Teacher’s Resource - Cape Breton (E)/files/Teacher's... · BLACKLINE MASTERS (Available on...

AUTHORS

Therese ForsytheB.Sc., B.Ed., M.Ed., M.Ed.

Annapolis Valley Regional School Board

Jason P. FullerHon. B.Sc., B.Ed., M.Ed.

Annapolis Valley Regional School Board

Dan GilfoyB.Sc., B.Ed.

Halifax Regional School Board

Jay SpeijerB.Eng., M.Sc.Ed., P.Eng.

Niagara District School Board

Daniel McDonaldB.Sc., B.Ed.

South Shore District School Board

Jodie MacIlreithB.A., B.Ed., M.Ed., M.Ed.


Debbie VassB.T.


Anna SpanikB.Sc., B.Ed., M.Ed.


Susan WilkieB.Sc., B.Ed., M.Ed.


Anne Burnham MacLeodB.Ed., DAUS

Fredericton, New Brunswick

Toronto Montréal Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco

St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid

Mexico City Milan New Delhi Santiago Seoul Singapore Sydney Taipei

Teacher’s Resource

COPIES OF THIS BOOK MAY BE OBTAINED BY CONTACTING:McGraw-Hill Ryerson Ltd.

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TOLL-FREE FAX:1-800-463-5885

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OR BY MAILING YOUR ORDER TO:McGraw-Hill RyersonOrder Department300 Water StreetWhitby, ON L1N 9B6

Please quote the ISBN andtitle when placing your order.

McGraw-Hill Ryerson Mathematics 9: Focus on Understanding Teacher’s Resource

Copyright © 2006, McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-HillCompanies. All rights reserved. No part of this publication may be reproduced ortransmitted in any form or by any means, or stored in a data base or retrieval system,without the prior written permission of McGraw-Hill Ryerson Limited, or, in the case ofphotocopying or other reprographic copying, a licence from The Canadian CopyrightLicensing Agency (Access Copyright). For an Access Copyright licence, visitwww.accesscopyright.ca or call toll free to 1-800-893-5777.

Any request for photocopying, recording, or taping of this publication shall be directedin writing to Access Copyright.

ISBN 0-07-096622-2

http://www.mcgrawhill.ca

1 2 3 4 5 6 7 8 9 10 XBS 0 9 8 7 6

Printed and bound in Canada

Care has been taken to trace ownership of copyright material contained in this text. Thepublishers will gladly accept any information that will enable them to rectify anyreference or credit in subsequent printings.

The Geometer’s Sketchpad®, Key Curriculum Press, 1150 65th Street, Emeryville, CA 94680, 1-800-995-MATH.

PUBLISHER: Linda AllisonPROJECT MANAGERS: Eileen Jung, Maggie CheverieDEVELOPMENTAL EDITOR: Bradley T. SmithMANAGER, EDITORIAL SERVICES: Crystal ShorttCOPY EDITOR: Loretta JohnsonEDITORIAL ASSISTANT: Erin HartleyMANAGER, PRODUCTION SERVICES: Yolanda PigdenPRODUCTION COORDINATOR: Andree DavisCOVER DESIGN: Dianna LittleART DIRECTION: Tom Dart/First Folio Resource Group Inc.ELECTRONIC PAGE MAKE-UP: Tom Dart, Greg Duhanney, Kim Hutchinson/First FolioResource Group, Inc.COVER IMAGE: Courtesy of Ron Erwin Photography

C O N T E N T S

Introduction to Teacher’s Resource ........................................................................xi

Program Overview and Philosophy............................................................................................xv

Mathematics at Home .....................................................................................................xvii

Literacy ..............................................................................................................................xix

Cooperative Learning........................................................................................................xxi

Mental Mathematics .......................................................................................................xxiii

Problem Solving .............................................................................................................xxxii

Technology ....................................................................................................................xxxiii

Correlations ....................................................................................................................xxxv

Grades 6–9 Continuum....................................................................................................xlii

Manipulatives, Materials, and Technology Tools .............................................................lvi

Assessment ...................................................................................................................................lix

Adaptations................................................................................................................................lxiii

Chapter 1 Number Sense ...............................................................................................2

Get Ready .......................................................................................................................................4

1.1 Real Numbers .......................................................................................................................8

1.2 Operations With Rational Numbers.................................................................................13

1.3 Applications of Square Roots ............................................................................................19

1.4 Working With Exponents ..................................................................................................24

1.5 Scientific Notation .............................................................................................................31

1.6 Matrices ..............................................................................................................................36

Review ..........................................................................................................................................40

Practice Test .................................................................................................................................42

Chapter 2 Patterns and Relations ..........................................................................46

Get Ready .....................................................................................................................................49

2.1 Represent Patterns in a Variety of Formats ......................................................................52

2.2 Interpret Linear and Non-Linear Relationships...............................................................58

Use Technology

Explore Relations Using The Geometer’s Sketchpad® ......................................................66

2.3 Discover the Slope of a Line..............................................................................................68

2.4 The Equation of a Line ......................................................................................................74

Use Technology

Explore the Properties of the Slope, m, and the y-intercept, b........................................81

2.5 Graphs of Horizontal and Vertical Lines ..........................................................................83

Review ..........................................................................................................................................88

Practice Test .................................................................................................................................90

Task: A Golden Investigation ......................................................................................................96

Introduction • MHR iii

Chapter 3 Equations and Inequalities...............................................................100

Get Ready ...................................................................................................................................102

3.1 Solve Single-Variable Equations......................................................................................105

3.2 Represent Sets Graphically and Symbolically.................................................................111

3.3 Solve Single-Variable Inequalities ...................................................................................116

3.4 Problem Solving with Linear Equations and Inequalities .............................................121

Review ........................................................................................................................................125

Practice Test ...............................................................................................................................127

Chapters 1-3 Review..................................................................................................................130

Chapter 4 Probability ..................................................................................................132

Get Ready ...................................................................................................................................134

4.1 Experimental and Theoretical Probability .....................................................................137

4.2 Dependent and Independent Events...............................................................................144

4.3 Solve Problems Involving Compound Events ................................................................151

4.4 Make Decisions Based on Probability or Judgment.......................................................155

Review ........................................................................................................................................160

Practice Test ...............................................................................................................................162

Task: Play Probability ................................................................................................................165

Chapter 5 Measurement.............................................................................................168

Get Ready ...................................................................................................................................170

5.1 Volume of Three-Dimensional Figures ..........................................................................174

5.2 Surface Area of Three-Dimensional Figures ..................................................................180

5.3 Solve Volume and Surface Area Problems ......................................................................185

Review ........................................................................................................................................191

Practice Test ...............................................................................................................................193

Chapter 6 Geometry .....................................................................................................196

Get Ready ...................................................................................................................................199

6.1 Create Unique Triangles ..................................................................................................202

6.2 Congruent Triangles ........................................................................................................207

6.3 Similar Triangles...............................................................................................................212

Use Technology

Identify Similar Triangles Using The Geometer’s Sketchpad® .......................................218

6.4 Properties of Transformations ........................................................................................219

6.5 Mapping Notation for Transformations.........................................................................224

6.6 Combinations of Transformations..................................................................................230

Review ........................................................................................................................................234

Practice Test ...............................................................................................................................236

Task: Golfing With Lorie Kane .................................................................................................239

Chapters 4-6 Review..................................................................................................................242

iv MHR • Introduction

Chapter 7 Polynomials................................................................................................244

Get Ready ...................................................................................................................................246

7.1 Add and Subtract Polynomials........................................................................................249

7.2 Common Factors .............................................................................................................255

7.3 Multiply a Monomial by a Polynomial ...........................................................................263

7.4 Multiply Two Binomials ..................................................................................................270

7.5 Polynomial Division ........................................................................................................278

7.6 Apply Algebraic Modelling ..............................................................................................285

Review ........................................................................................................................................289

Practice Test ...............................................................................................................................290

Chapter 8 Data Management ..................................................................................294

Get Ready ...................................................................................................................................296

8.1 Scatterplots .......................................................................................................................299

Use Technology

Find Lines of Best Fit Using a Graphing Calculator ......................................................306

8.2 Assess Data and Make Predictions ..................................................................................307

8.3 Display Data .....................................................................................................................312

8.4 Interpret Data...................................................................................................................316

Review ........................................................................................................................................321

Practice Test ...............................................................................................................................323

Task: Stock Market....................................................................................................................326

Chapters 1-8 Review..................................................................................................................330

Introduction • MHR v

BLACKLINE MASTERS

(Available on Mathematics 9: Focus on Understanding, Teacher’s Resource CD-ROM)

This package has generic resource masters, generic assessment masters, and chapter-specific

worksheets, assessment tools, and alternative activities.

Blackline masters worksheets are provided in WORD and PDF formats for the Get Ready and

each numbered section in a chapter. A Chapter Review and Practice Test are provided for

each chapter. Answers are given for all questions.

Also included are the rubrics for the Assessment questions, Chapter Problem Wrap-Ups,

and Tasks. These will assist you in keeping track of student achievement by chapter.

Masters are provided in support to some of the Discover the Math activities,

Check Your Understanding questions, and Assessment questions.

vi MHR • Introduction

The following Generic Resource Masters are

provided on the CD-ROM:

Master 01 Integer Number Lines

Master 02 Vertical Number Lines

Master 03 Horizontal Number Lines

Master 04 Square Dot Paper

Master 05 Isometric Dot Paper

Master 06 Centimetre Grid Paper

Master 07 Grid Paper

Master 08 Loops Game Card

Master 09 Mental Math Bingo 1



Master 12 Mental Math Bingo Sheet

Master 13 Basic Fact Practice

Master 14 Decimal Point Practice

The following Generic Assessment Masters are provided

on the CD-ROM:

Assessment Master 01 Assessment Recording Sheet

Assessment Master 02 Attitudes Assessment Checklist

Assessment Master 03 Portfolio Checklist

Assessment Master 04 Presentation Checklist

Assessment Master 05 Problem Solving Checklist

Assessment Master 06 Journal Assessment Rubric

Assessment Master 07 Group Work Assessment

Recording Sheet

Assessment Master 08 Group Work Assessment General

Scoring Record

Assessment Master 09 How I Work

Assessment Master 10 Self-Assessment Recording Sheet

Assessment Master 11 Self-Assessment Checklist

Assessment Master 12 My Progress as a Mathematician

Assessment Master 13 Teamwork Assessment

Assessment Master 14 My Progress as a Problem Solver

Assessment Master 15 Assessing Work in Progress

Assessment Master 16 Learning Skills Checklist

The following Generic Tech Master is provided on the

CD-ROM:

Technology Master 01 The Geometer’s Sketchpad®,

Version 4, The Basics

Introduction • MHR vii

CHAPTER 1 BLMs

Parent Letter BLM

1GR

Discover the Math BLMs

1.1 Alternate Discover

1.1 Blank Cards

1.1 Card Definitions

1.1 Real Numbers Chart

1.1 Venn Diagram

1.3 Square Roots Table

1.4 Exponent Table


1.5 Discover Technology Adaptation

1.5 Scientific Notation Tables


Extra Practice BLMs

Get Ready

1.1

1.2

1.2 Extension

1.3

1.4

1.5

1.6

1.6 Extension

Review

Practice Test

Answer Key

Assessment Questions Rubrics

1.1

1.2

1.3

1.4

1.5

1.6

Chapter Problem Wrap Up Rubric

Ch1 Prob Wrap Up

CHAPTER 2 BLMs

Parent Letter BLM

2GR


2.3

Extra Practice BLMs

Get Ready

2.1

2.2

2.3

2.4

2.5

Review

Practice Test

Answer Key

Assessment Questions BLMs

2.2

2.3

2.4

2.5


2.1

2.2

2.3

2.4

2.5


Ch2 Prob Wrap Up

Task Rubric

2 Task Rubric

CHAPTER 3 BLMs

Parent Letter BLM

3GR


3.1 Solve an Equation

3.2 Inequality Statements

3.3 Operations on an Inequality

3.3 Solution to Inequality

3.4 Systematic Trial

Check Your Understanding BLMs

3.1 Showing Steps

3.3 Showing Steps

Extra Practice BLMs

Get Ready

3.1

3.2

3.3

3.4

Review

Practice Test

Answer Key


3.1

3.3

3.4


3.1

3.2

3.3

3.4


Ch3 Prob Wrap Up

CHAPTER 4 BLMs

Parent Letter BLM

4GR


4.4 Spinner

Communicate the Key Ideas BLMs

4.4 Question 3

Extra Practice BLMs

Get Ready

4.1

4.2

4.3

4.4

Review

Practice Test

Answer Key


4.1

4.2

4.3

4.4


4.1

4.2

4.3

4.4


Ch4 Prob Wrap Up

Task Rubric

4 Task Rubric

viii MHR • Introduction

CHAPTER 5 BLMs

Parent Letter BLM

5GR

Extra Practice BLMs

Get Ready

5.1

5.2

5.3

Review

Practice Test

Answer Key


5.1

5.2

5.3


5.1

5.2

5.3


Ch5 Prob Wrap Up

CHAPTER 6 BLMs

Parent Letter BLM

6GR


6.1 Triangle Table


6.3 Data Table

6.3 Triangles





Example BLMs

Example 1

Extra Practice BLMs

Get Ready

6.1

6.2

6.3

6.4

6.5

6.6

Review

Practice Test

Answer Key


6.1

6.2

6.3

6.4

6.5

6.6


6.1

6.2

6.3

6.4

6.5

6.6


Ch6 Prob Wrap Up

Task Rubric

6 Task Rubric

Introduction • MHR ix

CHAPTER 7 BLMs

Parent Letter BLM

7GR

Get Ready BLM

Divisibility Rules

Extra Practice BLMs

Get Ready

7.1

7.2

7.3

7.4

7.5

7.6

Review

Practice Test

Answer Key


7.3

7.5

7.6


7.1

7.2

7.3

7.4

7.5

7.6


Ch7 Prob Wrap Up

CHAPTER 8 BLMs

Parent Letter BLM

8GR


8.4

Check Your Understanding BLMs

8.1 Scatterplots

Extra Practice BLMs

Get Ready

8.1

8.2

8.3

8.4

Review

Practice Test

Answer Key


8.1

8.2

8.3

8.4


8.1

8.2

8.3

8.4


Ch8 Prob Wrap Up

Task Rubric

8 Task Rubric

x MHR • Introduction

McGraw-Hill Ryerson Mathematics 9: Focus onUnderstanding Program Overview

The McGraw-Hill Ryerson Mathematics 9: Focus on Understanding program has three

components.

