jumpmath• Introduction to the

JUMP Workbooks and

Teaching Method

• Activities and

Extension Exercises

• Mental Math Exercises

• Blackline Masters

Teacher’s Guide: Workbook 7

JUMP Covers TE Grades 7 & 8 2001 1 10/5/07 4:04:49 PM

TABLE OF CONTENTS

Introduction......................................................................................................................1

Hints for Helping Students Who Have Fallen Behind.................................................... 16

Hints for Helping Students Who Finish Work Early ..................................................... 19

Hints for One-on-One Tutors ........................................................................................ 20

JUMP Math Instructional Approaches ........................................................................... 22

JUMP and the Process Standards for Mathematics ......................................................... 30

Mental Math Skills, Strategies and Exercises ................................................................... 32

Manual Notes, Activities and Extensions (Part 1) ........................................................... 52

Manual Notes, Activities and Extensions (Part 2) ......................................................... 158

Extra Worksheets and Blackline Masters ...................................................................... 223

Glossary ........................................................................................................................ 266

Copyright © 2006, JUMP Math

JUMP Math: Teacher's Guide for Workbook 7 – 10-digit ISBN 1-897120-28-1, 13-digit ISBN 978-7-897120-28-6 – Printed in Canada.

All rights reserved. No part of this publication may be reproduced in any way without the written permission of the copyright holder (except in accordance with the provisions of the Copyright Act). Use of these materials for profit is strictly prohibited.

page 1

Teacher’s Guide

Grade 7

INTRODUCTION TO THE JUMP WORKBOOKS – by John Mighton

Based on my work with hundreds of elementary students, spanning twenty years, I am convinced that all children, except possibly those who are so severely disabled that they would not be enrolled in a regular public school, can be led to think mathematically. (I say possibly because I haven’t worked with children who are outside the regular school system: it wouldn’t surprise me if these children were capable of more than people expect.) Even if I am wrong, the results of JUMP suggest that it is worth suspending judgment in individual cases. A teacher who expects a student to fail is almost certain to produce a failure. The method of teaching outlined in this book (or any method, for that matter) is more likely to succeed if it is applied with patience and an open mind.

If you are a teacher and you believe that some of the students in your class are not capable of leaning math, I recommend that you read The Myth of Ability: Nurturing Mathematical Talent in Every Child, and consult the JUMP website (at www.jumpmath.org) for testimonials from teachers who have tried the program and for a report on current research on the program.

You are more likely to help all your students if you teach with the following principles in mind:

1) If a student doesn’t understand your explanation, assume there is something lacking in your explanation, not in your student.

When a teacher leaves students behind in math, it is often because they have not looked closely enough at the way they teach. I often make mistakes in my lessons: sometimes I will go too fast for a student or skip steps inadvertently. I don’t consider myself a natural teacher. I know many teachers who are more charismatic or faster on their feet than I am. But I have had enormous success with students who were thought to be unteachable because if I happen to leave a student behind I always ask myself: What did I do wrong in that lesson? (And I usually find that my mistake lay in neglecting one of the principles listed below.)

I am aware that teachers work under difficult conditions, with over-sized classes and a growing number of responsibilities outside the classroom. None of the suggestions in this guide are intended as criticisms of teachers, who, in my opinion, are engaged in heroic work. I developed JUMP because I saw so many teachers struggling to teach math in large and diverse classrooms, with training and materials that were not designed to take account of the difficult conditions in those classrooms. My hope is that JUMP will make the jobs of some teachers easier and more enjoyable.

2) In mathematics, it is always possible to make a step easier.

A hundred years ago, researchers in logic discovered that virtually all of the concepts used by working mathematicians could be reduced to one of two extremely basic operations, namely, the operation of counting or the operation of grouping objects into sets. Most people are able to perform both of these operations before they enter kindergarten. It is surprising, therefore, that schools have managed to make mathematics a mystery to so many students.

A tutor once told me that one of her students, a girl in Grade 4, had refused to let her teach her how to divide. The girl said that the concept of division was much too hard for her and she would never consent to learn it. I suggested the tutor teach division as a kind of counting game. In the next lesson, without telling the girl she was about to learn how to divide, the tutor wrote in succession the numbers 15 and 5. Then she asked the child to count on her fingers by multiples of the second number, until she’d reached the first. After the child had repeated this operation with several other pairs of numbers, the tutor asked her to write down, in each case, the number of fingers she had raised when she stopped counting. For instance,

15 5 3

As soon as the student could find the answer to any such question quickly, the tutor wrote, in each example, a division sign between the first and second number, and an equal sign between the second and third.

15 ÷ 5 = 3

page 2

Teacher’s Guide

Grade 7

The student was surprised to find she had learned to divide in 10 minutes. (Of course, the tutor later explained to the student that 15 divided by five is three because you can add 5 three times to get 15: that’s what you see when you count on your fingers.)

In the exercises in the JUMP Workbook we have made an effort to break concepts and skills into steps that students will find easy to master. Fitting the full curriculum into 300 pages was not an easy task. Even in this new edition, where we have improved the layout, a few pages are more cramped than we would have liked and some pages do not provide enough practice or preparation. The worksheets are intended as models for teachers to improve upon: we hope you will take responsibility for providing students with warm-up questions and bonus questions (see below for a discussion of how to create these questions), and for filling in any gaps our materials wherever you find them. We have made a serious effort to introduce skills and concepts in small steps and in a coherent order, so a committed teacher should have no trouble seeing where they need to create extra questions for practice or where they need to fill in a missing step in the development of an idea.

3) With a weaker student, the second piece of information almost always drives out the first.

When a teacher introduces several pieces of information at the same time, students will often, in trying to comprehend the final item, 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). With weaker students, it is always more efficient to introduce one piece of information at a time.

I 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.

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. 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 ¾, 3 1/4, 4 1/4, 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 practice each step separately, he might never have learned the concept

As your weaker students learn to concentrate and approach their work with real excitement (which generally happens after several months if the early JUMP units are taught properly), you can begin to skip steps when teaching new material, or even challenge your students to figure out the steps themselves. But if students ever begin to struggle with this approach, it is best to go back to teaching in small steps.

4) Before you assign work, verify that all of your students have the skills they need to complete the work.

In our school system it is assumed that some students will always be left behind in mathematics. If a teacher is careful to break skills and concepts into steps that every student can understand, this needn’t happen. (JUMP has demonstrated this in dozens of classrooms.)

Before you assign a question from one of the JUMP workbooks you should verify that all of your students are prepared to answer the question without your help (or with minimal help). On most worksheets, only one or two new concepts or skills are introduced, so you should find it easy to verify that all of your students can answer the question. The worksheets are intended as final tests that you can give when you are certain all of your students understand the material.

Always give a short diagnostic quiz before you allow students to work on a worksheet. In general, a quiz should consist of four or five questions similar to the ones on the worksheet. Quizzes needn’t count for marks but students should complete quizzes by themselves, without talking to their neighbours (otherwise you won’t be able to verify if they know how to do the work independently). The quizzes will help you identify which

page 3

Teacher’s Guide

Grade 7

students need an extra review before you move on. If any of your students finish a quiz early, assign extra questions similar to the ones on the quiz.

If tutors are assisting in your lesson, have them walk around the class and mark the quizzes immediately. On days when you don’t have tutors, check the work of students who might need extra help first, then take up the answers to the quiz at the board with the entire class (or use peer tutors to help with the marking).

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 this guide. Write the bonus questions on the board (or have extra worksheets prepared and ask students to answer the questions in their notebooks. While students are working independently on the bonus questions, you can spend extra time with anyone who needs help.

During the first few months of the JUMP program, it is absolutely essential that you set aside five to ten minutes every few days to work with small groups of students who might need extra coaching before a lesson or an extra review after a lesson. As students catch up and become more confident about their abilities, they will need less of this extra help.

5) Raise the bar incrementally.

Any successes I have had with weaker students are almost entirely due to a technique I use which is, as a teacher once said about the JUMP method, “not exactly rocket science.” When a student has mastered a skill or concept, I simply raise the bar slightly by challenging them to answer a question that is only incrementally more difficult or complex than the questions I had previously assigned. I always make sure, when the student succeeds in meeting my challenge, that they know I am impressed. Sometimes I will even pretend I’m about to faint (students always laugh at this) or I will say “You got that question but you’ll never get the next one.” Students become very excited when they succeed in meeting a series of graduated challenges. And their excitement allows them to focus their attention enough to make the leaps I have described in The Myth of Ability. As I am not a psychologist I can’t say exactly why the method of teaching used in JUMP has such a remarkable effect on children who have trouble learning. But I am certain that the thrill of success and the intense mental effort required to remember complex rules, and to carry out long chains of computation and inference, helps open new pathways in their brains.

In designing the JUMP workbooks, I have made an effort to introduce only one or two skills per page, so you should find it easy to create bonus questions: just change the numbers in an existing question or add an extra element to a problem on a worksheet. For instance, if you have just taught students how to add a pair of three-digit numbers, you might ask students who finish early to add a pair of four- or five-digit numbers. This extra work is the key to the JUMP program. If you become excited when you assign more challenging questions, you will find that even students who previously had trouble focusing will race to finish their work so they can answer a bonus question too.

The bonus questions you create should generally be simple extensions of the material on the worksheet: if you create questions that are too hard or require too much background, you may have to help students who should be working independently. At times though, you will want to assign more challenging questions: that is why we have provided extension questions in this edition. Three years of in-class implementations of JUMP have shown that a teacher can always keep faster students engaged with extra work. But if, instead of assigning bonus questions, you allow some students to work ahead of others in the workbooks, you will never be able to build the momentum and excitement that comes when an entire class experiences success together.

6) Repetition and practice are essential.

Even mathematicians need constant practice to consolidate and remember skills and concepts. I discuss this point in more detail below, in the section entitled “Four Leading Ideas of Contemporary Education.”

page 4

Teacher’s Guide

Grade 7

7) Praise is essential.

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 and, because students having difficulty can get extra help from our tutors, you’ll find that you won’t be giving false encouragement. If you proceed using these steps, your students should be doing well on all their exercises. (This is one of the reasons kids love the program so much: for many, it’s a thrill to be doing well at math.)

In this vein, we hope that you won’t use labels such as “mild intellectual deficit” or “slow learner” as reasons for expecting a poor performance in math from particular children. We haven’t observed a student yet – even among scores of remedial students – who couldn’t learn math. When it is taught in steps, math is actually the subject in which children with attention deficits and learning disabilities can most easily succeed, and thereby develop the confidence and cognitive abilities they need to do well in other subjects. Rather than being the hardest subject, math can be the engine of learning for delayed students. This is one of JUMP’s cornerstone beliefs. If you disagree with this tenet, please reconsider your decision to use JUMP in your classroom. Our program will only be fully effective if you embrace the philosophy.

8) Make math a priority

I’ve occasionally met teachers who believe that because they survived school without knowing much math or without ever developing a love of the subject, they needn’t devote too much effort to teaching math in their own classes. There are two reasons why this attitude is harmful to students.

(i) It is easier to turn a good student into a bad student in mathematics than in any other subject: mathematical knowledge is cumulative; when students miss a step or fall behind they are often left behind permanently. Students who fall behind in mathematics tend to suffer throughout their academic careers and end up being cut off from many jobs and opportunities.

(ii) JUMP has shown that mathematics is a subject where students who have reading delays, attention deficits and other learning difficulties can experience immediate success (and the enthusiasm, confidence and sense of focus children gain from this success can quickly spill over into other subjects): In neglecting mathematics, a teacher neglects a tool that has the potential to transform the lives of weaker students.

JUMP and Current Philosophies of Education

Perhaps the most exciting development in JUMP this year has been the growth of our partnerships with dedicated educators. A number of teachers and administrators in Canada and the United States have demonstrated that schools can easily implement JUMP in classrooms or in after-school programs, without stretching their resources. (See our website for information about our partnerships.)

While JUMP has found many advocates among teachers, principals and parents, the program has met with skepticism or outright resistance from some educational theorists and administrators at boards and ministries of education. Some educators, who are not aware of the full scope of the program, seem to think that JUMP is a throw back to the kind of rote learning of mathematics that schools have tried to move away from. The Myth of Ability may have reinforced this opinion, as it advocates that students be led in small, rigorously laid out steps in the early part of a math program. In The Myth of Ability, I focused almost exclusively on the more mechanical side of JUMP program because I believe that teachers must be trained to break skills and concepts into the most basic atoms of perception and understanding. But I was also careful to stress that students are expected to work more independently and to discover and explain concepts on their own as they progress through the JUMP program. (This new edition of our grade-specific workbooks shows the scope of the program more fully: you will find many new activities, problem-based lessons and extension exercises in the Teacher’s Guide and in the Workbooks. Our enriched units, which are still in development, will complete the program by introducing young children to deeper mathematical investigations.)

I believe the educational debates that have raged for so many years in our schools have been so divisive and unfruitful because the basic terms of the debate have never been properly established. In particular, the word “conceptual” has

page 5

Teacher’s Guide

Grade 7

come to be defined too narrowly in math education, in a way that does not fully reflect the actual practice of mathematics or the way children (particularly children with learning disabilities) acquire concepts.

In my opinion, every side in the “math wars” has something of value to contribute to the debate. I have learned a great deal from educational philosophies that are different from my own, and I have tried to incorporate a variety of styles and approaches in new JUMP materials. But the various parties in the wars will never have a fruitful debate about best practices in math education, until educators examine the nature of mathematical concepts more carefully (after all, who wouldn’t want to teach math “conceptually?”).

I will examine four leading ideas about what it means to teach mathematics “conceptually.” These ideas have been adopted by many educators and school boards in North America. Each of the ideas is based on a reasonable insight about the way children learn. The ideas only become a matter of concern when they are held up as the only way to teach mathematics. When educators try to block JUMP in schools or claim that the program represents a return to rote learning, it is usually because they are not aware of the full scope of the program or because they have accepted one of these ideas uncritically in an extreme form.

NOTE: Since I wrote the first draft of this introduction a year ago, JUMP has begun to receive much more support from school boards and Ministries across Canada.

Four Leading Ideas of Contemporary Education

The First Idea: A teacher who neglects to use concrete materials (such as pattern-blocks or fraction-strips) whenever they introduce a mathematical idea is not teaching “conceptually.”

Pattern blocks, base ten materials and fraction strips, as well as three dimensional shapes such as prisms and pyramids, are very useful tools for teaching mathematics. These materials, or diagrams representing these materials, are used extensively in the JUMP workbooks. There are topics in elementary mathematics, such as the classification of three-dimensional solids, which are hard to teach without a physical model. But many topics in elementary mathematics can also be taught more abstractly, even at the same time as they are introduced with concrete materials. Young children, as early as Grade 1 or 2, can be taught to appreciate math as an algebraic or symbolic game that they can play sitting at their desks with no other tools than a pencil and a piece of paper. Dozens of JUMP implementations have shown that children enjoy and benefit from playing with mathematical rules and operations, even when those rules and operations are taught with scarcely any reference to a physical model (see The Myth of Ability and the JUMP website for details of research on this topic).

The idea that mathematical concepts must always be introduced with blocks and rods and pies, and that there is never any point in allowing children to play with mathematical symbols without having spent years playing with the things those symbols represent, is widespread in our schools. The idea is based on a serious misunderstanding of the nature of mathematical concepts and of the way mathematics connects to the world. The idea is also based on dubious assumptions about the way children acquire mathematical concepts.

Mathematics was invented for practical purposes: for counting sheep and measuring fields. In the modern world, through its applications in science and industry, mathematics is the source of virtually all of our material comforts. But mathematics became effective as a material tool primarily by becoming an abstract language in its own right.

Over the centuries, mathematicians have more often made discoveries by seeking to understand the logic or internal structure of that language, than by following their intuitions about the physical world. The nineteenth century mathematicians who discovered the laws of curved space did not intend to launch a revolution in the physical sciences, as happened when Einstein applied their ideas in the twentieth century. They simply wanted to make the axioms of geometry a little more concise. Richard Feynman, one of the great physicists of last century, once said: “I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.”

Einstein’s famous equation E = mc2 is clearly an abstract or symbolic representation of a physical law. But the floor plan of a house is also a symbolic representation: the floor plan is a set of lines drawn on flat paper that bears little

page 6

Teacher’s Guide

Grade 7

resemblance to the three dimensional house it represents. Similarly, the calculation a carpenter makes to determine how many nails are needed to build the house is entirely different from the act of counting out the nails. This is something we have lost sight of in our schools: mathematics, even in its most practical applications, in carpentry or finance or computer science, is fundamentally a game of inventing and manipulating symbols. And mathematical symbols, and the operations by which they are combined, are very different from the things they represent.

To understand this point, it helps to consider the operation of adding fractions. The operation is based on two rules:

i) If the denominators of a pair of fractions are the same, you add the fractions by adding the numerators (keeping the denominator the same).

ii) To make the denominators of a pair of fractions the same, you may multiply or divide the denominator of either fraction by any number as long as you do the same thing to the numerator.

These two rules have various physical representations: you can show children how the rules work by cutting up pieces of pie or by lining up fraction strips. But you can also teach children to add fractions without ever showing them a physical model of a fraction.

Of course I don’t advocate that children be taught mathematics without concrete materials. The JUMP workbooks are filled with exercises that show students how mathematical rules are embodied in physical models. But it is important to notice that the rules listed above don’t make any mention of pies or blocks or fraction strips. Everything you need to know to perform the operation of adding fractions is given in the rules. And the rules are simple enough that virtually any eight-year-old can learn to apply them flawlessly in a matter of weeks (this has been demonstrated conclusively in dozens of JUMP pilots). By focusing exclusively on models we have lost sight of how utterly easy it is for children to learn the individual steps of an operation (such as the addition of fractions) when those steps are isolated and taught one at a time.

An employee of a board of education once told me that research has proven that children should not be taught any operations with fractions until Grade 7. I’m not sure how research proved this, but I suspect that the research was based on fairly narrow assumptions about what children are capable of learning and on a limited understanding of the nature of mathematical concepts.

Contrary to current “research,” I believe that we should introduce kids to the symbolic game of mathematics at an early age. I can think of six reasons for doing so, which I give below.

(NOTE: The Fractions Unit is the only unit I have developed to date that is designed solely to introduce kids to math as a symbolic or algebraic game. The JUMP Workbooks (3 to 8) were developed for other purposes: they cover the regular elementary curriculum, so they introduce mathematical concepts in a fairly standard way, usually with concrete materials, although some sections provide enriched exercises or extra practice in following mathematical rules and operations. Eventually JUMP will develop enriched units that will allow kids explore the symbolic side of math in more depth.)

1) We underestimate children by assuming that they will only enjoy learning concepts that have obvious physical models or applications. While I wouldn’t discourage a teacher from serving pieces of pie or pizza to their class to illustrate a point about fractions, this is not the only way to get kids interested in math. Children will happily play a game with numbers or mathematical symbols, even if it has no obvious connection to the everyday world, as long as the game presents a series of interesting challenges, has clear rules and outcomes and if the person playing the game has a good chance of winning. Children are born to solve puzzles: in my experience, they are completely happy at school if they are allowed to exercise their minds and to show off to a caring adult. What children hate most is failure. They generally find mathematical rules and operations boring only because those things are often poorly taught, without passion, in a manner that produces very few winners.

2) Children acquire new languages more readily than adults. Mathematics itself is a kind of language, with its own rules and grammatical structures. Why not let them children become fluent in the language of mathematics at an

page 7

Teacher’s Guide

Grade 7

age when they are most ready to learn it? (Several JUMP instructors have noticed that Grade 1 and 2 students often learn the JUMP Fractions Unit as quickly as or more quickly than children who are much older).

3) Many fundamental mathematical concepts are not embodied in any concrete model. As early as Grade 7 students encounter concepts and operations that have no physical explanation.

Operations with negative numbers were first introduced in mathematics as a means for solving equations. For centuries, mathematicians multiplied negative numbers without knowing how to make sense of the operation. Leonard Euler, the greatest mathematician of the seventeen hundreds, said that negative multiplication shouldn’t be allowed because it was senseless.

It’s easy to see why a negative number times a positive number is a negative number. For instance, negative three times positive two is negative six: if you have a debt of three dollars and you double your debt, you end up with a debt of six dollars. But why should a negative times a negative equal positive? A math consultant once told me she explains negative multiplication as follows: “When you multiply negative two by negative three, you subtract a debt of 2 three times, which gives a gain of positive six”. But a clever student might say “You taught me that multiplication is a short form for repeated addition. Why now, when both factors are negative, does multiplication suddenly become a short form for repeated subtraction?” Telling a student to think about negative multiplication as repeated subtraction is not a bad way of helping them remember the rule, but unfortunately this approach does not explain the rule. There is no physical model we can point to to explain why the meaning of multiplication should change from repeated addition to repeated subtraction (or that guarantees that this way of thinking about multiplication is consistent with the rest of mathematics).

If the rule “A negative times a negative is a positive” has no physical basis, then why should we accept it as a rule of mathematics? And how is it that a rule that is not determined by any aspect of the physical world has proven to be so useful in physics and in other sciences? Mathematicians only found the answer to the first question in the eighteen hundreds. The second question remains a mystery.

To understand why a negative times a negative is a positive, it helps to look at the axioms of mathematics that govern the addition and multiplication of positive numbers. If you add the numbers three and five and then multiply the sum by two, the result is sixteen. But you get exactly the same result if you multiply three by two and five by two and then add the products:

(3 + 5) × 2 = 3 × 2 + 5 × 2

Sums and products of positive numbers always satisfy this simple equivalence (which is called the law of distribution). In the eighteen hundreds, mathematicians realized that if the law of distribution is to hold for negative numbers, then a negative times a negative must be a defined as a positive: otherwise the law produces nonsense (i.e. if you define a negative times a negative as a negative you can easily prove, using the distributive law, that the sum of any two negative numbers is zero). This is an example of what I meant when I said that mathematicians are more often led by the internal logic of mathematics than by physical intuition. Because negative numbers had proven to be so useful for solving problems, mathematicians decided to extend the distributive law (that holds for positive numbers) to negative numbers. But then they were forced to define negative multiplication in a particular way. The rule for negative multiplication has found countless applications in the physical world, even though there is no physical reason why it should work! This is one of the great mysteries of mathematics: how do rules that have no straightforward connection to the world (and that are arrived at by following the internal logic of mathematics) end up having such unreasonable effectiveness?

I always thought that I was a bit of an idiot in high school for not understanding negative multiplication (and even worse, the multiplication of complex or imaginary numbers). My teachers often implied that the rules for these operations had models or explanations, but I was never able to understand those explanations. If my teachers had told me that math is a powerful symbolic language in its own right, and that the world of our everyday experience is described by a tiny fragment of that language (as I later learned in university) I believe I would have found math somewhat easier and more interesting. The results of JUMP we have shown that young children have no fear of the symbolic side of mathematics: they are much more open minded, and more

page 8

Teacher’s Guide

Grade 7

fascinated by patterns and puzzles then most adults. If children were taught to excel at the symbolic game of math at an earlier age they wouldn’t encounter the problems that most students face in high school.

4) For some time now educators have advocated that we move away from the rote learning of rules and operations. This is a very positive development in education. Students should understand why rules work and how they are connected to the world. But unfortunately, in arguing against rote learning, some educators have set up a false dichotomy between mathematical rules and operations on the one hand and concepts and models on the other.

Not all concepts in mathematics are concrete (as the case of negative multiplication illustrates). And if a rule is taught without reference to a model, it is not necessarily taught in a rote way. Whenever a child sees a pattern in a rule, or applies a rule to a case they have never encountered, they are doing math conceptually, even if they haven’t consulted a model in their work (and even if they haven’t discovered the rule themselves). The fact that children should also be taught to see the connection between the rule and the model doesn’t take away from my point.

I read in an educational journal recently that when a child uses a rule to find an answer to a problem, the child isn’t thinking. I was surprised to learn this, as most of the work I did as a graduate student consisted of following rules. Many of the rules I learned in graduate school were so deep I doubt I could have discovered their applications on my own (especially not in the five and a half years it took me to get my masters and doctorate). But I was always proud of myself whenever I managed to use one of those rules to solve a problem that wasn’t exactly like the examples my professors had worked out on the board. Every time I used a rule to solve a problem I hadn’t seen before, I had the distinct impression that I was thinking. I find it hard to believe now that this was all an illusion!

Many teachers and educators have trouble recognizing that there is thought involved in following rules, because they are convinced that students must discover mathematical concepts in order to understand them (I will discuss this point below ) and because they believe that “conceptual” always means “having a model” or “being taught from a model.” I recently showed an influential educator the results of a JUMP pilot that I was very proud of: after a month of instruction, an entire Grade 3 class that I taught (including several slow learners) had scored over 90% on a Grade 7 test on operations with fractions. On seeing the tests the educator said they made her blood boil. I explained that many children had shown remarkable improvements in confidence and concentration after completing the unit. I also pointed out that the regular JUMP workbooks also teach the connection between the operation and the model: the Fractions Unit is just a brief excursion into the symbolic world of math. But I don’t think she heard anything I said. I expect she was so upset because I wasn’t supposed to be teaching fractions without models in Grade 3. This episode (and many other recent encounters) showed me the extent to which educators have come to associate mathematical concepts with concrete materials.

I recently came across the following question on a Grade 7 entrance exam for a school for gifted children:

If a ◊ b = a × b + 3, what does 4 ◊ 5 equal?

Most educators would probably say that this is a very good “conceptual” question for Grade 7 students. To solve the problem a student must see which symbols change and which ones remain the same on either side of the equal sign in the left hand equation. The letters a and b appear on both sides of the equal sign, but on the left hand side they are multiplied (then added to the number 3): once a student notices this they can see that the solution to the problem is 4 × 5 + 3 = 23. The ability to see patterns of this sort in an equation and to see what changes and what stays the same on either side of an equal sign are essential skills in algebra.

When I teach the JUMP Fractions Unit, I start by showing students how to add a pairs of fractions with the same denominators: you add the numerators of the fractions while keeping the denominator the same. But then, without further explanation, I ask students how they would add three fractions with the same denominator: in other words, I ask:

If 14 +

14 =

24 , what does

14 +

14 +

14 equal?

page 9

Teacher’s Guide

Grade 7

The logical structure of this question is very similar to the question from the enriched entrance exam: to find the answer, students have to notice that the number 4 remains unchanged in the denominators of the fractions and the numbers in the numerators are combined by addition. In my opinion, the question is “conceptual” in much that same that the way the question on the entrance exam is conceptual. Yet the educator whose blood boiled when she saw the Fractions Unit undoubtedly assumed that, because I hadn’t used manipulatives in teaching the unit, I was teaching in a rote way.

The exercises in the JUMP Fractions Unit contain a good deal of subtle conceptual work of the sort found in the example above: in virtually every question, students are required to see what changes and what stays the same in an equation, to recognize and generalize patterns, to follow chains of inference and to extend rules to new cases (for many students, it is the first time they have ever been motivated to direct their attention to these sorts of things at school). But because the questions in the Fractions Unit are not generally formulated in terms of pie diagrams and fraction strips, many educators have had trouble seeing any value in the Fractions Unit. (And no matter how often I point out that the regular JUMP worksheets contain lots of exercises with pattern blocks, pie diagrams and fraction strips, educators who believe that kids shouldn’t be taught any operations with fractions never seem to hear me.)

5) It can take a great deal of time (relative to the amount of learning that takes place) to conduct a lesson with manipulatives. While it is important that students receive some lessons with manipulatives, students often learn as much mathematics from drawing a simple picture as they do from playing with a manipulative. In mathematics, the ability to draw a picture or create a model in which only the essential features of a problem are represented is an essential skill.

Lessons with manipulatives must be very carefully designed to ensure that every student is engaged and none are left behind. In some of the inner city classes I have observed, I have seen children spend more time arguing over who had what colour of block or who had more blocks than they spent concentrating on the lesson. Students need to be confident, focused and motivated to do effective work with manipulatives. In JUMP we begin with the Fractions Unit (in which students are expected to work independently with pencil and paper) to allow students to develop the confidence and focus required for work with manipulatives.

If a teacher aims to engage all of their students (not just the ones who are more advanced than their peers), and if children must be confident and attentive to learn, then it seems obvious that the teacher must start their math program by assigning work that every student can complete without the help of their peers. When students work in groups with manipulatives, it is often hard to verify that every student has understood the lesson. The JUMP Fractions Unit is designed to allow teachers to identify and help students who need remediation immediately, so that every student gains the confidence they need to do more independent work.

6) Concrete materials do not, as is widely believed, display their interpretations on their surface. You can’t simply hand out a set of manipulatives to a group of children and expect the majority to use them to derive efficient rules and operations. Children usually need a great deal of guidance in order to deduce anything significant from playing with concrete materials.

The mathematical “opaqueness” of concrete materials was demonstrated quite strikingly by a recent anthropological discovery. Scientists found a tribe that has been catching and sharing great quantities of fish since prehistoric times, but the members of the tribe can’t say exactly how many fish they’ve caught when there are more than two fish in a net. This shows quite clearly that mathematical concepts don’t suddenly spring into a person’s mind when you slap a concrete material (like a fish) in their hands. Efficient rules and operations often take civilizations centuries to develop. So it’s not surprising that children need lots of practice with rules and operations, even if they have spent an enormous amount of time playing with blocks and rods.

The line between abstract and concrete thought is often rather fuzzy: even the simplest manipulatives and models do not provide transparent representations of mathematical concepts. I once saw kids in a remedial class reduced to tears when their teacher tried to introduce the operation of addition using base ten materials. When I showed the children how to add (and how to subtract, multiply and divide) by counting up on their fingers, they were able to perform the operations instantly. In my experience the hand is the most effective (and cheapest)

page 10

Teacher’s Guide

Grade 7

manipulative for students who have serious learning difficulties. When children perform operations by counting or skip counting on their fingers, they get a sense of the positions of the numbers in their body.

Of course children should eventually be weaned off of using their fingers: the JUMP Teacher’s Guide contains a number mental math tricks to help children learn their number facts. And base ten materials are very useful tools for teaching arithmetical operations to students who are confident and focused enough to use the materials. But in the early phases of a math program, I would recommend teaching weaker students who need to catch up to perform basic operations on their fingers.

I believe students would cover far more material in a year if we could find a better balance between symbolic and concrete work in our curriculum. Finding this balance may prove difficult, however, as schools are being pushed by educational experts to include more manipulatives in their mathematics programs. And increasingly, the research that “proves” that manipulatives are effective is being funded (directly or indirectly) by companies that sell textbooks and manipulatives. This is a rather alarming trend in education, particularly as much research in math education is not scientific and is often based on poor experimental designs and on rather startling leaps of logic.

NOTE: Over the past year I have found or developed a number of lessons with concrete materials that I believe students will enjoy and benefit from. I would encourage you to try these lessons (included in this guide), but bear these points in mind, particularly if you are working with inner-city children:

i) Weaker students can easily be left behind in activity-based lessons. Use the exercises on the worksheets that are keyed to a lesson to verify students have understood the lesson or that they have the skills and cognitive background necessary to start the lesson.

ii) Lessons with concrete materials can be time consuming. Students may be held back (particularly in their understanding of math as a symbolic activity) if you spend too much time on these kinds of lessons.

iii) Students should learn as early as possible to draw pictures or form mental images that embody only the essential features of a problem. A student who truly understands mathematics will eventually be able to solve most elementary problems using only a pencil and paper and their imagination.

I would encourage you to treat your classroom as a laboratory and to test a variety of approaches with your students until you have found a balance of approaches that serves your entire class. For more activities, see the following website: http://www.edu.gov.mb.ca/ks4/cur/math/index.html (accurate at time of printing). Also check our website (www.jumpmath.org) for details about any forthcoming enriched and problem solving units.

The Second Idea: A student who only partially understands a mathematical rule or concept, and who can’t always apply the concept or extend it to new cases consistently, understands nothing.

In the days when students were taught operations almost entirely by rote, the majority only partially understood the operations. Some educators who observed this state of affairs concluded that partial knowledge in mathematics is, in itself, always a bad thing. Rather than simply advocating that people be taught why operations work as well as how they are performed, these educators took the position that if you teach a student how to perform an operation without first teaching all of the concepts underlying the operation (or allowing the student to discover the operation) then you will prevent the student from ever learning those concepts properly in the future. This conclusion, however, is not supported by the actual practice of mathematics. Far from being bad, partial knowledge is the daily bread of every practising mathematician.

Mathematicians usually start their research by trying to master a small or artificially restricted area of knowledge. Often they will play with simplified systems of rules and operations, even before they have devised a physical model for the rules. Ideas seldom arrive full blown in mathematics: even after a mathematician discovers a new rule or operation, it can take generations before the rule is fully understood. And often it is the relentless practice with the rule, more than any physical intuition, that allows for the emergence of complete understanding. As one of the great mathematicians of the twentieth century, John von Neumann, said, understanding mathematics is largely a matter of getting used to things.

page 11

Teacher’s Guide

Grade 7

If we applied the standards and methods that are now used to teach children in elementary schools to graduate students in universities, very few students would ever complete their degrees. Children need to be given more practice using rules (so that they can “get used to” and gain a complete understanding of the rules) and they need more guidance when they fail to discover rules by themselves. Rules and concepts are often hard to separate: even in cases where the distinction is clear, the mastery of rules can help induce the understanding of concepts as much as the understanding of concepts supports the mastery of rules.

For a more complete discussion of this point, we need to look at an idea that is a close cousin to the idea that partial knowledge is always bad, namely...

The Third Idea: Children have definite stages of cognitive development in mathematics that can be precisely defined and accurately diagnosed and that must always be taken account of in introducing concepts. A child who can’t explain a concept fully or extend the concept to new cases is not developmentally ready to be introduced to the concept. Any effort to introduce a child to a concept before they are ready understand the concept in its entirety (or to discover the concept by themselves) is a violation of a child’s right to be taught at their developmental level.

This idea has done inestimable damage in remedial classes and to weaker students in general. I have worked with a great many Grade 6 and 7 students who were held back at a Grade 1 or 2 level in math because their teachers didn’t think they were cognitively or developmentally ready to learn more advanced material.

Having worked with hundreds of students who have struggled in math, I am convinced that the mind is more plastic than most psychologists and educators would allow (even after the first six years, which is when scientists have shown the brain is extremely plastic). I have seen dramatic changes in attitude and ability in very challenged students even after several weeks of work on the Fractions Unit (see The Myth of Ability and the JUMP website for details). In a recent survey, all of the teachers who used the fractions unit for the first time acknowledged afterward that they had underestimated (and in many cases greatly underestimated) the abilities of their weaker students in ten categories, including enthusiasm, willingness to ask for harder work, ability to keep up with faster students and ability to remember number facts.

Not long ago, in the 1960s, mathematicians and scientists began to notice a property of natural systems that had been overlooked since the dawn of science: namely that tiny changes of condition, even in stable systems, can have dramatic and often unpredictable effects. From stock markets to storm fronts, systems of any significant degree of complexity exhibit non-linear or chaotic behaviour. If one adds a reagent, one drop at a time, to a chemical solution, nothing may happen at all until, with the addition of a single drop, the whole mixture changes colour. And if, as a saying made current by chaos theory goes, a butterfly flaps its wings over the ocean, it can change the weather over New York.

As the brain is an immensely complicated organ, made up of billions of neurons, it would be surprising if it did not exhibit chaotic behaviour, even in its higher mental functions. Based on my work with children, I am convinced that new abilities can emerge suddenly and dramatically from a series of small conceptual advances, like the chemical solution that changes colour after one last drop of reagent. I have witnessed the same progression in dozens of students: a surprising leap forward, followed by a period where the student appears to have reached the limits of their abilities; then another tiny advance that precipitates another leap. One of my students, who was in a remedial Grade 5 class when he started JUMP, progressed so quickly that by Grade 7 he received a mark of 91% in a regular class (and his teacher told his mother he was now the smartest kid in the class). Another student, who couldn’t count by 2s in Grade 6, now regularly teaches herself new material from a difficult academic Grade 9 text.

A teacher will never induce the leaps I have described if they are unwilling to start adding the small drops of knowledge that will cause a student’s brain to reorganize itself. If the teacher waits, year after year, until the student is “developmentally ready” to discover or comprehend a concept in its entirety, the student will inevitably become bored and discouraged at being left behind, and the teacher will miss an opportunity to harness the enormous non-linear potential of the brain. This is what happens in far too many remedial classes. And this is why, in JUMP, we teach even the most challenged students to multiply on their fingers by 2s, 3s and 5s and then launch them into a Grade 7 unit with fractions whose denominators divide by those numbers. Students who complete the unit don’t

page 12

Teacher’s Guide

Grade 7

know how to add and subtract every type of fraction, nor do they understand fractions in great depth, but the effect of allowing them to completely master a small domain of knowledge is striking.

If a teacher only teaches concepts that students are ready to understand or explain in their entirety, then the teacher will not be able to use the method of “raising the bar incrementally” that I described earlier and that is the key to JUMP’s success with weaker students. In Ontario, students in Grade 3 are not expected to add pairs of numbers with more than three digits: I suppose this is because they are not developmentally ready to add larger numbers and because they haven’t spent enough time playing with concrete models of large numbers. But I have seen children in Grade 3 classes jump out of their seats with excitement when I’ve challenged them to extend the method for adding three-digit numbers to ten-digit numbers.

Whenever I challenge a class to add larger numbers, I start by teaching students who don’t know their addition facts how to add one-digit numbers by counting up on their fingers. I make sure that the digits of the numbers I write on the board are relatively small, so that every student has a chance of answering. As I write longer and longer numbers on the board, even the weakest students invariably start waving their hands and shouting “Oh, oh.” When they succeed in finding the sum of a pair of ten-digit numbers, they think they’ve conquered Mt. Everest. (I’ve found it is generally easier to generate a sense of real excitement about math in classrooms than in one-on-one tutorials: children love to be given the opportunity to succeed and show off in front of their peers.)

When Grade 3 students use a rule they have learned for adding three-digit numbers to add ten-digit numbers, they are behaving exactly like mathematicians: they see a pattern in a rule and they guess how the rule might work in more complex cases. Children needn’t wait until their teacher has purchased the right set of manipulatives or until they are developmentally ready before they can explore their hypotheses.

I still remember the impression left by a lesson my Grade 7 math teacher gave on Fermat’s Last Theorem. At the time I barely understood the concept of squares, let alone higher exponents. But I remember feeling that Fermat’s Theorem was very deep and mysterious and I remained fascinated with the theorem for the rest of my life.

By insisting that partial knowledge is always bad and that kids must always be taught according to their developmental level, educators risk removing any sense of enchantment from learning. Children would undoubtedly find mathematics and science more interesting if they were introduced to the deepest and most beautiful ideas in those fields at an early age. There are countless fascinating topics in pure and applied mathematics that only require elementary math, and that we needn’t wait until high school or university to teach.

To spark children’s imaginations, I have given several different lessons on theoretical computer science to students as early as Grade 3 (for details see The Myth of Ability). The students were able to complete the tasks I assigned them and they often asked me to extend the lessons. (JUMP is now developing enriched lessons on logic, problem solving, graph theory and topology and on applications of mathematics in biology, chemistry, physics, magic tricks, games, sports, and art.)

I start one of my lessons on computer science by showing students how to draw a picture of a theoretical model of a computer (called a finite state automata). Students then try to figure out what kind of patterns their “computer” will recognize by moving a penny around on their sketch like a counter on a board game. Following a suggestion of my daughter, I once gave kids in a Grade 3 class paper clips to hold their drawings in place on a cardboard folder. Rather than using a penny as a counter, the kids put pairs of fridge magnets on their drawings (one on the front and one on the back) and they used the back magnet to pull the front one around like a cursor. Many of the children mentioned this lesson in their thank you letters to JUMP: even though they only had a partial understanding of finite state automata, in their minds they had made real computers.

Representatives of a school board in Eastern Canada recently observed a JUMP lesson on how computers read binary codes. The lesson culminated in a mind reading trick that kids love. Afterward the teacher was barred from using JUMP in the class because the lesson hadn’t been taught “developmentally.”

NOTE: While partial knowledge isn’t necessarily bad, partial success is. Even when I introduce kids to ideas that they may only partially understand, I make sure that they are able to complete the exercises I give them. (However, if students are more motivated and confident, I will sometimes let them struggle more with an exercise: students

page 13

Teacher’s Guide

Grade 7

eventually need to learn that it’s natural to fail on occasion and that solving problems often takes a great deal of trial and error.)

The Fourth Idea: If a student is taught how to perform a mathematical operation, rather than discovering the method on their own, they are unlikely to ever understand the concepts underlying the operation.

I recently read a research paper in math education that found that many adults don’t know how to multiply or divide large numbers very well and many don’t understand the algorithms they were taught for performing those operations. Considering the way math was taught when I went to school, this news didn’t surprise me. But the conclusion the authors drew from their observations did. Rather than recommending that schools do a better job teaching operations, the authors claimed their data showed that standard methods for operations should not be emphasized in schools: instead children should be encouraged to develop their own methods of computation.

I certainly agreed with the authors that children should be encouraged to develop various non-standard tricks and “mental math” strategies for computation (and if they fail to discover these strategies they should be taught them). But it’s important to bear in mind that entire civilizations failed to discover the idea of zero as a place holder for division. If the Romans were incapable of developing an effective method of division over the course of eight centuries (just try dividing large numbers with Roman Numerals!) it seems a little unrealistic to expect a child to discover their own method in the course of a morning.

The idea that children have to discover an operation to understand it, like many ideas I have encountered in math education, is based on a reasonable idea that has simply been stretched too far. As a teacher I always encourage my students to make discoveries and extend their knowledge to new situations by themselves. But as a mathematician, I have a realistic idea what discovery means. I know, from my work as a student and as a researcher, that discoveries in mathematics are almost always made in tiny, painstaking steps.

My best teacher in high-school always had my classmates and me on the edge of our seats during his chemistry lessons. He led the class in steps, always giving us enough guidance to deduce the next step by ourselves. We always felt like we were on the verge of recreating the discoveries of the great chemists. But he didn’t expect us to discover the entire periodic table by ourselves. (Of course, if a class is ready to discover the periodic table, then by all means let them discover it: the goal of JUMP is to raise the level of students to the point where they can make interesting discoveries. Also, I would encourage a teacher to sometimes assign more difficult, open-ended exercises – as long as students who fail to make discoveries during the exercise are guided through the material afterwards.)

In the present educational climate, teachers will seldom verify that all of their students can perform an operation before they assign work that involves the operation. And students are rarely given enough practice or repetition to learn an operation properly. Students can easily reach Grade 9 now without anyone noticing that they have failed to discover even the most basic facts about numbers.

Some educators seem to assume that if a child discovers an operation or a concept they will always find it easy to apply the concept in new situations, and they will be able to recall the concept immediately, even if they haven’t had any opportunity to think about it for a year. This certainly does not reflect my experience as a mathematician. I have discovered original (and rather elementary) algorithms in knot theory that I only mastered after months of practice. And if you were to ask me how one of those algorithms works now, I would have to spend several weeks (of hard work) to remember the answer.

(Repetition and practice don’t have to be boring. If students are encouraged to discover and extend steps by themselves, if they are made to feel like they are meeting a series of challenges and if they are allowed to apply their knowledge to solve interesting problems, they will happily learn even the most challenging operations.)

JUMP has shown that children in Grade 2 can learn to perform operations with fractions flawlessly in less than a month and that children in Grade 3 will beg to stay for recess for lessons on theoretical computer science. Rather than compelling children to spend so much time attempting to discover rather mundane standard algorithms (or discover inferior versions of their own), why not guide children through the curriculum as quickly and efficiently as possible, and then allow them use the tools they have acquired to explore more substantial and more beautiful mathematics?

page 14

Teacher’s Guide

Grade 7

JUMP is a fledgling program with very limited resources. It may take years before we find the right balance between concrete and symbolic work, or between guided and independent work. But I think we have demonstrated one fact beyond a shadow of a doubt: it is possible to teach mathematics without leaving children behind. The results of JUMP have shown that we need to reassess current research in math education: in order to be called a “best practice” a new program must do far more than show that, on average, children in the program do a little better in math. No one would ever say, “It was a great day at school today, only one child starved.” Any program that claims to be a best practice must now demonstrate that it can take care of every child.

page 15

Teacher’s Guide

Grade 7

The Fractions Unit

To prepare your students to use this book, you should set aside 40 to 50 minutes a day for 3 weeks to teach them the material in the JUMP Fractions Unit. You may print individual copies of the unit from the JUMP website at no charge and you can order classroom sets (at cost) from the University of Toronto Press. NOTE: For large numbers, this option is cheaper than photocopying. The Fractions Unit has proven to be a remarkably effective tool for instilling a sense of confidence and enthusiasm about mathematics in students. The unit has helped many teachers discover a potential in their students that they might not otherwise have seen. In a recent survey, all of the teachers who used the Fractions Unit for the first time acknowledged afterwards that they had underestimated the abilities of some of their students. (For details of this study, see the JUMP website at www.jumpmath.org.)

The Fractions Unit is very different from the units in JUMP’s grade-specific material. These units follow the Ontario curriculum quite closely. The point of the Fractions Unit, however, is largely psychological: students who complete the unit and do well on the Advanced Fractions test show remarkable improvements in confidence concentration, and numerical ability. This has been demonstrated, even with the lowest remedial students, in a number of classrooms.

For a detailed account of the purpose of the Fractions Unit, please see the Introduction of the Teacher’s Manual for the Fractions Unit.

page 16

Teacher’s Guide

Grade 7

HINTS FOR HELPING STUDENTS WHO HAVE FALLEN BEHIND

In response to questions asked by teachers using the JUMP program, I have compiled some suggestions for helping students who are struggling with math. I hope you find the suggestions useful. (And I hope you don't find them impractical: I know, given the realities of the teaching profession, that it is often hard to keep your head above water.)

1. Teach 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. Students can certainly perform operations on a calculator, but they cannot begin to solve problems if they lack a sense of numbers: students need to be able to see patterns in numbers, and to make estimates and predictions about numbers, in order to have any success in mathematics. A calculator cannot provide these abilities.

It is much easier to teach students their number facts than is generally believed. In the “Mental Math” section of the JUMP Teacher’s Guide, you will find a number of effective strategies to help students learn their number facts (see, for instance, the section “How to Learn Your Times Tables in a Week”). After you have taught these strategies, I would recommend giving students who need extra practice daily two-minute drills and tests until they know their facts (you can give a student the same sheet repeatedly until they have memorized the facts on it – that way you don't have to do a lot of extra work preparing materials). You might also send home extra work or, whenever possible, ask parents to help their children memorize certain facts (don't overload the student – you might send home one times table or half a times table per night). Students might also quiz each other using flash cards. JUMP has shown that students will memorize material more quickly if their teacher is enthusiastic about their successes, no matter how small those successes may seem. (You might even have some kind of reward system or acknowledgment for facts learned.)

