ns8tg ch00 prelims 3rd b - cape breton 8_tr/teachers resource... · art direction: tom dart/first...

AUTHORS

Rona Chisholm-McLearyB.Sc., B.Ed., Graduate Diploma in Education

Technology, M.Ed.Halifax Regional School Board

Nancy FournierB.Sc., B.Ed.

Halifax Regional School Board

Daniel MacDonaldB.Sc., B.Ed.

Bridgewater, Nova Scotia

Jody MacllreithB.A., B.Ed., M.Ed., M.Ed.


David McKillopB.Sc., B.Ed., M.Ed.

Making Math Matter Inc.

Tess MillerBSc., B.Ed., M.Ed., Ph.D.

Queen’s University, Kingston, Ontario

Marilyn PriceB.Sc. in Ed., M.Ed.

Lower Sackville, Nova Scotia

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

District School Board of Niagara

Sandy SzetoB.Sc., B.Ed.

Toronto District School Board

Debby VassB.T.


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

The Nova Scotia Department of Education wishes to give a special thanks to Nancy Chisholm, of the

Department of Education, for her thoughtful reviews and advice on the use of technology for Grades 7 to 9.

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

WEB SITE:http://www.mcgrawhill.ca

E-MAIL:[email protected]

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

TOLL-FREE CALL:1-800-565-5758

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 RyersonMathematics 8: Focus on Understanding Teacher’s Resource

Copyright © 2008, 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-13: 978-0-07-096587-4ISBN-10: 0-07-096587-0

http://www.mcgrawhill.ca

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

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, CA94680, 1-800-995-MATH.

PUBLISHER: Linda AllisonPROJECT MANAGERS: Eileen Jung, Maggie CheverieDEVELOPMENTAL EDITORS: Bradley T. Smith, Julie Kretchman, Ingrid D’SilvaMANAGER, EDITORIAL SERVICES: Crystal ShorttCOPY EDITOR: Loretta JohnsonEDITORIAL ASSISTANT: Erin HartleyMANAGER, PRODUCTION SERVICES: Yolanda PigdenPRODUCTION COORDINATOR: Jennifer HallCOVER DESIGN: Dianna LittleART DIRECTION: Tom Dart/First Folio Resource Group Inc.ELECTRONIC PAGE MAKE-UP: Tom Dart, Kim Hutchinson, Adam Wood/First FolioResource Group, Inc.COVER IMAGE: Courtesy of Dick Killam, Tides Photography Inc.

C O N T E N T S

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

Program Overview and Philosophy...........................................................................................xiv

Approaches to Teaching Mathematics ..............................................................................xv

The Modern Classroom....................................................................................................xvi

Mathematics at Home ......................................................................................................xvi

Literacy ............................................................................................................................xviii

Cooperative Learning........................................................................................................xix

Estimation and Mental Mathematics...............................................................................xxi

Problem Solving ...............................................................................................................xxx

Technology.......................................................................................................................xxxi

Correlations....................................................................................................................xxxii

Grades 7-8 Continuum......................................................................................................xli

Manipulatives, Materials, and Technology Tools ..........................................................xlvii

Assessment .................................................................................................................................xlix

Adaptations..................................................................................................................................liii

Answers to Get Ready For Grade 8 ......................................................................lvii

Chapter 1: Squares, Square Roots, and Pythagoras......................................2

Get Ready .......................................................................................................................................5

1.1 Identify Perfect Squares and Related Patterns ....................................................................7

1.2 Find Square Roots ..............................................................................................................13

1.3 Discover the Pythagorean Relationship............................................................................18

Use Technology

Explore the Pythagorean Relationship Using The Geometer’s Sketchpad® .....................24

1.4 Apply the Pythagorean Relationship in Problem Solving ...............................................25

Review ..........................................................................................................................................29

Practice Test .................................................................................................................................31

Chapter 2: Fraction Operations ..............................................................................36

Get Ready .....................................................................................................................................38

2.1 Add and Subtract Fractions ..............................................................................................42

2.2 Multiply Fractions..............................................................................................................52

2.3 Divide Fractions .................................................................................................................61

2.4 Fractions and the Order of Operations ............................................................................69

Review ..........................................................................................................................................73

Practice Test .................................................................................................................................76

Task: Design a Park......................................................................................................................80

Introduction • MHR iii

Chapter 3: Geometry I...................................................................................................84

Get Ready .....................................................................................................................................86

3.1 Unique Triangles ................................................................................................................90

3.2 Prove Triangles are Congruent..........................................................................................96

3.3 Properties of Transformations .......................................................................................102

3.4 Regular Polygons..............................................................................................................112

Review ........................................................................................................................................120

Practice Test ...............................................................................................................................123

Chapters 1–3 Review ................................................................................................................127

Chapter 4: Ratio and Proportion .........................................................................130

Get Ready ...................................................................................................................................132

4.1 Fractions, Decimals, and Percents...................................................................................136

4.2 Explore and Apply Proportion, Ratio, and Rate ............................................................143

4.3 Solve Problems Involving Proportions, Ratios, and Rates.............................................149

Review ........................................................................................................................................154

Practice Test ...............................................................................................................................156

Chapter 5: Data Management and Probability ...........................................160

Get Ready ...................................................................................................................................162

5.1 Collect, Organize, and Use Data .....................................................................................166

5.2 Theoretical and Experimental Probabilities ...................................................................173

5.3 Explore the Effects on Mean, Median, and Mode..........................................................179

5.4 Construct and Interpret Box-and-Whisker Plots...........................................................184

5.5 Construct and Interpret Circle Graphs...........................................................................190

5.6 Construct and Interpret Scatterplots ..............................................................................196

Review ........................................................................................................................................202

Practice Test ...............................................................................................................................205

Task: Statistics in Everyday Life ................................................................................................209

Chapter 6: Rational Numbers.................................................................................212

Get Ready ...................................................................................................................................214

6.1 Negative Exponents..........................................................................................................218

6.2 Scientific Notation ...........................................................................................................225

6.3 Compare and Order Rational Numbers ........................................................................230

6.4 Operations With Rational Numbers...............................................................................236

6.5 Properties of Operations .................................................................................................242

Review .......................................................................................................................................247

Practice Test ...............................................................................................................................250

Chapters 4–6 Review .................................................................................................................255

iv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Chapter 7: Algebraic Expressions and Solving Equations ...................258

Get Ready ..................................................................................................................................260