S T U D E N T T E X T

The student text introduces topics in real-world contexts. In each section, Discoverthe Math activities encourage students to develop their own understanding of new

concepts. Worked Examples present solutions in a clear, step-by-step manner, and

then, the Communicate the Key Ideas summarize the new principles.

The text includes sections that can be used as assessment tools: ChapterReview, Practice Test, Chapter Problem, and Cumulative Review. Technology is

integrated throughout the program, and includes the use of scientific calculators,

graphing calculators, drawing software, and the Internet.

T E AC H E R ’ S R E S O U RC E

The teaching and assessment suggestions that are provided in this Teacher’s Resource

include

• sample responses for the Discover the Math questions

• sample responses for the Communicate the Key Ideas questions

• common student errors and suggested remedies

• sample responses and rubrics for the Assessment questions

S O LU T I O N S M A N UA L

The solutions manual provides full worked solutions for all questions in the num-

bered sections of the student text, as well as for questions in the Chapter Review,

Practice Test, and Cumulative Review features.

Introduction • MHR xi

An Introduction to Mathematics 9: Focus onUnderstanding Teacher’s Resource

The teaching notes for each chapter have the following structure:

Suggested Chapter Plan and Planning-Ahead Char t

This table provides an overview of each chapter at a glance, and specifies

• suggested timing for numbered sections and question numbers that most

students should be able to do

• any special materials and/or technology tools that may be needed

• related masters available on the CD-ROM

N u m b e re d S e c t i o n s

The side bar lists the following

• Specific Expectations that the section covers in whole or in part

• Materials needed for the section

• Technology Tools needed for the section

• Related Resources that are useful for extra practice, assessment, adaptations

Te a c h i n g N o te s

The key items include the following.

• Warm-Up exercises

• Answers for the Discover the Math questions let you know the expected

outcome of these activities.

• Teaching Suggestions give insights or point out connections that might not be

readily apparent on first read of the worked Examples.

• Sample responses for the Communicate the Key Ideas questions provide the

type of answers students are expected to give in this first assessment tool.

• Assessment suggestions give a variety of short assessment strategies or questions

that should be asked that can be used to assess the day’s learning

• A Question Planning Chart specifies questions to be assigned.

- Level 1: the minimum, usually knowledge questions, that all students should

be able to complete

- Level 2: questions that most students should attempt and be fairly successful

with

- Level 3: questions that extend the concepts or are more open problems to be

assigned with discretion

xii MHR • Introduction

E n d o f S e c t i o n / C h a p te r I t e m s

• Sample Solutions of a typical level 3 or 4 answer are provided for the

Assessment questions.

• Rubrics for the Assessment questions are provided. These can be

reproduced for discussion with students, so they understand what is expected on

the assessment activities.

• Rubrics are also provided for the Tasks that occur at the end of chapters 2, 4, 6,

and 8 in the student text.

• Answers to Making Connections and Puzzlers

The Teacher’s Resource CD-ROM also provides various editable masters, including:

• Generic Masters such as grid paper, nets

• Discover the Math Masters, scaffolded worksheets that support some of the

student text’s Discover the Math activities by helping students record the results

of their investigation in an organised format

• Blackline Masters of extra practice questions for

- each skill reviewed on the text’s Get Ready pages

- each numbered section

- each Chapter Review and Practice TestAnswers are included for the Blackline Master questions.

Introduction • MHR xiii

xiv MHR • Introduction

P RO G R A M OV E RV I E W A N D P H I LO S O P H Y

Mathematics 9: Focus on Understanding is an exciting new resource for the grade 9

student.

The Focus on Understanding program is designed to:

• provide full support in teaching the Atlantic Canada mathematics curriculum;

• enable and guide students’ progress from concrete to representational and then

to abstract thinking; and

• offer a diversity of options that collectively deliver student and teacher success.

During grades 7 to 9, most students are ready to progress from solely concrete

thinking toward more sophisticated forms of cognition, as shown in the diagram:

In Mathematics 9: Focus on Understanding, students start with the concrete where

appropriate. Once they have experience with this, they move to the semi-concrete.

Only when students are comfortable with the concrete and semi-concrete do they

begin to move toward the abstract.

Introduction • MHR xv

Concrete Thinking Representative Thinking Abstract Thinking

• typically work with physicalobjects

• focus of thinking is specific

• little or no reflection on thoughtprocesses

• able to solve very simple prob-lems

• sometimes called “semi-con-crete”

• typically work with diagrams

• thinking focus becoming moregeneral and systematic

• meta-cognitive thinking aboutthought processes begins to develop

• explore hypothetical or “what-if”thinking, with support

• able to solve moderately challenging problems

• use problem strategies effec-tively, with some guidance

• able to work with or withoutmaterials or diagrams

• thinking focus instinctively general and systematic

• meta-cognitive thinking is welldeveloped

• naturally explore hypothetical or“what-if” thinking

• able to solve problems thatextend or deepen thinking

• confidently select and adaptproblem strategies

Given the changes occurring during adolescence, school administrators and teachersneed to consider how best to match instruction to … the developing capabilities andvaried needs of intermediate students…

The Mathematics 9: Focus on Understanding program is based on a view that allstudents can be successful in mathematics… [It] reflects principles of effective practiceand research on how early adolescents learn, prerequisites for achieving a balancedapproach to mathematics.

Creating Pathways: Mathematical Success for Intermediate Learners, Folk,

McGraw-Hill Ryerson, 2004

Ap p ro a c h e s t o Te a c h i n g M at h e m at i c s

The concrete and abstract progression is exemplified in the following styles of

mathematics teaching.

At grade 9, students learn best by using a concrete, discovery oriented approach

to develop concepts. Once these concepts have been developed, a connectionist

approach helps students consolidate their learning.

At this level, some transmission-oriented learning is also useful. This variety of

approaches can be seen in the Mathematics 9: Focus on Understanding program design.

xvi MHR • Introduction

Transmission-Oriented Connectionist-Oriented Discovery-Oriented• teaching involves “delivering” the

curriculum

• focuses on procedures and routines

• emphasizes clear explanations and practice

• “chalk-and-talk”

• teaching involves helping students develop and apply theirown conceptual understandings

• focuses on different models and methods and the connec-tions among them

• emphasizes “problematic”challenges and teacher-studentdialogue

• “Van de Walle”

• teaching involves helping students learn by “doing”

• focuses on applying strategiesto practical problems and usingconcrete materials

• emphasizes student-determinedpacing

• “hands-on”

Feature Teaching Style(s) Supported

Chapter Problem connectionist

Discover the Math discovery, connectionist

Examples transmission, connectionist

Communicate the Key Ideas connectionist, transmission, discovery

Check Your Understanding transmission

Extend connectionist, transmission

Chapter Review transmission, connectionist

Task discovery, connectionist

The following assumptions and beliefs form the foundation of this textbook.

1) Mathematics learning is an active and constructive process.

2) Learners are individuals who bring a wide range of prior knowledgeand experiences, and who learn via various styles and different rates.

3) Learning is most likely when placed in meaningful contexts and inan environment that supports exploration, risk taking, and criticalthinking, and nurtures positive attitudes and sustained effort.

4) Learning is most effective when standards and expectations aremade clear and assessment and feedback are ongoing.

5) Learners benefit, both socially and intellectually, from a variety oflearning experiences, both independent and in collaboration withothers.

Department of Education, Nova Scotia, 2000

Th e M o d e r n C l a s s ro o m

The resources available in today’s classroom offer opportunities and challenges.

Indeed, the principal challenge––one that many teachers of mathematics are reluc-

tant to confront––is to teach successfully to the opportunities available.

Grouping

At one end of the scale, individual work provides an opportunity for students to

work on their own, at their own pace. At the other extreme, class discussion of prob-

lems and ideas creates a synergistic learning environment. In between, carefully

selected groups bring cooperative learning into play.

Manipulatives and Materials

Although many teachers feel unsure about teaching with manipulatives and other

concrete materials, many students find them a powerful way to learn. The

Mathematics 9: Focus on Understanding program supports the use of manipulatives,

but also helps teachers adapt to this kind of teaching. The notes in the Teacher

Resource provide suggestions for developing student understanding using semi-con-

crete materials such as diagrams and charts.

Technology

The scientific or graphing calculator is, or ought to be, a standard part of each stu-

dent’s mathematical toolbox. In the Mathematics 9: Focus on Understanding pro-

gram, scientific calculator keystrokes are provided in parallel with conventional

calculations.

Computer software, such as The Geometer’s Sketchpad®, provides a powerful

learning tool. The Focus on Understanding program supports use of such software as

an optional adjunct to class teaching. The Use Technology lessons offers alternative

activities using either the graphing calculator or The Geometer’s Sketchpad®. Teachers

enjoy maximum flexibility because they can teach some activities using manipula-

tives only, using software only, or with a combination of the two.

The Internet provides great opportunities to enhance learning, but it also raises

new dangers and concerns in teaching. As an integrated part of the Focus on

Understanding program, the McGraw-Hill Web site at www.mcgrawhill.ca/

links/math9NS offers safe and reliable links.

M AT H E M AT I C S AT H O M E

Research confirms that parents/guardians can profoundly influence the academic

success of their children (Department of Education, Nova Scotia, 2000). Parents can

be invaluable in convincing their children of the need to learn mathematics, espe-

cially when they understand a school’s mathematics program (NCTM, 2000). To

encourage this home and school connection, Focus on Understanding includes regu-

lar letters to parents.

This letter to parents is included as the first blackline master for each chapter. The

letter:

• provides parents with an overview of the material covered in the chapter;

• outlines the skills emphasized in the chapter;

Introduction • MHR xvii

Instructional practice thatincorporates a variety ofgrouping approachesenhances the richness oflearning for students.

Creating Pathways: MathematicalSuccess for Intermediate Learners,

Folk, McGraw-Hill Ryerson, 2004

The mathematics classroomneeds to be one in whichstudents are actively engagedeach day in the “doing ofmathematics.” …

The learning environment willbe one in which students andteachers make regular use ofmanipulative materials andtechnology…

Department of Education, Nova

Scotia, 2000

• explains how calculations are done (e.g., mentally, by hand, using a calculator);

• explains how students will be assessed (e.g., prepare using a chapter review, then

do a practice test before the actual test; complete a chapter problem wrap-up); and

• suggests some fun activities that parents can do at home with their children to

help them increase their understanding of mathematics.

Ideally, you could send this letter home as students start the Get Ready section for

each chapter. Encourage students and parents to understand that at-home activities

provide special times when parent(s) and child can work together to enjoy math.

Activities could fit into daily events and special interests. The chart below provides

some suggestions.

The Get Ready may be used as diagnostic tool that is assigned prior to beginning

a chapter. It may be a take-home activity or may be assigned during one class period

depending on students’ needs. It is expected to be a quick review of prerequisite skills

and not part of the core material of the chapter. Review of specific skills that need

improvement can be emphasized rather reviewing all skills listed in the Get Ready.

xviii MHR • Introduction

Home Activity/Interest Math Connection

chores predicting probability of selecting a specific chore from a job jar; estimating thecost of food placed in a grocery cart; using a calculator to keep track of grocery purchases

food halving and doubling recipes; estimating the fraction and percent of various ingredients in a trail mix; using geometric shapes to design a holiday dessert or display

meals working with fractions of various foods (e.g., pizzas, sandwiches, fruits, desserts);identifying patterns to calculate how many people can stand at various kitchenwork centre designs; using transformations to design a place mat or plan a specialtable design; working with probability to determine the number of combinationsfor a meal

music identifying fractions in various time signatures; researching the use of patterns inmusic; collecting and organizing data on various bands

outdoors identifying and classifying natural shapes; calculating area and perimeter of yardsor playgrounds; predicting probability of getting an orange flower at random froma collection of wildflowers; keeping data on the growth of a plant and calculatingmeasures of central tendency; keeping track of populations of insects or other plantor animal forms that reproduce exponentially; developing and answering Fermiproblems; using tiling patterns to design an outdoor patio or other area

pets feeding; keeping data on young pet’s daily mass and calculating measures of central tendency over time; using geometric shapes to design a special pet run or cage

sports/games calculating area and perimeter of play surfaces; identifying shapes; collecting dataand keeping personal statistics for specific activity; identifying patterns in team statistics and using them to make predictions; using transformations to create ateam logo or shirt

story time predicting probability of picking a certain magazine or book from a pile or familycollection; researching the use of geometric shapes in illustrations and page design

travelling identifying and collecting sign shapes; collecting data on gas consumption andfinding average kilometres per litre for a specific vehicle; using integers to showtravel East and West/North and South of a specific location; identifying three-dimensional figures and calculating volumes of space

During activities like these, students concentrate on process as they practise mathe-

matics and develop skills.

Other ways to involve parents/guardians include:

• providing clear and timely assessment information, especially when there is

evidence that a student may be at risk;

• recognizing and celebrating the first languages and cultures of students and

their families;

• having students explain to parents/guardians mathematical concepts learned in

class; and

• informing parents/guardians about what is happening in class and about

homework assignments.