Trying to do mathematics without knowing basic number facts is like trying to play the piano without knowing where the notes are: there are few things you could teach your students that will have a greater impact on their academic career than a familiarity with numbers.

2. Give Cumulative Reviews:

Even mathematicians constantly forget new material, including material they once understood completely. (I have forgotten things I discovered myself!) Children, like mathematicians, need a good deal of practice and frequent review in order to remember new material.

Giving reviews needn't create a lot of extra work for you. I would recommend that, once a month, you simply copy a selection of questions from the workbook units you have already covered onto a single sheet and Xerox the sheet for the class. Children rarely complain about reviewing questions they already did a month or more ago (and quite often they won't even remember they did those particular questions). The most you should do is change a few numbers or change the wording of the questions slightly. If you don't have time to mark the review sheets individually, you can take them up with the whole class (though I would recommend looking at the sheets of any students you think might need extra help or practice).

3. Make Mathematical Terms Part of Your Spelling Lessons and Post Mathematical Terms in the Classroom:

In some areas of math, in geometry for instance, 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. You might also create bulletin board or math wall with pictures and mathematical terms, so students can see the terms every day.

page 17

Teacher’s Guide

Grade 7

4. Find Five Minutes, Wherever Possible, to Help Weaker Students in Small Groups:

Whenever I have taught JUMP in a classroom for an extended time, I have found that I generally needed to set aside five minutes every few days to give extra review and preparation to the lowest four or five students in the class. (I usually teach these students in a small group while the other students are working on other activities.) Surprisingly, this is all it takes for the majority of students to keep up (of course, in extreme cases, it may not be enough).

I know, given current class sizes and the amount of paperwork teachers are burdened with, that it’s very hard for teachers to find extra time to devote to weaker students but, if you can find the time, you will see that it makes an enormous difference to these students and to the class in general. (By investing a little extra time in your weaker students, you may end up saving time as you won't have to deal so much with the extreme split in abilities that is common in most classes, or with the disruptive behaviour that students who have fallen behind often engage in.)

5. Teach Denser Pages in the Workbooks in Sections:

Fitting the full curriculum into 300 pages was not an easy task. Even in this new edition of the workbooks, where we have made an effort to improve the layout, several pages in our workbooks are more cramped than we would have liked, and some do not provide enough practice or preparation. If you feel a worksheet is too dense or introduces too many skills at once, assign only two or three questions from the worksheet at a time. Give your students extra practice before they attempt the questions on the page: you can create questions similar to the ones on the page by just changing the numbers or by changing the wording slightly.

6. Change Difficult Behaviour Using Success and Praise:

In my experience, difficult children respond much more quickly to praise and success than to criticism and threats. Of course, a teacher must be firm with students, and must establish clear rules and boundaries, but I've found it's generally easier to get kids to adhere to rules and to respect others if they feel admired and successful.

I have worked with hundreds of children with attention deficits and behavioural problems over the past 20 years (even in the correctional system), and I have had a great deal of success changing behaviour using a simple technique: if I encounter a student who I think might cause problems in a class I'll say: "You're very smart. I'd better give you something more challenging." Then I give the student a question that is only incrementally harder – or that only looks harder – than the one they are working on. For instance, if a student can add three fractions with the same denominator, I give them a question with four fractions. (I never give a challenge to a difficult student unless I'm certain they can do the question.) I always make sure, when the student succeeds in meeting my challenge, that they know I am impressed. Sometimes I even pretend to faint (students always laugh at this) or I will say: "You got that question but you'll never get the next one." Students become very excited when they succeed in meeting a series of graduated challenges. And their excitement allows them to focus their attention and make the leaps I have described in The Myth of Ability. (Of course you don't have to use my exact techniques: teachers find different ways to praise their students, but I think passion is essential.)

The technique of raising the bar is very simple but it seems to work universally: I have used it in inner-city schools, in behavioural classes and even in the detention system and I have yet to meet a student who didn't respond to it. Children universally enjoy exercising their minds and showing off to a caring adult.

Although JUMP covers the traditional curriculum, the program demands a radical change in the way teachers deliver the curriculum: JUMP is based on the idea that success is not a by-product of learning, it is the very foundation of learning. If you aren't willing to give difficult students graduated challenges that they can succeed at, and if you aren't willing to be excited at their successes, then you may leave those students behind unnecessarily.

In mathematics, it is extremely easy to raise the bar incrementally: I don't know of any other subject in which a teacher can break skills into such minute steps and can gage so precisely the size of the step and the student's readiness to attempt a new step. I believe there is no other subject in which it is easier to harness the attention and enthusiasm of difficult students.

page 18

Teacher’s Guide

Grade 7

I know that in a big class it's extremely hard to give attention to difficult students, but sometimes a few five-minute sessions spent giving a student a series of graduated challenges (that you know they can succeed at) can make all the difference to the student (and to your stress levels!).

(NOTE: Once students develop a sense of confidence in math and know how to work independently, you can sometimes allow them to struggle more with challenges: students need to eventually learn that it's natural to fail on occasion and that solving problems sometimes takes a great deal of trial and error.)

7. Isolate the Problem:

If your student is failing to perform an operation correctly, try to isolate the exact point or step at which they’re faltering. Then, rather than making the student do an entire question right from the beginning, give them a number of questions that have been worked out to the point where they have trouble and have them practice doing just that one step until they master it. For instance, when performing long division with a two-digit number, students sometimes guess a quotient that is too small:

3 3 3 46 198 46 198 46 198 – 138 60

One of the JUMP students was struggling with this step – even after many explanations, the student would forget what to do after performing the subtraction. Finally, the tutor wrote down a number of examples that had been worked out up to the subtraction and simply asked the student to check whether the remainder was larger than the divisor and, if so, to increase the quotient by one. The student quickly mastered this step and was then able to move on to doing the full question with ease.

NOTE: Students will also remember an operation better if they know why it works – the Manual Notes, Activities and Problem Solving sections of this guide contain exercises that will help students understand various operations.

60 is larger than 46

so the quotient 3 is

too small.

page 19

Teacher’s Guide

Grade 7

HINTS FOR HELPING STUDENTS WHO FINISH WORK EARLY

1. While this may seem counterintuitive, you will enable your fastest students to go further if you take care of the slowest. You can create a real sense of excitement about math in your classroom simply by convincing your weaker students that they can do well at the subject. You will cover more material in the year, and your stronger students will no longer have to hide their love of math for fear of appearing strange or different.

2. Assign students who finish work early bonus questions, or extension questions from this guide. Avoid singling out students who work on extension questions as the class geniuses, and, as much as possible, allow all of your students to try these questions (with hints and guidance if necessary). Students won’t generally notice or care if some students are working on harder problems, unless you make an issue of it. Your class will go much further, and some of your students may eventually surprise you, if you make them all feel like they are doing impressive work.

There will always be differences in ability and motivation between children, but those differences (particularly in speed) would probably not have much bearing on long term success in mathematics if schools were not so intent on making differences matter. Because a child’s level of confidence and sense of self will largely determine what they learn, teachers can easily create artificial differences in children by singling out some as superior and others as inferior. I’ve learned not to judge students too hastily: I’ve seen many slower students outpace faster ones as soon as they were given a little help or encouragement.

3. Even the most able students make mistakes, but sometimes it’s hard to convince a stronger student to write out the steps of a solution or calculation so you can see where they went wrong. If a student is reluctant to show their work, I will often say “I know you’re very clever, and you can do the steps in your head, but I can’t always keep up with you, so I need you to help me out and show me your steps occasionally.” I’ve also said. “Because you’re so clever, you may want to help a friend or a brother or sister with math one day, so you’ll need to know how to explain the steps.” I’ve found that students will generally show the steps they took to solve a problem if they know there are good reasons for doing so (and if they know I won’t always force them to write things out).

page 20

Teacher’s Guide

Grade 7

HINTS FOR ONE-ON-ONE TUTORS

1. Give a great deal of encouragement: When a student masters a new step, make sure you point out how well they are doing (without seeming like

you’re insane). Quite often you can make the steps in a problem so simple – and provide enough subtle hints – that the student can guess how to proceed. For instance, once they know how to add fractions with the same denominator, you might ask them if they can figure out what to do in this case:

45 –

15 = ?

If they guess correctly, point out that they are smart enough to figure out the rules for themselves. If they guess incorrectly, do your best to make them feel as if their answer made sense or was a good try.

2. Teach difficult concepts incrementally: If your student doesn’t understand a concept or rule fully, don’t assume they’re incapable of moving forward.

Often the confidence that students gain from performing and mastering simple operations makes them more open to understanding more difficult concepts. (Of course, it is always preferable to explain why a rule works first, if a student is ready for this.)

In working with a weaker student, it usually helps to give them an action to carry out, for instance: counting, crossing out a number, writing a number or symbol in a certain place, saying whether two numbers are the same, counting on their fingers by a certain number, or seeing if you reach another number. In the beginning, only ask your student to perform steps like these, steps they can’t possibly fail at. If you take this approach and give your student a great deal of encouragement, you will eventually begin to see conceptual abilities emerge. As your student becomes more focused and confident, you will be able to skip steps and challenge them to do more independent work.

3. Always make the student feel they’re working toward a clear goal: For instance, at the beginning of a lesson, you might say, “After this lesson (or a few lessons) you’ll be able to

add fractions well beyond your grade level.” You might even show them some questions from an advanced book that they’ll soon be able to do. (And then, when they do the questions, point out that they are in fact working at a higher level). The lesson should consist of a graduated series of challenges, and the student should be made aware of – and praised for – what they have accomplished at every step. You might, for instance, say: “I can’t believe you learned that step so quickly! Let’s try the next level of difficulty.” Once the student begins to feel that they can master any set of challenges in the lesson, they will carry that confidence back to their own classroom and begin picking up more at school.

4. Repetition is important: Do not introduce a new concept or step until the student has correctly completed a series of questions based on

the one you’ve just taught. Lesson should start with a review of the things you have covered in the last two or three weeks, and every homework assignment should have some questions (even one or two) based on the unit topics you’ve covered over the last few weeks or months.

5. Patience is essential: Initially, many of the JUMP students will have trouble paying attention, and some will not really enjoy doing

math. If you apply the principles outlined above, together with a great deal of patience, you’ll have a good chance of changing the way your students learn and think about themselves.

6. Be firm: Many of the students recommended for JUMP have rather short attention spans, and many have developed an

incredible facility for distracting their teachers. It is very easy to waste half a lesson chatting with your student. But remember: you are in charge of the lesson. You can establish a friendship with your student and you can certainly reward your student with lots of praise, but you have to be quite strict about getting the work done. If your student has trouble concentrating for long stretches, you might allow a short break in the middle of the lesson just let the student know they have to earn it. Your student should be aware that the point of the

page 21

Teacher’s Guide

Grade 7

lesson is not just to learn the material, but also to improve their concentration. Tell them that they’ll find it easier to focus if they practice: by doing their work without talking or being distracted, they will develop a kind of mental stamina, just as one develops physical stamina by exercising without a break.

7. Teach your student to listen: If you make each step simple enough and allow for repetition (making sure to tell your student how brilliant she

is for mastering the step), your student will gradually begin to have faith that they can understand what you are saying. As your student becomes confident, start asking them to repeat your explanation in their own words. Let them know that comprehension improves with practice. Ask them to practice these skills – listening with the faith that they can understand, repeating in their own words – at school.

8. Comprehensive homework is essential: Each week, you should give your student several pages of homework, including examples of everything they

have learned right from the beginning. Half the effectiveness of the program lies in the homework. It’s like piano scales: if the student has to recall and practice what they have learned right from the beginning, it becomes second nature to them and they’ll never forget it. If your student doesn’t bring their completed homework back, you should phone them during the following week to remind them. If it happens repeatedly, you should talk to your student’s parents or to JUMP.

9. In the Fractions Unit, use the homework to train memory: If your student is struggling with a section of the Fractions Unit, only cover the simplest questions of that type

(e.g. using twos, threes and fives) before moving on. The idea is to help the student master a small stock of very simple questions, then to repeat those same questions on the homework each week. After a month or so, you can go back and cover slightly harder questions. By moving through the unit in this way, you can maintain a sense of momentum: the work of mastering and remembering is accomplished through the repetition of the homework. For instance, with a slower student who is learning triple fractions, only teach them, initially, to solve 1/2 + 1/3 + 1/6 and 1/3 + 1/5 + 1/15. After a month or so of putting the same questions on their homework, you can then start to increase the level of difficulty by varying the order (1/6 + 1/2 + 1/3), increasing the numerators (5/6 + 1/2 + 2/3) or moving on to different denominators (1/2 + 1/4 + 1/8 ; 1/2 + 1/6 + 1/12), etc.

Don’t be surprised if your student constantly forgets what they’ve learned. If you repeat questions on the homework and give work that goes right back to the first lessons, you should soon notice an improvement in your student’s memory and concentration as well as in their grasp of the basics, such as the times tables of lower numbers.

page 22

Teacher’s Guide

Grade 7

JUMP MATH INSTRUCTIONAL APPROACHES – by Dr. Melanie Tait JUMP Math is based on the belief that with support and encouragement, all children will succeed at math. When teachers believe that all students can succeed, they will strive to establish a classroom environment where all students feel comfortable participating and taking risks.

In JUMP classrooms, if students don’t understand, the teacher must assume responsibility and find another way to explain the material. Three essential characteristics for a JUMP teacher are the ability to diagnose where students ”are at”, customize instruction to suit individual children, and improvise to meet their needs.

As Glickman (1991) writes:

“Effective teaching is not a set of generic practices, but instead is a set of context-driven decisions about teaching. Effective teachers do not use the same set of practices for every lesson . . . Instead, what effective teachers do is constantly reflect about their work, observe whether students are learning or not, and, then adjust their practice accordingly (p. 6).

JUMP teachers, like all excellent teachers, know their students well and use a variety of creative instructional strategies to meet their needs. JUMP teachers are constantly checking in with students to make sure everyone is moving forward. A JUMP class is a busy and interactive learning environment.

JUMP recognizes that teachers are skilled professionals with unique strengths and teaching preferences. Accordingly, the JUMP Math program is designed to accommodate a number of instructional approaches and strategies. Teachers are encouraged to vary instructional approaches and strategies to suit the class and the needs of individual students. JUMP’s approach is built on the belief that all children can learn when provided with the appropriate learning conditions in the classroom. Learning is supported through explicit instruction, interaction with the teacher and classmates, and independent learning and practice. The program is composed of well-defined learning objectives organized into smaller, sequentially organized units. Generally, unit consist of discrete topics which all students begin together. Students who do not satisfactorily complete a topic are given additional instruction until they succeed. Students who master the topic early engage in enrichment activities until the entire class can progress together.

In a JUMP classroom, the teacher employs a variety of instructional techniques, with frequent and specific feedback using diagnostic and formative assessment. Students require numerous feedback loops, based on small units of well-defined, appropriately-sequenced outcomes. Teachers assess student progress in a variety of ways and adjust their programs accordingly.

JUMP lessons are often dynamic: as soon as the teacher has explained or demonstrated a concept or operation, students are allowed to ‘show off’ their understanding through scaffolded tasks and quizzes that can be checked individually or taken up with the whole class. Students enjoy being able to apply their knowledge and they benefit from the immediate feedback. This way of teaching allows the teacher to assess what students know before moving on.

Fundamentals of JUMP Math Instruction There are several fundamental principles that guide effective JUMP instruction:

Establishing a safe environment. Teachers must create a safe environment where students can feel challenged without feeling threatened. This is perhaps especially important in math classrooms. Math anxiety can have a negative effect on children’s (and teachers’) self-confidence, enjoyment of math and motivation to learn math. (Tobias, 1980). JUMP is founded on the notion that given the right kind and amount of support and encouragement, all children, except perhaps those with severe brain damage, are able to learn mathematics. The Fractions Unit, for example, which is a wonderful way to begin the JUMP program, is designed to improve confidence, concentration and numerical ability while making math fun and interesting. JUMP teachers demonstrate, through their attitude to mathematics and their students, that it is important to persevere to solve problems and that people only learn through making errors. Patience, praise, encouragement and positive feedback are essential parts of the program.

page 23

Teacher’s Guide

Grade 7

Eliciting and encouraging participation. Teachers need to elicit participation in order to assess learning and to ensure that every student feels their contributions are valid and valued.

Some possible ways to elicit and encourage participation include:

• Share the expectation that all students will participate. • Remind everyone that all questions, answers and suggestions will be respected. • Ask if students understand before proceeding with the lesson. • Ask for volunteers at times and call on specific students at other times. • Ask for a contribution from someone who has not yet spoken. • Encourage quiet students (e.g., direct questions, pre-arrange questions). • Offer challenging and thought-provoking ideas for discussion. • Use open-ended questions. (What do you think about…? Why do you think it is important to…?) • Plan interactive activities (e.g., small-group discussions, solving problems in groups). • Give students time to think before they answer. • Rephrase your question. Use different wording give an example, or different examples. • Provide hints. • Show approval for student ideas (e.g., positive comments, praise for trying). • Answer questions in a meaningful way. • Incorporate student ideas into lessons.

Making connections explicit. Connecting new concepts to real life, other subject areas or other mathematical ideas may help students relate to the content and engage in the lesson. Seeing relationships helps students to understand mathematical concepts on a deeper level and to appreciate that mathematics is more than a set of isolated skills and concepts but rather something relevant and useful. JUMP explicitly make links between math skills and concepts from the different strands and ensures that all prerequisite knowledge is reviewed or retaught before going on to new material.

There are many ways to involve children in meaningful and relevant mathematics activities. Baking for a class fundraiser will require application of measurement skills and money concepts. Keeping track of the statistics in the NHL or World Cup calls for data management skills. To introduce a lesson on perimeter, students could be asked what they would need to know if they were responsible for installing a fence to go around the school playground and then actually measure the perimeter of the yard before working with standard algorithms.

Links between math and other subject areas, such as the arts, social studies and science, can easily be made. Children’s literature can also provide a lifelike context for math learning. Books provide children with experiences that they might otherwise not have, and many lend themselves beautifully to mathematical activities. JUMP materials also highlight the relevance of mathematics in careers and the media.

Diagnosing and responding. Once students are interested and paying attention, teachers need to determine where students “are at” in their comprehension and application of mathematics concepts. Core JUMP lessons begin with diagnostic checks of understanding. The teacher uses questions, discussion, student demonstrations on the board or diagnostic pencil and paper quizzes to verify understanding and determine the entry point for the lesson. Nothing should be assumed – for instance, before learning to find the perimeter of a rectangle, students must be able to add a sequence of numbers.

Gaps in understanding can only be addressed if they are identified. The teacher can continue to work with the others individually or in small groups by “buying time” through assigning interesting “bonus” questions which “raise the bar” incrementally. For example, if the teacher is verifying that students can add two-digit numbers with regrouping, those students who demonstrate competence with this skill can solve three-digit addition questions. This allows for one-on-one or small group tutorial time during the course of whole class lessons and is an important feature of a JUMP lesson.

Two characteristics of a good JUMP teacher are the abilities to improvise and customize to meet the needs of students. The students and their learning are more important than the lesson plan. Responding to the results of diagnostic questioning in an appropriate and timely way is crucial to the success of subsequent lessons. On occasion, this will mean that the teacher will need to go back to a previously presented concept for review or additional practice before moving forward.

page 24

Teacher’s Guide

Grade 7

Breaking concept down into discrete segments. To successfully use the JUMP method, teachers must break down new concepts and skills into small, sequential parts. The JUMP teacher demonstrates the concept or skill, explaining it as she goes and inviting student participation as appropriate. (“What should go here?” “What does this mean?” “Do you think this is right?” “How else could you do this?”) As the lesson progresses, she continues to constantly assesses whether concepts or procedures have been understood and then revisits the instruction using different words, examples or questions to help those who do not understand while providing “bonus” questions which extend the ideas in a manageable way for those who are ready to tackle them.

Guiding discovery. Guided discovery is another important element of the JUMP program. By using a variety of questioning styles, examples and activities, teachers lead students to understanding. Perhaps more importantly, though, teachers need to be able to simplify processes and procedures so that students are able to move forward from their current level towards discovering the pattern, rule or generalization. Guided discovery works best in a safe classroom environment because once students begin to trust the teacher and feel confident in their ability to progress, their attention and behaviour improve, they are more likely to take risks and their perseverance increases. The guided discovery strategy can be used with an entire class or with a small group or individual.

According to Mayer (2003), who believes that guided discovery tends to result in better long term retention and transfer of understanding and skills, guided discovery both encourages learners to search actively for how to apply rules and makes sure that the learner comes into contact with the rule to be learned. Teachers must give students appropriate and timely guidance, but must also have enough patience to let the learning process develop. Flexible thinking, a variety of readily available strategies and approaches, a willingness and ability to simplify problems and patience are valuable attributes for JUMP teachers.

Assigning independent practice. Most JUMP lessons end with independent practice in the form of assigned questions in the workbooks, problems to complete in the student’s math notebook or homework. This follow-up practice is an important component of the JUMP program. Independent practice helps students to retain and reinforce newly learned material as well a giving the teacher another way to assess learning and retention.

Instructional Strategies and Approaches

JUMP lessons lend themselves well to a variety of instructional strategies and approaches. Teachers are encouraged to develop and use these and other strategies as needed to respond to the needs of their students.

Explicit teaching is a core teaching strategy in a JUMP classroom. Topics and content are broken down into small parts and taught individually in a logical order. The teacher directs the learning, providing explanations, demonstration and modeling the skills and behaviours needed for success. Student listening and attention are important. Explicit teaching involves setting the stage for the lesson, telling students what they will be doing, showing them how to do it and guiding their application of the new learning through multiple opportunities for practice until independence is attained. This approach is modeled on our professional development videos.

Explanations and demonstrations are an important part of explicit teaching and key in JUMP instruction. The teachers explains the rule, procedure or process and then demonstrates how it is applied through examples and modeling. The demonstration provides the link between knowing about the rule to being able to use the rule. Research has shown that demonstrations are most effective when learners are able to follow them clearly and when brief explanations and discussion occur during the demonstration (Arends, 1998).

Concept formation enables students to develop and refine their ability to recall and discriminate between key ideas, to see commonalities and identify relationships, to formulate concepts and generalizations, to explain how they have organized data, and to present evidence to support their organization of the data involved.

Concept attainment is an indirect instruction strategy that used a structured inquiry procedure. Students figure out the attributes of a group or category that has been given to them by the teacher. To do so, they compare or contrast example that contain the attributes of the concept with those that do not.

page 25

Teacher’s Guide

Grade 7

Interactive instruction relies heavily on discussion and sharing among participants. As is illustrated in videos of JUMP classrooms, they are highly interactive learning environments and rely on various groupings, including whole class, small groups or pairs.

Cooperative learning is a useful interactive instructional strategy. Students work in groups which are carefully structured and monitored by the teacher. Specific work goals, time allotments, roles and sharing techniques are set by the teacher in order to ensure that all students are included and engaged. Think-pair-share, for example, is a strategy designed to increase classroom participation and “think” time while helping students clarify their thinking by explaining it to a partner. It is easy to use on the spur of the moment and in large class settings. Cooperative learning strategies can be used in a variety of ways in the JUMP classroom. Some examples of cooperative learning activities that work well in the JUMP classroom:

1. Have students turn to their partner and compare their answers to a problem.

2. Ask students to write a math problem, solve it, and then exchange problems with a partner. They check each other’s work and talk about it.

3. As an introduction to graphing, show the class pictures of different kinds of graphs from newspapers, magazines and other sources on the overhead projector. Before beginning to work in groups, the teacher reviews role expectations and what brainstorming means (giving as many ideas as possible with all ideas being acceptable).

Students are assigned to groups of four. Each person in the group has a role. The recorder writes down the thoughts of the group, the reporter shares the group’s ideas with the class, the timekeeper makes sure the group is on task and completes their work on time, and the encourages compliments group members on their participation and contributions. Groups are given three minutes to do their work and paper on which to record it. Students brainstorm different places in their live that they see graphs. The recorder writes down the ideas. The timekeeper makes sure the task is completed within three minutes. The encourager thanks group members for participating and compliments them on their contributions. The reporter shares the group’s list with the whole class.

The teacher creates a master list and facilitates a discussion about graphs and their purpose.

4. Have students play the 2-D sorting games in the Teacher’s Guide individually (see page 243), using their own sorting rules. Next, each student shares his sorting rule with a partner. The partners discuss their rules and come to some agreement about their criteria. They then share their ideas with another pair of partners. Work is done when everyone in the group has agreed on the sorting rule and can explain it to the teacher.

When using a cooperative learning approach, the teacher should verify that all students have understood the material before moving on. The JUMP worksheets can be used to assess what students have learned after a cooperative lesson.

Tips about cooperative learning strategies can be found at: http://olc.spsd.sk.ca/DE/PD/ instr/ strats/coop/index.html. Independent practice is an important part of the JUMP program. Following instruction, students are assigned practice questions from the student workbooks to consolidate their skills. Teachers can check the assigned work to make sure students have understood the lesson and are ready to move forward. Alternatively, if they have not understood the lesson, the teacher can decide how to reteach the content in a different way. It is suggested that JUMP questions be assigned to be completed at the end of the lesson or for homework.

Specific Instructional Techniques Wait time. Wait time is a key element of JUMP instruction. It 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, those 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

page 26

Teacher’s Guide

Grade 7

students and more frequent, unsolicited contributions. Teachers who increase their wait time tend to ask a greater variety of questions, are more likely to modify their instruction to accommodate students’ comments and questions and demonstrate higher expectations for their students’ success.

Increasing wait time has two apparent benefits for student learning (Tobin, 1987). First, it allows students more time to process and be actively engaged with the subject matter. Second, it appears to change the nature of teacher-student discussions and interactions.

Questioning. As in any classroom, questioning is an important instructional skill in a JUMP classroom. Strategic questioning helps teachers assess student learning, improves involvement and can help students deepen their understanding. In a JUMP lesson, questions are initially used to diagnosis levels of understanding. As the lesson progresses, the teacher uses questions to break down concepts and skills into smaller steps as needed, guiding student understanding incrementally. Questions can be used to elicit specific pieces of information or to stimulate thought and creativity. Socratic questions are useful to extend thinking and promote communication. They might begin with phrases like “Why do you think that?”, “How do you know that is true?”, “Is there another way of explaining that?”, “Could you give me an example?” (from http://set.lanl.gov/programs/CIF/Resource/Handouts/SocSampl.htm)

When questioning is used well:

• a high degree of student participation occurs as questions are widely distributed; • an appropriate mix of low and high level cognitive questions is used; • student understanding is increased; • student thinking is stimulated, directed, and extended; • feedback and appropriate reinforcement occur; • students’ critical thinking abilities are honed; and, • student creativity is fostered.

The teacher should begin by obtaining the attention of the students before the question is asked. The question should be addressed to the entire class before a specific student is asked to respond. Calls for responses should be distributed among volunteers and non-volunteers, and the teacher should encourage students to speak to the whole class when responding. However, the teacher must be sensitive to each student's willingness to speak publicly and never put a student on the spot.

Good questions should be carefully planned, clearly stated, and to the point in order to achieve specific objectives. Teacher understanding of questioning technique, wait time, and levels of questions is essential. Teachers should also understand that asking and responding to questions is viewed differently by different cultures. The teacher must be sensitive to the cultural needs of the students and aware of the effects of his or her own cultural perspective in questioning. In addition, teachers should realize that direct questioning might not be an appropriate technique for all students. (From http://olc.spsd.sk.ca/DE/PD/instr/questioning.html)

Scaffolding is the guidance, support and assistance a teacher or more competent learner provides to students that allows students to gain skill and understanding. It extends the range of what students would be able to do independently and is only used when needed. Scaffolding is a basic JUMP instructional skill that is ingrained both in the materials and in the lesson format.

Scaffolding involves several steps:

1. Task definition – what is the specific objective?

2. Establish a reasonable sequence

3. Model performance – demonstrate the learning strategy or skill while thinking aloud, explaining, answering one’s own questions

4. Provide prompts, cures, hints, links, partial solutions, guides and structures; ask leading questions; make connections obvious

5. Withdraw when the student is able to work independently

page 27

Teacher’s Guide

Grade 7

Demonstrating, explaining and questioning are all examples of scaffolding. As with questioning and other instructional strategies, scaffolding must be customized to suit individual students or groups of students. Pairing students and using cooperative learning strategies are two ways to provide scaffolding for students. Scaffolding needs to be tailored to meet the needs of specific students – it is intended to help the student move closer to being able to complete the task independently.

Scaffolding is demonstrated in the JUMP Guided Lessons and on the professional development video clips.

Drill and practice is an instructional technique that helps students learn the building blocks for more meaningful learning. Students are given opportunities to drill and practice their basic math facts throughout the program. Teachers are encouraged to assign homework to help students consolidate their skills and to provide information on student understanding for program planning. There are many web sites designed to help students learn and practice math facts (see partial list on page 29). Groupings in the JUMP Math Classroom Whole class instruction. JUMP lessons usually begin and end with a whole class grouping. The teacher sets the scene for the lesson with examples, questions or connections to previous lessons, other subject areas or real life experiences. At the end of the lesson, there is an opportunity to summarize what was covered, assign independent practice or set up the next lesson. As reported in Education for All: Report of the Expert Panel on Literacy and Numeracy Instruction (2005), research has shown that whole class instruction in mathematics is effective when both procedural skill and conceptual knowledge are explicitly targeted for instruction, and this type of instruction improves outcomes for children across ability and grade levels (Fuchs et al., 2002).

Individual work. There are several ways in which students may work as individuals during a JUMP lesson. For diagnostic and formative purposes, students are often asked to solve a problem or demonstrate a skill in their notebooks. If the need arises, students who need extra help are briefly supported individually by the teacher. Another routine part of JUMP lessons is the completion of independent practice based on the content of the lesson, either as a continuation of the lesson or as homework. There is an acknowledgement in the JUMP program that in order to properly determine student levels, some individual work and assessment is necessary. Pairs activities. Students are often encouraged to work with a partner using the Think-Pair-Share cooperative learning strategy. This enables them to share their thinking and discuss the concept or skill they are learning with a fellow learner. Peer support is also encouraged when appropriate. Students consolidate their skills when they are required to explain their reasoning to someone else. Small group work. Students work in small groups to solve problems, play games and discuss their work. They may also work in small tutorial groups on occasion when several students have similar questions or difficulties. Structured cooperative learning groups promote the participation of all students and encourage mutual support. Cooperative group responsibility and structural guidelines are very important to the success of this strategy. Students must understand that they are responsible for their own work and the work of the group as a whole. The group is only successful if everyone understands. Students must be willing to help if a group member asks for it or needs it. Another strategy is to allow small groups of students to ask the teacher questions only when everyone in the group has the same question. This encourages those who understand some aspect of the problem or skill to teach the others in the group. A useful place to learn more about cooperative learning and its benefits for students, including a self-guided tutorial and links to other sites, is: http://olc.spsd.sk.ca/DE/PD/instr/strats/coop/index.html Both pairs activities and small group work promote communication about mathematics. Through communication with their classmates, students are able to reflect upon and clarify their ideas, consolidate skills and deepen their understanding. Tutorial groups are structured by the teacher to support students who have similar needs or interests. When the teacher identifies a group of students who all need more practice or enrichment on a particular skill, she can work with that group separately from the rest of the class in a focused way to support their learning. This can often be

page 28

Teacher’s Guide

Grade 7

accomplished in just a few minutes while others are working independently or at another time, such as during the lunch period or after school if the teacher’s time permits. Classroom Management Classroom management is an important skill for all teachers. Teaching new material requires attentiveness. The teacher’s responsibility is to make sure everyone is following the lesson and respectful of others’ contributions. Eye contact, paying attention, taking turns, listening, participating and celebrating effort and success are all important aspects of a well-managed JUMP classroom. In classrooms where JUMP has been implemented, teachers often report that students are engaged and focused on the lessons and practice materials. A useful and interesting reference about classroom management is Classroom Teacher’s Survival Guide: Practical Strategies, Management Techniques, and Reproducibles for New and Experienced Teacher (Partin, R. L., 2005). Technology Links Web sites: There are a number of excellent web sites that support mathematical learning and problem-solving, as well as sites to help teachers plan interesting and creative lessons. An excellent user-friendly source of information and ideas for using the internet in teaching mathematics is Mathematics on the Internet: A Resource for K – 12 Teachers by J. A. Ameis. This guide includes help locating resources, planning lessons, engaging students in problem-solving and communication, as well as links to professional development in the areas of assessment, collaboration, and gender, multi-cultural, and special needs concerns. Virtual Manipulatives: An excellent article about using virtual manipulatives to support student learning can be found at http://my.nctm.org/eresources/view_media.asp?article_id=1902 This article explains the difference between different kinds of manipulative sites available on the internet, the advantages of using virtual manipulatives in the classroom, and questions to help teachers assess different sites. Math Central: http://mathcentral.uregina.ca/ This bilingual site, provided by the University of Regina, offers a number components for teachers, including resource sharing, lesson planning, teaching ideas, Teacher Talk, Quandaries and Queries, and Monthly Problems. National Council of Teachers of Mathematics: http://illuminations.nctm.org/ This very rich site contains resources for teachers and students, including activities organized by grade and links to other excellent sites. The Math Forum: http://www.mathforum.org/ This site provides resources for both teachers and students, including lessons, puzzles, problems and links to other valuable sites. The National Library of Virtual Manipulatives: http://nlvm.usu.edu/en/nav/vlibrary.html This site provides numerous engaging and useful activities for various grade levels which require students. Math Facts: http://home.indy.rr.com/lrobinson/mathfacts/mathfacts.html This site provides practice on math facts as well as links to other mathematics sites for students and teachers. A+ Math: http://www.aplusmath.com/ This site was developed to help students practice their math skills interactively. There are a variety of games and activities.

page 29

Teacher’s Guide

Grade 7

Census at School: http://www19.statcan.ca/r000_e.htm Census at School is an international online project that engages students from Grades 4 to 12 in statistical enquiry. The project began in the United Kingdom in 2000 and now includes participation from schools in Australia, Canada, New Zealand and South Africa. This project combines fun with learning, to the delight of hundreds of thousands of students around the world who have already participated. They discover how to use and interpret data about themselves as part of their classroom learning in math, social sciences or information technology. They also learn about the importance of the national census in providing essential information for planning education, health, transportation and many other services. Census at School offers students a golden opportunity to be involved in the collection and analysis of their own data and to experience what a census is like. Environment Canada: www.ec.gc.ca Bilingual; weather and environmental information. Stock Market: www.globeinvestor.com Up-to-the-minute Canadian stock market research and information. Statistics Canada: http://www.statscan.ca Canadian statistical data on a variety of topics, with useful information for teachers.

page 30

Teacher’s Guide

Grade 7

JUMP AND THE PROCESS STANDARDS FOR MATHEMATICS Problem-solving. Like many math programs, problem solving is the basis for JUMP Math. What distinguishes JUMP Math’s approach is the way that problem solving is taught and practiced. Prerequisite skills are identified and reviewed or retaught before students begin new sorts of problems. They are led through the stages of solving problems step by step in a logical and sequential way by the teacher who models each step. Students are given lots of practice before they tackle problems independently.

Problem solving strategies and activities are found throughout both the JUMP workbooks and Teacher’s Guide. Reasoning and proving. When students explain their reasoning to others, they are clarifying their own thinking as well as helping others to clarify theirs. As an integral part of the problem solving process, students must be able to explain their reasoning in a variety of ways, orally or on paper, using words, a picture, chart or model. In JUMP lessons, the teacher might ask students to explain to the whole class, to their small group or to a partner. In the workbooks, students frequently explain their work using words, diagrams, or pictures.

Reflecting. Students in JUMP classrooms are frequently asked to share how they solved a problem and to consider how others have solved it. They do this in the whole class setting, in small groups and pairs and individually, orally and in writing. Selecting tools and computational strategies. Students in JUMP classroom are encouraged to use a variety of methods to work with numbers and solve problems, including pencil and paper calculations, mental computations, estimation, calculators, models, drawings and manipulative materials. There is a specific section on Mental Math in the Teacher’s Guide (page 32).

To complete JUMP activities and solve problems, students must use tools (ruler, protractor, calculator), concrete materials and models (2- & 3-D shapes, base-ten materials for example), charts (hundreds chart, times tables), pictures, and other tools. Connecting. Connecting new concepts to real life, other subject areas or other mathematical ideas may help students relate to the content and engage in the lesson. Seeing relationships helps students to understand mathematical concepts on a deeper level and to appreciate that mathematics is more than a set of isolated skills and concepts but rather something relevant and useful.

Whenever new skills or concepts are introduce, JUMP teachers review or reteach the prerequisite skills necessary to move forward. This often entails explicitly making the links between and among the five strands in the curriculum. (Please see the JUMP Math: Teacher’s Manual for the Fraction Unit - Second Edition).

There are many ways to involve children in meaningful and relevant mathematics activities. Baking for a class fundraiser will require application of measurement skills and money concepts. Keeping track of statistics in the NHL or World Cup calls for data management skills. To introduce a lesson on perimeter, students could be asked what they would need to know if they were responsible for installing a fence to go around the school playground and then actually measure the perimeter of the yard before working with standard algorithms. Representing. The JUMP program includes numerous opportunities to represent mathematical ideas and relationships in a variety of ways. JUMP teachers explicitly teach and model mathematical notation, conventions and representations. As children learn and practice new concepts and skills, they are asked to represent their thinking and their work in different ways.

page 31

Teacher’s Guide

Grade 7

Communicating. Students in JUMP classrooms are encouraged to communicate frequently with the teacher and each other. Oral participation is a key component of the program and JUMP classrooms are typically highly interactive. Using a variety of questioning techniques, cooperative learning strategies and wait time, teachers ensure that all students are participating. Teachers model strategies while explaining their thinking out loud, teach appropriate symbols and vocabulary to facilitate written communication, encourage talk about the problem-solving process and encourage students to seek clarification or ask for help when they are unsure or do not understand.

Throughout the materials, students are asked to communicate their answers to problems using words, mathematical symbols, pictures, concrete materials or abstract models. Mathematics itself is a kind of language, with its own rules and grammatical structures, and math teaching and activities should help students become fluent in the language of mathematics. Works Cited: 1. Arends, R. Learning to Teach. New York: McGraw-Hill, 1998. 2. Fuchs, L. S., D. Fuchs, L. Yazdian and S.R. Powell. “Enhancing First-Grade Children’s Mathematical

Development with Peer-Assisted Learning Strategies.” School Psychology Review 31.4 (2002): 569-583. 3. Glickman, C. “Pretending Not to Know What We Know.” Educational Leadership 48.8 (1991): 4-10. 4. Mayer, R. E. Learning and Instruction. Upper Saddle River: Prentice Hall, 2003. 5. Tobias, Sheila. Overcoming Math Anxiety. Boston: Houghton Mifflin Company, 1980. 6. Tobin, Kenneth. “The Role of Wait Time in Higher Cognitive Level Learning.” Review of Educational

Research 57 Spring 1987: 69-95.

page 32

Mental Math There are few things you could teach your students that will have greater impact on their

academic careers than a sense of numbers. It is much easier to teach students their number facts

than is generally believed. In this section of the Teacher’s Guide, you will find a number of effective

strategies to help your students develop a facility with numbers.

Mental Math Skills: Addition and Subtraction

page 33

Teacher’s Guide

Grade 7

TEACHER:

If any of your students don’t know their addition and subtraction facts, teach them to add and subtract using their

fingers by the methods taught below. You should also reinforce basic facts using drills, games and flash 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 fingers 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 (Mental Math), 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)

To add 4 + 8, Grace says the greater number (8) with her fist closed. She counts up from 8, raising one

finger at a time. She stops when she has raised the number of fingers equal to the lesser number (4):

She said “12” when she raised her 4th finger, so: 4 + 8 = 12

To subtract 9 – 5, Grace says the lesser number (5) with her fist closed. She counts up from 5 raising

one finger at a time. She stops when she says the greater number (9):

She has raised 4 fingers when she stopped, so: 9 – 5 = 4

8 9 10 11 12

5 6 7 8 9

Mental Math Skills: Addition and Subtraction (continued)

page 34

Teacher’s Guide

Grade 7

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 finding the next even

number: i.e. 46 + 2 = 48, 48 + 2 = 50, etc. Knowing this, your students can easily add 2 to any

even number.

Name the next greatest even number:

a) 52 : ______ b) 64 : ______ c) 36 : ______ d) 22 : ______ e) 80 : ______

QUIZ

Name the next greatest even number:

a) 58 : ______ b) 68 : ______ c) 38 : ______ d) 48 : ______ e) 78 : ______

QUIZ

Mental Math Skills: Addition and Subtraction (continued)

page 35

Teacher’s Guide

Grade 7

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 finding the

preceding even number: i.e. 48 – 2 = 46, 46 – 2 = 44, etc.

Add:

a) 26 + 2 = ___ b) 82 + 2 = ___ c) 40 + 2 = ___ d) 58 + 2 = ___ e) 34 + 2 = ___

QUIZ

Name the preceding even number:

a) 48 : ______ b) 26 : ______ c) 34 : ______ d) 62 : ______ e) 78 : ______

QUIZ

Name the preceding even number:

a) 40 : ______ b) 60 : ______ c) 80 : ______ d) 50 : ______ e) 30 : ______

QUIZ

Subtract:

a) 58 – 2 = ___ b) 24 – 2 = ___ c) 36 – 2 = ___ d) 42 – 2 = ___ e) 60 – 2 = ___

QUIZ

Mental Math Skills: Addition and Subtraction (continued)

page 36

Teacher’s Guide

Grade 7

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 finding the next odd

number: i.e. 47 + 2 = 49, 49 + 2 = 51, etc. Knowing this, your students can easily add 2 to any

odd number.

Name the next greatest odd number:

a) 51 : ______ b) 65 : ______ c) 37 : ______ d) 23 : ______ e) 87 : ______

QUIZ

Name the next greatest odd number:

a) 59 : ______ b) 69 : ______ c) 39 : ______ d) 49 : ______ e) 79 : ______

QUIZ

Add:

a) 27 + 2 = ___ b) 83 + 2 = ___ c) 41 + 2 = ___ d) 59 + 2 = ___ e) 35 + 2 = ___

QUIZ

Mental Math Skills: Addition and Subtraction (continued)

page 37

Teacher’s Guide

Grade 7

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 finding the

preceding even number: i.e. 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 first adding 2, then 1 (e.g. 35 + 3 = 35 + 2 + 1). Subtract 3 from

a number by subtracting 2, then subtracting 1 (e.g. 35 – 3 = 35 – 2 – 1).

Name the preceding odd number:

a) 49 : ______ b) 27 : ______ c) 35 : ______ d) 63 : ______ e) 79 : ______

QUIZ

Name the preceding odd number:

a) 41 : ______ b) 61 : ______ c) 81 : ______ d) 51 : ______ e) 31 : ______

QUIZ

Subtract:

a) 59 – 2 = ___ b) 25 – 2 = ___ c) 37 – 2 = ___ d) 43 – 2 = ___ e) 61 – 2 = ___

QUIZ

Mental Math Skills: Addition and Subtraction (continued)

page 38

Teacher’s Guide

Grade 7

NOTE: All of the addition and subtraction tricks you teach your students should be reinforced with drills, flashcards

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 (e.g. 51 + 4 = 51 + 2 + 2). Subtract 4 from a number by subtracting

2 twice (e.g. 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 (i.e. 6 + 6 = 2 x 6). Students should either memorize the 2 times table or they should double numbers by counting on their fingers by 2’s.

Add a pair of numbers that differ by 1 by rewriting the larger number as 1 plus the smaller number (then

use doubling to find the sum): i.e. 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: i.e. 10 + 7 = 17.

Add 10 to any two-digit number by simply increasing the tens digit of the two-digit number by 1:

i.e. 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: i.e. 23 + 64 = 87.

SKILLS 15 and 16

To add 9 to a one-digit number, subtract 1 from the number and then add 10: i.e. 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: i.e. 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:

i.e. 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: i.e. 57 – 34 = 23.

Mental Math – Further Strategies

page 39

Teacher’s Guide

Grade 7

Further Mental Math Strategies

1. Your students should be able to explain how to use the strategies of “rounding the subtrahend (i.e.

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 - ____ + ____

b) 52 – 36 = 52 - ____ + ____

c) 76 – 49 = 76 - ____ + ____

d) 84 – 57 = 84 - ____ + ____

e) 61 – 29 = 61 - ____ + ____

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 = ____ + ____ = ____

b) 58 – 21 = ____ + ____ = ____

c) 43 – 17 = ____ + ____ = ____

d) 74 – 28 = ____ + ____ = ____

e) 93 – 64 = ____ + ____ = ____

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 = ____

b) 10 – 5 = ____

c) 14 – 7 = ____

d) 18 – 9 = ____

e) 16 – 8 = ____

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

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

The sum of counting up to the nearest ten and the original number is the difference.

Count from 50 until you reach the first number (62).

Count by ones from 45 to the nearest tens (50)

What method did we use here?

Mental Math Exercises

page 40

Teacher’s Guide

Grade 7

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 in the Teacher’s Guide (see note on Mental Math) 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 finding 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 finding 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 Math Exercises (continued)

page 41

Teacher’s Guide

Grade 7

9. Add:

REMEMBER: Adding 2 to an odd number is the same as finding 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 finding 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 in the Teacher’s Guide before you allow your students to answer Questions 13 and 14.

13. Add 3 to the number by adding 2, then adding 1 (e.g. 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 (e.g. 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?

Mental Math Exercises (continued)

page 42

Teacher’s Guide

Grade 7

TEACHER:

Teach skills 7 and 8 as outlined in the Teacher’s Guide.

17. Add 4 to the number by adding 2 twice (e.g. 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 (e.g. 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 in the Teacher’s Guide.

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 skill 11 as outlined in the Teacher’s Guide.

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 Exercises (continued)

page 43

Teacher’s Guide

Grade 7

TEACHER:

Teach skills 12, 13 and 14 as outlined in the Teacher’s Guide.

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 in the Teacher’s Guide.

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 in the Teacher’s Guide.

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 Math (Advanced)

page 44

Teacher’s Guide

Grade 7

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 first 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 =

I 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

II 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

III Sometimes in subtraction, it helps to think of a multiple of ten as a sum of 1 and a number

consisting entirely of 9’s (e.g. 100 = 1 + 99; 1000 = 1 + 999). You never have to borrow or

exchange when you are subtracting from a number consisting entirely of 9’s 1

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.

Mental Math Game: Modified Go Fish

page 45

Teacher’s Guide

Grade 7

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 first player takes it and lays it down along with the card from

their hand. The first player may then ask for another card. If the Player 2 doesn’t have the requested

card they say: “Go fish,” 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 first 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

finding 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 find

the pair that adds 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.