7.1 Add and Subtract Algebraic Expressions........................................................................263

7.2 Multiply Polynomial Expressions....................................................................................268

7.3 Solve Linear Equations ....................................................................................................272

Review ........................................................................................................................................277

Practice Test ...............................................................................................................................279

Chapter 8: Patterns and Relations .......................................................................284

Get Ready ...................................................................................................................................286

8.1 Explore Patterns and Relations .......................................................................................289

8.2 Linear and Non-Linear Relations....................................................................................298

8.3 Slope..................................................................................................................................305

8.4 Intersection of Two Lines on a Graph ............................................................................313

8.5 Analyse Relations .............................................................................................................319

Review ........................................................................................................................................326

Practice Test ...............................................................................................................................329

Task: Magic Squares...................................................................................................................333

Chapter 9: Geometry II...............................................................................................336

Get Ready ...................................................................................................................................338

9.1 Properties of Similar Figures ...........................................................................................341

9.2 Dilatations ........................................................................................................................350

9.3 Cones and Cylinders ........................................................................................................357

9.4 Draw Polyhedra................................................................................................................364

Review .......................................................................................................................................372

Practice Test ...............................................................................................................................375

Chapters 7–9 Review .................................................................................................................380

Chapter 10: Measurement.........................................................................................384

Get Ready ...................................................................................................................................386

10.1 Area and Perimeter of Quadrilaterals .............................................................................389

10.2 Area and Circumference of Circles .................................................................................396

10.3 Area of Composite Figures ..............................................................................................403

10.4 Surface Area of Three-Dimensional Figures ..................................................................411

10.5 Volume of Three-Dimensional Figures ..........................................................................417

Review ........................................................................................................................................423

Practice Test ...............................................................................................................................426

Introduction • MHR v

BLACKLINE MASTERS

(Available on Mathematics 8: 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.

Answers are given for all of these extra questions.

Also included are the rubrics for the Assessment questions, the 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 YourUnderstanding questions, and Assessment questions.

vi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

The following Generic Resource Masters are

provided on the CD-ROM:

Resource Master 01 Integer Number Lines

Resource Master 02 Vertical Number Lines

Resource Master 03 Horizontal Number

Lines

Resource Master 04 Square Dot Paper

Resource Master 05 Isometric Dot Paper

Resource Master 06 Centimetre Grid Paper

Resource Master 07 Grid Paper

Resource Master 08 Loops Game Cards

Resource Master 09 Mental Math Bingo

Questions 1


Questions 2


Questions 3


Sheet

Resource Master 13 Basic Fact Practice

Resource Master 14 Decimal Point Practice

Resource Master 15 Hundred Chart

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 Rubric

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 Technology Master is provided

on the CD-ROM:

Technology Master 01 The Geometer’s Sketchpad®,

Version 4, The Basics

Introduction • MHR vii

GET READY FORGRADE 8 BLMs

GR Finger Puppets

GR Math Spinner

CHAPTER 1 BLMs

Parent Letter BLM

1GR Parent Letter

Extra Practice BLMs

1GR Extra Practice

1.1 Extra Practice

1.2 Extra Practice

1.3 Extra Practice

1.4 Extra Practice

1R Extra Practice

1PT Chapter 1 Test

Ch1 EP Answer Key

Assessment Questions BLMs

1.2 Assessment Question



Assessment Questions Rubrics

1.2 Rubric

1.3 Rubric

1.4 Rubric

Chapter Problem Wrap-Up Rubric

1CP Chapter Problem Wrap-Up Rubric

CHAPTER 2 BLMs

Parent Letter BLM

2GR Parent Letter

Additional BLMs

2GR Paper Folding Fractions

2.2 Area Model Practice

2.4 Order of Operations Review

Ch2 BLM Answer Key

Extra Practice BLMs

2GR Extra Practice

2.1 Extra Practice

2.2 Extra Practice

2.3 Extra Practice

2.4 Extra Practice

2R Extra Practice

2PT Chapter 2 Test

Ch2 EP Answer Key







2.1 Rubric

2.2 Rubric

2.3 Rubric

2.4 Rubric



Task Rubric

2T Task Rubric

CHAPTER 3 BLMs

Parent Letter BLM

3GR Parent Letter

Discover the Math BLMs

3.3 Alternate DTM Part A Activity

3.4 DTM Polygon Table

3.4 Alternate DTM Activity

Check Your Understanding BLMs

3.4 CYU Table

Additional BLMs

3.4 Regular Polygons

Extra Practice BLMs

3GR Extra Practice

3.1 Extra Practice

3.2 Extra Practice

3.3 Extra Practice

3.4 Extra Practice

3R Extra Practice

3PT Chapter 3 Test

Ch3 EP Answer Key






3.1 Rubric

3.3 Rubric

3.4 Rubric



CHAPTER 4 BLMs

Parent Letter BLM

4GR Parent Letter


BLM 4.2 DTM Face

Extra Practice BLMs

4GR Extra Practice

4.1 Extra Practice

4.2 Extra Practice

4.3 Extra Practice

4R Extra Practice

4PT Chapter 4 Test

Ch4 EP Answer Key






4.1 Rubric

4.2 Rubric

4.3 Rubric



CHAPTER 5 BLMs

Parent Letter BLM

5GR Parent Letter


BLM 5.1 DTM Team Name Cards

BLM 5.1 DTM Tally Charts

BLM 5.2 DTM Contest Cards

Additional BLMs

BLM 5.5 Percent Circle

Extra Practice BLMs

5GR Extra Practice

5.1 Extra Practice

5.2 Extra Practice

5.3 Extra Practice

5.4 Extra Practice

5.5 Extra Practice

5.6 Extra Practice

5R Extra Practice

5PT Chapter 5 Test

Ch5 EP Answer Key









5.1 Rubric

5.2 Rubric

5.3 Rubric

5.4 Rubric

5.5 Rubric

5.6 Rubric



Task Rubric

5T Task Rubric

CHAPTER 6 BLMs

Parent Letter BLM

6GR Parent Letter

Additional BLMs

6GR Portfolio Performance Record

Sheet

6.3 Portfolio Snapshot sheet

6.4 Multiplication Table

6.5 Division Table

viii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Extra Practice BLMs

6GR Extra Practice

6.1 Extra Practice

6.2 Extra Practice

6.3 Extra Practice

6.4 Extra Practice

6.5 Extra Practice

6R Extra Practice

6PT Chapter 6 Test

Ch6 EP Answer Key







6.1 Rubric

6.2 Rubric

6.3 Rubric

6.4 Rubric

6.5 Rubric



CHAPTER 7 BLMs

Parent Letter BLM

7GR Parent Letter

Extra Practice BLMs

7GR Extra Practice

7.1 Extra Practice

7.2 Extra Practice

7.3 Extra Practice

7R Extra Practice

7PT Chapter 7 Test

Ch7 EP Answer Key






7.1 Rubric

7.2 Rubric

7.3 Rubric



CHAPTER 8 BLMs

Parent Letter BLM

8GR Parent Letter


8.1 Pattern Table

8.3 Alternate DTM Activity

Additional BLMs

8Task Equation Puzzle

Extra Practice BLMs

8GR Extra Practice

8.1 Extra Practice

8.2 Extra Practice

8.3 Extra Practice

8.4 Extra Practice

8.5 Extra Practice

8R Extra Practice

8PT Chapter 8 Test

Ch8 EP Answer Key








8.1 Rubric

8.2 Rubric

8.3 Rubric

8.4 Rubric

8.5 Rubric



Task Rubric

8T Task Rubric

Introduction • MHR ix

CHAPTER 9 BLMs

Parent Letter BLM

9GR Parent Letter


9.1 DTM Dilatation Table 1 and 2

9.3 DTM Pyramid Table

Extra Practice BLMs

9GR Extra Practice

9.1 Extra Practice

9.2 Extra Practice

9.3 Extra Practice

9.4 Extra Practice

9R Extra Practice

9PT Chapter 9 Test

Ch9 EP Answer Key





9.1 Rubric

9.2 Rubric

9.3 Rubric

9.4 Rubric



CHAPTER 10 BLMs

Parent Letter BLM

10GR Parent Letter

Additional BLMs

10.3 Rug Areas

Extra Practice BLMs

10GR Extra Practice

10.1 Extra Practice

10.2 Extra Practice

10.3 Extra Practice

10.4 Extra Practice

10.5 Extra Practice

10R Extra Practice

10PT Chapter 10 Test

Ch10 EP Answer Key








10.1 Rubric

10.2 Rubric

10.3 Rubric

10.4 Rubric

10.5 Rubric


10CP Chapter Problem Wrap-Up

Rubric

x MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

McGraw-Hill Ryerson Mathematics 8: Focus onUnderstanding Program Overview

The McGraw-Hill Ryerson Mathematics 8: 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: Review,

Practice Test, Chapter Problem, and Cumulative Review. Technology is integrated

throughout the program, and includes the use of calculators and graphing calcula-

tors, drawing and spreadsheet 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

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

Practice Test, and Cumulative Review features.

Introduction • MHR xi

An Introduction to Mathematics 8: Focus onUnderstanding Teacher’s Resource

The teaching notes for each chapter have the following structure:

C h a p te r P l a n n i n g C h a r t

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

• suggested timing for each section

• formative and summative assessment tools

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

• related blackline masters (BLMs) available on the CD-ROM

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

The introduction lists the following:

• Specific Curriculum Outcomes that the section covers in whole or in part.

• Link to Get Ready, prerequisite skills from the Get Ready that students need for

success with the section.

• Materials 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.

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

readily apparent on first read of the Discover the Math activities and worked

Examples.

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

outcome of these activities.

• 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.

• The 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 can complete fairly

successfully

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

assigned with discretion

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

Assessment questions.

• Rubrics for the Assessment questions are provided.

xii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Ad d i t i o n a l R u b r i c s a n d An s we r s

• Rubrics are provided for each Chapter Problem Wrap-Up.• Rubrics are provided for the Tasks that occur at the end of chapters 2, 5, and 8

in the student text. The score levels range from 1 to 4. A designation of No Mark

(NM) is to be assigned when the student fails to provide any evidence of

performance.

• Answers to Making Connections and Puzzlers are at the end of each numbered

section where they occur.

• Answers to all Check Your Understanding questions (except the Assessment

question) are at the end of each numbered section. Answers can be copied

and given to students, as there is no answer key in the student text.

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

• Generic Masters, such as grid paper and nets.

• Specific Masters, scaffolded worksheets that support some of the student text’s

Discover the Math and Communicate the Key ideas questions.

• Extra Practice Masters, extra practice questions for:

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

- each numbered section

- each Review and Practice Test• An Answer Key is included for the Extra Practice Master questions.

Introduction • MHR xiii

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 8: Focus on Understanding is an exciting new resource for the grade 8

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 8: 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.

xiv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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 simpleproblems

• sometimes called “semi-concrete”

• typically work with diagrams

• thinking focus becoming moregeneral and systematic

• meta-cognitive thinking aboutthought processes begins todevelop

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

• able to solve moderatelychallenging problems

• use problem strategieseffectively, with some guidance

• able to work with or withoutmaterials or diagrams

• thinking focus instinctivelygeneral 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 8: 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

Introduction • MHR xv

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 mathe-

matics teaching.

At grade 8, 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 8: Focus on Understanding program design.

Transmission-Oriented Connectionist-Oriented Discovery-Oriented• teaching involves

“delivering” the curriculum

• focuses on proceduresand routines

• emphasizes clearexplanations and practice

• “chalk-and-talk”

• teaching involves helping studentsdevelop and apply their ownconceptual understandings

• focuses on different models andmethods and the connections amongthem

• emphasizes “problematic” challengesand teacher-student dialogue

• “Van de Walle”

• teaching involves helpingstudents learn by “doing”

• focuses on applying strategies topractical 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 text-book.

1) Mathematics learning is an active and constructive process.

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

3) Learning is most likely when placed in meaningful contexts and in an environmentthat supports exploration, risk taking, and critical thinking, and nurtures positiveattitudes and sustained effort.

4) Learning is most effective when standards and expectations are made clear andassessment and feedback are ongoing.

5) Learners benefit, both socially and intellectually, from a variety of learningexperiences, both independent and in collaboration with others.

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

problems 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 8: 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-

concrete materials such as diagrams and charts.

Technology

The calculator is, or ought to be, a standard part of each student’s mathematical toolbox.

In the Mathematics 8: Focus on Understanding program, calculator keystrokes are

provided in parallel with some conventional calculations.

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

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

software as an optional adjunct to class teaching, where appropriate. The Use

Technology lessons offer alternative activities usingThe Geometer’s Sketchpad®.