L I T E R AC Y

Effective mathematics classrooms show students that math is everywhere in their

world. For example, students should see that knowledge of probability is useful when

learning about the electoral process in social studies class. Their work in graphing can

be used in science class. The journal entries they make about problem solving are also

language arts products. When connections such as these are made, students begin to

see that math is not an isolated subject but rather a vital part of everyday life.

The Reflect and Communicate the Key Ideas questions are opportunities for

students to explain and show their understanding of the mathematics. Answers to

these questions provided in the Teacher’s Resource are concise. However, it is expected

that students will provide full explanations when answering these questions.

L i t e ra c y Co n n e c t i o n s

These features give students the help they may need to understand a symbol, a

phrase, or a new word. They also provide suggestions for connecting to literacy, such

as developing organizers.

In the early chapters, you might ask students to discuss the boxes in small

groups. After that, students can use the features to support their learning, when

needed. Occasionally, the features could be a lead-in for discussing a concept. This

feature provides one more way for students to feel successful in mathematics.

O n g o i n g J o u r n a l A s s e s s m e nt

Journal work is an important part of the math program, as it helps students to write

about the mathematics they are learning, and allows them to communicate their feel-

ings and understanding about what they are learning and how mathematics relates

to the world around them.

Students have probably written journal entries in previous grades. At grade 9,

students should develop longer journals using the skills they have developed in lan-

guage arts and during grade 8.

If you find that journal responses are short and not well thought out, provide

some coaching along the following lines.

Journal entries need the following components:

• Introduction: The introduction can be a sentence or an entire paragraph. It

Introduction • MHR xix

should explain what the journal entry is going to be about and introduce the

discussion.

• Body: In this part, students “discuss” the subject matter. The body should be at

least several sentences, but as students progress through grade 9, it could be

expanded to more than three paragraphs. Diagrams, charts, and visuals could

also be included.

• Conclusion: This consists of a closing sentence or paragraph that wraps up the

discussion.

Take time to discuss the different types of journal entries that students have done.

Note that personal entries are for the benefit of students themselves. The other types

of entries, however, are meant to communicate with the teacher and will be assessed

as part of the mathematics work. The following chart shows the difference between

the types and provides some sample journal starters.

Throughout the textbook, ask students to choose one journal entry from each

chapter that they would like to share. If you wish, give them time to write a good

copy. Collect and assess the journal using Assessment Master 06 JournalAssessment Rubric. Reading the content of the journal will help you in the assess-

ment of students’ learning of the chapter content. You may wish to respond person-

ally to students’ thoughts and ideas.

xx MHR • Introduction

Personal EntriesEntries that Communicate to Teacher

Knowledge Connections Communication

Purpose • For the benefit ofthe student.

• Not to be read byanyone else unlessthe student offers toshare them.

• Used to discuss howthe student feelsabout their ability inmath or particularparts of the course.

• Show what stu-dents know andunderstand abouta topic or concept.

• Emphasize theconnections stu-dents are makingbetween themathematics theydo in the class-room and theirpersonal lives andthe world aroundthem.

• Provide reflec-tions on what stu-dents learnedduring a particu-lar section andwhat they think isimportant.

Sample Opener

• My biggest difficultyin mathematics is …

• The thing I like bestabout mathematicsis …

• My most memorablemathematics lessonwas …

• If I had to explain______________ tosomeone else, Iwould …

• The best way to____________ is to___________because …

• The differencebetween ______and _______ is …

• The most practicalplace I use_________ is …

• When I grow up,I’d like to be a_____________, sothe most impor-tant math I needto know is …

• I use _________ …

• I wish I had lis-tened more care-fully when theteacher wasexplaining …

• The most impor-tant thing Ilearned in thissection is …

CO O P E R AT I V E L E A R N I N G

Students learn effectively when they are actively engaged in the process of learning.

Most sections of Focus on Understanding begin with a hands-on activity that fosters

this approach. These activities are best done through cooperative learning during

which students work together—either with a partner or in a small group of three or

four—to complete the activity and develop generalizations about the topic or process.

Group learning such as this is an important aspect of a constructivist educa-

tional approach. It encourages interactions and increases chances for students to

communicate and learn from each other (Sternberg & Williams, 2002).

Teachers’ Role—In classrooms where students are adept at cooperative learn-

ing, the teacher becomes the facilitator, guide, and progress monitor. Until students

have reached that level of group cooperation, however, you as the teacher will need

to coach them in how to learn cooperatively. This may include:

• making sure that the materials are at hand and directions perfectly clear so that

students know what they are doing before starting group work;

• carefully structuring activities so that students can work together;

• providing coaching in how to provide peer feedback in a way that allows the

listener to hear and attend; and

• constantly monitoring student progress and providing assistance to groups

having problems either with group cooperation or the math at hand.

Types of Groups—The size of group you use may vary from activity to activ-

ity. Small-group settings allow students to take risks that they might not take in a

whole class (Van de Walle, 2000). Research suggests that small groups are fertile envi-

ronments for developing mathematical reasoning (Artz & Yaloz-Femia, 1999).

Results of international studies suggest that groups of mixed ability work well

in mathematics classrooms (Kilpatrick, Swafford, & Findell, 2001). If your class is

new to cooperative learning, you may wish to assign students to groups according to

the specific skills of each individual. For example, you might pair a student who is

talkative but weak in number sense and numeration with a quiet student who is

strong in those areas. You might pair a student who is weak in many parts of math-

ematics but has excellent spatial sense with a stronger mathematics student who has

poor spatial sense. In this way, student strengths and weaknesses complement each

other and peers have a better chance of recognizing the value of working together.

Cooperative Learning Skills—When coaching students about cooperative

learning, consider task skills and working relationship skills.

Introduction • MHR xxi

Task Skills Working Relationship Skills

• following directions

• communicating information and ideas

• seeking clarification

• ensuring that others understand

• actively listening to others

• staying on task

• encouraging others to contribute

• acknowledging and responding to the contribu-tions of others

• checking for agreement

• disagreeing in an agreeable way

• mediating disagreements within the group

• sharing

• showing appreciation for the efforts of others

Use class discussions, modelling, peer coaching, role-plays, and drama to provide

positive task skills. For example, you might role-play different ways to provide feed-

back and have a class discussion on which ones students like and why. You might dis-

cuss common group roles and how group members can use them. Make sure

students understand that the same person can play more than one role.

Ty p e s o f G ro u p s

Three group types are commonly used in the mathematics classroom.

Think/Pair/Share—This consists of having students individually think about a con-

cept, and then pick a partner to share their ideas. For example, students might work

on the Communicate the Key Ideas questions, and then choose a partner to discuss

the concepts with. Working together, the students could expand on what they under-

stood individually. In this way, they learn from each other, learn to respect each

other’s ideas, and learn to listen.

Cooperative Task Group—Task groups of two to four students can work on activi-

ties in the Discover the Math section. As a group, students can share their under-

standing of what is happening during the activity and how that relates to the

mathematics topic, at the same time as they develop group cooperation skills.

Jigsaw—Another common cooperative learning group is called a jigsaw. In this tech-

nique, individual group members are responsible for researching and understanding

a specific part of information for a project. Individual students then share what they

have learned so that the entire group gets information about all areas being studied.

For example, during data management, this type of group might have “experts” in

making various types of graphs using technology. Group members could then coach

each other in making each kind of graph.

xxii MHR • Introduction

Role Job Sample Comment

Leader • makes sure the group is on task and everyoneis participating

• pushes group to come to a decision

Let’s do this.

Can we decide...?

This is what I think we should do...Recorder • manages materials

• writes down data collected or measurementsmade

This is what I wrote down. Is that what youmean?

Presenter • presents the group’s results and conclusions

Organizer • watches time

• keeps on topic

• encourages getting the job done

Let’s get started.

Where should we start?

So far we’ve done the following...

Are we on topic?

What else do we need to do?Clarifier • checks that members understand and agree Does everyone understand?

So, what I hear you saying is...

Do you mean that...?

Another way of using the Jigsaw method is to assign “home” and “expert” groups

during a large project. For example, students researching the shapes on various

sports surfaces might have a home group of four in which each member is responsi-

ble for researching one of: soccer, baseball, hockey, or basketball. Individual members

could then move to “expert” groups. “Expert” groups would include all of the stu-

dents responsible for researching one of the sports.

Each of the “expert” groups would research their particular sport. Once the

information had been gathered and prepared for presentation, individual members

of the “expert” group would return to their “home” group and teach other members

about their sport.

M E N TA L M AT H E M AT I C S

A major goal of mathematics instruction for the 21st century is for students to make

sense of the mathematics in their lives. The development of all areas of mental math-

ematics is a major contributor to this comfort and understanding. Mental mathe-

matics is the mental manipulation of knowledge dealing with numbers, shapes, and

patterns to solve problems.

The diagram above shows the various components under the umbrella of Mental

Mathematics. All three are considered mental activities and interact with each other

to make the connections required for mathematics understanding. Estimation and

mental math are not topics that can be isolated as a unit of instruction; they must be

integrated throughout the study of mathematics.

Co m p u t at i o n a l E s t i m at i o n

Computational estimation refers to the approximate answers for calculations, a very

practical skill in today’s world. The development of estimation skills helps refine mental

computation skills, enhances number sense, and fosters confidence in math abilities, all

of which are key in problem solving. Over 80% of out-of-school problem solving situa-

tions involve mental computation and estimation (Reys and Reys, 1986).

Computational estimation does not mean guessing at answers. Rather, it

involves a host of computational strategies that are selected to suit the numbers

involved. The goal is to refine these strategies over time with regular practice, so that

estimates become more precise. The ultimate goal is for students to estimate auto-

matically and quickly when faced with a calculation. These estimations are a check

for reasonableness of solutions, to allow for recognition of errors on calculator dis-

plays, and provide learners with a strategy for checking their actual calculations.

MentalMathematics

Mental Imagery

Estimation(in computation and

in measurement)

MentalComputation

(precise answers)

Introduction • MHR xxiii

M e a s u re m e nt E s t i m at i o n

This skill relies on awareness of the measurement attributes (e.g., metre, kilometre,

litre, kilogram, hour). Just as computational estimation enhances number sense,

practice in measurement estimation enhances measurement sense.

A “referent” is a personal mental tool that students can develop for use in

thinking about measurement situations. Tools could include, the distance from

home to school, a 100 km trip, the capacity of a can of juice, the duration of 30 min-

utes, and the area of the math textbook cover. These referents develop with measure-

ment practice, and specifically with practice that encourages students to form these

frames of reference. Students can compare other measurements to these referents. By

doing so, they can gain a better understanding of what may be happening in a prob-

lem solving situation.

Help students develop referents by doing activities such as asking students to

use their fingers or hands to show such measurements as: 6 cm, 260 mm, 0.4 m, a 60°

angle, or 2000 cm3.

M e nt a l I m a g e r y

“Mental imagery” in mathematics refers to the images in the mind when one is doing

mathematics. It is these mental representations, or conceptual knowledge, that need

to be developed in all areas of mathematics. Capable math students “see” the math

and are able to perform mental maneuvers in order to make connections and solve

problems. These images are formed when students manipulate objects, explore num-

bers and their meanings, and talk about their learning. Students must be encouraged

to look into their mind’s eye and “think about their thinking.”

Asking, “What do you see in your mind’s eye” when asked to visualize, as for

example in the exercises below, forces students to think about the images they are

using to help them solve problems. Students are often surprised when fellow students

share their personal images; the discussion generated is very worthwhile.

Try these Mental Imaging Activities with your students.

xxiv MHR • Introduction

Example 1:

Draw the mental image you have for each of the following:

• 2–3

• 243 100 in relation to a million

• 75% of the questions on the page

• a 175º angle

• 0.56 m

• 36 cm

• 280 mm

• a 6 m � 10 m garden

• a 6.3 kg fish

• a 6 g fish

Example 2:

Use mental imagery to answer the following:

1. How many edges does a cube have?

2. If I am facing east, what direction is to my left?

3. What is the perimeter of a 90 cm � 30 cmshelf?

4. How many sides does a hexagonal pyramidhave?

5. Imagine a 5 cm cube. What is its volume?

6. You cut off one vertex on a cube. What shapeis left?

7. You cut the top off a square pyramid. Whatshape is exposed?

M e nt a l Co m p u t at i o n

Mental computation refers to an operation used to obtain the precise answer for a

calculation. Unlike traditional algorithms, which involve one method of calculation

for each operation, mental computations include a number of strategies––often in

combination with others––for finding the exact answer. These mental calculations

are often referred to as “Mental Math.” As with computational estimation, strategies

for mental computation develop in quantity and quality over time. A thorough

understanding of, and facility with, mental computation also allows students to solve

complicated multi-step problems without spending needless time figuring out cal-

culations and is a valuable prerequisite for proficiency with algebra. Students need

regular practice in these strategies.

Some Points Regarding Mental Mathematics

• Students must have a knowledge of the basic facts (addition and multiplication)

in order to estimate and calculate mentally. They learn the many strategies for

fact learning in elementary school. With practice, they eventually commit these

facts to memory. Without knowing the basic facts, it is unlikely that students

will ever attempt to employ any estimation or mental math strategies, as these

will be too tedious.

• The various estimation and mental calculation strategies must be taught and

best developed in context; opportunities must be provided for regular practice

of these strategies. Having students share their various strategies is vital, as it

provides possible options for classmates to add to their repertoire.

• Unlike the traditional paper-and-pencil algorithms, there are many mental

algorithms to learn. With the learning, however, comes a greater facility with

numbers. Key to the development of skills in mental math, is the understanding

of place value (number sense) and the number operations. This understanding

is enhanced when students make mental math a focus when calculating.