These numbers add to 10.

Mental Math Checklist

page 46

Teacher’s Guide

Grade 7

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

the pairs that

add to 5:

Can double

one-digit

numbers:

Mental Math Checklist

page 47

Teacher’s Guide

Grade 7

Student Name: Can add near

doubles

(i.e. 6 + 7 =

6 + 6 + 1):

Can add a

one-digit

number to any

multiple of 10

(i.e. 30 + 6 = 36):

Can add

any one-digit

number to a

number

ending in 9

(i.e. 29 + 7 =

30 + 6 = 36):

Can add one-

digit numbers by

“breaking” them

apart into pairs

that add to 10

(i.e. 7 + 5 =

7 + 3 + 2 =

10 + 2):

Can subtract

any multiple of

10 from 100

(i.e. 100 – 40 =

60):

Mental Math Checklist

page 48

Teacher’s Guide

Grade 7

Student can multiply and count by: Student Name: Can mentally

make change

from a dollar:

Can mentally

add any pair

of one-digit

numbers:

Can mentally

subtract any

pair of one-

digit numbers: 2 3 4 5 6 7 8 9

How to Learn Your Times Tables in a Week

page 49

Teacher’s Guide

Grade 7

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 difficulty seeing patterns in sequences and charts, solving proportions,

finding 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 five

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 JUMP 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 find the product 3 × 2, count by 2s until you have raised 3 fingers.

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 first 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 (i.e. 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:

You can find the product of 9 and any number by using the two facts given above. For instance, to find

9 × 7, follow these steps:

Step 1: 9 × 7 = __ __ 9 × 7 = __ __

Subtract 1 from the number Now you know the tens digit

you are multiplying by 9: 7 - 1 = 6 of the product.

9 × 4 = 36 9 × 8 = 72 9 × 2 = 18

3 is one less than 4

7 is one less than 8

1 is one less than 2

6

8

4

12 16

20

4

6

3

9 12

15

3

4

2

6 8

10

2

10

5

15 20

25

5

3 ×××× 2 = 6 2 4 6

How to Learn Your Times Tables in a Week (continued)

page 50

Teacher’s Guide

Grade 7

TEACHER:

• Make sure your students know how to subtract (by counting on their fingers if necessary) before you teach

them the trick for the nine times table.

• Give a test on Step 1 (above) before you move on.

Step 2:

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 find the ones digit of the first

five multiples of 8, by starting at 8 and counting

backwards by 2s.

8

6

4

2

0

Step 2: You can find the tens digit of the first

five 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 find the ones digit of the next

five multiples of 8 by repeating step 1:

8

6

4

2

0

Step 4: You can find the remaining tens digits by

starting at 4 and count up by 1s.

48

56

64

72

80

Practice 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

Count by 8 until you have 6 fingers up: 8, 16, 24, 32, 40, 48

So the missing digit is 9 – 6 = 3

(You can do the subtraction on your fingers if necessary)

9 × 7 = __ __

These two digits add to 9

9 × 7 = __ __ 6 6 3

How to Learn Your Times Tables in a Week (continued)

page 51

Teacher’s Guide

Grade 7

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:

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:

6 ×××× 6 = 36; 6 ×××× 7 = 42

Or, if you know 6 × 5, you can find 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 five facts:

2 ×××× 7 = 14; 3 ×××× 7 = 21; 4 ×××× 7 = 28; 5 ×××× 7 = 35; 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 = _____

6 × 4 = 4 + 4 + 4 + 4 + 4 + 4

5 fours plus one more 4

6 × 4 = 5 × 4 + 4 = 20 + 4 = 24

6 × 4 5 × 4

plus one more 4

page 52

Manual Notes Activities

Manual Notes

Activities and Problem Solving

Extensions

Workbook 7

page 53

Manual Notes We know that teachers have busy schedules, so we have included almost all of our hints for teaching

difficult material, as well as definitions and background material, on the worksheets themselves (so you

won’t have to keep consulting this manual). Most worksheets show you how to introduce concepts and

skills in steps, so we have only provided manual notes for a few worksheets where we couldn’t fit

everything on the page: these worksheets are marked with an .

Activities and Problem Solving For worksheets marked with an , we have provided activities, games and problem solving lessons

that you can use to introduce or reinforce skills and concepts covered on the worksheets. You may use a

worksheet keyed to a particular activity to verify that all of your students have the skills required to try the

activity, or as a diagnostic tool to verify that students have understood material introduced by the activity.

Extensions For worksheets marked with an , we have provided exercises that extends concepts introduced on

the worksheet. You may assign these exercises to students who finish their work early (although you

should try, as much as possible, to allow all of your students to attempt these exercises when you feel

they are ready). The exercises below are intended as samples, you can easily make up more of the

same type if you need them.

M

A

E

M

A

E

page 54

Workbook 7 Part 1

page 55

Teacher’s Guide

Grade 7

Patterns

PA7-1: Increasing Sequences page 1

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, I recommend the following method for recognizing gaps:

How far apart are 8 and 11?

Step 1: Say the lower number ("8") with your fist closed.

Step 2: Count up by ones, raising your thumb first, then one finger at a time until you have reached the

higher number (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 the method is foolproof.)

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. Here is one

approach you can use to help students find larger gaps between numbers.

Phase 1: Have your student memorize the gap between the number 10 and each of the numbers from

1 to 9: for instance, the gap between 8 and 10 is 2 (i.e. you need to add 2 to 8 to get 10). You

might make flash cards to help your student learn these facts (or have them play the Modified

Go Fish Game in the Mental Math section of this manual):

M

9 10 11

8

page 56

Teacher’s Guide

Grade 7

Front of card Back of card

You might also draw a picture of a number line to help your student visualize the gaps:

Phase 2: Have your student memorize the gap between 10 and each of the numbers from 11 to 19.

Again, you might use flashcards for this:

Front of card Back of card

Point out that the gap between 10 and any number from 11 to 19 is merely the ones digit of the

latter number (i.e. 16 minus 10 is 6, but 6 is just the ones digit of 16). If your student knows this

they will have no trouble recognizing the gap between 10 and any number from 11 to 19.

Phase 3: Your student can now find the gap between a particular number from 1 to 9 and a number

from 11 to 19 (say, between 7 and 15) as follows:

Step 1: Find the gap between 7 and 10 (by now, your student will know this is 3).

Step 2: Find the gap between 10 and 15 (your student will know this is 5).

Step 3: Add the two numbers you found in steps 1 and 2: 3 + 5 = 8. Hence 8 is the gap

between 7 and 15.

Show your student why this works with a picture:

Phase 4: Your student can use the method introduced in Phase 3 to find the gap between any pair of

two-digit numbers whose leading digits differ by 1. For instance, the gap between 47 and 55

is 8: starting at 47 you need to add 3 to get to 50 and then 5 to get to 55.

6 7 8 9 10 11 12

10 + ? = 17 10 + 7 = 17

6 7 8 9 10 11 12 13 14 15 16

+3 +5

46 47 48 49 50 51 52 53 54 55 56

+3 +5

8 + ? = 10 8 + 2 = 10

page 57

Teacher’s Guide

Grade 7

This method can ultimately be used to find the gap between any pair of two digit numbers. For instance:

to go from 36 to 72 on the number line, you add 4 to reach 40, then add 30 to reach 70, then add 2 to

reach 72: hence the gap between 36 and 72 is 4 + 30 + 2 = 36. (Before your student can attempt

questions of this sort they must be able to find the gap between pairs of numbers that have zeroes in

their units place: they should find the gap by subtracting the tens digits of the numbers in their heads: i.e.

the gap between 80 and 30 is 50, since 8 - 3 = 5.)

Do not discourage your student from counting on their fingers until they can add and subtract readily in

their heads. You should expect your student to answer all of the questions in this unit, even if they have

to rely on their fingers for help.

A weaker student can use their fingers to extend patterns 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 (i.e. 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 , 12 , ____ , ____

Step 3: Continue adding terms to extend the sequence.

3 , 6 , 9 , 12 , 15 , 18_

3 3

3 3

page 58

Teacher’s Guide

Grade 7

PA7-3: Extending Patterns and Identifying Rules page 3

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. 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, ___, ___, 48

3. One of these sequences was not made by a rule. Find the sequence and state the rules for the other

two sequences:

A 25, 20, 15, 10

B 6, 8, 10, 11

C 9, 12, 15, 18

4. Ask students to identify the starting numbers as well as the gap for the sequences below.

(For instance, the rule for the sequence 7, 10, 13, 16… is “Start at 7 and add 3.”)

a) 12, 15, 18, 21 b) 19, 17, 15, 13 c) 132, 136, 140, 144

5. Students should know the meaning of “term” and should be able to connect each term in a sequence

with its term number. (For instance, in the sequence 4, 10, 16, 22, the first term is 4, the second

term is 10, and so on.)

Here are some questions that will give students practice with this skill:

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, 125, 119, 113, 107

d) What operation was performed on each term in the sequence to make the next term:

2, 4, 8, 16… (Answer: Multiply the term by 2.)

6. From the description of each sequence, find the 4th term of the sequences:

a) Start at 5 and add 3.

b) Start at 40 and subtract 7.

E

page 59

Teacher’s Guide

Grade 7

PA7-4: Introduction to T-tables page 4

1. Give each student a set of blocks and ask them to build a sequence of figures that grows in a regular

way (i.e. according to some pattern) and that could be a model for a given T-Table. Here are some

sample T-tables you can use for this exercise:

PA7-5: T-tables page 6

1. Carol’s plant is 3 cm high and grows 5 cm per week.

Ron’s plant is 8 cm high and grows 3 cm per week.

How many weeks does it take before the plants will be at the same height?

PA7-7: 2-Dimensional Patterns page 9

1. Use a hundreds chart (see “Extra Worksheets and Blackline Masters,” page 257) to solve the

following problems:

a) I am a number between 30 and 100. My tens digit is 2 more than my ones.

b) I am an odd number between 35 and 50, the product of my digits is less than the sum.

c) Shade in the multiples of 9 on a hundreds chart. Describe the position of these numbers.

d) Add the ones digit and the tens digit of each multiple of 9. What do you notice?

e) What pattern do you see in the ones digits of the multiples of 9?

A

E

A

Figure

Number of Blocks

1 1

2 5

3 9

Figure

Number of Blocks

1 3

2 7

3 11

Figure Number of Blocks

1 4

2 6

3 8

page 60

Teacher’s Guide

Grade 7

2. Complete the multiplication chart. Describe the patterns you see in the rows, columns and diagonals

of the chart:

3. Shade any diagonal (from right to left) in a hundreds chart. What do you notice about the sum of the

digits of the numbers in the diagonal? (The sum of the digits is always the same) Can you explain

why this is the case?

4. Repeat question 1 by shading a diagonal from left to right. Students should notice that the sum of

the digits of the numbers in the diagonal increases by 2 as you move down the diagonal. Ask

students to explain why this happens. (Answer: As you move down the diagonal you move down

one row, which increases the tens digit by 1 and you move across one column, which increases the

ones digit by 1.)

PA7-8: Extensions page 10

1. Complete the magic square:

A pure 3 × 3 magic square places each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once in a 3 × 3 grid

in such a way that each row, column and diagonal add to the same number. In the following exercises,

we will make a pure 3 × 3 magic square:

2. By pairing numbers that add to 10, find 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.

3. Your answer to Question 2 tells you what all 3 rows add to. Show that each row (and hence column

and diagonal) must add to 15. This is called the magic number.

4. List all possible ways of adding 3 different numbers from 1, 2, 3, … 9 to total 15 (e.g., 2 + 4 + 9

works, but 3 + 3 + 9 and 6 + 9 do not). You should find 8 ways.

E

9 2 7

5 10

× 1 2 3 4 5 6

1

2

3

4

5

1 2

4

6

6

page 61

Teacher’s Guide

Grade 7

5. Look at a 3×3 grid. How many sets of numbers that add to 15 must the middle number be part of? Look at your list from Question 4 to determine which number must be in the middle.

6. Which numbers must be corner numbers? Why? 7. Make a pure 3 × 3 magic square:

8. Compare your magic square with those of other people. What transformations can you do to a magic

square to get another magic square?

You could then ask your students to make a magic square with the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10.

What will the new magic sum be? What if you use the numbers 3, 4, 5, 6, 7, 8, 9, 10, 11?

Make a T-table of magic numbers:

Numbers Used in the Magic Square Magic Number

1 – 9 15

2 – 10 18

3 – 11 21

4 – 12 24

Make sure your students understand that the second number in the first column isn’t necessary, so they could just make the T-table:

Least Number Used Magic Number

1 15

2 18

3 21

4 24

9. Predict the magic number for a magic square made with the 9 numbers: 67, 68, 69, 70, 71, 72, 73,

74, 75.

10. Sudoku is an increasingly popular mathematical game that is now a regular feature in many

newspapers. In the “Extra Worksheets and Blackline Masters” section of this guide, you will find

Sudoku suitable for children (page 237) with step-by-step instructions to find solutions. Once

students master this easier form of Sudoku, they can try the real thing.

page 62

Teacher’s Guide

Grade 7

11. Rearrange the numbers so that the number

in every row, diagonal and column add to 21:

12. Fill in the blanks in the Number Pyramid:

13. Pick one number from each row: Each number that you pick must be in a different column. Add the

numbers. Now repeat with a different set of selections. What do you notice about the two sums?

(Answer: They are the same.) Will this always happen? (Yes.) Can you explain why it happens?

EXPLANATION:

The first row is the 1 row, the second row the 6 row, the third is the 11 row, then the 16 row and the

21 row. One number is selected from each row. If you select a number in the, say, 16 row, you can

either pick 16 + 0, 16 + 1, 16 + 2, 16 + 3 or 16 + 4. No matter which row you pick from, you are

either adding 0, 1, 2, 3 or 4 to the first number from that row. Since you pick one number from each

column, you add 0 once, 1 once, 2 once, 3 once and 4 once, so the sum is 1 + 6 + 11 + 16 + 21 + 0

+ 1 + 2 + 3 + 4 = 65.

Perhaps teachers can give the students the hint to break up the numbers in each row as, for

example, 16 + 0, 16 + 1, 16 + 2, 16 + 3 and 16 + 4. Writing the 5 × 5 array like this might help the

students notice more patterns in the numbers they select. How often do the numbers 1, 6, 11, 16

and 21 get selected from the first part? And the numbers 0, 1, 2, 3 and 4 from the second part?

Giving the students a sheet with 8 or 12 arrays broken up like this might help them see the pattern

as they do different selections from each array:

1 1 + 1 1 + 2 1 + 3 1 + 4

6 6 + 1 6 + 2 6 + 3 6 + 4

11 11+1 11+2 11+3 11+4

16 16+1 16+2 16+3 16+4

21 21+1 21+2 21+3 21+4

As an activity, have students work in groups of 3 or 4 to make up their own 5 × 5 grid with the same

property that they can bring to a younger class and show off their magic trick to them. There are

different ways to portray this as a magic trick. One way is as follows. Have each group ask the

13 25

73

32

1 2 3 4 5

6 7 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

8

3 4 5

6 7 8

9 10 11

page 63

Teacher’s Guide

Grade 7

younger students to pick 5 numbers, one from each row and column, write them down and add

them. Ask them questions that make them think you’re using their information, but you’re really not –

example, is the third number’s ones digit bigger than the second number’s ones digit? and then just

to be funny ask at the end, “and when’s your birthday” or something silly and then give the answer

right away. Students can get creative with this. Each group can divvy up roles about who will ask

each question and who will answer each.

14. In a number pyramid, every number above the bottom row is the sum of the two numbers directly

below it, for example:

A missing number can sometimes be found using subtraction:

The number pyramids on the worksheets for this section can be solved directly by addition and

subtraction (students should start by finding a part of each pyramid where there is enough

information to fill in a missing number).

Here is a bonus question you can assign your students that

can be solved by trial and error:

Students should guess the number in the middle square of the bottom row, then fill in the squares in

the second row, checking to see if they add up to the number in the top square. Or students might

notice the following trick: adding the two numbers in the bottom row, subtracting their sum from the

number in the top row, then dividing the result by 2 gives the missing number in the bottom row.

To see why this trick works, notice that the missing number in the bottom row appears in the second

row twice, as in the following example, where the missing number is represented by the letter A:

As the number in the top row is the sum of the numbers in the middle row, we have:

15 = 4 + A + 5 + A or 15 = 2A + 9

2 3

2 3

5

3

7

3 4

7

15

4 A 5

4 + A 5 + A

15

4 A 5

15

4 5

page 64

Teacher’s Guide

Grade 7

Hence we have the rule stated above: to find the missing number A, subtract the sum of the two

numbers in the bottom row (4 + 5 = 9) from the number in the top row (15):

15 – 9 = 2A

Then divide the result by 2:

6 ÷ 2 = A so A = 3

This exercise can be used to introduce more advanced students to algebra.

NOTE: If students solve the pyramid above by guessing and checking, make sure previous guesses to inform

their next guesses. This is what makes it mathematical and not just random guessing. You might guide them in

this. For instance if they guess that the middle number is 6, they will see that the top number has to be 21, which

is too high. Should a number lower than 6 or higher than 6 be the next number? How much lower should their next

guess be?

Have your students do more examples of the same type to see if they can notice a pattern. If you

make up more examples for them, ensure that the top number is even if both the bottom numbers

are even or if they’re both odd; otherwise, the top number should be odd, as in the example. The top

number also needs to be bigger than the sum of the bottom numbers. Examples:

PA7-9: 2-Dimensional Patterns (Advanced) page 11

NOTE: For the activity on this worksheet your students will need a copy of the calendar sheet from the

“Extra Worksheets and Blackline Masters” section (page 256).

1. This activity supplements Question 4 on the worksheet:

Ask students to shade a 3 × 3 squares on their calendar and then add each set of 3 numbers that lie

along the directions shown in figure B. They will find that each sum is the same.

A

9

3 4

8

1 3

25

3 6

32

4 6

102

19 21

page 65

Teacher’s Guide

Grade 7

For instance, in the array shown on Figure C the sums are:

Ask students to explain why the sums are always the same. If they need a hint, rewrite all the

numbers in the 3 × 3 square as sums or differences involving the centre number in the square.

You can see from the pattern that the 3

numbers along any of the directions shown in

Figure B always add to 3 × the middle number

(12). This works for any 3 × 3 square on a

calendar.

2. Students could repeat the exercise in Questions 1 – 4 using a hundreds chart (page 257 in the

“Extra Worksheets and Blackline Masters” section) rather than a calendar. Ask students to say how

their answers to Questions 1 to 4 differ on a calendar and a hundreds chart.

The sequences introduced before section

PA7-16 have gaps that are constant:

E

12 - 8 12 - 7 12 - 6

12 - 1 12 12 + 1

12 + 6 12 + 7 12 + 8

Figure D

4 + 12 + 20 = 36

11 + 12 + 13 = 36

18 + 12 + 6 = 36

5 + 12 + 19 = 36

4 5 6

11 12 1313

18 19 20

Figure A Figure B

Figure C

Rule: Start at 3 and add 4

3 7 11 15

4 4 4

On worksheets PA7-16 to PA7-20, students

will find number sequences where the gap or

difference between terms changes:

Rule: Start at 3 and add 4

3 7 12 18

4 5 6

page 66

Teacher’s Guide

Grade 7

If you feel your students would enjoy a challenge, then you might assign them the questions below,

which all involve Pascal's Triangle. (You might wait until you have covered sections PA7-16 to PA7-20

before you assign Questions 1, 2, and 5, which all involve patterns with gaps that change.)

1. Describe the pattern in the numbers along the 3rd diagonal of Pascal’s triangle (repeat for the 4th diagonal). NOTE: The gap between the numbers in these diagonals increases in size in a regular way! For more

patterns where the gap increases see worksheets PA7-16 and PA7-17.

2. Try adding the numbers in the rows of Pascal’s triangle. What pattern do you see in the sums?

1 = 1

1 + 1 = 2

1 + 2 + 1 = 4

1 + 3 + 3 + 1 = 8

1 + 4 + 6 + 4 + 1 = 16

3. Start at the top of any (right to left) diagonal and move along the diagonal; adding the numbers you

encounter. Stop at any point you wish.

Example:

1

3

6

Where will you find the sum?

ANSWER: Just below and to the right of the last number you added:

1

3

6

10

4. How can you find the sum 1 + 2 + 3 + 4 + 5 quickly using Pascal’s triangle?

HINT: Use the pattern you found in Question 3.

5. Extend Pascal’s triangle up to 15 rows and shade the even numbers. What patterns do you see?

1 + 3 + 6 = 10

1 + 3 + 6 = 10

page 67

Teacher’s Guide

Grade 7

6. An application of Pascal’s triangle:

At a pizzeria you have a choice of 4 toppings: olives (O), tomatoes (T), peppers (P), and broccoli

(B). Ask your students to make an organized list of all the ways you could order a pizza if you were

allowed…

a) no toppings (There is only 1 way)

b) 1 topping (There are 4 ways: O, T, P, & B)

c) 2 toppings (There are 6 ways: BO, BP, BT, OP, OT, PT)

d) 3 toppings (There are 4 ways: BOD, BOT, BPT, OPT)

e) 4 toppings (There is only one way: BOPT)

Notice that the answers are the number in the 5th row of Pascal’s triangle. In general, the numbers in

the nth row of Pascal’s triangle tells you how many ways there are to choose n – 1 pizza toppings.

CHALLENGE: Ask students to figure out why this works. One way to see this is by adding another topping, say

mushrooms (M):

The number of pizzas with 2 of the 5 toppings

= # with mushrooms + # without mushrooms

= # of pizzas with 1 topping out of O, T, P and B (since the other topping will be M) + # of pizzas with

2 toppings out of O, T, P and B

= 4 (OM, TM, PM, BM) + 6 (OT, OP, OB, TP, TB, PB)

= 10

7.

Without extending Pascal’s triangle can you find the missing numbers in the 8th row? HINT: The first and last numbers in each row are 1. Also since the rows are symmetric, they can reduce their work

by half.

8. Find the missing letters:

HINT: This is an upside down variant of Pascal’s triangle – the two letters directly above a particular letter always

combine (by a fixed rule) to give the number. Notice that:

x + y = z x + z = y y + z = x

x + x = x y + y = y z + z = z

1

8

28

56

70

56

28

8

1

8th row

9th row

x

y

z

z

y y z z

x z x y x

z y x x z y

x y z y y x

page 68

Teacher’s Guide

Grade 7

PA7-10: Finding Rules for T-tables – Part I page 12

1. Students could make a design using concrete materials (as in Questions 3 and 4 in the worksheets)

and predict how many of each element in their design they would need to make 8 copies.

PA7-12: Finding Rules for Patterns page 17

Ask students to use blocks to make a sequence of shapes where the number of blocks in one part of the

shape stays the same, and the number of blocks in the other part increases directly with the position

number of the shape in the sequence (for examples, see question 3 on the worksheet). Students might

also try to describe the growth of their shapes in two different ways, as in question 5 on the worksheet.

PA7-13: Finding Rules for T-tables – Part II page 19

To discover a rule relating the input to the output of a T-table your students should first try adding or

subtracting a fixed number to the input. They should then try multiplying or dividing the input by a fixed

number.

If the input for the T-table increases by 1 and the output increases by a fixed amount, your student can

find the rule as follows:

EXAMPLE 1:

Input Output

2 7 3 10 4 13 5 16

Step 2:

If the output increases by a fixed amount X, while the input increases by 1, then you can derive the

output by multiplying the input by X and then adding or subtracting a fixed adjustment factor.

In the T-table above, the fixed amount of increase is 3.

A

A

M

By what amount does the output increase?

(In this case, the output increases by 3.)

Step 1:

page 69

Teacher’s Guide

Grade 7

Notice: Multiplying the input 2 by 3 gives 6, one less than the output 7.

Multiplying the input 3 by 3 gives 9, one less than the output 10.

Multiplying the input 4 by 3 gives 12, one less than the output 13, etc.

Hence the rule is: Multiply by 3 and add 1.

EXAMPLE 2:

Input Output

2 7

3 11

4 15

5 19

Step 2:

Notice: Multiplying the input 2 by 4 gives 8, one more than the output 7.

Multiplying the input 3 by 4 gives 12, one more than the output 11, etc.

Hence the rule is: Multiply by 4 and subtract 1.

If the input does not increase by 1 and the output does not increase by a fixed amount, your student

should find the rule for the T-table by guessing and checking.

NOTE: It is extremely important that your student learn to find rules for t-tables by the method outlined above as all of

the applications of T-tables in this unit involve finding rules.

Give your students a set of blocks and ask them to build a sequence of shapes that go with the tables in

Questions 2 and 3. Ask them if it is possible to find a part of each shapes in which the number of blocks

stays the same and another part in which the number of blocks increases directly with the position

number of the shape.

Your students will not always be able to do this: if the gap in the sequence is greater than the first term,

then it is not possible to build a sequence of shapes in which the number of blocks in part of each

shapes varies directly with the position number. Hence the kind of models that were used to illustrate

sequences in section PA7-12 cannot be built for all linear sequences of numbers (see for instance

Question 3 b) where the first term is 1 and the gap is 3). That is why it is important for students to

understand how to find a rule for linear sequences numerically by the method taught in this section other

than only relying on models. None of the math textbooks I have seen take account of this fact: they

present all sequences in terms of models (like the ones in PA7-12). Hence students do not learn to find

rules for more general linear sequences of the type in Question 3.

The output increases by 4

Step 1:

A

page 70

Teacher’s Guide

Grade 7

PA7-15: Two Ways of Describing Linear Patterns page 24

Students will be able to write a stepwise rule for the patterns in question 4 (for instance, the rule for 4 a is

start at 2 then add 1, then 2, then 3 and so on), but not a general rule. Students should recognize that

the gap in the patterns changes, so that the patterns are not linear. These non-linear patterns are studied

in PA7-16 and PA7-17.

1. Write a stepwise and general rule for the number of shaded rectangles in the pattern. How many

shaded rectangles will be in figure 8?

PA7-16: Patterns with Increasing & Decreasing Steps – Part I

page 25

1. The patterns below were made by multiplying successive terms by a fixed number and then adding

or subtracting a fixed number. Find the missing terms and state the rule for making the pattern.

Include the word term in your answer. (For instance, the rule for the first pattern below is “Start at 1.

Multiply each term by 2 and add 1.”)

a) 1, 3, 7, 15, 31, _____

b) 1, 4, 13, 40, _____

c) 2, 7, 22, 67, _____

d) 2, 3, 5, 9, _____

e) 1, 2, 5, 14, _____

M

E

Figure 1 Figure 2 Figure 3

E

page 71

Teacher’s Guide

Grade 7

PA7-17: Patterns with Increasing & Decreasing Steps – Part II page 26

1. Row Pascal's Triangle

Sum of Numbers in Each Row

of Pascal's Triangle

1 1 1

2 1 1 2

3 1 2 1 4

4 1 3 3 1 8

5 1 4 6 4 1 16

a) What pattern do you see in the sum of numbers in each row of Pascal's Triangle?

b) Use the pattern you found in 1 a) to find the sum of the numbers in the 8th row of Pascal's

Triangle.

2.

A restaurant has tables shaped like trapezoids.

2 people can sit along the longest side of a table, but only

one person can sit along each shorter side.

a) Draw a picture to show how many people could sit at 4 and 5 tables. Then, fill in the T-table.

b) Describe the pattern in the number of people. How does the step change?

c) Extend the pattern to find out how many people could sit at 8 tables.

3. a) The Ancient Greeks investigated numbers that could be arranged in geometric shapes.

The first four triangular numbers are shown in the figures.

i) Find the 5th and 6th triangular numbers by drawing a picture:

ii) Describe the pattern in the triangular numbers.

How does the step change?

iii) Find the 8th triangular number by extending the pattern you found in ii).

b) Repeat steps i) to iii) with the square numbers:

c) Add any two consecutive triangular numbers.

What property does the sum have?

E

Number

of Tables

Number of

People

1 3 6 10

1 4 9 16

page 72

Teacher’s Guide

Grade 7

PA7-17: Patterns (Advanced) page 28

1. Pick any two starting numbers and create a pattern using the rule for the Fibonacci sequence (i.e.

any term in the sequence is the sum of the two terms before it). For instance, if the starting numbers

are 4 and 5 the sequence is 4, 5, 9, 14, 23 … Create several Fibonacci-type sequences.

a) If your sequence starts with an even and an odd number, what type of number is every third

number in the sequence? What number is a factor of every 4th number?

b) Complete the Fibonacci sequences:

i) 2, ___, 10, ___, ___

ii) 3, ___, ___, 7, ___, ____, 31

iii) ___, ___, ___, ___, 13, 20

c) Ask students to make up Fibonacci puzzles like the ones in 1 b) above.

PA7-19: Predicting Patterns page 29

1. Put a dot in the centre of a polygon and draw a line from the centre to each vertex of the polygon.

How many line segments are there in each figure? Predict how many line segments there would be

in a hexagon and an octagon. Test your prediction.

E

E

page 73

Teacher’s Guide

Grade 7

Number Sense

NS7-1: Arrays and Factors page 31

To do some of the extensions in the Number Sense Unit, your students will need to know how to do long

multiplication. If some students don’t have this skill, assign them the review worksheets on multiplication

in the Blackline Masters. Your students should also know how to do long division. If they need to review

this concept, then you should cover the materials in Section NS7-43 and NS7-44 in the workbook

(see also the Manual Notes for these sections).

For a review of two-digit long division see the worksheets in the Blackline Masters.

Your students will have a great deal of difficulty with Number Sense if they don’t know their multiplication

facts. For students who need help remembering their facts, we would recommended:

i) That you do brief (one- to five-minute) regular reviews of the strategies for remembering

multiplication facts outlined in the Mental Math section of this Manual (see “How to Learn Your

Times Tables in a Week” on page 49).

ii) That you prepare a worksheet with all combinations of one-digit products and have students redo

the sheet until they have learned all (or most of) their facts.

iii) Provide students who don’t know these facts with multiplication tables.

If your students don’t know how to add and subtract multi-digit numbers, we suggest that you consult the

relevant sections of Part 1 of our Workbook 6.

1. The numbers that multiply to give a particular number are called the factors of the number. If you

draw all of the possible arrays for a number, as in Question 5 on the worksheet, then you have found

all of the possible factors of the number:

1 × 6 2 × 3 3 × 2 6 × 1

The factors of 6 are 1, 2, 3 and 6. Ask students to find all of the factors of 8, 12, 18 and 24. Ask

students to draw arrays for the numbers 2, 3, 5, 7 and 11. What do you notice? (There are only

2 arrays for each number.) Numbers that have only 2 distinct factors (1 and the number itself) are

called prime numbers: (i.e. 7 = 1 × 7 and 7 = 7 × 1 are the only factorizations of 7, and 1 and 7 are

the only factors of 7, so 7 is a prime.) Numbers that have more than 2 distinct factors are called

composites. Ask students if they can find the first 10 prime numbers. Then ask them to find the

longest string of consecutive composite numbers less than 30.

M

E

page 74

Teacher’s Guide

Grade 7

NS7-2: Organized Lists page 32

1. Ask your students to copy the shapes in below on cm2 grid paper.

They should cut the pieces out and try to assemble them into a rectangle. After your students have

worked on the puzzle, ask them if there is a systematic way they might have found the solution by

using the area of the small pieces. If they have trouble coming up with an answer, guide them with

the questions below.

a) What is the total area of the smaller pieces?

b) List all the rectangles with whole number side lengths that have that area.

c) Which of the rectangles from part b) can we eliminate as a possible answer?

d) Where does the 1×5 piece need to go? The 3×3 piece?

e) Now fit the pieces together.

For the pieces in Figure 1, it is probably just as easy to solve the puzzle by trial and error rather than

by using the area of the pieces. However, if students learn to eliminate possibilities by paying

attention to the area they will find more difficult puzzles, like the one in question 2, easier to

complete.

2. Use the same strategy to fit together these smaller rectangles to form a large rectangle.

a) What is the total area of the smaller pieces?

b) List all the rectangles with whole number side lengths that have that area.

c) Which of the rectangles from part b) can we eliminate as a possible answer? Why?

d) Where does the 1 × 12 piece need to go? The 3 × 3 and 3 × 5 pieces?

e) Now cut out the pieces and fit them together in your large rectangle.

f) How did organizing your information save time?

A

page 75

Teacher’s Guide

Grade 7

NS7-3: Prime Numbers and Composite Numbers page 33

1. Explore the patterns in the ones digits of the multiples of…

a) 2 and 8

b) 3 and 7

c) 4 and 6

What do you notice?

2. Extensions for question 6.

a) When were all multiples of 4 crossed out? Why? 6? 8? 9? 10?

b) What is the largest multiple of 11 that is less than 100? 11 × ____ = ____

c) What is the largest multiple of 12 that is less than 100? 12 × ____ = ____

d) What is the largest multiple of 13 that is less than 100? 13 × ____ = ____

e) Explain why any composite number less than 100 will have a number less than 10 as a factor.

f) Explain why any composite number less than 100 will have either 2, 3, 5 or 7 as a factor.

g) Explain why Eratosthenes’ Sieve works.

3. Are there more prime numbers less than 100 or more composite numbers? What strategy could you

use to answer the question without checking every number?

Solution: No even number except 2 is prime. Half the numbers from 3 to 100 are even. If you also

consider that all multiples of 5 are composite (and so on), clearly there are more composite numbers

than primes.

NS7-5: Prime Factorization page 35

Prime Factorizations and Organized Lists

Prime factorizations can make it easier to list all the factors of a number. For example, if 30 = 2 × 3 × 5,

then we can list all the factors of 30 by taking products of the prime factors, 1 at a time, 2 at a time, 3 at a

time, etc. until you have used up all the factors. (Don’t forget that 1 is always a factor).

1 2 3 5 2 × 3 2 × 5 3 × 5 2 × 3 × 5,

so 1, 2, 3, 5, 6, 10, 15, 30 are all the factors of 30.

List all factors of… a) 42 b) 105 c) 70 d) 770 e) 1430

E

A

page 76

Teacher’s Guide

Grade 7

Sometimes a number has a prime factor occurring more than once as in 12 = 2 × 2 × 3 or 18 = 2 × 3 × 3.

That means that when we list the factors, some will occur twice. The factors of 12 are: 1 2 3 2 × 3

2 × 3 3 × 3 2 × 3 × 3, so the factors of 12 are: 1, 2, 3, 6, 9 and 12. (We don’t have to list 6 twice).

List all factors of a) 18 b) 27 c) 75 c) 28 d) 135 e) 90

Show how to make a large rectangle by fitting together these smaller rectangles. Use the same strategy

as above.

TEACHER: Although not marked in the workbook, these extensions are for NS7-5.

1. Find the missing factor from each pair of multiplication statements:

a) 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 = 3 × 3 × 3 × 3 × 5 × 5 × 7 × ________

b) 22 × 222 × 2222 × 22222 = 11 × 111 × 1111 × 11111 × ________

c) 10 × 20 × 30 × 40 × 50 = 1 × 2 × 3 × 4 × 5 × ________

2. a) Draw factor trees for various prime numbers, e.g. 3, 7, 11, 13, 17, 19, etc.

b) What does the factor tree for a prime number always look like?

3. How many numbers less than 20 have a factorization tree with this branching pattern?

E

×

× ×

page 77

Teacher’s Guide

Grade 7

NS7-7: Factors, GCFs and LCMs page 37

ERRATA:

Question 2 should read: Find all the factors of each number below by dividing by the whole numbers in increasing

order (i.e. divide the number by 2, 3, 4, 5, 6 etc).

Question 5 should read: A number between 113 and 130 has 5 as factors. What is the number?

Before you assign the exercises on this page, please review long division (see manual note for NS7-1 in

this guide).

1. Complete the following chart:

Two Numbers Greatest

Common Factor Least Common Multiple

Product of the

Two Numbers

4 = 2 × 2

6 = 2 × 3 2 12 = 2 × 2 × 3 24 = 2 × 2 × 2 × 3

3, 5 1 15 = 3 × 5 15 = 3 × 5

4 = 2 × 2

12 = 2 × 2 × 3

3, 7

3

9 = 3 × 3

6 = 2 × 3

9 = 3 × 3

4 = 2 × 2

10 = 2 × 5

14 = 2 × 7

21 = 3 × 7

2, 17

44 = 4 × 11

99 = 9 × 11

Compare the greatest common factor, the least common multiple and the product of two numbers.

What pattern do you notice?

Check to see if this pattern continues to hold by checking five other examples of pairs of numbers.

By noticing a pattern, you have started the process that mathematicians use every day.

How can you, a mathematician, know for sure that the pattern always works? The question that

mathematicians need to ask all the time is: WHY does this pattern work? In the following exercises,

we will try to understand why this pattern holds.

TEACHER: Do not give the students the next worksheet until they have shown you the pattern.

A

M

page 78

Teacher’s Guide

Grade 7

i) Match the prime factors that are common to both numbers. Some pairs will not have any

matched primes. Circle the unmatched factors from both numbers.

a) 2 × 2 × 3 × 3 b) 2 × 3 × 7 × 7 c) 2 × 3 × 7 × 7 d) 2 × 2 × 7 × 13 e) 3 × 5 × 7

2 × 2 × 2 × 3 × 5 3 × 5 × 7 × 11 5 × 11 × 11 5 × 11 × 13 ×13 11 × 13

ii) Match the prime factors in both numbers and then find the prime factorizations of the LCM, GCF

and product of the two numbers.

a) 2 × 2 × 3 ×3

2 × 2 × 2 × 3 × 5

b) 2 × 3 × 3 LCM =

2 × 2 × 3 GCF =

Product =

iii) Answer the following questions by writing 0, 1 or 2 as appropriate. When finding the product,

LCM of GCF of two numbers how many times do we need to multiply….

a) The prime factors matched from both numbers?

b) The unmatched prime factors from the first number?

c) The unmatched prime factors from the second number?

Use your answer to the previous three questions to explain why the product of the least common

multiple and the greatest common factor of the two numbers is the same as the product of the two

numbers. When is the least common multiple equal to the product of two numbers?

Number 1 Number 2

Prime Factors

Matched from

both Numbers

Unmatched

Prime Factors

from Number 1

Unmatched

Prime Factors

from Number 2

4 6 2 2 3

3 5 3 5

4 12 2, 2 3

3 7

3 9

6 9

4 10

14 21

2 17

44 99

LCM = 2 × 2 × 2 × 3 × 3 × 5 GCF = 2 × 2 × 3 Product = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5

c) 2 × 3 × 3 × 7 LCM = 3 × 5 × 5 GCF =

Product =

Product LCM GCF

page 79

Teacher’s Guide

Grade 7

2. The Pool Factor (This activity adapted from work originally by Rich Cornwall.)

Look at a rectangular grid as a pool table with 4 corner pockets, but no side pockets.

Start in the bottom left-hand corner of the grid and hit a pool ball at a 45° angle. At some point, the

ball will fall into a pocket hole. We can:

a) Count the number of times the ball hits a side wall on its way (count both the starting and ending

points).

b) Count the number of squares the ball passes through on its trip.

c) Determine which hole the ball will finally fall into.

For the 4 × 6 grid, there are 5 hits, the ball passes through 12 squares and the final pocket is the top

left corner.

Complete the following chart by drawing pool tables of different dimensions on grid paper and drawing

the path of a pool ball hit at a 45° angle from the bottom left corner:

START

FINISH

page 80

Teacher’s Guide

Grade 7

Predict a rule for determining:

a) Number of hits from base and height

b) Number of squares passed through from base and height

Using b for base and h for height, express your answers from parts a) and b) in terms of variables.

Check your prediction with 5 other examples of base and height.

When does the ball land in:

a) The top right corner?

b) The bottom right corner?

c) The top left corner?

d) The bottom left corner?

3. Enriched Activity

As mathematicians, we cannot be satisfied with seeing that the pattern seems to hold – we need to

ask ourselves why this is happening.

Why does this pattern work? Look at the 4 by 6 grid and write another 4 by 6 grid beside it. Reflect

the path in the line between the two grids – what happens?

Base Height Number of

Hits

# of Squares

Passed Through Final Pocket

4 6 5 12 Top left

3 2

5 4

3 5

8 2

8 4

7 4

7 3

7 2

10 5

9 5

8 5

page 81

Teacher’s Guide

Grade 7

Do the same thing for the 3 by 5 grid, but using 3 copies of the grid instead of only 2.

Can you use this idea to see why the pattern holds?

TEACHERS: this activity is something that students should be allowed to play with and not expect themselves to

see the answer right away. It might even be a problem that they go back to on a weekly basis. This is what

mathematicians do every day – they go back to a problem they’re stuck on every once in a while.

NS7-8: Divisibility Tests page 38

1. True or false?

a) A number is divisible by 20 if it is divisible by both 4 and 5.

b) A number is divisible by 20 if it is divisible by both 2 and 10.

E

page 82

Teacher’s Guide

Grade 7

c) A number is divisible by 8 if it is divisible by both 2 and 4.

d) A number is divisible by 9 if it is divisible by both 3 and 3.

e) A number is divisible by 18 if it is divisible by both 2 and 9.

f) A number is divisible by 18 if it is divisible by both 3 and 6.

Make a prediction: when must a number that is divisible by two numbers be divisible by their

product.

2. The digits 1, 2, and 3 are each used once to form a 3-digit number.

a) List all the ways this can be done.

b) Use divisibility rules to determine which of the numbers from a) are divisible by…

i) 2 ii) 3 iii) 4 iv) 6 v) 12

HINT: A number is divisible by 6 if it is divisible by both 2 and 3 and a number is divisible by 12 if it is

divisible by both 3 and 4

3. A 2-digit number uses only digits from 1,2,3,and 4 at most once each.

a) List all the ways this can be done. HINT: There are 12 ways.

b) List all the numbers from your list above that are divisible by…

i) 4 ii) 3

4. Try to enter the digits 1, 2, 3, and 4 into the chart below to create numbers that satisfy the conditions

outlined in the brackets. One of them will not be possible.

a) b)

5. Enter the digits 2, 3, 4, 5 into the chart below to create numbers that satisfy the conditions outlined in

the brackets.

6. The digits 1,2,3,4,5 are each used once to compose a 5-digit number ABCDE such that the 3-digit

number ABC is divisible by 4, the 3-digit number BCD is divisible by 5 and the 3-digit number CDE is

divisible by 3. Find the number ABCDE. (from Crux Mathematicorum 30: No. 3 April 2004).

HINT: You may want to draw a diagram like questions 4 and 5 above.

A number that is divisible by 4

A number that is divisible by 3

A number that is divisible by 4

A number that is prime.

A number that is divisible by 3

A number that is divisible by 5

A number that is divisible by 4

A number that is divisible by 3

page 83

Teacher’s Guide

Grade 7

7. Start at 777 and count backwards by 3. Will you count the number 45? Students should realize that

you only count 45 if the difference of 777 minus 45 is divisible by 3.

8. Start at 7 and count by 3. Explain why 73 will be counted. ANSWER: The difference (73-7=66) is divisible by 3.

9. What is the largest 4 digit number in the sequence 1, 4, 7, 10…divisible by 3 with remainder 1:

a) 9995 b) 9996 c) 9997 d) 9999

10. A box contains less than 50 marbles. When the marbles are arranged in groups of 2, 3 or 5 there is

always one marble left over. How many marbles are in the box?

11. Raj subtracted 2 numbers and got 10. Anne multiplied the numbers and got 651. What are the numbers? HINT: Use the divisibility tests to find a one digit factor of 651. Use that factor to find other factors by long division. Look for two factors that differ by 10.

12. Give your students questions where they have to find mystery numbers using clues that involve the

divisibility conditions. For instance:

• I am a number between 15 and 25. I am divisible by 3 and 4.

• I am a number between 125 and 165. I am divisible by 3 and 5.

NS7-9: Proving the Divisibility Rules for 3, 4 and 9 page 40

TEACHER: Explain the terms assets, liabilities, expenses, & income before beginning this activity.

1. Assets are what you own, liabilities are what you owe. So Assets – Liabilities = Net Worth.

An accountant will add up the total value of each and if done correctly will find:

Assets – Liabilities – Net worth = 0

Bookkeepers need to examine their books each month to make sure that their books are balanced.

A book is balanced if they get 0 with the following formula:

Assets – Liabilities – Net worth = 0.

A

777 776 775 774 773 772 47 46 45 771

To count 45, this distance must be divisible by 3.

page 84

Teacher’s Guide

Grade 7

Suppose a book has the following totals:

a) Is this book balanced?

b) Anna verified that her books are balanced by putting 123 – 64 + 47 – 59 in her calculator. How

much would she be off by if she replaced:

i) Assets with $132? +$9_

ii) Liabilities with $46? –$18

iii) Liabilities with $54? ____

iv) Assets with $312? ____

v) Assets with $124? ____

vi) Assets with $103? ____

vii) Net worth with $95? ____

viii) Net worth $69? ____

Use a + sign if her calculation of Assets – Liabilities – Net worth will be bigger than 0 and a -

sign if her calculation will be less than 0.

c) List the mistakes made from getting only one digit wrong. How much was she off by in those

cases?

d) List the mistakes made from switching two digits. How much was she off by in those cases?

c) Anna noticed that when she switches two digits (instead of just getting one digit wrong), the amount she is off by has a special property. Find this property. Hint: Use the divisibility rules to find a special number that every mistake of this type is divisible by.

d) Anna decides that if the amount her answer is off by is divisible by this special number, then she

should look for a mistake where she switches two numbers. Find a mistake she could make by

only getting one digit wrong that would also be off by a multiple of this special number.

e) In each case, a bookkeeper makes a mistake and finds that Assets – Liabilities – Net Worth

does not give 0. For each value of Assets – Liabilities – Net Worth given below, decide whether

or not she should look for a mistake where she switches two numbers:

i) $45 ii) –$36 iii) $107 iv) $117 v) –$396 vi) –$226

NOTE: Bookkeepers regularly use this strategy to find their mistakes. If their books don't balance to 0, and the

answer they get is divisible by this special number, the mistake is almost always due to switching two numbers.

Assets Liabilities Net Worth

$123 $64 $59

page 85

Teacher’s Guide

Grade 7

2. Some mathematicians study codes in order to send messages in secret. They like to find problems

that are easy to do in one direction but hard to reverse, so that they can use the easy direction to

encode a message, but someone would have to use the hard direction to break the code and find

the message.

Factor each number and then verify by finding the product:

a) 38 b) 21 c) 45 d) 91 e) 143 f) 221

Which is easier, to factor 221 or to find the product of its prime factors? Which would you use to

encode the message – multiplying the two primes or factoring the product - and which would you

make someone use to break into the message? Why?

A mathematician wants to make codes from numbers with exactly two prime factors. Which of the

following “codes” would be easy to break without a computer? Which would take more work to break

without a computer?