Teachers enjoy maximum flexibility because they can teach some activities using

manipulatives 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 offers safe and reliable links for

students (www.mcgrawhill.ca/links/math8NS) and teachers (www.mcgrawhill.ca/

books/math8NS).

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,

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

To encourage this home and school connection, Focus on Understanding includes

regular 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;

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

xvi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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

Introduction • MHR xvii

• 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.

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

mathematics 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;

Home Activity/Interest Math Connection

chores predicting probability of selecting a specific chore from a job jar; estimating the cost of foodplaced 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 atrail mix; using geometric shapes to design a holiday dessert or display

meals working with fractions of various foods (e.g., pizzas, sandwiches, fruits, desserts); identifyingpatterns to calculate how many people can stand at various kitchen work centre designs;using transformations to design a place mat or plan a special table design; working withprobability to determine the number of combinations for a meal

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

outdoors identifying and classifying natural shapes; calculating area and perimeter of yards orplaygrounds; predicting probability of getting an orange flower at random from a collectionof wildflowers; keeping data on the growth of a plant and calculating measures of centraltendency; keeping track of populations of insects or other plant or animal forms thatreproduce exponentially; developing and answering Fermi problems; using tiling patterns todesign an outdoor patio or other area

pets feeding; keeping data on young pet’s daily mass and calculating measures of centraltendency 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 data andkeeping personal statistics for specific activity; identifying patterns in team statistics andusing them to make predictions; using transformations to create a team logo or shirt

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

travelling identifying and collecting sign shapes; collecting data on gas consumption and findingaverage kilometres per litre for a specific vehicle; using integers to show travel East andWest/North and South of a specific location; identifying three-dimensional figures andcalculating volumes of space

• 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.

Co m m u n i c at i o n M at h e m at i c a l l y

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

phrase, or a new word. They may 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

feelings 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. 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 Journal AssessmentRubric. Reading the content of the journal will help you in the assessment of

students’ learning of the chapter content. You may wish to respond personally to

students’ thoughts and ideas.

xviii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xix

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

learning, 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).

Personal Entries

Entries that Communicate to Teacher

Knowledge Connections Communication

Purpose • For the benefit of thestudent.

• Not to be read by anyoneelse unless the studentoffers to share them.

• Used to discuss how thestudent feels about theirability in math orparticular parts of thecourse.

• Show whatstudents know andunderstand about atopic or concept.

• Emphasize theconnectionsstudents aremaking betweenthe mathematicsthey do in theclassroom and theirpersonal lives andthe world aroundthem.

• Providereflections onwhat studentslearned during aparticular sectionand what theythink is important.

Sample Opener

• My biggest difficulty inmathematics is …

• The thing I like best about mathematics is …

• 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’dlike to be a_____________, sothe most importantmath I need toknow is …

• I use _________ …

• I wish I hadlistened morecarefully whenthe teacher wasexplaining …

• The mostimportant thing Ilearned in thissection is …

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.

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

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

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

xx MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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 contributions of others

• checking for agreement

• disagreeing in an agreeable way

• mediating disagreements within the group

• sharing

• showing appreciation for the efforts of others

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 you mean?

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 … ?

Introduction • MHR xxi

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

concept, 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

understood 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.

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 responsible

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

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

students 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.

E S T I M AT I O N A N D 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

mathematics is a major contributor to this comfort and understanding. Mental

mathematics is the mental manipulation of knowledge dealing with numbers,

shapes, and patterns to solve problems.

MentalMathematics

Mental Imagery

Estimation(in computation and

in measurement)

MentalComputation

(precise answers)

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 situations 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

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

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, and the area

of the math textbook cover. These referents develop with measurement 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 problem solving situation.

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

use their fingers or hands to show such measurements as: 10 cm, 250 mm, 0.5 m, a

90° angle, or 1000 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

numbers 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.

xxii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxiii

Try these Mental Imaging Activities with your students.

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

calculations 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.

Example 1:

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

• 3–4

• 60 in relation to one hundred

• 75% of the questions on the page

• a 120º angle

• 0.65 m

• 1000 cm

• 300 mm

• a 4 m � 4 m garden

• a 45.5-kg fish

• a 4.5-kg dog

Example 2:

Use mental imagery to answer the following:

1. How many edges does a rectangular prism have?

2. If I am facing south what direction is to my right?

3. What is the perimeter of a 20 cm � 30 cm shelf?

4. How many sides does a pentagonal pyramidhave?

5. Imagine an 8-cm cube. What is its volume?

6. You cut a cube in half along a diagonal. Whatsolid is left?

7. You cut the top off a cylinder. What shape isexposed?

• 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

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 written

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.

Problem situation

Use a computer

Use a calculator

Estimate

Calculation needed

Approximateanswer needed

Use mentalcalculation

Exact answerneeded

Use apaper-and-pencil

calculation

xxiv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxv

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?

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

programs from kindergarten through grade 12 should enable all students to compute

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

estimation 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 8

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

calculated 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 following

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 “nice” 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).

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 subtracting.

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.

750250

xxvi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxvii

Compensation Strategy for Subtraction

As with addition, subtract the “nice” number and add the difference. For example,

$3.27 – $0.98 = ($3.27 – $1.00) + $0.02 = $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.

“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 , , and . 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 “nice” combi-

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

the numbers to make the calculation easier. 4 � 250 = 1000 and 1000 multiplied 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.

110

11000

1100

110

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.

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

factor 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 example,

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 .110

14

15

xxviii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxix

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.

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 Resource Master 12 Mental MathBingo Sheet. 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 Resource 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 Resource Master 08 Loops Game Cards, and cut out the indi-

vidual 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 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 Resource Master 13 Basic Fact Practice and ResourceMaster 14 Decimal 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-

solving 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.

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

xxx MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxxi

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

Mathematics 8: 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 selected 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

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 8: Focus on Understanding uses specific technologies to engage students

in math inquiry, research, and problem solving. The Use Technology sections provide

students with hands-on experience in performing spreadsheet calculations and

constructing graphs. Students can also research a variety of topics using safe, reliable

Internet links offered through the McGraw-Hill Web site at www.mcgrawhill.ca/

links/math8NS.

The use of technology in instruction should further alter both the teaching and thelearning of mathematics. Computer software can be used effectively for classdemonstrations and independently by students to explore additional examples,perform independent investigations, generate and summarize data as part of a project,or complete assignments. Calculators and computers with appropriate softwaretransform the mathematics classroom into a laboratory much like the environment inmany science classes, where students use technology to investigate, conjecture, andverify their findings. In this setting, the teacher encourages experimentation andprovides opportunities for students to summarize ideas and establish connections withpreviously studied topics.

