real-life problem and to the mathematical problem:
a) 2 people share 5 marbles (groups of 2 and 3; 5 ÷ 2 = 2 R 1)
b) 2 people share 7 marbles (groups of 3 and 4; 7 ÷ 2 = 3 R 1)
c) 2 people share 9 marbles (groups of 4 and 5; 9 ÷ 2 = 4 R 1)
ASK: If 19 ÷ 2 = 9 R 1, how many marbles would each person get if 2 people shared 19 marbles?
Emphasize that we can use the mathematical definition of sharing as equally as possible even when the
answer isn’t exactly what we’re looking for. We just have to know how to adapt it to what we need.
5. Find the mystery number. I am between 22 and 38. I am a multiple of 5. When I am divided by 7 the
remainder is 2.
6. Have your students demonstrate two different ways of dividing…
a) 7 counters so that the remainder equals 1.
b) 17 counters so that the remainder equals 1.
Journal
The remainder is always smaller than the divisor because…
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NS3-67 Multiplication and Division and NS3-68 Multiplication and Division (Review)
Goal: Students will consolidate their learning on multiplication and division done so far.
Vocabulary: fact family, multiple, division, remainder, twice as many
Tell students that you have 13 apples and you have divided them into equal groups so that there is
one left over:
ASK: What division statement does this picture show? Then challenge students to find another way to divide
the apples so that there is only 1 left over. (Students might divide the apples into 2, 3 or 4 circles.) Take up
the various solutions.
Draw a 2 × 3 array on the board:
ASK: What multiplication sentence does this show? Challenge students to find another array that shows a
similar multiplication sentence but uses twice as many dots. (Review the phrase “twice as many” if needed.)
ASK: What addition sentence does this show:
ANSWER: 3 + 4 = 7. To guide students, ASK: How many dots are on the left side of the vertical line? How
many are on the right side? How many are there altogether?
Repeat with several similar examples. Then tell students that you are going to show them a more
complicated picture. This time the left side and the right side both show multiplication statements. Can they
tell which ones?
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The left side shows 3 × 2 and the right side shows 4 × 2. ASK: What does the whole picture show? What
multiplication statement is shown by all 7 circles? (7 × 2) Emphasize that this picture shows that seven sets
of 2 equals three sets of 2 plus four sets of 2. Do they think that 7 sets of 3 will equal three sets of 3 plus four
sets of three? Have students draw the picture to show this. Then have students change their picture, one
step at a time to show that:
a) seven sets of 3 equals two sets of 3 plus five sets of 3 (move the vertical line one place left – erase the
vertical line that exists and add a new one)
b) eight sets of 3 equals two sets of 3 plus six sets of 3 (add another circle to the right of the line)
c) ten sets of 3 equals four sets of 3 plus six sets of 3 (add two more circles to the left of the line)
d) ten sets of 4 equals four sets of 4 plus six sets of 4 (add one more dot in each circle)
e) ten sets of four equal three sets of 4 plus one set of 4 plus six sets of 4 (add another vertical line after the
third circle; do not erase the vertical line that exists)
Have students practise doing puzzles similar to question 10 on the worksheet.
Use the numbers 3, 4 and 6 once each to make the sentences true:
× + = 18 × + = 22
× + = 27 × – = 21
÷ + = 6 × – = 6
× ÷ = 2 + – = 7
Bonus: Have students make up a similar puzzle for a partner to solve. They can use any operation, but if
they use multiplication or division, it must only be the first operation used; addition and subtraction can be
either first or second. (This condition avoids potential problems with the order of operations, which they are
not required to know at this point.)
Bonus: Make 0 using the numbers 2, 3 and 6 once each in as many ways as you can.
(EXAMPLES: 2 × 3 – 6 = 0, 3 × 2 – 6 = 0, 6 ÷ 3 – 2, 6 ÷ 2 – 3) Note: Students should not be taught the order
of operations at this point. That 6 – 2 × 3 is also 0 should not be discussed.
The activity and extensions 5-8 below are designed to satisfy the Atlantic Curriculum.
Activity: This activity is best done after completing the extensions below. Play a game with your students
to see who (you or your class) can come up with the most multiplication or division questions that you could
solve in one or two steps just from knowing 5 × 6 = 30. Sample questions:
• 6 × 5 = 30
• 5 × 7 = 5 × 6 + 5 = 30 + 5 = 35
• 6 × 6 = 5 × 6 + 6 = 30 + 6 = 36
• 30 ÷ 5 = 6
• 30 ÷ 6 = 5
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• 60 ÷ 10 = 6
• 90 ÷ 15= 6
• 120 ÷ 20 = 6
• 60 ÷ 5= 12
• 5 × 12 = 60
• 10 × 6 = 60
• 4 × 6 = 30 – 6 = 24
Extensions:
1. Ask students to write a full fact family for a division problem (including 2 addition statements).
For example: The fact family for 14 ÷ 2 = 7 is 14 ÷ 7 = 2, 7 × 2 = 14, 2 × 7 = 14,
2 + 2 + 2 + 2 + 2 + 2 + 2 = 14, and 7 + 7 = 14.
2. Ask your students to complete the following story problems with their own numbers and solve the
problems.
a) Janice had _____ apples. She shared them equally with _____ friends. How many did each
person get?
b) Tim had _____ boxes. He placed _____ watermelons in each box. How many watermelons did he
have altogether?
3. The BLMs “Always, Sometimes, or Never True (Numbers)” and “Define a Number” will help students
sharpen their understanding of numbers.
4. (Adapted from the Atlantic Curriculum) Show students how 5 sets of 3 can be broken down into subsets
in various ways:
• 4 sets of 3 and 1 set of 3
• 3 sets of 3 and 2 sets of 3
• 5 sets of 2 and 5 sets of 1
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Challenge students to write multiplication and division statements that relate to these pictures. For example,
we might write:
• 4 × 3 + 1 × 3 = 5 × 3 or 12 ÷ 3 + 3 ÷ 3 = 15 ÷ 3
• 3 × 3 + 2 × 3 = 5 × 3 or 9 ÷ 3 + 6 ÷ 3 = 15 ÷ 3
• 5 × 2 + 5 × 1 = 5 × 3 or 10 ÷ 5 + 5 ÷ 5 = 15 ÷ 5
5. Ask students what happens to the product when you double one of the numbers in a
multiplication statement:
2 × 3 4 × 3 or 2 × 3 2 × 6
ANSWER: The product doubles.
What happens to the product when you multiply one of the factors by 3? (The product is multiplied by 3.)
Have students investigate what this means for division. For example, 2 × 3 = 6, so 2 × 6 = 12 becomes, in
terms of division: 6 ÷ 3 = 2, so 12 ÷ 6 = 2. Note that doubling both terms of the division statement will keep
the answer the same. Ask students how they could use doubling both terms to find 45 ÷ 5.
6. Remind students that, from Extension 5, the quotient remains the same when both the dividend and the
divisor are multiplied by 2 (or when both are multiplied by 3, or both multiplied by 4, and so on). Ask
students to investigate what happens to the quotient when the dividend is multiplied by 2 and the divisor
remains the same. (The quotient doubles.) What happens to the quotient when the divisor is multiplied
by 2 and the dividend remains the same? (The quotient is divided in half.) Tell students that 21 ÷ 7 = 3.
ASK: What is 42 ÷ 7? 84 ÷ 7? 168 ÷ 3? How does knowing 60 ÷ 6 help you to know 30 ÷ 6?
7. Teach students how to estimate products. Draw a number line as follows:
3 × 0 3 × 10
0 3 6 9 12 15 18 21 24 27 30
ASK: Is 3 × 7 closer to 3 × 0 or 3 × 10? Have a volunteer circle 3 × 7. Have volunteers circle 3 × 1,
3 × 9, 3 × 5. Then draw on the board the following number line:
4 × 0 4 × 10
0 40
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ASK: Is 4 × 8 closer to 4 × 0 or 4 × 10? Is 4 × 8 closer to 0 or 40? How much closer? Have a volunteer mark
where 4 × 8 occurs with an X. Then have another volunteer mark where they think 10, 20 and 30 occur.
Which two multiples of ten is 4 × 8 between? Which multiple of ten is 4 × 8 closest to? If you were to
estimate 4 × 8 to the nearest ten, which multiple of ten would you pick?
4 × 0 4 × 10
0 10 20 30 40
Have students use the same method and the following number line to estimate 3 × 17:
3 × 10 3 × 20
30 60
Where are 40 and 50?
8. Teach students how to estimate quotients. For example, draw the following number line:
20 40 60 80
5 10 15 20
Tell your students that the bottom numbers were all obtained from the top numbers in the same way.
Can they see how? (divide by 4) Bring to the students’ attention that the numbers on the bottom, just like
the numbers on the top, go up by a fixed amount. When the numbers on the top skip count by 20, and
you divide by 4 to get the numbers on the bottom, then the numbers on the bottom skip count by
20 ÷ 4 = 5. ASK: What do they think 35 ÷ 4 will be close to? Have students mark where 35 is on the top
and then use that to mark where they think 35 ÷ 4 will be on the bottom. What is the closest whole
number to 35 ÷ 4?
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NS3-69 Patterns Made with Repeated Addition Goal: Students will use their calculator to discover patterns made from repeated addition.
Prior Knowledge Required: Addition, patterns
Give each student a calculator. If available, photocopy a calculator onto an overhead transparency, so that
students can see all the buttons. Tell students to use their calculator to find 7 + 2. Have a volunteer show on
the overhead the buttons they pressed to find the answer (7, +, 2, =). Then have all students press the “=”
button again. What number showed up? Why? What operation did the calculator do to get from 9 to 11?
Again, have students press the “=” button. What number does the calculator show? Challenge students to
predict the next number the calculator will show when they press the “=” button, and then the next number.
Explain to students that pressing 7, +, 2, =, =, =, =, =, results in: 7 + 2 + 2 + 2 + 2 + 2.
Challenge students to find, by pressing “=” the correct number of times:
a) 3 + 2 + 2
b) 5 + 3 + 3 + 3
c) 3 + 5 + 5 + 5
d) 8 + 2 + 2 + 2 + 2 + 2 + 2
e) 7 + 3 + 3 + 3 + 3
f) 8 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9
Then show students how to record each step of the process:
8 8 + 9 = 17 8 + 9 + 9 = 26 8 + 9 + 9 + 9 = 35, and so on.
Write the sequence of results, all in a row:
8 17 26 35 44 53 62 71 80 89 98 107
Have students record the patterns of ones digits only:
8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7.
Can your students predict what the next ones digit will be? Have them explain their answer.
Have students record the patterns in the number of tens:
0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10.
Tell your students that you showed this pattern to someone who said that the numbers always increase by 1.
Tell them to look closely at the sequence and to tell you if that is correct. (No, the 8 is repeated) Have
students extend the pattern on their calculators to find out when the next repetition is. ASK: How many
numbers occur before the first repetition? How many numbers come before the second repetition? How
many numbers do they think will occur before the next repetition? Have students write what they think the
next ten terms of the sequence (of the number of tens) will be and then to check their answer using the
calculator.
Activity: Review the Skip Counting Machine activity from section NS3-13: Counting by 5s and 25s.
Have students build a skip counting by 9s machine, starting from any number. Students may also enjoy
building a skip counting by 11s machine.
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NS3-70 Counting by Dollars and Coins Goal: Students will use dollar and cent notation for money amounts that involve either only cents or only
dollars. Students will skip count to add money amounts that consist of a single type of coin or dollar bill.
Review the Canadian coins: pennies, nickels, dimes and quarters, and then introduce loonies and toonies.
Give students play coins. Ask them if the amount of money is shown on the coins. Does every coin have the
amount written on it? Is the value of the coin printed on both sides or just one? Ask them if there is a word
that shows the number is talking about money and not how many trees, cats, or houses. Tell them there are
two different words used to show money and challenge them to find both of them (they are written on the
coins). Ask them how cents and dollars are like units in measurement. Remind them that if something is 5
paper clips long, it is shorter than if it is 5 notebooks long. Tell them that if something is worth 3 cents, it is
worth less than if it is worth 3 dollars. A dollar is worth a hundred cents, so if Isobel has 2 dollars and Soren
has 30 cents, Isobel has more money even though 2 is less than 30. If Isobel is 2 m tall and Soren is 30 cm
tall, who is taller?
Write on the board:
“1 dollar = 100 cents and 2 dollars = 200 cents.” Have students look at the amount on their coins and
arrange them in order from least value to most value, keeping in mind that a dollar is worth a hundred cents.
Then have them arrange the coins in order from smallest to largest in size. Which coin is out of place?
To introduce the symbols $ and ¢, tell students that in math we use the words plus, minus, and equals a lot
and ask if there is a symbol we use instead of writing out the words all the time. Write on the board the words
“plus,” “minus,” and “equals” and have volunteers write the symbols used to show those words. Ask them if
there are words we use a lot when talking about money. Ask if they think those words should have a symbol
for them. Ask if anyone knows what the symbols are. Then write on the board, “$1 = 100¢ and $2 = 200¢.”
Teach students that when we count money in cents, we write the cent sign after the amount of cents, but
when we count money in dollars, we write the dollar sign before the amount of dollars. Have students
practice by using the correct notation for the following money amounts (do not include money amounts that
involve both dollars and cents).
a) five cents b) twelve dollars c) 9 cents d) 9 dollars e) 83 dollars f) 46 cents
Draw a circle and write “5¢” inside. ASK: Which coin does this represent? Repeat for several Canadian coin
values (EXAMPLES: 1¢, $1, 10¢, $2, 25¢)
Review skip counting. Then, have students skip count to find the total amount of money:
a) 5¢ 5¢ 5¢ 5¢
b) 10¢ 10¢ 10¢ 10¢
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c) 25¢ 25¢ 25¢ 25¢
d) 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢
e) $1 $1 $1 $1 $1 $1 $1
f) $2 $2 $2 $2 $2
To guide students, have them write the subtotal over each coin. For example:
5¢ 10¢ 15¢ 20¢
5¢ 5¢ 5¢ 5¢
Then teach students to skip count without writing the subtotals. Students should count how many coins there
are and then skip count on their fingers to find the total amount of money.
Present a problem: Jane emptied her piggy bank. She asked her elder brother John to help her count the
coins. He suggested she stack the coins of the same value in different stacks and count each stack
separately. Jane has:
19 pennies 8 nickels 7 dimes
3 quarters 6 loonies 3 toonies
How much money is in each stack? Ask students to write the amount in dollar notation when counting
loonies and toonies and to write the amount in cent notation when counting pennies, nickels, dimes and
quarters. Give them more practice questions, such as:
9 pennies 14 nickels 13 dimes
6 quarters 5 loonies 9 toonies
Then ask students how much money they have if they have 6 nickels. ASK: What do I skip count by? How do
I know when to stop? Repeat with several examples of a single type of coin. (EXAMPLES: 3 quarters, 5
dimes, 7 loonies, 4 nickels, 3 toonies).
Then tell students that they are only allowed to use one type of coin. ASK: How can I make 8 cents? (Use 8
pennies.) How can I make 10 cents? (Use 10 pennies or 2 nickels or 1 dime.) How many I make 20 cents?
25 cents? 50 cents? 3 dollars? 4 dollars? Students need only find 1 solution. Bonus: Faster students should
be challenged to find several solutions or even all solutions.
Then write the amount in words that you want the students to make, again using only 1 type of coin.
(EXAMPLES: eight cents, eighty cents, five dollars, forty-five cents, fifty cents) Bonus: Students can try to
make these amounts using more than one denomination. For example, five dollars can be made with a
toonie and three loonies or two toonies and a loonie, or twelve quarters and a toonie, and so on.
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Activities:
Heads and Tails Give students play money coins – one of each: penny, nickel, dime, quarter, loonie, and
toonie. Ask them to turn all the coins so that the heads side is up. Then tell them to take a white sheet of
paper and fold it in half so that the fold line separates the top to the bottom. They then unfold the paper and
place all the coins under the top half of the page and they rub a pencil over the paper so that they can see
the coin images. When this is done, they turn the coins around so that the tails side is showing, rearrange
the coins, place the coins under the bottom half of their sheet, and rub the pencil over the paper again. They
should then match each heads side with the tail of the same coin.
Pick the Right Coin Give students a bag of coins including pennies, nickels, dimes, and quarters. Ask
students to try to pick out a dime without looking. Repeat with picking a penny, a nickel and a quarter. What
characteristics are they looking for? What is the easiest coin to pick out? Why is it the easiest? What is the
hardest coin to pick out?
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NS3-71 Dollars and Cent Notation Goal: Students will express monetary values less than a dollar in dollar and cent notation.
Prior Knowledge Required: Place value to tenths and hundredths
Canadian coins
Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, tenths, notation, hundredths
Have students show how to make various amounts of money using dimes and pennies (EXAMPLES:
35¢ = dimes and pennies, 22¢, 25¢, 30¢, 36¢, 41¢)
Explain that there are two standard ways of representing money. The first is cent notation: you simply write
the number of cents or pennies that you have, followed by the ¢ sign. In dollar notation, “twenty-five cents”
is written $0.25. The first number to the right of the decimal represents the number of dimes and the next
digit to the right represents the number of pennies.
25¢ = $0.25
2 dimes 5 pennies
Have students write both the cent notation and the dollar (decimal) notation for each amount shown above.
Have students write the total value of a collection of coins both in dollar and cent notation (begin with only
one type of coin, then two types, then three types and finally four types of coins).
a) 25¢, 25¢ b) 10¢, 10¢, 10¢ c) 10¢, 10¢, 10¢, 5¢, 5¢, 5¢
d) 10¢, 10¢, 1¢, 1¢, 1¢, 1¢, 1¢, 1¢ e) 10¢, 10¢, 5¢, 1¢, 1¢, 1¢ f) 25¢, 25¢, 10¢, 10¢, 1¢
g) 25¢, 5¢, 5¢, 5¢, 1¢ h) 25¢, 25¢, 25¢, 10¢, 1¢, 1¢ i) 25¢, 10¢, 10¢, 5¢, 1¢, 1¢
j) 25¢, 10¢, 10¢, 10¢, 5¢, 1¢, 1¢, 1¢.
Then tell students that you have 7 nickels and 4 pennies. Show this on the board in a random arrangement:
5¢, 1¢, 1¢, 5¢, 5¢, 5¢, 5¢, 1¢, 5¢, 5¢, 1¢
Ask your students to write an addition sentence to show the total amount of money. Remind them that when
they have a long sequence of numbers to add, they should keep track as they go along by putting the total at
each stage in squares above the numbers:
6 7 12 17 22 27 28 33 38 39
5 + 1 + 1 + 5 + 5 + 5 + 5 + 1 + 5 + 5 + 1
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Then show the coins on the board, putting the nickels first. Tell them you want to add the nickels first and ask
what you should skip count by. Demonstrate doing this until you get to the pennies (5, 10, 15, 20, 25, 30, 35)
by crossing out the fives as you count them and then ask: Should I still skip count by 5s? Now what do I
count by? Demonstrate continuing the count (36, 37, 38, 39) by crossing out the pennies as you count them.
ASK: Did we get the same total both ways?
Repeat with pennies and dimes, then dimes, nickels and pennies and then quarters, dimes, nickels and
pennies, all arranged randomly and then in order. ASK: What if they didn’t have nickels or dimes or quarters,
and they only had pennies. How would that affect their counting? (It would make it a lot slower to count how
much money they have and it would make carrying the money around a lot heavier.)
Then introduce dollar and cent notation for money amounts that are at least a dollar:
100¢ = $1.00, 200¢ = $2.00.
Ask students how they would show, in dollar notation, the following various amounts: 300¢, 700¢, 900¢,
500¢. Bonus: 1200¢, 1000¢, 3000¢.
Then have students total the following amounts and then write the dollar notation for the total:
a) $1, $1, $1, $1, $1 b) $2, $2, $2, $2 c) $2, $2, $1, $1, $1
d) $5, $1, $1 e) $5, $5, $2, $2, $1 f) $10, $5, $5, $2, $2, $2, $1
Then tell students that 300¢ is written as $3.00 in dollar notation and 47¢ is written as $0.47 in dollar
notation. Challenge students to guess how 347¢ is written in dollar notation. (ANSWER: $3.47)
$3.47
dollars dimes pennies
Have students (volunteers at first and then individually) show how to write the following cent amounts in
dollar notation: 321¢, 21¢, 320¢, 301¢, 478¢, 408¢, 78¢, 470¢, 603¢, 57¢, 430¢, 541¢)
Activities:
1. Ask students to pretend that there is a vending machine which only takes loonies, dimes and pennies.
Have them make amounts using only these coins (EXAMPLE: 453¢, 278¢, 102¢, etc).
2. A Game for Two: The Change Machine
One player makes an amount using nickels and quarters. The other (“the machine”) has to change the
amount into loonies, dimes and pennies.
Extensions:
1. Discuss the difference and similarities between the dollar and cent notation and the metre and
centimetre notation. Tell students that 134 cm can equally be written as 1m 34 cm or as 1.34 m. But
134¢ can be written as $1.34 (not 1.34 $) but not as 1 $ 34 ¢.
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2. Have students write in symbols to make the following sentences correct (do the first one for them):
135¢ = $1 + 35¢
246 = 2 + 46
887 = 8 + 87
432 = 32 + 4
Remind students that the dollar sign goes to the left and the cents sign to the right of the number, but
you always say 3 dollars, not dollars 3. If students are comfortable with cm and m, have them fill in the
same number sentences with those units to make the sentences true and then discuss the comparison.
Then give only the number of cents (a 3-digit number) and have students break it up into dollars and
cents.
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NS3-72 Counting and Changing Units and NS3-73 Converting Between Dollar and Cent
Notation
Goal: Students will convert between dollar and cent notation.
Prior Knowledge Required: Understanding of dollar and cent notation
Knowledge of place value
Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, notation, convert
Changing from dollar to cent notation is easy—all you need to do is to remove the decimal point and dollar
sign, and then add the ¢ sign to the right of the number.
Changing from cent notation to dollar notation is a little trickier. Make sure students know that, when an
amount in cent notation has no tens digit (or a zero in the tens place), the corresponding amount in dollar
notation must have a zero in the dimes place.
EXAMPLE: 6¢ is written $.06 or $0.06, not $0.60.
0 dimes 6 pennies 6 dimes 0 pennies
Make a chart like the one shown below on the board and ask volunteers to help you fill it in.
Amount Amount in Cents Whole Dollars Dimes Pennies Amount in Dollars
3 toonies 800 8 0 0 $8.00
2 loonies
2 quarters
5 dimes
2 nickels
4 quarters
2 pennies
Students could also practice skip counting by quarters, toonies and other coins (both in dollar and in cent
notation).
$2.00, $4.00, ____, ____ $0.25, $0.50, ____, ____, ____, ____
200¢, 400¢, ____, ____ 25¢, 50¢, ____, ____, ____, ____
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Bonus: What coin is being used for skip counting?
$1.00, ____, ____, ____, $1.20
$2.00, ____, ____, ____, $3.00
You may wish to give your students some play money and to let them practice in pairs—each player picks 2
coins and then both students individually write the total amount of the 4 coins in dollar and cent notation.
Partners then compare answers.
Assessment
1. Write the total amount in dollar and cent notation:
a)
Total amount = _____ ¢ = $ ______
b)
Total amount = _____ ¢ = $ ______
2. Convert between dollar and cent notations:
$ .24 = ___¢, _____=82¢, _____ = 6¢, $12.03 = _____, $120.30 = ______
Bonus:
3. Change these numbers to dollar notation: 273258¢, 1234567890¢.
4. Change $245.56 to cent notation. Try these amounts: $76.34, $12.03, $120.30, $123.52, $3789.49.
Challenge students to make various money amounts using exactly two coins (EXAMPLES: 6¢, 10¢, 11¢,
15¢, 20¢, 30¢, 35¢, 50¢, Bonus: $2, $3, $1.05, $2.25)
Then challenge students to make various money amounts using exactly three coins (EXAMPLES: 7¢, 12¢,
15¢, 27¢, 30¢, 31¢, 40¢, 51¢, 75¢, Bonus: $3, $4, $5, $6, $2.10, $1.10, $1.15, $1.11)
Challenge students to make various money amounts using exactly four coins (EXAMPLES: 8¢, 13¢, 22¢,
30¢, 40¢, 45¢, 60¢, 65¢, 70¢, 76¢, 80¢, 81¢, 85¢, Bonus: $4, $5, $6, $7, $8, $1.40, $2.10, $3.25, $4.30,
$3.50, $5.10)
25¢ 10¢ $2 5¢ 1¢ 1¢
25¢ $1 10¢ 5¢ 1¢
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Activities:
1. Guess the hidden coins Player 1 hides several coins: for example, a loonie, a toonie and a dime or
three loonies and a dime or a loonie, a toonie and two nickels for $3.10. Player 1 tells her partner how
many coins she has and the value of the coins in dollar notation. The partner has to guess which coins
she has. There may be more than one possibility for the answer. The player should try to use money and
coin amounts that have more than one possibility to increase the chances of an incorrect guess.
Students might discuss whether it is a good strategy for this game to use the least number of coins. (It is
not, because the least number of coins is always unique and quite easy to guess.) Challenge students to
change the rules of the game so that it is better to use the least number of coins. Then play the game.
2. Money Matching Memory Game. (See the BLM “Money Matching Memory Game”) Students play this
game in pairs. Each student takes a turn flipping over two cards. If the cards match, that player gets to
keep the pair and takes another turn. Students will have to remember which cards are placed where and
also match up identical amounts written in dollar and cent notation.
3. Each pair should have the BLM “Adding or Trading Game” as a game board and each player should
have a different token to use as their playing piece. They will also need a die to know how many pieces
to move forward. When they roll, they move forward the correct number of squares and receive the coin
shown on the board. When both players are at the end of the board (not necessarily by the exact amount
shown on the die), they count up their money – the player with the most amount of money wins.
Variation: The player with the fewest coins wins; players may trade from a shared bank for equal value
coins at the end of the game to try to have fewer coins.
4. Trading Game Have students work in pairs with different goals. Give each player 10 pennies, 4 nickels,
and 1 dime. Player One’s goal is to get 10 coins and Player Two’s goal is to get 20 coins. They must only
trade for equivalent values. To ensure this is always the case, have students add their money at the
beginning and periodically. They should always have 40¢. Note that both players will achieve their goals
at the same time, so they should cooperate.
Variation: Player One’s goal: 18 coins; Player Two’s goal: 12 coins.
Variation: Give each player 5 pennies, 5 dimes, 4 nickels, and 1 quarter. Player 1 tries to get 20 coins –
the other person will then try to get 10 coins. Students should repeatedly check that they have a total of
$1 or 100¢.
Variation: Give each player 2 loonies, 3 quarters, 10 dimes and 7 nickels, so that each player has 22
coins. Player 1 aims for 11 coins while Player 2 aims for 33 coins. Students should repeatedly check that
they have a total of $4.10.
Extensions:
1. Provide the BLM “Smallest Number of Coins Chart.”
2. Give students the BLM “Dimes, Pennies, and Base Ten Materials” to show them the relationship
between dimes and base ten blocks, and pennies and ones blocks.