HINT: Your divisibility rules to decide whether the numbers have small primes as factors.

a) 46 b) 221 c) 323 d) 194 e) 96 f) 145 g) 141 h) 2231

To build a message that is hard to break, would you use numbers that are products of two large

prime numbers or numbers that are products of a small prime number and a large prime number?

Why?

NOTE TO TEACHERS: After your students hand in this activity, tell them that to ensure that people cannot break

codes even using the powerful computers available today, mathematicians will use products of prime numbers

that are hundreds of digits long!

3. Applying the divisibility rule for dividing by 11. Before doing this activity do the extension in this

section on the divisibility rule for dividing by 11.

Either supply children with data of publisher numbers or have them do a Language Arts project

pretending they are a writer looking for a publisher. They write children’s books and want to decide

which publisher to use. Find the following information: The publishing company’s name and location,

whether they are a large or small company, and 5 best-selling books by that publisher. Which

company do you like best and why?

Find the ISBN #’s of all the books you found. Write down all the observations you notice about the

ISBN #s. How many digits are there? When do two books have several digits in common and where

are those digits located? What do you notice about the last digit that’s different from the other digits?

Each ISBN number has a portion of it assigned to the publisher. Is there a limit to the number of

books a publishing company can publish before it needs to apply for a new number? If it costs the

publishing company money to apply for a new number, should a large company that publishes many

books ask for a large number or a small one? Why?

page 86

Teacher’s Guide

Grade 7

The first digit of a 10-digit ISBN number is the region number, then the next group of numbers is the

number assigned to the publisher. The next group is the book number within that publishing

company. The single last digit is determined from the previous nine numbers. There is a formula to

obtain the last digit of the ISBN number from the previous nine digits. This formula allows the

computer to verify that the ISBN number entered in is correct. For example, the ISBN number

0-451-52838-7, if incorrectly entered as 0-452-52838-7 would no longer satisfy the formula and the

computer would tell you that there is an error. You could then retype in the number. Before looking

at the formula, we investigate the question:

How can you turn a 9-digit number into a single digit? And which mistakes would this method be

able to identify for us?

Method 1: Add up all the digits. Repeat the process until you end up with a 1-digit number. Put this

as your check digit.

Example: 0-451-52838 would become 36 and then 9, so the check digit would be 9. If I accidentally

typed in the following numbers instead, would the computer tell me there’s a mistake?

Why or why not?

a) 0-452-52838-9 b) 0-451-62838-9 c) 0-951-52838-9 d) 0-451-32838-9

e) 0-451-52888-9 f) 0-451-52838-8 g) 0-451-52837-9 h) 0-541-52838-9

i) 0-451-25838-9 j) 0-451-52388-9 k) 0-455-12838-9

What type of mistake do you think this method will identify? Which type of mistake will it not identify?

Method 2: Take the remainder of the 9-digit number when dividing by 10.

Example: 0-451-52838 would have check digit 8. If I accidentally typed in the following numbers

instead, would the computer tell me there’s a mistake? Why or why not?

a) 0-452-52838-8 b) 0-451-62838-8 c) 0-951-52838-8 d) 0-451-32838-8

e) 0-451-52888-8 f) 0-451-52838-6 g) 0-451-52837-8 h) 0-541-52838-8

i) 0-451-25838-8 j) 0-451-52388-8 k) 0-455-12838-8

What type of mistake do you think this method will identify? Which type of mistake will it not identify?

page 87

Teacher’s Guide

Grade 7

Method 3: Add all the digits of the 9-digit number and then take the remainder when dividing by 10.

Example: 0-451-52838 would become 36 and then 6, so the check digit would be 6. If I accidentally

typed in the following numbers instead, would the computer tell me there’s a mistake?

Why or why not?

a) 0-452-52838-6 b) 0-451-62838-6 c) 0-951-52838-6 d) 0-451-32838-6

e) 0-451-52888-6 f) 0-451-52838-8 g) 0-451-52837-6 h) 0-541-52838-6

i) 0-451-25838-6 j) 0-451-52388-6 k) 0-455-12838-6

What type of mistake do you think this method will identify? Which type of mistake will it not identify?

Method 4: Since adding all the digits doesn’t help to correct a mistake where I accidentally switch

two digits, I decide to try something a bit different. Find the first number multiplied by 1, the second

number multiplied by 2, the third number multiplied by 3, and continue until you have the ninth

number multiplied by 9. Then add the results. Call this number the SUM #. Take the remainder of

the SUM# when divided by 10 and call this number the check digit.

Example: If 0-451-52838, we do the following:

The SUM # of all the products is as follows: = 0+8+15+4+25+12+56+24+72

= (72+8) + (56+24) + (25+15) + 4 + 12

= 80 + 80 + 40 + 16

= 216

The remainder of the SUM # when dividing by 10 is 6, so the check digit is 6.

What type of mistake do you think this method will identify? Which type of mistake will it not identify?

Method 5: The actual check digit of an ISBN number is obtained by finding the SUM # and then

taking the remainder when dividing by a secret number. Find this secret number by writing down

several ISBN numbers and calculating their SUM #s.

0 × 1 0

4 × 2 8

5 × 3 15

1 × 4 4

5 × 6 30

2 × 7 14

8 × 8 64

3 × 9 27

8 × 9 72

page 88

Teacher’s Guide

Grade 7

Note that 123 is divisible by 3, so remainder is 1. 3 cannot be the secret number.

However, 120 is divisible by 5, so 123 has remainder 3. 5 might be the secret number. The same

secret number needs to work for all ISBN numbers.

Predict when X is the last digit of an ISBN #.

Verify your prediction by finding books in your classroom with ISBN # having last digit X.

Mathematicians have shown that this method will always find single-digit mistakes as well as

mistakes that interchange two digits.

Write an instruction manual explaining how to obtain the last digit of an ISBN number given the first

nine digits. Be sure to include examples where the last digit is an X.

Think of as many other places where it would be a good idea to have a “check digit” or “password

number”. Are there other situations where you wouldn’t want just any number to be a valid number?

Encourage them to ask their parents and to jot down a list and be prepared to give an oral

explanation for each thing on their list.

You are a publisher in Toronto, Ontario, Canada and have been assigned publisher number 2345.

You are going to publish your 837th book.

1) Find your region number.

2) Write down the first nine digits of the ISBN of your book.

HINT: you might need to put a 0 in front of the book number to make a total of nine digits.

3) Explain why you wouldn’t put a 0 at the end of the book number to make the nine digits.

4) Make up your own rule for getting the last digit. Explain your rule in words and give an example

using your 837th book of how you obtained the last digit.

ISBN number SUM # Remainder Check digit

0-451-53006-3 124 3 3

1-4000-3185-0 0

1-55278-011-2 2

3-8228-3135-2 2

3-8228-1628-0 0

8-876-24126-4 4

1-85797-801-3 3

0-451-52838-7 7

3-7913-2922-7 7

0-553-21175-7 7

page 89

Teacher’s Guide

Grade 7

TEACHER: Although not marked in the workbook, these extensions are for NS7-9.

1. Find a divisibility rule for dividing by 11.

a) Find the closest multiple of 11 to…

i) 10 ii) 100 iii) 1000 iv) 10000

b) Extend the pattern. Find the closest multiple of 11 to 10000000 and to 100000000.

c) List all the 2-digit numbers that are multiples of 11. How can you tell immediately if a 2-digit

number is a multiple of 11?

d) Any 3-digit number can be written as follows:

431 = 100×4 + 10×3 + 1

= 99×4 + 4 +11×3 – 3 + 1

= 99×4 +11×3 +4 – 3 + 1

Is 431 divisible by 11? How do you know?

e) Rewrite the following numbers as shown above. Which ones are divisible by 11?:

i) 778 ii) 162 iii) 450 iv)273 v)982

f) Write down a divisibility rule for dividing 3-digit numbers by 11.

g) Write down a rule for finding the remainder when a 3-digit number is divided by 11.

NS7-10: Powers page 41

NOTE: this activity is best done only after doing an activity such as the one below (Powers of Two) one that gives

students familiarity with the fact that any number can be written as a sum of powers of 2.

Ancient Egyptian mathematicians noticed that every number can be written as a sum of powers of 2. For example, 217 = 128 + 64 + 16 + 8 +1 = 27 + 26 + 24 + 23 + 20. The Ancient Egyptians used this observation to multiply any number by any other number as follows: To calculate 37 x 217, start with 37 and double it continually: 37 x 1 = 37 37 x 2 = 74 37 x 4 = 148 37 x 8 = 296 37 x 16 = 592 37 x 32 = 1184 37 x 64 = 2368 37 x 128 = 4736

Since … 217 = 128 + 64 + 16 + 8 +1,

37 x 217 = 37 x (128 + 64 + 16 + 8 +1)

A

E

page 90

Teacher’s Guide

Grade 7

= 37 x 128 + 37 x 64 + 37 x 16 + 37 x 8 + 37 x 1 = 4736 + 2368 + 592 + 296 + 37 = 8029.

1. Use the ancient method of multiplying to do the following questions and check your answer by using

the standard algorithm: 2. When you do these questions, you have a choice. For example, when multiplying

15 x 19, you could either write 15 = 1 + 2 + 4 + 8 and find 19 x 1 + 19 x 2 + 19 x 4 + 19 x 8 or you could write 19 = 1 + 2+ 16 and find 15 x 1 + 15 x 2 + 15 x 16. Do each of the above questions using the other way and decide which way you like better for each question and why.

3. State one thing you like about the ancient Egyptian method of multiplying. 4. State one thing you don’t like about the ancient Egyptian method of multiplying. 5. Powers of Two

Look at the “magic cards” below:

Card #1 Card #2 Card #3 1 3 5 7 2 3 6 7 4 5 6 7 9 11 13 15 10 11 14 15 12 13 14 15 17 19 21 23 18 19 22 23 20 21 22 23 25 27 29 31 26 27 30 31 28 29 30 31 Card #4 Card #5 8 9 10 11 16 17 18 19 12 13 14 15 20 21 22 23 24 25 26 27 24 25 26 27 28 29 30 31 28 29 30 31

Here’s the magic trick: You pick a number between 1 and 31. I ask you if it’s on card 1 and you say yes or no. Then I ask about card 2, then card 3, then card 4 and finally card 5. In each case, you answer yes or no. I can then tell you, even without looking at the cards, what your number is, even though I don’t have all the cards memorized. How do I do it?

Example: If you tell me that your number is on cards 1, 2 and 4, but not on 3 and 5, then I know that your number is 11.

Of course, there is a trick! To find it, answer the following questions.

15 x 19

13 x 27

37 x 29

page 91

Teacher’s Guide

Grade 7

a) Which numbers are only on one card?

______ is only on card # ______. ______ is only on card # ______.

______ is only on card # ______.

______ is only on card # ______.

______ is only on card # ______.

b) What pattern do you see in the numbers that are only on one card? Write the

numbers you found as exponents.

c) For each number that is on more than one card, write in the chart below the

numbers you found in question 1 that are on the same card.

d) What pattern do you notice when comparing the number occurring on more than one card to the

numbers from question 1 which occur on the same card?

e) Without looking at the cards or your chart, which card numbers is 19 on? 23? 29?

f) Without looking at the cards or your chart, which number is on cards 1, 3 and 4, but not on 2

and 5?

g) If I tell you that I am thinking of a number that is on cards 1, 3 and 5, but not cards 2 and 4.

What number am I thinking of?

h) Make up 6 cards with numbers from 1,,,, 63 by using the same trick.

i) Which numbers between 1 and 63 can be written as a sum of powers of 2? Do you think this

pattern continues to hold?

Number occurring

on more than one

card

Numbers from

question 1

which occur on

the same card

Number

occurring on

more than one

card

Numbers from

question 1

which occur on

the same card

3 1, 2 19

5 1, 4 20

6 2, 4 21

7 1, 2, 4 22

9 1, 8 23

10 24

11 25

12 26

13 27

14 28

15 29

17 30

18 31

page 92

Teacher’s Guide

Grade 7

1. How many 8’s must I add together to get a sum equal to 83?

2. How many 8×8 squares do I need to have in order to have a total area of 83?

3. Find 3 consecutive numbers which add to 60.

Hint: How can you use 60÷3?

4. a) 103 = _____

b) 203 = _____

c) Find the whole number which, when cubed, is closest to 1320. Hint: Will the

number be closer to 10 or 20?

d) The product of three consecutive whole numbers is 1320. Find these

numbers.

e) The product of three consecutive whole numbers is 7980. Find these

numbers.

5. The greatest common factor of 56 and 57 is: ______

6. The lowest common multiple of 56 and 57 is: ______

7. Extension for Question 8.

You saw in question 8 that 35 = 27 × 9 > 25 × 5 = 53. Break up the following powers to determine

which is bigger:

a) 73 or 34 b) 28 and 53 c) 212 and 55

Find the largest power of 2 that is smaller than 57.

8. Which is bigger:

a) 24 or 10

b) 28 =(2 × 2 × 2 × 2) × (2 × 2 × 2 × 2) or 10 × 10 = 102

c) 212 =(2 × 2 × 2 × 2) × (2 × 2 × 2 × 2) × (2 × 2 × 2 × 2) and 10 × 10 × 10 = 103

d) 23 and 10

e) 26 = (2 × 2 × 2) × (2 × 2 × 2) and 10 × 10 = 102

f) 29 and 103

g) 212 and 104

Without calculating it, determine how many digits the number 212 has.

E

page 93

Teacher’s Guide

Grade 7

9. Which is bigger between:

a) 53 and 102

b) 56 = (5 × 5 × 5) × (5 × 5 × 5) and 10 × 10 × 10 × 10 = 104

c) 59 and 106

d) 512 and 108

e) 54 and 103

f) 58 and 106

g) 512 and 109.

Without calculating it, determine how many digits the number 512 has.

NS7-11: Equal Parts of Regions and Lines page 43

1. Draw a picture to solve the puzzle. There are 7 triangles and squares. 27 of the figures are triangles,

37 are shaded. 2 triangles are shaded.

2. What fraction of each figure is shaded?

a) b)

c) d) e)

E

page 94

Teacher’s Guide

Grade 7

NS7-12: Equal Parts of a Set page 44

1. What fraction of an hour has passed since ? Reduce if possible.

a) b) c) d)

2. What word do you get when you combine…

a) the first 23 of sun and the first

12 of person? ANSWER: “SUPER”

b) the first 12 of grease and the first

12 of ends?

c) the first 13 of trance and the last

34 of luck?

d) the first 12 of wood and the last

23 of arm?

e) Try making up your own questions like this.

3. Would you take equal parts of a set or a region to find:

a) … what fraction of your class are girls? ________________

b) … what fraction of the pie has not been eaten? ______________

c) … what fraction of the pieces of the pie are small pieces?

d) … what fraction of a cookie is flour? ________________

e) … what fraction of a cookie recipe’s ingredients are dry ingredients? _________

f) … what fraction of the floor is white? ________________

4. A mayor of a small town wants to decide what fraction of the town’s streets are paved. What would

she be measuring if she took the fraction of a set? What would she be measuring if she took the

fraction of a region? If you were the mayor, which one would you use? Why?

E

9:15 9:07 9:40 9:30

9:00

page 95

Teacher’s Guide

Grade 7

NS7-15: Mixed and Improper Fractions page 47

1. Students should know how to count forwards by halves, thirds, quarters, and tenths beyond 1.

Ask students to complete the patterns:

a) 14 ,

24 ,

34 , ____, ____, ____, ____, ____

b) 2 24 , 2

34 , ____, ____, ____, ____, ____

c) 13 ,

23 , ____, ____, ____, ____, ____

d) 710 ,

810 , ____, ____, ____, ____, ____

2. Write the following fractions in order:

3 14 ,

274 ,

114 , 2

14 ,

364 , 4

34

3. Figure A is a model of 1 whole and Figure B is a model of

52 .

a) Ori says that Figure A represents more pizza than

Figure B. Is he correct? How do you know? b) Ori says that because Figure A represents more pizza

than Figure B, 1 whole must be more than 52 . What is

wrong with his reasoning?

NS7-17: More Mixed and Improper Fractions page 49

1. Give students counters to make a model of the following problem:

Postcards come in packs of 4. How many packs would you need to buy to send 15 postcards? Write

a mixed and improper fraction for the number of packs you would use.

Students could use a counter of a particular colour to represent the post cards they have used and a

counter of a different colour to represent the cards left over.

After they have made their model, students could fill in the following chart.

E

A

B

A

page 96

Teacher’s Guide

Grade 7

Number of Postcards

15

Number of packs of 4

postcards (improper fraction) 154

Number of packs of 4

postcards (mixed fraction) 3 34

Here is another sample problem students could try:

Juice cans come in boxes of 6. How many boxes would you bring if you needed 20 cans?

What fraction of the boxes would you use?

NS7-18: Fractions of Whole Numbers page 50

TEACHER: Although not marked in the workbook, these activities are for NS7-18.

The force of gravity is different on each planet in the solar system. Your weight depends on which planet

you’re on. On the moon, the gravitational pull is only about 1/6 of what it is on Earth, so if you weigh 60

kg on Earth, you would only weigh 10 kg on the moon – although your mass would still be the same.

Body Sun Mercury Venus Earth Moon Mars Jupiter Saturn Uranus Neptune Pluto

Gravitational pull as fraction of Earth’s gravity.

271920

821

111 1

16

821 2 411

2325

2225 1 3

22 342

1. Which body’s gravitational pull (other than Earth) is closest to that of Earth?

a) List the bodies in order from least gravitational pull to highest gravitational pull.

b) List the bodies in order from smallest to largest. You will need to do some research for this.

c) Compare your lists in parts a) and b). What do you notice?

2. Of the nine planets, on which one do you think it would be the hardest to move around on?

The easiest?

3. Scientists have discovered that by raising certain animals from birth in a lab with higher gravitational

pull, the animals become stronger. On which planet would an animal growing up on that planet be

the strongest? The weakest?

4. Match the words according to how they would fit in the following sentence:

A(n) __________________ weighs about 350 kg on ____________________.

Car Pluto

Elephant Sun

Sumo Wrestler Moon

Baby Neptune

Model: 4 postcards in each package

One left over

A

page 97

Teacher’s Guide

Grade 7

5. You want to move each of the following objects to a location that will make them weigh as close to

30 kg as you can. State where you should move each object and justify your answer with

calculations:

a) A person weighing 75 kg.

b) a 1L bottle of water weighing 1 kg.

c) a bike weighing 12 kg.

d) a motorcycle weighing 200 kg.

1. How many 18 sized pieces of pizza would be contained in…

a) 1 18 pizzas b)

118 pizzas c)

74 pizzas d) 3

12 pizzas e) 2

14 pizzas f) 5 pizzas

2. If a penny weighs 2 13 grams, how much will 100 pennies weigh? Express your answer as a mixed

fraction of grams.

3. Tina drinks 56 of a bottle of water each day. How much does she drink in 4 days? In 6 days? In 7

days?

4. If a 1500 m race takes 2 14 laps of a track, how many laps would a 500 m race take?

5. If 1 lane of a swimming pool is 38 the length of an Olympic swimming pool, how many lengths of an

Olympic swimming pool does Tina swim if she swims the lane 5 times?

6. Rita wants to buy a CD and finds the price with discounts at 3 different stores. Her options are $15

with 25 off the price, $14 with

37 off and $12 with

720 off. Find all three prices and decide which one you

would recommend.

NS7-19: Equivalent Fractions page 52

Anne makes a model of 25 using 15 squares as follows:

First she makes a model of 25 using shaded and unshaded squares, leaving as much space as possible

between the squares:

E

A

page 98

Teacher’s Guide

Grade 7

Step 1:

25 of the squares are shaded. She then adds squares one at a time until she has placed 15 squares:

Step 2:

Step 3:

From the picture, Anne can see that 25 of a set of 15 squares is equivalent to

615 of the set.

1. Give students blocks of 2 colours and have them make models of fractions of whole numbers using

the method described at the top of the worksheet. Here are some fractions they might try:

a) 35 of 15 b)

34 of 16 c)

35 of 20 d)

27 of 21

2. Ask students to draw 4 boxes of equal length on grid paper and shade 1 box.

Point out to students that 14 of the area of the boxes is shaded. Now ask students to draw the same

set of boxes, but in each box to draw a line dividing the box into 2 parts.

Now 28 of the area is shaded. Repeat the exercise, dividing the boxes into 3 equal parts, (roughly:

the sketch doesn’t have to be perfectly accurate), then 4 parts, then five parts.

Point out to your students that while the appearance of the fraction changes, the same amount of

area is represented.

14 ,

28 ,

312 ,

416 ,

520 all represent the same amount: They are equivalent fractions.

Ask students how each of the denominators in the fractions above can be generated from the initial

fraction of 14 . Answer: each denominator is a multiple of the denominator 4 in the original fraction:

8 = 2 × 4; 12 = 3 × 4; 16 = 4 × 4; 20 = 5 × 4;

312 of the area is shaded

416 of the area is shaded

520 of the area is shaded

page 99

Teacher’s Guide

Grade 7

Then ask students how each fraction could be generated from the original fraction. Answer:

multiplying the numerator and denominator of the original fraction by the same number:

14 =

28 ;

14 =

312 ;

14 =

416 ;

14 =

520 ;

Point out that multiplying the top and bottom of the original fraction by any given number, say 5,

corresponds to cutting each box into that number of pieces.

The fractions 14 ,

28 ,

312 ,

416 … form a family of equivalent fractions. Notice that no whole number

greater than 1 will divide into both the numerator and denominator of 14 :

14 is said to be reduced to

lowest terms. By multiplying the top and bottom of a reduced fraction by various whole numbers, you

can generate an entire fraction family. For instance 25 generates the family

25 =

410 ;

25 =

615 ;

25 =

820 ;

Ask students to use the patterns in numerators and denominators of the equivalent fractions below to

fill in the missing numbers.

a) 3 = 26 =

39 =

4 = 15

b) 5 = 610 =

9 =

1220 = 15

c) 3 = 8 =

912 =

1216 = 15

NS7-21: More Equivalent Fractions page 54

1. Find a fraction from the family 23 …

a)…whose denominator is 3 more than its numerator.

b)…whose denominator is 5 more than its numerator.

c)…whose numerator is a multiple of 3.

× 2 × 2

× 3 × 3

× 5 × 5

× 4 × 4

14

× 5 there are 5 pieces in each box × 5 there are 4 × 5 pieces altogether.

5 pieces each box

4 × 5 = 20 pieces altogether

× 3 × 3

× 2 × 2

× 4 × 4

E

page 100

Teacher’s Guide

Grade 7

d)…whose denominator is a multiple of 5

e)…whose denominator is 4 less than twice its numerator

2. Find a fraction from the family 12 whose numerator is prime and whose denominator is a multiple of 7.

3. If 23 of the players on a soccer team are boys and the team has 10 boys, how many players are on

the team?

4. Katie ate 14 of a pizza. If she ate 3 pieces, how many pieces did the pizza have?

5. True or false. Explain your answer.

a) A proper fraction can never be in the same family as an improper fraction.

b) A mixed fraction can never be in the same family as an improper fraction.

c) If two fractions are in the same family, one denominator must be a multiple of the other.

d) Any fraction in the same family as 715 must have a denominator that is a multiple of 15.

e) If a proper fraction (other than 1 or 0) has a denominator of 15, then any fraction in the same

family must have a denominator which is a multiple of either 3 or 5.

NS7-23: Further Addition and Subtraction page 56

1. Fill in the missing numbers.

HINT: There is a trick to this puzzle: Notice what the numbers on the left hand side add up to.

a) 510

+ 510

= 513

+ 13

b) 37 +

47 =

49 +

9

c) 710 +

310 =

135 - 5

d) 1099 +

5099 +

3999 =

150101 - 101

2. Rita picked 15 of a basket of berries, Melanie picked

25 of a basket, Paul picked

35 and Katie picked

45 . Did all their berries fit into two baskets?

E

page 101

Teacher’s Guide

Grade 7

3. Find 15 +

210 +

315 +

12 +

12 . Find an easy way to do this.

4. Predict the next two sums in this series:

1 + 12 = _____ ; 1 +

12 +

14 =_____ ; 1 +

12 +

14 +

18 = _____ ;

NS7-25: Adding and Subtracting Fractions page 59

1. Sally raises $1000 for three charities and decides to give 14 to a homeless shelter,

38 to a medical

supply organization to third world countries and the rest to a local food bank. Decide how much

money she gave to the local food bank by:

a) first finding the fraction of her money spent on the local food bank;

b) first finding the amount of money spent on the other two organizations.

2. In a student election, 14 of the students voted for Melanie,

16 for Katie,

13 for Rita and the rest voted for

Paul. Who won the election?

3. Write the numbers 1, 2, 4, and 8 into the fractions below to make the greatest and least sum possible. REMEMBER: You can only use each number once for each sum!

____ + ____ =

NS7-27: Comparing and Ordering Fractions page 64

1. Tom uses 43 of a cup of flour to make bread and Allan uses 2

12 cups. Who uses more flour?

2. Fill in the missing numbers in the sequence:

16 ,

13 ,

12 , ,

56 ,

HINT: Change all fractions to equivalent fractions with denominator 6.

E

E

page 102

Teacher’s Guide

Grade 7

3. Put the numbers into the correct boxes to complete the sequence:

410 ,

12 ,

35 ,

910 ,

15 , 1

4. Find the next two numbers in the pattern: 13 ,

26 ,

14 ,

312 ,

15 ,

420

HINT: Can you find any equivalent pairs of fractions? Reduce.

5. The number π is less than 4 but greater than 3. Use the fact that 3< π <4 to put the following

fractions in order from largest to smallest:

1π

14π

3π2

10π3

1 π

2 13π

6. Order the following numbers from largest to smallest.

1 π

3

9 π2

7. Comparing fractions using the same numerator.

The fraction 37 means that there are 7 pieces in a whole and we are choosing 3 of them. Each piece

is thus 1 seventh of a whole. The 7 tells you how big each piece is and the 3 tells you how many

pieces you are choosing.

Nadia compares 27 to

29 by thinking that if she has 2 pieces from a pie, she would get more pie if it

was only cut into 7 instead of into 9. So, bigger pieces and the same number of pieces implies a

bigger fraction. Use Nadia’s method to compare the following fractions. In each pair, circle the

biggest fraction.

a) 112

and 113

b) 720

and 715

c) 1113

and 1198

d) 9921001

and 992999

8. Order each group of fractions by finding i) a common denominator and ii) a common numerator.

110

310

710

810

page 103

Teacher’s Guide

Grade 7

a) 34 , 45 b)

74 , 53 c) 1

34 , 85 d)

34, 512

e ) 23, 35, 47

9. a) Order the following fractions by finding a common numerator or a common denominator –

which ever is easiest: 14,

25,

36

b) Continue the sequence in part a) by finding the next two fractions.

c) Look at your pattern. Which do you expect to be bigger: 197200 or

198201?

d) How much less than 1 is each fraction in your sequence from parts a) and b). Can you see why

the fractions in the sequence become larger?

10. Write a fraction between

a) 5760

and 6457

b) 5047

and 6057

c) 1 35 and

127 d)

1315

and 1415

e ) 78 and

89

NS7-30: Decimal Tenths and Hundredths page 67

1. Students often make mistakes in comparing decimals where one of the decimals is expressed in

tenths and the other in hundredths. (For instance they will say that .17 is greater than .2). The

following activity will help students understand the relation between tenths and hundreds.

Give each student a set of play-money dimes and pennies. Explain that a dime is a tenth of a dollar

(which is why it is written as $0.10) and a penny is a hundredth of a dollar (which is why it is written

as $0.01). Ask students to make models of the amounts in the left hand column of the chart and to

write as many names for the amounts as they can think of in the right hand columns (sample

answers are provided in the italics).

Amount

Amount in

Pennies

Decimal Names

(in words)

Decimal names

(in numbers)

2 dimes 20 pennies 2 tenths (of a dollar)

20 hundredths

.2

.20

3 pennies 3 pennies 3 hundredths .03

4 dimes and 3

pennies 43 pennies

4 tenths and 3 hundredths

43 hundredths

.43

.43

You should also write various amounts of money on the board and have students make models of the

amounts (i.e. make models of .3 dollars, .27 dollars, .07 dollars etc). Also challenge students to make

M

page 104

Teacher’s Guide

Grade 7

models of amounts that have 2 different decimal representations (i.e. 2 dimes can be written as .2

dollars or .20 dollars).

When you feel your students are able to translate between money and decimal notation ask them to

say whether they would rather have .2 dollars or .17 dollars. In their answers, students should say

exactly how many pennies each amount represents (i.e. they must articulate that .2 represents 20

pennies and so it is actually the larger amount).

Amount

(in dollars)

Amount

(in pennies)

.2

.15

1. Students can learn to count forwards and backwards by decimal tenths using dimes. Ask students to

complete the following patterns using dimes to help them count. (Point out that a number such as 2.7, while not standard dollar notation, can be thought of as “2 dollars and 7 dimes”. Ask students to practice saying the money amounts in the sequences below out loud as they count up. For instance, for the sequence 2.7, 2.8, ____, ____, students would say “2 dollars and 7 dimes, 2 dollars and 8 dimes, 2 dollars and 9 dimes…” etc. This will help them see that the next term in the sequence is 3 dollars.

a) .2 , .3 , .4 , ____, ____, ____

b) .7 , .8 , .9 , ____, ____, ____

c) 2.7 , 2.8 , 2.9 , ____, ____, ____

d) 1.4 , 1.3 , 1.2 , ____, ____, ____

2. Ask students to fill in the blank hundreds chart, counting by hundredths rather than ones.

.01 .02 .03 .04 .05 .06 .07 .08 .09 .10

.11 .12 .13 .14 .15 .16 .17 .18 .19 .20

.21 .22 .23 .24 .25 .26 .27 .28 .29 .30

Ask your students if they can find the following patterns in their charts (they should describe the position

of the patterns using the words “columns” and “rows”).

For extra practice, ask students to fill in the right hand column of the chart and then circle the greater amount. (Create several charts of this sort for your students.)

.45

.55

.65

.75

.68

.78

.88

.98

.14

.25

.34

.45

.12

.23

.34

.45

A

page 105

Teacher’s Guide

Grade 7

BONUS: Can you fill in the missing numbers in the patterns below without looking at the hundreds chart?

3. Give your students a set of base ten blocks. Tell them the hundreds block is the whole, so that a

tens block represents .1 and the ones block represents .01.

Show as many different decimals as you can by combining blocks.

Write a decimal for each answer.

Solution: 1.0, 1.1, 1.01, 1.11, 0.1, 0.01, 0.11

NOTE: One way to prepare students for this sort of exercise is to hold up combinations of blocks and ask them to

write the corresponding decimal in their notebook. For instance, if you held up the hundreds block (which

represents a unit) and the tens block (which represents a tenth) they would write 1.1.

4. Using base ten blocks show a decimal…

a) greater than .7

b) less than 1.2

c) between 1 and 2

d) between 1.53 and 1.55

e) with tenths digit equal to its ones digit

f) with hundredths digit one more than its tenths digit

5. Create models of 2 numbers so that...

a) the first number is 4 tenths greater than the first.

b) The first number has tenths digit 4 and is twice as large as the second number.

1. Write the decimals in order: 22.2 2.22 2.02 20.2 200 2.20 .220 .202

2. Write the decimals in order: .8 .4 .07 .17 .32 .85 .3

3. Explain how you would change 3.2 dm into cm.

E

.65

.75

.95

.33

.53

.64

page 106

Teacher’s Guide

Grade 7

4. Explain why it is important to keep track of the decimal place in keeping track of sums like:

12.50 + 3.2 + 5.832

5. Write in the decimal by estimating (rather than carrying out the operation):

a) 27.25 + 832.5 = 8 5 9 7 5

b) 57.23 × 2.5 = 1 4 3 0 7 5

c) 989.2 × 3.6 = 3 5 6 1 1 2

6. Without adding, how can you tell whether the sum is greater or less than 435?

9.5 + .37 + 407.63

7. Find any pairs of numbers that add to give a whole number:

6.3 2.3 1.4 3.6 2.9 4.7 1.1

8. Find 100 ÷ 8 on a calculator. How do you know from the decimal that there is a remainder?

NS7-31: Fractions and Decimals page 69

1. People have been trying to find approximations to π for thousands of years. In the table below are

some of the earliest approximations to π.

Approximate Year Fraction used to

Approximate π Region

Equivalent Decimal

Approximation to π

2000 B.C. 3 18 Babylon

2000 B.C. 25681 Egypt

250 B.C. 3 1071 < π < 3

17 Greece

150 A.D. 377120 Egypt

450 A.D. 355113 China

530 A.D. 39271250 India

a) Complete the chart by using a calculator.

b) Use your calculator to write π correctly to as many decimal places as you can. Press the π key

on your calculator.

c) To how many decimal places was Egypt’s 2000 B.C. estimate of π accurate? Their 150 A.D.

estimate?

d) India once estimated π as 10 . Do you think this was before or after 530 A.D.? Why?

e) Which of the regions above knew π to the most correct decimal places? The least?

A

page 107

Teacher’s Guide

Grade 7

f) Put the regions and dates in order from most accurate to least accurate. Do later dates tend to

be more accurate?

g) Which region, Egypt (150 A.D.) or Greece, had the more accurate approximation?

NS7-34: Comparing Fractions and Decimals page 72

1. Write the first two fractions in each group as a decimal and then round the third fraction to one

decimal place. Do not use a calculator.

a) 820

= ________ 620

= ________ 720

≈ __________

b) 1830

= ________ 1530

= ________ 1730

≈ __________

c) 615

= ________ 915

= ________ 715

≈ __________

d) 310

= ________ 315

= ________ 311

≈ __________

e) 918

= ________ 930

= ________ 919

≈ __________

2. Write each fraction as a decimal rounded to one decimal place. Do not use a calculator.

a) 724

b) 827

c) 3140

d) 1925

e) 541

f) 1631

g) 821

h) 2839

i) 3437

j) 297

k) 9597

l) 95101

3. Write a fraction between the two fractions given.

a) 15 and

13 b)

13 and

23

c) 47 and

67 d)

47 and

57

e) 710

and 810

f) 15100

and 21100

g) 17100

and 18100

h) 1125

and 1225

4. Write a fraction between the two decimals by first converting the decimals to fractions.

a) .1 and .3 b) .4 and .5 c) .17 and .23 d) .56 and .58 e ) .11 and .12

E

page 108

Teacher’s Guide

Grade 7

NS7-36: Adding Hundredths page 75

–

TEACHER: Although not marked in the workbook, these Activities are for NS7-36.

Give students a set of base ten blocks and assign the following problems.

1. Start with these blocks:

• Add 2 blocks so that the sum (or total) is between 3.4 and 3.48

• Write a decimal for the amount you added.

2. Take these blocks:

• Add 2 blocks so that the sum (or total) is between 2.47 and 2.63

• Write a decimal for the amount you added.

3. Take these blocks:

• Add 2 blocks so that the sum (or total) is between 2.51 and 2.6

• Write a decimal for the amount you added.

NS7-38: Operations with Decimals page 77

TEACHER: Although not marked in the workbook, these Activities are for NS7-38.

Repeat the exercise of the preceding section with the following instructions. 1. Take these blocks:

• Take away 2 blocks so the result (the difference) is between 1.21 and 1.35

• Write a decimal for the amount you took away.

2. Take these blocks:

A

A

page 109

Teacher’s Guide

Grade 7

• Take away 3 blocks so the result (the difference) is between 2.17 and 2.43

• Write a decimal for the amount you took away.

TEACHER: Make up more problems of this sort.

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):

a)

Thousands Block

b)

Hundreds Block

c)

Tens Block

d)

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.

Harder: (You might want to wait until students have finished more of the number sense unit

before you assign these questions.)

d) Show 315 using exactly 36 blocks.

e) I am a 2 digit number: Use 6 blocks to make me. Use twice as many tens blocks as ones

blocks.

f) I am greater than 20 and less than 30. My ones digit is one more than my tens digit.

g) I am a 3 digit number. My digits are all the same: use 9 blocks to make me.

h) I am a 2 digit number. My tens digit is 5 more than my ones digit: use 7 blocks to make me.

i) 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. What would I be if I was represented

by 7 blocks?

j) Show 1123 using exactly 16 blocks. (There are 2 answers.)

k) I am a 4 digit number. My digits are all the same: use 12 blocks to make me.

2. Wrap Up (Visualizing base ten materials)

Ask your students to imagine choosing some base 10 blocks…

Instructions:

a) You have more tens than ones. What might your number be? (More than one answer is

possible)

b) You have the same number of ones and tens blocks. What might your number be? (Or give

harder questions).

c) You have twice as many tens blocks as ones blocks. What two digit numbers could you

have?

d) You have six more ones than tens. What might your number be?

e) You have one set of blocks that make the number 13 and one set of blocks that make the

number 22. Could you have the same number of blocks in both sets?

page 110

Teacher’s Guide

Grade 7

f) You have one set of blocks that make the number 23 and one set of blocks that make the

number 16. Could you have the same number of blocks in both sets?

g) You have an equal number of ones, tens, and hundreds and twice as many thousands as

hundreds. What might your number be?

NOTE: If some of the questions are too hard to solve by visualization, let students sketch base 10 models.

3. Ask students to explain and show with base 10 blocks the meaning of each digit in a number with all

digits the same, for instance 3333.

NS7-40: Multiplying Decimals by 100 and 1000 page 79

1. Teach your students to multiply by higher multiples of 10:

a) 67 × 10 000 b) 385 × 100 000

c) 274 × 100 000 d) 39 × 1 000 000

e) 80 × 1 000 000 f) 502 × 100 000 000

NS7-42: Dividing Decimals by 10 and 100 page 81

1. In each case, compare your answers to the 2 questions. What do you notice?

a) 346 ÷ 10 =_______ 346 × 0.1 =_______ b) 29 ÷ 10 =_______ 29 × 0.1 =____

c) 1700 ÷ 10 =______ 1700 × 0.1 =______ d) 13 ÷ 100 =______ 13 × 0.01 =_____

2. In each question circle the pair that gives the same product or quotient:

a) 17 × 0.53 0.17 × 53 1.7 × 53 b) 3 × 0.3 30 × 3 30 × 0.03

c) 0.7 × 395 7 × 3.95 0.07 × 395 d) 3 × 31.27 30 × 312.7 30 × 3.127

e) 5.2 ÷ 1.4 52 ÷ 1.4 52 ÷ 14 f) 34.5 ÷ 6.3 345 ÷ 63 3.45 ÷ 63

g) 3.8 ÷ 11.5 38 ÷ 1.15 38 ÷ 115 h) 7.4 ÷ 60 7.4 ÷ 0.6 74 ÷ 6

E

E

page 111

Teacher’s Guide

Grade 7

NS7-43: Long Division – 2-Digit by 1-Digit page 82

1. Give each student a set of base ten blocks and 3 containers large enough to hold the blocks. Ask

students to make a model of 74 using the blocks. Point out that each of the tens blocks they used to

make the model is made up of 10 smaller ones blocks. Then ask them to show how they would

divide the 74 ones blocks in their model into the three containers as evenly as possible. (Students

could play this game in teams, with each team scoring points for the right answer.)

Students should see that they can solve the problem as follows:

Step 1: Make a representation of 74.

Step 2: Divide the tens blocks among the three containers as evenly as possible.

Students should see that they can only place 2 tens blocks per container, with one left over

(7 ÷ 3 = 2 R1). Point out to students that when they perform the standard long division algorithm, they

are simply keeping track of the steps they followed in dividing up the tens blocks.

Step 3: Students should recognize that they can only divide up the remaining units if they regroup

the tens block as ten ones.

In the long division algorithm, this step is equivalent to bringing down the number in the ones column.

) 74 3

2

Regroup the tens block as 10 ones and put all your ones together – now you have 14 ones.

6 14

) 74 3

2 You can put 2 tens blocks in each container.

You have 7 tens blocks to place. 6 1

You want to divide the blocks evenly into 3 containers.

You can only place 6 tens

blocks 2 × 3 = 6 There is one tens block left over.

A

page 112

Teacher’s Guide

Grade 7

Step 4: Divide the 14 ones among the 3 containers.

In the algorithm:

After placing the 2 tens blocks and 4 ones blocks into each of the containers you have placed 24

ones blocks altogether (and this is exactly the number you get on top of the division sign). Play the

game with different numbers. Keep track of the steps students take with the blocks and write the

equivalent step in the long division algorithm on the board. Ask students to come to the board to

explain how the step they took matches the step in the algorithm.

NS7-44: Long Division – 3 and 4-Digit by 1-Digit page 86

If some students are having trouble with the multiplication tables, you can teach them an algorithm for

long division that only requires multiplying by 1, 2 and 5. (Note that an easy way to multiply by 5 is to

multiply by 10 and then divide by 2). For example, to do 25 into 7626, they only need to know that 25×1

= 25, 25×2 = 50 and 25×5 = 125. Then, the algorithm would be

_____

25 7626

5000 200

2626

2500 100

126

125 5

1 R 305

M

There are 2 left over.

) 74 3

24 You can put 4 ones blocks in each container (14 ÷ 3 = 4)

6 14 12 2

You can place 12 ones altogether (4 × 3 = 12)

) _______

36 234652

180000 5000

54652

36000 1000

18652

18000 500

652

360 10

292

180 5

112

72 2

40

36 1

4 R 6518

)

page 113

Teacher’s Guide

Grade 7

1. Students could repeat the exercise of the last section with hundreds blocks. Ask them to model the

division question 852 ÷ 3 by putting base ten blocks into 3 containers.

Here is a different approach to the long division algorithm that is worth showing the students after they

are comfortable with base 10 blocks. They need to be comfortable with their full multiplication tables. It

will increase their understanding of the algorithm and give them a greater appreciation of how a

mathematician might discover the algorithm in the first place.

The algorithm will first find the number of digits in the answer and then find the digits one at a time.

Requiring students to think about the number of digits in the answer will require them to understand the

algorithm well enough to be able to estimate the answer.

2. Sally knows that 3×10=30, 3×100=300 and 3×1000=3000. She knows that 157 ÷ 3 will have 2

digits because 3×10<157<3×100, so the answer will be at least 10, but less than 100. How many

digits does each answer have:

a) 931 ÷ 3 b) 147 ÷ 3 c) 23 ÷ 3 d) 300 ÷ 3 e) 1465 ÷ 3 f) 84 ÷ 3

This method of estimation can be extended to dividing decimals by whole numbers.

Complete the chart and then use it to answer the questions below:

3 × .001 = _______

3 × .01 = _______

3 × .1 = _______

3 × 1 = _______

3 × 10 = _______

3 × 100 = _______

3 × 1000 = _______

a) _______ < 174 ÷ 3 < _______

b) _______ < 0.14 ÷ 3 < _______

c) _______ < 72.6 ÷ 3 < _______

d) _______ < 1127 ÷ 3 < _______

e) _______ < 2.56 ÷ 3 < _______

f) _______ < 3.8 ÷ 3 < _______

g) _______ < 0.045 ÷ 3 < _______

A

Think: 3×10 < 174< 3×100 10

0.01

100

0.1

page 114

Teacher’s Guide

Grade 7

Complete the chart and then use it to answer the questions:

1.7 ×. 001 = _______

1.7 × .01 = _______

1.7 × .1 = _______

1.7 × 1 = _______

1.7 × 10 = _______

1.7 × 100 = _______

1.7 × 1000 = _______

a) _______ < 174 ÷ 1.7 < _______

b) _______ < 0.14 ÷ 1.7 < _______

c) _______ < 72.6 ÷ 1.7 < _______

d) _______ < 1127 ÷ 1.7 < _______

e) _______ < 2.56 ÷ 1.7 < _______

f) _______ < 3.8 ÷ 1.7 < _______

g) _______ < 0.045 ÷ 1.7 < _______

Put in the decimal point so that the number is between 10 and 100. Add 0s where necessary.

a) 27341 b) 321 c) 5 d) 12 e) 1371

Put in the decimal point so that the number is between 0.1 and 1. Add 0s where necessary.

a) 27341 b) 321 c) 5 d) 12 e) 1371

Put in the decimal point so that the number is between the two given values. HINT: In some questions you will have to add zeroes to the right or left of the number in order to add the decimal point

in a given place value.

a) 642 b) 52 c) 546 d) 4567 0.01 and 0.1 0.01 and 0.1 1 and 10 0.1 and 1

e) 5 f) 84 g) 3176 h) 27643 10 and 100 0.001 and 0.01 0.01 and 0.1 100 and 1000

i) 14 j) 7 k) 27 l) 37 100 and 1000 0.1 and 1 0.1 and 1 10 and 100

2. How many digits does each answer have:

a) 279 ÷ 13 b) 2357 ÷ 24 c) 8793 ÷ 7 d) 7458 ÷ 76 e) 3 543 567 ÷ 432

Think: 1.7×10 < 174< 1.7×100 100 10

0.01 0.1 Think: 1.7×.01 < 0. 14< 1.7×.1

page 115

Teacher’s Guide

Grade 7

3 Now that Sally knows that 3 into 157 has two digits, she wants to know what the first digit is. Sally

thinks: 3 × 50 < 157 < 3 × 60, so the answer must be at least 50 but less than 60. That means the

first digit is a 5. Find the first digit in each case.

Example: 931 ÷ 3

3 into 931 has 3 digits, so compare 3 × 100, 3 × 200, etc.

Then 3 × 300 < 931 < 3 × 400, so the answer is between 300 and 400; the first digit is 3.

Here are some questions to practice with:

a) 147 ÷ 3 b) 23 ÷ 3 c) 300 ÷ 3

d) 1465 ÷ 3 e) 84 ÷ 3 f) 279 ÷ 13

g) 2357 ÷ 24 h) 8793 ÷ 7 i) 7458 ÷ 76

j) 4732 ÷ 163 k) 63 045 ÷ 264 l) 3 543 567 ÷ 432

4 Sally now knows that there are two digits and she knows the first digit. She is almost done! She only

needs to know the second digit. This step is the key to the algorithm. It turns the problem into an

easier one. Since she knows that 3×50=150, she just has to find out how many times 3 divides into

157 – 150 = 7. If 3 divides into 150, fifty times and into 7 twice with remainder 1, then 3 divides into

157, fifty -two times with remainder 1.

This algorithm can be used with decimals too. In fact, now that students have a good idea of how to

estimate their answers, they can easily determine where to put the decimal place in their answer.

5. How many digits before the decimal point will each answer have:

a) 23.5 ÷ 4 b) 76.3 ÷ 25 c) 43.5 ÷ 5 d) 8.4325 ÷ 6 e) 43 567.34 ÷ 13

If the answer is less than 1, (some questions should be given to ensure the students can tell when

the answer will be less than 1), then they need to determine how many 0s will occur after the

decimal point, i.e. when will the first non-zero digit appear.