• Mental math strategies are flexible; one needs to select one that is appropriate

for the numbers in the computation. Practice should be in the form of

practising the strategy itself, selecting appropriate strategies for a variety of

computation examples, and using the strategies in problem solving situations.

• Although students should not be pressured with time constraints when first

learning a mental math strategy, it is beneficial to provide timed tests once they

have some facility at mental computation. If too much time is provided, many

students will resort to the traditional algorithm, and will not use mental

strategy.

• Mental math algorithms are used with whole numbers, fractions, and decimal

numbers.

• Sometimes mental math strategies are used in conjunction with paper-and-

pencil tasks. The questions are rewritten to make the calculation easier.

• The ultimate goal of mental mathematics is for students to estimate for

reasonableness, and to look for opportunities to calculate mentally.

• Encourage students to refer to the strategies by their name (for example, front-

end strategy). Once the strategies have been taught, post them around the room

for the students. Have students write problems in which a mental strategy would

Introduction • MHR xxv

be the appropriate computation. Share these problems with the class.

• Students need to identify why particular procedures work; they should not be

taught computation “tricks” without understanding.

• Those who are skilled in using mental mathematics will be able to transfer,

relate, and apply mental strategies to paper-and-pencil tasks.

Keep in Mind

Practice in classrooms has traditionally been in the form of asking students to write

the answers to questions presented orally. This is particularly challenging for students

who are primarily visual learners. Although we are sometimes faced with computa-

tions of numbers we cannot see, most often the numbers are written down. This

makes it easier to select a strategy. In daily life, we see the numbers when solving writ-

ten problems (e.g., when checking calculations on a bill or invoice, when determining

what to leave for tips, when calculating discounted prices from a price tag). Provide

students with mental math practice that is sometimes oral and sometimes visual.

Capable students of mathematics are comfortable with numbers. This comfort

means that the students see patterns in numbers and intuitively know how they

relate to each other and how they will behave in computational situations. Because

of their comfort with numbers, these students have developed strong skills in estima-

tion and mental math. Because of this, their understanding of number is further

strengthened. We say they have “number sense.” This sense of number develops grad-

ually and varies as a result of exploring numbers, visualizing them in a variety of

contexts, and relating them in ways that are not limited by traditional algorithms.

The position of the National Council of Teachers of Mathematics (NCTM) on

how to proceed when faced with a problem that requires a calculation is best

explained with this chart.

The chart tells us that, given a problem requiring calculation, students should ask

themselves the following questions:

• Is an approximate answer adequate or do I need the precise answer?

• If an estimate is sufficient, what estimation strategy best suits the numbers

provided?

• If an exact answer is needed, can I use a mental strategy to solve it?

• If the numbers don’t lend themselves to a mental strategy, can I do the

calculation using a paper-and-pencil method?

• If the calculation is too complex, I will use a calculator. What is a good estimate

for the answer?

Problem situation

Use a computer

Use a calculator

Estimate

Calculation needed

Approximateanswer needed

Use mentalcalculation

Exact answerneeded

Use apaper-and-pencil

calculation

xxvi MHR • Introduction

NCTM’s Number and Operations Standard for Grades 6–8 states that, “Instructional

programs from kindergarten through grade 12 should enable all students to com-

pute fluently and make reasonable estimates” (Principles and Standards for School

Mathematics, 2000). Whether the students select an estimation strategy, a mental

strategy, a paper-and-pencil method, or use the calculator, they must use their esti-

mation skills to judge the reasonableness of any answer.

In Nova Scotia, for Grades 1–9, it is expected that students will be engaged in

five minutes of mental math each day. The Department of Education has created a

professional development package that includes DVDs on mental math and a yearly

plan for each grade level. Use the appropriate sections of the DVD and the grade 9

yearly plan to help you create your mental math program.

M e nt a l M at h St rat e g i e s

Addition

Break Up the Numbers Strategy

This strategy is used when regrouping is required. One of the addends is broken up

into its expanded form and added in parts to the other addend. For example, 57 � 38

might be calculated in this way: 57 � 30 is 87 and 8 more is 95.

Front-End (left-to-right) Strategy

This commonly used strategy involves adding the front-end digits and proceeding to

the right, keeping a running total in your head. For example, 124 � 235 might be cal-

culated in the following way: Three hundred (100 � 200), fifty (20 � 30) nine (4 � 5).

Rounding for Estimation

Rounding involves substituting one or more numbers with “friendlier” numbers with

which to work. For example, 784 � 326 might be rounded as 800 � 300, or 1100.

Front-End Estimation

This strategy involves adding from the left and then grouping the numbers in order

to adjust the estimate. For example, 5239 � 2667 might be calculated in the follow-

ing way: Seven thousand (5000 � 2000), eight hundred (600 � 200)––no, make that

900 (39 and 67 is about another hundred). That’s about 7900.

Compatible Number Strategy

Compatible numbers are number pairs that go together to make “friendly” numbers.

That is, numbers that are easy to work with. To add 78 � 25, for example, you might

add 75 � 25 to make 100, and then add 3 to make 103.

Near Compatible Estimation

Knowledge of the compatible numbers that are used for mental calculations is used

for estimation. For example, in estimating 76 � 45 � 19 � 26 � 52, one might do the

following mental calculation: 76 � 26 and 52 � 45 sum to about 100. Add the 19; the

answer is about 219.

Balancing Strategy

A variation of the compatible number strategy, this strategy involves taking one or

more from one addend and adding it to the other. For example, 68 � 57 becomes

70 � 55 (add 2 to 68, take 2 from the 57).

Introduction • MHR xxvii

Clustering in Estimation

Clustering involves grouping addends and determining the average. For example,

when estimating 53 � 47 � 48 � 58 � 52, notice that the addends cluster around 50.

The estimate would be 250 (5 � 50).

Special Tens Strategy

In the early grades, students learn the number pairs that total ten––1 and 9, 2 and 8,

3 and 7, and so on. These can be extended to such combinations as 10 and 90, 300

and 700, 6000 and 4000, etc.

Compensation Strategy

In this strategy, you substitute a compatible number for one of the numbers so that

you can more easily compute mentally. For example, in doing the calculation

47 � 29, one might think (47 � 30) � 1.

Consecutive Numbers Strategy

When adding three consecutive numbers, the sum is three times the middle number.

Subtraction

Compatible Number Estimation

Knowledge of compatible numbers can be used to find an estimate when subtract-

ing. Look for the near compatible pairs. For example, when subtracting 1014 � 766,

one might think of the pairing.

Front-End Strategy

When there is no need to carry, simply subtract from left to right. To subtract

368 � 125, think 300 � 100 � 200, 60 � 20 � 40, 8 � 5 � 3. The answer is 243.

Front-End Estimation

For questions with no carrying in the highest two place values, simply subtract those

place values for a quick estimation. For example, the answer to $465.98 � $345.77

is about $120.00.

Compatible Numbers Strategy

This works well for powers of 10. Think what number will make the power of 10. For

example, to subtract 100 � 54, think what goes with 54 to make 100. The answer

is 46.

Equal Additions Strategy for Subtraction

This strategy avoids regrouping. You add the same number to both the subtrahend

and minuend to provide a “friendly” number for subtracting, then subtract. For

example, to subtract 84 � 58, add 2 to both numbers to give 86 � 60. This can be

done mentally. The answer is 26.

Compensation Strategy for Subtraction

As with addition, subtract the “friendly” number and add the difference. For exam-

ple, $3.27 � $0.98 � ($3.27 � $1.00) � $0.2 � $2.29.

“Counting On” Strategy for Subtraction

Visualize the numbers on a number line. For example, 110 � 44. You need 6 to make

50 from 44, then 50 to make 100, then another 10. The answer is 66.

750250

xxviii MHR • Introduction

“Counting On” Estimation

“Counting On” can also be used for estimation. For example, to estimate 894 � 652,

think that 652 � 200 gives about 850. Then another 50 gives about 900. The differ-

ence is about 250.

Multiplication

Multiplying by 10, 100, and 1000 Strategy

Instead of counting zeroes and adding them on, students use the concept of annex-

ing zeroes. For example, multiplying tens by tens gives hundreds; tens by hundreds

gives thousands; hundreds by hundreds results in ten thousands; and thousands by

thousands results in millions.

Multiplying by 0.1, 0.01, and 0.001 Strategy

Students need to realize that these decimals represent , , . They

should think about groups of 10’s, 100’s, and 1000’s.

Compatible Factors Strategy

This strategy involves using the Associative Property and looking for “friendly” com-

binations to multiply. For example, in multiplying 4 � 76 � 250, one might

rearrange the numbers to make the calculation easier. 4 � 250 � 1000 and 1000 mul-

tiplied by 76 gives 76 000.

Make Compatible Factors Strategy

Students show the numbers as their factors and then regroup to develop numbers

that are easier to work with. For example, 16 � 75 can be written as 4 � 4 � 3 � 25.

4 � 25 � 100; 4 � 3 � 12. The answer is 1200.

Squaring Numbers Strategy

Students learn that there is a pattern when squaring numbers that end in 5. For

example, the answer always ends with 25.

Round to Estimate Multiplication

Use rounding to estimate factors with two digits. For example, when multiplying

58 � 32, round to 60 � 30. The answer is about 1800.

Percentage/Fraction Connection

To find common percentages, think of the percentage as a fraction and divide by the

denominator. For example, 50% of $25 is half of $25. Divide by 2. The answer is $12.50.

Estimating Percent Using 1%, 10%, and 100%

As in multiplying by 0.1, students need to consider that they are looking for of

the number.

Front-End Multiplication Strategy

This is usually used when one factor is a single digit and there is no regrouping. For

example, 3 � 2313 � 6000 � 900 � 30 � 9 � 6939.

Compensation Strategy for Multiplication

As with addition and subtraction, work with “friendly” numbers. For example,

5 � 29 � 5 � 30 � 5 � 145.

Double and Halve Strategy

Make numbers easier to multiply by doubling one factor and halving the other to

provide a “nice” number. For example, 16 � 35 � 8 � 70 � 560.

110

11000

1100

110

Introduction • MHR xxix

xxx MHR • Introduction

Multiplying by 11 Strategy

Have students look for a pattern in the product. They will see that, in answers to

questions such as 44 � 11, the first number of the answer is the tens digit of the fac-

tor that is not 11, the middle number is the sum of the two numbers of the factor

that is not 11, and the final number is the ones digit of the factor that is not 11. The

answer is 484.

Further Multiplying by 11 Strategy

When the sum of the middle number above is greater than 9, add the remainder to

the tens digit of the factor that is not 11 and proceed as above. So 84 � 11 � 924.

Division

The Percentage/Fraction Connection

Students learn that a knowledge of common fractions is helpful when calculating

percentages. For example, 20% is and 25% is . So, to find 20%, divide by 5; for

25%, divide by 4, etc.

Break Dividend Into Parts Strategy

For many simple computations, divide the dividend into parts and divide. For exam-

ple, 1515 � 5 � (1500 � 5) � (15 � 5) � 300 � 3 � 303.

Double and Halve Estimation

Double both numbers of the dividend to get “friendly” numbers and then estimate.

For example, 72 � 3.5. 72 doubled is about 140. 3.5 doubled is 7. The answer is

approximately 20.

Double and Halve Strategy

This can be used to simplify dividing. For example, 48 � 5 is the same as 96 � 10

(9.6).

“Think Multiplication” Estimation

For example, to divide 2088 by 7, think what number you multiply 7 by to get

approximately 2088. Seven times 300 is 2100.

Dividing by 10, 100, and 1000

Students learn when dividing by powers of 10 occurs, the place value of the last digit

of the dividend changes according to the divisor. For example, dividing tens by tens

gives units; hundreds by tens gives tens; thousand by tens gives hundreds, and thou-

sands by hundreds gives tens, and so on. They should also understand that they can

write an equivalent multiplication statement using the decimals. For example,

dividing by 10 is the same as multiplying by .

Dividing by 0.1, 0.01, and 0.001

Students should recognize that when dividing by powers of 10 with negative expo-

nents they can write an equivalent multiplication statement using powers of 10. For

example, dividing by 0.1 is the same as multiplying by 10.

Common Zeroes

You can factor out powers of ten from the dividend and divisor for an expression that is

easier to calculate. For example, 3600 � 120 is the same as 360 � 12. The answer is 30.

110

14

15

Never Divide by 5 Again!

Have students use the double and halve strategy to simplify all division by 5. For

example, 520 � 5 is the same as 1040 � 10. The answer is 104.

M e nt a l M at h G a m e s

Try these games with your students to enhance their Mental Mathematics skills!

Mental Math Bingo

Provide each student with a single 5 � 5 grid from Master 12 Mental Math BingoSheet. Direct students to randomly place the numbers you read in the 5 � 5 grid.

These numbers are answers to mental math questions. Make an overhead of the

questions you want to use from one of Masters 09–11 Mental Math BingoQuestions 1 to 3. Display the questions one at a time, randomly, for 5–10 seconds.

Students scan their grids for what they think is the correct answer. When they think

they have found the correct answer, students cross off the answer with an X and write

the question number in the box. The winner is the first person to get 5 correct

answers in a row crossed off, either vertically, horizontally, or diagonally.

Go Fish

This game is for 3 or 4 players and follows the general rules of FISH. Using blank

index cards, make at least 30 pairs of compatible number cards (e.g., one with 36, and

one with 64, or one with 3.6, the other with 6.4). You may wish to have students fish

for 1000, 100, 10, or even 1. The object of the game is to get rid of all your cards first.