Curriculum and Evaluation Standards for School Mathematics, NCTM, 1989

CO R R E L AT I O N S

Strand/Outcome Chapter/Section Pages Assessment

Number Concepts/Number and Relationship Operations

Specific Curriculum Outcome

A1 model and link variousrepresentations of square root of anumber

1.11.3 2Task

14–19,25–35

102–103

Summative

BLM 2T Task Rubric

A2 recognize perfect squaresbetween 1 and 144 and applypatterns related to them

1.11.22Task

14–19,20–24

102–103

Summative

BLM 2T Task Rubric

A3 distinguish between an exactsquare root of a number and itsdecimal approximation

1.2

2Task

20–24

102–103

Summative

BLM 2T Task Rubric

A4 find the square root of anynumber, using an appropriatemethod

1.2

1CP Wrap-Up

2Task

20–24

45

102–103

Formative

BLM 1.2 Rubric

Summative

BLM 1CP Chapter Problem Wrap-UpRubric

BLM 2T Task Rubric

A5 demonstrate and explain themeaning of negative exponents forbase ten

6.1 246–253 Formative

BLM 6.1 Rubric

A6 represent any number written inscientific notation in standard form,and vice versa

6.2

8Task

254–261

380–381

Formative

BLM 6.2 Rubric

Summative

BLM 8T Task Rubric

A7 compare and order integers andpositive and negative rationalnumbers (in decimal and fractionalforms)

6.3

6CP Wrap-Up

262–267

287

Formative

BLM 6.3 Rubric

Summative


A8 represent and apply fractionalpercents, and percents greater than100, in fraction or decimal form, andvice versa

4.1 148–157

A9 solve proportion problems thatinvolve equivalent ratios and rates

4.2 158–167

xxxii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Introduction • MHR xxxiii




B1 demonstrate an understanding ofthe properties of operations withintegers and positive and negativerational numbers (in decimal andfractional forms)

6.5

6CP Wrap-Up

8Task

276–283

287

380–381

Formative

BLM 6.5 Rubric

Summative


BLM 8T Task Rubric

B2 solve problems involvingproportions, using a variety ofmethods

4.2

4.3

4CP Wrap-Up

158–167

168–177

181

Formative

BLM 4.2 Rubric

Summative


B3 create and solve problems whichinvolve finding a, b, or c in therelationship a% of b = c, usingestimation and calculation

4.1

4.3

148–157

168–177

Formative

BLM 4.1 Rubric

BLM 4.3 Rubric

B4 apply percentage increase anddecrease in problem situations

4.1

4.3

4CP Wrap-Up

6CP Wrap-Up

148–157

168–177

181

287

Summative



B5 add and subtract fractionsconcretely, pictorially, andsymbolically

2.1

2CP Wrap-Up

2Task

56–69

101

102–103

Formative

BLM 5.1 Rubric

Summative


BLM 2T Task Rubric

B6 add and subtract fractionsmentally, when appropriate

2.1

2CP Wrap-Up

2Task

56–69

101

102–103

Summative


BLM 2T Task Rubric

B7 multiply fractions concretely,pictorially, can symbolically

2.2

2Task

70–81

102–103

Formative

BLM 2.2 Rubric

Summative

BLM 2T Task Rubric

B8 divide fractions concretely,pictorially, and symbolically


BLM 2.3 Rubric

B9 estimate and mentally computeproducts and quotients involvingfractions

2.2

2.3

2CP Wrap-Up

2Task

70–81

82–91

101

102–103

Formative

BLM 2.2 Rubric

Summative


BLM 2T Task Rubric

xxxiv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource




B10 apply the order of operations tofraction computations, using bothpencil and paper and the calculator

2.42CP Wrap-Up

92–97,101

Formative

BLM 2.4 Rubric

Summative


B11 model, solve, and createproblems involving fractions inmeaningful contexts

2.1

2.2

2.3

2.4

2CP Wrap-Up

2Task

56–69,

70–81

82–91

92–97

101

102–103

Summative


BLM 2T Task Rubric

B12 add, subtract, multiply, and dividepositive and negative decimalnumbers with and without thecalculator

6.4

6CP Wrap-Up

8Task

268–275

287

380–381

Formative

BLM 6.4 Rubric

Summative

BLM 8T Task Rubric

B13 solve and create problemsinvolving addition, subtraction,multiplication, and division ofpositive and negative numbers

7.1

7CP Wrap-Up

8Task

296–304

325

380–381

Formative

BLM 6.4 Rubric

Summative

BLM 8T Task Rubric

B14 add and subtract algebraic termsconcretely, pictorially, and symbolicallyto solve simple algebraic problems

7.1

7CP Wrap-Up

8Task

296–304

325

380–381

Formative

BLM 7.1 Rubric

Summative


BLM 8T Task Rubric

B15 explore addition and subtractionof polynomial expressions, concretelyand pictorially

7.1

8Task

296–304

380–381

Formative

BLM 7.1 Rubric

Summative

BLM 8T Task Rubric

B16 demonstrate an understandingof multiplication of a polynomial by ascalar, concretely, pictorially, andsymbolically


BLM 7.2 Rubric

Introduction • MHR xxxv


Patterns and Relations


C1 represent patterns andrelationships in a variety of formatsand use these representations topredict unknown values

8.1

8.5

8Task

330–340

368–375

380–381

Formative

BLM 8.1 Rubric

Summative

BLM 8T Task Rubric

C2 interpret graphs that representlinear and non-linear data

8.2

8.5

8CP Wrap-Up

341–347

368–375

379

Formative

BLM 8.2 Rubric

BLM 8.5 Rubric

Summative


C3 construct and analyse tables andgraphs to describe how change inone quantity affects a relatedquantity


BLM 8.2 Rubric

C4 link visual characteristics of slopewith its numerical value bycomparing vertical change withhorizontal change


BLM 8.3 Rubric

C5 solve problems involving theintersection of two lines on a graph


BLM 8.4 Rubric

C6 solve and verify simple linearequations algebraically

7.3,

7CP Wrap-Up

8Task

310–321

325

380–381

Formative

BLM 7.3 Rubric

Summative

BLM78CP Chapter Problem Wrap-UpRubric

BLM 8T Task Rubric

C7 create and solve problems, usinglinear equations

7.3 310–321

xxxvi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource


Shape and Space (Measurement)