3. (Atlantic Curriculum A5.11) Mary won $5000 in a contest. If she wants all her prize money in $10 bills,
how many would she receive?
4. (Atlantic Curriculum A4.4) Pretend that you won three thousand dollars. How many hundred dollar bills
would that be?
5. (Atlantic Curriculum A4.5) Martin said the car cost thirty-four hundred dollars, while Sam said he thought
it cost over three thousand dollars. Are they disagreeing? Explain.
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NS3-74 Canadian Bills and Coins Goal: Students will identify Canadian coins by name and denomination, and express their values in dollar
and cent notation. They will identify correct forms of writing amounts of Canadian currency.
Prior Knowledge Required: Dollar and cent notation
Vocabulary: penny, toonie, nickel, dollar notation, dime, cent notation, quarter, currency, loonie
Remind students that when writing in dollar notation, the number of full dollars is written to the left of the
decimal point. There is no limit to how many place value columns this can extend to. Demonstrate by writing
$2.00, $22.00, $222.00 and $2222222222222.00 on the board.
However, there are only two place value columns to the right of the decimal place. Remind students that, if
you have a number of cents which is only a single digit (say 3), in dollar notation a zero is placed in the
dimes column.
Ask the class to invent some ways to write money amounts in incorrect notation. Allow them to come up
with a wide variety of ideas and welcome silly answers. Something that looks like this: 54$, is incorrect. Add
several examples yourself, for instance:
2.89$, $26.989, $67¢, ¢45, ¢576, 37.58¢, ¢67.89, $12.34¢, $1 35
Review the names and values of Canadian coins and bills. Discuss the images depicted on the coins. Point
out that the animal on the quarter is a caribou (not a moose) and that the dime shows the Bluenose, a type
of sailing ship called a schooner.
Discuss with students the relationships between various coins and bills.
Ask students how many…
a) dimes are in a loonie, toonie, or a 5 dollar bill.
b) nickels are in a loonie or toonie.
c) quarters are in a loonie, toonie, 5 or 10 dollar bill.
d) toonies are in a 10 or 20 dollar bill.
Students could use play money as manipulatives for the above problems.
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Activities:
1. As a class or in small groups, have students visit the Canadian Currency Museum website and their
exhibit on Canada’s Coins. This has great information about the history and symbolism of Canada’s
coins. You could have students write a short report about one of the symbols on a Canadian coin and
why they are important enough to appear on a Canadian coin.
http://www.bankofcanada.ca/currencymuseum/eng/learning/canadascoins.php
2. Cross-curricular connection: Students could design coins that represent important symbols of the
First Nation peoples in Upper Canada in the years around 1800. Ask them to explain what image they
chose and why they think the symbol could be important.
3. (Atlantic Curriculum A4) Have students play “Race For a Loonie.” Ask each student to repeatedly
toss a die and count out pennies on a mat. Ten pennies are exchanged for a dime, and ten dimes for
a loonie. You may wish to have the students toss the die onto a plate to prevent the die from going
too far.
4. (Atlantic Curriculum A4.2) Have the students use a mat with sections marked off for $1, 10¢ and 1¢.
Ask them to toss two dice, find the sum, and place the total on the mat. Have them exchange 10
pennies (or 10 counters in the 1¢ section) for one dime (or one counter in the 10¢ section) and
continue until they have reached a dollar.
Extensions:
1. The Royal Canadian Mint has great resources available on their website. Their Currency Timeline might
be of great interest to the students. This outlines the history of Canadian settlement and development
and all the varieties of currency that have been used from the early 16th century to the present. Use this
as a resource and have students research other kinds of coins (denominations, forms, images, etc.),
that have existed in Canada’s past.
www.mint.ca/teach
Project Ideas
Choose a coin and find the following information on the mint.ca/teach web-site.
• What are the different images that have appeared on this coin and what did they
commemorate?
• Who drew the design for the coin?
• When was the coin introduced?
• Were there changes in the metal used to make the coin? When and why were these changes
made?
2. If you have any students who have lived in or travelled to other countries, have them bring in samples
of the other currency as a ‘show and tell’ for the class. Discuss the different images and shapes of the
coins and bills.
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NS3-75 Adding Money Goal: Students will be able to add money amounts using traditional algorithms.
Prior Knowledge Required: Adding three-digit numbers
Adding using regrouping
Converting between dollar and cent notation
Familiarity with Canadian currency
Vocabulary: penny, quarter, nickel, loonie, dime, toonie, cent, adding, dollar , regrouping
Review adding 2-digit numbers. Demonstrate the steps: line up the numbers correctly, then add the digits in
each column starting from the right. Start with some examples that would not require regrouping, and then
move on to some which do (EXAMPLES: 15 + 23, 62 + 23, 38 + 46).
Use volunteers to pass to 3-digit addition questions, first without regrouping, then with regrouping.
SAMPLE QUESTIONS:
545 + 123, 123 + 345, then 132 + 259, 234 + 556, 578 + 789, 346 + 397.
Tell your students that you want to add $5.45 + $1.23. ASK: How many cents are in $5.45? (545¢) How
many cents are in $1.23? (123¢) How many cents is that altogether? (668¢) What is that in dollar notation?
($6.68) Show on the board:
$5.45 545¢
+ $1.23 123¢
$6.68 668¢
When adding money, the difference is in the lining up—the decimal point is lined up over the decimal point.
ASK: Are the one dollars lined up over the one dollars? The ten dollars over the ten dollars? The dimes
over the dimes? The pennies over the pennies? Tell them that if the decimal point is lined up, all the other
digits must be lined up correctly too, since the decimal point is between the ones and the dimes. Students
can model regrouping of terms using play money: for instance, in $2.33 + $2.74 they will have to group 10
dimes as a dollar.
Students should complete a number of problems in their notebooks. Some SAMPLE PROBLEMS:
a) $5.08 + $1.51 b) $3.13 + $2.98 c) $1.74 + $5.22
d) $3.95 + $4.28 e) $1.79 + $2.83
Ask volunteers to help you solve several word problems, such as: Julie spent $4.98 for a T-shirt and $3.78
for a sandwich. How much did she spend in total?
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Students should also practice finding the value of sets of coins and bills and writing the amount in dollar
notation, such as:
There are 19 pennies, 23 nickels and 7 quarters in Jane’s piggybank. How much money does
she have?
Helen has a five-dollar bill, 1 toonie and 9 quarters in her pocket. How much money does she have?
Randy paid 2 toonies, a loonie, 5 quarters and 7 dimes for a parrot. How much did his parrot cost?
A mango fruit costs 69¢. I have a toonie. How many mangos can I buy?
If I add a dime, will it suffice for another one?
Have students do the following questions individually.
1. Add:
a) $7.25 b) $3.89 c) $5.08 d) $5.37 e) $2.86 + $5.17
+$1.64 +$6.23 +$3.87 +$2.79
2. Sheila saved 2 toonies, 4 dimes and 8 pennies from babysitting. Her brother Noah saved 4 loonies and
6 quarters from mowing a lawn.
a) Who has saved more money?
b) They want to share money to buy a present for their mother. How much money do
they have together?
c) They’ve chosen a teapot for $9.99. Do they have enough money?
Bring in fliers from local businesses. Ask students to select gifts to buy for a friend or relative as a
birthday gift. They must choose at least two items. They have a $10.00 budget. What is the total
cost of their gifts?
Extension: Fill in the missing information in the story problem and then solve the problem.
a) Betty bought _____ pairs of shoes for _____ each. How much did she spend?
b) Una bought ____ apples for _____ each. How much did she spend?
c) Bertrand bought _____ brooms for _____ each. How much did he spend?
d) Blake bought _____ comic books for _____ each. How much did he spend?
Bonus: Have students make up their own story problem and have a partner solve it.
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NS3-76 Subtracting Money Goal: Students will subtract money amounts using traditional algorithms
Prior Knowledge Required: Subtraction of 3-digit numbers
Subtracting using regrouping
Converting between dollar and cent notation
Familiarity with Canadian currency
Vocabulary: penny, toonie, nickel, cent, dime, dollar, quarter, subtracting, loonie, regrouping
Review 2-digit and 3-digit subtraction. Start with some examples that would not require regrouping: 45 − 23,
78 − 67, 234 − 123, 678 − 354.
Show some examples on the board of numbers lined up correctly or incorrectly and have students decide
which ones are done correctly.
Demonstrate the steps: line up the numbers correctly, subtract the digits in each column starting from the
right. Move onto questions that require regrouping, EXAMPLE: 86 − 27, 567 − 38, 782 − 127, 673 − 185,
467 − 369.
Tell your students that you want to subtract $0.38 from $5.67. ASK: How many cents are in $5.67? (567)
How many cents are in $0.38? (38) What is 567 – 38?
567¢
– 38¢
529¢
ASK: What is 529¢ in dollar notation? ($5.29) Show on the board:
567¢ $5.67
– 38¢ – $0.38
529¢ $5.29
When subtracting money, the difference is in the lining up—the decimal point is lined up over the decimal
point. ASK: Are the one dollars lined up over the one dollars? The dimes over the dimes? The pennies over
the pennies? Remind them that if the decimal point is lined up, all the other digits must be lined up correctly
too, since the decimal point is between the ones and the dimes. Students can model regrouping of terms
using play money: for instance, in $2.74 – $2.36 they will have to group a dime as ten pennies.
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Students should complete a number of problems in their notebooks. For EXAMPLE:
Subtract:
a) $8.89 b) $9.00 c) $4.00 d) $8.37 e) $8.47 - $0.62
−$1.64 −$7.23 −$3.87 −$5.79
Bonus: $28.14 – 17.43
ASK: Alyson went to a grocery store with $10.00. She would buy buns for $1.69, ice cream for $3.99 and
tomatoes for $2.50. Does she have enough money? If yes, how much change will she get?
Next, teach your students the following fast way of subtracting from powers of 10 (numbers such as 10,
100, 1 000 and so on) to help them avoid regrouping:
For example, you can subtract any money amount from a dollar by taking the amount away from 99¢ and
then adding one cent to the result.
1.00 .99 + one cent .99 + .01
− .57 = −.57 = −.57
.42 + .01 = .43 = 43¢
As another example, you can subtract any money amount from $10.00 by taking the amount away from
$9.99 and adding one cent to the result.
10.00 9.99 + .01
− 8.63 = − 8.63
1.36 + .01 = 1.37
NOTE: If students know how to subtract any one-digit number from 9, then they can easily perform the
subtractions shown above mentally. To reinforce this skill have students play the Modified Go Fish game
(in the MENTAL MATH section) using 9 as the target number.
Literature Connection
Alexander, Who Used to Be Rich Last Sunday. By Judith Viorst.
Last Sunday, Alexander’s grandparents gave him a dollar—and he was rich. There were so many things
that he could do with all of that money!
He could buy as much gum as he wanted, or even a walkie-talkie, if he saved enough. But somehow the
money began to disappear…
A great activity would be stopping to calculate how much money Alexander is left with every time he ends
up spending money.
Students could even write their own stories about Alexander creating a subtraction problem of their own.
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Extensions:
1. a) Ella paid for a bottle of water with $2 and received 35¢ in change. How much did the water cost?
b) Jordan paid for some spring rolls with $5 and received 47¢ in change. How much did his meal cost?
c) Paige paid for a shirt with $10 and received 68¢ in change. How much did the shirt cost?
d) Geoff paid for a hockey stick with $10 and received 54¢ in change. How much did the hockey
stick cost?
2. Fill in the missing information in the story problem and then solve the problem.
a) Kyle spent $4.90 for a notebook and pencils. He bought 5 pencils for ______. How much did the
notebook cost?
b) Sally spent $6.50 for a bottle of juice and 3 apples. The apples cost ______How much did the
juice cost?
c) Clarke spent $9.70 for 2 novels and a dictionary. The ______ cost ______. How much did the
dictionary cost?
d) Mary spent $16.40 for 2 movie tickets and a small bag of popcorn. ___________________. How
much did her popcorn cost?
3. (Atlantic Curriculum B4.1) Count back the change for $5.00, if the bill totalled $3.59.
4. (Atlantic Curriculum B4.3) Write a subtraction problem that includes $1.40 and 16¢. Solve the problem.
Number Sense Teacher’s Guide Workbook 3:2 67 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-77 Estimating Goal: Students will use rounding to estimate money amounts.
Prior Knowledge Required: Rounding
Canadian bills and coins
adding and subtracting money
dollars and cents
word problems
Vocabulary: rounding, estimating, estimate, about
Begin with a demonstration. Bring in a handful of change. Put it down on a table at the front of the class.
Tell your students that you would like to buy a magazine that costs $2.50. Do they think that you have
enough money? How could they find out?
Review rounding. Remind the class that rounding involves changing numbers to an amount that is close to
the original, but which is easier to use in mental calculations. As an example, ask what is easier to add:
8 + 9 or 10 + 10.
Explain to the class that they will be doing two kinds of rounding—rounding to the nearest 10¢ and rounding
to the nearest dollar. To round to the nearest 10¢ look at the digit in the pennies column. If the pennies digit
is less than 5, you round down. ASK: For which pennies digits would you round down? 7? 0? 3? 4? 9?
(round down for 0, 1, 2, 3 or 4) If the pennies digit is 5 or more, you round up. For which ones digits would
you round up? (5, 6, 7, 8 and 9) For example 33¢ would round down to 30¢, but 38¢ would round up to 40¢.
Use volunteers to round: 39¢, 56¢, 52¢, 75¢, 60¢, 44¢.
Model rounding to the nearest dollar. In this case, the number to look at is in the tenths place (the dimes
place). If an amount has 50¢ or more, round up. If it has less than 50¢, round down. So, $1.54 would round
up to $2.00. $1.45 would round down to $1.00. Use volunteers to round: $1.39, $2.56, $3.50, $4.75, $0.60,
$0.49.
Give several problems to show how rounding can be used for estimation:
Make an estimate and then find the exact amount:
a) Dana has $5.27. Tor has $2.38. How much more money does Dana have than Tor?
b) Mary has $3.74. Sheryl has $5.33. How much money do they have altogether?
a) Jason has saved $9.95. Does he have enough money to buy a book for $4.96 and a binder for $5.99?
Why is rounding not helpful here?
Ask students to explain why rounding to the nearest dollar isn’t helpful for the following question:
“Millicent has $2.15. Richard has $1.97. About how much more money does Millicent have than Richard?”
(Both round to $2. Rounding to the nearest 10 cents helps.)
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Make sure students understand that they can use rounding to check the reasonableness of answers. Ask
students to explain how they know that $2.52 + $3.95 = $9.47 can’t be correct.
Have students solve the following problems individually.
1. Make an estimate and then find the exact amount: Molina has $7.89. Vinijaa has $5.77. How much
more money does Molina have?
2. Benjamin spent $2.94 on pop, $4.85 on vegetables and dip and $2.15 on bagels. About how much did
he spend altogether?
Activities:
1. Ask students to estimate the total value of a particular denomination (EXAMPLE: dimes, loonies, etc.)
that would be needed to cover their desk or a book. Students could use play money to test their
predictions. Students could also trace an outline of their hand and predict the value of dimes that would
cover their hand and then test their prediction.
2. Place a handful of play money bills and coins on a table and cover up the amount after students have
had a chance to look at it 5 or 10 seconds. Ask students to estimate the total amount. Alternatively, you
might ask students to estimate how much extra money would be needed to make a particular amount
(say $5 or $10).
1. Pairs of students practice estimating by using play money coins. Player 1 places ten to fifteen play
money coins on a table. Player 2 estimates the amount of money before counting the coins.
Extensions:
1. What is the best way to round when you are adding two numbers: to round both to the nearest dollar, or
to round one up and one down?
Explore which method gives the best answer for the following amounts:
$2.56 + $3.68 $4.55 + $4.57 $6.61 + $1.05
Students might notice that, when two numbers have cent values that are both close to 50¢ and that are
both greater than 50¢ or both less than 50¢, rounding one number up and one down gives a better
result than the standard rounding technique. For instance, rounding $2.57 and $3.54 to the nearest
dollar gives an estimated sum of $7.00 ($2.57 + $3.54 ≈ $3.00 + $4.00), whereas rounding one
number up and the other down gives an estimated sum of $6.00, which is closer to the actual total.
2. (Adapted from Atlantic Curriculum B10)
a) Popsicles cost 10¢ each. How many popsicles could you buy with a loonie? (ANSWER: 10)
b) Popsicles cost 20¢ each. How many popsicles could you buy with a loonie? (ANSWER: 5)
c) Popsicles cost 14¢ each. About how many popsicles could you buy with a loonie? Will the answer
be closer to 5 or 10? If I want to buy a popsicle for myself and five others, will a loonie be enough?
d) If erasers are on sale for 19¢, how many would you estimate you could buy with a loonie?
e) If I have a loonie, can I buy 3 party favours that cost 29¢ each?
f) If I have a loonie, can I buy 5 packages of stickers that cost 21¢ each?
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NS3-78 Equal Parts Goal: Students will verbally express fractions given as diagrams or numerical notation.
Prior Knowledge Required: The ability to count
Fraction of an area
Ordinal numbers
Vocabulary: part, numerator, whole, denominator, fraction, area
Draw:
Ask your students how many circles are shaded.
Draw then ask them again how many circles are shaded.
Explain that the whole circle is no longer shaded. Ask your students if they know the word for a number that
is not a whole number, but is only part of a whole number. [Fraction.]
Explain that a fraction has a top and bottom number, then ask your students if they know what the numbers
represent. Explain that the shaded fraction of the circle is written as 34 (pronounce this as “three over four” for
now). What does the 3 represent? What does the 4 represent? Draw more fractions and ask students who
understand the significance of the numbers to identify the fractions without explaining it to the rest of the
class.
14
34
24
24
Draw examples with different denominators. Ask: What does the top number of the fraction represent? [The
number of shaded parts.] What does the bottom number of the fraction represent? [The number of parts in a
whole.]
If some students say that the bottom number is counting the number of parts altogether, tell them that from
what they’ve seen so far, that’s a good answer, but later we will see improper fractions where we have more
than one whole pie, so if each pie has 4 pieces and we have 2 pies, there are 8 pieces altogether, but we still
write the bottom number as 4. For example, if you bought 2 pies and 3 pieces were eaten from each pie, you
could say that 64 of a pie was eaten. (You could also say that
68 of what you bought was eaten, but that would
change the whole to 2 pies instead of 1 pie). Explain that fractions aren’t generally pronounced as they are.
Ask your students if they know the expressions for “three over four.” [Three-fourths or three-quarters.] Then
ask if they know the expression for 12 . Draw a half-shaded circle to encourage answers.
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Draw several diagrams and have students express the fraction for the shaded parts of each whole by
writing the top and the bottom numbers.
Gradually increase the number of parts in each whole and shaded parts.
Ask them to explain their methods (skip counting or multiplication, for example) for determining the total
number of squares. At first, shade parts in an orderly way to facilitate a count. Then shade parts randomly,
but never exceed more than 20 shaded parts, and start with a small number of parts.
Ask students if they know which number—the top or bottom—is called the numerator. [Top.]
ASK: Does anyone know what the bottom number is called? [The denominator.] Which number—the
numerator or denominator—expresses the amount of equal parts in the whole?
Have students shade the correct number of parts to illustrate the following fractions.
16
35
47
Extension: A sport played by witches and wizards on brooms regulates that the players must fly higher
than 5 m above the ground over certain parts of the field (shown as shaded). Over what fraction of the field
must the players fly higher than 5 m?
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NS3-79 Models of Fractions Goal: Students will name fractions given as diagrams or in numerical notation. Students will understand
that parts of a whole must be equal to determine a fraction that one or more parts represent.
Prior Knowledge Required: Expressing fractions
Fraction of an area
Fraction of a length
Vocabulary: numerator, denominator, fraction, part, whole
Illustrate these fractions — 12 ,
13 ,
14 ,
15 ,
16 ,
17 ,
18 — with circles.
Ask students if they know the proper way to pronounce these fractions. Remind them of ordinal numbers.
Say: If Sally is first in line, Tom is second in line and Rita is in line behind Tom, in what place is Rita? If Bilal
is in line behind Rita, in what place is Bilal? Continue to the eighth position. Explain that most of the ordinal
numbers—except for first and second—are also used for fractions. No one refers to half of a pie as one-
second of a pie, but we do say one-third, one-fourth, one-fifth, and so on.
Then ask your students to pronounce these fractions: 1
11 , 124 ,
113 ,
119 ,
1100 ,
192 .
If your students are comfortable with ordinal numbers up to a hundred, 1
92 could lead to some confusion,
since “first” and “second” are usually unused when dealing with fractions. In this case, the fraction is
expressed as “one ninety-second of a whole.” Explain that fractions with a numerator larger than one are
expressed the same way, with the numerator followed by the ordinal number. For example, 311 is expressed
as “three elevenths”. The ordinal number is pluralized when the numerator is greater than one, i.e., one
eleventh, two elevenths, three elevenths, and so on. Some ESL students might find it helpful to contrast this
with how we say 200 = two hundred, not two hundreds.
Ask your students to pronounce these fractions: 314 ,
295 ,
17100 ,
9495 ,
6183 ,
4151 ,
3052 .
Include fractions on spelling tests by writing the numeric fraction on the board.
Ask your students if they have ever been given a fraction of something (like food) instead of the whole, and
gather their responses. Bring a banana (or some easily broken piece of food) to class. Break it in two very
unequal pieces. Say: This is one of two pieces. Is this half the banana? Why not? Emphasize that the parts
have to be equal for either of the two pieces to be a half. Then draw the following rectangle.
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Ask your students if they think the rectangle is divided in half. Explain that the fraction 12 not only expresses
one of two parts, but it more specifically expresses one of two equal parts. Draw numerous examples of this
fraction, some that are equal and some that are not, and ask volunteers to mark the diagrams as correct or
incorrect.
���� x x �
Ask: Which diagram illustrates one fourth? What’s wrong with the other diagram? Isn’t one of the four pieces
still shaded?
Then draw the following diagram on the board:
Ask volunteers to come to the board to show 34 in different ways. Then challenge a volunteer to draw the
circle themselves and to show 34 in yet a different way. Have students practice individually drawing a circle
divided into fourths in their notebooks and to find as many ways of showing 24 as they can.
Then show students the four ways of showing 14 :
Challenge students to find another way of showing 14 in a circle. Draw the following pictures on the board to
help them:
Have volunteers draw their own circle divided into fourths to show yet another way of modelling 1/4. You
could help them get started by drawing one line for them:
Then challenge students to individually show in their notebooks many ways of dividing a pie into halves, and
then into thirds (they should find a pie drawn in the workbook that is already divided into thirds to help them)
and then into eighths.
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Bonus: Cut the following pies into thirds:
Activity: After students have completed exercise 6 and are familiar with the words “half,” “third,” “fourth,”
and so on, ask them to identify fractions (i.e. of pizzas, of objects in the classroom such as the blackboard,
the carpet, their desk, their pencil, and of diagrams) using phrases such as “one half,” “two thirds,” etc.
Extension: Have students ask their French teacher if ordinal numbers are used for fractions in French,
and have them tell you the answer next class.
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NS3-80 Fractions of a Region or Length Goal: Students will understand that parts of a whole must be equal to determine an entire measurement,
when given only a fraction of the entire measurement.
Prior Knowledge Required: Expressing fractions
Fraction of an area
Fraction of a length
Vocabulary: numerator, denominator, fraction, part, whole
Explain that it’s not just shapes like circles and squares and triangles that can be divided into fractions, but
anything that can be divided into equal parts. Draw a line and ask if a line can be divided into equal parts.
Ask a volunteer to guess where the line would be divided in half. Then ask the class to suggest a way of
checking how close the volunteer’s guess is. Have a volunteer measure the length of each part. Is one part
longer? How much longer? Challenge students to discover a way to check that the two halves are equal
without using a ruler, only a pencil and paper. [On a separate sheet of paper, mark the length of one side of
the divided line. Compare that length with the other side of the divided line by sliding the paper over. Are
they the same length?]
Have students draw lines in their notebooks and then ask a partner to guess where the line would be
divided in half. They can then check their partner’s work.
ASK: What fraction of this line is double?
Say: The double line is one part of the line. How many parts, equal to that one, are in the whole linne,
including the double line? [5, so the double line is 15 of the whole line.]
Mark the length of the double line on a separate sheet of paper. Compare that length to the entire line to
determine how many of those lengths make up the whole line. Repeat with more examples.
Then ask students to express the fraction of shaded squares in each of the following rectangles.
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Have them compare the top and bottom rows of rectangles.
ASK: Are the same fraction of the rectangles shaded in both rows? Explain. If you were given the
rectangles without square divisions, how would you determine the shaded fraction? What could be used to
mark the parts of the rectangle? What if you didn’t have a ruler? Have them work as partners to solve the
problem. Suggest that they mark the length of one square unit on a separate sheet of paper, and then use
that length to mark additional square units.
Prepare several strips of paper with one unit shaded, and have students determine the shaded fraction
without using a pencil or ruler. Only allow them to fold the paper. For instance they could fold the following
pieces of paper into 3, 4 and 5 equal parts respectively.
Draw a shaded square and ask students to extend it so the shaded part becomes half the size
of the extended rectangle.
Repeat this exercise for squares that are one-third and one-quarter the size of extended rectangles. ASK:
How many equal parts are needed? [Three for one-third, four for one-quarter.]
How many parts do you already have? [1.] So how many more equal parts are needed?
[Two for one-third, three for one-quarter.]
Activities:
After students have completed Question 3 from worksheet NS3-80, ask them to use a ruler to draw
rectangles in which:
a) 13 of the area is shaded b)
34 of the area is shaded
Extensions
1. On grid paper draw a rectangle with a width of 2 boxes and a length of 3 boxes. Shade 13 of the boxes.
2. On grid paper draw a rectangle with a width of 2 boxes and a length of 5 boxes. Shade 15 of the boxes.
3. a) Sketch a pie and cut it into fourths. How can it be cut into eighths?
b) Sketch a pie and cut it into thirds. How can it be cut into sixths?
4. Ask students to identify several fractions they see in the classroom and name them (e.g. about one third
of the chalk board is covered with writing, the room is about 4 fifths wide as it is long).
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NS3-81 Equals Parts of a Set Goal: Students will understand that fractions can represent equal parts of a set.
Prior Knowledge Required: Fractions as equal parts of a whole
Review equal parts of a whole. Tell your students that the whole for a fraction might not be a shape like a
circle or square. Tell them that the whole can be anything that can be divided into equal parts. Brainstorm
with the class other things that the whole might be: a line, an angle, a container, apples, oranges, amounts
of flour for a recipe. Tell them that the whole could even be a group of people. For example, the grade 3
students in this class is a whole set and I can ask questions like: what fraction of students in this class are
girls? What fraction of students in this class are eight years old? What fraction of students wear glasses?
What do I need to know to find the fraction of students who are girls? (The total number of students and the
number of girls). Which number do I put on top: the total number of students or the number of girls? (the
number of girls). Does anyone know what the top number is called? (the numerator) Does anyone know
what the bottom number is called? (the denominator) What number is the denominator? (the total number of
students). What fraction of students in this class are girls? (Ensure that they say the correct name for the
fraction, for example: “eleven twentieths” instead of “eleven over twenty”) Tell them that the girls and boys
don’t have to be the same size; they are still equal parts of a set. Ask students to answer: What fraction of
their family is older than 8? Younger than 8? Female? Male? Some of these fractions, for some students,
will have numerator 0, and this should be pointed out. Avoid asking questions that
will lead them to fractions with a denominator of 0 (For example, the question “What fraction of your siblings
are male?” will lead some students to say 0/0).