Eg. 0.01 ÷ 17 � 17 × 0.1 = 1.7, 17 × 0.01= 0.17 17 × 0.001=0.017, 17 × 0.0001 = 0.0017, so the

answer is between 0.0001 and 0.001, so there will be three 0s before the first non-zero digit after the

decimal point.

It’s not any harder to do long division by a decimal number instead of a whole number.

6. How many digits before the decimal point will each answer have:

a) 23 ÷ 1.7 b) 76.3 ÷ 2.5 c) 456 ÷ 37.2 d) 435 ÷ 0.5 e) 33 ÷ 0.02

page 116

Teacher’s Guide

Grade 7

NS7-45: Dividing Decimals by Whole Numbers page 88

This is a difficult worksheet. Provide an example where each step is done thoroughly before asking the

students to do these worksheets. Also, make sure that the students master each step before allowing

them to proceed.

NS7-48: Concepts in Decimals page 92

1. Under which deal do you pay less for each apple: 4 apples for $1.99 or 6 apples for $3.49?

2. On a map, 1 cm represents 35 km. Two towns are 3.7 cm apart on the map.

How far apart are the towns?

3. It took Cindy 20 minutes to finish her homework: she spent 35 of the time on math and

15 of the time

on history.

a) How many minutes did she spend on math and history?

b) How many minutes did she spend on other subjects?

c) What percent of the time did she spend on other subjects?

E

M

page 117

Teacher’s Guide

Grade 7

Measurement

ME7-1: Measuring Lengths page 93

1. Students can use benchmarks, such as the width of their thumb or index finger or little finger

(approximately 1 cm) or the width of their hand with fingers spread slightly (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 other non-standard units, such as the diameter of a penny (which is about 2 cm)

to estimate or measure lengths. Give students play money pennies and have them 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’.)

ME7-2: Estimating Lengths page 94

1. Draw each object to the measure given:

a) A shoe, 6 cm long b) A tree, 5 cm high c) A glass, 3 cm tall

2. Draw a triangle on grid paper and measure its sides to the nearest cm.

3. Ask students to name an object in the classroom that they think would have length…

a) 60 mm b) 300 mm c) 100 cm d) 200 cm

4. Students will find estimating with large numbers in mm is easier if they mentally change the

measurements in mm to cm (by dividing by 10) and then use the width of their index finger or little

finger, which is about 1 cm, to estimate. Students might also change the measurements in cm to

metres and then use their arm span (or the length of their leg) to estimate in metres.

5. 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 the

three ways shown above.

A

A

page 118

Teacher’s Guide

Grade 7

6. Ask students to measure and compare the lengths of various body parts using a string and a ruler.

Students could investigate the following questions:

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?

Encourage your students to predict the answers before they perform the measurements.

7. Hold up a metre stick and point to various positions on the stick. Ask students to say whether the

positions are closer to 0 metres, 12 metre, or 1 metre. When students estimate and perform the

measurements in question 1 on the worksheet, ask them to give their estimates to the nearest

half metre and their answers to the nearest cm. (Make sure students understand that half a metre

is 50 cm.)

8. On a school walking trip, ask students to say when they think they have walked half a kilometre or

1 kilometre.

9. Ask students if they have taken any trips to nearby towns. Ask them to estimate how many

kilometers away the towns are and then have them check the actual distances by measuring the

distances on a map with a ruler, and then converting their measurements to kilometers using the

scale on the map.

10. “Could you…?” Challenge:

Here are some questions you could ask your students to help them practice estimating.

For part c below, you should teach your students how to multiply by multiples of ten

(i.e. 30 × 600 = 1800).

a. (Warm Up): Could you…fit all the students in your school onto three school buses?

Solution:

Ask – How many students fit in one school bus?

Ask – How many students in the school?

Estimate – Round the numbers.

Answer – Unlikely (unless it is a very small school) that you could fit all the students in your school

in three school buses.

b. (Warm Up): Could you…walk up 100 steps from your classroom to the main office?

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.

c. (Harder): Could you…reach as high as the CN Tower with all the rulers in your school?

Solution:

Research – What is the height of the CN Tower? The CN Tower is about 555 metres tall. This is

equal to 55 500 cm.

page 119

Teacher’s Guide

Grade 7

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.)

d. Could you…stack 100 pennies to be as high as the school?

e. Could you…fit 500 ice cubes into a dishwasher?

f. Could you…count every hair on your head in a week?

g. Could you…stack 100 school buses on top of each other so that the pile would be taller than

Mount Everest?

h. Could you…place 500 toothpicks in a line around the entire perimeter of your school?

i. Could you…give everyone at your school a granola bar if you had an entire refrigerator

full of them?

j. Could you…fill up a bathtub with 1000 packets of ketchup?

11. Big Numbers:

Here are some questions you could assign or discuss with students to give them practice calculating

and estimating with large numbers. Your students will need to know how to multiply whole numbers

by multiples of 10.

a. (Warm Up): There are 60 seconds in 1 minute. How many seconds are there in 1 year?

Solution:

Start with seconds: How many seconds in one minute? 60 seconds

How many minutes in a day? 60 minutes = 1 hour; 1 day = 24 hours

24 × 60 = 1440 minutes in one day

1440 × 60 = 86 400 seconds in one day

Finally, multiply the number of seconds in one day by the number of days in one year:

86 400 seconds in one day × 365 days in one year = 31 536 000 seconds in one year!

In one year there are more than 31 million seconds!

page 120

Teacher’s Guide

Grade 7

b. How many minutes have you been alive?

Solution:

Start with the date the student was born and the current date.

Example: Born August 29, 1995. Current date is September 16, 2005.

From August 29 to the end of the month we add three days. From September 1, 1995 to

September 1, 2005 is 10 years. The current date is September 16, so we add: 3 days + 10 years

+ 16 days = 19 days

The student has been alive for about 10 years and 19 days in total.

How many minutes in one day? 24 × 60 = 1440 minutes in one day = 1440 minutes in one day

How many minutes has the student been alive for?

1 440 minutes in one day × 365 days in one year = 525 600 minutes in one year

525 600 minutes in one year × 10 years; Add: 1440 × 19 (extra days) = 27 360

5 256 000 minutes in 10 years

+ 27 360 minutes in 19 days

5 283 360 minutes in the student’s life so far!

NOTE: For a more accurate answer you should find out how many leap years the student has lived through and

add 1 day or 1440 minutes for each leap year.

c. According to the Guinness Book of World Records, the tallest tree ever measured was a

eucalyptus tree discovered in Watt’s River in Victoria, Australia in 1872 by a forester named

William Ferguson. The eucalyptus tree was 132.6 m tall. The Dyerville Giant, a coastal redwood tree, found in the Humboldt Redwoods State Park in

California, USA is named as the tallest tree of modern times. This tree was 1600 years old and 113.4 metres high when it fell in March 1991.

The thickness of a dime is 1 mm. How many dimes would need to be stacked to be as high as

each of the tallest trees in history? How much would all these dimes be worth?

Solution:

Convert each tree height from metres to millimetres.

132.6 m × 1000 = 132 600

113.4 m × 1000 = 113 400

The thickness of a dime is 1 mm. We can stack 132 600 dimes to be as high as the Eucalyptus tree

and 113 400 dimes to be as high as the Redwood tree.

The total values are 1 326 000¢ and 1 134 000¢ or $13 260 and $11 340.

page 121

Teacher’s Guide

Grade 7

d. Ellen MacAurthur from France was the first woman to travel around the world on a sailboat by

herself. Her boat was named the Kingfisher and it could only carry one person. Ellen had

dehydrated food, and a few changes of clothing, as well as e-mail access on board her boat. It took her a total of 94 days, 4 hours, 25 minutes, and 40 seconds to complete her journey

around the world. Ellen had to be alert at all times so she could not take breaks to sleep. Amazingly, she would

take 40 minute long naps, 10 times a day during her trip. About how many hours of Ellen’s trip sailing around the world was she sleeping?

Solution:

40 minutes × 10 naps per day = 400 minutes of sleep per day

400 minutes of sleep ÷ 60 minutes per hour = approximately 7 hours of sleep each day

7 hours × 94 days = 658 hours

Ellen slept approximately 660 hours during her trip around the world.

e. “Wranny” is what Andy Martell from Toronto, Ontario, Canada calls his creation made from

Saran wrap. He is marked down in the Guinness Book of World Records for having created the

largest ball made entirely from Saran wrap. Measured in February 2003, the ball had a

circumference of 137 cm and weighed 20.4 kg.

A jelly bean weighs 0.5 grams. How many jelly beans do we need to equal the weight of

“Wranny”, Andy’s famous Saran wrap ball?

Solution:

The Saran wrap ball weighs 20.4 kg = 20.4 × 1000 = 20 400 grams

One jelly bean weights 0.5 grams, so 2 jelly beans would weigh 1 gram.

Therefore 20 400 × 2 = 40 800 jelly beans would weigh the same as the Saran wrap ball.

f. The world’s longest pencil was created in Malaysia by the company Feber-Castell. The pencil is

19.75 m long, and has a diameter of 0.8 m.

a) How many regular pencils do you need to stack end to end to be as high as the largest

pencil in the world?

b) Could you put your hand around the largest pencil in the world?

Solution:

a) Convert the height of the largest pencil to cm: 19.75 × 100 = 1 975 cm

It would take about 1 975 ÷ 10 = 197.5 ≈ 198 pencils to equal the length of the world’s

longest pencil.

b) Have your students brainstorm how to answer this question. Measure the length of the palm of

their hand (this should be around 10 cm to 12 cm). Convert the diameter of the pencil to cm:

0.8 m × 100 = 80 cm

page 122

Teacher’s Guide

Grade 7

Then approximate the circumference of a pencil. The circumference of a circle is approximately

3 times the diameter: 3× 80 =240 cm

g. The largest flag created was flown in Brasilia, Brazil in August 1998. The flag has a width of 70

m and a height of 100 m.

What is the area of the largest flag?

Measure the area of a regular piece of paper. How many regular pieces of paper would be

needed to cover the entire flag?

Solution:

The area of the largest flag is: 70 × 100 = 7 000 m2 The dimensions of a regular piece of paper are: about 30 cm × 20 cm = .3 m × .2 m = .06 m2

The area of a regular pieces of paper is: 3 m × .2 m = .06 m2 But 100 000 × .06 = 6000 (which is fairly close to 7000). At least 100 000 pieces of regular paper would be needed to cover the largest flag in the world. NOTE: All the data collected for questions c to g is from the Guinness Book of World Records website:

www.guinessworldrecords.com

ME7-4: Converting Units: From Larger to Smaller page 96

Integrate math with language arts by getting the students to associate the word for centimetre with 100,

decimetre with 10 and millimetre with 1000.

How many cents are in a dollar?

How many years are in a century?

How many centimetres are in a metre?

What do you think the root word “cent” means?

Think of the words, “double,” “triple,” and “quadruple”. What do you think the word “centuple” means?

How many years are in a decade?

How many decimeters are in a metre?

What is a decathlon?

E

page 123

Teacher’s Guide

Grade 7

What do you think the root word “deca” means?

December is now the 12th month of the calendar. It was originally the ____ month.

Think of the French word for seven. Which month do you think was originally the seventh month?

How many millimetres are in a metre?

What is a millennium?

What do you think the root word “milli” means?

ME7-6: Fractions of Units page 98

1. Write each measurement as a fraction (in lowest terms) of a metre.

a) 20 cm b) 50 cm c) 75 cm d) 70 cm

e) 40 cm f) 25 cm g) 180cm h) 300 cm

2. By writing each measurement as a fraction of a metre, decide which areas represent more than 1 m2:

a) 20 cm × 300 cm EXAMPLE: 20 cm × 300 cm = 1

5 m × 3 m =

3

5 m

2 < 1

b) 40 cm × 200 cm

c) 60 cm × 300 cm

d) 80 cm × 150 cm

3. By writing each measurement as a fraction, decide which volumes represent more than 1 m3.

a) 20 cm × 300 cm × 40 cm

b) 50 cm × 300 cm × 80 cm

c) 80 cm × 150 cm × 40 cm

d) 60 cm × 120 cm × 150 cm

e) 365 cm × 35 cm × 80 cm

ME7-7: Mixed Measurements page 99

Given any decimal representation of a measurement, your students should be able to say immediately

what unit of measurement each digit of the decimal represents.

As a warm up exercise you might ask student what unit of measurement is represented by the tenths

digit of a particular measurement. For instance:

E

M

page 124

Teacher’s Guide

Grade 7

$2.57 (The tenths are dimes.)

2.57 m (The tenths are dm.)

2.57 dm (The tenths are cm.)

Then you could ask about the hundredths digit.

$2.57 (The tenths are pennies.)

2.57 m (The tenths are cm.)

2.57 dm (The tenths are mm.)

HINT: To help students with this you could write a copy of the stairway on Worksheet ME7-4 on the board.

ME7-10: Changing Units page 103

Make sure your students understand that you need more of a smaller unit to fill the same amount of

space as a larger unit. That’s why, in changing from a larger to a smaller unit, you multiply (by some

multiple of ten, depending on which unit you are in and which you are changing into).

For instance, there are 10 mm in each cm, so to change .7 cm to mm you multiply by 10 (which, as

demonstrated in section NS7-39, shifts the decimal one place right). Changing from a smaller unit to a

larger unit you need fewer of the larger unit, so you divide.

Which is a greater length: 3 786 000 mm or 3.8 km?

ME7-11: Changing Units (Advanced) page 105

1. John has a strip of paper 1 dm long. He folds the strip of paper so that it has a crease in its centre.

What measurements can John make in cm using the strip?

2. Write a measurement in decimeters that is between…

a) 320 and 437 mm b) 507 and 622 mm c) 1 12 metres and 1

34 metres

d) 3 cm and 4 cm e) 47 mm and 48 mm f) 5000 mm and 6000 mm g) 2.8 m and 2.9 m

3. Which insect travels faster: an insect moving 42 cm per second on an insect moving 24 m per hour?

M

E

E

page 125

Teacher’s Guide

Grade 7

ME7-13: Investigations page 108

1. Students might investigate the following question: Can you draw a rectangle on grid paper (with sides

that are a whole number of units) with perimeter:

a) 7 units

b) an odd number

Both are impossible. Students should try to explain why this is the case.

(If you prefer, students could use a geoboard rather than grid paper.)

Try drawing shapes on grid paper with each measurement below as perimeter. Each side of your

shape must have a whole number side length. Write down in the chart how many you can draw.

If you can’t draw any, put 0.

Perimeter 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of shapes

For which perimeters can you draw at least one shape. For which perimeters can you not draw any

shapes? Can you explain why?

Students could also investigate the following question: Start with a square and add squares one at a time so that each added square has at least one edge in common with another square in the shape. EXAMPLE: Using this procedure, could you ever build a shape with odd perimeter? ANSWER: No. Every time you add a square you either

i) Cover up 1 edge and expose 3 new edges (perimeter increases by 2 units). ii) Cover up 2 edges and expose 2 new edges (perimeter stays the same). iii) Cover up 3 edges and expose 1 new edge (perimeter decreases by 2 units).

A

�

�

�

�

�

�

page 126

Teacher’s Guide

Grade 7

2. Group Activity: Find all triangles with perimeters as given in the chart below using whole number side

lengths. Use toothpicks to help you. Then draw your triangles in the chart. Once you’ve made a

triangle with toothpicks, you should draw your triangle in the chart and then re-use the toothpicks.

a) How many toothpicks do you need to do this activity?

Perimeter (# of toothpicks)

Triangles

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

b) Can you make a triangle with perimeter 4 toothpicks? Why or why not?

c) Can you make a triangle with sides 3 toothpicks, 3 toothpicks, 7 toothpicks? Why or why not?

d) Is there a rule for how big the largest side can be compared to the other two sides? If so,

what is it? (The length of the longest side cannot be greater than or equal to the sum of the

lengths of the other two sides.)

e) Which of the following can be the side lengths of a triangle? Justify your answer.

35 cm, 10 cm, 50 cm 17 cm, 80 cm, 90 cm 5 m, 15 m, 0.1 km

72 dm, 84 dm, 1 m 72 cm, 84 cm, 1 m 4 mm, 54 cm, 55 cm

1. a) For which shapes can you add a square to make the perimeter smaller?

E

page 127

Teacher’s Guide

Grade 7

b) Draw two different shapes, one where you can make the perimeter smaller by adding a square

and one where you cannot.

2. a) Find the perimeter of each shape below:

b) Draw two more shapes with the same perimeter.

3. a) Each rectangle of the figure has dimensions

2 cm by 8 cm. What is the perimeter of the figure?

Is there a fast way to find the perimeter?

b) Note that the perimeter of a 4 × 32 rectangle containing the

figure is exactly the same as the perimeter of the original figure.

(So the perimeter is 8 + 32 + 8 + 32.)

Why are the perimeter of the rectangle and the original figure the same?

4. To reinforce the skills just learned about changing units:

Which shape has the largest perimeter? (Rectangles are not drawn to scale.)

a)

b)

5. A triangle has perimeter 12 cm. The length of each side is a whole number. How many different

triangles can you find that satisfy these conditions?

6. A tetrominoe is a figure made of 4 squares in which every square has at least one side in common

with another square.

a) Find all non congruent tetrominoes. Which tetrominoe has the least perimeter?

b) Repeat the exercise with pentominoes (figures made with 5 squares).

tetrominoe Not tetrominoes

3 m 32 dm 4 dm 3 dm

10 cm 12 cm

5 dm 40 cm

A

page 128

Teacher’s Guide

Grade 7

Probability and Data Management

PDM7-2: Bar Graphs page 111

1. Excellent background for teachers is available from Statistics Canada. The Statistics Canada site has

an extensive discussion of bar graphs, with examples, at

http://www.statcan.ca/english/edu/power/ch9/bargraph/bar.htm

1. Using the bar graph in Question 2 on the worksheet, fill in the blanks.

a) Bobby saw _____ times as many fish as eagles.

b) Bobby saw half as many ________ as ________.

c) Bobby saw 9 more mallards than _________.

d) In total, Bobby saw 7 _________ and _________.

2. Plot the following data on a bar graph:

Earnings From Paper Route

Pat $75

Samir $150

Ali $50

What scale should you use?

How much more money did Samir earn than Pat and Ali combined?

3. Draw a bar graph to show the following situation. Gita earned $20.00 babysitting. Anne earned 12

more dollars babysitting than Carmen. Carmen earned 16 more dollars than Gita. What scale would

you use?

PDM7-3: More Bar Graphs page 112

1. Look at the bar graph from question 1. Was the number of students who got A more or less than the

number of students who got B? How can you tell this without even knowing the scale? Show the

exact same data on the graph in Figure 1 below. Why does a quick glance make it look as though the

number of students with B was 3 times the number of students with A? Why is it important that each

square represent the same number of students on the whole axis?

M

E

E

page 129

Teacher’s Guide

Grade 7

E

D

C

B

A

1 2 3 4 5 6

2. Determine the values

of the other bars on

the graphs:

3. a) Make a bar graph to illustrate the area of

the five Great Lakes. Include a title, labels,

and scale.

b) Make up three questions about your bar graph and answer them.

4. Sally drew a table of how she spends her 24 hour day. Draw a bar graph to show her data.

5. Make up three questions you could answer by using a bar graph.

6. Have students work in groups and look up on the internet the starting times for movies at a movie

theater near their home. Tally the results for each hourly interval from 12:00 – 9PM: 12:00-12:59,

1:00-1:59, …, 8:00-9:00. Make a bar graph of results. Write a report on which time intervals should

require more staff to work the snack bar and how many people they would have working at the snack

bar during each time interval. Present their results to the class. Groups of 4 should be able to present

results for each day of the week. Different groups can be responsible for different movie theatres.

Great Lake Superior Michigan Huron Erie Ontario

Area (km2) 82 000 57 000 60 000 26 000 19 000

Activity School Chores Homework Recreation Sleep Eat Other

Time (hour) 7 1 1 3.5 8 2 1.5

b)

A

B

24

a)

A B C

60

c)

120

A B C

Figure 1

page 130

Teacher’s Guide

Grade 7

PDM7-4: Further Bar Graphs page 113

1. A company claims shoe A is much more popular than shoe B. Is the company's claim reasonable?

PDM7-5: Double Bar Graphs page 114

1. To help students with 1 d), show students how to use a ruler to get at least approximately accurate

measurements for each company’s sales.

2. For question 2, have students brainstorm questions they could ask and answer about the graph.

3. Have students brainstorm things they could compare using a double bar graph that they haven’t

already compared.

PDM7-6: Stem and Leaf Plots page 116

1. For teachers, the Statistics Canada site has an extensive discussion of stem and leaf plots, with

examples at http://www.statcan.ca/english/edu/power/ch8/plots.htm

1. What number has both stem and leaf 0?_________

E

M

M

E

30

28

26

24

A B C

Kind of shoe.

percent of sales

page 131

Teacher’s Guide

Grade 7

PDM7-8: More Stem and Leaf Plots page 119

1. Ask students how they think they should make a stem and leaf plot for data using decimals, such as

this data set: 57.4 60.2 59.8 62.1 58.5 57.6 60.4 62.7

PDM7-9: Line Graphs page 120

TEACHER: Although not marked in the workbook, this manual note is for PDM7-9.

1. Excellent background for teachers is available from Statistics Canada. The Statistics Canada site has

an extensive discussion of line graphs, with examples, at:

http://www.statcan.ca/english/edu/power/ch9/linegraph/line.htm

1. Measure the temperature twice a week for a month and plot your measurements on a line graph.

Describe your data. What trends do you notice?

2. The following chart shows the average rainfall each month in different parts of the world:

Average Monthly Rainfall or Precipitation (in cm)

Coniferous

Forests Tundra Grasslands Rainforest

January 25 10 100 120

February 20 10 100 120

March 25 10 100 120

April 35 10 20 120

May 45 13 10 120

June 50 18 5 120

July 60 18 5 120

August 55 13 5 120

September 50 10 10 120

October 40 10 20 120

November 40 10 60 120

December 35 5 100 120

GRAPH

a) Which graph best matches

the data in each column?

Write the letter in the space

provided under each column:

E

E

M

A.

Month

B.

C.

Month

D.

Month

Month

page 132

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

b) Describe any trends you see in the graphs. How do you account for the trends?

c) Sketch a graph that you think would represent the average monthly temperature where you live.

3. Draw a line graph showing the temperature during the early morning:

Time 4:00 5:00 6:00 7:00 8:00 9:00 10:00

Temperature 2° 3° 5° 7° 8° 11° 13°

What is the rise in temperature from: 5:00 to 6:00? 7:00 to 10:00? 6:30 to 8:00? 5:30 to 9:30?

7:30 to 10:00? Errata for PDM7-11: The three line graphs at the top of page 122 in the workbook should have scales from 200 to 1000

(i.e. 100 is an error and should be replaced by 1000 in each case).

PDM7-12: Interpreting Line Graphs page 123

It is easy to integrate language arts lessons with the kind of exercise given below – have the students make up their own stories as creative writing projects, then exchange with a partner and draw each other’s graphs.

1. a) Which graph best describes the story below. The important words and phrases have been

underlined for you.

A hiker is walking in the woods, sees a bear and starts running back home. Then she stops

behind a tree to catch her breath. The bear doesn’t seem to be following her, so she walks

slowly, carefully away. When she gets to the end of the woods, she runs home. Which graph

best fits this story?

a) b) c)

E

M

500 m

400 m

300 m

200 m

100 m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

time in minutes

Graph A

0

page 133

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

a) How far from home did the hiker start? b) How far from home was the hiker when she stopped to catch her breath?

PDM7-13: Drawing Line Graphs page 124

1. Tina wrote 10 math tests this year, each worth 100 marks. Her marks were:

a) Draw an accurate line graph showing all her marks. Title and label your graph.

b) Draw a line graph showing only tests 1, 3, 5, 7 and 9.

c) What trend does the second graph suggest that the first one does not?

d) Tina wants to show her parents how much her math mark is improving. If she shows her parents

the second graph, do you think she is being honest? Explain.

1. Extension for question 3. In 2004, the Ontario government decided to increase minimum wage to

$8.00. Why do you think they chose to do it in stages instead of all at once?

If Tina works full-time (40 hours a week) at minimum wage, how much would she have made for the

whole year in 2003? How much will she make in 2008?

Test 1 2 3 4 5 6 7 8 9 10

Mark 60 50 65 55 70 30 75 45 80 65

A

E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

500 m

400 m

300 m

200 m

100 m

time in minutes

Graph B

0

page 134

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

PDM7-14: Circle Graphs page 125

1. Circle graphs can be used to show what fraction of something has a specific property. Start with one

or two examples, then brainstorm. Examples may include:

What fraction of…

…employees work full time?

…my spending money is on clothes?

…donations received by a charity are spent on fundraising?

…Maple Leafs’ goals occur on the power play?

…fairy tale heroes are female?

2. Make sure students know that two circle graphs can represent the same data even though they look

different.

3. Excellent background for teachers is available from Statistics Canada. The Statistics Canada site

has an extensive discussion of circle graphs, with examples, at:

http://www.statcan.ca/english/edu/power/ch9/piecharts/pie.htm

PDM7-15: More Circle Graphs page 126

1. Make sure that students understand that when two angles are marked with the same symbol (for

instance, a small circle) they are the same. It may help to point out the similarity to the notation used

to indicate parallel lines. More than one symbol is used when there are more than one group of

angles or lines that are equal or parallel.

M

M

14

18

14

18 1

4

14

18

18

14

14

page 135

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

PDM7-16: Interpreting and Drawing Circle Graphs page 127

Errata for PDM7-16: The circle graph in Question 1d)

on page 127 of the workbook should be:

1. Make sure students are familiar with the steps needed to draw a circle graph. Example: In a grade 7 class, 10 students walk to school, 5 travel by bus, 5 bicycle, and 5 skateboard.

Step 1: Find the total number of students (25) Step 2: Express each piece of data as a fraction of the total (reduce to lowest terms)

1025 = 25 walk

525 = 15 bus

525 = 15 bicycle

525 = 15 skateboard

Step 3: Change each fraction to an equivalent fraction out of 360.

25 =

?360

25 =

144360

The part of the circle graph that represents the students

who walked to school should subtend an angle of 144°.

15 =

?360

15 =

72360

The parts of the graph that represent the students who walked, bused, or skateboarded to school

should each subtend angles of 72°. Step 4: Draw a circle and then draw a radius.

Step 5: Use a protractor to construct a radius for each of the angles you found in step 3.

×72

×72

×72

×72

144°°°° 144°°°°

72°°°°

144°°°°

72°°°°

72°°°° 72°°°°

M

× ×

×

R

B

R

Y

B B

×

Y

instead of: × ×

×

R

B

R

Y

B B

Y ×

page 136

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

1. The following question is based on an actual reasoning mistake on a web page made by adults. An

opinion poll asks people to strongly agree, agree, disagree or strongly disagree with an opinion. Draw

a graph representing the answers of a group of adults who were surveyed as follows:

a) What fraction of people surveyed: Agreed? ______ Disagreed? _______

Strongly agreed? _______ Strongly Disagreed? _______

b) What fraction of people surveyed either agree or strongly agree? ____________

c) The survey concludes: “Not counting the lunatic 1/10 of people who strongly disagree, only 4/10

of people disagree with us.” Are they right? Explain?

Answer: Out of 10 people, you would expect 3 to agree, 2 to strongly agree, 4 to disagree and 1 to

strongly disagree. Not counting the 1 who disagrees, 4/9 disagree. This is slightly more than 4/10.

2. Ask students how the following circle graph is misleading. (Students should be able to estimate what

the angles should look like).

(For instance 35 is greater than

12 , but the part marked H covers less than

12 the circle. Also

15 is

double 110 so the parts marked B and S should cover twice as much area as the part marked 0.)

Favourite sport

H: Hockey 35

B: Baseball 15

Ask students why it is important to measure each angle accurately when drawing a circle graph.

A

S: Soccer 15

O: Other 110

H

B

O

S

Agree

Strongly

Agree

Disagree

Strongly

Disagree

page 137

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

3. Ask students to make up a joke for homework. Tell them you will read through them and choose

the five best ones for a math project. Make up a ballot sheet with the five best jokes on it.

Students each vote for 1 joke. Then walk them through making a circle graph.

1. The following data shows the number of deaths due to each type of recreation in one year.

Boating 80 Swimming 40 Biking 36 Jet skiing 4

a) Make a frequency table showing the fraction of deaths each recreation represents.

b) Draw a circle graph showing the data.

c) Can you conclude that biking is more dangerous than jet skiing? Why or why not? What extra

information would be relevant?

PDM7-17: Comparing Graphs page 128

1. In Activity 2 below, students use the Internet to look up census and survey information. If you wish to

discuss bias in a census or survey with your students, you might look ahead to sections PDM7-22

and 23. Four concepts are presented in these sections, focused mainly on secondary data:

• A census may provide more reliable data than a survey, but in many cases is not practical as it is

expensive and time-consuming.

• Surveys and experiments should use representative samples, not biased samples.

• The larger the sample size, the more accurate the results are likely to be.

• People can play tricks with data.

1. All students should bring in a sample of numerical or graphical data from a newspaper or magazine

to be used in a class discussion. For some classes, the teacher might provide all students with a

copy of some recently published data.

E

M

A

page 138

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

2. Find your own data on the Internet. Choose what kid of graph you want to use to represent this data.

Here are some sites you can explore:

Site address Comments

http://www.statcan.ca/start.html This is the main Statistics Canada site. To get to a wide

selection of summary tables, under Latest Indicators

click on More in Canadian Statistics. Several tables from

this site are reproduced below.

www.censusatschool.ca Statistics Canada participates in the Census at School

project, a free international classroom project for

students aged 8 to 18. They complete a brief online

survey, analyze their class results and compare

themselves with students in Canada and other countries.

The site has background information and complete

instructions for teachers.

http://www.statcan.ca/english/ Estat/licence.htm

E-Stat, a branch of Statistics Canada, is free to

registered educational institutions. It is a data

warehouse for registered schools, offering Canadian

data on a wide range of social and economic topics, as

well as some international data. A user-guide is

available for download.

http://www.rom.on.ca/ontario/risk.php At this site there is easily accessible information about

plants and animals currently at risk in Ontario

http://www.corusmedia.com/ytv/index.asp This is the site of YTV. Click on “Research,” then select

summary reports on a variety of topics. (This site could

be a useful reference for classes doing a survey.)

http://www.cyberschoolbus.un.org/ Choose “Country at a Glance” to find physical and

population data, or InfoNation to produce graphs that

compare countries, whatever category of data you

choose.

http://unstats.un.org/unsd/demographic/ products/socind/

By choosing PublicationsStamps/Database/Statistics

Division/Databases/Social Indicators, you can see a

wide variety of tables that include every member

country of the United Nations

http://www.nhlpa.com/ The NHL Players Association site has more than enough

data for any conceivable student project.

http://www.ec.go.ca Environment Canada has a wealth of statistical data

about weather and the environment.

http://www.globeinvestor.com The Globe and Mail keeps track of Canadian stock

market information and other economic data.

page 139

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Much of the information available on the Internet is not numerical data that is suitable for statistical or

graphical analysis by students. Above are listed a few sites where numerical data is available. The

second in the list, the Census at Schools project, would fulfil any curriculum requirement to “collect

data via computer networking.”

A 2006 Census Teacher’s Kit is available free from Statistics Canada at

http://www22.statcan.ca/ccr06/ccr06_004_e.htm. It includes

• 8 teacher-ready activities

• background about the census

• an introduction to census data that may be useful in schools

• an alternative approach to several subject areas.

• Select which individual pieces of data (or portion of the table) you want to use.

• Graph the data – choose an appropriate kind of graph and make it by hand or by using software.

PDM7-18: Concepts in Data Management page 129

1. For question 3, recommend to students that they check off each type of graph as they use it to help

them with filling in the rest. They should do the ones they’re sure about first.

2. You may wish to talk about various technologies that can be used to draw graphs (e.g. Microsoft

Excel, Appleworks). See the Activities below for instructions to create graphs in Excel.

3. Statistics Canada participates in the Census at School project, a free international classroom project

for students aged 8 to 18. They complete a brief online survey, analyze their class results and

compare themselves with students in Canada and other countries. The site has background

information and complete instructions. See www.censusatschool.ca.

1. Using Excel to create a circle graph.

A class conducted a survey to find out how

students got to school.

Their results are shown in the table.

M

page 140

Teacher’s Guide

Grade 7

A. Input this table into a spreadsheet. Select the six cells that show the data (from Walk at the top-

left to 8 at the bottom-right).

B. From the menu, select Insert/Chart and under Chart Type click on Pie. Click on Next.

C. Now choose either of the following:

- if you now click on Finish, you could print your pie chart, then add the title later by hand.

- if you now click again on Next, you can input a title for the graph, then click on Finish.

D. Print your pie graph.

2. Using Excel to create a bar graph.

A. Input the above data into a spreadsheet. Select all cells from “Day” at the top-left to “756” at the

bottom-right.

B. From the menu, select Insert/Chart, and from Chart Type list select Column or Bar. Then click on

Next.

C. Choose Next again. Then fill in

- the title “Audience Size for 7 p.m. Shows,”

- the Category (X) axis: “Days of the week.”

- the Value (Y) axis “Number of people.”

D. Click Next. In this box choose the second option under “Place chart in” , so the graph will be in

the same page as the data.

E. Click Finish.

F. Make adjustments to title, labels. axes, size of graph by selecting and editing them. When you are

finished, print your graph.

Students can use spreadsheet software such as AppleWorks or Excel, or statistical software such as

Fathom™. to make graphs. Instructions here refer to Excel – you may want to produce more specific

instructions for the specifics of your school software, or you may want to use technology-savvy students

to help the others.

Using such technology for making circle graphs is simple and time-efficient.

Bar graphs are also easy to produce. However, formatting graphs – choosing font sizes. axis labels that

fit into the space, colours, symbols, and legends can be time-consuming. Some students will love playing

with the parameters. For those who don’t, suggest that they print the graph, then modify or add labels,

legends, and titles by hand.

page 141

Teacher’s Guide

Grade 7

Geometry

G7-1: Introduction to Angles page 131

1. Use a geoboard with elastics to make…

a) a right angle

Example:

b) an angle less than a right

angle

c) an angle greater than a

right angle

2. Use a geoboard with elastics to make a figure with…

a) no right angles

Example:

b) 1 right angle

Example:

c) 2 right angles

Example:

1. Copy the shapes onto grid paper and mark any right, acute, and obtuse angles as in exercise 3 on

the worksheet. Which shape has one internal right angle?

Angles that are greater than 180° are called reflexive angles. There are 5 reflexive angles inside

Figure B. Can you find them?

2. Draw your own shape on grid paper and mark any right, acute, obtuse and reflexive angles.

E

A

A B

page 142

Teacher’s Guide

Grade 7

3. Have your students compare the size of angles on the pattern block

pieces (see Blackline Masters) by superimposing pieces (students

could put the angles in order according to size).

For instance they might notice that there are two angles on the

trapezoid that are greater than the angles in the square and two that

are less than the angles in the square.

They might also notice that the large angles in the trapezoid are equal

in size to the angle in two of the triangles.

Ask your students how they know all the angles in the triangle are the same – they could show this

by superimposing two triangles and rotating one of the triangles.

G7-2: Measuring Angles page 132

1. How many different angles can you create by putting 2 tangram pieces (or two pattern blocks)

together? Measure the angles with a protractor. NOTE: If you don't have tangram and pattern block shapes we have provided blackline masters for the shape

which you can photocopy and have students cut out.

EXAMPLE: With the small triangle and the square of the tangram you can create an angle of 135º.

Can you create an angle of 150º using pattern blocks? (Answer: Yes, use the square and the

trapezoid, or the equilateral triangle.)

1. a) Draw a shape on grid paper, (a trapezoid for instance) and measure all its angles.

b) Draw a triangle and measure all of the angles in the triangle. Add up the angles. Repeat with

several other triangles. What do you notice?

2. Construct the following figures using a ruler and protractor.

a) a square b) a right angled triangle

A

E

135º

page 143

Teacher’s Guide

Grade 7

G7-4: Angles in Triangles and Polygons page 136

1. Check to see if the angles in each triangle in question 2 add up to 180°. If they don’t, your

measurements were off by a small amount.

2. Can a triangle have 2 obtuse angles? Explain.

3. Which figure below has…

a) all acute angles? b) all obtuse angles? c) some acute and some obtuse angles?

G7-5: Classifying Triangles page 137

1. Give students a ruler, scissors, and a set of straws. Have them measure and cut a set of straws of

the following lengths.

Question:

a) How many (distinct) right angle triangles can you make using the straws.

Answer: 2 distinct triangles with sides of length 3, 4, 5 and 6, 8, 10.

E

A

10 cm

9 cm

8 cm

6 cm

5 cm

5 cm

5 cm

4 cm

4 cm

3 cm

page 144

Teacher’s Guide

Grade 7

b) How many isosceles triangles can you make using the straws?

Answer: 9 triangles with sides of length:

3, 4, 4; 3, 5, 5; 4, 5, 5; 5, 5, 5; 5, 4, 4; 6, 4, 4; 6, 5, 5; 8, 5, 5; and 9, 5, 5

NOTE: Ask students why side lengths 8, 4, 4; 9, 4, 4; 10, 4, 4; and 10, 5, 5 won’t work (see answer to

question e)

c) Can you make an equilateral triangle?

Answer: Yes, with side lengths 5, 5, 5.

d) Show an example of an…

i) obtuse triangle (one example 3, 4, 6) ii) a scalene triangle (one example 8, 9, 10)

iii) acute triangle (one example 4, 5, 6)

e) Can you give a rule for determining the sets of straws that will make a triangle and those that

won’t?

Rule: A set of 3 straws will only make a triangle if the sum of the lengths of the two shortest

straws is greater than the length of the longest (otherwise the two shorter straws will not

meet). Here is an example of a set that doesn’t work (since 4 + 3 is less than 10).

2. On a geoboard, dot paper, or grid paper create…

a) an isosceles triangle

b) an isosceles triangle with a right angle

c) an acute scalene triangle

d) an obtuse scalene triangle

3. Students could investigate question 3 using a geoboard, dot paper, or grid paper.

1. Choose one name from each category to describe the

triangle:

2. A triangle can be acute, obtuse, or right and it can be equilateral, isosceles or scalene. By combining

these properties, there are nine combinations, but some are not possible to draw. Have students

investigate all nine combinations to see which ones are actually possible.

Category A Category B

Acute

Obtuse

Right Angled

Scalene

Equilateral

Isosceles

E

A

B

10 cm

4 cm 3 cm

page 145

Teacher’s Guide

Grade 7

G7-9: Perpendicular Bisectors page 142

Cut out the circle. Fold the circle in half so that A goes into B. Look at the line that the crease in your fold

makes. Is it

a) a bisector of angle C?

b) a perpendicular bisector of line segment AB?

c) A diameter of the circle?

d) A radius of the circle?

Fold the circle in half again, this time making A meet C. What two properties will the crease fold have?

Repeat making B meet C.

What point in the circle will all three perpendicular bisectors meet at?

Given a triangle ABC, how can you use perpendicular bisectors to help you draw the circle going through

the three points A, B and C? Demonstrate your method on the following triangle. Draw the perpendicular

bisectors by hand – do not cut and fold the triangle.

Investigation. Draw several right triangles and their circles around them.

Label the point at the right angle C and label the other two points A and B.

Fold the paper along the side AB. Draw a pencil mark where C folds into.

You will have to press hard enough with your pencil that the mark goes

through the fold to the paper. What do you notice? Will this be true if you fold

along other sides? Try it and see!

NOTE: The mark they make will still lie on the circle if the angle is 90°°°°.

A A

C

B

A

C B

page 146

Teacher’s Guide

Grade 7

G7-11: Constructing Parallel Lines page 145

1. Find a picture in a magazine or newspaper that has a pair of parallel lines. Mark the lines with a

coloured pencil.

2. Ask your students if they can draw a parallelogram that's not a rectangle using only a ruler and a right

angled triangle.

Solution:

G7-12: Constructions with a Compass page 146

Here are some methods for constructing angle bisectors and perpendicular bisectors without a compass.

Angle Bisectors

1. a) Draw an acute angle on a blank sheet of paper (not in your notebook as you will need to fold

paper).

b) Draw an angle bisector using a plastic right triangle.

A

AA

Step 1 Draw one side of the

parallelogram.

Step 2 Draw two of the parallel sides using the triangle.

Step 3 Use the ruler to complete the figure.

A

Step 1:

You should have an acute angle drawn on your paper.

A

Step 2:

Draw a line from B as shown.

B

page 147

Teacher’s Guide

Grade 7

c) Check your answer by folding the paper along AD. Does line AB meet line AC?

d) Draw an obtuse angle and repeat steps b) and c).

2. You are given an angle and a transparent mirror. How can you find the angle bisector?

HINT: An angle bisector is a line of symmetry.

3. You are given an angle and a ruler. How can you find the one angle bisector?

Answer: Measure equal distances from the vertex on both rays of the angle and draw perpendicular

lines.

4. Perpendicular bisectors

a) Draw a line dark enough that you can see through the paper.

Fold the paper so that A meets B. What line has your crease made? Answer (perpendicular

bisector). Use a ruler and protractor to check your answer.

b) Draw another line dark enough so you can see through the page. Fold the paper so that you

can find a line perpendicular to AB, but does not bisect AB. How can you use this process to

find a line parallel to AB? How can you use a ruler or any right angle to find a line parallel too

AB?

5. You are given a line and a point not on the line.

a) Use a ruler or any right angle to draw a line through P that is perpendicular to AB. Check your

answer by paper folding.

b) Find a line through P that is parallel to AB.

c) Place a mirror passing through P so that the line AB is its own mirror image. What angle does

the mirror make with the line AB? Answer: 90°

NOTE: When students find perpendicular bisectors using a compass, they should verify using a ruler that the point is

really the midpoint and the line is really perpendicular to the given line.

Step 3:

Draw a line from C as shown.

A B

C D

Step 4:

Draw a line through A and D as shown

A B

C D

A B

A B

P

page 148

Teacher’s Guide

Grade 7

1. E D

F A B C

Line EB bisects angle ABD and Line FB bisects angle DBC.

a) Find each angle by using a protractor:

ABE = _____ EBD = ______ DBF = ______ FBC = _______.

b) Find angle EBF without using a protractor and then verify your answer using a protractor.

2. In each diagram below, draw angle bisector EB of angle ABD and angle bisector BF of angle DBC by

using a compass. In each case, what is angle EBF? Why? (It is always 90°?) Verify with a protractor:

NOTE: Students don't have to draw exact copies of the lines shown. No matter how they position line DC,

angle EBF will be a 90°°°° angle. Students could prove this by algebra (as shown below) after they study the section

on algebra in Part 2 of the workbook:

x + x + y + y = 180° 2x + 2y = 180° 2 (x + y) = 180° x + y = 90°

E

A B C

E

D F

y

y x

x

A B C

D

A

D

B C

B C

D

A B C A

D

page 149

Teacher’s Guide

Grade 7

3. Find the centre of a circle using a compass and a straight-edge. You may use a ruler but not the

markings.

SOLUTION:

Draw any two chords as shown. Construct perpendicular bisectors of the chords; the bisectors will

meet at the centre of the circle.

G7-13: Congruency page 148

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. Ask students how they moved a particular piece onto

another one to check congruency. (Did they slide the piece? Did they have to rotate or flip the

piece?)

2. Give students a set of square tiles and ask them to build all the non-congruent shapes they can find

using exactly 4 square blocks where each square must have at least one side with another square

(and sides may not partially overlap). These shapes are called tetrominoes.

Solution:

3. In addition to the exercises on the worksheet students could try this: Make 2 shapes on a geoboard.

Use the pins to help you say why they are not congruent.

Example:

A

The two shapes are not congruent: one has a larger base (you need 4 pins to make the base).

page 150

Teacher’s Guide

Grade 7

4. On grid paper, draw two non-congruent figures with:

a) the same perimeter

b) the same area

c) the same shape

1. How many non-congruent shapes can you make by adding a square to the figure?

2. How many non-congruent shapes can you make by removing one square from the 3 × 3 array?

Answer: 3 shapes

3. How many non-congruent shapes can you make by removing 2 squares from the 3 × 3 array?

Answer: 8 shapes

Encourage your students to proceed systematically in looking for the answer. For instance they

might start by finding all the shapes they can make after they have removed a corner square

Then they could try removing a middle square on the outside of the figure:

(Notice the last 2 shapes have been crossed out because they are already on the list for the previous

figure.)

Finally, they could try removing the middle square (but all of the shapes that can be made after

removing the middle square are already listed).

4. Make as many non-congruent shapes as you can using 5 square blocks where each square must

share at least one side with another square (and sides may not partially overlap).

Try to organize your search systematically. How many of your shapes have a line of symmetry?

Which shape has the greatest perimeter?

E

page 151

Teacher’s Guide

Grade 7

G7-14: Exploring Congruency page 149

1. Use toothpicks cut into sticks of lengths 1 cm, 2 cm, 3 cm, 4 cm. Have students construct polygons

with the toothpicks using tape or modeling clay. Then try to bend them into a non-congruent polygon

with the same side lengths. Which ones can they bend more easily? Why? What is preventing them

from bending? If you are a scientist and want to build strong structures, would you want your shapes

to bend easily or to be difficult to bend? Why? Look at bridge structures and structures from ancient

civilizations that have been built. What shape is most common – a triangle, a quadrilateral, a

pentagon or a hexagon? Why?

1. Using straight-edge and compass, try to draw two non-congruent polygons with the following side

lengths:

a) 2 cm, 2 cm, 3 cm, 3 cm b) 1 cm, 2 cm, 3 cm, 4 cm

c) 3 cm, 3 cm, 3 cm, 3 cm d) 3 cm, 3 cm, 3 cm

e) 2 cm, 3 cm, 4 cm f) 2 cm, 2 cm, 3 cm

g) 2 cm, 2 cm, 3 cm, 3 cm, 4 cm, h) 2 cm, 2 cm, 2 cm, 2 cm, 2 cm, 2 cm

When can you draw two non-congruent polygons with the same side lengths? When can you not do

this? Check your guess with two other examples, one where you think it will work and one where you

think it will not

G7-15: Exploring Congruency (Advanced) page 150

In question 5, if the students want to give only 3 measurements, they need to either give 3 side lengths

or they need to be specific about where the angles and sides they state are in relation to each other.

Without that, you can even have 5 measurements given (3 angles and 2 side lengths) and it won’t be

enough – see below.

1. Adapted From OISE/UT Math Trail Assignment

A restaurant called Jammz Bistro and Bar in the underground PATH in downtown Toronto has many

windows of unusual shapes. One of them is a trapezoid as shown below. You need to order over the

A

E

M

A

page 152

Teacher’s Guide

Grade 7

phone a replacement for this window. You tell the person that the window is a trapezoid, but you

need to give more information to ensure that the replacement window is the exact shape and size.