Determine who will deal and who will go first. Deal six cards to each of the

players. The first player asks a selected player if he has a particular card that would

be a compatible number. For example, a player holding a 46 card would ask, “Do you

have 54?” If he does, he makes a pair that he places on the table. He will continue with

his turn until he is told to “Go Fish,” at which time he takes another card from the

pile of remaining cards. Play continues in a clockwise fashion. Once someone has

paired all his cards, the game is over, and the other players add up the numbers on

their cards. Their totals are recorded and added to previous totals. Once someone has

reached 500 (or 50, or 5) they are out of the game.

Loops

A “loop” is a fun way to practise mental math strategies. The questions are designed

so that any card can be the beginning card, and the last card “loops” back to the first

card. Teachers can design these loops for whatever skill needs practising.

Make a copy of Master 08 Loops Game Cards, and cut out the individual cards

with scissors. Cards are dealt out to small groups or to the whole class. One student

reads a card (e.g., “I have 45. Who has this multiplied by 16?”). The player with the

number 720 on a card reads the next card (e.g., “I have 720. Who has this divided by

80?”). Play continues until the last card is read, looping back to 45 (e.g., “I have 98.

Who has this minus 8 and divided by 2?”).

If choosing to play as a whole class activity, have students who have already

read their card(s) record the answers on paper for the rest of the questions. Their

challenge is to record before the answer is read out. This will ensure that they continue

Introduction • MHR xxxi

to practise the strategies throughout the game. You may wish to pair students up for

whole class loop games.

Some students, particularly visual learners, find applying mental strategies dif-

ficult when they cannot see the numbers. As each card is read, write the number on

the board. When designing a loop, start with a number and using the “I have, who

has” pattern, make as many cards as you require. The last card you make must have

the number on your first card as an answer.

You may also wish to use Master 13 Basic Fact Practice and Master 14Decimal Point Practice with students who need this type of reinforcement.

P RO B L E M S O LV I N G

Problem solving is an integral part of mathematics learning. The National Council

of Teachers of Mathematics recommends that problem solving should be the focus

of all aspects of mathematics teaching because it encompasses skills and functions,

which are an important part of everyday life.

Problem solving is, however, more than a vehicle for teaching and reinforcing math-

ematical knowledge and helping to meet everyday challenges. It is also a skill that can

enhance logical reasoning. Individuals can no longer function optimally in society by

just knowing the rules to follow to obtain a correct answer. They also need to be able

to decide through a process of logical deduction what algorithm, if any, a situation

requires, and sometimes need to be able to develop their own rules in a situation

where an algorithm cannot be directly applied. For these reasons problem solving

can be developed as a valuable skill in itself, a way of thinking, rather than just the

means to an end of finding the correct answer.

However, true problem solving involves much more than solving word or story

problems that accompany a new skill or concept in a textbook. True problem-solv-

ing tasks occur in a context where the solution path is not readily apparent; students

have to identify the problem, decide on the solution method, and then implement it.

The problem-based learning approach is the focus of this program. In

Mathematics 9: Focus on Understanding, a variety of problem solving opportunities

are provided for students:

• Each chapter begins with an investigation of a real-life problem. The Chapter

Problem is then revisited multiple times through engaging word problems in the

Check Your Understanding section.

• At the end of every two chapters, students are presented with a Task where the

solution path is not readily apparent and where solving the problem requires

more than just merely applying a familiar procedure. These cross-curricular

tasks require students to apply what they have learned in the two previous

NCTM Problem-Solving Standard

Instructional programs should enable all students to––

• Build new mathematical knowledge through problem solving

• Solve problems that arise in mathematics and in other contexts

• Apply and adapt a variety of appropriate strategies to solve problems

• Monitor and reflect on the process of mathematical problem solving

Solving problems is not only agoal of learning mathematicsbut also a major means ofdoing so. Students shouldhave frequent opportunities to formulate, grapple with,and solve complex problemsthat require a significantamount of effort and shouldthen be encouraged to reflecton their thinking

National Council of Teachers ofMathematics, 2000

xxxii MHR • Introduction

chapters to solve real-life, broad-based problems.

• In the Extend section of Check Your Understanding section and in the Extended

Response section at the end of every chapter, there are problems that challenge

higher levels of thinking and extend thinking beyond the curriculum.

T E C H N O LO G Y

Mathematics 9: Focus on Understanding uses specific technologies to engage students

in math inquiry, research, and problem solving.

The use of technology as an alternative method of carrying out the Discover

the Math activities provides students with hands-on experience in creating graphs

and constructing and manipulating geometric figures.

The main software program used in Mathematics 9: Focus on Understanding is

The Geometer’s Sketchpad® 4.

It is also suggested that E-STAT (Web-based url: estat.statcan.ca) be accessed to

gather real data for the Data Management chapter.

The use of technology ininstruction should further alter both the teaching andthe learning of mathematics.Computer software can beused effectively for classdemonstrations andindependently by students to explore additionalexamples, performindependent investigations,generate and summarize data as part of a project, orcomplete assignments.Calculators and computerswith appropriate softwaretransform the mathematicsclassroom into a laboratorymuch like the environment inmany science classes, wherestudents use technology toinvestigate, conjecture, andverify their findings. In thissetting, the teacher encouragesexperimentation and providesopportunities for students tosummarize ideas and establishconnections with previouslystudied topics.

Curriculum and Evaluation Standards

for School Mathematics, NCTM, 1989

Introduction • MHR xxxiii

xxxiv MHR • Introduction

CO R R E L AT I O N S

Introduction • MHR xxxv

Strand/Outcome Chapter/Section Pages Assessment

Number Concepts/Number and Relationship Operations

Specific Curriculum Outcome

A1 solve problems involving squareroot and principal square root

1.3 30–35 FormativeBLM 1.3 Assessment Question Rubric

SummativeBLM Chapter 1 Problem Wrap Up Rubric

A2 graph, and write in symbols andin words, the solution set for equa-tions and inequalities involving inte-gers and other real numbers

3.2, 3.3, 3.4 144–163 FormativeBLM 3.2 Assessment Question Rubric

BLM 3.3 Assessment Question Rubric

BLM 3.4 Assessment Question Rubric


A3 demonstrate an understanding ofthe meaning and uses of irrationalnumbers



A4 demonstrate an understanding ofthe interrelationships of subsets ofreal numbers



A5 compare and order real numbers 1.5 44–51 FormativeBLM 1.5 Assessment Question Rubric

SummativeBLM Chapter 1 Problem Wrap Rubric

A6 represent problem situationsusing matrices






B1 model, solve, and create problemsinvolving real numbers

1.1, 1.2 14–29 FormativeBLM 1.1 Assessment Question RubricBLM 1.2 Assessment Question Rubric


B2 add, subtract, multiply, and dividerational numbers in fractional anddecimal forms, using the most appro-priate method

1.2Task 1/2

22–29125

FormativeBLM 1.2 Assessment Question Rubric

SummativeBLM Chapter 1 Problem Wrap Up RubricBLM Chapter 2 Task Rubric




B3 apply the order of operations inrational number computations



B4 demonstrate an understandingof, and apply the exponent laws for,integral exponents



B5 model, solve, and create prob-lems involving numbers expressedin scientific notation



B6 determine the reasonableness ofresults in problem situations involv-ing square roots, rational numbers,and numbers written in scientificnotation

1.3, 1.5 30–35,44–59

FormativeBLM 1.3 Assessment Question RubricBLM 1.5 Assessment Question Rubric


B7 model, solve, and create prob-lems involving the matrix operationsof addition, subtraction, and scalarmultiplication



B8 add and subtract polynomialexpressions symbolically to solveproblems



B9 factor algebraic expressions withcommon monomial factors, con-cretely, pictorially, and symbolically

7.2, 7.3, 7.4, 7.6 332–351,357–361

FormativeBLM 7.2 Assessment Question RubricBLM 7.3 Assessment Question RubricBLM 7.4 Assessment Question RubricBLM 7.6 Assessment Question Rubric


B10 recognize that the dimensionsof a rectangular area model of apolynomial are its factors

7.2, 7.3, 7.4, 7.6 332–351,357–361



B11 find products of two monomi-als, a monomial and a polynomial,and two binomials, concretely, picto-rially, and symbolically

7.2, 7.3, 7.4, 7.6 332–351,357–361



xxxvi MHR • Introduction

Introduction • MHR xxxvii




B12 find quotients of polynomialswith monomial divisors



B13 evaluate polynomial expres-sions



B14 demonstrate an understandingof the applicability of commutative,associative, distributive, identity, andinverse properties to operationsinvolving algebraic expressions



B15 select and use appropriatestrategies in problem situations

throughout allchapters

14–423


Patterns and Relations


C1 represent patterns and relation-ships in a variety of formats and usethese representations to predict andjustify unknown values

2.1Task 1/2

72–79125



C2 interpret graphs that representlinear and non-linear data

2.2, UseTechnology

80–93 FormativeBLM 2.2 Assessment Question Rubric


C3 construct and analyse tables andgraphs to describe how changes inone quantity affect a related quan-tity



C4 determine the equations of linesby obtaining their slopes and y-intercepts from graphs, and sketchgraphs of equations using y-inter-cepts and slopes

2.4, UseTechnology



xxxviii MHR • Introduction




C5 explain the connections amongdifferent representations of patternsand relationships

2.1

8.2

72–79386–393



C6 solve single-variable equationsalgebraically, and verify the solutions



C7 solve first-degree single-variableinequalities algebraically, verify thesolutions, and display them on num-ber lines



C8 solve and create problemsinvolving linear equations andinequalities




Shape and Space (Measurement)


D1 solve indirect measurementproblems by connecting rates andslopes

2.3

8.1, 8.2



D2 solve measurement problemsinvolving conversion among SI units

5.1, 5.2, 5.3 226–251 FormativeBLM 5.1 Assessment Question RubricBLM 5.2 Assessment Question RubricBLM 5.3 Assessment Question Rubric


D3 relate the volumes of pyramidsand cones to the volumes of corre-sponding prisms and cylinders



D4 estimate, measure, and calculatedimensions, volumes, and surfaceareas of pyramids, cones, andspheres in problem situations

5.2, 5.3 236–251 FormativeBLM 5.2 Assessment Question RubricBLM 5.3 Assessment Question RubricSummativeBLM Chapter 5 Problem Wrap Up Rubric

D5 demonstrate an understandingof and apply proportions within sim-ilar triangles

6.3, UseTechnology



Introduction • MHR xxxix


Shape and Space (Geometry)


E1 investigate, and demonstrate anunderstanding of, the minimum suf-ficient conditions to product uniquetriangles



E2 investigate, and demonstrate anunderstanding of, the properties of,and the minimum sufficient conditionto, guarantee congruent triangles



E3 make informal deductions, usingcongruent triangle and angle prop-erties

6.2

Task5/6

268–273315



E4 demonstrate an understandingof and apply the properties of simi-lar triangles

6.3, UseTechnology



E5 relate congruence and similarityof triangles

6.3, UseTechnology



E6 use mapping notation to repre-sent transformations of geometricfigures, and interpret such notations

6.4

Task5/6

284–289

315



E7 analyse and represent combina-tions, using mapping notation

6.6

Task5/6

300–307

315



E8 investigate, determine, and applythe effects of transformations ofgeometric figures on congruence,similarity, and orientation

6.5Task 5/6

290–299315



xl MHR • Introduction


Data Management and Probability


F1 describe characteristics of possi-ble relationships shown in scatter-plots

8.1, UseTechnology



F2 sketch lines of best fit and deter-mine their equations

8.1, UseTechnology



F3 sketch curves of best fit for rela-tionships that appear to be non-lin-ear



F4 select, defend, and use the mostappropriate methods for displayingdata

8.3

Task 7/8

394–399417



F5 draw inferences and make pre-dictions based on data analysis anddata displays

8.4

Task 7/8

400–407

417



F6 demonstrate an understandingof the role of data management insociety

throughout chapter 8

372–407

F7 evaluate arguments and interpre-tations that are based on data analysis

8.4

Task 7/8

400–407417



Introduction • MHR xli




G1 make predictions of probabilitiesinvolving dependent and independ-ent events by designing and con-ducting experiments andsimulations

4.2, 4.3

Task 3/4

188–201

215



G2 determine theoretical probabili-ties of independent and dependentevents

4.2, 4.3

Task 3/4

188–201

215



G3 demonstrate an understandingof how experimental and theoreticalprobabilities are related



G4 recognize and explain why deci-sions based on probabilities may becombinations of theoretical calcula-tions, experimental results, and sub-jective judgments



Grade 6 Grade 7 Grade 8 Grade 9

Number Concepts

Number and Number Relationship OperationsGeneral Curriculum Outcome A:

A1 represent large num-bers in a variety of forms

A3 rewrite large numbersfrom standard form to sci-entific notation and viceversa

A6 represent any numberwritten in scientific nota-tion in standard form, andvice versa

A5 demonstrate andexplain the meaning ofnegative exponents forbase ten

A1 model and use power,base, and exponent torepresent repeated multi-plication

A1 model and link vari-ous representations ofsquare root of a number

A2 recognize perfectsquares between 1 and144 and apply patternsrelated to them

A3 distinguish betweenan exact square root andits decimal approximation

A4 find the square root ofany number, using anappropriate method

A1 solve problems involv-ing square root and prin-cipal square root

A2 graph, and write insymbols and in words, thesolution set for equationsand inequalities involvingintegers and other realnumbers

A3 demonstrate anunderstanding of themeaning and uses of irra-tional numbers

A4 demonstrate anunderstanding of theinterrelationships of sub-sets of real numbers

A5 compare and orderreal numbers

A6 represent problem sit-uations using matrices

G R A D E S 6 – 9 CO N T I N U U M

xlii MHR • Introduction

Introduction • MHR xliii


Number Concepts

Number and Number Relationship OperationsGeneral Curriculum Outcome A:

A2 represent fractionsand decimals

A9 compare and orderproper and improperfractions, mixed number,and decimal numbers