Then draw pictures of shapes with two attributes changing:
a)
ASK: What fraction of these shapes are shaded?
What fraction are circles?
What fraction of the circles are shaded?
b)
ASK: What fraction of these shapes are shaded? What fraction are unshaded? What fraction are
squares? What fraction are triangles? Bonus: What fraction of the triangles are shaded? What fraction
of the squares are shaded? What fraction of the squares are not shaded?
Have students write fraction statements in their notebooks for similar pictures.
Then tell your students that you have five squares and circles. Some are shaded and some are not. Have
students draw shapes that fit the puzzles:
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a) 25 of the shapes are squares.
25 of the shapes are shaded. One circle is shaded.
SOLUTION:
b) 35 of the shapes are squares.
25 of the shapes are shaded. No circle is shaded.
c) 35 of the shapes are squares.
35 of the shapes are shaded.
13 of the squares are shaded.
Ask some word problems:
A basketball team played 5 games and won 2 of them. What fraction of the games
did the team win?
A basketball team won 3 games and lost 1 game. How many games did they play altogether? What
fraction of their games did they win?
Bonus: A basketball team won 4 games, lost 1 game and tied 2 games. How many games did
they play? What fraction of their games did they win?
Activities:
1. On a geoboard, have students enclose a given
fraction of the pegs with an elastic.
For instance, 1025 .
2. (Adapted from Atlantic Curriculum Grade 4) Have the students “shake and spill” a number of two-
coloured counters and ask them to name the fraction that represents the red counters.
Extensions:
1. There are 8 figures in total. 12 of the figures are squares. The rest of the shapes are triangles.
58 of the figures are shaded. One triangle is shaded. How many squares are shaded?
2. a) Complete the following sentences (by writing a fraction in the first blank):
ii) _____ of the children in my family are _____ ii) _____ of the children in my class are _____
b) Make up your own sentence like the ones above in a).
3. Then draw the following picture.
ASK: How many pieces are shaded? (1) How many pieces are there altogether? (4)
Are one quarter of the pieces shaded? (yes) Is 14 of the shape shaded? (No, because
the four pieces do not have equal area.) Note that, just like boys and girls don’t have
to be the same size to be equal parts of a set, the pieces don’t have to be the same
size to be equal parts of a set.
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NS3-82 Parts and Wholes Goal: Students will compare fractions with the same denominator. Students will compare fractions with
the same numerator. Students will divide shapes into equal parts to determine what fraction of the shapes
are shaded.
Prior Knowledge Required: Fractions as area
Parts of a whole must be equal to determine a fraction that one or
more parts represent.
Comparing fractions with the same denominator.
Draw on the board:
Have students name the fractions shaded and then have them say which circle has more shaded. ASK: Which is
more: one fourth of the circle or three fourths of the circle? Then draw different shapes on the board:
14
34
14
34
Have volunteers show the fractions on the board by shading and ASK: Is three quarters of something
always more than one quarter of the same thing? Is three quarters of a metre longer or shorter than a
quarter of a metre? Is three quarters of a dollar more money or less money than a quarter of a dollar? Is
three fourths of an orange more or less than one fourth of the orange? If possible, bring in an actual orange,
or use a paper circle, cut into quarters. Put one quarter aside and count the remainder: one quarter, two
quarters, three quarters. ASK: Which is more: one quarter or three quarters? Is three fourths of the class
more or less people than one fourth of the class? If three fourths of the class have brown eyes and one
quarter of the class have blue eyes, do more people have brown eyes or blue eyes?
Tell students that if you consider fractions of the same whole—no matter what whole you’re referring to—
three quarters of the whole is always more than one quarter of that whole, so mathematicians say that the
fraction 34 is greater than the fraction
14 . Ask students if they remember what symbol goes in between:
34
14 (< or >).
Remind them that the inequality sign is like the mouth of a hungry person who wants to eat more of the
pasta but has to choose between three quarters of it or one quarter of it. The sign opens toward the bigger
number:
34
14 or
14
34 .
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Ensure that students understand that three of anything is always more than one of them. ASK: Are three
fifths more than one fifth?
Count as you colour in the fifths: one fifth, two fifths, three fifths.
ASK: Are three eighths more than one eighth? (Draw a picture of two pies, one with one eighth shaded and
the other with three eighths shaded). Which is more: five eighths or seven eighths? Five apples or seven
apples? $5 or $7? If Sally gets one sixth and Tony gets two sixths, who gets more? If Sally gets three sixths
and Tony gets two sixths, who gets more?
Have volunteers circle the largest fraction: 35 ,
25 ;
17 ,
47 ;
36 ,
56 ;
58 ,
38 .
Have students solve the following problems individually.
19 or
29 ?
112 or
212 ?
153 or
253 ?
1100 or
2100 ?
1807 or
2807 ?
29 or
59 ?
311 or
411 ?
911 or
811 ?
3587 or
4387 ?
91102 or
52102 ?
Bonus: 7 432
25 401 or 869
25 401 ? 52 645
4 567 341 or 54 154
4 567 341 ?
Comparing fractions with the same numerator.
Draw on the board:
12
13
14
15
Have a volunteer colour the first part of each strip of paper and then ask students which fraction shows the
most: 12 ,
13 ,
14 or
15 . Ask: Do you think one sixth of this fraction strip will be more or less than one fifth of it?
Will one eighth be more or less than one tenth? Then colour the second fifth and ASK: Which is more: two
fifths or one half. ASK: Two is more than one; why aren’t two fifths more than one half? (The fifth-sized
pieces are smaller than the half-sized pieces. It’s like saying two pencils are longer than one desk because
2 is more than 1 – show the two pencils next to your desk).
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Then draw two circles the same size on the board:
Have volunteers shade one part of each circle. ASK: What fractions are represented? Which fraction
represents more? Repeat with different shapes:
12
14
12
14
12
14
Ask: Is one half of something always more than one quarter of the same thing? Is half a metre longer or
shorter than a quarter of a metre? Is half an hour more or less time than a quarter of an hour? Is half a dollar
more money or less money than a quarter of a dollar? Is half an orange more or less than a fourth of the
orange? Is half the class more or less than a quarter of the class? If half the class has brown eyes and a
quarter of the class has green eyes, do more people have brown eyes or green eyes?
Tell students that no matter what quantity you have, half of the quantity is always more than a fourth of it, so
mathematicians say that the fraction 12 is greater than the fraction
14 . Ask students if they remember what
symbol goes in between: 12
14 (< or >). What symbol goes in between now?
14
12
Tell your students that you are going to try to trick them with this next question so they will have to listen
carefully. Then ASK: Is half a minute longer or shorter than a quarter of an hour? Is half a centimetre longer
or shorter than a quarter of a metre? Is half of Stick A longer or shorter than a quarter of Stick B?
Stick A:
Stick B:
Ask: Is a half always bigger than a quarter?
Allow everyone who wishes to attempt to articulate an answer. Summarize by saying: A half of something is
always more than a quarter of the same thing. But if we compare different things, a half of something might
very well be less than a quarter of something else. When mathematicians say that 12 >
14 , they mean that a
half of something is always more than a quarter of the same thing; it doesn’t matter what you take as your
whole, as long as it’s the same whole for both fractions.
Draw the following strips on the board:
Ask students to name the fractions and then to tell you which is more.
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Have students draw the same fractions in their notebooks but with circles instead of strips.
Is 34 still more than
38 ? (yes, as long as the circles are the same size)
Bonus:
Show the same fractions using a line of length 8 cm.
Ask students: If you cut the same strip into more and more pieces of the same size, what happens to the
size of each piece?
Draw the following picture on the board to help them:
1 big piece
2 pieces in one whole
3 pieces in one whole
4 pieces in one whole
Many pieces in one whole
Ask: Do you think that 1 third of a pie is more or less pie than 1 fifth of the same pie? Would you rather have
one piece when it’s cut into 3 pieces or 5 pieces? Which way will you get more? Ask a volunteer to show how
we write that mathematically (13 >
15 ).
Do you think 2 thirds of a pie is more or less than 2 fifths of the same pie? Would you rather have two pieces
when the pie is cut into 3 pieces or 5 pieces? Which way will get you more? Ask a volunteer to show how we
write that mathematically (23 >
25 ).
If you get 7 pieces, would you rather the pie be cut into 20 pieces or 30? Which way will get you more pie?
How do we write that mathematically? (720 >
730 ).
Ask the students to answer individually: Which is greater?
a) 12 or
17 b)
13 or
14 ? c)
19 or
16 ? d) Bonus:
1132 or
1147
e) 37 or
38 f)
419 or
415 g)
822 or
825 h) Bonus:
132234 or
132198
SAY: Two fractions have the same numerator and different denominators. How can you tell which fraction is
bigger? Why? Summarize by saying that the same number of pieces gives more when the pieces are
bigger. The numerator tells you the number of pieces, so when the numerator is the same, you just look at
the denominator.
The bigger the denominator, the more pieces you have to share between and the smaller the portion you
get. So bigger denominators give smaller fractions when the numerators are the same.
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SAY: If two fractions have the same denominator and different numerators, how can you tell which fraction
is bigger? Why? Summarize by saying that if the denominators are the same, the size of the pieces are the
same. So just as 2 pieces of the same size are more than 1 piece of that size, 84 pieces of the same size
are more than 76 pieces of that size.
Emphasize that students can’t do this sort of comparison if the denominators are not the same. ASK: Would
you rather 2 fifths of a pie or 1 half? Draw the following picture to help them:
Tell your students that when the denominators and numerators of the fractions are different, they will have
to compare the fractions by drawing a picture or by using other methods that they will learn in later grades.
The same fraction of the same thing are always equal.
Draw:
ASK: What fraction of the first square is shaded? What fraction of the second square is shaded? Are the
squares the same size? Are the shaded parts the same size?
Repeat for various pairs of shapes:
Challenge students to find various ways of dividing these shapes into quarters.
EXAMPLES:
Dividing shapes into equal parts.
Draw on the board the shaded strips from before:
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ASK: Is the same amount shaded on each strip? Is the same fraction of the whole strip shaded in each
case? How do you know? Then draw two hexagons as follows:
ASK: Is the same amount shaded on each hexagon? What fraction of each hexagon is shaded? Then
challenge students to find the fraction shaded by drawing their own lines to divide the shapes into
equal parts:
Give your students pattern blocks. Ask them to make a rhombus from the triangles. How many triangles do
they need? What fraction of a rhombus is a triangle? Challenge them to find:
a) What fraction of a hexagon is the rhombus? The trapezoid? The triangle?
b) What fraction of a trapezoid is the triangle?
Activities:
1. Give each group of 3 students 3 large congruent shapes, but cut differently into the same number of
equal parts. EXAMPLE:
Have students shade one part of each shape and then cover, in a single layer, the shaded part with as
many small counters as they can. Did each quarter of the shape require the same number of counters, at
least approximately?
2. Have students toss several coins (or two-colour counters). What fraction of the coins came up heads?
What fraction of the coins came up tails? What do the two numerators add to? (the denominator) Why do
they think that happened? Will it always happen?
3. Ask students to enclose a fraction of the pegs on a geoboard and then to write the fraction of pegs that
are not enclosed. Have students investigate with many examples, so that they see that the two
numerators always add up to the denominator. For extra practice with fractions that add to 1, use the
2-page BLM “Fractions That Add to 1.”
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Extensions:
1. a) What fraction of a tens block is a ones block?
b) What fraction of a tens block is 3 ones blocks?
c) What fraction of a hundreds block is a tens block?
d) What fraction of a hundreds block is 4 tens blocks?
e) What fraction of a hundreds block is 32 ones blocks?
f) What fraction of a hundreds block is 3 tens blocks and 2 ones blocks?
2. On a geoboard, show 3 different ways to divide the area of the board into 2 equal parts.
EXAMPLES:
3. Give each student a set of pattern blocks. Ask them to identify the whole of a figure
given a part.
a) If the pattern block triangle is 16 of a pattern block, what is the whole?
Answer: The hexagon.
b) If the pattern block triangle is 13 of a pattern block, what is the whole?
Answer: The trapezoid.
c) If the pattern block triangle is 12 of a pattern block, what is the whole?
Answer: The rhombus.
d) If the rhombus is 16 of a set of same-shaped pattern blocks, what is the whole?
Answer: 2 hexagons or 6 rhombuses or 12 triangles or 4 trapezoids.
2. Draw each shape below onto cm grid paper so that each square takes up a 4 cm by 4 cm square. Have
students find the area of each shaded piece:
Bonus:
To help students count half squares and whole squares, see ME3-30.
5. The pattern block triangle represents 14 . What might the whole be?
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6. (From Atlantic Curriculum A3.9) This shape is 12 of a larger one. What could the larger one look like?
How many different possibilities can you find?
a) b) c) d)
NOTE: If you prefer, you could assign these shapes on grid paper (rather than dot paper or a
geoboard).
7. Students can construct a figure using the pattern block shapes and then determine what fraction of the
shapes is covered by the pattern block triangle.
8. What fraction of the figure is covered by…
a) The shaded triangle
b) The small square
9. This extension is best done after Activity 2 above. Compare the following fractions by comparing how
much of a whole pie is left if the following amounts are eaten: 34 or
45 . Emphasize that the fraction with
a bigger piece left-over is the smaller fraction.
10. Write the following fractions in order from least to greatest. Explain how you found the order. 13 ,
12 ,
23 ,
34 ,
18
HINT: Use the ideas from Extension 9 above.
11. Fold a strip of paper 3 times to create eighths. Write the following fractions over top of the
corresponding folds:
18
38
12
78 Bonus:
14
12. Give each student three strips of paper. Ask them to fold the strips to divide one strip into halves, one
into quarters, one into eighths. Use the strips to find a fraction between
a) 38 and
58 (one answer is
12 ) b)
14 and
24 (one answer is
38 ) c)
58 and
78 (one answer is
38 )
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13. Have students fold a strip of paper (the same length as they folded in EXTENSION 12 into thirds by
guessing and checking. Students should number their guesses.
EXAMPLE:
Try folding here too short, so try a little further
1st guess
2nd
guess
Is 13 a good answer for any part of Extension 12? How about
23 ?
14. Why is 23 greater than
25 ? Explain.
15. Why is it easy to compare 25 and
212 ? Explain.
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NS3-83 Sharing and Fractions Goal: Students will find fractions of numbers. Students will see examples of equivalent fractions.
Prior Knowledge Required: Fractions as area
Fractions of a set
Finding fractions of whole numbers.
Brainstorm the types of things students can find fractions of (circles, squares, pies, pizzas, groups of
people, angles, hours, minutes, years, lengths, areas, capacities, apples).
Brainstorm some types of situations in which it wouldn’t make sense to talk about fractions. For example:
Can you say 3 12 people went skiing? I folded the sheet of paper 4
14 times?
Explain to your students that it makes sense to talk about fractions of almost anything, even people and
folds of paper, if the context is right: EXAMPLE: Half of her is covered in blue paint; half the fold is covered
in ink. Then teach them that they can take fractions of numbers as well. ASK: If I have 6 hats and keep half
for myself and give half to a friend, how many do I keep? If I have 6 apples and half of them are red, how
many are red? If I have a pie cut into 6 pieces and half the pieces are eaten, how many are eaten? If I have
a rope 6 metres long and I cut it in half, how long is each piece? Tell your students that no matter what you
have 6 of, half is always 3. Tell them that mathematicians express this by saying that the number 3 is half of
the number 6.
Tell your students that they can find 12 of 6 by drawing two circles. Put one dot in each circle until you have
placed 6 dots.
2 dots 4 dots 6 dots
Now, one half of the dots are in the first circle and one half of the dots are in the second circle. So 12 of 6
is 3. Have students use this method to find 12 of a) 10 b) 8 c) 14
Then have students draw 3 circles to find 13 of a) 6 b) 12 c) 9 d) 15 e) 3 f) 18.
Then have students write a division statement for each picture a) to f) above. (6 ÷ 3 = 2 and 12 ÷ 3 = 4 and
so on)
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Then have students draw a picture and write a division statement for each fraction of a number:
a) 12 of 8 b)
13 of 12 c)
14 of 8 d)
12 of 10
e) 15 of 10 f)
15 of 20 g)
14 of 12 h)
15 of 15
Write a division statement to find the fraction of the number without using pictures.
a) 12 of 22 b)
14 of 84 c)
15 of 100 d)
110 of 70
Teach your students to see the connection between the fact that 6 is 3 twos and the fact that 13 of 6 is 2. The
exercise below will help with this:
Complete the number statement using the words “twos”, “threes”, “fours” or “fives”. Then draw a picture
and complete the fraction statements. (The first one is done for you.)
Number Statement Picture Fraction Statement
a) 6 = 3 twos 13 of 6 = 2
23 of 6 = 4
b) 12 = 4 ________ 14 of 12 =
24 of 12 =
34 of 12 =
c) 15 = 3 ________ 13 of 15 =
23 of 15 =
Equivalent fractions
Tell your students that sometimes two fractions that look different can mean the same thing. Show the
following pictures:
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ASK: What fraction does each picture show? (12 ,
24 ,
48 and
36 ) ASK: Is the same amount shaded in each
picture? Ask students to find other fractions that mean the same as 12 . ASK: If a pie had 10 pieces, how
many would be half? What fraction with denominator 10 means the same thing as 12 ?
Have students find fractions that mean the same as 12 and have given denominators:
12 = 6
12 = 12
12 = 8
12 = 10
12 = 20
12 = 14
Then have students find fractions that mean the same as 12 and have given numerators:
12 =
2
12 =
5
12 =
3
12 =
4
12 =
8
12 =
50
Finally, mix up questions of both types:
12 =
6
12 = 6
12 =
10
12 = 10
12 = 16
12 =
16
Activities:
1. Have students compare using fraction strips 12 and
24 ,
34 and
68 ,
14 and
28 ,
12 and
48 ,
24 and
48 .
2. Students can find fractions of a whole number using the following method: Find 12 of 12.
STEP 1 – Make a model of 12 using 12 yellow counters
STEP 2 – Replace yellow counters one at a time with red counters until an equal number of the
counters are red and yellow.
Ask students to use this method to find 12 of 6, 8, 10, 14, etc.
3. Students can find 12 of 6 by drawing rows of dots. Put 2 dots in each row until you have placed 6 dots.
STEP 1 STEP 2 STEP 3
There are 3 dots in each column, so 3 is 12 of 6. Have students find
12 of 10, 12, 8 and 16 using this
method. Students might also find 13 of various numbers by drawing rows of dots with 3 dots in each row.
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Extensions:
1. How many months are in:
a) 12 year? b)
13 year? c)
14 years?
2. How many minutes are in:
a) 13 of an hour? b)
12 of an hour? c)
14 of an hour?
3. How many hours are in:
a) 12 of a day? b)
13 of a day? c)
14 of a day?
4. Give your students counters to model the following problems.
a) 2 is 13 of a set. How many are in the set? b) 3 is
14 of a set. How many are in the set?
5. Show students how to find 23 of 6 dots.
STEP 1. Find 13 of 6 dots.
3 dots 6 dots
So 13 of 6 is 2.
STEP 2. Multiply by 2.
Each circle has 13 of 6 dots, so 2 circles have
23 of 6 dots.
23 of 6 is 4. Have students use this method
to find:
a) 23 of 9 b)
34 of 12 c)
23 of 12 d)
25 of 10 e)
35 of 15
6. Have students investigate.
a) 34 of 4 b)
45 of 5 c)
37 of 7 Bonus: What is
13215 of 215?
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7. (Atlantic Curriculum A3.4) Tell the student that Lee and Teddy bought their mother a gift for Christmas
which cost $20. Lee paid 34 of the cost, and Teddy paid the balance. Ask: How much did each pay?
Provide coloured counters to help him/her solve the problem.
8. (Adapted from Atlantic Curriculum A3.5)
a) Pair each student with a partner to solve this problem: Eight-year-old Samantha, whose birthday is
January 25th, said, “I can’t wait until I’m 8 and 1112 .” Ask: Why was she excited?
b) Natalia said that she will turn 8 and 13 on Christmas day. When is her birthday?
9. (Atlantic Curriculum A3.6) Ask the student to tell why, whenever you see a model of 13 , there is always a
model of 23 associated with it.
10. (From Atlantic Curriculum A3.8) You have 8 coins. Half of them are pennies. More than 18 of them
are quarters. The others are nickels. Use coins to represent the situation. How much money might
you have?
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NS3-84 Comparing Fractions Goal: Students will decide when a picture shows more than half or less than half, about a half, or
about a third or about a fourth.
Prior Knowledge Required: Comparing fractions
Using models to represent fractions
Vocabulary: less than, more than, half, close to
Ask students to tell you whether the shaded fraction is more than half or less than half and how they can tell:
Then repeat the exercise, but this time have a volunteer extend one of the border lines of the shaded region
to show half. Then give several similar problems for students to do individually, and tell them to imagine
where the line would be extended.
Ask students to tell you whether the shaded fraction is less than 13 , more than
13 but less than
23 or more than
23 and have students explain how they can tell:
Then repeat the exercise, but this time have a volunteer use one of the border lines of the shaded region to
show the pie cut into thirds. Then give several similar problems for students to do individually, and tell them
to imagine where the lines would be to divide the pies into thirds.
Repeat with fractions that show less than a quarter, between one and two quarters, between two and three
quarters and more than three quarters.
Draw some fraction strips on the board:
12
13
14
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Give students fraction strips with a part shaded. EXAMPLES:
Have students estimate what fraction of the strips are shaded. Challenge the students to find a way to check
their answer by folding the fraction strips.
Then ask: Which figure shows about 13 ?
Then show on a strip of paper the following circles:
Ask students to estimate what fraction of the circles are shaded. ASK: How can we check by folding the
paper? Demonstrate folding the paper and then circling the 4 groups of 3 circles. ASK: How many circles are
shaded? (3) How many groups of 3 circles are there altogether? What fraction of the circles are shaded?
(one group out of four groups are shaded, so one fourth or one quarter of the circles are shaded. Show
students how they can do this without folding.
ASK: How many circles are shaded? (2) Demonstrate circling groups of 2 circles until you have circled all the
circles. ASK: How many groups of 2 circles are there? (5) What fraction of the circles is shaded? (one group
out of five groups are shaded, so one fifth of the circles is shaded. Give students several such problems to
practise with.
Activities:
1. Oranges or Apples
Bring oranges or apples into the class and cut some in fourths and some in half and some in eighths. Ask
students to decide which is more between different fractions: for example, one half or three eighths.
Encourage them to put three eighths together so that they can see that it is not quite half.
2. Mark off 13 and
23 of a clean plastic cylinder with scotch tape. Partially fill the cylinder with water and turn
the cylinder so students can’t see the tape marks. Ask them whether the cylinder is closer to 13 or
23 full.
Then turn the cylinder so that students can see the tape marks. Did they guess correctly? Repeat with
different amounts of liquid
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Extension:
Ordering Fraction Strips
Use the BLM “Shaded Fraction Strips” and give each student three fraction strips to compare. They can tape
the fraction strips to a coloured piece of paper in order from smallest fraction to largest fraction and write
their conclusions on the same paper. If they are familiar with the < and > symbols for “less than” and “greater
than” students can write their answers in the form 38 <
12 <
23 . Otherwise, they can write sentences: “
38 is less
than 12 ” and “
12 is less than
23 .”
Literature/Cross-Curricular Connections:
Apple Fractions by J. Pallotta Different apples are used to teach kids all about fractions. Students will learn
to divide apples into halves, thirds, fourths, and more. See the above activity.
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NS3-85 Fractions Greater than One Goal: Students will model fractions greater than 1 by using pictures.
Prior Knowledge Required: comparing fractions, using models to represent fractions
Vocabulary: less than, more than
Tell students that beads come in packs of 4. Ask volunteers to show, by modelling beads with circles:
a) one whole pack, b) one half of a pack, c) 2 packs, d) one pack and another half of a pack.
Tell students that, rather than saying one pack and then another half of a pack, we describe this as one and
a half packs. Ask a volunteer to draw what they think two and a half packs will look like.
Then tell students that beads come in packs of 6. Ask students to show individually in their notebooks: a) one
half of a pack, b) one and a half packs, c) two packs, d) two and a half packs, e) three and a half packs.
Then tell students that beads come in packs of 3. Ask students to show individually in their notebooks: a) one
third of a pack, b) two thirds of a pack, c) one and a third packs, d) two and a third packs, e) four and two
thirds packs.
NOTE: You might also demonstrate what it means to have “one and a half” or “two and a half” of something
using an area model: for instance, one and a half pizzas.
Extension: Teach students that one whole can be written as a fraction in many different ways.
Have students name the fractions shaded. Tell them that they are all one whole and write 1 = 44 and 1 =
66 .
Then have a student volunteer to fill in the blanks: 9 7
Then repeat with larger numbers and have students fill in the denominator (give them only the numerator).
Repeat the above exercises with fractions that show two wholes. Ensure that students understand that to
find the numerator, they double the denominator, or to find the denominator, they take half of the numerator.
Gradually increase the denominators to make them more difficult to double: 3, 7, 23, 34, 36, 52, 47, 74, 78.
1 = 1 =
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NS3-86 Puzzles and Problems Goal: students will consolidate and apply their learning about fractions.
Prior Knowledge Required: fractions as equal parts of an area or set
fractions that look different but mean the same amount
comparing fractions
fractions greater than one
fractions of numbers
Vocabulary: less than, more than
This worksheet is review. It can be used as an assessment.
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NS3-87 Decimal Tenths Goal: Students will learn the notation for decimal tenths. Students will add decimal tenths with sum up
to 1.0. Students will recognize and be able to write 10 tenths as 1.0.
Prior Knowledge Required: Fractions Thinking of different items as a whole (EXAMPLE: a pie, a hundreds block)
Place value
0 as a place holder
Vocabulary: decimal, decimal tenth, decimal point
Draw the following pictures on the board and ask students to show the fraction
110 in each picture:
Tell students that mathematicians invented decimals as another way to write tenths: One tenth ( 1
10 ) is
written as 0.1 or just .1. Two tenths ( 2
10 ) is written as 0.2 or just .2. Ask a volunteer to write 310 in decimal
notation. (.3 or 0.3) Ask if there is another way to write it. (0.3 or .3) Then have students write the following
fractions as decimals:
a) 710 b)
810 c)
910 d)
510 e)
610 f)
410
In their notebooks, have students rewrite each addition statement using decimal notation:
a) 310 +
110 =
410 b)
210 +
510 =
710 c)
210 +
310 =
510 d)
410 +
210 =
610
Bonus:
Include subtraction problems such as:
a) 710 –
310 =
410 b)
910 –
410 =
510 c)
310 –
110 =
210 d)
610 –
310 =
310
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Draw on the board:
ASK: What fraction does this show? (410 ) What decimal does this show? (0.4 or .4)
Repeat with the following pictures:
Have students write the fractions and decimals for similar pictures independently, in their notebooks.