Is it enough to give:

a) The lengths a, b, c, d of the four sides?

b) The angles A and B and sides a, d?

c) The angles A, B and sides a, c?

d) The angles A, B and sides b, d?

e) The sides a, b, c and angle B?

f) The sides a, b, c and angle A?

g) The sides a, b, c, d and angle A?

If yes, explain your answer. If no, draw two different trapezoids with the given sides and angles

equal.

Investigate other combinations.

2. Congruent triangles and reflection:

Reflect point C in the line AB. Call the point C'. Is triangle ABC congruent to triangle ABC'? How do you

know? Join C and C'. Do you see a perpendicular bisector anywhere? If so, where?

B C C' A

A= 60°°°° a= 7 cm

a=10 cm D=120°°°°

c = 4.5 cm

C=110°°°°

B=70°°°°

b = 6 cm

page 153

Teacher’s Guide

Grade 7

3. By using reflection in each side of the triangle, draw 3 different triangles that are congruent to triangle

ABC: A B C

4. Understanding the compass constructions using congruency:

Look at the steps on p. 146 of your workbook. Explain why QS = QT. Call the intersection of the two

arcs M. Explain why TM = SM. Join Q to M. Can you explain why triangle TMQ is congruent to

triangle SMQ? Which angle does angle TQM correspond to in triangle SMQ? Explain how you know

that you’ve found the angle bisector.

Look at the steps on p. 147 of your workbook. Call the point where the two arcs meet C and the

point where the line ℓ meets AB, D. Then explain why triangle ACD is congruent to triangle BCD.

What angle in triangle BCD does angle ADC correspond to? What do angle ADC and its

corresponding angle add to? Can you explain why the line ℓ is perpendicular to AB? Why is it the

perpendicular bisector?

Draw two non-congruent triangles with:

a) A 30° and a 60° angle. What is the third angle in each of your triangles?

b) A 30° and a 50° angle and a side of length 5 cm. What is the third angle in each of your triangles?

c) Two sides of lengths 3 cm, 5 cm and a 90° angle. What is the third side in each of your triangles?

If two triangles have their corresponding sides equal, they are congruent, so 3 measurements are

enough. Can you find an example of 4 measurements that are not enough? (Answer: 3 angles

and a side are not enough for congruency – 2 angles and 2 sides are also not enough, but this is

harder to see. In fact, this gives an example of 5 measurements (3 angles and 2 sides) that are

not enough. If your students are familiar with similar triangles and with ratios, you could show

them the following two triangles on the board: 2 ¼ cm, 3 cm, 4 cm and 3 cm, 4 cm, 5 1/3 cm and

get them to verify that the triangles are similar (all three angles are equal or the ratio of side

lengths are equal) and two of the side lengths are shared – they can then try to construct other

examples to solidify their understanding).

E

page 154

Teacher’s Guide

Grade 7

G7-16: Symmetry page 151

1. 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.

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.

In the picture below, the dotted line cuts the figure into two parts that have the same size and shape.

But the two halves do not coincide when you fold the figure along the line: they are not mirror

images of each other in the line. Hence, according to the second definition, this line is not a line of

symmetry:

1. Draw a shape that has a horizontal and vertical line of symmetry, but no diagonal line.

2. How many lines of symmetry does an oval have?

3. What geometrical shape has an infinite number of lines of symmetry? (Answer: The circle)

4. Sudha drew a mirror line on a square. Then she drew

a reflection of the corner of the square in the mirror

line and shaded both the corner and its reflection.:

a) What shape did she shade?

b) Draw a large square on grid paper. Draw any mirror line on the square and

reflect part of the square in the line. Shade both the part to its reflection. What shape did you

shade?

c) Can you place the mirror line so that the two parts of the square on either side of the mirror line

so that the part you shaded is…

i) A rectangle ii) A hexagon (a shape with 6 sides) iii) An octagon (a shape with 8 sides)

5. Complete the figure so that it will have a line of symmetry and will have the least area.

M

E

page 155

Teacher’s Guide

Grade 7

6. Add a square to each figure so that it has a line of symmetry.

Ask students to create more difficult versions of this problem with more complex shapes (and that involve

adding more than one square).

7. For an exercise in finding lines of symmetry in flags see the Blackline Masters.

G7-18: Special Quadrilaterals page 153

1. For question 3, no trapezoids are parallelograms and no parallelograms are trapezoids because a

trapezoid must have exactly 1 pair of parallel sides – some textbooks define trapezoid to have at

least 1 pair of parallel sides, in which case, some trapezoids would be parallelograms and all

parallelograms would be trapezoids.

1. Special Quadrilaterals and Number Squares (Adapted from Minds on Math 7, pp. 44, 45)

Look at the number square below and copy it into your workbook:

Draw several quadrilaterals with corners in the middle of four of the squares and then add the numbers

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

M

A

page 156

Teacher’s Guide

Grade 7

The numbers in the first square add to 34 and the numbers in the second square add to 25.

Complete the following chart, using several quadrilaterals of each shape:

Corner Numbers

Sum of Corner

Numbers Square Rhombus

Parallelo-gram

Rectangle Trapezoid Other

2,5,12,15 34 X X

1,2,11,14 28 X

What patterns do you notice? Investigate using other symmetrical 4 by 4 arrangements of the

numbers 1 through 16, or try a 5 by 5 grid or 6 by 6 or a 4 by 4 grid from a calendar or a hundreds

chart. To create other charts, students may go diagonally or spiral or left to right and then right to left

alternating, etc. Allow students to be creative in deciding their own investigation.

2. a) How many trapezoids are in the figure? (There are 2 hidden trapezoids plus the 8 small

trapezoids.)

b) How many parallelograms are in the figure? (There are 4, each consisting of 2 trapezoids.)

3. Arrange circular blocks in the shape shown. How many squares (with the circles as vertices) can you

find in the figure? (There are 19 hidden squares altogether.)

NOTE: Give your students the bold square as an example.

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

9 like this 4 like this 4 like this 2 like this

page 157

Teacher’s Guide

Grade 7

G7-19: Sorting and Classifying Shapes page 154

1. 2-D Shape Sorting Game:

Give each student (or team of students) a deck of shape cards and a deck of property cards. (These

cards are in the Blackline Masters section.) 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 students a copy of the Venn diagram sheet in the Blackline Masters section (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 using the Venn Diagrams. Give 1 point for each shape

that is placed in the correct region of the Venn diagram.

3. See the Extra Worksheets section of this manual for worksheets for the game called

Always/Sometimes/Never True (Shapes). This game will help students sharpen their understanding

of two-dimensional shapes.

G7-20: Optical Illusions page 157

1. Students can be encouraged to think about why the optical illusions trick them. They might do some

research on optical illusions or find other examples of illusions where they would have to measure

the shapes in the illusion and use their knowledge of geometry to determine the reality underlying the

illusion.

A

M

page 158

Workbook 7 Part 2

page 159

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Patterns

PA7-21: Substitution page 159

1. Students could try 2-variable substitutions.

For instance, find the value of each expression for x = 2 and y = 3.

a) 5x + 4y b) 6x – 2y c) 9xy

2. A rectangle has area xy. Find the area if x = 5, y = 7.

3. A triangle has area 12 bh. Find the area if b = 8 and h = 3.

PA7-23: More Expressions page 162

1. Write an expression using x for the unknown:

a) Write an expression for the cost of a pizza (x) divided among 4 people (x ÷ 4).

b) Pizza costs $5 per student and drinks cost $2. Write an expression for the total cost of a meal

for x students (5x + 2x).

2. Write an expression for area and perimeter of each rectangle:

a)

b)

3. The expression 3 × 5 is short for repeated addition: 3 × 5 = 5 + 5 + 5.

Similarly 3x is short for x + x + x.

a) Write the following expressions as repeated addition:

i) 5n

ii) 4x iii) 7y

E

x

y

E

x

y

5

x

page 160

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

b) Write each sum as a product

c) Can you use what you learned in parts a) and b) to write 2x + 3x in a single form?

Answer: 2x + 3x = x + x + x + x + x = 5x

d) Simplify: i) 8x + 2x ii) 9x + 4x + 3x

PA7-24: Equations page 163

1. Students can create models for an equation that involves addition. A square could stand for the

unknown and a set of circles could be used to model the terms of the equation. For instance the

equation

has the model

If the students think of the square as having a particular weight, then solving the equation becomes

equivalent to finding the weight of the square in the form of the circles.

Ask students to make a model with squares and circles to solve the following problems. (They

should draw a picture of their model with a balance scale as in the figures above.) Then they should

explain how many circles they would remove to find the weight of the square.

i) x + x + x + x = (4x)

ii) n + n + n + n+ n + n

iii) m + m + m iv) y + y + y+ y+ y+ y+ y

2x 3x

A

+ 2 = 7

=

How many circles are needed to balance the square?

You can find the answer by removing all the circles from the left hand scale and an equivalent number of circles from the other side.

The square has the same weight as 5 circles.

page 161

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

a) n + 2 = 8 b) n + 3 = 10 c) n + 5 = 9

Ask students to translate the following pictures into equations using the letter n as the unknown.

a)

b)

2. Ask students to translate each story problem below into an equation. They should also model

problem a) with squares and circles.

a) Carl has 7 stickers. He has 2 more stickers than John. How many stickers does John have?

Solution: Let n stand for the number of stickers that John has. Carl has 2 more stickers, so you would have to add 2 to the number of stickers John has. So the correct equation is:

n + 2 = 7 b) Katie has 10 stickers. She has 3 fewer stickers than Laura.

Solution: Let n be the number of stickers that Laura has. Katie has 3 fewer stickers than Laura so you need to subtract 3 from the number of stickers that Laura has. So the correct equation is:

n – 3 = 10 3. Give your students a copy of a times table. Ask them to write an equation that would allow them to

find the numbers in a particular column of the times table given the row number. For instance, to

find any number in the 5s column of the times table you multiply the row number by 5: Each number

in the 5s column is given by the algebraic expression 5 × n where n is the row number. Ask

students to write an algebraic expression for the numbers in a given row.

4. In the magic trick below, the magician can always predict the result of a sequence of operations

performed on any chosen number. Try the trick with students, then encourage them to figure out

how it works using a block to stand in for the mystery number (give lots of hints).

The Trick

The Algebra

Pick any number Use a square block to represent the mystery number.

Add 4 Use 4 circles to represent the 4 ones that were added.

page 162

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Multiply by 2

Create 2 sets of blocks to show the doubling.

Subtract 2

Take away 2 circles to show the subtraction.

Divide by 2

Remove one set of blocks to show the division.

Subtract the mystery number

Remove the square.

The answer is 3!

No matter what number you choose, after performing the operations in the magic trick, you will

always get the number 3. The model above shows why the trick works.

Encourage students to make up their own trick of the same type.

For extra problems with equations, see worksheet in the Blackline Masters.

PA7-26: Solving Equations by Working Backwards page 165

1. Solve the following equations:

a) x + 2.1 = 7.8

b) x – 27.32 = 57.06

c) x

2.5 = 7

d) 3x = 2.1

E

E

page 163

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

PA7-27: Two Step Problems page 167

1. As a warm up for the section give your students the following kind of puzzles. Have the class work

together to find the answers:

Examples: a) I am a number. If you multiply me by 2 and add 5 the result is 15 --- what number am I?

b) I am a number if you multiply me by 8, increase me by 4 and then divide me by 3 the result is 12.

The students should see that they can find the original number by working backwards from the final answer. For example, b): multiply 12 by 3 ( = 36)

subtract 4 (= 32) divide by 8 ( = 4) Then the original number is 4.

Students should be encourage to check their answer:

2. Students could play the following game:

Step 1:

A student takes a small number of paper bags or containers with lids with an equal number of counters in each container. The student also selects some counters to be left outside the containers.

Step 2: The student tells their partner the total number of counters they placed both in the container and outside the containers. (For the example above, the student would tell their partner "I've placed 9 counters altogether".)

Step 3: The partner has to figure out (without looking in the containers) how many counters are in each container. Students will naturally work backwards to solve the puzzle. For the example above, they might reason as follows: "There are 3 counters outside the containers, so there are 9 – 3 = 6 inside. There are 3 containers, so there are 6 ÷ 3 = 2 counters in each container."

Once students become good at this game, show them the connection with the algebra: • There are x counters in each container, so there are 3x counters inside the three containers. • There are 3x + 3 counters altogether. • I know there are 9 counters altogether, so 3x + 3 = 9 • The number of counters inside the containers (3x) equals 9 minus the number outside:

3x = 9 – 3 3x = 6

• There are 3 containers so I divide 6 by 3: x = 6 ÷ 3 x = 2

M

4 × 8 = 32 32 + 4 = 36 36 ÷ 3 = 12

page 164

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

1. TEACHER: Review the method of simplifying expressions shown in extension exercise 3 for PA7-23

(i.e. 2x + 3x = 5x). Then ask your students to solve the following equations:

a) 2x + 3x = 25 b) 7x + 2x = 18

c) 3x + 8x = 10 + 12 d) 2x + 2x + 2x + 7 = 19

PA7-28: More Two Step Problems page 169

Applying pattern rules to problem solving:

Ask your students to try adding 1 + 2 + 3 + 4 + … + 100. How would they approach the problem? Would

they just start adding and hope to finish quickly? Would they look for a pattern? The mathematician

Gauss came up with a clever answer that avoided both methods. He noticed that you could write this as

1 + 2 + 3 + … + 48 + 49 + 50

+100 + 99 + 98 + + 53 + 52 + 51

101 + 101 + 101 + … + 101 + 101 + 101

Ask your students how many 101s there are in the sum? How do they know?

Answer: there are fifty 101s. So the sum is: 101 × 50 = 101 x 100

2

Ask them to find the sum of the numbers from 1 to 10 using this method, from 1 to 20, and from 1 to 70.

What happens if they find the sum from 1 to 25? What makes this problem different? There are at least

two approaches here. They could find the sum from 1 to 24 and then add 25. Or they can find the sum

from 1 to 25 directly. Tell them to do it both ways.

1 + 2 + 3 + ... + 12 1 + 2 + 3 + ... + 12 + 13

24 + 23 + 22 + …+ 13 25 + 24 + 23 + … + 14_____

25 + 25 + 25 + … + 25 26 + 26 + 26 + … + 26 + 13

25 × 12 + 25 = 25 × 13

How many 26s are being added? Find the answer as a multiplication and addition statement.

26 × 12 + 13 = 13 × 24 + 13 = 13 × 25 = 25 × 26

2

E

E

page 165

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Note that in any case, the answer is always (N)(N+1)

2 where N is the number of terms. It doesn’t matter

whether N is even or odd.

Where do pattern rules come in?

Ask the students to add: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 without actually doing the additions.

There are several possible ways to look at this:

(1 + 2+ 3 + … + 10) × 2

OR 2 + 4 + 6 + 8 + 10

+ 20 + 18 + 16 + 14 + 12

22 + 22 + 22 + 22 + 22 = 22 × 5

OR (1 + 3 + 5 + 7 + … + 19) + 10 and then try to find 1 + 3 + 5 + … + 19. (This is actually an easier

pattern to notice if they’re familiar with the square numbers).

Note that ALL of these methods require noticing that the number of terms is 10.

Since your students know how to write algebraic expressions for linear patterns, they will be able to find

the number of terms in any sum.

Example: How many terms are in the sum 19 + 21 + 23 +…+ 83?

Step 1: Make a T-table

Term Number Term

1 19

2 21

3 23

Step 2: Write a rule for the table

2 × Term Number + 17 or 2n + 17

Step 3: The last term is 83. We can find the term number as

2n + 17 = 83

2n = 83 – 17

2n = 66

n = 33

Step 4: There are 33 terms in the sum. So there are 16 pairs plus one in the middle. The middle (or

17th) term is: 2 (17) + 17 = 51

19 + 21 + 23 … + 49 + 51

83 + 81 + 79 … + 53

102 +102 +102 +102 + 51

The pairs all add to 102. So the sum is 16 × 102 + 51 = 33 × 51 = 1683

page 166

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

Practice questions:

7+8+9+… + 59

44+46+48+… + 96

35+37+39+ …+ 105

32+35+38+…+104

123 + 127 + 131 + … + 203

PA7-30: Algebraic Puzzles page 171

1. Students could solve the puzzles in questions 1 and 2 using blocks. Give your students two

containers to represent the two balance scales, and ask them to place a selection of blocks that

match the picture into the container. Students must solve the problem by isolating the block that

represents the solution (for instance, in question 1 on the worksheet they must isolate a square).

Students are allowed to add or take away blocks from the containers but they must follow two rules:

Rule 1: You may add or remove one or more blocks from one container as long as you add or

remove the same number of blocks from the other container. (This mirrors the algebraic

rule that whatever you add or subtract from one side of an equation you must add or

subtract from the other side.) EXAMPLE:

Rule 2: If all the blocks in one container are of a particular type and all blocks in the other container

are of a particular type and if you can group the blocks in each container (equally into

exactly the same number of sets) you may remove all but one of the sets of blocks from

each container. EXAMPLE:

In the picture above, the blocks are grouped into three equal sets of squares and three

equal sets of triangles. Each square must weigh the same as 4 triangles. So you can

remove all but one group of squares and triangles without unbalancing the scale.

A

If 2 blocks are taken from the left side of the scale, then 2 blocks must be taken from the right hand side as well.

The blocks are placed into 3 sets on either side. 2 of the sets are removed from the scale.

page 167

Teacher’s Guide

Grade 7

Teacher’s Guide

Grade 7

(This rule mirrors the algebraic rule that you may divide both sides of an equation by the same

number.) Ask students to draw a series of pictures that show the contents of their containers as

blocks are added or removed.

PA7-35: Concepts in Patterns and Algebra page 179

1. The first term of a sequence of numbers is 2. Each term after the first is obtained by multiplying the

preceding term by 5 then subtracting 6. What is the 5th term of the sequence?

2. If 4 � 3 = (4 × 3) + (4 + 3)

and 2 � 5 = (2 × 5) + (2 + 5),

calculate 7 � 9.

3. Find the sum of the squares and then describe the pattern in the sum:

12 + 22 =

22 + 32 =

32 + 42 =

42 + 52 =

4. Kyle paid $22 for a taxi ride. The initial charge was $2 and he drove for 5 minutes. What was the

charge per minute?

5. Challenging

The numbers greater than 1 are arranged in an array, in which the columns are numbered 1 to 5 as shown:

a) Describe the patterns in each column. How do you

know that the number 2004 does not appear in

column (1) or (5)?

b) In which column does 2004 appear?

c) In which columns will the perfect squares appear?

HINT: Complete the chart:

Perfect square: 1 4 9 16 25 36 49 64

Remainder when divided by 8: 1 4 1 0

Perfect square: 81 100 121 144 169 196 225 256

Remainder when divided by 8:

What pattern do you notice? What numbers can be the remainder of a perfect square divided by 8? Now look at

the columns in the original chart and look for the patterns when you divide the numbers in the column by 8.

E

· · ·

(3)

2 3 5 4

6 7 8 9

10 11 12 13

14 15 16 17

(1) (2) (4) (5) (3)

2

page 168

Teacher’s Guide

Grade 7

Number Sense

NS7-52: Word Problems page 184

1. Neptune orbits the sun 3 times in the same time it takes Pluto to orbit the sun 2 times. How many

orbits does Pluto complete while Neptune orbits 12 times?

2. Two cm on a map represents 5 km in real life. If a lake is 6 cm long on the map, what is its actual

size? (Here the quantities compared are cm and km.)

NOTE: To solve the question below, you will need to reduce the ratio given to lowest terms.

3. There are 6 boys for every 10 girls on a school trip. If there are 35 girls, how many boys are there?

NS7-53: Further Ratios page 185

1. Have students draw a scale drawing of their living room on paper. Mark what measurement each cm

represents.

2. Golden Ratio

The golden ratio, about 1.62, occurs often in human design and architecture and also in nature.

Find the ratio of length to width in each rectangle below. You will have to do some measuring.

a) Several picture frames

b) a standard sheet of paper

c) your notebook

d) a television screen

e) a computer monitor

f) a postcard

g) a birthday card

h) a standard sheet of paper folded in half

i) an index card

Which ones are close to golden ratios?

A

E

page 169

Teacher’s Guide

Grade 7

Draw several rectangles and determine which ones are close to the golden ratio. Many people

believe that rectangles with length to width ratio close to the golden ratio are nicer to look at. Do you

agree?

Leonardo Da Vinci used the golden ratio in his

paintings. Look at the “Mona Lisa”, taken from

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm.

Draw a rectangle around her face. Are the length and

width in a golden ratio? Divide the rectangle by using

her eyes as a horizontal divider. Can you find another

golden ratio?

The Parthenon in Athens was built in about 440BC. A scale drawing is shown below, as found on

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm. How many golden ratios can you find?

The Fibonacci sequence starts 1 1 2 3 5 8 13 21. Find the next three terms. Find the ratio of one term to

the previous term for the first 11 terms:

1:1 2:1 3:2 5:3 8:5 13:8 21:13 1 2 1.5 1.67

What ratio do these numbers appear to be getting close to?

E

page 170

Teacher’s Guide

Grade 7

NS7-57: Comparing Decimals, Fractions & Percents page 190

1. Write the percent to which the fraction is closest.

a) 35 ______ b)

45 ______ c)

25 ______ d)

210 ______

e) 110 ______ f)

410 ______ g)

910 ______ h)

425 ______

i) 1120 ______ j)

1620 ______ k)

3740 ______ l)

112 ______

NS7-58: Finding Percents page 191

1. 5% is half of 10%. Find 5% of the following numbers by first finding 10% then dividing by 2 (using

long division on a separate piece of paper):

a) 80 b) 16 c) 72 d) 50 e) 3.2 f) 2.34

2. Find 15% of the following numbers by finding 10% and 5% and then adding. (Use a separate piece

of paper for your rough work.)

a) 60 b) 240 c) 12 d) 7.2 e) 3.80 f) 6.10

3. Taking 1% of a number is the same as dividing the number by 100. (The decimal shifts 2 left.)

Find 1% of:

a) 27 b) 3.2 c) 773 d) 12.3 e) 68

E

10% 25% 50% 75% 100%

E

page 171

Teacher’s Guide

Grade 7

NS7-59: Finding Percents (Advanced) page 192

1. Which is greater – 60% of 70 or 70% of 60?

2. Investigate: what happens to the area of a rectangle when you increase its length by 50% and

reduce its width by 50%? Does it increase or decrease? By what percentage? Will the answer

change if you increase the width by 50% and decrease the length by 50%?

3. What happens if you increase the length by 40% and reduce the width by 40%? What happens if

each dimension is changed by 30% instead?

4. If your students are familiar with the volume of a cylinder: Investigate: what happens to the volume

of a cylinder when you increase the diameter by 50% and reduce its height by 50%? Does it

increase or decrease? By what percentage? What happens if you increase the height and reduce

the diameter instead – will the answer change?

NS7-60: Percent: Word Problems page 193

A ratio is a comparison of 2 numbers. The ratio of the number of vowels to consonants in the word

“ratio” is 3:2 (a ratio can also be written in words “3 to 2” or as a fraction 32 ).

1. The chart shows the fraction or percent of stamps that children have collected from various countries.

Canada England Other Countries

Brian’s

Collection 23%

35

Faith’s

Collection 34 7%

Andrew’s

Collection 12 30%

Which child has collected the most stamps from countries other than Canada and England?

E

M

E

page 172

Teacher’s Guide

Grade 7

2. Mr. Bates buys…

• 5 single-scoop ice cream cones for $1.45 each

• 3 double-scoop ice cream cones for $2.65 each

A tax of 15% is added to the cost of the cones. Mr. Bates pays with a $20.00 bill. How much

change does he receive? Show your work.

3. Find the missing percent of each child's collection that comes from other countries. The first one has

been done for you: HINT: Change all fractions to percents.

a) Anne’s Collection

Canada USA Other

40%

12

= 50%

12

10%

b) Brian’s Collection

Canada England Other

80% 110

c) Juan’s Collection

Mexico USA Other

12

40%

d) Lanre’s Collection

Canada Nigeria Other

22% 35

e) Faith’s Collection

Jamaica Canada Other

34

15%

f) Carlo’s Collection

France Italy Other

34

10%

4. There are 15 boys and 12 girls in a class. 34 of the girls have black hair, and 60% of the boys have

black hair. How many children have black hair?

5. Kevin has 360 hockey cards. 30% are Toronto Maple Leaf cards, 12 are Edmonton Oilers cards,

and the rest are Calgary Flames cards. How many of each type of card does he have?

6. Income Tax

Countries have to tax their citizens in order to pay for public services like health care. Usually this is

done by making each adult pay a percentage of their yearly income. A country is deciding how to tax

their citizens. There are two different suggestions:

Suggestion A: Tax everyone at 20% of their yearly income.

Suggestion B: Do not tax people who make less than $20 000 a year. Tax people who make at least

$20 000 but less than $40 000 at 15% of their yearly income. Tax people who make at least $40 000

but less than $70 000 at 25% of their yearly income. Tax people who make more than $70 000 at

50% of their yearly income.

page 173

Teacher’s Guide

Grade 7

Fill in the table below:

Income before taxes Income after taxes using Suggestion A

Income after taxes using Suggestion B

$5 000 $5 000 × 80/100 = $4 000 $5 000 $10 000 $15 000 $19 900 $20 000 $25 000 $30 000 $30 000×80/100 = $24 000 $30 000×85/100 = $25 500 $35 000 $39 000 $40 000 $50 000 $60 000 $69 000 $70 000 $71 000 $100 000 $1 000 000 $100 000 000

What are the advantages and disadvantages of suggestion A? Of suggestion B? Research to find

out how the Canadian government actually charges taxes. In what way is the system similar to

suggestion A? In what way is it similar to suggestion B?

7. Writing a cheque

On a cheque, the amount being paid is written both numerically and in words:

PAY TO THE : ________________________________________________ ORDER OF

twenty-seven ----------------------------------------------------------------------- 99100

___________________________

a) Why do we have to write a number on a cheque if we already have the numerical writing?

b) Give two examples of how the numerical writing could be easily tampered with to make a

significantly larger amount. HINT: What number can easily be added to the beginning?

c) What do you think the purpose is of drawing the line after the written words "twenty-seven"?

d) If the line wasn't there, how could the cheque be changed to make it a substantially greater

amount?

HINT: 1000 is often written as 1,000. So could you change $27.99 to $27, 990.99 and change twenty seven 99

100 to twenty seven thousand nine hundred and ninety

99100

.

$27.99

page 174

Teacher’s Guide

Grade 7

e) I walk into a department store and want to buy a shirt for $29.99, another shirt for $35.99 and

pants for $39.99. These prices are listed before taxes, so I need to add 15% to the cost of each

item. Determine how much I need to fill out the cheque for and then complete the cheque below.

PAY TO THE : ________________________________________________ ORDER OF

twenty-seven ----------------------------------------------------------------------- 99100

___________________________

8. Percent recommended daily value

The Nutrition Facts label on a cottage cheese container says the amount in the container and the

percent of the recommended daily value. Find the recommended daily value for each item.

For example, to find what value 2 g is 3% of, we need:

2 = (3100 ) × what number? The number must be 2 ÷ (

3100

) = 2 ÷ .03 or about 67 g.

Amount Percent of Daily

Value Amount Recommended as

Daily Value

Fat 2 g 3 % 2 ÷ 3100 = 67

Saturated and Trans fat

1.1 g 6 %

Repeat using a different food label. Do both labels give the same recommended daily value? If not,

are the numbers close? What could cause any differences?

9. Heartbeats

Scientists are trying to develop an artificial heart that is capable of beating one billion times. What

percentage of a normal heart’s lifetime beating is this? You need to decide how long a normal heart

lasts for in minutes and how fast it beats in beats per minute. Assuming the artificial heart beats at

the same rate as a normal heart, about how long will the artificial heart last (in years)?

10. Sales Tax

The Sales tax in Ontario (including both provincial and national taxes) is 14%. This means that if a

store wants to make $1.00 from selling a product, it needs to put the price 14% higher, which is

$1.00×14/100, or $1.00×0.14 = $0.14 = 14¢ higher. So the total price becomes $1.14. Find the total

price if the store’s original price is as given:

Original Price 14% of Original Price New Price

$20.00 $20×0.14 = $2.80 $22.80 $32.00 $49.99 $4.00 $2449.99

$27.99

page 175

Teacher’s Guide

Grade 7

Rita looks at the chart and says, “I can do the same thing in one step instead of two. I know that

$20.00 = $20.00×1 so $20.00 + $20.00×0.14 = $20.00×1 + $20×0.14

= $20.00×(1+0.14)

= $20.00×1.14.

What property of multiplication is Rita using?

Use Rita’s method to fill in the chart. Do you get the same answers?

Original Price New Price

$20.00 $20.00×1.14 = $22.80 $32.00 $49.99 $4.00 $2 449.99

Sometimes, a product has no national taxes, only the 6% provincial taxes. Calculate the new price

for each of the original prices given above if the product only has the provincial tax.

Sales tax in Ontario was recently decreased from 15% to 14%. How much money is saved on a

purchase of $3? $25? $350? $3 499? What percentage of the original price is saved?

Melanie sees a book on sale at 20% off. The sales clerk calculates the price in two different ways.

1) by adding 14% to the original price and then taking 20% off

2) by taking 20% off and then adding 14% to the discounted price.

Melanie thinks she’ll save money by using the second method because she’s paying taxes on a

smaller amount. Is she right? Calculate the final price using both methods for various original prices.

Explain why she does or does not save money.

Original Price Method 1 Price Method 2 Price

$25.00 $4.00 $37.00 $22.49 $7.99

11. Price Increases

If a price increases by 10% of the original price, the final price would be 110% of the original price.

If the price of a 20¢ candy increases by 25%, what is the final price?

If the price of a 50¢ candy increased to 60¢, by what percent of the original price did the price

increase? Think: The price increased by 10¢. 10¢ is what percent of the original price?

What percent of the original price would the final price be after:

A 30% increase? A 75% increase? A 100% increase? A 200% increase?

page 176

Teacher’s Guide

Grade 7

A 30% decrease? A 75% decrease?

Can a price decrease by 200%?

The price increases by 60% and then by 40%. Has it increased by more or less than 100%? What if

the price increases first by 40% and then by 60%?

A 200% increase is the same as a 50% increase followed by a _______% increase.

NS7-61: Fractions, Ratios and Percents page 195

To solve questions involving fractions, ratios and percents your students need to be able to recognize the

part and the whole (and to make a fraction showing the ratio of the part to the whole).

1. What fraction of the original price would the final price be after:

A 50% decrease? A 20% decrease? A 50% increase? An 80% increase? A 110% increase?

2. When you compare 2 numbers, you can estimate what fraction (or percent) the numbers make by

changing one or both of the numbers slightly. If you increase the numerator, you should increase the

denominator. Example A: 5 out of 11 is close to 5 out of 10, which is close to 12 or 50%.

Example B: Is 9 out of 22 closer to 8 out of 2 or 10 out of 25. How do you know?

The chart shows the lengths of calves and adult whales (in feet). Approximately what fraction and

what percent of the adult length is the length of the calf?

Type of Whale Killer Humpback Narwhal Fin Backed Sei

Calf Length (feet) 7 16 5 22 16

Adult Length (feet) 15 50 15 70 60

3. There are 3 apartment buildings in a block:

• Apartment ‘A’ has 50 suites.

• Apartment ‘B’ has 50% more suites than apartment ‘A’

• Apartment ‘C’ has twice as many suites as apartment ‘B’.

a) How many suites do apartment ‘B’ and apartment ‘C’ have?

b) How many suites do all three apartments have altogether?

4. Tom spent $500 on furniture: he spent 310 of the money on a chair, $50.00 on a table and the rest

on a sofa. What fraction and what percent of the $500.00 did he spend on each item?

M

E

page 177

Teacher’s Guide

Grade 7

5. The peel of a banana weighs 18 of the total weight of a banana.

If you buy 4 kg of bananas at $0.60 per kg…

a) How much do you pay for the peel?

b) How much do you pay for the part you eat?

NS7-62: Further Fractions, Ratios and Percents page 196

1. Ron, Fatima and Chyann went to the store. One person had $20, one had $60 and one had $36.

At the store, one person spent 12 of their money, one spent

23 and one spent 25%. Ron spent $10

and Fatima spent $9. Chyann had $20 left. How much money did each person go to the store with?

2. 98% of Antarctica is covered in ice. What fraction of Antarctica is not covered in ice?

3. A ball is dropped from a height of 100 m.

Each time it hits the ground, it bounces 35 of the height it fell from. How high did it bounce…

a) on the first bounce? b) on the second bounce?

4. Measure the angles and complete the chart. Compare the results of both charts. How should you

compare the numbers to find out which regions have the most and least amount of wealth? NOTE: Latin America includes South America, Central America, and Mexico.

Region Population Percent

Africa

Europe

USA & Canada

Latin America

Oceania

Asia

Total 6 500 000 000 100%

E

Europe

Africa

USA & Canada

Asia

Latin America

Oceania

World Population Breakdown

page 178

Teacher’s Guide

Grade 7

Region Wealth Percent

Africa

Europe

USA & Canada

Latin America

Oceania

Asia

Total $20 000 000 000 000 100%

Does the distribution of the world’s population match the distribution of the world’s wealth? Does the

distribution seem fair? Adapted from “Rethinking Mathematics: Teaching social justice by The Numbers”.

NS7-63: Order of Operations page 198

1. Using four 4’s each time, make expressions for each value from 0 through 10. You may use

brackets and any of the four operations. 2. Find a) 1 × 2 + 3 × 4 b) 1 + 2 × 3 + 4 c) 1 × (2 + 3) × 4 d) (1 + 2) × (3 + 4) 3. Find a) 3 × 4 ÷ 6 b) 3 × (4 ÷ 6) c) 3 ÷ 4 × 6 d) 3 ÷ (4 × 6) e) 3 ÷ 6 × 4 Which two answers were the same? Why?

4. Find a) 3 + 4 - 6 b) 3 + (4 - 6) c) 3 – 4 + 6 d) 3 - (4 + 6)

5. Which two answers were the same? Why?

6. Using the first three integers in each row, show how to get the fourth number by using brackets and

operations (+,-,×,÷). a) –8, 3, 4 -20 b) –10, 2,-9 4 c) 7, -11, 7 0 d) –6, 1, 15 -3

E

World Wealth Breakdown

Europe

Africa

USA & Canada

Asia

Latin America

Oceania

page 179

Teacher’s Guide

Grade 7

NS7-64: Square Roots page 199

1. Evaluate both sides and put >,< or = as appropriate.

a) 9+16 _____ 9 + 16 b) 9x16 _____ 9x 16

c) 25-9 _____ 25 - 9 d) 36÷9 _____ 36÷ 9

e) 36+64 _____ 36 + 64 e) 100×64 _____ 100 × 64

g) 100-64 _____ 100 - 64 h) 100÷64 _____ 100 ÷ 64

2. A number is a perfect square if its square root is a whole number. Round to the nearest whole

number by finding a perfect square close to the number under the square root sign.

Eg. 99 ≈ 100 = 10.

a) 10 ≈ _____ b) 37 ≈ _____ c) 50 ≈ _____

d) 15 ≈ _____ e) 120 ≈ _____ f) 145 ≈ _____ 3. Use a pattern to find the missing numbers:

a) 121 =11 b) 12321 = 121 c) 1234321 = _____

d) 123454321 = ______ e) 12345678987654321 = __________________

4. Do you think the pattern in question 3 continues to hold? Why or why not?

5. A number is a perfect square if its square root is a whole number. Do you get a perfect square if …

a) … you add two perfect squares?

b) … you multiply two perfect squares?

c) … you divide two perfect squares?

d) … you subtract two perfect squares?

E

page 180

Teacher’s Guide

Grade 7

NS7-68: More Integers page 203

1. The 3×3 grid is a magic square because each row, column and diagonal add to the same number (in

this case, 15). Construct a magic square from the integers:

- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4. What does each row, column and diagonal add to?

6 7 2

1 5 9

8 3 4

NS7-70: Subtracting Integers page 205

NS7-70 Subtracting Integers (from OISE/UT)

1. I start at sea level. If given a helium balloon, I move one metre up from sea level and if given a

sandbag, I move one metre down from sea level. Record where I am after each move and write an

appropriate addition or subtraction statement. Use a number line if it helps with the addition and

subtraction.

a) Someone gives me 3 helium balloons 0 + (+3) = +3 I am 3 m above sea level.

b) Someone gives me 4 sandbags +3 + (- 4) = -1

I am 1 m below sea level.

c) Someone takes away 2 helium balloons -1 - (+2) = - 3 I am 3 m below sea level.

d) Someone gives me 1 sandbag e) Someone gives me 7 helium balloons f) Someone takes away 3 sandbags g) Someone takes away 4 helium balloons

E

E

page 181

Teacher’s Guide

Grade 7

h) Someone gives me one helium balloon i) Someone gives me 2 sandbags j) Someone takes away 3 sandbags.

2. Time Zones

Because of the earth’s rotation, different parts of the earth have daylight at different times, so one

part of the earth will be having nighttime while another part will be having daytime. When it is 3 PM

in one place, it might be 1 AM in another place.

Integers are used to show whether a particular place is earlier or later than London, England. Athens

is +2 and hence is 2 hours later than London. Halifax is –4, which means that in Halifax, it is 4 hours

earlier than in London.

A circus is traveling the world. Each time they enter a different time zone, they need to change their

watches. Use a subtraction statement to determine how they should change their watches after each

move.

London 0 Toronto - 5

Rome +1 Vancouver - 8

Moscow +3 South Africa +2

Halifax - 4 Ethiopia +3

a) They travel from Toronto to Moscow. +3 - (- 5) = +8

They should change their watches to be 8 hours later.

b) They travel from Moscow to Rome. +1 - (+3) = - 2

They should change their watches to be 2 hours earlier.

c) They travel from Rome to Vancouver.

d) They travel from Vancouver to South Africa.

e) They travel from South Africa to Ethiopia.

f) They travel from Ethiopia to Halifax.

g) They travel from Halifax back to Toronto.

3. Count the total number of hours they moved their watches earlier and count the total number of

hours they moved their watches later. Explain why these numbers should be the same.

page 182

Teacher’s Guide

Grade 7

NS7-74: Concepts in Integers page 209

An NHL hockey player’s + / – rating is determined as follows. If his team scores while he is on the ice,

+ 1 is added to his rating; if the other team scores while he is on the ice, – 1 is added to his rating.

Look at the statistics below and answer the following questions.

The following data is from www.nhl.com:

Name Position A PTS + / –

Mats Sundin Centre 31 47 78 7

Bryan McCabe Defense 19 49 68 – 1

Tomas Kaberle Defense 9 58 67 – 1

Darcy Tucker Left-wing 28 33 61 – 12

Jason Allison Centre 17 43 60 – 18

Alexander Steen Centre 18 27 45 – 9

Kyle Wellwood Centre 11 34 45 0

Jeff O'Neill Right-wing 19 19 38 – 19

Alexei Ponikarovsky Left-wing 21 17 38 15

Nik Antropov Centre 12 19 31 13

Chad Kilger Left-wing 17 11 28 – 6

Matt Stajan Centre 15 12 27 5

Tie Domi Right-wing 5 11 16 – 10

Alexander Khavanov Defense 6 6 12 – 11

Aki Berg Defense 0 8 8 – 5

Clarke Wilm Centre 1 7 8 – 15

1. What does a positive +/– rating mean? What does a negative +/– rating mean?

2. How many players had a positive +/– rating? How many players had a negative +/– rating?

3. If the players were listed in order from best +/– rating to worst +/– rating, write down the first five

names and the last five names in the order they would occur.

4. Who had the best +/– rating and who had the worst +/– rating on the team? How much did their

ratings differ by?

5. You are coaching the Toronto Maple Leafs and decide to mix the team up during practice. You play

Jason Allison, Tie Domi, Alexei Ponikarovsky, Bryan McCabe and Alexander Khavanov against Mats

Sundin, Jeff O’Neill, Chad Kilger, Matt Stajan and Tomas Kaberle. Based on only the sum total of

each “team’s” +/– rating, who do you expect to win: Allison’s team or Sundin’s?

6. Make up the +/– rating all-star team for the Toronto Maple Leafs by choosing the best forwards

(one centre, one right-wing, one left-wing) and the two best defensemen.

7. In the above list, no goalies are listed. In fact, goalies do not keep track of +/– ratings. Why is this?

E

G

page 183

Teacher’s Guide

Grade 7

NS7-77: Multiplying Decimals by Decimals page 212

1. The cost of painting a median line on a highway is $124.60 per kilometre. If the highway from

Edmonton to Calgary is 301.3 km long, how much will it cost to paint the median line on the

highway. 2. Find each product by first changing each decimal to an improper fraction and using the method for

multiplying fractions. Then change your answers back into a decimal.

a) 1.3 × 4.7 = 1310 ×

4710 =

611100 = 6.11 b) 1.5 × 3.2 =

c) 3.6 × 4.25 = d) 32.5 × 4.8 = e) 4.01 × 25.2 = f) 4.235 × 1.1 =

3. Look at your answers to question 2. Does the same rule hold that you found in question 3 of your

workbook? Use your observation to multiply the following decimals:

a) 1.5 × 2.4 b) 3.2 × 0.46 c) 4.56 × 1.01 d) 0.3 × 8.4 e) 5.23 × 0.47

NS7-81: Multistep Problems page 217

1. a) A newspaper charges $687.99 plus 14% sales tax for a full page advertisement per day. How

much would a 2/3-page advertisement cost to run for a week.

b) The same newspaper charges $0.15 a word to run an employment advertisement for a day. How

much would it cost to run the following ad for a week?

Cook wanted at fast food Indian restaurant.

Experience an asset. Fax resume to 555-1234

Don't forget to add 14% sales tax.

c) Another newspaper charges 3¢ per character, not including spaces. In which newspaper would

the "cook wanted" advertisement be cheaper?

2. The price of a soccer ball is $8.00.

If the price rises by $0.25 each year, how much will the ball cost in 10 years?

E

E

page 184

Teacher’s Guide

Grade 7

3. Janice earned $28.35 on Monday. On Thursday, she spent

$17.52 for a shirt. She now has $32.23.

How much money did she have before she started work Monday?

4. Anthony’s taxi service charges $2.50 for the first kilometre and

$1.50 for each additional km.

If Bob paid $17.50 in total, how many km did he travel in the taxi?

HINT: Guess and check.

page 185

Teacher’s Guide

Grade 7

Measurement

ME7-14: Mass page 219

1. A monarch butterfly has a mass of about 500 mg.

How many monarch butterflies would have a mass of about a gram? ______________________

How many would have a mass of about a kilogram? ___________________________________

2. Write in the missing masses to balance the scales. The masses on the right hand scale are equal in

each question:

a) b)

3. Solve the following word problems involving grams and kilograms:

a) The cost of shipping a package is $15.00 for each kilogram shipped.

How much does it cost to ship a package that has a mass of 14 kilograms?

b) There are 35 mg of calcium in a vitamin pill.

How many mg of calcium would you consume if you had a vitamin pill every day of the week?

c) There are 675 salmon in the pond, and each has a mass of approximately two kilograms.

What is the total mass of all the salmon in the pond?

1. Let students feel the weight of objects that are close to 1 g or 1 kg. Help them establish referents for

1 g (i.e. a paperclip, a dime) and 1 kg (i.e. a one litre bottle of water) that they can use to make

estimates. Students should use scales to record the weight of objects in the class. (Have them

estimate before they weigh.)

2. Weigh an empty container, then weigh the container again with some water in it. Subtract the two

weights to find the weight of the water. Repeat this with a different shaped 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.

E

A

10 kg 10 kg 10 kg 9 g 9 g

page 186

Teacher’s Guide

Grade 7

3. 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 g of water weigh? (How much does 1 g of water weigh?)

4. When Duncan stands on a scale the arrow points to 45 kg. When he stands on the scale and holds

his cat, the arrow points to 50 kg. How could he use these two measurements to find the weight

of his cat?

ME7-15: Capacity page 220

1. You will need: graduated cylinders holding 100 mL of liquid (3 per group)

Water, corn syrup and vegetable oil (100 mL per group)

Two separate bins to pour water and oil into.

A scale to weigh the liquid in the graduated cylinders.

a) Weigh an empty graduated cylinder.

b) Put 100 mL of water in the graduated cylinder.

c) Weigh the water in the cylinder.

d) How much does 100 mL of water weigh? Remember to subtract the weight of the cylinder:

e) How much would 1 mL of water weigh?

f) Do you think other liquids will weigh the same amount?

g) Try measuring vegetable oil and corn syrup by the same method. Leave each liquid in the

graduated cylinder. 1 mL of vegetable oil weighs _______. 1 mL of corn syrup weighs

________.

h) Heavier liquids will sink to the bottom if put in the same container as a lighter liquid. If you put 40

mL of vegetable oil in the same cylinder as 40 mL of water, which liquid do you expect to see on

the bottom? Try it and see – pour 60 mL of the water into the water bin and 60 mL of the

vegetable oil into the oil bin and then put the remaining water into the cylinder with oil. Record

your results. Do you think it matters which liquid was in the container first?

i) 60 mL of vegetable oil weigh more than 30 mL of water. Will 60 mL of vegetable oil sink when

put in a container with 30 mL of water? Make a prediction and then test your answer. You will

need to use the liquids from your bins. Record your results.

2. See the worksheet on Volume and Capacity in the Blackline Masters.

3. Collect 5 containers (cups, cans, bottles, pails) of different sizes:

a) Estimate the capacity of each container.

b) Measure and record the capacity of each container.

c) Order the measurements from greatest to least.

d) Compare your measurements with your estimates.

A

page 187

Teacher’s Guide

Grade 7

4. a) Weigh a container that will hold 1 L of water.

b) Measure 1 L of water and pour it into the container. Weigh the container with the water in it.

c) How can you calculate the mass of 1 L of water using the 2 measurements you made?

d) What do you notice about the mass of the water (in g) and the volume of the container in (mL)?

ME7-17: Area in Square Centimetres page 223

1. Gift wrapping: You want to wrap a birthday gift in a 20 cm × 30 cm × 40 cm box.

a) What shape of wrapping paper will you start with? b) How many different ways can you wrap this box? Hint: There are 3 different bases for your box - 20××××30,

20××××40 and 30××××40. For each base, how many ways can you wrap the box?

c) Which way uses the most paper? The least paper? Why don’t they all use the same amount of

paper? Assume that you always use the least amount of paper possible.

1. Ask students to construct rectangles of the same area, but with different lengths and widths, on a

geoboard, or with square tiles on grid paper. How many different rectangles can they make with

area 8 cm2 ?

2. Students could try to make as many shapes as possible with an 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.

Students could do the same exercise on a geoboard.