A3 write and interpretratios, comparing part-to-part and part-to-whole

A10 illustrate, explain,and express ratios, frac-tions, decimals, and per-cents in alternative forms

A4 demonstrate anunderstanding of equiva-lent ratios

A10 illustrate, explain,and express ratios, frac-tions, decimals, and per-cents in alternative forms

A9 solve proportionproblems that involveequivalent ratios andrates

A5 demonstrate anunderstanding of theconcept of percent as aratio

A11 demonstrate num-ber sense for percent

A8 represent and applyfractional percents, andpercents greater than100, in fraction or decimalform and vice versa

A6 demonstrate anunderstanding of themeaning of a negativeinteger

A12 represent integers(including zero) concretely,pictorially, symbolically,using a variety of models

A13 compare and orderintegers

A7 compare and orderintegers and positive andnegative rational num-bers (in decimal and frac-tional forms)

A7 read and write wholenumbers in a variety offorms

A2 rename numbersamong exponential, stan-dard, and expanded forms

A8 demonstrate anunderstanding of theplace value systemA9 relate fractional anddecimal forms of num-bers

A7 apply patterning inrenaming numbers fromfractions and mixed num-bers to decimal numbers

A8 rename single-digitand double-digit repeat-ing decimals to fractionsthrough the use of pat-terns and use these pat-terns to make predictions

A10 determine factorsand common factors

A4 solve and create prob-lems involving commonfactors and greatest com-mon factors

A5 solve and create prob-lems involving commonmultiples and least com-mon multiples

A11 distinguish betweenprime and compositenumbers

A6 develop and applydivisibility rules for 3, 4, 6,and 9

xliv MHR • Introduction


Number Concepts

Number and Number Relationship OperationsGeneral Curriculum Outcome B:

B1 compute products ofwhole numbers and deci-mals

B2 model and calculatethe products of two deci-mal numbers

B3 compute quotients ofwhole numbers and deci-mals

B4 model and calculatethe quotients of two deci-mals

B2 use mental mathstrategies for calculationsinvolving integers anddecimal numbers

B3 demonstrate anunderstanding of theproperties of operationswith decimal numbersand integers

B5 apply the order ofoperations for problemsinvolving whole and deci-mal numbers

B1 demonstrate anunderstanding of theproperties of operationswith integers and positiveand negative rationalnumbers (in decimal andfractional forms)

B1 model, solve, and cre-ate problems involvingreal numbers

B2 add, subtract, multiply,and divide real numbersin fractional and decimalforms, using the mostappropriate method

B3 apply the order ofoperations in rationalnumber computations

B5 add and subtract sim-ple fractions using mod-els

B6 estimate the sum ordifference of fractionswhen appropriate

B5 add and subtract frac-tions concretely, pictori-ally, and symbolically

B6 add and subtract frac-tions mentally, whenappropriate

B11 model solve, and cre-ate problems involvingfractions in meaningfulcontexts

B1 model, solve, and cre-ate problems involvingreal numbers


B7 multiply mentally afraction by a whole num-ber and vice versa

B7 multiply fractions con-cretely, pictorially, andsymbolically

B8 divide fractions con-cretely, pictorially, andsymbolically

B9 estimate and mentallycompute products andquotients involving frac-tions

B10 apply the order ofoperations to fractioncomputations, using bothpencil and paper and thecalculator


B6 demonstrate anunderstanding of thefunction nature of input-output situations

Introduction • MHR xlv


Number Concepts


B16 create and evaluatesimple variable expres-sions by recognizing thatthe four operations applyin the same way as theydo for numerical expres-sions

B17 distinguish betweenlike and unlike terms

B18 add and subtract liketerms by recognizing theparallel with numericalsituations, using concreteand pictorial models

B14 add and subtractalgebraic terms con-cretely, pictorially, andsymbolically to solve sim-ple algebraic problems

B15 explore addition andsubtraction of polynomialexpressions, concretelyand pictorially

B8 add and subtract poly-nomial expressions sym-bolically to solveproblems

B13 evaluate polynomialexpressions

B16 demonstrate anunderstanding of multi-plication of a polynomialby a scalar, concretely, pic-torially, and symbolically

B9 factor algebraicexpressions with com-mon monomial factors,concretely, pictorially, andsymbolically

B10 recognize that thedimensions of a rectangu-lar area model of a poly-nomial are its factors

B11 find products of twomonomials, a monomialand a polynomial, andtwo binomials, concretely,pictorially, and symboli-cally

B12 find quotients ofpolynomials with mono-mial divisors

B14 demonstrate anunderstanding of theapplicability of commuta-tive, associative, distribu-tive, identity, and inverseproperties to operationsinvolving algebraicexpressions

xlvi MHR • Introduction


Number Concepts


B7 solve and create rele-vant addition, subtrac-tion, multiplication, anddivision problems involv-ing whole numbers

B11 add and subtractintegers concretely, picto-rially, and symbolically tosolve problems

B12 multiply integersconcretely, pictorially, andsymbolically to solveproblems

B13 divide integers con-cretely, pictorially, andsymbolically to solveproblems

B14 solve and pose prob-lems which utilize addi-tion, subtraction,multiplication, and divi-sion of integers

B15 apply the order ofoperations to integers

B4 demonstrate anunderstanding of, andapply the exponent lawsfor, integral exponents

B5 model, solve, and cre-ate problems involvingnumbers expressed in sci-entific notation

B6 determine the reason-ableness of results inproblem solving situa-tions involving squareroots, rational numbers,and numbers written inscientific notation

B7 model, solve, and cre-ate problems involvingthe matrix operations ofaddition, subtraction, andscalar multiplication

B8 solve and create rele-vant addition, subtrac-tion, multiplication, anddivision problems involv-ing decimals

B13 solve and createproblems involving addi-tion, subtraction, multipli-cation, and division ofpositive and negativedecimal numbers

B12 add, subtract, multi-ply and divide positiveand negative decimalnumbers with and with-out the calculator

Introduction • MHR xlvii


Number Concepts


B8 estimate and deter-mine percent when giventhe part and the whole

B9 estimate and deter-mine the percent of anumber

B10 create and solveproblems that involve theuse of a percent

B2 solve problems involv-ing proportions, using avariety of methods

B3 create and solve prob-lems which involvingfinding a, b, or c in therelationship a% of b = c,using estimation and cal-culation

B4 apply percentageincrease and decrease inproblem situations

B9 estimate products andquotients involving wholenumbers only, wholenumbers and decimals,and decimals only

B1 use estimation strate-gies to assess and justifythe reasonableness of cal-culation results for inte-gers and decimalnumbers

B10 divide numbers by0.1, 0.01, and 0,001 men-tally

B2 use mental mathstrategies for calculationsinvolving integers anddecimal numbers

B11 calculate sums anddifferences in relevantcontexts by using themost appropriate method

B4 determine and use themost appropriate compu-tational method in prob-lem situations involvingwhole numbers and/ordecimals

B12 calculate productsand quotients in relevantcontexts by using themost appropriate method

B4 determine and use themost appropriate compu-tational method in prob-lem situations involvingwhole numbers and/ordecimals

B15 select and use appro-priate strategies in prob-lem situations

xlviii MHR • Introduction



General Curriculum Outcome C:

C1 solve problems involv-ing patterns

C1 describe a pattern,using written and spokenlanguage and tables andgraphs

C2 summarize simple pat-terns, using constants,variables, algebraicexpressions, and usethem in making predic-tions

C1 represent patternsand relationships in avariety of formats and usethese representations topredict unknown values

C1 represent patternsand relationships in avariety of formats and usethese representations topredict and justifyunknown values

C5 explain the connec-tions among differentrepresentations of pat-terns and relationships

C2 use patterns toexplore division by 0.1,0.01, and 0.001

C2 interpret graphs thatrepresent linear and non-linear data

C2 interpret graphs thatrepresent linear and non-linear data

C3 recognize and explainhow changes in base orheight, affect areas of rec-tangles, parallelograms, ortrianglesC4 recognize and explainhow changes in height,depth or length affectvolumes of rectangularprismsC5 recognize and explainhow a change in oneterm of a ratio affects theother term

C9 construct and analysegraphs to show howchange in one quantityaffects a related quantity

C3 construct and analysetables and graphs todescribe how change inone quantity affects arelated quantity

C3 construct and analysetables and graphs todescribe how changes inone quantity affect arelated quantityC4 determine the equa-tions of lines by obtainingtheir slopes and y-inter-cepts from graphs, andsketch graphs of equa-tions using y-interceptsand slopes

C6 recognize equivalentratios using tables andgraphsC7 represent square andtriangular numbers con-cretely, pictorially andsymbolically

Introduction • MHR xlix



General Curriculum Outcome C:

C8 solve simple linearequations using openframes

C4 solve one- and two-step single-variable linearequations, using system-atic trial

C5 illustrate the solutionfor one- and two-step lin-ear equations, using con-crete materials anddiagrams

C6 graph linear equa-tions, using a table of val-ues

C7 interpolate andextrapolate number val-ues from a given graph

C8 determine if anordered pair is a solutionto a linear equation

C4 link visual characteris-tics of slope with itsnumerical value by com-paring vertical changewith horizontal change

C5 solve problems involv-ing the intersection oftwo lines on a graph

C6 solve and verify simplelinear equations alge-braically

C7 create and solve prob-lems, using linear equa-tions

C6 solve single-variableequations algebraically,and verify the solutions

C7 solve first-degree sin-gle-variable inequalitiesalgebraically, verify thesolutions, and displaythem on their numberlines

C8 solve and create prob-lems involving linearequations and inequali-ties

C9 demonstrate anunderstanding of the useof letters to replace openframes

C3 explain the differencebetween algebraicexpressions and algebraicequations


Shape and Space

General Curriculum Outcome D:

D1 use the relationshipamong particular SI unitsto compare objects

D1 identify, use, and con-vert among the SI units tomeasure, estimate, andsolve problems that relateto length, area, volume,and capacity

D2 solve measurementproblems, using appropri-ate SI units

D1 solving indirect meas-urement problems by con-necting rates and slopes

D2 solve measurementproblems involving con-version among SI units

D2 describe mass meas-urement in tonnesD3 demonstrate anunderstanding of therelationship betweencapacity and volume

D1 identify, use, and con-vert among the SI units tomeasure, estimate, andsolve problems that relateto length, area, volumeand capacity

D3 relate the volumes ofpyramids and cones tothe volumes of corre-sponding prisms andcylinders

D4 estimate, measure,and calculate dimensions,volumes, and surfaceareas of pyramids, cones,and spheres in problemsituations

l MHR • Introduction


Shape and Space


D3 develop and use rateas a tool for solving indi-rect measurement prob-lems in a variety ofcontextsD4 construct and analysegraphs to show change inone quantity affects arelated quantityD5 demonstrate anunderstanding of therelationships amongdiameter, radii, and cir-cumference of circles, anduse the relationships tosolve problems

D4 estimate and measureangles using a protractorD5 draw angles of a givensize

D5 demonstrate anunderstanding of andapply proportions withinsimilar triangles

D6 solve measurementproblems involvinglength, capacity, area, vol-ume, mass and time

D2 apply concepts andskills related to time inproblem situations

D1 solve indirect meas-urement problems, usingproportions

D3 estimate area of cir-cles

D4 develop and use theformula for the area of acircle

D7 demonstrate anunderstanding of therelationships among thebases, height, and area ofparallelograms

D5 describe patterns andgeneralize the relation-ships between areas andperimeters of quadrilater-als, and areas and circum-ferences of circles

D8 demonstrate anunderstanding of the rela-tionship between the areaof a triangle and the areaof a related parallelogram

D6 calculate the areas ofcomposite figures

Introduction • MHR li


Shape and Space


D9 demonstrate anunderstanding of therelationships between thethree dimensions of rec-tangular prisms and vol-ume and surface area

D7 estimate and calculatevolumes and surfaceareas of right prisms andcylinders

D8 measure and calculatevolumes and surface areaof composite 3-D shapesD9 demonstrate anunderstanding of thePythagorean relationship,using models


Shape and Space

General Curriculum Outcome E:

E1 describe and representthe various cross-sectionsof cone, cylinders, pyra-mids, and prismsE2 make and interpretorthographic drawings of3-D shapes made withcubes

E1 demonstrate whethera set of orthographicviews, a mat plan, and anisometric drawing canrepresent more than one3-D shape

E2 examine and drawrepresentations of 3-Dshapes to determinewhat is necessary to pro-duce unique shapes

E1 decide and justifywhich combinations oftriangle classifications arepossible, through con-struction using materialsand/or technology

E1 investigate, anddemonstrate an under-standing of, the minimumsufficient conditions toproduce unique triangles

E2 investigate, anddemonstrate an under-standing of, the proper-ties of, and the minimumsufficient conditions to,guarantee congruent tri-angles

lii MHR • Introduction


Shape and Space


E3 make and apply gen-eralizations about thesum of the angles in trian-gles and quadrilaterals

E2 determine and userelationships betweenangle measures and sidelengths in triangles

E7 explain, using a model,why the sum of the meas-ures of the angles of a tri-angle is 180°.