Then ask students to convert the following decimals to fractions, and to draw models in
their notebooks:
a) 0.3 b) .8 c) .9 d) 0.2
Demonstrate the first one for them:
0.3 = 3
10
Have students write addition statements, using fractions and decimals, for each picture:
+ = + = + =
Draw on the board:
0 110
210
310
410
510
610
710
810
910 1
Have students count out loud with you from 0 to 1 by tenths: zero, one tenth, two tenths, … nine
tenths, one.
Then have a volunteer write the equivalent decimal for 1
10 on top of the number line:
0.1
0 1
10 210
310
410
510
610
810
910 1
Continue in random order until all the equivalent decimals have been added to the number line.
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Then have students write, in their notebooks, the equivalent decimals and fractions for the spots marked on
these number lines:
a) b)
0 1 0 1
c)
0 1
d)
0 A 1 B C 2 D 3
Have volunteers mark the location of the following numbers on the number line with an X and the
corresponding letter.
A. 0.7 B. 2 710 C. 1.40 D.
810 E. 1
910
0 1 2 3
Invite any students who don’t volunteer to participate. Help them with prompts and questions such as: Is
the number more than 1 or less than 1? How do you know? Is the number between 1 and 2 or between
2 and 3? How do you know?
Tell your students that there are 2 different ways of saying the number 1.4. We can say “one decimal four” or
“one and four tenths”. Both are correct.
Have students write the following numbers as decimals:
a) four tenths b) one and six tenths c) three and one tenth d) two and five tenths
Have students write the following decimals as words:
a) 1.2 b) 2.1 c) 3.4 d) 7.3 e) 9.1 f) 2.9
Have students place the following fractions on the number line from 0 to 3:
A. three tenths B. two and five tenths C. one and seven tenths
D. one decimal two E. two decimal eight
Which of the numbers (from A, B, C, D and E above) are less than 1? Which are more than 1 and less
than 2? Which are more than 2? Is 2.3 more or less than 1.8? How do you know? Which two whole numbers
is 2.3 between? Which two whole numbers is 1.8 between?
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Teach students to count forwards and backwards by decimal tenths using dimes. ASK: How many dimes
make up a dollar? (10) What fraction of a dollar is a dime? (one tenth) Tell students that we can write the
dime as .1 dollars, since .1 is just another way of writing 110 . ASK: What fraction of a dollar are 2 dimes?
How would we write that using decimal notation? What fraction of a dollar are…
a) 7 dimes? b) 3 dimes? c) 9 dimes? d) 6 dimes? e) 10 dimes?
Have students write all the fractions as decimals.
ASK: How many dimes are in … a) $0.70 b) $0.20 c) $0.60 d) $0.30
ASK: How many tenths of a dollar are in … a) $0.70? b) $0.20 c) $0.60 d) $0.80 e) $0.90
Activity: (From Atlantic Curriculum A8)
Decimal War. Prepare a deck of cards with numbers such as 0.1, 0.2, …, 0.9, 1.0, 1.1, …,
1.9, 2.0, 2.1, …, 2.9 for each pair of students. Each student gets half the deck. They both turn over one
card at a time. The student with the card showing the greater number keeps both cards. Play continues
until someone has all the cards. Variation: Give each pair two identical sets of cards so that ties are
possible.
You might choose to have students play the same game with cards numbered 1 through 29 instead of 0.1
through 2.9 and have them compare the two games. Notice that 1.1 is greater than 0.9 precisely because
11 (tenths) is greater than 9 (tenths).
Extensions:
1. Put the following sequence on the board: .1, .3, .5, _____ and have students extend the pattern.
2. If your students are familiar with equivalent fractions, have them convert each fraction to an equivalent
fraction with denominator 10 and then to a decimal:
a) 25 b)
12 c)
45
3. If your students are familiar with equivalent fractions, have them rewrite each addition or subtraction
statement using decimal notation by first changing all fractions to an equivalent fraction with
denominator 10:
a) 12 +
15 =
710 b)
12 +
25 =
910 c)
35 –
12 =
110 d)
12 –
15 =
310
4. Teach students numbers with two decimal places and where to find the tenths.
ASK: How many dimes are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35
ASK: How many tenths of a dollar are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35
ASK: Where do you see the number of tenths in each number:
a) 0.78 b) 0.93 c) 0.21 d) 0.35
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5. (From Atlantic Curriculum A8)
a) Identify the decimals and then put them in order:
A.
B.
C.
D.
b) Model the decimals and then write and model a decimal that is between the two decimals:
3.4 and 3.8 2.8 and 3.1 1.4 and 3.9 0.5 and 1.2 1.9 and 3.2
c) Complete the pattern: 2.9, _____, 3.1, 3.2, _____, 3.4
6. (From Atlantic Curriculum A7)
a) Which number is larger: 2.9 or 4.2? How do you know?
b) Which number is larger: 6.2 or 40? How do you know?
c) Which number is 0.2 more than 0.4? Use a number line or ten frame to help you.
d) Which number is 0.2 less than 1? Use a number line or ten frame to help you.
e) Teach students that decimals are equivalent to fractions with denominator 10 and just as they can
take a fraction of a set, they can take a decimal of a set. Ask students to circle about 0.4 (or 410 ) of
the dots.
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NS3-88 Word Problems (Warm Up) Goal: Students will decide when to use addition, subtraction, multiplication or division in word problems
given in point-form notation.
Prior Knowledge Required: addition, subtraction, multiplication, division
Ask students whether the underlined words make them think of adding or subtracting:
1) Three more joined.
2) Two children left to go skipping.
3) There are five altogether.
4) She took five away.
5) There are seven in total.
6) How many are leftover?
7) How many cookies are left?
8) How many altogether?
9) How many are not red?
10) How many more apples than oranges are there?
11) How many fewer days are in a week than in a month?
12) How many days are in a month and a week altogether?
13) How much longer is a school bus than a car?
Have students write down the important words in each question and then the symbol (+ or –) that it makes
you think of:
a) How many apples were sold altogether?
b) How many more red apples than green apples were sold?
c) How many apples were not red?
d) How many stickers were collected altogether?
e) How many stickers were not from Canada?
f) How much longer is a ruler than my pencil?
g) How long are my ruler and pencil when placed end to end?
Tell students that you went to a zoo and saw 14 spotted animals and 9 striped animals and 6 plain animals.
ASK:
a) How many animals are there altogether? 14 + 9 + 6 = 29
b) How many animals are not plain? (29 – 6 = 23)
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c) How many spotted and striped animals are there? (Have students discuss two different solutions:
14 + 9 = 23, or 29 – 6 = 23)
d) How many animals are not striped? Discuss the different solutions.
e) How many more spotted animals are there than plain animals?
f) How many fewer plain animals are there than striped animals?
g) Make up your own question and have a partner solve it, then check the partner’s solution.
Tell students that you went to a different zoo and saw 12 spotted animals and 15 striped animals and 34
animals altogether. ASK:
a) How many spotted and striped animals are there altogether?
b) How many more striped animals are there than spotted animals?
c) How many plain animals are there? (I.e. How many animals are there that are NOT spotted or striped?)
d) How many animals are not spotted? (Discuss the two solutions here.)
e) How many animals are striped or plain?
Then ask students to decide between addition and multiplication and how they know:
a) There are 4 pages in each chapter. There are 5 chapters. How many pages are there altogether?
b) There are 7 pages in Chapter 1, 4 pages in Chapter 2 and 8 pages in Chapter 3. How many pages are
there altogether?
c) There are three bookshelves. The shelves have 9, 4, and 6 books each. How many books are there
altogether?
d) There are three bookshelves. There are 6 books on each shelf. How many books are there altogether?
Ask students to tell you how they know whether to add or multiply. Emphasize that when the same number is
on each shelf or in each page, or on each whatever, then they know to multiply (they could add too, but
multiplication is less work).
Have students solve the following multiplication word problems:
a) There are 5 bookshelves and 3 books on each shelf. How many books are there altogether?
b) There are 100 cm in each metre. How many cm are in 3 metres?
c) There are 3 medals given in each event. How many medals are given in 7 events?
d) Bonus: There are 52 weeks in each year. If Jenny turned 8 today, how many weeks until she turns 10?
Remind students how to decide between multiplication and division (see NS3-65):
Number of sets × Number of objects in each set = Total number of objects
Have students decide which two pieces of information they are given (between the number of sets, the
number of objects in each set and the total number of objects) and which piece they need to determine.
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a) 30 boxes. 6 shelves. How many boxes on each shelf?
6 × _________ = 30
number of sets × number of objects in each set = total number of objects
b) 30 books. 6 books on each shelf. How many shelves?
_______ × 6 = 30
number of sets × number of objects in each set = total number of objects
c) 3 books on each shelf. 5 shelves. How many books?
5 × 3 = _______
number of sets × number of objects in each set = total number of objects
When they are given the total number, they should divide. When they have to find the total number, they
should multiply. Give students more practice with this skill.
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NS3-89 Word Problems and NS3-90 Planning a Party and NS3-91 Additional Problems
Goal: Students will solve word problems involving addition, subtraction, multiplication and division.
Prior Knowledge Required: addition, subtraction, multiplication and division, word problems
Vocabulary: sum, difference
Most of these worksheets provide review and extra practice.
For Question 5 on NS3-89, remind students how to list in an organized way the pairs of numbers that sum to
a given number. They could then look at all their pairs to see if there is one pair with the correct difference.
Extension: After all students have completed Question 6 on NS3-89, discuss the various solutions. For
example, some students might multiply 79¢ by 5 to find the total and then subtract the total from $5.00. Other
students might calculate the change from $1 for each pack (21¢) and then multiply by 5. This works because
the total number of dollars is the same as the number of packs.
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NS3-92 Charts Goal: Students will use charts to find possibilities.
Prior Knowledge Required: addition, subtraction, multiplication, division
Tell students that a restaurant has orange juice or apple juice to drink and eggs or pancakes or French toast
for breakfast to eat. If we want to choose one of each, what are the possible choices? Have students
volunteer possibilities. Then tell them that you want an organized way to make sure that you don’t miss any
choices. Write down orange juice on the board and ask how many choices to eat you can have if you pick
orange juice to drink. What are those choices? Write on the board:
Orange juice, eggs
Orange juice, pancakes
Orange juice, French toast
Apple juice,
Apple juice,
Apple juice,
ASK: Why did I write down apple juice three times? Have a volunteer finish the chart. ASK: How many
choices are there altogether? Did we find all of them? How do you know? Could I have organized my chart
differently? Write on the board:
Eggs,
Eggs,
Pancakes,
Pancakes,
French toast,
French toast,
ASK: How am I organizing my chart now? Why did I write each choice twice instead of 3 times? Then have a
volunteer finish the chart.
Repeat with several similar examples, always keeping two choices for one option and three choices for
the other.
Then show students a dart board where they get 2 points for hitting the board but 5 points for hitting the
centre of the board:
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Ask students to make a chart to show all the scores they could get by throwing the dart twice. Assume that
they will never miss the board, so they never get 0 points (if they miss the board, they get another throw).
Help get them started:
1st
dart 2nd
dart Total Score
2
2
ASK: Why did I write 2 twice?
Have students individually finish the chart in their notebooks?
Bonus:
Assume that they can get a score of 0 by missing the board. Help them get started:
ASK: Why did I write each first score three times?
1st
dart 2nd
dart Total Score
0
0
0
2
2
2
5
5
5
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NS3-93 Arrangements and Combinations and NS3-94 Arrangements and Combinations
(Advanced)
Goal: Students will learn problem-solving.
Ask: I want to make a 2-digit number that uses the digits 1 and 2 each once. How many different numbers
can I make? (2) What are they? 12 and 21. What 2-digit numbers can I make using the digits 3 and 5 each
once? 4 and 7? 2 and 9?
Ask students to make 2-digit numbers:
a) an even number using the digits 3 and 4.
b) An odd number using the digits 5 and 6.
c) A number greater than 40 using the digits 1 and 7.
d) A number less than 30 whose tens digit is four times smaller than its ones digit.
e) A number divisible by 5 using the digits 3 and 5.
f) A number divisible by 5 using the digits 0 and 6.
Ask students to find a 3-digit number that uses each of the digits 1, 2 and 3 once. Write down all their
answers and do not stop until they found all of them. Encourage them to write the 6 different numbers in an
organized list. Repeat with the digits 2, 5 and 7 and then with the digits 3, 4, and 8 and then with the digits 4,
6 and 7. Ask them how they are organizing their list (for example, to write all the numbers using digits 1,2
and 3 each once, they can start with numbers having hundreds digit 1, then numbers with hundreds digit 2
and then numbers with hundreds digit 3; there will be 2 of each) Ask how does the organization make it easy
to know whether they have all of them or not?
Ask students to make 3-digit numbers:
a) An even number using the digits 3, 4 and 5
b) An odd number using the digits 4, 5 and 6.
c) A number greater than 400 using the digits 2, 3, and 6.
d) An even number greater than 400 using the digits 3, 4 and 5.
e) An odd number less than 200 using the digits 1, 3 and 4.
f) An odd number divisible by 5 using the digits 0, 3 and 5.
Draw on the board:
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Show students how to arrange these circles in a row in 2 different ways: or
Then draw a third colour and have them find different ways to arrange the circles in a row. Have
volunteers come up to show the different ways. After all 6 ways are shown, tell your students that it is a good
idea to reflect back and ask if there is an organized way to find all 6 ways. If we want to make sure we found
all of them and didn’t leave out any, how could we start? (start by deciding what colour the first one is) Let’s
say the first one is white. What can the second one be? (solid or striped) Have a volunteer show the 2
possibilities that start with the white circle. Have another volunteer show the 2 possibilities that start with the
solid circle and another volunteer show the 2 possibilities that start with the striped circle.
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NS3-95 Guess and Check Goal: Students will learn how to improve previous guesses to get closer to an answer.
Bring in a large strip of paper and ask a volunteer to guess where half is. Ask the class: Should the next
guess be further or not as far? How can they tell? Repeat until half is found. (NOTE: One way of deciding if
the guess is too far or not far enough is to directly compare the two halves either by folding the paper or
tracing one part of the line and placing it over the other part.
Then write on the board: 23 + __ __ = 58. Have students guess an answer that they think is close to the
correct answer. Students might say 30 or 35 or 40, for example. Then ASK: Is the guess too high or too low?
Should I guess a larger number or a smaller number next? Continue until students have found the correct
answer.
Then write on the board: 47 + __ __ = 74. Repeat the process. ASK: How am I improving my guess each
time? Would guessing and checking be a good strategy if I didn’t improve my guess each time? What if I just
randomly tried a bunch of numbers and hoped to eventually get it right—is that a good strategy?
Ask students if they know of any games that use the guess and check strategy by improving the guess each
time? (warm and cold, the person tells you whether you are looking close or far away from the object and
you adjust your search accordingly; guessing a number between 1 and 20 that the other person is thinking of
and they tell you whether you are too high or too low) Which games do they know that use guessing without
improving their guess each time? (Hide and seek unless you are given clues such as noise) Which game
usually takes longer to find the correct answer? Games like guess a number between 1 and 20 or games like
hide and seek? Encourage students to share times when hide and seek took a long time precisely because
guessing the possibilities didn’t help with eliminating any option except the one they just checked.
Then write on the board: 2 ___ + ___ 4 = 65. ASK: How is this problem different from: 47 + __ __ = 74?
Emphasize that now they are guessing digits of different numbers instead of guessing the whole number.
Tell students that it might seem a bit overwhelming to try to guess two different numbers at the same time.
One way they can do this is to guess something for one of the numbers and then change the other number
repeatedly until you get close to the answer. For example: 20 + ____ 4 is about 65. We can try 1 and see
that 20 + 14 is 34, which is too low, so our next guess should be higher. Continue in this way until students
are confident they found the closest number they can: 20 + 44 = 64, is about 65.
Then have students adjust the first number until they get 65: 21 + 44 = 65. Repeat this with several examples
where no regrouping is required in the adding of 2-digit numbers. (EXAMPLES: 3 __ + __ 4 = 89,
4 __ + __2 = 59) Bonus: 2 __ 4 + __ 3 ___ = 796.
Then write on the board: 3 __ + __ 7 = 85 Again have students guess a number to fill in: 30 + __7 is about
85. (30 + 57 = 87 and 30 + 47 = 77) Since 87 is closer to 85, students might think that 5 is the best digit to
use. However, they are not allowed to change the 7 to a 5, so they would have to reduce the 30. This would
require changing the 3 to a 2, which they are also not allowed to do. Emphasize that 30 is the smallest the
first number can be, so whatever digit we put in front of the 7 cannot make the answer become more than
Number Sense Teacher’s Guide Workbook 3:2 111 Copyright © 2007, JUMP Math For sample use only – not for sale.
85. Since 87 is too high, the second number must be 47. Once we found the second number, the problem
becomes 3 __ + 47 = 85. This can again be done by guessing and checking. Give students practice with
more examples of this sort (EXAMPLES: 1__ + __8 = 81, 3__ + __ 9 = 56.)
Then tell students that you are looking for two numbers that add to 10 and have a difference of 2. Ask
students for some numbers they know that have a difference of 2. Is there a way of listing numbers with
difference 2 so that their sums always increase? (1 and 3, 2 and 4, 3 and 5, and so on). So we can guess
a pair and check to see if the sum is too high or too low. We then know whether to go further in the list
or earlier in the list. For example, if we try 5 and 7, we know that the sum is too high, so try 4 and 6.
We’re done.
Have students find two numbers with difference two that add to:
a) 16 b) 24 c) 22 d) 48 e) 56
Bonus: Find two numbers that add to 234 and have a difference of 2. Students might notice the pattern that
the answer is always very near to half of the given number. For example 7 and 9 are close to 8, which is half
of 16.
Ask the class to list in order the first ten pairs of numbers with difference 3: (1 and 4, 2 and 5, 3 and 6, … ,
10 and 13). Tell students that you are looking for a pair of numbers with difference 3 that add to 37. Have a
volunteer guess a number they think will be close to the right number. For example 15 + 18 is 33, which is
too low. Then try 16 + 19 = 35, again too low. Then try 17 + 20 = 37; done.
Repeat for various such examples.
Cross-Curricular Connection: If you have a piano available and a tape recorder, record yourself playing
several piano notes with enough time in between so that rewinding between notes is easy. Take your
students to a room with a piano. Play your first recorded note once and tell your students that you want to
guess which note this was. Demonstrate guessing a note on the piano right in the middle. ASK: Should my
next guess be higher or lower? Where should my next guess be – to the right or to the left? Rewind and play
the note again. If the next guess should be higher, ASK: A lot higher or a little higher? How about here?
Repeat several times until they guess it correctly. If your students have trouble listening to musical notes, put
tape on certain keys (say 10 keys apart) and tell your students that the note you played is one of the taped
keys. This will make it easier to guess and check. As your students become more comfortable, you can
gradually move the tape pieces closer together.
Extensions:
1. Find two numbers that:
a) Have a difference of 9 and add to 9;
b) Have a difference of 9 and multiply to 10
2. Refer back to finding pairs of numbers that add to 10 and have a difference of 2. Ask students if, instead
of starting with numbers that differ by 2, could they have started by listing numbers that add to 10. What
is an ordered way of listing numbers that add to 10? (Start with the first number 1 and increase the first
number by 1 each time: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5) Then notice that the differences
decrease so if you start by guessing 2 and 8, your next guess needs to be further in the list.
Number Sense Teacher’s Guide Workbook 3:2 112 Copyright © 2007, JUMP Math For sample use only – not for sale.
NS3-96 Puzzles Goal: students will improve their problem-solving ability.
Prior Knowledge Required: adding two or three 1-digit numbers, trading for coins of equal value.
Write the following numbers on the board: 1, 2, 3, 4, 5, 6. Tell your students that they are only allowed to use
each digit once for each question below. There may be more than one solution for each part.
a) + = 4 (ANSWER: 1, 3)
b) + = 6 (ANSWER: (1, 5 or 2, 4)
Ask students to place numbers from 1 to 6, at most once each, so that any pair of numbers joined by a
straight line add to 6:
c) + = 8 (ANSWER: 2, 6 or 3, 5)
Repeat the puzzle above with numbers joined by a straight line adding to 8 instead of 6.
d) + = 7 (ANSWER: 1, 2, 4)
e) + = 8 (ANSWER: 1, 2, 5 or 1, 3, 4)
Take up both answers for part e) with the class. Then tell the students to solve the following problem
using the numbers from 1 to 5 each once so that both of the diagonal sums equal 8:
Number Sense Teacher’s Guide Workbook 3:2 113 Copyright © 2007, JUMP Math For sample use only – not for sale.
ASK: Which number is in both sums? Where does that number need to be placed? How do you know? Can
any other number be at the top? Why not?
Then challenge your students to use the numbers from 1 to 5 each once so that both lines of 3 numbers
add to 8:
f) + + = 9 (ANSWER: 1, 2, 6 or 1, 3, 5 or 2, 3, 4)
Challenge students to find all possible solutions, then have them solve the following problem using the
numbers from 1 to 6 each once so that all of the edge sums equal 9:
ASK: Which numbers need to be in the corners? Why?
e) Repeat part d) with + + = 10 (ANSWER: 1, 3, 6 or 1, 4, 5 or 2, 3, 5)
Now tell students that you have 3 dimes, 5 nickels and 5 pennies. ASK: How can this be evenly split
among 2 people. Draw the coins or write their letters (D, D, D, N, N, N, N, N, P, P P, P, P) and 2 circles
to divide them into. Divide the dimes first, using nickels if necessary:
D, N, N D, D D, N, N D, D
N, N N, P, P
P, P, P
N, N, N, P, P, P, P, P
Workbook 3 - Number Sense, Part 2 1BLACKLINE MASTERS
Adding or Trading Game _________________________________________________2
Always, Sometimes, or Never True (Numbers) ________________________________3
Define a Number _______________________________________________________4
Dimes, Pennies, and Base Ten Materials _____________________________________5
Fractions That Add to 1 __________________________________________________6
Money Matching Memory Game ___________________________________________8
Number Lines to Twenty _________________________________________________9
Shaded Fraction Strips __________________________________________________10
Smallest Number of Coins Chart __________________________________________12
NS3 Part 2: BLM List
Adding or Trading Game
END 1¢ 5¢ 1¢
1¢ 5¢ 1¢ 5¢10¢ 5¢
1¢ 5¢ 1¢25¢ 10¢ 1¢
START 5¢ 10¢ 1¢ 10¢25¢ 1¢ 1¢
1¢
1¢
10¢5¢
1¢
1¢
10¢ 25¢
1¢
1¢
10¢1¢10¢25¢
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Always, Sometimes or Never True (Numbers)
Choose a statement from the chart above and say whether it is always true, sometimes true, or never true. Give reasons for your answer.
1. What statement did you choose? Statement Letter
This statement is…
Always True Sometimes True Never True
Explain:
2. Choose a statement that is sometimes true, and reword it so that it is always true.
What statement did you choose? Statement Letter
Your reworded statement:
3. Repeat the exercise with another statement.
AA 4-digit number is greater than
a 3-digit number.
BThe product of two multiples
of 5 is odd.
CIf you multiply a 2-digit number by a 1-digit number, the answer
will be a 2-digit number.
DIf you multiply a number by zero,
the answer will be zero.
EWhen you subtract a 1-digit
number from a multiple of 100 you will have to regroup.
FThe product of two
even numbers is even.
GWhen you divide, the remainder
is less than the number you divide by.
HThe product of two numbers
is greater than the sum.
IIf you have two fractions, the
one with the smaller denominator
is the larger fraction.
JMultiples of 8 end in
even numbers.
KTenths are larger than
hundredths.
L10 thousands is the same
as 10 thousandths.
MMultiples of 5 are divisible by 2.
NThe product of 0
and a number is 0.
OA number that ends in an
even number is divisible by 2.
Workbook 3 - Number Sense, Part 2 3BLACKLINE MASTERS
Define a Number
1. Name a number that statement D applies to:
2. Name a number that statement C and O apply to:
3. Name three numbers that statements N and A apply to: , ,
4. a) Name a number that statements B, D, G and O apply to:
b) Name a number that statements D, L and O apply to:
5. a) Which statements apply to both the number 22 and the number 30?
b) Which statements apply to both the number 12 and the number 32?
6. Can you find four numbers that statement Q applies to? , , ,
AThe number is even.
BThe number is odd.
CYou can count to the
number by 4s.
DYou can count to the
number by 3s.
EYou can count to the
number by 25s.
FYou can count to the
number by 100s.
GIf you multiplied the
number by 5, the product would be larger than 100.
HThe number has 3 digits.
IThe ones digit is one less than the tens digit. The ones digit is
5.
JThe number has 3 or more
digits.
KThe sum of its digits
is greater than 9.
LThe number has 2 digits
and the ones digit is greater than the tens digit.
MThe number is less than 40.
NIf you rolled two dice and
added the numbers together, you could get the number.
OThe number is less than 25.
PThe ones digit of this number
is divisible by 3.
QYou can get this number by
multiplying another number by itself (EXAMPLE: 9 = 3 × 3).
RThe ones digit of this number
is more than the tens digit.
Each statement describes at least one whole number between 1 and 100.
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Dimes, Pennies, and Base Ten Materials
Show each amount using tens blocks and ones blocks.
Look at the pictures.
Fill in the blanks.
The number of dimes is equal to the number of ___________
blocks.
The number of pennies is equal to the number of ___________
blocks.
tens or ones
tens or ones
Workbook 3 - Number Sense, Part 2 5BLACKLINE MASTERS
Fractions That Add to 1
What fraction is shaded?
What fraction is not shaded?
shaded
not shaded
shaded
not shaded
shaded
not shaded
shaded
not shaded
shaded
not shaded
shaded
not shaded
shaded
not shaded
shaded
not shaded
1
4
3
4
6 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Colour in the rest to make 1 whole.
How much did you colour?
+ = 1 whole 1
4
3
4
+ = 1 whole 1
3
+ = 1 whole 1
6
+ = 1 whole 1
9
+ = 1 whole 1
5
Fractions That Add to 1 (continued)
Workbook 3 - Number Sense, Part 2 7BLACKLINE MASTERS
Money Matching Memory Game
$0.75 75¢ $7.50
750¢ 20¢ $0.20
200¢ $2 $1
1¢ 100¢ $0.01
$2.02 $2.20 22¢
202¢ 220¢ $0.22
8 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Nu
mb
er L
ines
to
Tw
enty
10
11
12
13
14
15
16
17
18
19
20
98
76
54
32
10
10
11
12
13
14
15
16
17
18
19
20
98
76
54
32
10
10
11
12
13
14
15
16
17
18
19
20
98
76
54
32
10
10
11
12
13
14
15
16
17
18
19
20
98
76
54
32
10
Workbook 3 - Number Sense, Part 2 9BLACKLINE MASTERS
Shaded Fraction Strips
10 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Shaded Fraction Strips (continued)
Workbook 3 - Number Sense, Part 2 11BLACKLINE MASTERS
Smallest Number of Coins Chart
Use the smallest number of coins to make each amount.
HINT: Use as many quarters as you can first, then dimes, then nickels, and then pennies.