D1 solve indirect measurementproblems, using proportions

4.2

4.3

158–167

168–177

D2 solve measurement problems,using appropriate SI units

10.1

10.2

10.3

10.4

10.5

10CP Wrap-Up

430–435

436–443

444–455

456–463

464–473

477

Summative


D3 estimate areas of circles 10.2 436–443

D4 develop and use the formula forthe area of a circle


BLM 10.2 Rubric

D5 describe patterns and generalizethe relationships between areas andperimeters of quadrilaterals, andareas and circumferences of circles

10.1

10.2

10.3

430–435

436–443

444–455

Formative

BLM 10.1 Rubric

BLM 10.3 Rubric

D6 calculate the areas of compositefigures

10.3

10CP Wrap-Up

444–455

477

Formative

BLM 10.3 Rubric

Summative


D7 estimate and calculate volumesand surface areas of right prisms andcylinders

10.4,

10.5

10CP Wrap-Up

456–463

464–473

477

Formative

BLM 10.4 Rubric

BLM 10.5 Rubric

Summative


D8 measure and calculate volumesand surface areas of composite 3-Dshapes

10.4

10.5

10CP Wrap-Up

456–463

464–473

477

Formative

BLM 10.5 Rubric

Summative


D9 demonstrate an understanding ofthe Pythagorean relationship, usingmodels

1.3

1.4

1CP Wrap-Up

2Task

25–35

36–41

45

102–103

Formative

BLM 1.3 Rubric

Summative


BLM 2T Task Rubric

D10 apply the Pythagoreanrelationship in problem situations

1.4

1CP Wrap-Up

2Task

36–41

45

102–103

Formative

BLM 1.4 Rubric

Summative


BLM 2T Task Rubric

Introduction • MHR xxxvii

Strand/Expectation Chapter/Section Pages Assessment

Shape and Space (Geometry)

General Outcomes

E1 make and apply informaldeductions about the minimumsufficient conditions to guarantee theuniqueness of a triangle and thecongruency of two triangles

3.1

3.2

108–113

114–120

Formative

BLM 3.1 Rubric

E2 make and apply generalizationsabout the properties of rotations anddilatations, and use dilatations inperspective drawings of various 2-Dshapes

3.3

9.1

9.4

9CP Wrap-Up

121–130

388–394

410–417

421

Formative

BLM 3.3 Rubric

BLM 9.1 Rubric

Summative


E3 make and apply generalizationsabout the properties of similar 2-Dshapes

9.2

9CP Wrap-Up

395–402

421

Formative

BLM 9.2 Rubric

Summative


E4 perform various 2-D constructionsand apply the properties oftransformations to theseconstructions

3.3

3CP Wrap-Up

121–130

141

Summative


E5 recognize, name, describe, andmake and apply generalizationsabout the properties of prisms,pyramids, cylinders, and cones

3.4

9.3

9CP Wrap-Up

131–137

403–409

421

Formative

BLM 3.3 Rubric

BLM 3.4 Rubric

BLM 9.3 Rubric

Summative


E6 draw isometric and orthographicviews of 3-D shapes and construct 3-D models from these views

9.4

9CP Wrap-Up

410–417

421

Formative

BLM 9.4 Rubric

Summative


xxxviii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource


Data Management and Probability

General Outcome

F1 demonstrate an understanding ofthe variability of repeated samples ofthe same population

5.1 188–197

F2 develop and apply the concept ofrandomness

5.1

5CP Wrap-Up

188–197

237

Formative

BLM 5.1 Rubric

Summative


F3 construct and interpret circlegraphs

5.5

5Task

220–225

238–239

Formative

BLM 5.5 Rubric

Summative

BLM 5T Task Rubric

F4 construct and interpret scatterplots and determine a line of best fitby inspection

5.6, 5CP

Wrap-Up

5Task

226–233

237

238–239

Formative

BLM 5.6 Rubric

Summative


BLM 5T Task Rubric

F5 construct and interpret box-and-whisker plots

5.4

5CP Wrap-Up

5Task

213–219

237

238–239

Formative

BLM 5.4 Rubric

Summative


BLM 5T Task Rubric

F6 extrapolate and interpolateinformation from graphs

5.6

5CP Wrap-Up

5Task

226–233

237

238–239

Formative

BLM 5.6 Rubric

Summative


BLM 5T Task Rubric

F7 determine the effect of variationsin data on the mean, median, andmode


BLM 5.3 Rubric

F8 develop and conduct statisticsprojects to solve problems

5Task 238–239 Summative

BLM 5T Task Rubric

F9 evaluate data interpretations thatare based on graphs and tables

throughout ch5 186–233

Introduction • MHR xxxix



General Outcome

G1 conduct experiments andsimulations to find probabilities ofsingle and complementary events

5.1

5CP Wrap-Up

188–197

237

Summative


G2 determine theoreticalprobabilities of single andcomplementary events


BLM 5.2 Rubric

G3 compare experimental andtheoretical probabilities

5.2

5CP Wrap-Up

198–205

237

Formative

BLM 5.2 Rubric

Summative


G4 demonstrate an understanding ofhow data is used to establish broadprobability patterns

throughout ch5 186–233

xl MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Grade 7 Grade 8


General Curriculum Outcome A:

A1 model and use power, base, and exponent torepresent repeated multiplication

A2 recognize perfect squares between 1 and 144 andapply patterns related to them

A2 rename numbers among exponential, standard,and expanded forms

A1 model and link various representations of squareroot of a number

A3 distinguish between an exact square root of anumber and its decimal approximation

A4 find the square root of any n umber, using anappropriate method

A3 rewrite large numbers from standard form toscientific notation and vice versa

A5 demonstrate and explain the meaning of negativeexponents for base ten

A6 represent any number written in scientific notationin standard form, and vice versa

A4 solve and create problems involving commonfactors and greatest common factors (GCF)

A5 solve and create problems involving commonmultiples and least common multiples (LCM)

A6 develop and apply divisibility rules for 3, 4, 6, and 9

A7 apply patterning in renaming numbers fromfractions and mixed numbers to decimal numbers

A8 rename single-digit and double-digit repeatingdecimals to fractions through the use of patterns anduse these patterns to make predictions

A9 compare and order proper and improper fractions,mixed number, and decimal numbers