E3 make informal deduc-tions, using congruent tri-angle and angleproperties

E4 demonstrate anunderstanding of andapply the properties ofsimilar triangles

E5 relate congruence andsimilarity of triangles

E4 apply angle pair rela-tionships to find missingangle measures

E6 apply angle relation-ships to find angle meas-ures

E4 make and apply gen-eralizations about thediagonal properties oftrapezoids, kites, parallelo-grams and rhombi

E4 analyse polygons todetermine their proper-ties and interrelationships

E3 construct angle bisec-tors and perpendicularbisectors, using a varietyof methodsE5 identify, construct,classify, and use anglepair relationships pertain-ing to parallel lines andnon-parallel lines andtheir transversals

E5 sort the members ofthe quadrilateral “family”under proper headings

E6 recognize, name,describe and representsimilar figures

E8 sketch and build 3-Dobjects, using a variety ofmaterials and informationabout the objects

Introduction • MHR liii


Shape and Space


E7 make generalizationsabout the planes of sym-metry of 3-D shapesE8 make generalizationsabout the rotational sym-metry property of allmembers of the quadri-lateral “family” and of reg-ular polygons

E9 draw, describe, andapply translations, reflec-tions, and rotations, andtheir combinations, andidentify and use the prop-erties associated withthese transformations

E9 recognize and repre-sent dilatation images of2-D figures and connectto similar figures

E5 represent, analyse,describe, and apply dilata-tions

E10 predict and repre-sent the result of combin-ing transformations

E10 create and describedesigns using translation,rotation, and reflection

E3 draw, describe, andapply transformations of3-D shapes

E6 use mapping notationto represent transforma-tions of geometric fig-ures, and interpret suchnotations

E7 analyse and representcombinations of transfor-mations, using mappingnotation

E8 investigate, determine,and apply the effects oftransformations of geo-metric figures on congru-ence, similarity, andorientation



General Curriculum Outcome F:

F1 choose and evaluateappropriate samples fordata collection

F2 formulate questionsfor investigation from rel-evant contexts

F3 select, defend, and useappropriate data collec-tion methods and evalu-ate issues to beconsidered when collect-ing data

F1 demonstrate anunderstanding of the vari-ability of repeated sam-ples of the samepopulation

F8 develop and conductstatistics projects to solveproblems

F2 identify various typesof data sources

F1 communicate throughexample the distinctionbetween biassed andunbiassed sampling, andfirst- and second-handdata

liv MHR • Introduction



General Curriculum Outcome F:

F3 plot coordinates infour quadrants

F2 develop and apply theconcept of randomness

F4 use bar graphs, doublebar graphs and stem-and-leaf plots to display data

F4 construct a histogram

F5 construct appropriatedata displays, groupingdata where appropriateand taking into consider-ation the nature of thedata

F5 construct and inter-pret box-and-whiskerplots

F5 use circle graphs torepresent proportions

F3 construct and inter-pret circle graphs

F6 interpret data repre-sented in scatter plots

F4 construct and inter-pret scatter plots anddetermine a line of bestfit by inspection

F6 extrapolate and inter-polate information fromgraphs

F1 describe characteris-tics of possible relation-ships shown in scatterplots

F2 sketch lines of best fitand determine theirequations

F3 sketch curves of bestfit for relationships thatappear to be non-linear

F7 make inferences fromdata displays

F6 read and make infer-ences for grouped andungrouped data displays

F7 formulate statisticsprojects to explore cur-rent issues from withinmathematics, other sub-ject areas, or the world ofstudents

F9 draw inferences andmake predictions basedon the variability of datasets, using range and theexamination of outliers,gaps, and clusters

F9 evaluate data interpre-tations that are based ongraphs and tables

F4 select, defend, and usethe most appropriatemethods for displayingdata

F5 draw inferences andmake predictions basedon data analysis and datadisplays

F7 evaluate argumentsand interpretations thatare based on data analy-sis

F8 demonstrate anunderstanding of the dif-ference between mean,median, and mode

F8 determine measuresof central tendency andhow they are affected bydata presentations andfluctuations

F7 determine the effectof variations in data onthe mean, median, andmode

F9 explore relevant issuesfor which data collectionassists in reaching conclu-sions

F6 demonstrate anunderstanding of the roleof data management insociety

Introduction • MHR lv



General Curriculum Outcome G:

G1 conduct simple simu-lations to determineprobabilities

G2 solve probabilityproblems, using simula-tions and by conductingexperiments

G1 conduct experimentsand simulations to findprobabilities of single andcomplimentary events

G1 make predictions ofprobabilities involvingdependent and inde-pendent events bydesigning and conduct-ing experiments and sim-ulations

G2 evaluate the reliabilityof sampling resultsG3 analyse simple proba-bilistic claims

G5 compare experimen-tal results with theoreticalresults

G3 compare experimen-tal and theoretical proba-bilities

G3 demonstrate anunderstanding of howexperimental and theo-retical probabilities arerelated

G4 determine theoreticalprobabilities

G3 identify all possibleoutcomes of two inde-pendent events, usingtree diagrams and areamodels

G6 use fractions, deci-mals, and percents asnumerical expressions todescribe probability

G2 determine theoreticalprobabilities of single andcomplimentary events

G2 determine theoreticalprobabilities of independ-ent and dependentevents

G5 identify events thatmight be associated witha particular theoreticalprobability

G1 identify situations forwhich the probability

would be near 0, 1–4

, 1–2

, 3–4

and 1

G4 demonstrate anunderstanding of howdata is used to establishbroad probability pat-terns

G4 recognize and explainwhy decisions based onprobabilities may be com-binations of theoreticalcalculations, experimentalresults, and subjectivearguments

G4 create and solve prob-lems, using the numericaldefinition of probability

M A N I P U L AT I V E S , M AT E R I A L S , A N D T E C H N O LO G Y TO O L S

Manipulatives/Materials

Used inChapter/Section

Available fromMcGraw-Hill Ryerson

ISBNSuggestedQuantity

Algebra tiles Ch 3 Get Ready, 3.1,Ch 3 Review, Ch 7

Balls (tennis ball,racquetball,basketball)

8.1

Balloons Ch 4 MakingConnections, 8.3

Calculators,scientific preferably

Get Ready for Grade9, Ch 1, Ch 2 GetReady, Ch 4 MakingConnections, Ch 5,6.3, 7.6, Ch 8 GetReady, 8.1, 8.2, 8.4,Ch 8 Review, Ch 8Practice Test

Calculators,graphing

2.4, 8.1

Cards, playing Ch 4 Get Ready

Card stock 5.2Class set ofenvelopescontaining 3 sidelengths and 3 anglemeasures

6.1

Compasses 1.1, 5.2, 6.4, 6.6, Ch 6Review, Ch 6Practice Test

Student SAFE-Tcompass, durable,plastic, draws circlesfrom 1 cm to 25 cmdiameter, in 5 mmincrements

0-322-07104-6

Computers 2.2, 2.4, 6.1*, 6.3, 6.4*,6.5*, 6.6*

Construction paper,white

8.1

Counters Ch 2 Get Ready, Ch2 Chapter Problem,4.1

Two-colour counters:red on one side,white on other

0-322-05539-3: setof 2000-322-05540-7:bucket of 400

one class set per two classes

Cubes, centimetre,linking

2.1 Centicubes(interlocking)

0-322-06776-6: setof 1000-322-06777-4: setof 5000-322-06778-2: setof 1000


Fraction strips Get Ready for Grade9

lvi MHR • Introduction

* = optional

Introduction • MHR lvii

* = optional





Hand lenses 8.3

Ink pads 8.3

Markers 5.2, 8.1

Markers, non-per-manent

2.2

Masking tape 2.3

Nets (of cube, rec-tangular prism, tri-angular prism,cylinder)

Ch 5 Get Ready

Number cubes Ch 4 Get Ready,Ch 4 Task

Number cubes,six sided, wooden

1-56107-704-6: setof 12

Number line, lami-nated

1.1, Ch 3 Get Ready

Oranges 5.2

Paper 6.1

Paper, centimetregrid

Get Ready for Grade9

See Generic Masters

Paper, grid Ch 2, Ch 6 GetReady, 6.4, 6.5, 6.6,Ch 6 Review,Ch 6 Practice Test,Ch 6 Task, 8.1, 8.2,Ch 8 Review,Ch 8 Practice Test

See Generic Masters

Paper, isometric dot Get Ready for Grade9

See Generic Masters

Paper, square dot Ch 6 Get Ready

Paper, tracing 6.2,Ch 6 MakingConnections,6.6

Paper, waxed 5.2

Paper bags 4.1, 4.2

Paper clips 4.4

Pattern blocks Ch 2 Get Ready,Ch 6 MakingConnections

Pattern blocks: wood or plastic, 1 cm thick

0-322-05566-0: plas-tic, bucket of 2500-322-05567-9:wood, bucket of 250


Pencil crayons 4.2*, 4.3*, 4.4, Ch 5/6Task

Pencils Get Ready for Grade9, 4.4, 6.1

Pipe cleaners 6.1*

lviii MHR • Introduction





Protractors 4.4, 5.2, 6.1, 6.2, 6.3,Ch 6 Review, Ch 6Practice Test

Protractoropen centre, raisedgraduations measure up to 180° angles and15 cm lines

0-322-06816-9: setof 10

Pyramid figures Ch 8 Get Ready

Relational geometric solids

Ch 5 Get Ready, 5.1

Rice (or sand orupopped popcorn)

5.1

Rulers Get Ready for Grade9, 1.1, 2.3, Ch 2MakingConnections, 4.4, Ch4 MakingConnections, 5.1,5.2, 6.2, 6.3, 6.4, 6.6

Ruler, 30 cm, clear,plastic, measure incentimetres andinches

0-322-07126-7

Scissors Get Ready for Grade9, 5.2

Software, TheGeometer’sSketchpad®

2.2, 2.4, 6.1*, 6.3, 6.4*,6.5*, 6.6*

String Ch 4 MakingConnections, 5.2, 8.1

Tape Get Ready for Grade9, 5.2, 8.1

Tape, measuring,metric, two-sided

2.3, 5.1, 8.1

Tiles, colour Ch 4 Get Ready, 4.1,4.2, Ch 4 Review

Colour tiles 2.5 cm square plastictiles; 100 each of red,yellow, blue, greenOverhead colour tiles

0-322-06873-8:bucket of 2800-322-06768-5:bucket of 4000-322-06769-3: setof 48


Tissues 8.3

* = optional

A S S E S S M E N T

The primary purpose of assessment is to improve student learning. Assessment data

helps teachers determine the instructional needs of students throughout the learn-

ing process. Some assessment data is used for the evaluation of students for the pur-

pose of reporting.

Assessment must be purposeful and inclusive for all students. It should be

appropriately varied to reflect learning styles of students and be clearly communi-

cated with students and parents. Assessment can be used diagnostically to determine

prior knowledge, formatively to inform instructional planning, and in a summative

manner to determine how well the students have achieved the expectations at the

end of a learning cycle.

D i a g n o s t i c A s s e s s m e nt

Assessment for diagnostic purposes can determine where individual students will

need support and will help to determine where the classroom time needs to be spent.

Mathematics 9: Focus on Understanding provides the teacher with diagnostic

support at the start of the text and the beginning of every chapter.

• The Get Ready for Grade 9 section at the beginning of the student text (pp. 2–7)

provides a Diagnostic Tool for teachers to assess student readiness for grade 9.

The Teacher’s Resource has checklists to be used with this tool.

• Get Ready reviews at the beginning of each chapter provide coaching on

essential concepts and skills needed for the upcoming chapter.

• For students needing support beyond the Get Ready, additional blackline

masters are provided in this Teacher’s Resource that both develop conceptual

understanding and improve procedural efficiency.

Diagnostic support is also provided at the start of every section.

• Each section begins with an introduction to facilitate open discussion in the

classroom.

• Each Discover the Math activity starts with a question that stimulates prior

knowledge and allows teachers to monitor students’ readiness.

Fo r m at i ve A s s e s s m e nt

Formative assessment tools are provided throughout the text and Teacher’s Resource.

Formative assessment allows teachers to determine students’ strengths and weaknesses

and guide their class towards improvement within lessons and chapters. Mathematics 9:

Focus on Understanding provides BLMs for student use that complement the text in areas

where formative assessment indicates that students need support.

The chapter opener, visual, and the introduction to the Chapter Problem at the

beginning of each chapter provide opportunities for teachers to do a rough forma-

tive assessment of student awareness of the chapter content.

Within each lesson

• Reflect questions allow the teacher to determine if the student has developed the

conceptual understanding and/or skills that were the goal of the Discover the Math.

• Communicate the Key Ideas offers teachers an opportunity to determine

students’ understanding of concepts through conversations and written work.

Introduction • MHR lix

• Check Your Understanding allows teachers to monitor students’ procedural

skills, their application of procedures, their ability to communicate their

understanding of concepts, and their ability to solve problems relating to the

Communicate the Key Ideas section.

• Assessment questions, with accompanying rubrics, target the key ideas of

the section. These questions have been designed so that the key concepts of a

lesson may be assessed. Each question has several parts of differing levels of

difficulty so all students will have success with at least some parts of the

question.

• Chapter Problem revisits provide opportunities to verify that students are

developing the skills and understanding they need to complete the Chapter

Problem Wrap-Up.

• Extend questions are aimed at Level 3 and 4 performances as indicated in the

rubrics.

• Journal opportunities allow teachers insight into students’ thinking at key

locations.

• Specific Problem Solving strategies are embedded in appropriate sections

throughout the book, allowing formative assessment of students’ ability to solve

problems.

• Chapter Reviews and Cumulative Reviews provide an opportunity to assess

Knowledge/Understanding, Application, Communication, and Problem Solving

S u m m at i ve A s s e s s m e nt

Summative data is used for both planning and evaluation.

• A Practice Test in each chapter assess students’ achievement of the expectations

in the areas of Reasoning, Connecting, Communication, and Problem Solving.

• The Chapter Problem provides a problem solving opportunity using an open-

ended question format that is revisited several times in the chapter. This

assessment can be used to evaluate students’ understanding of the expectations

under the categories of Reasoning, Connecting, Communication, and Problem

Solving.

• Making Connections activities provide rich summative opportunities that

involve connections to some other strands and subject areas.