10¢25¢ 5¢ 1¢ TOTAL
8¢
35¢
30¢
26¢
52¢
61¢
71¢
12¢ 1 2 3
12 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Measurement Teacher’s Guide Workbook 3:2 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-18 Analogue Clock Faces Goal: Students will order the numbers on clock faces.
Prior Knowledge Required: Familiarity with clocks
Understanding of ordinal numbers
Vocabulary: clock, analogue, quarters, sequence, before, after, first, second, third
ASK: What is a clock? How do we use clocks? What does a clock measure? Record the students’ ideas.
Draw a circle on the board or on chart paper with boxes where the numbers on a clock would be. (See
below for initial example.) Ask students to tell how many numbers are usually on a clock’s face. Show
students exactly what part of the clock is referred to as its face. Explain that the clock face is divided into
four parts called quarters. The easiest way to draw a clock face is to first put the numbers 12, 3, 6, and 9 in
the proper places. Pointing at the box where the 12 should be, ask students to tell which number fits into
that spot. Have a volunteer come to the board and write in the 12. Repeat with the numbers 3, 6, and 9, in
any order.
Next, show a similar clock face where the 12 and the 6 have been added, but the 3 and the 9 are missing.
Have another volunteer insert the missing numbers.
Finally, remove all four “key” numbers and encourage a volunteer to use the preceding clocks and the clock
in the classroom (if there is one) to fill in all four missing numbers.
Now ask students what numbers come before and after the 3. Write those numbers on the clock face.
Repeat for the 6 and the 9. Ask students what comes before the 12. Fill in the 11. Now discuss with
students what number usually comes after the number 12. Most should say 13. Some will say 1, based on
their prior knowledge of clocks. Explain that the clock face shows only 12 hours and then starts again. You
may wish to explain at this point that a day consists of 24 hours and that it is divided into two periods called
a.m. and p.m. (a.m. starts at midnight and ends at noon, while p.m. starts at noon and ends at midnight).
Before moving on to the next lesson, ensure all students understand that a clock tells time sequentially—
one hour follows another.
12
9 3
Measurement Teacher’s Guide Workbook 3:2 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Brainstorm with students everything they know about time. Record all information on a chart.
2. Working in small groups, have students search through magazines to find and cut out pictures of
watches, clocks, timers, etc. They can use the pictures to create a collage. Remind students to give
their collage a title. Students who are ready to read and write the time could write the time shown on
each clock underneath the picture.
3. The BLM “Analogue Clock Faces” shows the clock face divided into quarters, with 15-minute intervals
noted for students who need an extra visual support.
4. http://www.learningplanet.com/act/tw/index.asp?contentid=410
In this interactive online activity, students can fill in the missing numbers on the clock face.
Extensions:
1. Students can research the origins of the clock and why it measures time in 12-hour increments.
2. Challenge students to order times on the half-hour! Show clock faces with these times: 2:30, 4:30, 1:30,
12:30.
3. Challenge students to order times on the hour and the half-hour. Clock faces might show 12:00, 1:30,
5:30, 3:00, and 11:30.
Journal:
Describe a clock (what it looks like and how it works). Draw a picture to go along with your explanation.
Measurement Teacher’s Guide Workbook 3:2 3 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-19 Hands on an Analogue Clock Goal: Students will identify the hour and the minute hands on an analogue clock.
Prior Knowledge Required: Familiarity with a clock face
Familiarity with the hands on a clock
Vocabulary: clock face, clock hands, hour, minute
Show students a clock and draw their attention to the two hands. ASK: How are the hands the same?
How are they different? Do you know what each hand is called? You can use a graphic organizer, such
as a Venn diagram, to record students’ answers. Once all ideas have been recorded, summarize key
points for the class. Make sure students understand the key difference—the minute hand is longer than the
hour hand.
Next, show the hands at different positions on the clock, and have students identify which hand is pointing
where. Ask questions such as: Which hand is pointing at 12, minute or hour? Which hand is pointing at 3?
Now put the minute hand at the quarter hours only (12, 3, 6, and 9) and ASK: What number is the minute
hand pointing at? Once students are comfortable with these basics, show the minute hand at different five-
minute intervals. Keep the hour hand at 12 while you change the position of the minute hand.
Activity: Students can make their own clock. Each student will need a paper plate, some sturdy paper
and a paper fastener.
The plate is the face of your clock. Draw or paint the numbers on the plate. Cut out a large and a small
arrow for the hands. Use a paper fastener to affix them to the centre of the clock. The clock can be used to
show different times during this and the next lessons.
Extensions: Project:
Which non-digital clocks do not have hands? (Examples for students to investigate: hourglass, sundial,
water clock) Choose one of the clocks without hands. How does it show time? Where was it used and why?
Journal:
The hour and the minute hands are similar and different because…
Measurement Teacher’s Guide Workbook 3:2 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-20 Telling Time The Hour Hand Goal: Students will identify hours on an analogue clock.
Prior Knowledge Required: Familiarity with a clock face
Familiarity with the hands on a clock
Ability to differentiate the two hands on a clock
Vocabulary: clock face, clock hands, hour, minute
Review the previous lesson. Draw two clock faces—one with the hour hand pointing at 2, and one with the
hour hand pointing at 3 (leave off the minute hand). Ask students which comes first (assuming they are
focusing only on a 12-hour period), two o’clock or three o’clock. Repeat with six o’clock and five o’clock.
Continue with more examples until students can readily identify which time comes first. Finish with twelve
o’clock and one o’clock.
Next, draw three clock faces without the minute hand, showing three, two, and one o’clock, and have
volunteers determine which came first. Ask them to order the times using 1st, 2
nd, and 3
rd. Repeat with more
sets of three times.
Remind your students that the hour hand shows the hour. Ask your students to tell you what hour it is in the
clocks you have drawn. Write the answers in two forms: “The hour is 3” and “3:00.”
ASK: How do people answer the question, What time is it? Record the answers students have heard on the
board in 2 columns—times “past the hour” and times “to the hour”
EXAMPLES:
Ten minutes after three Twenty minutes before two
Quarter past five Five to ten
Half past one Quarter to eight
Eight thirty-five
…and so on
Start with the times “past the hour.” ASK: The time is twenty minutes after three. Which hour has just
passed? (three o’clock) So the hour is 3. Is it 3:00? No, it’s after three, so the minutes are different. Where
is the hour hand pointing? Explain that the hour hand is not pointing directly at 3; it has already started
moving towards 4. Draw an analogue clock without the minute hand and with the hour hand pointing
between 3 and 4. Draw several more clock faces without the minute hand and with the hour hand pointing
between hours. Ask your students to write the hour for each time in two ways: Hour:___ and ___: .
Repeat the exercise, but draw the minute hand as well. Students do not need to write the minutes yet.
Measurement Teacher’s Guide Workbook 3:2 5 Copyright © 2007, JUMP Math For sample use only – not for sale.
As a challenge, you might draw several clock faces without the hands, give several times in verbal form
(e.g., five past ten) and in digital form (e.g., 4:15), and ask your students to draw the hour hand for the
times. Students can also use the clocks they created during the activity in ME3-19.
Extension: The time is ten minutes before two. Is it already two o’clock? Which hour has passed? What
is the hour?
Write the hour for these times:
Five minutes to ten
Twenty minutes before two
Fifteen before twelve
Five minutes to nine
Measurement Teacher’s Guide Workbook 3:2 6 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-21 Telling Time Five-Minute Intervals Goal: Students will identify minutes (in five-minute intervals) on an analogue clock.
Prior Knowledge Required: Familiarity with a clock face
Familiarity with the hands on a clock
Ability to differentiate the two hands on a clock
Skip counting by 5s
Vocabulary: clock face, clock hands, hour, minute
Review skip counting by 5s. Draw a close-up of part of a clock face on the board so that students can see
the minutes clearly. Let students count the number of minutes between each pair of numbers.
Remind your students that when a minute hand moves from one number to the next, five minutes have
passed. Ask your students where the minute hand points at the round hour, that is, when it is exactly three
o’clock or four o’clock (or any o’clock). Draw a minute hand pointing at 1 and ASK: How many numbers did
the minute hand move from the round hour? How many minutes have passed since the round hour? Write
“5” outside the clock face, beside the 1, and continue moving the hand and skip counting by 5s until you
reach 60. Explain to your students that there are 60 minutes in one hour, so when one hour ends and a new
hour begins, we start back at 0 minutes.
Draw several clocks with only a minute hand pointing at different numbers. Ask your students to skip count
by 5s around the clock, writing down the minutes as they go, until they reach the position of the hand. Ask
your students to tell you how many minutes have passed since the round hour on each clock.
Draw several clocks with both hands and ask your students to tell how many minutes past the hour it is, and to write that down in this form: :45.
Activity: Each student will need two dice and a clock face with moveable hands (e.g., the play clock
made in the activity in ME3-19). Each student rolls the dice, adds the results, and points the minute hand
towards the sum. For example, if a student rolls 3 and 4, he should set the minute hand pointing at 7,
i.e. 35 minutes. Then the student writes down the minutes past the hour given by the minute hand.
Measurement Teacher’s Guide Workbook 3:2 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-22 Telling Time Putting It Together! Goal: Students will tell time (in five-minute intervals) on an analogue clock.
Prior Knowledge Required: Familiarity with a clock face
Familiarity with the hands on a clock
Ability to differentiate the two hands on a clock
Skip counting by 5s
Vocabulary: clock face, clock hands, hour, minute
Review the previous two lessons. Tell your students that today they will tell both the hour and the minutes,
like grown-ups. Explain that, to tell time, we look at the hour hand first and then the minute hand. Draw
many clock faces showing various times (five-minute intervals only) and ask students to say and to write the
time. The should write the time in both digital form (using numbers) and verbal form (using words).
Encourage your students to say the time in various ways, such as “five past three,” “ten minutes after
twelve,” and “four and twenty-five minutes”.
Assessment:
What time is it? Write each time in digital and in verbal form: What time is it? Draw the hands on an analogue clock and write these times in verbal form: 12:50 2:45 1:05
Activities:
1. Each student will need two dice and a clock face with moveable hands (e.g., the play clock made in the
activity in ME3-19). The student rolls the dice, adds the results, and sets the hour hand pointing at the
sum. The student rolls the dice again, adds the results, and sets the minute hand pointing at the sum.
Then the student writes down the time. For example, if a student rolls 7 at the first roll and 9 at the
second roll, the clock should be set at 7:45.
2. Each pair of students will need a die and a clock face with moveable hands (e.g., the play clock made in
the activity in ME3-19). Player 1 rolls the die three times. He or she adds the results of the first two rolls
and writes them down as the hour, then multiplies the result of the third roll by 10 and chooses whether
or not to add 5 for the minutes. (If the third roll is 4, the minutes could be :40 or :45. If the third roll is 6,
the player should write :00 or :05 instead of :60 or :65.) Player 2 has to set the play clock to this time.
12
6
9 3
1
2
8
7
4
5
11
10
12
6
9 3
1
2
8
7
4
5
11
10
12
6
9 3
1
2
8
7
4
5
11
10
12
6
9 3
1
2
8
7
4
5
11
10
Measurement Teacher’s Guide Workbook 3:2 8 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-23 Digital Clock Faces Goal: Students will tell time (in five-minute intervals) using both digital and analogue clocks.
Prior Knowledge Required: Familiarity with a clock face
Familiarity with the hands on a clock
Ability to differentiate the two hands on a clock
Skip counting by 5s
Vocabulary: clock face, clock hands, hour, minute, digital, analogue
Show students an analogue and a digital clock (perhaps an alarm clock). ASK: How are these clocks
similar? How are they different?
Next, review the two ways students have learned to write time. Have volunteers write various times using
numbers (e.g., 12:00) and words (e.g., twelve o’clock). Sample times: 12:30, 12:15, 3:45, 2:50, 6:10.
Ask students to look at the digital clock face again, and to compare it to the times they’ve written. What
does the time on the digital clock face look like? (The time on the digital clock faces matches one of the
ways students have learned to write time.)
Show one o’clock on an analogue clock. Ask a volunteer to show what the time would look like on a digital
clock. Repeat with several times, to give students opportunities for practice. Then do the reverse: state and
write digital times and have students show the times on the analogue clock.
Activities:
1. Draw the following table and have students fill in the blanks.
2. On the BLM “Time: Digital Clock Faces,” students are asked to match the digital time to the
analogue clock.
3. Use the BLM “Time Memory Game” to play a game. Directions: Arrange the cards face down in a
rectangular array. Players take turns turning over pairs of cards. If the times on the cards match, the
player lays the pair aside. If the times do not match, the cards are turned face down again. The player
with the largest number of pairs wins.
1 hour ago now 1 hour later
3:15
7:45
11:30
1:45
8:00
Measurement Teacher’s Guide Workbook 3:2 9 Copyright © 2007, JUMP Math For sample use only – not for sale.
Extensions:
1. Ask students to calculate how much time passed between the given times (assume the times are both
a.m. or both p.m.).
a) 7:20 and 7:25 b) 10:30 and 10:45 c) 8:20 and 8:40
d) 1:10 and 1:30 e) 3:45 and 3:50 f) 2:00 and 2:20
g) 3:15 and 3:55 h) 11:05 and 11:40 i) 10:15 and 10:45
HARDER:
j) 5:40 and 6:05 k) 6:45 and 7:20 l) 8:30 and 9:20
2. At various times during the day, ask your students to record the time on the classroom analogue clock
digitally. At the end of the day, ask them to calculate the amount of time that passed between each
reading.
Literature Connection:
Telling Time: How to tell time on digital and analogue clocks by J. Older
This book introduces the how and why of analogue and digital clocks. Read this as an introduction to time
and to the relationship between digital to analogue.
Journal:
List the differences and similarities between how we record digital and analogue times.
Measurement Teacher’s Guide Workbook 3:2 10 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-24 Timelines Goal: Students will find elapsed time using number lines.
Prior Knowledge Required: Thorough understanding of the concept of time
Ability to differentiate between long periods of time and short
periods of time
Understanding of hours and minutes
Ability to write the time
Knowledge of the number line
Vocabulary: passed, elapsed, hours, minutes
Draw this number line on the board:
Ask students to tell what they know about number lines and how they have used them in the past. (Adding,
subtracting, skip counting, and measuring are some of the possible answers.)
Draw a leap starting at nine o’clock and ending at ten o’clock:
ASK: How many hours have passed? How can you tell? Add another leap from ten o’clock to eleven o’clock
and repeat the question. Continue until students are comfortable using the number line to count the number
of hours that have passed.
Next, using the same number line, draw a leap from eleven o’clock to noon. Ask students how many hours
have passed. Add a leap to one o’clock. Repeat the question. Point out that although the numbers are not
sequential, the hours are. (The number 1 does not follow the number 12, but one o’clock comes after twelve
o’clock.) Students should focus on the number of leaps to help them determine how much time has passed.
Draw a new number line using half-hour increments. Ask how much time has passed with this leap:
Students should reply a half-hour or thirty minutes. Repeat the exercise, adding to the number of leaps
incrementally. Then use a different starting time, such as 10:30, and repeat the exercise to ensure that
students are using the leaps to help them count the half-hour increments that have passed.
9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00
9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00
9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 1:00
Measurement Teacher’s Guide Workbook 3:2 11 Copyright © 2007, JUMP Math For sample use only – not for sale.
3:50 3:55 4:00 6:00 8:00 9:00 9:05 7:00 5:00
Finally, repeat the exercises with five-minute increments. Let your students tell how much time passed from
9:15 to 9:40, from 9:45 to 10:05, from 9:55 to 10:25, and so on.
Assess the understanding before proceeding to more complicated number lines. Do not continue until all
students are able to find elapsed time by counting forward using an equally divided number line.
Assessment:
Sindi played hockey from 9:45 to 10:20. How long did she play?
Draw this number line:
Tell students you want to calculate how much time passed between 3:50 and 9:05. Ask your students to
count by 5s until they get to the hour. How much time elapsed? (10 minutes) Draw a leap from 3:50 to 4:00
and label it “10 min.” Ask a volunteer to count by hours from 4:00 to 9:00. Draw another leap from 4:00 to
9:00 and label it “5 hours.” Ask another volunteer to draw a third leap from 9:00 to 9:05 and to label the arrow
“5 min.” Ask a volunteer to add up the labels to get the total time that passed from 3:50 to 9:05. Students will
need more practice with problems of this type.
SAMPLE PROBLEMS:
• Rita worked from 8:00 to 4:05. How long did she work?
• Sam’s birthday party started at 12:45 and ended at 3:10. How long was his birthday party?
Make sure all students are able to find longer period of elapsed time using number lines with leaps of
different length before proceeding to the next material.
Show 7:15 on a play clock or a clock drawn on the board. SAY: This is the time Eve wakes up. She eats
breakfast at 7:40. How much time passes between the time Eve wakes up and the time she eats breakfast?
Turn the hand slowly and ask a volunteer to count the minutes on the clock by 5s. Give your students several
practice questions, such as:
Eve brushes her teeth at 7:30 and gets to school at 8:15. How much time elapsed?
Eve arrived at school at 8:15. The math lesson started at 9:05. How much time elapsed?
Let your students solve such problems by counting by 5s or by drawing number lines marked off in five-
minute intervals. For harder questions involving times with two different hours (e.g., 4:15 and 5:35), suggest
that students use number lines. Invite volunteers to present their solutions.
9:40 9:45 9:50 9:55 10:00 10:05 10:10 10:15 10:20 10:25 10:30
Measurement Teacher’s Guide Workbook 3:2 12 Copyright © 2007, JUMP Math For sample use only – not for sale.
SAMPLE PROBLEMS:
• Eve put a cake in the oven at 7:55. The cake should bake for 25 minutes. When should she take the
cake out?
• The art lesson starts at 1:30 and lasts for 55 minutes. When does it end?
• The TV show is on from 8:15 till 8:55. How long is the show?
Assessment: Cyril has to catch a school bus at 8:25. He woke up at 7:40. How much time does he have before the
bus leaves?
Bonus:
A witch is cooking a potion. The potion turns purple at 3:45. Twenty minutes after it turns purple, the witch
has to add snake heads. When should the witch add the snake heads? The snake heads should stay in the
potion for 35 minutes. Then the witch should stir the potion quickly 7 times clockwise, remove the heads,
and let the potion boil for 35 minutes more. After this, the potion will be ready. When should the witch take
the cauldron from the fire?
Activity:
http://www.shodor.org/interactivate/activities/ElapsedTime/
This website illustrates skip counting in “See” mode and lets students check their own skip counting in
“Guess” mode.
Extension:
Ask students to calculate how much time passed between the given times.
a) 7:25 a.m. and 11:25 a.m. b) 10:05 a.m. and 11:45 a.m. c) 4:20 p.m. and 8:50 p.m.
d) 3:10 p.m. and 8:30 p.m. e) 1:45 a.m. and 7:50 a.m. f) 2:00 p.m. and 8:20 p.m.
g) 9:00 p.m. and 1:30 a.m. h) 11:15 a.m. and 3:15 p.m. i) 11:00 p.m. and 3:35 a.m.
Measurement Teacher’s Guide Workbook 3:2 13 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-25 Intervals of Time Goal: Students will express time intervals in different units.
Prior Knowledge Required: Thorough understanding of the concept of time
Ability to differentiate between long periods of time and short
periods of time
Understanding of hours and minutes
Ability to write the time
Vocabulary: passed, hours, decade, century, centuries
Ask your students to suppose that a relative is coming to visit in two weeks and two days. How many days is
this? Students might reason as follows: A week is 7 days long, so in 2 weeks there are 7 + 7 = 14 days. The
relative is coming in two weeks and two days; since 14 + 2 = 16, the relative will arrive in 16 days. Give your
students more time intervals to express in days, such as 3 weeks and 4 days, 4 weeks and 1 day, and 6
weeks and 5 days.
Suggest that your students use models, such as dots grouped in 7s, to find out how many weeks are in 19
days. Students can draw 19 dots and circle sets of 7 dots to see that there are 2 full sets (2 weeks) and 5
dots (5 days) left over.
Ask your students to convert the following time intervals into weeks and days:
14 days 17 days 21 days 24 days 29 days 27 days
ASK: How many minutes are in one hour? In two hours? Three hours? Record the numbers in a table. Write
two time intervals and ask your students to tell which one is longer. Start with simple pairs of intervals, such
as 20 minutes and two hours, 40 minutes and one hour, 65 minutes and one hour, 60 minutes and two
hours. Continue with harder pairs, such as 90 minutes and one hour, 90 minutes and two hours, 130
minutes and two hours. Ask your students to explain how they know which interval is longer. Explain to your
students that when they compare two measurements in different units, it is convenient to convert both
measurements into smaller units. For example, to compare 145 minutes to 2 hours and 10 minutes,
students should convert the second time interval to minutes: 2 hours + 10 minutes = 120 minutes + 10
minutes = 130 minutes). As a final series of challenges, ask your students to compare more such pairs of
intervals.
SAMPLE INTERVALS:
1 hour and 40 minutes and 2 hours
3 hours and 50 minutes and 225 minutes
three periods of 45 minutes and 2 hours
Measurement Teacher’s Guide Workbook 3:2 14 Copyright © 2007, JUMP Math For sample use only – not for sale.
Explain to your students that longer periods of time are measured not only in years, but also in decades and
centuries. Write the singular and plural forms of “decade” and “century” on the board. Explain that a decade
is 10 years long, and a century is 100 years long. Ask your students to give an example of a period of time
that is measured in decades or in centuries (e.g., age of a country, age of a tree, time since the Middle
Ages). List several periods of time and ask your students to write them in centuries and/or decades.
SAMPLE PROBLEMS:
50 years = _____ decades 110 years = _____ decades
90 decades = _____ years = _____ centuries 1 century = _____ decades
300 years = _____ centuries = _____ decades 210 years = _____ centuries + _____
decades
Assessment:
Convert:
7 weeks = _____ days
31 days = _____ weeks and _____ days
50 decades = _____ years = _____ centuries
2 centuries = _____ decades = _____ years
500 years = _____ centuries = _____ decades
Measurement Teacher’s Guide Workbook 3:2 15 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-26 Estimating Time Intervals Goal: Students will estimate various time intervals.
Prior Knowledge Required: Thorough understanding of the concept of time
Ability to differentiate between long periods of time and short
periods of time
Understanding of hours and minutes
Ability to write the time
Vocabulary: decade, century (centuries), hour, minute, second, week, month
Ask your students to name some units people use to measure time. Record the answers on the board. Ask
students which units they would use to measure these intervals of time:
• the length of Christmas holidays
• the length of summer holidays
• the age of a person
• the length of a music lesson
• the length of a movie
• the age of a baby that is not yet one month old
Next, ask students if they know what unit of measurement is used to count an amount of time smaller than a
minute. Students may say seconds, nanoseconds, milliseconds, etc. Accept all correct answers. Explain
that seconds are a very short amount of time. There are sixty seconds in one minute and sixty minutes in
one hour. Draw a long line on the board and tell student that it represents one hour. Ask a volunteer to draw
a line that represents one minute, and ask another volunteer to draw a line that represents one second. The
final drawing may look something like this:
An hour _____________________________________________________
A minute ____
A second _
Ask a volunteer to clap 5 times, to do 5 sit-ups, and to run around a desk. Measure the time each activity
takes with a stopwatch, to help students develop a better understanding of how long one second is.
Divide your students into 7 groups. Give each group a sheet of paper, and assign each group a unit for
measuring time from the vocabulary for this lesson. Ask each group to list 10 time intervals that would
normally be measured in their unit. To decide in which order the lists will be shared with the class, ask a
volunteer to order the units of time from least to greatest.
Ask your students to estimate the time it takes to do various daily activities, such as sleeping, washing,
brushing teeth, eating, studying, walking a dog, etc. To help students estimate the time, you might ask
questions like: When do you go to bed? When do you get up? How much time elapsed?
Measurement Teacher’s Guide Workbook 3:2 16 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Ask students to estimate how long it will take to…
a) write 100 words
b) walk up a flight of stairs in the school
c) read a chapter of a book
d) count to 50
e) clap your hands 30 times
f) sing the national anthem
g) do 20 jumping jacks
(Students should check some of their estimates using a clock with a second hand.)
2. Students can use the BLM “Time: How Long is a Minute?” to record activities that can be done in one
minute. Students will need a stopwatch to time themselves doing various activities.
Extensions:
1. How many…
a) seconds are in a minute?
b) minutes are in an hour?
c) hours are in a day?
d) days are in a week?
e) weeks are in a month?
f) years are in a decade?
2. How many…
a) seconds are in 3 minutes?
b) minutes are in 2 hours?
3. If a ones block (in base ten materials) represents 1 year, which block would represent…
a) a decade?
b) a century?
Measurement Teacher’s Guide Workbook 3:2 17 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-27 Cumulative Review
This worksheet is a review worksheet for time and money. Review the basic concepts of money before
assigning it.
Measurement Teacher’s Guide Workbook 3:2 18 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-28 Area Goal: Students will measure area in non-standard units.
Prior Knowledge Required: Concept of a surface
Concepts of size (big and small)
Spatial sense
Vocabulary: surface, area, big, small
ASK: What is a surface? How would you measure the size of a surface? Record and discuss students’
ideas.
Introduce the term “area” and explain that to measure the area of a surface, you can cover it with same-
sized units, such as squares, and count them. Draw a rectangle on the board and affix some squares to it,
so that many gaps are left. Ask your students if this is a good way to cover the shape. Repeat with
overlapping squares. Is this better? Repeat with squares that cover much more space than the shape itself.
Invite volunteers to affix the squares in the right way (close to but not overlapping one another and only on
the shape).
Draw this shape on the board or on graph chart paper.
ASK: What is the area of this shape in squares? (2)
Next, draw this shape and ask what its area is.
Ask students which shape has the larger area and have them explain how they know.
Put more and more complex shapes on the board, and challenge students to find their areas. Here are
some examples (see the worksheet for more):
For the more complex shapes, students will have to keep track of the squares they have counted. Ask
students how they can keep track. They may use the reading pattern, count in rows, check boxes off as
they count and tally, etc. The more methodical they are about the process, the easier it will be to find the
area of the more complex shapes.
Measurement Teacher’s Guide Workbook 3:2 19 Copyright © 2007, JUMP Math For sample use only – not for sale.
Repeat the sequence of exercises above for shapes made up of triangles. For each shape, ASK: What is
the area of this shape in triangles?
Draw a shape that is not divided into squares or triangles and ask your students to estimate how many
squares might cover the shape. Let them check and record the actual area. To help your students make
good estimates, draw a shape that can be covered by squares or triangles, ask your students to estimate
the area, then affix the squares or triangles one by one and ask your students after each shape is added if
they would like to revise their estimate. Continue until the whole shape is covered.
Draw two different shapes with the same area.
Invite a student to measure the area of one of the shapes with squares. Then ask your students to estimate
and then measure the area of the second shape. Record the measurements and the estimate. Students will
see that the shapes have the same area. Now provide your students with a different measurement unit,
such as a right-angled triangle that has half the area of the square. Ask a volunteer to measure the area of
the first shape with triangles. Record the area, so that your students have an opportunity to notice the
pattern (the area in triangles is twice the area in squares). ASK: What will be the area of the second shape
in triangles? Let your students check their predictions.