3. Do a research project to see if most people instinctively know the best way to wrap a gift in order to

use the least amount of paper. Have students design a test to give to grade 8 students from a

different class. For example, students could give them 3 boxes to wrap of different sizes and exactly

enough wrapping paper to use for all three of them if they do it properly. Or the grade 8 students

could be given a roll of paper and just told to use the least amount of wrapping paper they can, using

their intuition. For the purpose of this exercise, newspaper can be used instead of wrapping paper for

environmental reasons.

E

A

page 188

Teacher’s Guide

Grade 7

ME7-19: Area & Perimeter page 225

1. Describe a situation in which you would have to measure area or perimeter. (For instance, to cover a

bulletin board, or make a border for a picture.) Make up a problem based on the situation.

2. A rectangle has a area of 20cm2 and length 5 cm. What is its perimeter?

3. The shape has perimeter 24 cm. What is its area?

4. Find the area:

1. Research project – find applications of perimeter-area ratio. Students should make a list of at least 5

places where perimeter-area ratio is used in a real-life context. Demonstrate using the following

web-site. http://www.colostate.edu/Depts/Entomology/courses/en507/papers_1997/ellingson.html

Demonstrate using Powerpoint if available or using an overhead transparency otherwise. After

instructing students to search for “perimeter-area ratio,” and finding the above website on the

second page, students can be shown how to do a computer search for the word perimeter inside the

article (usually ctrl + F for “Find”). Get a student to read the sentence out loud. Ask if anyone knows

what “emigration” means. If not, get a dictionary, look it up and instruct them that this is what they

are expected to do for their research project. After reading out the definition, continue searching the

article and then get them to read the sentence with the word perimeter as well as the next sentence.

Ask them if they know what empirical means, look it up if necessary and then ask them to sum up

how the perimeter-area ratio is important for people studying insect habitats. To earn a level 3,

students will need to say how the perimeter-area ratio is used (in this case, to study insect habitats

and how often insects leave the habitat) and whether a higher or lower perimeter-area ratio is

beneficial. A level 4 student will either say why the perimeter-area ratio is better small or large or

why we need to study the application of perimeter-area ratio (eg. I found that perimeter-area ratio is

used to study insect habitats and how often insects leave the habitat. A greater perimeter-area ratio

will mean that insects will leave the habitat more often because there will be a greater chance of

reaching the outside of the shape if there is a higher perimeter-area ratio. OR We need to

understand how the insect’s environment affects the insect’s behaviour because we as humans

A

E

10 4

8

20

14

21

12

page 189

Teacher’s Guide

Grade 7

often influence the insect’s environment by our actions. All students should cite the web-site they

found that suggests the importance of the perimeter-area ratio. Students should show an

understanding of what is being said by using their own words.

Students should find at least 3 other, completely different, places where perimeter-area ratio is used.

Web-sites found from a quick google search:

www.in.gov/judiciary/opinions/previous/wpd/06150401.tgf.doc - perimeter:area ratio of buildings

applicable to calculating taxes. This web-site sometimes uses “PAR” to mean perimeter-area ratio.

Here, a smaller ratio means that the building space is being used more efficiently (less walls per

square foot) so would be taxed less.

http://lenr-canr.org/acrobat/WarnerJelectrolys.pdf - high perimeter-area ratio is ideal for thermal

power generation

http://phoenix.gov/PARKS/sonoran2.pdf

www.fs.fed.us/ne/fia/studies/LDS/fragmetrics_define.doc

2. Research project – Adapted from “Tessellating Emus: Activities for Middle School Math Grade 7”.

Students need to research to find the amount of space they must leave for each emu according to

government regulations. They could do this by finding a Canadian emu farmer on the internet and

email the person or by using the library. If you want to keep 6 emus on your farm, what area do you

require? Why do you think the government has regulations about this?

Fencing comes in 1 m sections. Using the grids on graph paper to represent 1 m2, show as many

different rectangular pens as you can that will hold all your emus.

Each metre of fencing costs $3 and grass seed costs $10 for each square metre. Fill in the chart to

decide how much each rectangular pen will cost.

Length Width Perimeter (m) Area (m2) Cost ($)

Which pen do you recommend? You find out that emus need to be kept in 3 different pens – one for male adults, one for female adults and one for babies. You then need to use some of the fencing for borders rather than only perimeter. You have two male adults, two female adults and two babies. Look at your best rectangular pen from the chart above. Find 4 different ways of dividing the pen into 3 equal areas. Which method uses the least total amount of fencing? Try at least two other rectangular pens from your chart above to make sure you have made the best possible choice.

page 190

Teacher’s Guide

Grade 7

Think: if one pen uses less exterior fencing, does it necessarily use less total fencing? Why or why not?

3. Research project

If one litre of paint covers 7 m2, how much paint would someone need to paint a wall of dimensions 6m by 3 m? What if the wall has a door that is 2 m high by 80 cm wide? What if the wall also has a window that is a 1 m by 1 m square? A room has a closet that is 1 metre deep, 2 metres wide and 2.5 metres high. I want to paint the sides, back and top of the closet, but not the floor. How much paint do I need? Design a room that has at least two windows, a door and a closet. Calculate the amount of paint needed in order to paint it if you want to paint the closet walls (including the ceiling), all the walls in the room, but not the ceiling or the floor and not the door or windows. What other things would you have to paint around? A heater? Electrical outlets? Anything else? 4. Research project – You want to build a rectangular swimming pool with largest possible area. You have 300 square tiles, each 3 dm × 3 dm, that you want to place around the pool as a border. What should the perimeter of the pool be? Remember: tiles need to be placed at the corners as well, so the perimeter won’t quite be 300 × 3 dm. Find as many different possible rectangles as you can, remembering that the length and width must both be a multiple of 3 dm, and record their perimeters and areas in the chart below.

Length Width Perimeter (dm) Area (dm

2)

Which pool will have the largest area? How does your pool compare to an Olympic size pool in area? In shape? Do most existing pools maximize the area given the perimeter? Why or why not?

ME7-21: Area of Parallelograms page 228

The area of a parallelogram is found by multiplying the length of the base of the parallelogram by the

height of the parallelogram (A = l × h). You can derive this formula quite easily by following exercise 1 on

the worksheet.

M

page 191

Teacher’s Guide

Grade 7

ME7-22: Area of Triangles page 229

Have students try the exercises on the worksheets by cutting out parallelograms and triangles on grid paper as in question 2.

ME7-28: Subdividing Composite Shapes (Advanced) page 235

1. Find the area of the shaded trapezoid by …

a) … using the formula for the area of a trapezoid. b) … finding the area of the rectangle and subtracting the areas of the two triangles.

2. Find the area of the shaded trapezoid by …

a) … using the formula for the area of a trapezoid. b) … not using the formula for the area of a trapezoid. 2 cm 3 cm 4 cm

3 cm

3. Find the area of the shaded parallelogram by using the area of the large rectangle that contains the parallelogram and small rectangles that result when the parallelogram is removed: 3 cm 1 cm 2 cm = + _____________ = ______________ + ____________

5 cm 3 cm

7 cm

2 cm

A

E

page 192

Teacher’s Guide

Grade 7

Find the area of each parallelogram and then complete the chart keeping the multiplication statements in your answer.

Parallelogram Area of large rectangle Area of small rectangle Area of parallelogram

7 2

3

9 × 3 = 27 cm2 7 × 3 = 7 cm2

9 × 3 – 7 × 3 = (9 – 7) × 3 cm2

= 2 × 3 cm2

= 6 cm2

4 3 8

5 5 1

6 1 10

9 17 2 5

4. Find the area:

ME7-32: Concepts in Surface Area page 239

1. A rectangular prism has a square base of area 9 cm2 and a surface area of 78 cm2.

What are the dimensions of the prism?

E

5 5

10

3

4

page 193

Teacher’s Guide

Grade 7

2. Before doing the following extension to question 6, make sure your students know how to find the

square root of a perfect square if you know its prime factorization: eg. The square root of

26 × 34 × 710 is 23 × 32 × 75. Give them lots of practice. Example of practice question: Make a chart of

each whole number from 1 to 15 with its square and the prime factorization of both numbers. Then

ask them, given a prime factorization of a whole number such as 63 = 3×3×7, to find the prime

factorization of 632. Then ask them to recognize which prime factorizations are perfect squares.

(The exponents must all be even). They are then ready to find the prime factorization of a square

root given the prime factorization of a perfect square. After lots of practice with this, you could give

your students the following extension.

Your students could do question 6 by listing factors of, say, 18 and then checking the pairs of

numbers that multiply to 18 against the other given information. Another way to do the question uses

the following clever trick that often appears in math contests. Find 18 × 12 × 6 in terms of a, b and c.

(Answer: the square of their product.) Use the prime factorization of 18 × 12 × 6 to find a×b×c. If

a×b×c = 36 and b×c = 6, what is a? How can you find b? c? Use this trick to solve the following more

difficult questions:

Edges a, b, and c have lengths that are whole numbers. The surface area of each face is written

directly on the face. What are some possible lengths for edges a, b, and c?

a) b) 540 384 270 450 384

ME7-33: Volume page 240

1. Give students the following front, top, and side views of a structure and ask them if they can build it

using interlocking cubes. Then ask them to determine the volume of their structure.

a)

Front

Top

Side

b)

Front

Top

Side

A

256

a

a

b

c

c

b

page 194

Teacher’s Guide

Grade 7

2. BONUS: Give your students a very simple front, top, and side view and ask them if they can calculate

the volume without building the shape. “An example might be… Front Top Side

5. Which front top and side view are not possible? Front Top Side Front Top Side

ME7-36: Surface Area and Volume page 244

1. Find the surface area and volume of each of the following right rectangular prisms. a) b) c) 5 cm 10 cm 15 cm 6 cm 9 cm 2 cm 4 cm 6 cm Surface area: ___________ Surface area: ___________ Surface area: ___________ Volume: _______________ Volume: ______________ Volume: _____________

2. How does the surface area of a right rectangular prism change when each side length is multiplied

by the same amount? How does the volume of a right rectangular prism change when each side

length is multiplied by the same amount. 3. If each dimension is enlarged by 50%, what percentage does the surface area and volume increase

by? HINT: a 50% increase is the same as multiplying by what factor?

E

E

3 cm

page 195

Teacher’s Guide

Grade 7

4. Find the surface area and volume of the tissue box shown below: 22 cm 12 cm 12 cm 7 cm 5. You are designing a cereal box for a cereal company. The box needs to have a volume of 2000 cm3.

There are many possible boxes you could make with this volume. a) Verify that the 3 sets of measurements for the boxes below each have a volume of 2000 cm3. 1 × 1 × 2000 2 × 25 × 40 5 × 25 × 16 b) Calculate the surface area of each box above. c) If it costs 25¢ for each cm2 for the material to make the box, which box would you recommend? d) Find three more boxes with the same volume and calculate the surface area of each. Now

which box would you recommend? e) The cereal company wants the front of the box to be at least 20 cm wide and 20 cm high and

the depth of the box to be at least 4 cm. Find two boxes satisfying these conditions that each have a volume of 2000 cm3. Which box would you recommend?

Calculate the surface area and volume of several sizes of milk cartons: 250 mL, 500 mL, 1L and 2 L. Find the prices at a supermarket for the four different sizes of milk cartons. Make sure you keep everything the same except size (eg. Brand, fat content, lactose-free?, etc.) Enter your data into a chart with headings: capacity (mL), cost ($), surface area (cm2), volume (cm3). How are capacity and volume related? Does twice as much milk cost twice as much? Does twice as much milk use twice the packaging? Graph your data using mL on the x-axis and cost on the y-axis and then draw another graph using mL on the x-axis and surface area on the y-axis. List several reasons why someone might wish to buy a smaller milk carton instead of a larger, less expensive milk carton. [Answers may include: not enough cash, not enough space in fridge, won’t use the full 2 L before it goes bad – perhaps because going on vacation, or lives alone, or only needs a small amount for a recipe, etc.]

5 cm

A

page 196

Teacher’s Guide

Grade 7

Probability and Data Management

PDM7-21: Exploring the Mean, Mode and Median (Advanced) page 257

1. Janice is sales manager in a department store. She must maintain average (mean) daily sales of at

least $8500. Sales for Monday, Tuesday, Wednesday and Thursday are $7530, $8470, $6550 and

$7155. The store is not open on Sunday. What sales will Janice need to make on Friday and

Saturday to come in over the target? 2. The table shows several different stores and their prices for the same pair of shoes.

Store A B C D E F G Price ($) 83 85 84 86 86 82 81

Store B claims that their prices are lower than average. What “average” could they use to make this

statement true (mean, mode or median)? Do you think their claim is misleading? Why? 3. You wrote 6 history tests so far and you will have three more.

Your marks were: 75, 72, 81, 62, 65, 75.

a) If you want a mode of at least 75, what must you do on the last 3 tests?

b) If you want a mean of at least 70, what must you do on the last 3 tests?

Use algebra to find missing data.

4. The difference between two numbers is 20. The mean is 50. What are the two numbers?

a) Call the smaller number x. The larger number is x + _____.

b) The mean of the two numbers is 50, so the sum of the two numbers is

50 × ____= _____

c) x + x + ____ = _____

d) Find x.

e) Remember that x was the smaller number. What is the larger number? 5. Wasim’s mark on the first two tests are 92 and 93. He wants to have at least a 90 average on the first

3 tests. Let x be the mark he needs on the third test.

a) x + 92 + 93 = 90 × ___ = _____ x = ____ 6. Bob wants an average of 80 an all 6 tests. His marks on the first 5 tests were 72, 86, 92, 63 and 77.

What must he get on the last test? 7. The mean weight of three dogs, Tippy, Pat and Baxter, is 25 kg. Tippy weighs 31 kg. Pat and Baxter

both weigh the same amount. What is Pat’s mass.

E

page 197

Teacher’s Guide

Grade 7

8. If you got 80, 93 and 91 on 3 history tests, what do you need on the next test to average 90?

9. If you got 80, 93 and 91 on 3 history tests, what do you need to average on the next two tests to

average 90 on the first five tests?

PDM7-22: Using the Mean, Mode and Median page 258

1. Sally's first eight math test scores are (out of 20):

13 14 18 16 14 13 14 18

a) Calculate her mean score, median , and mode.

b) Sally's father tells her that if she increases her average score to 1620 after the next two tests, he

will give her $200. By "average", do you think he means the mode, the mean or the median? Why?

c) How can she increase her average to 1620 .

2. Find the mean.

a) 2 4 4 6 9

b) 2 2 4 4 4 4 6 6 9 9

c) 2 2 2 4 4 4 4 4 4 6 6 6 9 9 9

d) 2 2 2 2 4 4 4 4 4 4 4 4 6 6 6 6 9 9 9 9 3. Find the mean.

a) 3 3 4 6 6 (___+___+___+___+___) ÷ 5 = _____

b) 3 3 3 3 4 4 6 6 6 6 (___+___+___+___+___) ×2 ÷ 10 = _____

c) 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 (___+___+___+___+___) × 3 ÷ 15 = _____

d) 2 5 4 1 (____+____+____+____) ÷____ = _____

e) 2 2 2 5 5 5 4 4 4 1 1 1 (____+____+____+____) ×___÷___ = _____ 4. Does repeating each data value the same number of times change the mean? Explain your answer. 5. Investigate what happens with the mode and the median. Do the answers change? Make up your

own data to check.

6. Sally surveyed 20 families on her street to

find the number of cars they have.

She displays her result in both a frequency

table and a circle graph.

E

0 car 2 cars

1 car

15

25

25

Frequency

4 8 8

# of cars

0 1 2

page 198

Teacher’s Guide

Grade 7

To find the mean, Sally uses the frequency table:

(0 + 0 + 0 + 0 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2) ÷ 20

Tina uses the circle graph and pretends there are only 5 families:

(0 + 1 + 1 + 2 + 2 ) ÷ 5

a) What answers do they get?

b) Whose method do you like better? Why?

c) If Sally accidentally divides 24 by 10 instead of by 20, she would get a mean of 2.4. How can

she tell immediately that this is wrong?

7. Sally has a pen pal in a third world country. Her pen pal gives her a circle graph of the income of

people in that country. 1

100 make $1 000 000 000/year, 2

100 make $30 000 per year, and 97

100 make

$50 /year. Show this in a circle graph.

Sally says that to find her friend’s country’s average income, she can pretend that only 100 people

live in that country and take the average of $1 000 000 000, $30 000, $30 000, $50, $50, $50, … ,

$50 (97 fifties)

a) Is Sally right? Explain.

b) Find the average income of everyone in her pen pal’s country? (Find mean, median and mode).

c) How much affect did the 97 fifties have on the country’s mean income? On the mode? On the

median?

d) Based only on the median annual income, is this a country that you would want to live in?

Based on the mode? Based on the mean?

e) Would you move to this country? Why?

f) If you want to move to a country with a “good” average income, would you use the median, the

mode or the mean to determine the average? Explain your answer.

A family drives at 110 km/h for 6 hours and then 60 km/h for 4 hours.

a) How far did the family drive in the first 6 hours? In the last 4 hours? Altogether?

b) Mean speed = distance travelled / time travelled = ________ km/h

c) Is the mean speed closer to 110 km/h or 60 km/h? Why does this make sense?

PDM7-23: Bias in a Sample or Census page 260

Investigation: Which gives results that better represent the whole population – a large sample or a small sample?

A

page 199

Teacher’s Guide

Grade 7

Group work: The marks for everyone in the class on a math test were: 72 73 87 64 71 38 91 60 74 83 48 85 66 55 73 42 97 59 74 77 For the entire population of the class, the mean is 64, 45, mode are 73, 74 and the median is 72.5. In this exercise, students will pick 3 marks at random three times and then 10 marks at random twice, and compare the mean, median and mode of their samples with those for the whole class. To pick numbers randomly, students can either use graphing calculators, or http://www.random.org/sform.html (this one will generate the numbers 1 to 20 in random order. Students can then pick the first 3 for a sample of 3 or they can pick the first 10 for a sample of 10) or students can put the 20 marks in a bowl and pick 3 or 10 without looking and then mix the bowl well after each sample is replaced. If students use the internet, it is important that the random number generator they use does not allow the same number to be chosen twice. (Surveys do not ask the same person twice when they randomly select people). Students will then complete the following chart:

Sample size Mean Mode Median

3 3 3 10 10 20 (whole class)

For the small samples of 3 marks, were the mean, mode and median close to the actual mean, mode and median? For the larger samples of 10 marks?

Sometimes, the wording of a question can affect the results of the survey, even when the sample is representative.

1. To ask whether school uniforms should be required, two questions are proposed.

A: Do you think that students should be allowed to express themselves by what they wear? B: Do you think that equality among students is important enough to require consistent student

uniforms?

a) Which question do you think was proposed by someone in favour of school uniforms? Which was proposed by someone against school uniforms?

b) Write down a question they could ask instead that does not already suggest an answer.

2. A town council is thinking of selling a city park and allowing a department store to build in its place. Two groups ask different questions:

A: Are you in favour of having a new store that will provide jobs for 50 people in our town? B: Are you in favour of keeping our parks quiet and peaceful?

E

page 200

Teacher’s Guide

Grade 7

a) Which question do you think was proposed by someone in favour of selling the park? Which was proposed by someone against selling the park?

b) Write down a question they could ask instead that does not already suggest an answer.

3. Student council is required to choose music and order food for events. Most people enjoy the music, but not the food. Look at the two surveys below:

a) What is the same about the two surveys? b) What is different about the two surveys? c) Which survey is more likely to suggest that student council is doing a good job? Why? d) How can order of the questions affect the results of a survey?

4. Sample size can also affect the outcome of a survey.

I want to know if a coin is fair. If I toss a coin once and it is heads, can I conclude that it is more likely to come up heads than tails? What if I toss it 3 times and get 2 heads? What if I toss it 300 times and get 280 heads?

A friend tells me that most people in Canada are against the death penalty. Can I make this conclusion if I ask one random person and he or she is against the death penalty? What if I ask 3 random people and 2 of them are against the death penalty? What if I ask 300 random people and 280 of them are against the death penalty?

When a random sample of the population is chosen, is it more likely to be representative of the whole population if it is a large sample or a small sample? Why?

5. For a more advanced exercise in exploring bias, see the worksheet Bias (Advanced) in the

Blackline Masters.

PDM7-24: Designing, Displaying and Analysing a Survey page 262

1. Ask your students to represent their survey results in 2 different ways and explain the advantages and short comings of each different representation.

E

Survey A: 1. Do you enjoy the music at the events?

2. Is student council doing a good job?

3. Do you enjoy the food at events?

Survey A: 1. Do you enjoy the food at the events?

2. Is student council doing a good job?

3. Do you enjoy the music at events?

page 201

Teacher’s Guide

Grade 7

PDM7-30: Compound Events page 271

Cereal boxes come with either a picture of a cat or a dog inside. You win a prize if you collect one of each picture. Two out of every three boxes have a cat and one out of every three has a dog. Draw a tree diagram to find the probability that you will not win a prize if you buy: a) 2 boxes b) 3 boxes Do you have a better chance of winning a prize with this game or with the one in your workbook? What if 3 out of every 4 boxes have a cat – would this make it easier or harder to win? Why?

PDM7-31:Expectation page 272

1. Sally asks her classmates when their birthdays are. There are 366 possible outcomes.

a) Is the outcome of Aug. 31 equally probable to the outcome Feb. 29th, more probable or less

probable? Why?

b) Feb. 29th is one outcome out of 366. Is the probability of being born on Feb. 29th equal to 1/366?

Why or why not?

c) Find the probability of being born on Feb. 29th. HINT: how many days are there in any 4 consecutive years? How many of those days are Feb. 29

th?

d) Assume that Toronto has 4 500 000 people. How many people in Toronto would you expect to

have their birthday as Feb. 29th? 2. Look at the following question.

In what year did women in Canada gain the right to vote? a) 1915 b) 1916 c) 1917 d) 1918 e) 1919

a) If you guess randomly, what is the probability of answering correctly?

b) On a test of 30 similar questions, how many questions would you expect to guess correctly?

Incorrectly? Assume you answer each question blindly.

c) You get a point for each correct answer and –1/4 for each incorrect answer. What do you expect

your final score to be?

E

E

page 202

Teacher’s Guide

Grade 7

PDM7-32: Describing Probability page 275

The following exercise can either be given verbally to the whole class or can be given as an individual exercise for each student.

Describe each event as certain, likely, even, unlikely or impossible.

a) You will live to be 30 years old. b) You will live to be 100 years old. c) You will become older. d) You will become younger. e) You will eat breakfast tomorrow. f) You will be let out early for recess this afternoon. g) There will be no school tomorrow. h) It will rain next Tuesday. i) It will snow today. j) You will have lunch today.

PDM7-34: Organized Lists and Tree Diagrams page 279

1. Before doing Question 5 on this worksheet, it would be beneficial to students to play the following

game.

Each student receives a game board as follows and a pair of dice: Eleven runners are entering a race. Each runner is given a number from 2-12. They move one place

forward every time their number is the total number rolled on a pair of dice. Which number wins the

race?

2 3 4 5 6 7 8 9 10 11 12

E

A

k) You will blink at least once in the next hour. l) You will fall asleep in the next ten minutes. m) There will be no school on Christmas day. n) The Leafs will win the Stanley Cup this year.

page 203

Teacher’s Guide

Grade 7

Students will see the experimental result that 7 occurs most often before they see it theoretically. Then they will do the mathematics to understand why it occurs. Most students should see 7 win the race. They will also see that the person most likely to win won’t always win.

2. After doing Question 5, your students can do the following activity to demonstrate their

understanding.

When playing snakes and ladders, which numbers would be easiest to land on? (Think of multiples of numbers that occur often). Which numbers would be hardest to land on? You want to design a snakes and ladders game on the following game board:

21 22 23 24 25

20 19 18 17 16

11 12 13 14 15

10 9 8 7 6

1 2 3 4 5

Work in pairs. Design a game that you think would be easy to win and then another game that would be difficult to win. Check your prediction by playing your games.

1. You have 50 coins with a total value of $1.00. If you lose one coin what is the chance it is a quarter?

PDM7-35: Applications of Probability page 280

1. Use probability to decode messages.

The Roman emperor Julius Caesar developed a way to encode messages known as the Caesar cipher. His method was to shift the alphabet and replace each letter by its shifted letter. A shift of 3 letters would be:

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 D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

A

E

page 204

Teacher’s Guide

Grade 7

The following news headline was recently reported by BBC:

US man survives chocolate ordeal Using the shift of 3 letters, the message would begin:

XV pdq vxuylyhv Finish encoding the message. Tally the occurrence of each letter in the original message and in the encoded text

Letter 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

Original

Encoded

Which letters occur most often in the original message? Which letters occur most often in the encoded message? Why?

In the English language, the letter occurring most often is almost always “e,” especially for large samples of the language.

Practice using the following website to encode and decode messages that you make up: http://rumkin.com/tools/cipher/caesar.php

If you encode a message with a shift of 3, you will need to decode the message using a shift of 26 – 3 = 23 to get back your original message. What shift would you need to use to decode a message that was encoded using a shift of 7? A shift of 16? A shift of 24?

To decode the following message, first determine which letter occurs most often by inputting the following text in the web-site: http://www.central.edu/homepages/LintonT/classes/spring01/cryptography/java/textanalyzer.html or search for “letter frequency” to find a different web site.

G 21-ekgx-urj AY sgt ktjkj av ot nuyvozgr glzkx yvktjotm zcu nuaxy zxgvvkj ot g bgz ul

inuiurgzk, vuroik ot Coyiutyot ygoj ut Lxojge. Znk sgt ygoj nk ngj iroshkj otzu znk zgtq hkluxk

hkiusotm zxgvvkj cgoyz-jkkv ot inuiurgzk, vuroik inokl Xgtje Hkxtkx zurj GV tkcy gmktie.

Nuckbkx, uznkx xkvuxzy yammkyz nk cgy yzoxxotm znk inuiurgzk cnkt nk lkrr ot. Xkyiak

cuxqkxy gtj yzgll gz znk Jkhkroy Iuxvuxgzout aykj iuiug-hazzkx zu znot uaz znk inuiurgzk gtj

varr nos lxkk. "Oz cgy vxkzze znoiq. Oz cgy boxzagrre roqk waoiqygtj," Igvzgot Hkxtkx ygoj.

"Oz'y znk loxyz zosk O'bk kbkx nkgxj ul gteznotm roqk znoy," nk gjjkj. Znk cuxqkx ygoj noy

gtqrky ckxk yuxk glzkx znk otiojktz, gtj nk cgy zgqkt zu g ruigr nuyvozgr cnkxk nk oy

xkiubkxotm. Znk giiojktz otburbkj jgxq inuiurgzk. Which letter do you think represents the letter “e”? Why? What shift do you think was used? What shift do you think will decode the message? Check your answer by using the http://rumkin.com/tools/cipher/caesar.php web site.

page 205

Teacher’s Guide

Grade 7

The above message decodes as:

"A 21-year-old US man ended up in hospital after spending two hours trapped in a vat of chocolate, police in Wisconsin said on Friday. The man said he had climbed into the tank before becoming trapped waist-deep in chocolate, police chief Randy Berner told AP news agency. However, other reports suggest he was stirring the chocolate when he fell in. Rescue workers and staff at the Debelis Corporation used cocoa-butter to thin out the chocolate and pull him free. "It was pretty thick. It was virtually like quicksand," Captain Berner said. "It's the first time I've ever heard of anything like this," he added. The worker said his ankles were sore after the incident, and he was taken to a local hospital where he is recovering. The accident involved dark chocolate".

Give your students a coded message in French. Tell them to use a sample of a piece of French writing to determine the letter that “should” occur most often. Why is it better to use a large sample than a small sample? How does using a large sample make your experimental probability closer to the actual probability of each letter occurring? Then use the internet to determine which letter actually does occur most often in the sample given. Decode the message. Use an internet translator to translate the message. An example of an internet translator is: http://www.google.com/language_tools

2. What is the probability of winning the Lotto 6/49?

In the Lotto 6/49, balls numbered 1, 2, 3, 4, …, 49 are randomly mixed in a machine. 6 balls are then chosen at random from the machine. Balls are not put back, so the same number cannot be chosen twice. You can pick your own numbers, so even if more tickets are sold than the number of different tickets, it’s possible that no-one wins, because many people might have bought the same ticket. The probability of winning doesn’t depend on how many tickets are sold, but the amount you win does. This is the opposite of a raffle.

a) If the Lotto 6/49 picks 6 different numbers from 1, 2, 3, …, 49, explain what the Lotto 2/7

would be. b) The only ticket in the Lotto 2/2 is: 1 2. The Lotto 2/3 has 3 tickets: 1 2 1 3 2 3. Use an

organized list to find the tickets for: The Lotto 2/4 The Lotto 2/5 The Lotto 2/6 The Lotto 2/7

HINT: It is a good idea to start by listing all the tickets with 1 and then list the tickets from the rest that have 2, etc.

c) Complete the following chart:

N Lotto # of tickets Probability of winning N x (N-1)

2 2/2 1 1 2

3 2/3 3 13

6

4 2/4 6 16

12

5 2/5

6

7

page 206

Teacher’s Guide

Grade 7

d) By comparing the last two columns of the chart in question 3, write down a rule for the probability of winning the Lotto 2/N in terms of N.

e) Find:

i) The probability of winning the Lotto 2/10. ii) The probability of winning the Lotto 2/49.

f) Mathematicians have shown that the chances of winning the Lottos 2/49, 3/49 and 4/49 are as follows:

Lotto Probability of Winning

2/49 1 x 2_ 49 x 48

3/49 1 x 2 x 3_ 49 x 48 x47

4/49 1 x 2 x 3 x 4_ 49 x 48 x 47 x 46

Extend the pattern to predict the chances of winning the Lotto 6/49. 3. According to www.lotterycanada.com, the odds of winning the Lotto 6/49 are 1 in 13 983 816. Verify this using your prediction.

There are less than 14 000 000 different tickets, so if you have $28 000 000, you have enough

money to buy all the tickets, guaranteeing that you will win. If the jackpot is $40 000 000, would you

buy all the tickets? Why or why not?

Find the probability of winning for each of the following Lottos:

� Lotto 1/7

� Lotto 2/7

� Lotto 3/7

� Lotto 4/7

� Lotto 5/7

� Lotto 6/7

Order your answers from greatest probability of winning to lowest probability of winning.

Use your intuition and your answers to questions 3 and 9 to decide which of the following lotteries

you think will have a greater chance, an equal chance or a lesser chance of winning than the

Lotto 6/49. (Do not attempt to calculate the actual probabilities.) In each case, explain your answer:

a) Lotto 7/49

b) Lotto 5/49

c) Lotto 6/48

d) Lotto 6/50

e) Lotto 43/49

HINT: 6 + 43 = 49

page 207

Teacher’s Guide

Grade 7

1. Scientists can use probability to estimate the number of fish in a lake. They catch 100 fish, tag them

and release them back into the lake. A week later, they catch 200 fish. 40 of them are tagged.

a) What is the probability that a fish chosen at random is tagged?

b) We know there are 100 tagged fish in the lake. What fraction of the total population do those

100 fish form?

c) Estimate the number of fish in the lake.

d) Why would scientists be interested in counting the number of fish in a lake?

2. The colour of a flower is determined by its genes. A red rose, for example, has two R genes, a white

rose has two r genes and a pink rose has one R gene and one r gene.

If two pink roses crossbreed, the “child rose” can be either red, white or pink. The possible results

are shown in the table below: Gene from parent 1: R r Gene from parent 2: R r R r Resulting gene: RR Rr Rr rr Resulting colour: Red Pink Pink White

a) When two pink roses crossbreed, what is the probability the resulting rose will be pink?

b) When a pink rose crossbreeds with a red rose, what is the probability the resulting rose will be

red? Pink? White?

c) When a pink rose crossbreeds with a white rose, what is the probability that the resulting rose

will be pink? Red? White?

d) What happens when a red rose crossbreeds with a white rose?

E

page 208

Teacher’s Guide

Grade 7

Geometry

G7-24: Alternate Angles page 285

1. Find the missing angles. The first question is done for you.

HINT: Fill in as many missing angles as you can.

a) b) c)

∠1 = 50º

∠2 = 130º

∠1 = ________

∠2 = ________

∠1 = ________

∠2 = ________

d) e) f)

∠1 = ________

∠2 = ________

∠3 = ________

∠1 = ________

∠2 = ________

∠3 = ________

∠1 = ________

∠2 = ________

∠3 = ________

g) h) i)

∠1 = ________

∠2 = ________

∠3 = ________

∠4 = ________

∠1 = ________

∠2 = ________

∠3 = ________

∠4 = ________

∠1 = ________

∠2 = ________

∠3 = ________

∠4 = ________

E

1

55º 65º

2 50º

2 1

1

75º

3

45º

2 119º

1

2 3

3 65º

2 1

4

57º

70º

1

2

53º

30º 95º

1

2

3

1

35º 55º

2 3 4 60º 3

2

1

39º 4

page 209

Teacher’s Guide

Grade 7

G7-25: Angles in a Triangle page 286

1.

Without measuring the triangles, match each triangle with its description:

________ isosceles, right-angled ________ obtuse scalene ________ equilateral

________ obtuse isosceles ________ scalene right-angled

2. Measure each side length for each triangle and answer the following questions:

a) b)

c) d)

3. Look at the length of the sides and the sizes of the angles opposite those sides in question 1. Write

down a rule they seem to follow:

4. Either sketch an example or state that it does not exist:

AB = ________mm AC = ________mm BC = ________mm

The longest side is opposite the _____º angle.

The shortest side is opposite the _____º angle.

AB = ________mm AC = ________mm BC = ________mm

The longest side is opposite the _____º angle.

The shortest sides are opposite the ___º angles.

E

AB = ________mm AC = ________mm BC = ________mm

The longest side is opposite the _____º angle.

The shortest side is opposite the _____º angle.

15º 155º

10º

A

B

C

30º

A

B C

60º AB = ________mm AC = ________mm BC = ________mm

60º

80º

A

B C 40º

30º 30º

120º

A

B C

The longest side is opposite the _____º angle.

The shortest side is opposite the _____º angle.

A B C D D E

page 210

Teacher’s Guide

Grade 7

a) an isosceles obtuse triangle b) an equilateral obtuse triangle

c) an equilateral right triangle d) a scalene right triangle

e) a scalene acute triangle f) an isosceles right triangle

g) an isosceles right triangle where the shortest side is unique.

HINT: Can the two longest sides both be opposite a right angle?

5. Without drawing the triangles, determine which of the following are not possible to be the side lengths

of a right triangle. Explain your answer.

a) 4, 4, 4 b) 3, 4, 5 c) 2, 3, 3 d) 5, 12, 12 e) 5, 12, 13 f) 7, 7, 8

1. Look at question 2. Sum all the angles in each triangle. What do you notice? Draw any triangle on a

blank sheet of paper. Cut it out. Then rip the corners and line up the angles. What do you get?

Repeat the exercise with different quadrilaterals – a parallelogram, a trapezoid, a square, an

arbitrary quadrilateral. What is the sum of the angles in a triangle? In a quadrilateral? Can all the

angles in a trapezoid be acute? Obtuse? What about a parallelogram? Any quadrilateral?

2. Three students were asked to find the sum of the angles in a hexagon without using a protractor. These are the drawings they came up with:

Cathy: Mike Laura

A

A C B

The angles will always add to 180°

B

C A

page 211

Teacher’s Guide

Grade 7

Cathy says the answer is 180°×4, Mike says the answer is 180°×5 - 180° and Laura says the answer

is 180°×6 - 360°. a) What numerical answers do they each get?

b) Explain Cathy's method.

c) Explain Mike's method.

d) Explain Laura's method.

e) Sketch a hexagon and use all three methods to find the sum of the angles. Do you get the same answer as Cathy, Mike and Laura?

f) Predict the sum of the angles in a polygon with N sides. Explain your answer. Draw several examples of polygons to show your answer.

G7-27: Introduction to Coordinate Systems page 288

1. Secret Squares

Player 1 draws a 4 x 4 grid as shown and picks a square. Player 2 tries to guess

the square by giving its coordinates.

Each time Player 2 guesses, Player 1 writes the distance between the guessed

square and the hidden square.

For instance, if Player 1 has chosen square B2 (�) and Player 2 guesses C4,

Player 1 writes 3 in the guessed square. (Distances on the grid are counted

horizontally and vertically, never diagonally.)

The game ends when Player 2 guesses the correct square.

2. TEACHER:

As a warm-up for the Secret Squares game above, have the class (as a whole) try to guess the locations of each

hidden square from the information given in the grids below. In 2 grids, not enough information is given (have

students mark all possible locations for the hidden square) and in three grids too much information is given (have

your students identify one piece of redundant information):

A

2

2

3

A B C D

4

3

3

2

1

�

A B C D

4

3

2

1

A B C D

4

3

2

1

2 1

A B C D

4

3

2

1

A B C D

4

3

2

1 2

2 4

A B C D

4

3

2

1

1

3

A B C D

4

3

2

1

1

4

A B C D

4

3

2

1

1

2

2 A B C D

4

3

2

1

2

3

3

2

page 212

Teacher’s Guide

Grade 7

G7-28: Plotting Points in Coordinate Systems page 289

1. Draw a coordinate system on grid paper. Draw the line joining each pair of points and find the midpoint of the line segment. a) (2,7) (10,7) b) (3,5) (7,5) c) (4,-2) (6,-2) d) (-2,4) (6,4) e) (-5,-4) (3,-4) (6,7)_ _____ _______ _______ ________ 2. Use the coordinate system you drew to answer the following questions.

a) How long is the line segment between (3,4) and (3,-4)? b) What is the midpoint of the line segment between (3,4) and (3,-4)? c) A horizontal line is 7 units long and starts at (-3,-3). What is the other endpoint? d) The midpoint of a line segment is (3,3). One of its endpoints is (3,-1). What is the other

endpoint? 3. Use your grid paper and a ruler to find the midpoint of the line segments joining each pair of points. a) (3,7) (7,7) b) (3,1) (7,7) c) (4,8) (4,10) d) (3,8) (6,10) e) (-2,6) (4,8) 4. Can you see a pattern for how to determine the midpoint of a line segment if you know both coordinates of the endpoints? Predict the midpoint of each line segment below and then check your answer using grid paper. a) (-2,5) (0,9) b) (-3,-1) (1,-7) c) (-2,6) (4,2) d) (3,-1) (6,3) e) (7,5) (-4,1)

G7-30: Translations

page 292

1. A shape has coordinates A (2,1) B(6,1) C(3,5) D(5,5). Under a translation vertex A moved to

position (7,8). Give the coordinates of the other vertices under the translation.

A

E

page 213

Teacher’s Guide

Grade 7

G7-31: Reflections page 295

Example: Reflect point P(1, 3) through the mirror line ℓ.

Step 1: Draw a dotted line through point P perpendicular to line ℓ.

Label the point where the dotted line meets the line X.

Step 2: Locate a point P′ on the dotted line such that ( P′ is the same

distance from the mirror line ℓ as P ): PX = P′X.

P′ is the image of P after reflecting through line ℓ.

1. Reflect the set of points P, Q and R through the mirror line ℓ. Label the image points P′, Q′ and R′.

a) b) c)

P ( , ) P′ ( , )

Q ( , ) Q′ ( , )

R ( , ) R′ ( , )

P ( , ) P′ ( , )

Q ( , ) Q′ ( , )

R ( , ) R′ ( , )

P ( , ) P′ ( , )

Q ( , ) Q′ ( , )

R ( , ) R′ ( , )

2. Reflect the figure by first reflecting the points through line ℓ:

a) b) c)

A′ ( , ) B′ ( , )

A′ ( , ) B′ ( , )

A′ ( , ) B′ ( , )

C′ ( , ) D′ ( , )

Q

P

Y

–1

–2

–3

–4

–4 –3 –2 –1 0 1 2 3 4

4

3

2

1

R

ℓ

X

Q

P

Y

X

–1

–2

–3

–4

–4 –3 –2 –1 0 1 2 3 4

4

3

2

1

R

ℓ

X

ℓ

Q

P

Y

–4 –3 –2 –1 0 1 2 3 4

4

3

2

1

R

–1

–2

–3

–4

–4 –3 –2 –1 0 1 2 3 4

4

3

2

1

–1 –2

–3

–4

P′(3, 1)

P(1, 3)

45º

Y ℓ

X

X

Y

X

A

ℓ

C

B

–1

–2

–3

–4

4

3

2

1

–4 –3 –2 –1 0 1 2 3 4

B

Y

X

A

ℓ

C

–4 –3 –2 –1 0 1 2 3 4 –1

–2

–3

–4

4

3

2

1

B

Y

X

A

ℓ

D C

4

3

2

1

–1

–2

–3

–4

–4 –3 –2 –1 0 1 2 3 4

A

page 214

Teacher’s Guide

Grade 7

3. What do you notice about the x and y coordinates of each point and its image in questions 2a), b) and c)?

1. Draw a triangle with vertices A (1,1), B (4,2), and C (3,3). Draw a vertical mirror line through the

points (5,0) and (5,5). Reflect the triangle in the mirror line.

2. Students could try to copy and reflect a shape in a slant line: for example.

or

3. If you want to reflect a point in a line so that its x-coordinate doesn’t change, which line should you

reflect in? 4. If you want to reflect a point in the x-axis so that the y-coordinate doesn’t change, what does the y-

coordinate have to be?

G7-32: Rotations page 297

1. Draw a figure on grid paper. Draw a dot on one of its corners. Show what the figure would look like

if you rotated it a quarter turn clockwise around the dot.

2. Step 1 - Draw a trapezoid on grid paper and highlight one of its sides as shown:

Step 2 - Use a protractor to rotate the line 120° clockwise:

Step 3 - Draw the trapezoid in the new position:

HINT: You will have to measure the sides and angles of the trapezoid to reconstruct it.

E

E

page 215

Teacher’s Guide

Grade 7

3. Rotate an equilateral triangle 60° clockwise around one of its vertices. What do you notice?

4. Rotate each shape 180°

around centre P by

showing the final position

of the figure.

Use the line to help you:

5. Rotate each shape 180°

around centre P:

HINT: First highlight an edge

of the figure and rotate the

edge (as in Question 1).

6. Rotate each shape 90° around 7. Rotate each shape 90° around the point in the

point P in the direction shown: direction shown: HINT: First highlight a line on the

figure and rotate the line 90°.

8. Write 90° beside the figure (1 or 2) that was made by rotating the original figure 90° counter

clockwise. Then write 180° beside the figure that was made by rotating the original figure 180°.

a) b) c)

9. Write 180° beside the figure (1 or 2) that was made by rotating the original figure 180°. Then write ‘R’

on the figure that was made by a reflection. Mark the centre of the rotation and draw a mirror line for

the reflection:

a) b) c)

2

1

2

1 2

1

2

1

1

2

1

2

P P

P P

P P P P

P P P

P

page 216

Teacher’s Guide

Grade 7

10. a) Translate Figure A 4 units left and 2 units down. Label the image B.

b) Turn Figure B 90° clockwise around point (5,2).

Label the image C.

c) Write the ordered pairs of the vertices of C.

( ______ , ______ ) ( ______ , ______ )

( ______ , ______ )

11. a) Reflect the Figure A in line L. Label the image B.

b) Rotate Figure B 180° around point (7,2). Label

the image C.

c) Write the ordered pairs of the vertices of C.

( ______ , ______ ) ( ______ , ______ )

( ______ , ______ ) ( ______ , ______ )

12. a) Name 5 pairs of congruent shapes in the picture.

b) Draw a mirror line on the picture that would allow you to

reflect Shape E onto Shape G.

c) For which shapes can you check congruency using

only a slide?

d) For which shapes can you check congruency using:

(i) a rotation and slide? (ii) a reflection and a slide?

13. A shape has coordinates A(1,2) B(1,4) C(4,2). Under a rotation point C moves to position (3,7).

Which vertex in the shape was used as the center of rotation and what was the rotation?

14. Draw a coordinate grid on grid paper. Draw a triangle with vertices A (1,5), B(4,5), C(4,7). Draw a

vertical mirror line through the points (5,0) and (5,7). Reflect the triangle in the mirror line. Then

rotate the triangle 14 turn counter clockwise around the image of vertex C. Write the coordinates of

the vertices of the new triangle.

0 1 2 3 4 5 6 7 8 9 10

8

7

6

5

4

3

2

1

A

0 1 2 3 4 5 6 7 8 9 10

8

7

6

5

4

3

2

1

A

L

A B

C D

E F G

H J I

page 217

Teacher’s Guide

Grade 7

1. To help students visualize the effect of a rotation, have them make a small flag 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 shapes similar to the ones on the worksheet and trace the images of the

shapes after a rotation.

2. For an activity on reflecting, rotating and translating a triangle, see the extra worksheets section of

this manual.

G7-33: Transformations page 300

1. Rotate each figure around the point P:

a)

90° clockwise

b)

90° counter

clockwise

c)

90° clockwise

d)

90° clockwise

e)

180° counter

clockwise

f)

90° clockwise

g)

180° clockwise

h)

90° clockwise

2. Reflect each figure in the mirror line: 3. Slide each figure 1 unit right:

a) b) a) b)

E

A

M

M

P

P

P P

P

P P

page 218

Teacher’s Guide

Grade 7

4. Extend each pattern. Then describe the two transformations used to create the pattern. Draw in any

mirror lines or points of rotation:

a)

b)

c)

5. Trace and cut out the shape below. Make a pattern by...

a) Sliding the shape repeatedly one unit right:

b) Reflecting the shape repeatedly in the mirror lines:

c) Rotating the shape repeatedly 180° around the dots:

6. Each of the patterns below was made by repeating a transformation or a combination of

transformations. On a separate piece of paper, use the words “slide”, “rotation” or “reflection” to

describe how the shape moves from...

Position 1 to 2 Position 2 to 3 Position 3 to 4 and Position 4 to 5.

a) b)

2 3 4 5

1

1

2 3

4 5

page 219

Teacher’s Guide

Grade 7

c)

d)

e) One of the patterns above can be described in two different ways. Which pattern is it? And

which two single transformations could each produce the pattern?

7. Draw a shape on grid paper and make your own pattern by a combination of slides, rotations and

reflections. Explain which transformations you used in your pattern.

8. Draw a coordinate grid on grid paper. Draw a trapezoid with vertices A (1,5), B(4,5), C(3,6), D(2,6).

Rotate the trapezoid 90° counter clockwise around point D. Then slide the image 3 units right and 1

down. Write the coordinates of the vertices of the new trapezoid.

G7-37: Concepts in Similarity page 305

1. Students should understand that corresponding sides in similar figures are all in the same

proportion. For instance, if rectangles A and B are similar and the width of B is 5 times the width of

A, then the length of B must also be 5 times the length of A.