A7 compare and order integers and positive andnegative rational numbers (in decimal and fractionalforms)

A10 illustrate, explain, and express ratios, fractions,decimals, and percents in alternative forms

A8 represent and apply fractional percents, andpercents greater than 100, in fraction or decimal form,and vice versa

A9 solve proportion problems that involve equivalentratios and rates

A11 demonstrate number sense for percent

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

A13 compare and order integers A7 compare and order integers and positive andnegative rational numbers (in decimal and fractionalforms)

G R A D E S 7 – 8 CO N T I N U U M

Introduction • MHR xli

Grade 7 Grade 8


General Curriculum Outcome B:

B1 use estimation strategies to assess and justify thereasonableness of calculation results for integers anddecimal numbers

B2 use mental math strategies for calculationsinvolving integers and decimal numbers

B3 demonstrate an understanding of the propertiesof operations with decimal numbers and integers

B1 demonstrate an understanding of the propertiesof operations with integers and positive and negativerational numbers (in decimal and fractional forms)

B4 determine and use the most appropriatecomputational method in problem situationsinvolving whole numbers and/or decimals

B5 apply the order of operations for problemsinvolving whole and decimal numbers

B5 add and subtract fractions concretely, pictorially,and symbolically

B6 estimate the sum or difference of fractions whenappropriate

B6 add and subtract fractions mentally, whenappropriate

B9 estimate and mentally compute products andquotients involving fractions

B7 multiply mentally a fraction by a whole numberand vice versa

B7 multiply fractions concretely, pictorially, cansymbolically

B8 divide fractions concretely, pictorially, andsymbolically

B10 apply the order of operations to fractioncomputations, using both pencil and paper and thecalculator

B11 model, solve, and create problems involvingfractions in meaningful contexts

B8 estimate and determine percent when given thepart and the whole

B9 estimate and determine the percent of a number

B10 create and solve problems that involve the use ofpercent

B3 create and solve problems which involve finding a,b, or c in the relationship a% of b � c, using estimationand calculation

B4 apply percentage increase and decrease inproblem situations

B2 solve problems involving proportions, using avariety of methods

xlii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Grade 7 Grade 8


General Curriculum Outcome B:

B11 add and subtract integers concretely, pictorially,and symbolically to solve problems

B12 multiply integers concretely, pictorially, andsymbolically to solve problems

B13 divide integers concretely, pictorially, andsymbolically to solve problems

B12 add, subtract, multiply, and divide positive andnegative decimal numbers with and without thecalculator

B14 solve and pose problems which utilize addition,subtraction, multiplication, and division of integers

B13 solve and create problems involving addition,subtraction, multiplication, and division of positiveand negative numbers

B15 apply the order of operations to integers

B16 create and evaluate simple variable expressionsby recognizing that the four operations apply in thesame way as they do for numerical expressions

B17 distinguish between like and unlike terms

B18 add and subtract like terms by recognizing theparallel with numerical situations, using concrete andpictorial models

B14 add and subtract algebraic terms concretely,pictorially, and symbolically to solve simple algebraicproblems

B15 explore addition and subtraction of polynomialexpressions, concretely and pictorially

B16 demonstrate an understanding of multiplicationof a polynomial by a scalar, concretely, pictorially, andsymbolically

Introduction • MHR xliii

Grade 7 Grade 8

Patterns and Relations

General Curriculum Outcome C:

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

C2 summarize simple patterns, using constants,variables, algebraic expressions, and equations, anduse them in making predictions

C1 represent patterns and relationships in a variety of formats and use these representations to predictunknown values

C3 explain the difference between algebraicexpressions and algebraic equations

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

C5 illustrate the solution for one- and two-step linearequations, using concrete materials and diagrams

C6 solve and verify simple linear equationsalgebraically

C7 create and solve problems, using linear equations

C6 graph linear equations, using a table of values

C7 interpolate and extrapolate number values from a given graph

C8 determine if an ordered pair is a solution to alinear equation

C9 construct and analyse graphs to show howchange in one quantity affects a related quantity

C2 interpret graphs that represent linear and non-linear data

C3 construct and analyse tables and graphs todescribe how change in one quantity affects a relatedquantity

C4 link visual characteristics of slope with itsnumerical value by comparing vertical change withhorizontal change

C5 solve problems involving the intersection of twolines on a graph

xliv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Grade 7 Grade 8

Shape and Space (Measurement)

General Curriculum Outcome D:

D1 identify, use, and convert among the SI units tomeasure, estimate, and solve problems that relate tolength, area, volume, and capacity

D2 solve measurement problems, using appropriateSI units

D2 apply concepts and skills related to time inproblem situations

D3 develop and use rate as a tool for solving indirectmeasurement problems in a variety of contexts

D1 solve indirect measurement problems, usingproportions

D4 construct and analyse graphs to show change inone quantity affects a related quantity

D5 demonstrate an understanding of the relationshipsamong diameter, radii, and circumference of circles,and use the relationships to solve problems

D3 estimate areas of circles

D4 develop and use the formula for the area of acircle

D5 describe patterns and generalize the relationshipsbetween areas and perimeters of quadrilaterals, andareas and circumferences of circles

D6 calculate the areas of composite figures

D7 estimate and calculate volumes and surface areasof right prisms and cylinders

D8 measure and calculate volumes and surface areasof composite 3-D shapes

D9 demonstrate an understanding of thePythagorean relationship, using models

D10 apply the Pythagorean relationship in problemsituations

Introduction • MHR xlv

Grade 7 Grade 8

Shape and Space (Geometry)

General Curriculum Outcome E:

E1 recognize, name, describe, and construct polygons

E2 predict and generate polygons that can be formedwith a transformation or composition oftransformations of a given polygon

E2 make and apply generalizations about theproperties of rotations and dilatations, and usedilatations in perspective drawings of various 2-Dshapes

E3 make and apply generalizations about theproperties of regular polygons

E3 make and apply generalizations about theproperties of similar 2-D shapes

E4 perform various 2-D constructions and apply theproperties of transformations to these constructions

E4 make and apply generalizations abouttessellations of polygons

E5 construct polyhedra using one type of regularpolygonal face, and describe and name the resultingPlatonic Solids

E5 recognize, name, describe, and make and applygeneralizations about the properties of prisms,pyramids, cylinders, and cones

E6 draw isometric and orthographic views of 3-Dshapes and construct 3-D models from these views