Activities:
1. Guide students to name or find objects in the classroom that they can use to illustrate an understanding
of area. EXAMPLES: “The blackboard has a big area.” “The cover of this book has a smaller area
than...” “The wall has a bigger area than...”
2. Students can choose a unit of measurement and prove that one object has a bigger area than another.
They can record this in their journals. Encourage them to use some form of graphic organizer, such as
a T-table with the headings “big area” and “small area,” to record their results.
Extensions:
1. Order the two shapes in QUESTION 8 on the worksheet according to their area in triangles. Does the
order change when you measure their area in rhombuses?
2. Have students draw a square with a perimeter of 12 cubes. Ask them to find the length of each side of
the square and then the area. Have them predict whether the area of a square with perimeter 12 cubes
would be the same as the area of a rectangle with perimeter 12 cubes. Have them check their
predictions.
Journal:
Area is….
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ME3-29 Area in Square Centimetres Goal: Students will measure the area of shapes on grids in square centimetres.
Prior Knowledge Required: Concept of a surface
Area
Concepts of size (big and small)
Spatial sense
Measuring and drawing lines with a ruler
Perimeter
Vocabulary: surface, area, big, small, centimetre, square centimetre, centimetres squared (cm2),
2-dimensional, perimeter, rectangle
ASK: What units do we use to measure length? Accept all correct answers. Draw a shape on the board and
ask students what units they would use to measure its length—metres, centimetres, or kilometres. Would
they measure the length in linking cubes? Suppose they have to explain how large the rectangle is to a
person that has never seen linking cubes. Would they still use linking cubes or would they prefer a more
standard unit? Ask your students if they think that there might be a standard unit for area as well.
Explain to your students that area is often measured in units called “centimetres squared” or cm2. Show
students an example of a square centimetre, that is, a square whose sides are all 1 cm long.
Draw several rectangles and other shapes (EXAMPLE: L-shape, E-shape) on the board and subdivide them
into equal squares (or draw the shapes on a grid on the board). Label the side length of one square “1 cm”
Ask volunteers to count the number of squares in each shape and write the area in cm2.
Then draw several more rectangles and mark their sides at regular intervals, as shown below.
Ask volunteers to divide the rectangles into squares by joining the marks using a metre stick. Ask more
volunteers to calculate the area of these rectangles.
Ask students to draw their own shapes on grid paper and to find the area and perimeter for each one.
Activities:
1. Students could try to make as many shapes as possible with area 6 units (or squares) on a geoboard.
For a challenge, students could try making shapes with half squares. For an extra challenge, require
that the shapes have at least one line of symmetry. For instance, the shapes below have area 6 units
and a single line of symmetry.
Measurement Teacher’s Guide Workbook 3:2 21 Copyright © 2007, JUMP Math For sample use only – not for sale.
2. Students work in pairs. One student draws a shape on grid paper, and the other calculates the area and
the perimeter.
Extension: Sketch the shape below (at left) on centimetre grid paper. What is its area in cm2? (16) Now
calculate the area using a different unit: a 2 cm × 2 cm square (see below right). What is its area in these
units? What happens to your measurement of area when you double the side lengths of the square you are
measuring with? (The area measurement decreases by a factor of 4.) If the area of a shape is 20 cm2, what
would its area be in 2 cm × 2 cm squares?
2 cm × 2 cm
square unit
Measurement Teacher’s Guide Workbook 3:2 22 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-30 Half Squares Goal: Students will find the area of shapes built with whole squares and half squares (i.e., triangles).
Prior Knowledge Required: Division by 2
Area
Vocabulary: area
Before the lesson, cut a square into two triangles by cutting across the diagonal. Now you have 2 congruent
right triangles whose base and height are the same length. Hold up the 2 triangles and show students that
they are congruent. Then demonstrate how you can put the triangles together to form a square. SAY: If I
know that the base and height of the triangles is 2 cm, what is the area of the square? (2 × 2 = 4 cm2) What
part of the square does each triangle cover? (half) What area does it have? (2 cm2)
Draw this shape on the board:
Tell students the square has area 1 cm2. ASK: What is the area of the whole shape? How did you figure
that out? Invite students to draw another shape with the same area.
Draw several shapes made up of whole squares and half squares (i.e., triangles) on a grid and ask students
to calculate the shapes’ area in terms of whole squares. Then draw a shape with 3 half squares and ASK:
What is the area of this shape? (one and a half squares) Ask students to draw shapes using both whole
squares and half squares. They should swap their drawings with a partner and calculate the area of their
partner’s shapes. Students can then check each others’ area calculations.
Draw several shapes made with even an number of half squares, say four. Ask your students to find the
area in triangles. ASK: If you have a shape built from four half squares, what is its area in whole squares?
What did you do to calculate the area in whole squares? (divided the number of triangles by 2) What is the
area of a shape made of 10 half squares? 200 half squares? 100 half squares and 100 full squares?
Assessment Calculate the area of the shape.
Height 2
Base 2
Measurement Teacher’s Guide Workbook 3:2 23 Copyright © 2007, JUMP Math For sample use only – not for sale.
Bonus:
1. Calculate the area of the shape.
2. Write your name on grid paper using only squares and half squares. What is its total area? Compare
your name with a partner’s. Whose name has the greater area? Who in the class has the name with the
greatest area? The least area?
Activities:
1. Students use geoboards or dot paper (see the BLM “Dot Paper”) to create shapes with a given area
using whole squares. Then students make shapes that include a particular number of half squares.
(EXAMPLE: Build a shape that has an area of 8 squares and contains 8 half squares.)
2. Draw a 6 × 6 quilt pattern using whole squares and half squares. Use shading or colour to create a
design on your quilt. Calculate the area of the shaded or coloured squares. VARIATION: Use different
colours and calculate the area covered by each colour separately.
Extension: If it takes Sandra two seconds to colour one (triangle or half square). How long will it
take her to colour the whole shape from Bonus 1?
Measurement Teacher’s Guide Workbook 3:2 24 Copyright © 2007, JUMP Math For sample use only – not for sale.
ME3-31 Puzzles and Problems ME3-32 Investigating Units of Area These are review worksheets that can be complemented with the following activities and extension.
Activities:
1. Students can make up their own problems like those in QUESTIONS 2 and 3 on worksheet ME3-31.
Partners can use graph paper or a geoboard to solve each other’s problems.
2. A rectangle has area 6 cm2 and perimeter 10 cm. What is the length of the rectangle?
Extension:
History of Measurement: An Investigation
Weights and measures are some of the earliest tools invented by humans.
Length: The first measurements were based on lengths of parts of the human body (for example, the width
of a thumb) or distances between them (for example, the distance between the end of an outstretched arm
and the chin). The Egyptian cubit was based on the length of the arm from elbow to outstretched finger tip.
Roman soldiers measured marching distances by counting paces, the distance from the heel of one foot to
the heel of the same foot when it next touched the ground.
Capacity: To compare the capacities of containers such as gourds or clay or metal vessels, they were filled
with plant seeds which were then counted.
Mass: The first weights were seeds and stones. The Egyptians and the Greeks used a wheat seed as the
smallest unit of weight. Today's carat (used to measure the mass of gems) is based on the weight of a
carob seed, used by the Arabs. The Babylonians compared the weight of an object with a set of special
stones and used different stones for weighing different things.
Standard Units
People and seeds come in different sizes. For measurements to be useful, everyone
needs to be using exactly the same unit. The units need to be standard. Often, kings
or queens decided what a standard for a measurement would be. These standards
spread through trade and commerce.
Metric System
About 200 years ago, the French created the metric system. This system of
measurement is now used throughout the world. It includes measurements for
length, capacity, volume, and weight. The metric system is based on 10s. To
change a measurement from one unit to another, you just move the decimal point.
For example, 300 centimetres = 3.00 metres.
U N
I T
S
Measurement Teacher’s Guide Workbook 3:2 25 Copyright © 2007, JUMP Math For sample use only – not for sale.
1. With a partner, choose a part of the body (for example, length from elbow to fingertip, length of foot) as
a non-standard unit of measure for length. Write down a description for your measurement unit and
choose a name for your unit (e.g., the “stretch”: as far as you can stretch your thumb and index finger).
Choose 5 items to measure using your unit. Each partner will estimate the length of each item in their
unit of measurement, measure each item, and record the measurement (e.g., 4 stretches). Partners
should then compare their measurements.
2. Repeat the activity in Question 1 for capacity (for example, you could measure the capacity of a
container using ones blocks or linking cubes).
3. Repeat the activity in Question 1 for mass (for example, you could measure the mass of an object using
pennies or unsharpened pencils).
Class Projects:
1. Students can each measure a single length (for example, the width of the classroom) using a non-
standard measure, such as a cubit, then compare their measurements. They can then repeat the
activity using metres and centimetres.
2. Students can bring in stones of 3 approximate sizes—large (about 3 cm wide), medium (about 1 cm
wide), and small (3–5 mm wide)—and use them to measure the mass of various items. They can
construct a simple scale using a coat hanger with small foil pie plates hung from each end.
Workbook 3 - Measurement, Part 2 1BLACKLINE MASTERS
Analogue Clock Faces ___________________________________________________2
Dot Paper _____________________________________________________________3
Time Memory Game _____________________________________________________4
Time: Digital Clock Faces _________________________________________________5
Time: How Long is a Minute? ______________________________________________6
ME3 Part 2: BLM List
1st quarter
2nd quarter
4th quarter
3rd quarter
Analogue Clock FacesThe hour and minute hands m
ove in th
is dire
ction
.:00
12
6
11
5
1
48
7
9
10 2
:45
:30
:153
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Dot Paper
Workbook 3 - Measurement, Part 2 3BLACKLINE MASTERS
Time Memory Game
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
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21
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21
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9 3
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9 3
1011
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21
7
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9 3
1011
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9 3
1011
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9 3
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78
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9 3
1011
54
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8
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
4 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Time: Digital Clock Faces
A digital clock face looks like this . It’s a quarter past 3.
It is exactly the same as on an analogue clock.
12
6
9 3
1011
54
21
7
8
Match the analogue clock faces to the digital times.
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
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6
9 3
1011
54
21
7
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6
9 3
1011
54
21
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6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
12
6
9 3
1011
54
21
7
8
Workbook 3 - Measurement, Part 2 5BLACKLINE MASTERS
Time: How Long is a Minute?What can you do in a minute? Have your teacher time you.
How many times can you write your name?
How many jumping jacks can you do?
What number can you count to? Can you write the alphabet?
How many sit-ups can you do? How many times can you blink?
6 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Probability & Data Management Teacher’s Guide Workbook 3:2 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
A Note about Terms in Probability
Terms and definitions for methods of representing data (bar graphs, etc.) as well as concepts in
probability have been included on the worksheets. We give a general summary of the concepts underlying
probability below.
Outcomes and Events
A simple action (such as “rolling a die,” “flipping a coin,” or “spinning a spinner”) has various possible results.
These results are called the outcomes of the action. If you flip a coin, the outcomes are: “You flip a head,”
and “You flip a tail.” If you identify a particular outcome or set of outcomes of an action (such as “rolling a
six,” “rolling an even number,” “tossing a head,” or “spinning red”) you identify an event.
In probability theory, the terms outcome and event have very precise meanings. But in elementary texts, the
term outcome is occasionally misused.
On the spinner below there are three possible outcomes:
The outcomes on the spinner are:
1. The pointer lands in the blue region.
2. The pointer lands in the red region.
3. The pointer lands in the green region.
Some textbooks will identify the outcomes as:
1. You spin blue.
2. You spin red.
3. You spin green.
Identifying outcomes by colour only can cause confusion when two or more regions of a spinner are painted
the same colour. Here is a spinner with four coloured regions:
R B
• G
R B
• B B
Probability & Data Management Teacher’s Guide Workbook 3:2 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
There are 4 outcomes of spinning the spinner:
1. The pointer lands in the blue region at top right.
2. The pointer lands in the blue region at bottom right.
3. The pointer lands in the blue region at bottom left.
4. The pointer lands in the red region.
This is clearly not the same as saying that the outcomes are:
1. You spin red.
2. You spin blue.
“You spin red” and “You spin blue” are events, not outcomes. To assess the relative likelihood of spinning
red or blue, students must recognize that the pointer may land in four distinct regions of the spinner (so that
there are four possible outcomes). In three of the four outcomes, the spinner lands in a blue region. Hence,
the event “You spin blue” is more likely than the event “You spin red.”
The outcomes for Question 1 e) on Worksheet PDM3-15 are “The spinner lands in region 1,” “The spinner
lands in region 2,” and “The spinner lands in region 3.” A student may write something more concise, such as
“You spin a 1,” “You spin a 2,” “You spin a 3.” Accept these answers, as long as your student knows that
when different regions of a spinner have the same number or colour, each region must be counted as a
distinct outcome. (To avoid confusion, we only use the term outcome on worksheets when the regions of the
spinner are uniquely coloured or labelled. When the same colour or label appears more than once on the
spinner, we use phrases like “ways of spinning red” instead of “outcomes.”)
Probability & Data Management Teacher’s Guide Workbook 3:2 3 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-15 Outcomes
Goals: Students will identify the possible outcomes of various events.
Prior Knowledge Required: Experience in playing with a spinner and rolling a die.
Vocabulary: outcome
Tell students that today they will start learning how to predict the future! Hold up a die and ask students to
predict what will happen when you roll it. Can it land on a vertex? On an edge? No, the die will land on one of
its sides. Ask students to predict which number you will roll. Then roll the die (more than once, if necessary)
to show that the prediction about landing on a side works, but the number students picked does not
necessary come. Explain that the possible results of rolling the die are called outcomes. To predict the future
students must learn to identify which outcomes of various actions are more likely to happen and which are
not. But first, students must learn to identify outcomes correctly.
Hold up a coin and ASK: What are the possible outcomes of tossing a coin? How many outcomes are there?
Show a spinner and a set of marbles. What are the possible outcomes of spinning the spinner or picking a
marble with your eyes closed? Ask students to identify the possible outcomes of a soccer game. How many
outcomes are there? (3 outcomes: team A wins, team B wins, a draw)
ASK: You have to make a spinner with 5 possible outcomes. How would you do this? Invite volunteers to
draw possible spinners. Then draw the spinner below, shade each region with a different colour, and ASK:
How many outcomes are there for this spinner?
Are all the outcomes bound to come equally, or is there an outcome that might happen more often than the
other ones? Which colour are you most likely to spin? Draw the second below and ASK: How many
outcomes does the second spinner have? (4) Will the pointer ever be in the grey region? (no, never)
Activity: Your students will need a spinner like the first spinner above, with all regions coloured
differently. Students spin the spinner 10 times and tally the results. Which colour occurs the greatest number
of times?
Probability & Data Management Teacher’s Guide Workbook 3:2 4 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-16 Even Chances Goals: Students will identify situations where the chances of an event are even.
Prior Knowledge Required: Outcome, Half of a number, Visual representations of fractions
Vocabulary: outcome, even chances, draw
Review with your students how they can find half of a number, half of a pie, half of a set of objects. You
might wish to use Questions 1–5 on worksheet PDM3-16 for this review.
Draw six squares on the board. Invite a volunteer to circle half of the squares. Ask your students to shade
less than half of the squares red. Draw a different set of six squares and ask your students if they can shade
a different number of squares blue, so that the number of blue squares will be still less than half of six.
ASK: What is half of eight? I need a number that is less than half of eight—which numbers fit this
description? Repeat these questions for different even numbers. Then draw even numbers of shapes and
ask students to find several ways to shade more than half of the shapes. Have volunteers help you solve
several problems like the following:
� I have 10 marbles. Half of them are red. How many marbles are red?
� I have 12 marbles and 6 of them are green. How many of my marbles are green: half, less than half, or
more than half?
� I had 14 marbles. I lost 6 of them. Did I lose more than half, less than half, or exactly half?
Questions 6–9 on worksheet PDM3-16 can be used to assess the understanding of the concept.
Hold up a coin. ASK: What are the possible outcomes of flipping this coin? How many outcomes are there?
Which is more likely—flipping a head or a tail? Explain that the chances are the same—you have even
chances of flipping a head or a tail. Write the term even chances on the board and explain that the chances
of an event are even when the event happens in exactly half of the outcomes. Flipping a tail is 1 out of 2
possible outcomes, and 1 is half of 2. ASK: How many outcomes are there when you roll a die? (6) How
many outcomes are numbers that are more than 3? (3) Since half of the outcomes are numbers greater than
3, you have even chances of rolling a number greater than 3 (and even chances of rolling a number that is
3 or less).
SAY: We have 8 marbles in a box. I take out 1 marble (without looking!). How many outcomes are possible?
(8, regardless of the colour of the marbles) What is half of 8? I would like to have even chances of taking out
a green marble. How many marbles should be green? (exactly 4) Does it matter what colour the other
marbles are? (No, provided they’re not green.) Invite a volunteer to draw a collection of 8 marbles (or to
create a collection using actual marbles, if available) that gives even chances of drawing a green marble.
If the collection uses only 2 colours, ask another volunteer to make a collection that uses at least 3 colours
but still gives even chances of drawing a green marble.
Probability & Data Management Teacher’s Guide Workbook 3:2 5 Copyright © 2007, JUMP Math For sample use only – not for sale.
Draw a spinner:
ASK: Which part of the spinner is shaded green? What are the possible outcomes for this spinner? How
many outcomes are there? Which part (or fraction) of the outcomes is “The pointer lands in a green region”?
Are the chances of spinning green even?
Repeat this exercise with the spinner below. Emphasize, if necessary, that this spinner has 4 possible
outcomes because it has 4 different regions, but 2 of the 4—half—are green.
Draw 3 spinners:
Ask students to identify the outcomes for each spinner. ASK: In which spinners do you have even chances of
spinning red? (the one on the left and the one on the right) Draw more spinners on the board and ask
students to identify the spinners where you have even chances of spinning red.
Ask students to describe an event with even chances for rolling a die. (Possible answers: roll a number that
is 3 or less; roll an even number; roll 2, 3, or 5.) Ask students to describe another event with even chances.
Encourage students to think of examples that do not involve rolling dice, spinning spinners, or drawing
marbles. (EXAMPLE: Several pairs of boots are in a closet. I pick a boot in the dark. It is either a left boot or
right boot. So “I pick a left boot” has even chances.)
Assessment:
1. Circle the spinners where you have even chances of spinning red.
G R • Y
G R
• R B
G R
•
Y G
• G B
B
R •
G
R
• Y
B
G R
• B
G R
• R
B R
B
G R • Y
G R •
R
B
Probability & Data Management Teacher’s Guide Workbook 3:2 6 Copyright © 2007, JUMP Math For sample use only – not for sale.
2. Draw a collection of 6 marbles that gives even chances of picking a green marble.
3. Draw a collection of 8 marbles of at least 3 colours that gives even chances of picking a yellow marble.
Activity: Divide students into 3 groups. Let each group make one of the 3 spinners below, spin it 24
times, and record the results. Then have groups make a bar graph of the results.
ASK: Did your group spin red in more than half, less than half, or exactly half of the spins? Is this what
you expected? Discuss as a class.
Extension: Complete each statement by writing “more than half” or “less than half.”
a) 2 is ___________________ of 5
b) 3 is ___________________ of 7
c) 6 is ___________________ of 13 d) 7 is ___________________ of 11
e) 11 is ___________________ of 15 f) 5 is ___________________ of 11
G R
• R B
G R
• B R
G R
• B
R B
Probability & Data Management Teacher’s Guide Workbook 3:2 7 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-17 Even, Likely and Unlikely Goals: Students will describe the chances of events as even, likely, and unlikely.
Prior Knowledge Required: Outcome, Half of a number, Visual representations of fractions
Vocabulary: outcome, even chances
Have students make some predictions: ask them to tell you if the following events are likely or unlikely.
� The sun will rise tomorrow.
� The teacher will give the answers to the test before giving the test.
� An alien will walk into the class in the next minute.
� We will have lunch in half an hour.
� It will rain tomorrow.
� It will snow in June.
Invite students to name some events and have other students tell if the events are likely or unlikely. You
could ask students to compare the likelihood of events. For example, it is unlikely to snow in June, but it is
more unlikely that an alien will walk into the class!
Draw the spinner below and ask students if it is likely that the spinner will point to red. Conduct an
experiment with 12 spins. Use a tally chart to record the results. Explain to your students that
mathematicians call an event likely if it is expected to happen more than half the time and unlikely if it
is expected to happen less than half the time. From the last lesson, they know that an event with even
chances is expected to happen exactly half the time. Write all three terms on the board. Using the tally chart,
describe the chances of spinning red, blue, and green in these terms.
Describe several events (see Examples below) and ask students to count the total number of outcomes and
the number of outcomes that suit the event, and to tell whether the event is likely, unlikely, or has even
chances. Remember: If the event matches more than half the outcomes, then that event is likely—chances
are it will happen more than half the time. If the event matches less than half the outcomes, it is unlikely.
EXAMPLES:
There are 4 pairs of boots in a dark closet—2 black, 2 brown. Events:
� Pull out a right boot � Pull out a black boot
� Pull out a brown left boot � Pull out a boot that is either
� Pull out a boot that is not left brown black or right brown
G R
• B
Probability & Data Management Teacher’s Guide Workbook 3:2 8 Copyright © 2007, JUMP Math For sample use only – not for sale.
A model (draw 8 shoes and colour them) could be helpful for students who have trouble counting outcomes.
Events: Spin green. Events: Spin blue.
Spin purple. Spin purple.
Assessment:
1. Are the chances likely, unlikely, or even?
a) Pull a black sock from a box with 6 green socks and 4 black socks.
b) Pull a penny from a pocket with 5 pennies, 4 dimes, and 1 nickel.
2. Harold rolls a die. Give an example of a likely event and an unlikely event for Harold. Both events should
be possible.
P
B
P Y
P
P P
B P G
GB G
B
B
Probability & Data Management Teacher’s Guide Workbook 3:2 9 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-18 Describing Probability Goals: Students will describe the likelihood of events.
Prior Knowledge Required: Outcome, Half of a number, Likely, Unlikely,
Vocabulary: outcome, even chances, likely, unlikely, certain, impossible
Review the meaning of the terms likely and unlikely with your students. Write the terms on the board. Ask
your students which word people would use to describe an event like meeting a live dinosaur in the street.
Can that happen at all? Add the word impossible to the list. Ask your students which words describe an
event that will definitely happen, like rolling a number less than 7 on a die. Add the word certain to the list.
Ask students to describe the following events as likely, unlikely, certain, or impossible:
� It will rain on September 12
� It will snow on July 15
� There will be a math test before the end of the month
� You will grow wings
� Pull a green sock from a drawer with 10 green socks and 2 red socks
� Pull a $5 bill from a wallet with three $20 bills and one $5 bill
� Roll a number greater than 0 on a die
� Meet a green panther
NOTE: Although students might use the word impossible to describe the likelihood of meeting a dinosaur,
this event is not necessarily impossible (scientists might find a way to clone dinosaurs in the future). The only
events that are strictly impossible are events that are contradictory—like rolling a number greater than 6 on a
regular die. You might discuss this distinction with students.
Ask students to give examples of various events and explain whether they are likely, unlikely, certain, or
impossible. Encourage students to think of events using marbles, dice, money, and other objects, as well
events from daily life, such as meeting a tiger or an astronaut on the way to school.
Draw the spinners below on the board and ask your students to describe the following events as likely,
unlikely, having even chances, certain, or impossible for each spinner:
� Spinning green
� Spinning red
� Spinning blue
� Spinning yellow
G R
• B
G R
• • B G Y
• B
G R
• B R
Probability & Data Management Teacher’s Guide Workbook 3:2 10 Copyright © 2007, JUMP Math For sample use only – not for sale.
Ask students to draw a spinner to match this description:
1. You are likely to spin yellow.
2. You are unlikely to spin green.
3. It is impossible to spin blue.
Show students a collection of marbles or coloured counters:
ASK: What are your chances of picking green? Which colour are you most likely to pick? Which colour is less
likely to be picked, yellow or red? So which colour is least likely to be picked?
Assessment:
Draw a collection of marbles of at least three colours so that:
1. You are likely to draw a green marble.
2. You are unlikely to draw a yellow marble.
3. It is impossible to draw a purple marble.
Bonus:
Design 3 spinners with different number of regions that fit this description:
1. You are likely to spin yellow.
2. You are unlikely to spin green.
3. It is impossible to spin blue.
Activity: Have students flip a coin 10 times and keep a tally of the number of heads. It is likely that not
every student flipped heads exactly half the time. Ask students to identify the result that was furthest from the
expected number of 5 heads. Then add up the total number of heads from all the tallies and the total number
of tosses (10 × number of students in class). The overall proportion of heads should be closer to half of the
total number of tosses. Explain to students that the more trials you conduct (i.e., the more times you repeat
an experiment) the more closely the actual outcome will match the expected outcome for an event.
Extension: Invent or describe a game where a certain player’s chance of winning is very close to certain.
What are the chances of the other player(s) winning?
R B B G G G G G Y Y
Y
Probability & Data Management Teacher’s Guide Workbook 3:2 11 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-19 Describing Probability (Advanced) Goals: Students will describe and compare the likelihood of events using vocabulary words.
Prior Knowledge Required: Outcome, Half of a number, Likely, Unlikely
Vocabulary: outcome, even chances, likely, unlikely, certain, impossible, most likely, very unlikely
Review the previous lesson.
Draw the following collection of marbles and a probability line on the board:
ASK: How many marbles are in the collection? How many marbles of each colour do we have? Which colour
are you most likely to pick? Is your chance of picking green even, likely or unlikely? Which colour is less
likely to be picked: yellow or red? Why? Which colour is least likely to be picked? Ask your students to mark
on the line the probability of picking…
� a green marble � a blue marble � a yellow marble
� a red marble � a pink marble � a marble of any colour
Ask your students to give examples of various mathematical or real-life events. For each one, ASK: Where
on the probability line would you put this event? Why?
R B B G G G G G Y G R B
impossible certain even unlikely likely
Probability & Data Management Teacher’s Guide Workbook 3:2 12 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-20 Fair Games Goals: Students will use the concept of even chances in games.
Prior Knowledge Required: Outcome, Half of a number, Even chances
Vocabulary: outcome, even chances, likely, unlikely, certain, impossible, most likely, very unlikely,
fair game
SAY: I would like to play a game with you. The rules of the game are simple. I will spin a spinner. If I get red,
I win; if I get blue, the class wins. Ask students if they agree to play by these rules. Now show them the
spinner. Do they still want to play? Why not?
Write the term fair game on the board. Ask students to explain what they think this term might mean.
Encourage students to use math vocabulary in their explanations. Point out that in a fair game, both players
have equal chances, or are equally likely, to win.
Give the rules for another game: There are 6 marbles in a box. If I draw a red marble, I win; if I draw a blue
marble, the class wins. To make the game fair, how many blue marbles should be in the box? How many red
marbles?
Vary the game: If I draw a red marble, I win, but if I draw any other colour, the class wins. If 2 of the 6
marbles are red, who has more chances of winning, me or the class? What if 5 marbles are red? What
should be in the box to make the game fair? Encourage the class to think of more than one solution.