• Tasks are open-ended investigations with rubrics and exemplars. Most cover at

least two strands.

lx MHR • Introduction

Po r t fo l i o A s s e s s m e nt

Student-selected portfolios provide a powerful platform for assessing students’

mathematicalthinking. Portfolios:

• help teachers assess students’ growth and mathematical understanding

• provide insight into students’ self-awareness about their own progress; and

• help parents understand their child’s growth.

Mathematics 9: Focus on Understanding has many components that provide ideal

portfolio items. Inclusion of all or any of these chapter items provides insight into a

student’s progress in a non-threatening, formative manner. These items include:

• student responses to the Chapter Opener;

• student responses to Chapter Problem Wrap-Up assignments;

• answers to Reflect questions, which provide early opportunities for students to

construct knowledge about the section content;

• answers to Communicate the Key Ideas questions, which allow students to

explore their initial understanding of concepts;

• answers to Assessment questions, which are designed to show student

achievement in each section of the text

• solutions to Chapter Problems, which provide helpful scaffolding for students

who need additional direction;

• Journal responses, which show student understanding of the chapter skills and

process; and

• Task assignments, which show student understanding across several chapters

and strands.

Introduction • MHR lxi

A s s e s s m e nt M a s te r s

As well as the Assessment question rubrics, Chapter Problem Wrap-Up rubrics,

and Task rubrics provided with the chapter-specific BLMs, the Focus on

Understanding program has a wide variety of generic assessment BLMs. These BLMs

will allow you to effectively monitor student progress and evaluate instructional

needs.

Generic Assessment BLM Purpose

Assessment Master 1Assessment Recording Sheet

This generic chart can be used to organize your comments forassessmentof student observations, journals, portfolios, andpresentations.

Assessment Master 2Attitudes Assessment Checklist

This checklist will allow you to assess a student’s attitude ashe/she works on a task.

Assessment Master 3Portfolio Checklist

This checklist will allow you to assess students’ portfolios.

Assessment Master 4Presentation Checklist

This checklist will allow you assess students’ oral and writtenpresentations.

Assessment Master 5Problem Solving Checklist

This checklist will allow you to assess students’ problem-solvingskills.

Assessment Master 6Journal Assessment Rubric

This rubric will allow you to evaluate students’ journal entries.

Assessment Master 7Group Work Assessment Recording Sheet

This sheet will allow you to record comments as students workon group tasks.

Assessment Master 8Group Work Assessment GeneralScoring Rubric

This rubric will allow you to assess students’ group-related work.

Assessment Master 9How I Work

This sheet will allow students to self-assess their ownindependent and group work.

Assessment Master 10Self-Assessment Recording Sheet

This sheet will allow students to self-assess their understandingof chapter material.

Assessment Master 11Self-Assessment Checklist

This checklist will allow students to self-assess theirunderstanding of chapter material.

Assessment Master 12My Progress as a Mathematician

This checklist will allow students to self-assess theirunderstanding of mathematics, in general.

Assessment Master 13Teamwork Assessment

This worksheet will allow students to evaluate their work as partof a team.

Assessment Master 14My Progress as a Problem Solver

This checklist will allow students to self-assess their own ability atsolving problems.

Assessment Master 15Assessing Work in Progress

This sheet will allow student groups to assess their progress asthey work to complete a task.

Assessment Master 16Learning Skills Checklist

This checklist will allow you to assess a student’s work habits andlearning skills.

lxii MHR • Introduction

A D A P TAT I O N S

Mathematics 9: Focus on Understanding considers a broad range of needs and learn-

ing styles, including those of students requiring adaptations, students with limited

proficiency in English, and gifted learners.

• Excellent visuals and multiple representations of concepts and instructions

support visual learners, ESL students, and struggling readers.

• Literacy Connection boxes and key terms bolded, highlighted, and defined in

the margin support struggling readers and promote mathematics literacy for all

learners.

• Relevant contexts including multi-cultural examples that engage students and

provide a purpose for the mathematics being learned.

• Extend questions and math games provide additional challenges for giftedlearners.

• Making Connections activities provide additional opportunities for hands-on

and minds-on learning.

This Teacher’s Resource provides support in addressing multiple intelligences and

learning styles, though additional activities, Student Success Masters, Adaptation

suggestions, ESL support, and Interventions strategies.

R e a c h i n g Al l St u d e nt s

Students may experience difficulty meeting provincial standards for a variety of rea-

sons. General cognitive delays, social-emotional issues, behavioural difficulties,

health-related factors, and extended or sporadic absences from instruction underlie

the math difficulties experienced by some students. However, these factors do not

explain the challenges other students encounter. For these students, math difficulties

are usually related to three key areas.

Three Key Areas Underlying Math Difficulties

Language

Students with language learning difficulties demonstrate difficulty reading and

understanding math vocabulary and math story problems, and determining saliency

(e.g., picking out the most important details from irrelevant information).

Processing information that is presented using oral or written language is often dif-

ficult for these students, who may be more efficient learners when information is

presented in a non-verbal, visual format. Diagrams and pictorial representations of

math concepts are usually more meaningful to these students than lengthy verbal or

written descriptions.

Visual/Perceptual/Spatial/Motor

Some students demonstrate difficulties understanding and processing information

that is presented visually and in a non-verbal format. Language support to supple-

ment and make sense of visually presented information is often beneficial (e.g., ver-

bal explanation of a visual chart). Visual, perceptual, spatial, and motor difficulties

may be evident in students’ written output, as well as in their ability to process visu-

ally inputted information. Difficulties with near and far point copying, accurately

aligning numbers in columns, properly sequencing numbers, and illegible hand

Introduction • MHR lxiii

writing are examples of output difficulties in this area.

Memory (Short Term Memory, Working Memory, and Long Term Memory)

Students with short term memory difficulties find it hard to remember what they

have just heard or seen (e.g., auditory short term memory, visual short term mem-

ory). A weak working or active memory makes it difficult for students to hold infor-

mation in their short term memory and manipulate it (e.g., hold what they have just

heard and then perform a mathematical operation with that information). For oth-

ers, the retrieval of information from long term memory (e.g., remembering num-

ber facts and previously taught formulae) is difficult. Students with long term

memory difficulties may also have difficulty storing information in their long term

memory, as well as retrieving it.

I n d i v i d u a l Pro g ra m P l a n s ( I P P ) a n d Ad a p t at i o n s

An Individual Program Plan (IPP) is to be developed and implemented for all stu-

dents for whom the provincial outcomes are not applicable or attainable. The IPP is

a program that is curriculum-based but focuses on the student’s strengths and needs.

Developing an IPP is a well-defined process involving the principal, teachers, parent,

and student. Addressing a student’s need for an IPP falls outside the scope of this

Teacher’s Resource.

Adaptations

Adaptations do not change the provincial outcomes. Rather, an adaptation to a stu-

dent’s program alters the “how,” “when,” or “where” the student is taught or assessed

without changing curriculum expectations.

This Teacher’s Resource provides suggested adaptations based on the student’s

identified area of difficulty, and groups these accommodations under the following

three classifications:

• Presentation/Instructional adaptations refer to changes in teaching strategies

that allow the student to access the curriculum.

• Organizational/Environmental adaptations refer to changes that are required

to the classroom and/or school environment.

• Assessment adaptations refer to changes that are required in order for the

student to demonstrate learning.

The chart outlines the differences between adaptations and an IPP.

Some students may require a combination of adaptations and an IPP.

lxiv MHR • Introduction

Suggested Math Adaptations

The following three charts provide adaptations for the three key areas underlying

math difficulties.

Chart I provides adaptations for students with language difficulties.

Chart II provides adaptations for students with visual, perceptual, spatial, and/or

motor difficulties.

Chart III provides adaptations for students with memory difficulties.

Adaptations have been grouped under the headings of presentation/instructional,

organizational/environmental, and assessment.

Chart I Adaptations for Students with Language Difficulties

Introduction • MHR lxv

Adaptations IPP

There are no changes to public school outcomes.

The teaching strategies are developed in one or moreof the following areas:• presentation• assessment/evaluation• motivation• environment• class organization• resources

Adaptations are not noted on the student’s reportcard and/or transcript.

Adaptations are documented in the student’scumulative record card.

An IPP may involve any or all of the following:• implementing the same general curriculum

outcomes but at a significantly different outcomelevel than expected for the grade level

• deleting specific curriculum outcomes when thedeleted outcomes are necessary to develop anunderstanding of the general curriculum outcome

• where needed, providing programming foroutcomes that are not part of Nova Scotia’s publicschool program (e.g., behaviour programming, lifeskills)

• adding new outcomes where students requireenrichment

An IPP is indicated on the student’s report cardand/or transcript. A copy of the IPP is filed in thestudent’s cumulative record file.

Presentation/Instructional Organizational/Environmental Assessment

• pre-teach vocabulary

• give concise, step-by-step directions

• teach students to look for cluewords, highlight these words

• use visual models

• use visual representations toaccompany word problems

• encourage students to look for common patterns in word problems

• have students make useof a math journal

• provide reference charts withoperations and formulaestated simply

• post reference charts with mathvocabulary

• reinforce learning with visual aids and manipulatives

• using a visual format, poststrategies for problem solving

• use a peer tutor, buddy system,or pair reading

• read instructions/wordproblemsto student on tests

• extend time lines

• offer choice of assessmentformats (e.g. portfolios,individual contracts)

Chart II Adaptations for Students with Visual /Perceptual /Spatial /Motor Difficulties

Chart III Adaptations for Students with Memory Difficulties

Adaptations for Enrichment

Some students benefit from having their programs enriched by extending their learn-

ing and emphasizing higher-order thinking skills. For the purposes of this resource

manual, the term “enrichment” will be applied to activities that enrich and extend a

student’s program. Enrichment may also take the form of adding new outcomes to a

student’s IPP. The program planning team for an IPP should include the principal,

vice-principal, teachers and parents, and may include the student. Adapting a stu-

dent’s program for enrichment falls beyond the scope of this Teacher’s Resource.

lxvi MHR • Introduction


• reduce copying

• provide worksheets

• provide graph paper

• provide concrete examples

• allow use of number lines

• provide a math journal

• encourage and teach self-talk strategies

• chunk learning and tasks

• reduce visual bombardment

• provide a work carrel or workarea that is not visuallydistracting

• allow rest periods and breaks

• provide various print formats(e.g., large print, high contrast,braille)

• provide graph paper for tests


• provide consumable tests

• reduce the number ofquestions required toindicate competency

• provide a scribe whenlengthy written answers arerequired


• regularly review concepts

• activate prior knowledge

• teach mnemonic strategies(e.g., BEDMAS)

• teach visualization strategies

• provide a math journal

• allow use of multiplication tables

• colour-code steps in sequence

• teach functional math conceptsrelated to daily living

• make available reference chartswith commonly used facts,formulae, and steps for problem-solving

• allow use of a calculator

• use games and computerprograms for drill and repetition

• allow to use multiplicationcharts

• allow to use other referencecharts as appropriate

• allow to use calculators


• present one concept/typeof question at a time

Adaptations for Enrichment

Adaptations for ESL Students

For ESL students, language issues are pervasive throughout all subject areas, includ-

ing math. Non-math words are often more problematic for ESL students because

understanding the meaning of these words is often taken for granted. Everyday-lan-

guage is laden with vocabulary, comparative forms, figurative speech and complex

language structures that are not explained. By contrast, key words in math are usu-

ally highlighted in the text and carefully explained by the teacher.

Adaptations to the programs of ESL students do not change the curriculum

expectations. An adaptation to a student’s program alters the “how,” “when,” or

“where,” the student is taught or assessed.

Adaptations for ESL Students

Learning-Disabled Students

A student with a learning disability usually suffers from an inability to think, listen,

speak, write, spell, or calculate that is not obviously caused by any mental or physi-

cal disability. There seems to be a lag in the developmental process and/or a delay in

the maturation of the central nervous system.

AdaptationsProviding simplified presentations, repetitions, more specific examples, or breaking

content blocks into simpler sections may help in minor cases of learning disability.

Introduction • MHR lxvii


• structure learning activities to develophigher-order thinking skills (analysis,synthesis, and evaluation)

• provide open-ended questions

• value learner’s own interests and learningstyle, and allow for as much student inputinto program options as possible

• encourage students to link learning towider applications

• encourage learners to reflect onthe process of their own learning

• encourage and reward creativity

• avoid repetitive tasks and activities

• encourage a stimulatingenvironment that invitesexploration of mathematicalconcepts

• display pictures of rolemodels who excel inmathematics

• provide access to computerprograms that extendlearning

• reduce the number ofquestions to allow time formore demanding ones

• allow for opportunities todemonstrate learning in non-traditional formats

• pose more questions thatrequire higher-level thinkingskills (analysis, synthesis andevaluation)

• reward creativity


• pre-teach vocabulary

• explain colloquial expressions andfigurative speech

• review comparative forms ofadjectives

• display reference charts withmathematical terms and language

• encourage personal mathdictionaries/journals with mathterms and formulae

• allow access to personalmath dictionaries

• read instructions to studentand clarify terms

• allow additional time

At-Risk Students

“At-risk” students are in danger of completing their schooling without adequate skill

development to function effectively in society. Risk factors include low achievement,

retention, behaviour problems, poor attendance, low socioeconomic status, and

attendance at schools with large numbers of poor students.

Adaptations

Neither failing such students nor putting them in pullout programs has produced

much gain in achievement, but there are certain approaches that do help.

• Allow students to proceed at their own pace through a well-defined series of

instructional objectives.

• Place students in small, mixed-ability learning groups to master material first

presented by teacher. Teams are then rewarded based on the individual learning

of all team members.

• Have students serve as peer tutors, as well as being tutored. This helps to raise

their self-esteem and make them feel they have something to contribute.

• Involve students in learning about something that is relevant to them, such as

money management or wise shopping.

• Get parents involved in their child’s learning as much as possible.

lxviii MHR • Introduction

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