Your students might notice that if the side lengths of a figure all increase by a factor of n, then the

area increases by a factor of n × n. For instance, if you make each of the sides of a rectangle 3

times longer, the area will be 9 times greater.

One way to prove that two figures are not similar is to show that a pair of corresponding sides in the

two figures is not in the same ratio as another corresponding pair of sides. For instance, the figures

below are not similar because the side marked with an X in B is 3 times longer than the

corresponding side marked with an X in A, whereas the side marked with a circle in B is only 2 times

longer than the corresponding side marked with a circle in A (it should be 3 times longer).

M

2 3 4 5 1

1 2 3 4 5

3 cm 3 cm 2 cm 2 cm

2 cm

8 cm

× 9 cm 9 cm

6 cm 6 cm

4 cm

24 cm

×

page 220

Teacher’s Guide

Grade 7

1. A magnifying glass magnifies objects by a factor of 10.

a) How large would a 1 cm line segment be under the magnifying glass?

b) How large would a 5° angle be under the magnifying glass?

c) Draw a triangle with side lengths 8 mm, 3 mm and 9 mm. Draw what it would look like under the

magnifying glass. Are the two triangles similar?

G7-42: Prisms and Pyramids page 312

1. In each face of a hexagonal prism, how many pairs of sides are parallel? (How many sides intersect

at a vertex of any prism? Explain how you know.)

2. Sketch a skeleton of a prism or pyramid. For example:

Ask students to answer true or false to the following questions:

• My bases are all triangles.

• I have more vertices than edges.

• All of my faces are congruent

• Etc.

3. How many faces, edges, and vertices would a pyramid with a ten-sided base have?

4. Give a rule for calculating the number of edges in a pyramid that has a base with n sides. (Use n in

your answer).

Solution: A pyramid with n edges in the base also has n vertices in the base. Attached to each vertex

in the base there is one non-base edge. Hence there are n non-base edges and n base edges

altogether. Therefore there are 2 × n edges in a pyramid with n base edges.

E

E

page 221

Teacher’s Guide

Grade 7

5. Repeat question 2 for a prism that has a base with sides.

Answer: There are 3 × n edges in a prism with n base edges.

6. Select a prism and say how many pairs of parallel edges each face has.

1. Give students a set of 3-D shapes. Ask students to place each shape—base downward—on a piece

of paper and trace the base. (That way you can verify that each student knows how to find the base.)

Students should write the name of the figure beside the base and indicate whether the figure has one

or two bases (or whether the faces of the figure are congruent, as in a cube). If students have trouble

spelling the names of the figures you should write them on the board.

2. After students have constructed the pyramids and prisms from the Backline Masters nets, ask them

to sketch the nets from memory. You might also ask them to sketch what they think the net for a

hexagonal pyramid would look like.

3. Predict whether the net shown will make a pyramid. Copy the shape onto grid paper.

Cut the shape out and try to construct a pyramid. Was your prediction correct?

4. On grid paper, draw as many nets for a cube as you can. Then cut your nets out and see if they

make cubes.

5. Sketch the net for a prism by

rolling a 3-D prism on paper and

tracing its faces. For reference, use

the steps shown (at right)

for a pentagonal prism:

6. Draw the net for a pyramid using a method similar to the one in Question 10.

7. Students should try to sketch their own nets for rectangular prisms and cubes on grid paper (as in

questions 3 and 4). They should cut out their nets and tape them together to see if they make prisms

or cubes.

G7-43: Faces, Edges and Vertices page 313

1. Give your students the 3-D shapes shown on the worksheet or have them construct the shapes from

the nets in the teacher’s manual. Students should fill in the chart after examining the shapes.

A

A

Step 1: Trace one of the non-base faces.

Step 2: Roll the shape onto each

of its bases and trace the bases.

Step 3: Roll the shape onto each of its remaining rectangular

faces and trace each face.

page 222

Teacher’s Guide

Grade 7

2. Show your students an example of a cone and a cylinder. Explain that a cone has 1 curved surface

and 1 flat surface, while a cylinder has 2 flat surfaces and 1 curved surface. Ask students to find as

many examples of pyramids, prisms, cones, and cylinders in the classroom as they can.

3. After students have constructed the pyramids and prisms from the Blackline Master nets, ask them to

sketch the nets from memory. You might also ask them to sketch what they think the net for a

hexagonal pyramid would look like.

1. a) Pick 3 shapes and trace each of their faces (e.g. if you picked a square pyramid, trace all 5

of the faces on the pyramid – even if some of them are congruent). Compare the number of

faces in your tracings with those in your chart from Question 1: do you have the right number of

faces? Be sure to organize your work neatly so you can tell which faces go with which shapes.

b) Underneath the faces for each shape, answer the following questions:

(i) How many different-shaped faces does this 3D shape have? What are they?

(ii) Circle the face (or faces) that form the base of the 3D shape. Copy and complete this

sentence: “The base of a ___________________ is a _______________.”

2. Pick two 3-D shapes and say how they are similar and how they are different.

3. How many faces are on…

a) the outside of the figure? (including the hidden faces)

b) the interior of the figure?

4. On a cube draw a line from vertex A to B, then a line from B to C. What is the measure of ∠ABC.

HINT: Imagine drawing a line from vertex A to C on the hidden face. What kind of triangle is ABC?

Answer: ABC is an equilateral triangle. So ABC = 60°

5. Each edge of the cube was made with 5 cm of wire.

How much wire was needed to make the cube?

(Don’t forget the hidden edges.)

E

C

A

B

page 223

Extra Worksheets and Blackline Masters

Slides, Rotations and Reflections

page 224

Teacher’s Guide

Grade 7

Trace and cut out the triangle. Place the triangle in Position 1 and move it to

Position 2 by one of the transformations:

� Slide (1 unit right or left)

� ¼ Turn (clockwise / counter-clockwise around P)

� Reflection (in line ‘M’)

1. Describe the transformation used:

a) b) c)

____________________

____________________

____________________

d) e) f)

____________________ ____________________ ____________________

2. Using either a slide, a reflection or a turn, move the triangle from

Position 1 to a Position 2 of your choice. Add the second triangle to

the diagram and identify the transformation you used below:

________________________________________________________

3. Put the triangle (from above) in Position 1 in the grid. In your notebook describe how you can move

the figure from Position 1 to Position 2 using one transformation.

a) b)

___________________________________ ___________________________________

M

P

1 2

M

P

1

2

M

P

1 2

M

P

1

2

M

P

2 1

M

P

1 2

M

P

1

Line 2

P

2

1

Line 1

Line 2

P

2

Line 1

1

Slides, Rotations and Reflections (continued)

page 225

Teacher’s Guide

Grade 7

c) d)

___________________________________ ___________________________________

4. Describe how the figure moved from Position 1 to Position 2 by using two transformations. (Some

questions have more than one answer – try to find them.)

a) b)

___________________________________

___________________________________

___________________________________

___________________________________

c) d)

___________________________________

___________________________________

___________________________________

___________________________________

2

Line 2

P

1

Line 1

Line 2

P

1

Line 1

2

Line 2

P 1

2

Line 1

Line 1

Line 2

P

2

1

Line 2

P

1

Line 1

2

Line 2

P

1

2

Line 1

Preparation for Rotations

page 226

Teacher’s Guide

Grade 7

TEACHER:

Review the meaning of the terms “clockwise” and “counter clockwise” for this lesson.

1. Name the fractions shown in the pictures:

a)

b) c) d)

e)

f) g) h)

2. The picture shows how far the hand of the clock has turned. Shade the part of the circle the

hand has moved across. Then, in the box, write what fraction of a turn the hand has turned:

HINT: What fraction of the circle did you shade?

a) b) c) d)

turn clockwise

turn clockwise

turn clockwise

turn clockwise

e) f) g) h)

turn clockwise

turn clockwise

turn counter

clockwise

turn clockwise

i) j) k) l)

start

finish

start

finish

start

finish

start finish

start finish

start

finish

start finish

start finish

start finish

start

finish

finish

start

start

finish

clockwise counter clockwise

Multiplying – 2-Digit by 2-Digit

page 227

Teacher’s Guide

Grade 7

Grace multiplies 26 × 28 by splitting the product into a sum

of two smaller products:

26 × 28 = (6 × 28) + (20 × 28)

= 168 + 560

= 728

She keeps track of the steps of the multiplication using a chart:

1. Practice the first step of the multiplication:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

2. Practice the second step of the multiplication:

a)

b)

c)

d)

e)

2 4

1

×

3 6

2

×

3

×

6 2

4

×

3

×

1 6

3

×

3 7

1

×

2 5

4

×

3 6

4

×

2 4

5

×

3 4

2

×

3 3 3 6

4

5

3 3 6

6

8

3

28

6

20 26

6 × 28

20 × 28

Step 1

She multiplies 6 × 28:

2 8

2 6 ×

2 8

2 6 ×

8

4

2 8

2 6 ×

6 8 1

4

Step 2

Grace multiplies 20 × 28.

(Notice that she starts by writing

a 0 in the ones place because she

is multiplying by 20.)

2 8

2 6 ×

6 8 1

4 1

6 0

2 8

2 6 ×

6 8 1

4

0

2 8

2 6 ×

6 8 1

4 1

6 0

3 4

4 3

×

0

2

1

1

8

0

1

5

2 4

×

2

4 6 9

6 2

×

3

8

1

1

5 6

3 6

×

3

6

3

3

6 7

2 5

×

3

5

3

3

Multiplying – 2-Digit by 2-Digit (continued)

page 228

Teacher’s Guide

Grade 7

3. Practice the first two steps of the multiplication:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

4. Complete the multiplication by adding the numbers in the last two rows of the chart:

a)

b)

c)

d)

e)

5. Multiply:

a) b)

c)

d)

e)

6. Multiply the following in your notebook:

a) 35 × 23 b) 64 × 81 c) 25 × 43 d) 42 × 87 e) 13 × 94 f) 28 × 37

Step 3

Grace completes the multiplication by taking the products of 6 × 28 and 20 × 28 and adding them together,

as seen below in Question 4 a).

3

2

5

6

×

1

3

3

7

×

3

5

2

4

×

5 2 ×

7 4

3 4 ×

5 4

3 2 ×

8 7

2-Digit Division

page 229

Teacher’s Guide

Grade 7

1. To divide a 3-digit dividend by a 2-digit divisor, start by estimating

how many times the divisor goes into the divided as shown below:

Complete Step 1 for EVERY question before moving onto Step 2.

Step 1: Round the DIVISOR to the nearest ten and enter that number

in the oval provided.

Step 2: Count by the leading digit of the rounded divisor to see how many

times it goes into the DIVIDEND. Write your answer in the square.

a) 21 195

b) 19 142

c) 29 243

d) 42 353

e) 48 265

f) 41 256

g) 49 378

h) 32 268

i) 62 274

j) 29 196

k) 28 195

2. Next, multiply the divisor by the quotient:

Step 3: Multiply the DIVISOR by the quotient.

Step 4: Write the product underneath the DIVIDEND.

a) 41 256

b) 28 195

c) 19 142

d) 21 195

e) 62 274

f) 49 378

g) 29 196

h) 29 243

i) 42 353

j) 32 268

k) 48 265

) e.g. 18 122

108

6 quotient

divisor dividend

Step 1: Round 18 → 20

Step 2: Find out how many times 20

goes into 122, by skip counting or

checking how many times 2 goes

into 12 (= 6).

)

20 6

e.g. 18 122

)

)

)

)

)

)

)

)

)

)

)

) ) 6 6

7

) ) ) 9 4

) ) ) 7 6 8

) ) ) 8 8 5

Step 3: 18 (not 20) × 6 108

Step 4: Write 108 under the

dividend (122).

2-Digit Division (continued)

page 230

Teacher’s Guide

Grade 7

3. Complete Step 5 for EVERY question before moving onto Step 6:

Step 5: Subtract.

Step 6: Write the remainder beside the QUOTIENT.

8 a) 42 353 – 336

5 b) 48 265 – 240

7 c) 49 378 – 343

8 d) 32 268 – 256

4 e) 62 274 – 248

6 f) 29 196 – 174

6 g) 28 195 – 168

7 h) 19 142 – 133

6 i) 41 256 – 246

8 j) 29 243 – 232

9 k) 21 195 – 189

Answer the following questions in your notebook:

4. Step 1: Round the DIVISOR to the nearest ten and enter that number in the square provided.

Step 2: Count by the leading digit of the rounded divisor to see how many times it goes into the DIVIDEND.

Step 3: Multiply the DIVISOR by the quotient.

Step 4: Write the product underneath the DIVIDEND.

Step 5: Subtract.

Step 6: Write the remainder beside the QUOTIENT.

a) 21 156

b) 38 249

c) 49 358

d) 47 326

e) 94 419

f) 61 559

g) 28 192 h) 28 219 i) 92 293

) ) )

) ) )

) ) )

Step 5

) e.g. 18 1 2 2

– 108

14

6 R 14 1 1

Step 6

) ) )

) )

) ) )

) ) )

2-Digit Division – Correcting Your Estimate

page 231

Teacher’s Guide

Grade 7

1. For each question, say whether the

estimate was too high or too low:

6 a) 17 135 – 102

6 b) 23 129 – 138

6 c) 17 121 – 102

4 d) 26 149 – 104

9 e) 44 362 – 396

6 f) 24 126 – 144

2. In the space provided, correct the questions

by calculating with the new estimate:

6 a) 24 126 24 126 – 144 negative number!

TOO HIGH!

4 b) 26 149 26 149 – 104 45 45 > 26

TOO LOW!

6 c) 17 135 17 135 – 102 33 33 > 17

TOO LOW!

8 d) 34 263 34 263 – 272 negative number!

TOO HIGH!

6 e) 17 121 17 121 – 102 19 19 > 17

TOO LOW!

6 f) 23 129 23 129 – 138 negative number!

TOO HIGH!

9 g) 44 362 44 362 – 396 negative number!

TOO HIGH!

3. Answer the following questions in your notebook: Remember to check whether your estimate is “too high,” “too low” or “just right.”

a) 23 196

b) 24 189

c) 63 539

d) 48 6452

e) 86 4677 f) 76 8460 g) 62 2486 h) 36 4175

7 6 R 18 e.g. 23 156 23 156 – 161 – 138

negative number! 18

ESTIMATE TOO HIGH!

) )

7 is too high so use 6

) )

) ) ) ) ) )

) ) ) ) ) )

)

) ) ) )

) ) ) )

e.g. 7 6 23 156 16 123 – 161 – 96

negative number! 27 but 27 > 16

ESTIMATE TOO HIGH! ESTIMATE TOO LOW!

) ) )

) )

)

) )

Volume and Capacity

page 232

Teacher’s Guide

Grade 7

Recall that the capacity of a container is how much it can hold. For instance, the capacity of a

large bottle of water is 1 L.

1 mL is equivalent to 1 cm3 and 1 L = 1000 cm3

1. Audrey places a layer of centicubes in the bottom of a small glass box:

a) How many centicubes are in the box now? _______________

b) What is the volume occupied by one layer of centicubes?

______________________

c) In the one layer, there are 3 rows of 4 centicubes.

Write a multiplication statement for

the volume of one layer of centicubes:

______________________

d) Write a multiplication statement for the

volume that would be occupied if Audrey

place two layers of centicubes in the box:

______________________

e) Write a multiplication statement for the volume of three layers of centicubes: ________

f) What is the volume of the glass box? _______________________________________

g) What is the capacity of the box? ___________________________________________

Answer the remaining questions in your notebook.

2. a) Write a multiplication statement for the number of centicubes in each picture:

i) ii) iii)

b) Each picture in Part a) shows the number of centicubes needed to cover the base of a

rectangular prism that is 5 cm high.

Write a multiplication statement for the volume of each prism.

c) If you know the length, width and height of a rectangular prism, how do you calculate its

volume?

3. Write one possible set of lengths, widths and heights for a rectangular prism with the volume:

a) 12 cm3 b) 8 cm3 c) 18 m3 d) 24 m3

3 cm

3 cm 4 cm

Multiplying Whole Numbers by 0.1, 0.01 & 0.001

page 233

Teacher’s Guide

Grade 7

Recall that multiplying a whole number by a decimal has the same effect as shifting the decimal. For example: .01 × 5 = .05 OR .01 × 5 = .01 × 5 = .05

1. Answer the following questions in your notebook:

a) 5 × 0.1 = b) 62 × 0.1 = c) 85 × 0.1 = d) 16 × 0.1 =

e) 246 × 0.1 = f) 645 × 0.1 = g) 754 × 0.1 = h) 951 × 0.1 =

i) 1154 × 0.1 = j) 187 × 0.1 = k) 3954 × 0.1 = l) 12 784 × 0.1 =

2. Answer the following questions in your notebook:

a) 4 × 0.01 = b) 45 × 0.01 = c) 26 × 0.01 = d) 78 × 0.01 =

e) 264 × 0.01 = f) 856 × 0.01 = g) 776 × 0.01 = h) 422 × 0.01 =

i) 1956 × 0.01 = j) 134 × 0.01 = k) 7584 × 0.01 = l) 12 444 × 0.01 =

3. Answer the following questions in your notebook:

a) 3 × 0.001 = b) 61 × 0.001 = c) 76 × 0.001 = d) 34 × 0.001 =

e) 128 × 0.001 = f) 657 × 0.001 = g) 237 × 0.001 = h) 567 × 0.001 =

i) 5647 × 0.001 = j) 654 × 0.001 = k) 2348 × 0.001 = l) 36 559 × 0.001 =

4. Answer the following questions in your notebook:

a) 8 × 0.01 = b) 65 × 0.001 = c) 27 × 0.1 = d) 82 × 0.01 =

e) 645 × 0.1 = f) 872 × 0.01 = g) 364 × 0.001 = h) 229 × 0.1 =

i) 6488 × 0.01 = j) 1599 × 0.001 = k) 7481 × 0.001 = l) 34 122 × 0.01 =

m) 2178 × 0.01 = n) 64 788 × 0.1 = o) 26 944 × 0.01 = p) 98 756 × 0.1 =

6 × 0.1 = .6 or 0.6 (decimal point moves ONE to the left)

6 × 0.01 = .06 or 0.06 (decimal point moves TWO to the left)

6 × 0.001 = .006 or 0.006 (decimal point moves THREE to the left)

Shift the decimal TWO places

Probability: Problems and Puzzles

page 234

Teacher’s Guide

Grade 7

1. If you rolled the tetrahedral die ( ) 100 times, how many times would you expect to roll the

number 3? Hint: Use ratios.

a) b) c)

_______ times _______ times _______ times 2. If you used the spinner 50 times, how many times would you expect to spin yellow?

a) b) c)

_______ times _______ times _______ times

3. If you rolled the cube ( ) 60 times, how many times would you expect to roll the number 2?

a) b) c)

_______ times _______ times _______ times

4. Reduce the following fractions to lowest terms:

a) 25100

= b) 70100

= c) 4050 = d)

75100

= e) 2050 = f)

4060 =

5. Colour the spinner red and green (or write R & G) in a way that is most likely to give the result. HINT: You will need to make a fraction from the information you are given and reduce the result.

a) You spun red 30 out of 50 times:

b) You spun green 75 out of 100 times:

c) You spun red 20 out of 60 times:

6. a) The probability of spinning blue on a spinner is 13 . If you used the spinner 100 times about how

many times would you expect to spin blue? HINT: Use long division and ignore the remainder.

b) The probability of spinning yellow on a spinner is 14 . If you used the spinner 70 times about how

many times would you expect to spin yellow?

2 1 3

2

3 3 3

1

2 3 3

1

Y ●

Y B

R R Y

● Y R

B Y

● Y R

R B B

1 1 3 3

2

1

2 1 2 2

2

2

2 1 2 3

3

3

● ● •

Bias (Advanced)

page 235

Teacher’s Guide

Grade 7

Sometimes how the survey is done can affect who is included in a sample and so can

affect the bias in a sample.

1. A school opinion poll randomly selects parents of enrolled students, calls them and asks them their

opinion on the statement, “School uniforms should be mandatory.”

The parents are asked to state whether they strongly agree, agree, neither agree nor disagree,

disagree or strongly disagree. They display their results in a circle graph:

Another survey sends newsletters home to parents and asks them to call back stating whether they

agree or disagree with the same statement. They display their results in a circle graph:

a) Look at the circle graph for the phone survey (Graph 1).

Measure the angle for the part of the graph that: i) strongly agree ii) have a strong opinion

What fraction of people with a strong opinion strongly agreed with the statement?

Look at the circle graph for the call-us-back survey (Graph 2). Measure the angle for the pat of

the graph that represents the people who agree.

What fraction of people calling back agreed with the statement?

b) Do you think the call-us-back survey reflects most people’s opinion, or just those with a really

strong opinion? Why?

c) Rita says that the call-us-back survey is better because people who just agree or disagree don’t

really care that much about the issue anyway and so shouldn’t be counted. Katie says that the

phone survey is better because it takes everyone’s opinion into account. What do you think?

Disagree

Agree

Graph 2

Agree

Disagree

Strongly Disagree

Neutral

Strongly Agree Graph 1

Always, Sometimes, or Never True (Numbers)

page 236

Teacher’s Guide

Grade 7

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 ___________

Is the statement…. 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.

A If you multiply a 3 digit number by a one digit number, the answer will be a three digit number.

B If you subtract a three digit number from 999 you will not have to regroup.

C The product of two numbers is greater than the sum.

D If you divide a number by itself the answer will be 1.

E The product of 0 and a number is 0.

F Mixed fractions are larger than improper fractions.

G The product of 2 even numbers is an even number.

H The product of 2 odd numbers is an odd number.

I A number that ends with an even number is divisible by 4.

J When you round to the nearest thousands place, only the thousands digit changes.

K When you divide, the remainder is less than the number you are dividing.

L The sum of the digits of a multiple of 3 is divisible by 3.

M The multiples of 5 are divisible by 2.

N Improper fractions are greater than 1.

O If you have two fractions the one with the smaller denominator is the larger fraction.

Mini Sudoku

page 237

Teacher’s Guide

Grade 7

In the Sudoku pattern blocks, 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). Each Sudoku puzzle only has one solution.

Example:

Here’s how you can find the numbers in the shaded second column:

The 2 and 4 are given so we have to decide where to place 1 and 3.

There is already a 3 in the third row of the puzzle so we must place a 3 in the first row of the shaded

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)

d)

2.

a)

b)

c)

d)

1

2

4

3

2

4

3

1

1

3

2 4 1 3 4

2

3 2 1 4

1

3

4 2 4

3 4 1 4

4

3

2

2 1

3 1

4 4

3

1

4

4 1

3 2

3 2

3

4 2

4 2

4 4

1

2

3

1

1

4

1

2

4

3

2

4 3 4

1

4 3

2

2

Mini Sudoku (continued)

page 238

Teacher’s Guide

Grade 7

Try these Sudoku Challenges with numbers from 1 to 6. The same rules and strategies apply!

BONUS:

3.

4.

5.

6.

4

6

5

3

5

2

1

2 3

4

6

5

1

2

1

3

5

1

2

3

6

4

2

6

4

5 1 3

4

3

6

5

2

5

3

6

5

1

3

1 2

1

4

4

6

2

1

3 4 6 2 5

4

2

5

4

2 3

1

5

6

1

5

3

6

4

1

6

3 2

Sudoku – Warm Up

page 239

Teacher’s Guide

Grade 7

1. Each row, column or box should contain the numbers 1, 2, 3, and 4. Find the missing number in

each set:

a)

b) c) d) e) f)

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)

1 3 4

1 3

4

3

1

2

4 1 2 2

4

3 4 2

1

2 3 4 4

3

1

3

1

2 1 2 3

3

4 2

3

2

4

3

4

2

2 4 3 3

4 2

4

1

3

2 1

3

1

4 3

1 2 4

1

3

2

1

2 3 1

2

2

4

3

4

4

2

1

Sudoku – Warm Up (continued)

page 240

Teacher’s Guide

Grade 7

d)

e) f)

BONUS:

Can you find 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!

3

4 2

2 4

1

4 2

1

1

2

2

4 1 2

4

1 4

3

2

4

1

4

3

2

3

2 4

1

1 3

2 2

1

4

3

1

4

1

4 3

3 4

1

2 4

1

4

1

3

1

4

2

2

3

3

2

2

1 3 4

2 3

1

4

4

2

2

2

3

4

1

4

3

1 4

2

4

3

2

1

1

A New Strategy – Looking at the 1s, 2s, etc.

page 241

Teacher’s Guide

Grade 7

The following 6 × 6 grid shows some of the numbers from a Sudoku puzzle. The following strategy is

illustrated with one, but can be used with any number:

Look at the shaded square. It looks like it could contain either 1, 2, 5 or 6. But none of the other squares

in the grid can contain a 1 – can you see why not? Since there must be a 1 in the 2 × 3 box, the shaded

square must contain a 1. Use this strategy in the following exercises.

1. The following grids show some of the numbers from 6 × 6 Sudoku puzzles. Fill in all the 1s:

a) b) c)

2. Write in any more numbers you can in the above puzzles. There is not enough information to finish

them. Remember to only put in numbers that must be in each square – do not guess.

3. Fill in as many numbers as you can using this new strategy. Don’t forget to use the old strategy too!

One of these can be done completely. Which one?

a) b) c)

4. Copy Question 3 c) into your notebook. Do this question without using this new strategy. Which

way do you like better? NOTE: Although the strategy was illustrated only with 1s, your students will need to use the strategy will all the

numbers in order to do these puzzles.

1

4

31

1

41

4

4 1

1 4

4

4

4

1

6

2

5

1

1

4

1

2 3

56

5

2

4

5

5

5

1

1

2

3

3 4

5

62

3

3

5

1

4

4

1

1 2

6

Sudoku – The Real Thing

page 242

Teacher’s Guide

Grade 7

Try these Sudoku puzzles in the original format 9 × 9.

You must fill 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! ** symmetry.

5

4

1

7

9 4

9

8 9

6

4 1

4

6

5 4

3 7

8

3

9

6

8

4

1

8

1

3 1

2

2

3 5

7

2

7

2

6

7

6

3

2

8

9

8

7 5

6

4

4 1

3

7

5 8

6

3

4

1

2

3 9

3 1

9

6

8

7

4

1

1

8

2

5

6

7

5

2-D Shape Sorting Game

page 243

Teacher’s Guide

Grade 7

2-D Shape Sorting Game (continued)

page 244

Teacher’s Guide

Grade 7

2-D Shape Sorting Game (continued)

page 245

Teacher’s Guide

Grade 7

2-D Shape Sorting Game (continued)

page 246

Teacher’s Guide

Grade 7

Three or more vertices

Four or more

vertices

No lines of symmetry

Hexagon

Three or more

sides

More than one

line of symmetry

2-D Shape Sorting Game (continued)

page 247

Teacher’s Guide

Grade 7

Two or more right angles

No right angles

Polygon

Two or more acute angles

Quadrilateral

Equilateral

2-D Shape Sorting Game (continued)

page 248

Teacher’s Guide

Grade 7

No obtuse angles

3 pairs of parallel sides

Not equilateral

One or more acute angles

Hexagon

Triangles

3-D Shape Sorting Game

page 249

Teacher’s Guide

Grade 7

3-D Shape Sorting Game (continued)

page 250

Teacher’s Guide

Grade 7

3-D Shape Sorting Game (continued)

page 251

Teacher’s Guide

Grade 7

3-D Shape Sorting Game (continued)

page 252

Teacher’s Guide

Grade 7

Triangular

shaped base

Square shaped

base

One or more rectangular

faces

One or more triangular

shaped faces

One or more

curved surfaces

8 or more vertices

3-D Shape Sorting Game (continued)

page 253

Teacher’s Guide

Grade 7

All faces congruent

Six or fewer vertices

Pyramids

Prisms

More than one

base

Exactly 12 edges

3-D Shape Sorting Game (continued)

page 254

Teacher’s Guide

Grade 7

More than 4

congruent faces

Two or more faces that are not bases

Not a pyramid

Not a prism

Base has more than 4 sides

Less than 4 congruent

faces

Venn Diagram for the Shape Sorting Game

page 255

Teacher’s Guide

Grade 7

Calendars

page 256

Teacher’s Guide

Grade 7

Name of Month

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

Hundreds Charts

page 257

Teacher’s Guide

Grade 7

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

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

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

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

Pattern Blocks

page 258

Teacher’s Guide

Grade 7

Triangles Squares Rhombuses

Trapezoids

Hexagons

Dot Paper

page 259

Teacher’s Guide

Grade 7

Isometric Dot Paper

page 260

Teacher’s Guide

Grade 7

Nets for 3-D Shapes

page 261

Teacher’s Guide

Grade 7

Triangular

Pyramid

Square

Pyramid

Nets for 3-D Shapes (continued)

page 262

Teacher’s Guide

Grade 7

Cube

Triangular

Prism

Nets for 3-D Shapes (continued)

page 263

Teacher’s Guide

Grade 7

Pentagonal

Prism

Pentagonal

Pyramid

Define a Number (Advanced)

page 264

Teacher’s Guide

Grade 7

Each statement describes at least one whole number between 1 and 100:

A The number is even. B The number is odd. C You can count to the number

by 2's.

D You can count to the number

by 3's.

E You can count to the number

by 5's.

F You can count to the number

by 10's.

G The number is greater than 15. H The number is less than 25. I The number has 1 digit.

J The number has 2 digits. K The number has two digits that

are the same.

L The number has a zero in it.

M The number is 7. N The number is less than 7. O You can see the number on the

face of a clock.

P The units digit is smaller than 6. Q The sum of its digits is less

than 9.

R The number is divisible by 2.

1. Write a number that statement A applies to: _____

2. Statements B, D, H and J all apply to the number 15.

Which other statements also apply to the number 15? _____________

3. Choose a number between 1 and 25 (besides 15).

Find all the statements that apply to that number.

Your Number _____ Statements that apply to that number _______________________

4. a) Which statements apply to both 3 and 18? ________________________

b) Which statements apply to both 7 and 11? ________________________

5. a) Can you find a number that statements A, E, J, L and R apply to?

NOTE: There may be more than one answer.

________________________________________________________________________

b) Can you find a number that applies to statements B, D, H and Q apply to?

________________________________________________________________________

c) Can you find a number that applies to statements C, K, H and P apply to? (22)

________________________________________________________________________

Symmetry

page 265

Teacher’s Guide

Grade 7

1. The following are flags from the nations indicated:

• Are there any lines of symmetry in any of these flags? HINT: Some flags may contain more than one line of symmetry.

• Draw the line or lines of symmetry in each flag.

BONUS:

In which continent are these countries?

a) Bahamas

c) Cameroon

b) India

d) Honduras

EXTENSION:

2. Can you find the type of symmetry in the flag of United Kingdom shown below? What is another

name for United Kingdom? In which continent is this country?

Answer the following question in your notebook.

3. Go to www.countryreports.org and examine the maps of the countries of your choice:

a) Find at least two countries (from different regions) with their maps having a horizontal line

of symmetry.

b) Find at least two countries (from different regions) with their maps having a vertical line

of symmetry.

Grade 7: Glossary

page 266

Teacher’s Guide

Grade 7

a.m. a time period that is in the morning, from 12 o’clock midnight until 12 o’clock noon

acute angle an angle that is less than a right

angle acute-angled triangle a triangle in which all

three angles are acute angles add to find the total when combining two or

more numbers together adjacent sides in a polygon, a pair of sides that

meet at a vertex alternate angles pairs of equal angles that are

formed on opposite sides of a transversal between two parallel lines

altitude of a triangle a perpendicular line from

a vertex to the side opposite the vertex angle the amount of a turn measured in degrees angle bisector a line that divides an angle into

two equal parts approximate a value made to a given precision

(as in a number of decimal places), but not the exact value

area the amount of space occupied by the face

or surface of an object arms the lines that form an angle array an arrangement of things (for example,

objects, symbols, or numbers) in rows and columns

attribute a characteristic (for example, colour,

size, shape) average found by adding a whole data set and

dividing the sum by the number of values it contains (also called the mean)

average speed found by dividing the total

distance traveled by the amount of time spent traveling

bar graph a graph that uses horizontal or

vertical bars to display a relationship between two countable variables

base angle an angle where one of the arms is the base of the shape

base of an exponent the factor that is repeated base of a pyramid the non-triangular face of a

pyramid; (if it is a triangular pyramid, any face can be a base)

bases of a prism two identical parallel faces of

a prism base ten materials materials used to represent

ones (ones squares or cubes), tens (tens strips or rods), hundreds (hundreds squares or flats), and thousands (thousands cubes)

bias an emphasis on characteristics that are not

typical of the entire population capacity the amount a container can hold

cartesian coordinate system a grid that is extended to include negative integers

categories a way to organize groups of data census the process of obtaining information

about every member of a population centimetre (cm) a unit of measurement used to

describe length, height, or thickness

cent notation a way to express an amount of money (for example, 40¢)

certain if an event must happen (for example,

when you roll a die it is certain that you will roll one of the numbers from 1 to 6)

circle graph (see pie graph) circumference the distance around the outside

of a circle clockwise a circular motion of an object in the

same direction to the movement of the hands of a clock

column numbers or symbols placed one above

another common denominator a number that is a

multiple of the denominators of two or more fractions

Grade 7: Glossary (continued)

page 267

Teacher’s Guide

Grade 7

common factor a number that is a factor of a group of numbers

complimentary angles angles whose sum

is 90º composite number a natural number that has

more than two factors composite shapes shapes created by

combining 2 or more polygons cone a 3-dimensional shape with one edge

and one vertex; a circular base and a curved edge

congruent a term used to describe shapes

if they are the same size and shape; congruent shapes can be different colours or shades and can have different orientations

consecutive numbers numbers that occur one

after the other on a number line coordinate system a grid with labeled rows and

columns, used to describe the location of a dot or object, for example the dot is at (A,3)

core the part of a pattern that repeats corresponding angles congruent angles

located on the same side of a transversal counter-clockwise a circular motion in the

opposite direction to the movement of the hands of a clock

cube a block that has six equal-sided square

faces cubic number the product of 3 identical factors cylinder a 3-dimensional object with 2 parallel

circular bases data facts or information database a collection of information that has

been systematically organized for easy access and analysis

decimal a short form for tenths (for example,

0.2) or hundredths (for example, 0.02), and so on

decimetre (dm) a unit of measurement used to describe length, height, or thickness; equal to 10 cm

decreasing sequence a sequence where each

number is less than the one before it denominator the number on the bottom portion

of a fraction; tells you how many equal parts are in a whole

diagonal things (for example, objects, symbols,

or numbers) that are in a line from one corner to another corner

diameter a line segment that passes through the center of a circle and has its endpoints on the circumference

difference the “gap” between two numbers; the

remainder left after subtraction digit a single character in a numeral or number

system dilation a transformation that generates an

image that is similar to the original (same proportional shape but different dimensions)

dimensions the length, width, and/or height of a

geometric figure divide to find how many times one number

contains another number dividend in a division problem the number that

is being divided or shared divisible by containing a number a specific

number of times without having a remainder (for example, 15 is divisible by 5 and 3)

divisor in a division problem, the number that is

divided into another number dollar notation a way to express an amount of

money (for example, $4.50) edge where two faces of a 3-D figure meet enlargements a dilation where the image is

larger than the original equally likely when two or more events have

the same chance of occurring

Grade 7: Glossary (continued)

page 268

Teacher’s Guide

Grade 7

equation a statement showing the equality of two expressions

equilateral a term used to describe a polygon

with sides that are all the same length equilateral triangle a triangle that has all sides

of equal length equivalent fractions fractions that represent

the same amount, but have different denominators (for example, 24 =

36 )

estimate a guess or calculation of an

approximate number even number the numbers you say when

counting by 2s (starting at 0) expanded form a way to write a number that

shows the place value of each digit (for example, 27 in expanded form can be written as 2 tens + 7 ones, or 20 + 7)

expectation when you think a specific outcome

will happen

exponent a condensed expression of a multiplication statement where a number written above and to the right of a factor indicates how many times that factor should be repeated

expression a mathematical statement of numbers and/or variables connected by operators

face the flat surface of a 3-D figure factor tree a diagram representing the factors of

a given number factors whole numbers that are multiplied to

give a product

fair used to describe a game of chance if both players have the same chance of winning

first-hand data data you collect yourself

(for example, by taking measurements, conducting experiments, or conducting surveys)

flip the reflection of an object or shape across a line; the image is a mirror-image of the original object or shape

formula a mathematical expression designed to express a fact or rule

fraction a number used to name a part of a set

or a region frequency the number of repetitions of a certain phenomenon in a given space or time gram (g) a unit of measurement used to

describe mass greater than a term used to describe a number

that is higher in value than another number greatest common factor the largest number

that is a factor of two or more numbers group of data similar data that are considered

together, such as the colour of your hair and your friends’ hair colours

growing pattern a pattern in which each term is

greater than the previous term height a measurement perpendicular to the

base of a shape, used to measure area or volume

hexagon a polygon with six sides horizontal oriented parallel to the horizon

horizontal axis (x axis) the horizonal line on a coordinate grid (going from left to right)

hour hand the short hand on a clock that tells

what hour it is image the figure that is the result of a

transformation impossible if an event cannot happen (for

example, rolling a “7” on a die) improper fraction a fraction that has a

numerator that is larger than the denominator; this represents more than a whole

increasing sequence a sequence where each

number is greater than the one before it

Grade 7: Glossary (continued)

page 269

Teacher’s Guide

Grade 7

integer a positive or negative whole number or zero integers the set of numbers which includes zero and both positive and negative whole numbers interior alternate angles a pair of

supplementary angles formed between two parallel lines on the same side of a transversal

intersecting lines lines that share a common point in space

isosceles triangle a triangle with two sides of

equal length key tells what the scale of a pictograph is so you

know how many each symbol represents kilogram (kg) a unit of measurement used to

describe mass; equal to 1 000 g kilometre (km) a unit of measurement for

length; equal to 1 000 m leaf a number’s right-most digit less than a term used to describe a number that

is lower in value than another number lexigraphic order the order used to organize

words in a dictionary or a phonebook likely if an event will probably happen

line an infinitely long and straight geometrical object

line graph a graph that uses a line to show how

data change over time

line segment a portion of a line between two end points

line symmetry when an object matches itself when reflected in a line

litre (L) a unit of measurement used to describe

capacity lowest common multiple (LCM) the smallest

common multiple of two or more numbers (for example, the LCM of 6 and 9 is 18)

lowest terms a fraction that is reduced so the only whole number that will divide into its numerator and denominator is 1

map a picture that represents a place (for

example, Canada or an imaginary place) mass measures the amount of substance, or

matter, in a thing mean found by adding a whole data set and

dividing the sum by the number of values it contains (also called average)

median the middle number or the mean of the

two middle numbers in an ordered data set (for example, the median of 1,3,4,6 and 7 is 4)

metre (m) a unit of measurement used to

describe length, height, or thickness; equal to 100 cm

midpoint the point exactly halfway between two

set points milligram (mg) a unit of measurement used to

describe mass; 1 000 times smaller than 1 g millilitre (mL) a unit of measurement used to

describe capacity; 1 000 times smaller than 1 L

millimetre (mm) a unit of measurement used to

describe length, height, or thickness; equal to 0.1 cm

minute hand the long hand on a clock that tells

the number of minutes past the hour; there are 60 minutes in 1 hour

mixed fraction a mixture of a whole number

and a fraction mode the number in a data set that appears the

most frequently model a physical representation (for example,

using base ten materials to represent a number)

more than a term used to describe a number

that is higher in value than another number multiple the product of a given whole number

and another whole number

Grade 7: Glossary (continued)

page 270

Teacher’s Guide

Grade 7

multiple of 2 a product of multiplying a number by 2

multiple of 3 a product of multiplying a number

by 3 multiply to find the total of a number times

another number natural numbers the set of regular counting

numbers, starting from 1 and counting up {1, 2, 3, 4, 5...}

negative numbers the set of numbers less than

0; denoted by a minus sign placed in front of the number

non-congruent a term used to describe 2-D

shapes that are not the same size and / or shape

number line a line with numbers marked at

equal intervals, used to help with skip counting numerator the number on the top portion of a

fraction; tells you how many parts are counted

obtuse angle an angle that is greater than a

right angle obtuse-angled triangle a triangle that has one

obtuse angle octagon a polygon with 8 sides odd number the numbers you say when

counting by 2s (starting at 1); numbers that are not even

operation a mathematical action or process

(adding, subtracting, multiplying, dividing, etc.)

opposite in a quadrilateral, a pair of sides that

do not meet at a vertex

opposite angles pairs of equal angles that are formed when lines intersect

opposite integers two integers where the sum

is 0

order of operations the rules for carrying out different operations in a mathematical expression in the correct sequence

ordinal number a word that describes the

position of an object (for example, first, second, third, fourth, fifth, sixth, seventh, eighth, ninth)

outcomes the different ways an event can turn out

p.m. a time period that is in the afternoon, from

12 o’clock noon until 12 o’clock midnight parallel lines lines that are straight and always

the same distance apart parallelogram a quadrilateral with two pairs of

parallel sides pattern (repeating pattern) the same repeating

group of objects, numbers, or attributes pentagon a polygon with five sides percent a ratio that compares a number to 100;

the term means “out of 100” perfect square the result of a whole number

multiplied by itself perimeter the distance around the outside of a

shape perpendicular bisector the line perpendicular

to a given line segment and which divides it into two equal parts

perpendicular lines two lines that intersect at

90º angles pictograph a way to record and count data

using symbols pie chart (see pie graph) pie graph a way to display data involving a

circle divided into parts (also called a circle graph and a pie chart)

piece of data a specific item of data, such as

the colour of a person’s hair or a type of fish polygon a 2-D shape with sides that are all

straight lines

Grade 7: Glossary (continued)

page 271

Teacher’s Guide

Grade 7

polyhedron any 3-dimensional object where all the sides are polygons

population a set of people, objects, or other

specimens that are being studied power a short form for repeated multiplication prime factorization the expression of a

composite number as a product of prime numbers

prime number a number that has two factors

(no more, no less), itself and 1 prism a 3-D figure with two congruent polygon-

shaped bases and the rest of the faces, parallelograms

probability how likely it is that an outcome will

happen product the result from multiplying two or more

numbers together property an attribute or characteristic of a thing

(for example, the number of edges of a shape, the number of vertices of a shape)

protractor a semi-circle with 180 subdivisions

(called degrees) around its circumference; used to measure an angle

pyramid a 3-D figure with a polygon as a base

and triangular faces that taper to one point quadrilateral a polygon with four sides quotient the result from dividing one number by

another number radius a line segment from the centre of a circle

to a point on the circumference range the difference between the lowest number

and highest number of a data set rate two quantities that are measured in different

but proportional units ratio a comparison of two numbers; written in

ratio form (for example, 4:5) or fraction form

(for example, 45 )

ray an object that extends infinitely from a single point in a single direction

rectangle a quadrilateral with four right angles reduction a dilation where the image is smaller

than the original reflection an object or shape that is a mirror-

image of the original object

reflexive angle any angle between 180º and 360º

regroup to exchange one place value for

another place value (for example, 10 ones squares for 1 tens strip)

regular polygon any polygon where all the sides are equal in length and all angles are equal in size

remainder the number left over after dividing or

subtracting (for example, 10 ÷ 3 = 3 R1) repeating decimal a decimal number where the

digits to the right of the decimal create a repeating pattern

rhombus a quadrilateral with equal sides right angle a 90º angle, also called a square

corner; found in many places, including the corners of squares, rectangles, and some triangles

right prism a prism where the non-base faces

are rectangles right-angled triangle a triangle that has one

right angle, or square corner rotation a circular motion of an object to a new

position row items arranged in a horizontal line sample/sampling a portion of a population that

is examined as representing the whole scale a ratio (or relationship) between two sets

of measurements scale of a graph the numbers along an axis

of a graph indicating what each interval represents

Grade 7: Glossary (continued)

page 272

Teacher’s Guide

Grade 7

scalene triangle a triangle with no two sides of equal length

scatter plot a way to display data; individual

pieces of data are graphed separately as dots

second hand the longest hand on a clock that

tells what second it is; there are 60 seconds in 1 minute

second-hand data data collected by someone

else (for example, in books and magazines, on the Internet)

sequence an ordered set of numbers set a group of like objects SI notation an international standard for writing

information side a boundary of a 2-D shape (for example,

one of the line segments that form the boundary of a polygon)

similar two shapes that have proportional

dimensions, but are not congruent simplest form a fraction or ratio where both

numbers have no common factors skip counting counting by a number

(for example, 2s, 3s, 4s) by “skipping” over the numbers in between

slide the movement of a shape along a straight

line with no turning speed the rate of motion or the distance

covered in a certain time square a quadrilateral with equal sides and four

right angles square centimetre (cm

2) a unit of measurement

used to describe area square number the product of 2 identical

factors square root a number that creates a square

number when multiplied by itself stem all of a number’s digits except its right-

most digit

stem and leaf plot a way of displaying data which separates the stem of each numer from the leaf

straight angle an angle of 180º subtract to take away one or more numbers

from another number sum the result from adding two or more

numbers together supplementary angle two angles that add up

to 180º surface area the total area of the surface of an

object as measured in the number of square units needed to cover a given object

symbol an object used to represent a larger

number in a pictograph (for example,

1 � means 5 flowers) symmetrical shape a shape that has either a

line of symmetry or rotational symmetry tally a way to record and count data; each

stroke represents “1” and every fifth stroke is made sideways to make counting easier (for example, //// = 5)

term an element of a pattern or sequence term number a number describing the location

of a given number in a sequence tessellation a pattern made up of one or more

shapes that completely covers a surface (without any gaps or overlaps)

three dimensional (3-D) measured in three

directions, such as length, width and height (for example, a cube, cylinder, cone, sphere)

tonne used to measure mass; equal to 1 000 kg transversal a line that intersects with 2 or more

lines trapezoid a quadrilateral with one pair of

parallel sides tree diagram uses branches (lines) to show all

the possible choices or outcomes triangle a polygon with three sides

Grade 7: Glossary (continued)

page 273

Teacher’s Guide

Grade 7

triangular having a face that is a triangle (for example, a pyramid with a three-sided base)

T-table a chart used to find a relationship

between two sequences of numbers turn a circular motion of an object to a new

position two-dimensional (2-D) measured in two

directions, such as length and width (for example, a circle, square, triangle, rectangle)

unlikely if an event will probably not happen variable a letter or symbol that stands for one or

more numbers Venn diagram a diagram that involves two

circles, used to organize things according to attributes; the areas of the circles that overlap represent items that share both attributes

vertex a point on a shape where two or more

sides or edges meet; the plural of vertex – vertices

vertical situated or being at right angles to the

horizon; upright vertical axis (y axis) the vertical line on a

coordinate grid (going up and down) volume the amount of space taken up by a

three-dimensional object zero principle the principle that the sum of

two opposite numbers will always be 0 (see opposite integers)