Assessment: Two players are spinning this spinner. Invent rules of play to ensure that the game is fair.
R
• B
R • B
W
Probability & Data Management Teacher’s Guide Workbook 3:2 13 Copyright © 2007, JUMP Math For sample use only – not for sale.
Activities:
1. Pair up students and give each pair a container with 2 red counters and 4 blue counters inside. Either
student can pick counters from the container (no peeking!). Player 1 wins if the counter drawn is red and
Player 2 wins if the counter drawn is blue. Ask students to explain whether the game is fair or not. Have
students play the game 20 times (replacing the counter each time) and keep a tally of who wins each
time. Ask students if the results are what they expected.
2. Ask pairs to keep track of who wins and who loses in 20 repetitions of the following game: Players take
turns rolling a die. Player 1 wins if the number rolled is a 1 or a 6; Player 2 wins otherwise. ASK: Is this
game fair? Are the results what you expected?
Extensions:
1. How are the games in Activities 1 and 2 similar? Is the first game less fair than the second? (From the
point of view of probability, the two games are identical. In both games, Player 2 has 4 out of 6 chances
of winning: the probability of Player 2 winning is 46 .)
2. Introduce the concept of equal likelihood.
Draw this spinner on the board:
ASK: What is more likely, to spin red or to spin green? Green or blue? Red or something else? What is
more likely to happen, rolling 2 or 3 on a die? These pairs of events are equally likely to happen.
SAY: A game is fair if both players are equally likely to win. Is the following game fair?
Players spin the spinner above. If they spin green, Player 1 wins. If they spin blue, Player 2 wins. If they
spin red, it is a draw. (The game is fair, since both players have equal chances of winning—the
probability of winning is 14 for both.)
G R
• B
Probability & Data Management Teacher’s Guide Workbook 3:2 14 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-21 Experiments and Expectations Goals: Students will compare theoretical and experimental probability.
Prior Knowledge Required: Outcome, Half of a number, Even chances
Vocabulary: outcome, even chances, fair game
Draw a spinner on the board:
ASK: Which part of the spinner is coloured red? How can you describe the chances of spinning red? (even
chances) How many outcomes are there for spinning this spinner? (4) Which part of the outcomes is
“spinning red”? (2 out of 4, or half) If I spin the spinner 4 times, how many times would I expect to get red?
What if I spin it 10 times? 20 times? Give your students spinners like the one shown, and ask them to spin
the spinner 10 times and to tally the results. Did everybody get 5 reds? (No) What was the most common
number of red spins? What was the smallest number of red spins? The largest number?
Pool the results for the whole class. Compare the class results with the individual results using graphs or
other visual representations. For example, individuals could make bar graphs of their results (showing the
number of times they spun red, blue, and green) and you could put all the results together in a class bar
graph. Discuss with your students the difference between the class results and the individual results. ASK:
Which results are nearer to the prediction? Which data do you think is more reliable: the individual data or
the group data? Why? Point out that the more results you have, the closer you get to the expected outcome.
The number of reds spun by the class will be closer to half of the total (the expected outcome) than many of
the individual results.
Repeat with several more experiments, such as:
� Flip the coin 10 times. How many times will you get a head?
� Draw a marble from a box with 6 marbles, 3 blue and 3 green. Return the marble to the box. Repeat
12 times. How many times do you expect to draw a blue marble?
� Draw a marble from a box with 6 marbles: 3 blue, 1 yellow, and 2 green. Return the marble to the box.
Repeat 12 times. How many times do you expect to draw a blue marble?
Ask your students to design an experiment where they would expect to get a certain result 10 times in 20
repetitions (in other words, the result has even chances).
Extension: See BLM “Shape Spinner.”
G R
• B
R
Probability & Data Management Teacher’s Guide Workbook 3:2 15 Copyright © 2007, JUMP Math For sample use only – not for sale.
PDM3-22 Cumulative Review
PDM3-22 is a review worksheet for Probability and Data Management, parts 1 and 2.
Extension: Write the numbers from 1 to 10 on ten cards (one number per card). If you select 6 cards at
random, ask students to think about the probability of any one or more of the following events:
� The sum of the numbers will be greater than 60.
� All the numbers will be even.
� Two numbers will be neighbours.
� No numbers will be neighbours.
� The sum of the numbers will be less than 12.
� The sum of the numbers will be greater than 25.
Ask students to predict whether each event is certain, impossible, likely, or unlikely. Invite individuals, pairs,
or small groups to conduct 10 experiments to check their predictions. Combine the individual results to get
class results. How do the class results compare to the individual results? How do the class results help
students to confirm (or revise) their predictions? Can students explain why some of the events are certain or
impossible?
Workbook 3 - Probability & Data Management, Part 2 1BLACKLINE MASTERS
Shape Spinner _________________________________________________________2
PDM3 Part 2: BLM List
Shape Spinner
You will be asked to spin
the spinner until it lands on
the same shape 10 times.
Predict which shape this
will be:
My Prediction My Partner’s Prediction
Create a spinner like the one below.
Test your predictions. Take turns spinning with your partner.
The first player with 10 tallies in one column wins!
Record the data in these charts.
My Chart My Partner’s Chart
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
Sample use only - not for sale
Geometry Teacher’s Guide Workbook 3:2 1 Copyright © 2007, JUMP Math For sample use only – not for sale.
G3-18 Introduction to Coordinate Systems Goal: Students will identify columns and rows of coordinate systems given their numbers.
Prior Knowledge Required: Ability to count
Horizontal
Vertical
Vocabulary: coordinate system, row, column
To illustrate the concept of a coordinate system, you can start with the following card trick.
1. First, deal out nine cards—face up—in the arrangement shown below:
Row 3
Row 2
Row 1
Column 1 Column 2 Column 3
2. Next, ask your students to select a card in the array and then tell you only what column it’s in (they
shouldn’t identify the card!).
3. Gather up the cards, with the three cards in the column your students selected on the top of the deck.
4. Deal the cards face up in another 3 × 3 array making sure the top three cards of the deck end up in the
top row of the array.
5. Ask your students to tell you what column their card is in now. The top card in that column (i.e., the card
in row 3) is their card, which you can now identify!
Geometry Teacher’s Guide Workbook 3:2 2 Copyright © 2007, JUMP Math For sample use only – not for sale.
6. Repeat the trick several times and ask your students to try to figure out how it works. You might give
them hints by telling them to watch how you place the cards, or even by repeating the trick with a
2 × 2 array.
When your students understand how the trick works, you can ask the following questions:
• Would there be any point to the trick if the person picking the card told the person performing the trick
both the row number and the column number of the card they had selected? Clearly there would be no
trick if the performer knew both numbers. Two pieces of information are enough to unambiguously
identify a position in an array or graph. This is why graphs are such an efficient means of representation;
two numbers can identify any location in two-dimensional space (in other words, on a flat sheet of
paper). This discovery, made over 300 years ago by the French mathematician René Descartes,
was one of the simplest and most revolutionary steps in the history of mathematics and science.
His idea of representing position using numbers underlies virtually all modern mathematics, science,
and technology.
• Will the trick work with a larger array? Have students try the trick with a 4 × 4 array. They should see that
as long as the array is square (with an equal number of rows and columns), the trick works for any
number of cards. Ask your student to explain why this is so and why the trick doesn't work if the array
isn't square (for instance, try it with 2 columns and 6 rows).
• Will the trick work if the person picking the card tells the performer which row the card is in rather than
which column? What does the performer have to do differently in this case? Have your students show
you how the new trick would be performed. The fact that the trick works equally well in both cases
illustrates a very deep principle of invariance in mathematics. In a square array, there is no real
difference between the rows and columns. In fact, if you rotate the array by a quarter turn, the rows
become columns and vice versa. More generally, once you fix an origin in space, it doesn't matter how
you set up your grid (the lines representing the rows and columns). In all cases you need only two
numbers to identify a position.
• How many numbers would be required to represent the position of an object relative to an origin in three-
dimensional space? (The answer is three. Think of the origin as being situated on a plane, or a flat piece
of paper, that has a grid or graph on it. You need two numbers to tell you how to travel from the origin
along the grid lines on the plane to situate yourself directly above or below the object, and one more
number to tell you how far you have to travel up or down from the plane to reach the object.)
Now draw a 3 × 3 array and number the columns and rows:
1 2 3
3
2
1
C R O W
L U
M N
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The diagram next to the array illustrates the orientation of columns (horizontal) and rows (vertical). Point out
a row and a column in the array, and stress that we order rows from bottom to top, and columns from right to
left. Ask several volunteers to locate the third column, second row, etc. Then ask your students to complete
the worksheet for this lesson.
Assessment:
Join the dots in the given column and row:
a) Column 3, Row 2 b) Column 1, Row 3 c) Column 3, Row 3
Bonus:
Which letters of the alphabet can be written on a grid and described in terms of rows and columns only?
(See Question 4 on the worksheet.) Which numbers can be written this way?
Extension:
The card trick above can be modified for non-square arrays if one makes an extra rearrangement. Deal out
an array of 3 columns and 9 rows. Have a student select a card and tell you what column it’s in. Re-deal the
cards so that all nine cards from the chosen column land in the top three rows of the new array. Ask the
student to tell you what column their card is in now, and re-deal the top three cards in that column into the
top row of a new array. Once the student tells you what column their card is in, you can identify the card—it
will be the top card in that column.
This version of the trick illustrates a powerful general principle in science and mathematics: when you are
looking for a solution to a problem, it is often possible to eliminate a great many possibilities by asking a well-
formulated question. In the card trick, one is able to single out one of 27 possibilities by asking only three
questions. Repeat the trick, asking your students how many possibilities are eliminated by the first question
(18), by the second question (6), and by the third (2).
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G3-19 Coordinate Systems Goal: Students will locate a point in an array given its column and row numbers.
Prior Knowledge Required: Horizontal
Vertical
Columns and rows in coordinate systems
Vocabulary: coordinate system, row, column
Review the previous lesson. Explain to your students that today you will give them a point in the array and
they will have to identify the column and the row. Draw several arrays of dots on the board, join the points in
a column and a row in each array, and ask the students to identify the highlighted row and column. NOTE:
You can create bonus questions using larger arrays, i.e., arrays with more rows and columns.
Draw more arrays and circle a dot in each. Ask your students to identify the column and the row for each dot.
Students that have trouble identifying the column and the row should first highlight them. Then reverse the
task—give the column and the row and ask students to find the dot.
Assessment:
1. Circle the dot where the two lines meet:
a) Column 2, Row 3 b) Column 3, Row 1 c) Column 1, Row 1
2. Identify the proper column and row for the circled dot:
a) b) c) d)
Column ____
Row ____
Column ____
Row ____
Column ____
Row ____
Column ____
Row ____
Activity:
Ball Game The students are the points in a coordinate system. Ask each student to say which column and
row they are in. Give one of the students a ball and identify a point on the array with a column number and a
row number. The student with the ball has to toss it to the student at the given point. Students can continue
tossing the ball around by calling out coordinates rather than names.
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G3-20 Introduction to Slides (or Translations) Goal: Students will slide a dot on a grid.
Prior Knowledge Required: Ability to count
Distinguish between right and left
Vocabulary: slide, translation
For this lesson, a magnetic board with a grid on it (or an overhead projector with a grid drawn on a
transparent slide) would be helpful. Let your students practice sliding a dot, in the form of a small circular
magnet, right and left, then up and down. Students should be able to identify how far a dot slid in a particular
direction (how many squares, or units) and also be able to slide a dot a given distance (e.g., 2 units up, 5
units left, 4 units right).
If any students have difficulty distinguishing between right and left, write the letters L and R on the left and
right sides of the board or overhead. You can also spend some time teaching them to distinguish between
left and right:
• Have students raise their right hand, then their left. (Show them that their left index finger and thumb
makes a correct L, whereas their right index finger and thumb makes a backwards L.)
• Draw two figures on the board and have students identify the one that is on the right and the one that is
on the left. Repeat.
After students can slide a dot in a given direction, show them how to slide a dot in a combination of
directions. EXAMPLE: Slide the dot 4 units up then 1 unit left. Slide the dot 3 units right then 3 units down.
Assessment:
Slide the dot:
a) 3 units right; 3 units up b) 6 units left; 3 units down c) 7 units left; 2 units up
Activities:
1. Ball Game The students are points on the grid, and you give directions such as “The ball slides 3 units
to the right.” The student with the ball has to toss it to the right “point” on the grid.
2. In the schoolyard, draw a grid on the ground. Ask your students to move a certain number of units,
in various combinations of directions, by hopping from point to point in the grid.
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G3-21 Slides Goal: Students will slide a shape on a grid.
Prior Knowledge Required: Slide a dot on a grid
Distinguish between right and left
Vocabulary: slide, translation
Tell your students the following story. You might use two actual figures to demonstrate the movements in
the story.
Suppose you have a pair of two-dimensional figures and you wish to place one of the figures on top of the
other. But the figures are very heavy—and very hot—sheets of metal. You need to program a robot to move
the sheets, and to write the program you have to divide the process into very simple steps. It is always
possible to move a figure into any position in space using some combination of the following three
movements:
1. You may slide the figure in a straight line (without allowing it to turn):
SLIDE or TRANSLATION
2. You may turn the figure around some fixed point (usually on the figure):
TURN or ROTATION
3. You may flip the figure over:
FLIP or REFLECTION
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Two figures are congruent if the figures can be made to coincide by some sequence of flips, slides, and
turns. For instance, the figures in the picture below can be brought into alignment by rotating the right hand
figure counter-clockwise a quarter turn around the highlighted point (also a vertex), then sliding it to the left.
It is not always possible to align two figures using only slides and turns. To align the figures below you must,
at some point, flip one of the figures:
One way to flip a figure is to reflect the figure through a line that passes through an edge or a vertex of
the figure.
Tell your students that today you are going to teach them about slides. Show students the following picture
and ask them how far the rectangle slid to the right. Ask for several answers and record them on the board.
You may even call for a vote.
Students might say the shape moved anywhere between 1 and 7 units right. Take a rectangular block and
perform the actual slide, counting the units with the students. The correct answer is 4.
Show another picture:
This figure has a dot on its corner. How much did it slide? This time it is easier to describe the slide—just use
the benchmark dot on the corner. Check with the block.
Show a third picture:
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Is this a slide? No, this is a slide together with a rotation. You cannot slide this block from one position to the
other without turning it.
Activity: Give your students a set of pattern blocks or Pentamino pieces and ask them to trace a shape
on dot paper so that at least one of the corners of the shape touches a dot (use BLMs if needed).
Ask students to slide the shape a given number of units in one or more directions. After the slide, trace the
pattern block again.
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G3-22 Slides (Advanced) Goal: Students will slide a shape on a grid and describe the slide.
Prior Knowledge Required: Slide a dot on a grid
Distinguish between right and left
Vocabulary: slide, translation, translation arrow
Draw a shape on a grid on the board and perform a slide, say 3 units right and 2 units up. Draw a translation
arrow as shown on the worksheet. Ask your students if they can describe the slide you’ve made. If they have
trouble describing the slide, suggest that they look at how the vertex of the figure moved (as shown by the
transition arrow). To help students describe the slide, you might tell them that the grid lines represent streets
and they have to explain to a truck driver how to get from the location at the tail of the arrow to the location at
the tip of the arrow. The arrow shows the direction as the crow flies, but the truck has to follow the streets.
Make sure your students know that a slide is also called a translation. Ensure that students also understand
that a shape and its image under a translation are congruent.
Extensions
1. Slide the figures however you want, and then describe the slide.
2. Describe a move made by a chess knight as a slide. Describe some typical moves of other chess pieces,
such as a pawn or a rook (castle).
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G3-23 Slides on a Grid
Goal: Students will describe slides on a grid.
Prior Knowledge Required: Slide a dot on a grid
Distinguish between right and left
Vocabulary: slide, translation, translation arrow
Draw a hockey rink on a magnetic grid (or use the overhead projector) and place several “players” on the
intersections of grid lines, as shown below. You might use a small magnet to show the passes of the puck.
Ask your students to describe the following passes:
From Player 1 to Player 2: ____ units up
From Player 2 to Player 3: ____ units _________
From Player 3 to Player 4: ____ unit _________ and ___ units down
From Player 4 to Player 5 ____ units _________ and ___ units _________
From Player 5 to Goalie: ____ units _________ and ___ units _________
Pass the puck between other players to provide your students with more practice. Suggest that your students
move the players and describe more passes. Students can also answer questions like these:
• Player 3 passes the puck 5 units right and 3 units down. Who receives the pass?
• Player 5 wants to pass the puck to Player 4. How many units left and how many units down should the
puck go?
1
2 3
4
5 G
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G3-24 Coordinate Systems and Maps Goal: Students will describe and perform slides on a grid.
Prior Knowledge Required: Slides
Coordinate systems
Vocabulary: slide, translation, translation arrow, row, column, coordinate system, coordinates
Assign a letter to each row of desks in your class and a number to each column. Ask your students to give
the coordinates of their desks. Then invite students to write a short message to a classmate and to include
the recipient’s “address” in coordinates. A volunteer “postman” then delivers the letters. The postman has to
describe how the letter moved (two to the front and one to the left, for example). Reverse the roles: the
postman delivers the letter and the addressee has to say which slide the letter performed. Let several
students perform the role of postman.
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G3-25 Mapping Exercise Goal: Students will use coordinates to describe and find points on a map.
Prior Knowledge Required: Slides
Coordinate systems
Vocabulary: slide, translation, translation arrow, row, column, coordinate system, coordinates
Place a transparency with a map of Saskatchewan on the overhead projector (see the BLM “Map of
Saskatchewan”). Ask volunteers to find the cities on the map and to answer these questions:
• What are the coordinates of Saskatoon?
• What are the coordinates of Regina?
• What are the coordinates of Uranium City?
• What are the coordinates of Prince Albert?
• What can you find in the square A4? D5? D1?
Ask more questions of this sort.
Activity:
Memory Game
Students will need a grid and 1 to 4 small objects (such as play money of different values or beads of
different colours). The objects are placed on intersections of the grid. Player 1 slides one of the objects while
Player 2’s back is turned, and Player 2 then has to guess which object was moved and how (describe the
slide). Students will have to study the board and memorize the original coordinates of the objects. They can
then compare the coordinates after the objects were moved with the coordinates before the objects were
moved to determine exactly which object moved and how.
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G3-26 Flips Goal: Students will reflect simple shapes through vertical and horizontal lines parallel to their sides.
Prior Knowledge Required: Symmetry
Vocabulary: symmetry, line of symmetry, horizontal, vertical
Draw a scalene right-angled triangle and a line parallel to one of the legs of the triangle on a transparency
and project them on the board. Invite a volunteer to trace the shape and the line on the board. Flip the
transparency so that the line coincides with the line in the drawing and the projected triangle is a reflection of
the triangle in the drawing. Explain to your students that the movement that you performed is called a flip or
reflection of the triangle over the line. Ask your students if they remember a name that refers to these two
triangles together (congruent, symmetrical). What is the line called? (line of symmetry)
Give your students an assortment of Pentamino pieces. Ask them to trace each piece on grid paper (see
BLM section), draw a mirror line through a side of the piece, flip the shape over the line, and then trace the
flipped piece. As a challenge, ask them to draw the mirror line parallel to, but not touching, one of the sides
and repeat the exercise. Students should use the squares of the grid to determine the right distance between
the mirror line and the flipped shape.
Activities:
1. Students could create their own shape and trace the shape before and after a flip.
2. Find-a-Flip Game
Divide your students into groups of 2-5 players each. Each group will need two copies of the BLM
“Find-a-Flip Game.” Let the students cut out the cards. Players shuffle the deck, deal out 4 cards for
each player, and lay a card face up on the table. If a player has a card that is a reflection of the card on
the table over one of its sides, he or she can lay the card on the table and pick a new card from the deck:
Players take turns placing cards on the table until they have a square:
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The player that placed the last card in the square obtains 1 point.
If a player does not have a card that is a reflection of one of the cards on the table, the player picks cards
from the deck until he or she can either add a card to the shape on the table or create a square of
shapes from his or her own cards. A player that does not have any more cards in hand picks 4 cards
from the deck. If a player runs out of cards and there are no more cards in the deck, the player exits the
game and obtains 2 points. The player with the greatest number of points wins.
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G3-27 Reflections Goal: Students will reflect simple shapes on a grid through vertical and horizontal lines.
Prior Knowledge Required: Symmetry
Vocabulary: symmetry, line of symmetry, horizontal, vertical
Give your students an assortment of Pentamino pieces. Ask them to trace each piece on grid paper, draw a
mirror line through a side of the piece, and then draw the reflection of the piece in the mirror line. Students
could check if they have drawn the image correctly by flipping the Pentamino piece over the mirror line and
seeing if it matches the image. Remind your students that a flip is also called a reflection. Students should
notice that each vertex on the original shape is the same distance from the mirror line as the corresponding
vertex of the image. Let your students practise reflecting shapes with partners: one student draws a shape of
no more than 10 squares and chooses the mirror line; the partner has to reflect the shape over the given
mirror line.
ADVANCED GAME: One student draws a shape of no more than 10 squares and its reflection in a mirror
line, but misplaces one of the squares in the reflection. The partner has to correct the mistake.
Extension:
Students could try to copy and reflect a shape in a slant line.
EXAMPLES:
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G3-28 Flips and Slides Goal: Students will distinguish between flips (reflections) and slides (translations).
Prior Knowledge Required: Symmetry
Reflections
Slides
Vocabulary: symmetry, line of symmetry, horizontal, vertical, slide, reflection, flip, translation
Draw a parallelogram on a grid on the board. Ask your students which transformations they learnt to do.
Slide the parallelogram several units right and ASK: What did I do with the shape? Ask your students to
describe the slide. Reflect the initial parallelogram over a horizontal line and ask them to describe the
transformation. Repeat with several more shapes, including a shape with a horizontal line of symmetry, such
as a square or rectangle (so that the horizontal reflection does not change the shape). Discuss with your
students which transformations could take this shape onto its image. Check with a cut-out of the shape that
both a slide and a reflection can bring the shape onto its image.
Activity:
Divide your students into groups of 2–5 players each. Each group will need two copies of the BLM “Find-a-
Flip Game.” Let the students cut out the cards. Players shuffle the deck, deal out 4 cards for each player,
and lay a card face up on the table. If a player has a card that is a slide or a reflection of the card on the table
over one of its sides, he or she adds the card to the table (and picks a new card from the deck):
or or or
Players take turns placing cards until they have a square or rectangle of area 4 (or more) cards. The player
that placed the last card in the rectangle obtains a number of points equal to the area of the rectangle. Check
with the students which rectangles are possible (They are: 4 × 1, 2 × 2, 2 × 3, 2 × 4. The rectangle 5 × 1
seems possible but isn’t because the previous player will have claimed the 4 × 1 rectangle.) A player is not
allowed to place a card so that there will not be enough cards to make a rectangle. (There are only 8 cards
with shapes that can go into the same rectangle.)
This is not a rectangle.
This shape is not allowed – you will not have enough cards to make a rectangle
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If a player does not have a card that is a reflection or a slide of one of the cards on the table, the player picks
cards from the deck until he or she can either add a card to the shape on the table or make a rectangle from
the shapes on his or her own cards. A player that does not have any more cards picks 4 cards from the deck.
If there are no more cards left in the deck, the player exits the game and obtains 2 points. The player with the
greatest number of points wins.
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G3-29 Turns Goal: Students will describe rotations that are multiples of a quarter turn.
Prior Knowledge Required: Fractions: 12 ,
14 ,
34
Clockwise, counter-clockwise
Vocabulary: rotation, clockwise, counter-clockwise
Review the division of the analogue clock face into quarters. If a large clock is available, put the hands of the
clock so that they are perpendicular and ask your students to identify the angle between them. Move the
hands into various positions, such as 3:00, 9:00, or 3:30, but also 1:20 or 10:38. Review the meaning of the
terms clockwise and counter-clockwise using the clock and also by drawing arrows on the board. Ask
volunteers to rotate the minute hand, clockwise and counter-clockwise, a full turn, a half turn, and a quarter
turn. You might also ask your students to be the clocks: each student stands with an arm outstretched and
turns clockwise (CW) or counter-clockwise (CCW) according to your instructions.
Review with your students these fractions of a circle: 14 ,
12 ,
34 . Let them both name shaded fractions of a
circle and shade fractions of a circle.
Sample fractions to name:
Sample fractions to shade:
Shade: 14
12
34
34
14
12
Draw pairs of arrows on the board, such as those shown below, and ask your students to shade the part
between the arrows. Ask them to lay a pencil on the “start” arrow and to turn it to the “finish” arrow. How
much has the pencil turned?
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___ turn CCW ___ turn CW ___ turn CW ___ turn CCW
Students who have trouble deciding which part of the circle is shaded might extend the hands of the clock to
the other side of the central dot to divide the circle into quarters. Repeat, but this time let your students
decide whether the turn was clockwise or counter-clockwise.
Bonus:
How much has the arrow turned?
___ turn CW ___ turn CCW ___ turn CCW ___ turn CW
start
finish start finish
start
finish start
finish
start
finish
start
finish
start finish
start
finish
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G3-30 Rotations Goal: Students will describe and perform rotations that are multiples of a quarter turn.
Prior Knowledge Required: Fractions: 12 ,
14 ,
34
Clockwise, counter-clockwise
Vocabulary: rotation, clockwise, counter-clockwise
Review the previous lesson with the students.
Draw several clocks on the board as shown below and ask your students to tell you how far and in which
direction each hand moved from start to finish:
12 turn CCW _______________ _______________ _______________
Then draw examples with only one arrow and ask students to turn the arrow:
a) 12 turn CCW b)
14 turn CCW c)
14 turn CW d)
34 turn CCW
Assessment
1. Describe the rotation of the arrow:
2. Show the position of the arrow after each turn:
a) 14 turn CW b)
34 turn CW c)
34 turn CCW d)
14 turn CW
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G3-31 Rotations (Advanced) Goal: Students will describe and perform rotations that are multiples of a quarter turn.
Prior Knowledge Required: Fractions: 12 ,
14 ,
34
Clockwise, counter-clockwise
Vocabulary: rotation, clockwise, counter-clockwise
Review the previous lesson by drawing several arrows or clock hands. To help your students visualize the
effect of a rotation on a shape, have them make a small flag (as in Question 1 on the worksheet) by taping a
triangular piece of paper to a straw. Ask students to rotate the flag and trace its image after the rotation.
Students could also cut out other shapes used on the worksheet and trace the images of these shapes after
a rotation. Then ask your students to trace or draw a figure, decide on a rotation, and draw the new shape
without a prop. Students could also practise rotating pattern blocks or Pentamino shapes (around vertices of
the shapes) on a grid.
Assessment: Draw the shape after each turn:
a) 14 turn CW b)
12 turn CW c)
14 turn CCW d)
34 turn CW
Activities:
1. NOTE: This activity was adapted from the Atlantic curriculum.
Trace a pattern block trapezoid on a sheet of paper. Rotate the block around one of the vertices a half-
turn clockwise and trace it again. Repeat with a half-turn counter-clockwise. What do you notice? Repeat
with a different shape (not necessarily a pattern block).
2. Play another version of the Find-a-Flip game. Each group of 2–4 students will need two copies of any
three rows of shapes on the BLM “Find-a-Flip Game.” Let your students play the game as in Activity 2 of
G3-26, but this time each card placed should be a quarter turn rotation of the adjacent cards.
Extension: Using pattern blocks or cardboard polygons, trace a figure on a sheet of paper. Then choose
a vertex and rotate the shape around the vertex a half turn. Trace the figure again. Would you get the same
result if you had reflected the figure? (Right-angled trapezoids are particularly interesting in this case.)
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G3-32 Flips, Slides and Turns Goal: Students will distinguish between flips (reflections), slides (translations), and turns (rotations).
Prior Knowledge Required: Fractions: 12 ,
14 ,
34
Clockwise, counter-clockwise
Reflections. rotations, and slides
Vocabulary: rotation, turn, clockwise, counter-clockwise, reflection, flip, slide, translation
Show your students several pairs of shapes and ask if one shape was moved onto the other by a flip, a turn,
or a slide. Start with non-symmetric shapes, such as an L-shape and a right-angled trapezoid, and continue
to shapes that have a center of symmetry or a line of symmetry, such as parallelograms and kites.
Encourage students to give multiple answers. For example, these two pairs of shapes could be moved by a
slide as well as by a reflection (kite) or a rotation (parallelogram).
Repeat with shapes that have more than one line of symmetry, such as a rectangle and a rhombus. Invite
volunteers to trace the shapes on tracing paper and to check the answers.
As a challenge, present the following pair of figures:
Ask your students if the second shape was obtained by rotating, reflecting, or sliding the first shape. Let your
students try all three moves. Students should see that none of the three moves produces the second shape.
Give your students a cut-out of the shape and ask a volunteer to check if the figures are congruent. (ASK:
Maybe the figures are different? Maybe this is not the same figure?) Ask the students to follow the
movements of the volunteer closely, so that they can spot which transformation or transformations are
performed. Repeat several times if needed, doing one movement at a time, until all the students understand
that the figure has to be both rotated and reflected.
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Invalid placement – this shape is a clockwise turn of the shape
on the left, but it cannot be obtained from the shape above it by
a rotation, slide, or reflection over the common side.
Good placement – this shape is a counter-clockwise turn of the
shape on the left and a reflection of the shape above over the
common side.
Activities:
1. NOTE: This activity was adapted from the Atlantic curriculum. Students will need grid paper
and tracing paper.
Draw a rectangle on a piece of grid paper. Trace the rectangle on the tracing paper. Keep the tracing
paper on the grid paper, choose a vertex of the rectangle, and press a pencil point firmly to this point.
Rotate the shape on the tracing paper around the fixed point a quarter turn clockwise. Mark the vertices
of the rotated shape using a sharp pencil. Remove the tracing paper and join the vertices of the image.
Compare the shapes. Could you move the first rectangle onto the second rectangle by a slide? A
reflection?
Repeat this activity using a half turn, a quarter turn counter-clockwise, and three quarters of a turn
clockwise. Compare the images. What do you notice?
Repeat the activity with a square, a rhombus, and a parallelogram.
2. Ask students to create their own shape and move it from the start box to the finish box in Question 2 on
the worksheet. Students should describe the flips, slides, and turns they used to move their shape.
3. Play another version of the Find-a-Flip Game. Each group of 2–4 students will need two copies of any
three rows of shapes on the BLM “Find-a-Flip Game.” Let your students play the game as in the Activity
of G3-28, but this time each card placed could be a reflection, a quarter turn, or a slide of the shapes on
the adjacent cards.
Students should name the transformation they used.
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G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids.
Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons
Vocabulary: edge, vertex, vertices, face, pyramid, skeleton, base, triangular, rectangular, pentagonal,
hexagonal
Start with a riddle: “You have 6 toothpicks. Make 4 triangles with them. The toothpicks must touch each other
only at the ends.” Let your students try to solve the riddle using toothpicks and modelling clay to hold the
toothpicks together at the vertices of the triangles. The answer, of course, is the triangular pyramid. You
might give your students the hint that the solution is three-dimensional.
Sketch a rectangular pyramid on the board and shade the base. Ask volunteers to mark the edges and
the vertices (count them and make a tally chart). Write the words “base,” “edge,” “vertex,” and “vertices” on
the board.
Give your students modelling clay and toothpicks. Show them how to make a pyramid. Start with a base,
then add an edge to each vertex of the base and join the edges at a point. The students should make
triangular, square, and pentagonal pyramids. Then let them fill in the chart and answer the questions on the
worksheet. After finishing the worksheet, they may check their prediction for the hexagonal pyramid by
making one.
Tell your students that the shapes they have built are called “skeletons” of pyramids. You might write the
following “equation” on the board: Skeleton = Edges + Vertices. As animal skeletons are covered with flesh
and skin, the skeleton of a pyramid can be covered with paper, glass, or other substances and will have
faces. Show a pyramid (with faces) and write the word “faces” on the board as well.
Assessment: Add a row to the chart on the worksheet for a pyramid with a heptagonal (7-sided) base,
and fill it in.
Activity: Build skeletons of pyramids using marshmallows and toothpicks or straws.
Extensions
1. How many faces, edges, and vertices would a pyramid with a 10-sided base have?
2. Ask your students to bring to class pyramids or pictures of pyramids (e.g., Egypt, Mexico, Japan,
entrance to Louvre in Paris) that they can find at home. You can use these pyramids in lesson G3-38:
Edges, Vertices and Faces.
3. PROJECT: Ask your students to learn about a pyramidal structure and give a presentation about it—
what was the structure used for, when and where was it built, why does it have the pyramidal form?
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G3-34 Building Prisms Goal: Students will build skeletons of prisms and describe properties of prisms.
Prior Knowledge Required: Polygons: triangles, quadrilaterals, rectangles, pentagons, hexagons
Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, triangular, rectangular,
pentagonal, hexagonal
Give each student several pattern block triangles and ask them to place the triangles on the table one on top
of the other, aligning the sides. Ask your students what shape the stack has. What does this shape look like
from above? (triangle) What does it look like from the side? (rectangle) Explain that mathematicians call this
3-D shape a triangular prism. Let your students build prisms from other pattern blocks. Point out the faces,
the edges, and the vertices. Explain that the shape that was used to build the prism (triangle, for example) is
called the base. Sketch a prism on the board and shade the bases. Ask volunteers to mark the edges and
the vertices (count them and make a tally chart). Write the words “base,” “faces,” “edges,” “vertex,” and
“vertices” on the board.
Give your students modelling clay and toothpicks. Show them how to make a prism. First make two copies of
the base, and then join each vertex on one base to a vertex on the other base with an edge. Students should
make triangular prisms, pentagonal prisms, and a cube. Let them fill in the chart and answer the questions
on the worksheet. After finishing the worksheet, they may check their prediction for the hexagonal prism by
making one.
ASK: What have you built? (skeletons of prisms) What do we call the “bones” of your skeletons? (edges)
What do the skeletons need to become prisms? (faces)
Assessment: Add a row to the chart on the worksheet for the prism with a heptagonal (7-sided) base,
and fill it in.
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G3-35 Edges, Vertices and Faces Goal: Students will identify vertices, edges (including hidden edges), and faces (side, back, front, top, and
bottom) in the drawings of 3-D shapes.
Prior Knowledge Required: Count edges of polygons
Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape
Remind your students that lots of 3-D shapes in the world around us are either pyramids or prisms. As an
example, you might show them a photo of the pyramids in Egypt.
Hold up a 3-D shape and draw a picture of the shape on the board. Write “3-D shape” next to your drawing.
Ask volunteers to identify the edges, the faces, and the vertices on the shape itself and on the drawing, and
write the terms “edge,” “face,” and “vertex” on the board. Remind your students that the plural of “vertex” is
“vertices.”
Your students will need the skeletons of the cubes they made during the last two lessons. Give each student
2 squares made of paper and 4 squares made of transparent material. Ask them to add
• the paper (non-transparent) squares as the bottom face and the back face;
• the transparent squares as the top, front, and side faces.
It is a good idea to show the students how to add faces on a larger model before they work on their own
models. Add the faces one at a time, emphasizing the position and name of each one.
ASK: Which edges of the cube do you see only through the transparent paper. If the transparent faces were
made of paper, would you see these edges? (no) The edges that would be invisible if all the faces were non-
transparent are called the “hidden edges.” On a two-dimensional drawing of a cube these hidden edges are
marked with dotted lines.
Extension: Ask students to hold or place their cubes in various positions and to look at them from
different angles (on the table, on the floor seen from above, slightly above their heads, and so on.) Ask
students to describe what the faces look like when seen from different angles (they look like a square, a
parallelogram, etc.). The outline of the shape itself can look like a square, a rectangle, a hexagon, a
trapezoid, and a rhombus.
Geometry Teacher’s Guide Workbook 3:2 27 Copyright © 2007, JUMP Math For sample use only – not for sale.
G3-36 Pyramid Nets Goal: Students will build pyramids from nets.
Prior Knowledge Required: Count edges of polygons
Polygons: triangle, square, pentagon, hexagon
Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape
Show your students nets for hexagonal and square pyramids. ASK: What do these drawings have in
common? What is different? Let your students count the faces in the net. What shape do they have? How
many shapes of each kind? If your students do not mention that each net has one face that is different from
the others, point that out and ask what this face is called. (the base) Ask your students what the net for a
triangular pyramid might look like.
Ask students to complete the worksheet for this lesson. They can use the nets on the worksheet to answer
Question 1. Alternatively, you can give them copies of pyramid nets (see BLM section) to cut and fold. Let
your students count the edges, the vertices, and the faces of the shapes they make.
Draw a net for a triangular pyramid on the board and ask your students to count the number of edges on the
net. Is it the same as the number of edges in the 3-D pyramid? Let the students explain. (There are nine
edges on the net and only six edges in the shape. The six edges on the outside of the net are glued in pairs,
so they produce three edges in the 3-D shape. The three internal edges of the net together with the other
three produce six edges in the pyramid.) Repeat with a square pyramid. As a challenge, draw a net of a
pentagonal pyramid and ask your students to tell from the net how many edges are in the 3-D shape.
Extensions:
1. ASK: What happens if you cut one of the side faces in a pyramid net and try to glue it at some other
place? Students might actually cut off one of the triangles and reattach it to another edge. You might
draw the following examples on the board. Will the net still fold into a pyramid if the face is glued in this
position?
no yes no
2. Describe the nets for different shapes. Describe the shape of each face and count the number of faces of
a given shape. Draw a freehand sketch of all the faces that make up a particular 3-D shape.
For example, the parts of a square prism are and the net is .
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G3-37 Prism Nets Goal: Students will build prisms from nets.
Prior Knowledge Required: Count edges of polygons
Polygons: triangle, square, pentagon, hexagon
Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape
Repeat the previous lesson for prism nets.
Activities:
1. After your students have constructed the pyramids and prisms from the nets on worksheets G3-36 and
G3-37, or the BLM section, ask them to sketch the nets from memory. They can test their nets by
making the shapes. You might also ask students to sketch and test nets for pentagonal and hexagonal
pyramids and prisms.
2. Give students square or pentagonal pyramids and ask them to trace the faces on a piece of paper, so
that they create a net. Ask them to cut out the nets they have drawn. Let them cut off faces of the net and
reattach the faces at different places. Will the new net fold into the same pyramid? Which edges are
places where you would want to re-glue the faces and which are not? Repeat this exercise with a prism.
This activity is important because it lets students explore various ways to create nets for the same solid
rather than memorizing a single net shape.
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G3-38 Prisms and Pyramids Goal: Students will compare prisms and pyramids.
Prior Knowledge Required: Count edges of polygons
Polygons: triangle, square, pentagon, hexagon
Prisms
Pyramids
Vocabulary: edge, vertex, vertices, face, skeleton, base, 3-D shape, prism, pyramid, triangular, square,
rectangular, pentagonal, hexagonal
Divide your students into groups. Give each group several 3-D shapes, so that each group has some
rectangular and triangular pyramids, rectangular and triangular prisms, and a cube. Ask your students to
count the faces of the shapes. If some students are having trouble keeping track of the number of faces, they
might mark each face with a chalk dot or a small sticker. Ask your students to count the edges and vertices
on the 3-D shapes as well (They might shade edges with chalk and mark vertices with stickers.) Ask them to
write the results of their count in the table on the worksheet (see Question 2).
Draw a pentagonal pyramid and a triangular prism on the board and let volunteers count the edges, faces,
and vertices of these figures. Ask them to mark the edges and circle the vertices as they count.
Review the difference between a skeleton and a 3-D shape. Show your students a pentagon made of
toothpicks and modelling clay. SAY: I want to create a skeleton of a pyramid. What do I need to add to this
shape to turn it into a skeleton? How many more vertices (balls of clay) and edges (toothpicks) do I need?
Invite a volunteer to add the vertex and the edges. Show your students another pentagon as above and ask
what is needed to turn it into a prism. Ask the students to compare the skeletons. What do they have in
common? What is different? What was different in the way they built the skeletons?
Activities:
1. Show your students an example of a cone and a cylinder. Explain that a cone has one curved surface
and one flat surface, while a cylinder has two flat surfaces and one curved surface. Ask students to find
as many examples of pyramids, prisms, cones, and cylinders in the classroom as they can.
2. Ask your students to bring to class objects that are prisms, pyramids, cubes, cylinders, cones. Create a
collection of such shapes for future use. EXAMPLES: paper cylinders, glasses, boxes (sometimes you
can find boxes or tins in the shape of cylinders or hexagonal prisms), juice or milk cartons (these are
pentagonal prisms).
3. From the Atlantic curriculum: Put a variety of prisms and pyramids in a bag. Have the students, using
only their sense of touch, describe the shape and name it before bringing it out of the bag to check.
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4. Review the notion of congruency. Then give each student a pentagonal pyramid, a triangular prism, and
a cube and ask them to do the following:
a) Place each shape—base down—on a piece of paper and trace the base. (That way you can verify
that each student knows how to find the base.)
b) Write the name of the figure beside the base and indicate whether the figure has one or two bases.
c) If all faces of the figure are congruent, indicate this.
Extension:
Ask your students to add the number of faces and vertices of a cube and subtract the number of edges
(6 faces + 8 vertices – 12 edges). The result is 2. What happens if you do that to another solid? (The result
will again be 2. This fact is known as Euler’s formula and was discovered by the great Swiss mathematician
Leonard Euler in the 18th century.)
Let your students construct shapes from Polydrons. They should make both regular 3-D shapes, like prisms
and pyramids, and irregular “shapes,” like animals and buildings. Count the faces, edges, and vertices.
Check if Euler’s formula holds.
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G3-39 Drawing Pyramids and Prisms Goal: Students will draw skeletons of prisms and pyramids.
Prior Knowledge Required: Count edges of polygons
Polygons: triangle, square, pentagon, hexagon
Prisms
Pyramids
Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape
Show your students how they can draw a picture of a cube on dot paper.
STEP 1 Draw a 2 × 2 square that will become the front face.
STEP 2 Draw another 2 × 2 square so that the centre of the first square is a corner of the second square.
STEP 3 Join the vertices with lines as shown.
STEP 1 STEP 2 STEP 3
Follow the same three steps to draw the second diagram below on the board and ASK: What is the
difference between this cube and the previous one (the 2 × 2 × 2 cube)? How are Steps 1 and 2 performed
differently? Draw the remaining two diagrams below. ASK: What is the difference between the three shapes?
How do we see that in the drawing? How is Step 2 performed differently in each drawing?
Geometry Teacher’s Guide Workbook 3:2 32 Copyright © 2007, JUMP Math For sample use only – not for sale.
ASK: What would you do in Step 2 to draw a very long rectangular prism? Invite a volunteer to draw it. If the
drawing does not look good (i.e., if edges overlap), suggest to students that the corner of the back face not
sit on the diagonal:
rather than
Suggest that your students use the same method to draw other prisms, such as triangular or pentagonal
prisms. In this case students should draw bases in STEPS 1 and 2. EXAMPLES:
ASK: What did you draw, prisms or skeletons? As a challenge, students might change some of the lines in
the finished skeletons to dotted lines to show hidden edges and faces.
Review with your students the fact that a pyramid has one base and a point opposite to it. Show your
students this method for drawing a triangular pyramid: draw a base, choose a point, and join the vertices of
the base to the point. Let your students practise drawing various prisms and pyramids.
Activity: After students have learned to sketch pyramids and prisms, ask them to draw a picture that
includes some buildings that have those shapes. For instance, they might draw a scene from Ancient Egypt
(after consulting picture books or non-fiction books for historical details for their drawing).
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Extension: Show your students how to draw a skeleton of a prism standing on a base rather than on one
of the rectangular faces. To see the distortion of the base in this position, suggest that your students hold a
pattern block horizontally, slightly below eye level. They should see that the polygon in the base appears
shorter and wider from this angle than when you look at it head on. To draw a prism standing on a base,
draw the base wider than it is when viewed head on, then draw the second (top) base directly above the
bottom base and join the vertices to produce the side faces.
EXAMPLE:
Geometry Teacher’s Guide Workbook 3:2 34 Copyright © 2007, JUMP Math For sample use only – not for sale.
G3-40 Properties of Pyramids and Prisms Goal: Students will compare prisms and pyramids systematically.
Prior Knowledge Required: Faces, edges, vertices of 3-D shapes
Polygons: triangle, square, pentagon, hexagon
Prisms
Pyramids
Vocabulary: edge, vertex, vertices, face, skeleton, base, 3-D shape, prism, pyramid, triangular, square,
rectangular, pentagonal, hexagonal, cone, cylinder
Show your students how you can make a cone and a cylinder from a piece of paper. ASK: Where have you
seen this shape before? (ice cream cone, clown hat, etc) Does a cone remind you of some other geometric
shape that we have studied? (a pyramid) If you have an octagonal pyramid (or a pyramid with more than 8
sides in the base), show it to the students and ASK: What do these shapes have in common? Repeat with a
cylinder and an octagonal prism.
Give your students pentagonal pyramids and prisms or have them make these shapes using the nets from
the BLM. Ask your students to draw the following chart in their notebooks and to fill it in, using the shapes.
Property Pentagonal
Prism
Pentagonal
Pyramid Same? Different?
Number of faces
Shape of base
Number of bases
Number of faces
that are not bases
Shape of faces
that are not bases
Number of
edges
Number of
vertices
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When students are finished, invite a volunteer to use the information in the table to write a paragraph
comparing the two shapes. Repeat with another pair of shapes, such as a rectangular prism and a
pentagonal prism.
Draw several shapes on the board and ask your students which 3-D shape they make.
SAMPLES:
This would be a good time for students to try the Activity.
Assessment:
1. a) Make a property chart for rectangular and triangular prisms.
b) Use the chart to write a paragraph comparing rectangular and triangular prisms.
2. Who am I?
a) I have only rectangular faces.
b) I have 8 faces, and 6 of them are rectangles.
c) I am a prism with 9 edges and 5 faces.
d) I have one circular base.
Activity:
A Game in Pairs Your students might need a table of the number of faces, edges, and vertices in prisms and pyramids (see G3-33 and G3-34, or G3-38). One player gives a description of a shape; the other has to name the shape.
ADVANCED: The player gives only two of the three possible pieces of information: number of edges,
number of vertices, number of faces.
EXAMPLES:
12 edges and 6 faces (a rectangular prism)
12 edges and 7 faces (a hexagonal pyramid)
Extension: Have students complete the BLM “Word Search Puzzle (3-D Shapes).”
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G3-41 Sorting 3-D Shapes Goal: Students will sort prisms and pyramids according to their properties.
Prior Knowledge Required: Venn diagrams
Prisms and pyramids
Vocabulary: edge, vertex, vertices, face, skeleton, base, 3-D shape, prism, pyramid, triangular, square,
rectangular, pentagonal, hexagonal
Give individual students or small groups a deck of shape cards and a deck of property cards. These cards
are in the BLM section of the guide. (If you have enough 3-D shapes in your classroom, students can use
actual shapes instead of the cards.) Let them play the following games:
3-D Shape Sorting Game
Each student flips over a property card and then sorts the shapes into two piles according to whether a
shape on a card has the property or not. Students get a point for each card that is in the correct pile. (If you
prefer, you could choose a single property for the class and have everyone sort the shapes using that
property.)
Once students have mastered this sorting game they can play the next game.
3-D Venn Diagram Game
Give each student a copy of the BLM “Venn Diagram” (or have students create their own Venn diagram on a
sheet of construction paper or Bristol board). Ask students to choose two property cards and place one
beside each circle of the Venn diagram. Students should then sort their shape cards into the Venn diagram.
Give 1 point for each shape that is placed in the correct region of the Venn diagram.
Assessment:
Sort the shapes below into a Venn diagram according to these properties:
1. One or more rectangular faces
2. One or more triangular faces
A B C D E F
Extension: Draw a Venn diagram to sort the shapes on the worksheet according to these properties:
1. Pyramid
2. One or more triangular faces
What do you notice about your Venn diagram? Explain why part of one of the circles is empty.
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G3-42 Classifying Shapes and Making Patterns Goal: Students will review the entire Geometry unit and make patterns with transformations.
Prior Knowledge Required: Venn diagrams
Polygons
Slides, reflections, and rotations
Attributes
Extending patterns
Vocabulary: edge, vertex, vertices, triangle, square, rectangle, pentagon, hexagon, slide, turn, flip,
translation, rotation, reflection
This is a review worksheet.
Activity: Students will need a spinner as shown and 8 cards (4 copies of one shape and 4 copies
of the flipped shape) from the BLM “Find-a-Flip Game.” Player 1 spins the spinner so that Player 2 does
not see the result. Player 1 builds a pattern with the cards using the transformation shown by the spinner
and shows the pattern to his or her partner. Player 2 has to determine which transformation was used to
make the pattern.
Extension:
Use the shapes from Question 1 on worksheet G3-42 for this problem:
Each chart identifies shapes with given attributes. Can you figure out what the two attributes are?
a) b)
Shapes with Attribute 1 Shapes with Attribute 2
B, D
A, D
Shapes with Attribute 1
Shapes with Attribute 2
F, C
F, E
Flip Turn
• Slide
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G3-43 Geometry in the World Goal: Students will see applications of geometry in real life.
Prior Knowledge Required: Slides, reflections, and rotations
Symmetry
3-D shapes
Vocabulary: edge, vertex, vertices, face, triangle, square, rectangle, pentagon, hexagon, octagon, slide,
turn, flip, translation, rotation, reflection, prism, pyramid
The worksheet G3-43: Geometry in the World is an extension and review worksheet. Students can also
complete one or more of these cross-curricular projects:
Symmetry
1. Flags and Coats of Arms of Canadian provinces/cities—Which ones have lines of symmetry? Which
ones have more than one line of symmetry? (None!)
2. Flags of the world—Make a list of countries with flags that have two lines of symmetry.
3. Coats of Arms of Soccer/Baseball/Hockey clubs—Which ones have lines of symmetry? Which ones have
more than one line of symmetry?
4. Cultural Diversity: Alphabets—Find letters in a non-Latin alphabet (e.g., Greek, Arabic, Hindi, Hebrew,
Korean) that have lines of symmetry. Are there any letters or symbols that have more than one line of
symmetry?
5. Snowflakes—Make several designs of snowflakes. How many lines of symmetry do your
snowflakes have?
Geometry in Everyday Life
1. Bee hives and hexagons—Research why bees build hexagonal shapes in the hive. Why don’t they use
rectangular or triangular shapes?
2. Geometrical floor patterns—Use reflections and rotations to create a pattern design.
3. Traffic signs—Which geometric shapes are used in various traffic signs?
4. A robot is used to draw letters. The robot understands the following commands:
• Draw a line from the current position to the point ___ units up/down and ___ units right/left.
• Move ___ units up/down and ___ units right/left without drawing a line.
Give the robot directions to draw various letters of the alphabet. Can you make the robot
write your name?
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Geometry and History
1. Ancient Egypt—Which geometric shapes were used in ancient Egyptian buildings?
2. Ancient Egypt: The Pyramid of Khufu—How many right angles can you find in this pyramid? Find more
interesting facts about this or other Egyptian pyramids.
3. Ancient Maya—Which geometric shapes were used in ancient Mayan temples?
Geometry Teacher’s Guide Workbook 3:2 40 Copyright © 2007, JUMP Math For sample use only – not for sale.
G3-44 Problems and Puzzles
The worksheet G3-44: Problems and Puzzles is a review worksheet and may be used for extra practice.
Workbook 3 - Geometry, Part 2 1BLACKLINE MASTERS
3-D Shape Sorting Game _________________________________________________2
Dot Paper _____________________________________________________________6
Find-a-Flip Game _______________________________________________________7
Grid Paper _____________________________________________________________8
Map of Saskatchewan ___________________________________________________9
Nets for 3-D Shapes ____________________________________________________10
Pattern Blocks _________________________________________________________13
Pentamino Pieces ______________________________________________________14
Venn Diagram _________________________________________________________15
Word Search Puzzle (3-D Shapes) _________________________________________16
G3 Part 2: BLM List
3-D Shape Sorting Game
2 TEACHER’S GUIDE Copyright © 2007, JUMP Math
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3-D Shape Sorting Game (continued)
Workbook 3 - Geometry, Part 2 3BLACKLINE MASTERS
Four or more
triangular-
shaped faces
Square-
shaped base
Fewer than
six faces
More than
four faces
Triangular-
shaped base
Two or more
square-
shaped faces
3-D Shape Sorting Game (continued)
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Ten or more
edges
Four or more
vertices
Pyramids
Six or fewer
vertices
Exactly
twelve edges
Prisms
3-D Shape Sorting Game (continued)
Workbook 3 - Geometry, Part 2 5BLACKLINE MASTERS
Dot Paper
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Find-a-Flip Game
Workbook 3 - Geometry, Part 2 7BLACKLINE MASTERS
Grid Paper (1 cm)
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Map of Saskatchewan
Workbook 3 - Geometry, Part 2 9BLACKLINE MASTERS
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SquarePyramid
Nets for 3-D Shapes
Tri
an
gu
lar
Pyra
mid
Workbook 3 - Geometry, Part 2 11BLACKLINE MASTERS
Nets for 3-D Shapes (continued)
Cube
Tri
angula
rP
rism
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Nets for 3-D Shapes (continued)
Pentagonal Pyramid
Pentagonal
Prism
Pattern Blocks
Triangles
Squares
Rhombuses
Trapezoids
Hexagons
Workbook 3 - Geometry, Part 2 13BLACKLINE MASTERS
Pentamino Pieces
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Venn Diagram
Workbook 3 - Geometry, Part 2 15BLACKLINE MASTERS
Word Search Puzzle (3-D Shapes)
WORDS TO SEARCH:
base pyramid
edge rectangular
face skeleton
hexagonal triangular
net vertex
pentagonal vertices
prism
r s k e l e t o n o
a e e s a b g e m a
l i v l n o t d r r
u p v a o p r a e n
g e e n g a t r p g
n v r o a l d n y i
a e t g t m s i r p
t r i a n g u l a r
c t c x e c a f m g
e e e e p e a m i u
r x s h l c n p d l
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