E6 construct semi-regular polyhedra and describeand name the resulting solids, and demonstrate anunderstanding about their relationships to thePlatonic Solids

E7 make and apply generalizations about anglerelationships

E8 make and apply generalizations about thecommutativity of transformations

E9 make and apply informal deductions about theminimum sufficient conditions to guarantee that agiven triangle is of a particular type

E1 make and apply informal deductions about theminimum sufficient conditions to guarantee theuniqueness of a triangle and the congruency of twotriangles

E10 make informal deductions about the minimumsufficient conditions to guarantee that a givenquadrilateral is of a particular type, and to understandformal definitions of the various members of thequadrilateral family

xlvi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Grade 7 Grade 8


General Curriculum Outcome F:

F1 communicate through example the distinctionbetween biased and unbiased sampling, and first andsecond-hand data

F2 formulate questions for investigation from relevantcontexts

F3 select, defend, and use appropriate data collectionmethods and evaluate issues to be considered whencollecting data

F1 demonstrate an understanding of the variability ofrepeated samples of the same population

F2 develop and apply the concept of randomness

F4 construct a histogram F3 construct and interpret circle graphs

F4 construct and interpret scatter plots anddetermine a line of best fit by inspection

F5 construct and interpret box-and-whisker plotsF5 construct appropriate data displays, grouping datawhere appropriate and taking into consideration thenature of the data

F6 read and make inferences for grouped andungrouped data displays

F9 evaluate data interpretations that are based ongraphs and tables

F7 formulate statistics projects to explore currentissues from within mathematics, other subject areas,or the world of students

F8 develop and conduct statistics projects to solveproblems

F8 determine measures of central tendency and howthey are affected by data presentations andfluctuations

F7 determine the effect of variations in data on themean, median, and mode

F9 draw inferences and make predictions based onthe variability of data sets, using range and theexamination of outliers, gaps, and clusters

F6 extrapolate and interpolate information fromgraphs

Grade 7 Grade 8


General Curriculum Outcome G:

G1 identify situations for which the probability would

be near 0, 1–4

, 1–2

, 3–4

, and 1

G2 solve probability problems, using simulations andby conducting experiments

G1 conduct experiments and simulations to findprobabilities of single and complementary

G3 identify all possible outcomes of two independentevents, using tree diagrams and area models

G4 create and solve problems, using the numericaldefinition of probability

G5 compare experimental results with theoreticalresults

G2 determine theoretical probabilities of single andcomplementary events

G3 compare experimental and theoretical probabilities

G4 demonstrate an understanding of how data isused to establish broad probability patterns

G6 use fractions, decimals, and percents as numericalexpressions to describe probability

Introduction • MHR xlvii

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 ISBN

SuggestedQuantity

Algebra tiles Chapter 7

Bags 5.15.2

Base-10 blocks, fillable 10.5*

Base-10 materials 1.210.5

beakers 8.5

Bullseye compasses 1.3Chapter 35.510.2

Calculators 2Task6.2, 6.3, 6.4, 6.5*8Task10GR

Cans, cylindrical 10.5

Cardboard or constructionpaper

5Task6.4*

Centimetre cubes 9.4

Colour tiles 8.1, 8.2, 8.410.3

Colour tiles2.5 cm square plastictiles; 100 each of red,yellow, blue, green

Overhead colour tiles

0-322-06873-8:bucket of 280

0-322-06769-3:set of 48

one class setper twoclasses

Coloured counters 4.25.1

Two-colour counters:red on one side, whiteon other

0-322-05539-3:set of 200


Coloured pencils 1.22.24.15.56.48.49.1

Compasses, set of 5.5*10.2

Student SAFE-Tcompass, durable,plastic, draws circlesfrom 1 cm to 25 cmdiameter, in 5 mmincrements

0-322-07104-6

Computers

† = with Internet access

1Use Tech

5Task†

8Task

* = optional

xlviii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource




SuggestedQuantity

Dot paper, isometric 9.1, 9CP, Ch 7-9Review

See MathematicsBlackline MastersGrades P to 9

Dot paper, isometric,centimetre

9.4 See MathematicsBlackline MastersGrades P to 9

Dot paper, square 2.23R, Ch 1-3 Review9.1, 9R, 9CP, 9PT,Ch7-9 Review


Dot paper, square,centimetre

1.1, 1.23.3


Erlenmeyer flasks 8.5

Excel® spreadsheetsoftware

8Task

Fraction blocks 2.1*

Fraction Factory pieces 2.1 51-piece set of plasticrectangular fractionpieces

0-322-05575-X:15 sets

The Geometer’s Sketchpad®software

1Use Tech

The Geometry Template® 3.4*

Geoblocks 9.410.4

Geoboards 1.110.1

Geometric solids, fillable 10.5

Geostrips® 3.1, 3.4

Graduated cylinders 8.510.5

Graph paper 5.6, 5R, 5TaskChapter 8Ch 7-9 Review

Grid paper 1.33GR

See Resource Masters

Grid paper, centimetre 1.2*, 1.34.25.4Chapter 10


Grids, 10 � 10 4.1, 4.3 See MathematicsBlackline MastersGrades P to 9

Index cards 1.3

* = optional

Introduction • MHR xlix

* = optional




SuggestedQuantity

Linking cubes, centimetre 4.18.1, 8.2, 8.3, 8.49.410.1, 10.5

Centicubes(interlocking)

0-322-06777-4:set of 500

0-322-06778-2:set of 1000

one class setper two classes

Math textbooks 5.6

Measuring scoops 8.5

Metre sticks 5.4

Newspapers, magazines 5Task

Pan balances 10.5 Number balance with20 plastic, 5-g masses

0-322-06824-X

Paper 2.2, 2.3

Pattern blocks 2.1*8.19.1

Pattern blocks: woodor plastic, 1 cm thick

0-322-05566-0:plastic, bucket of 250

0-322-05567-9:wood, bucket of 250


Pattern block trianglepaper

3.1

Pencils 5.5

Polydron® pieces 9.3

Power Polygons® 9.1

Protractors 1.3*Chapter 35.5*9.2

Protractoropen centre, raisedgraduations measureup to 180° angles and15 cm lines

0-322-06816-9:set of 10

Rulers(† in millimetres)

1.2† , 1.32.2, 2Task3.2†, 3.3†, 3.4†5.4, 5.5*, 5.66.48.2, 8.3, 89.2†10.4

Ruler, 30 cm, clear,plastic, measure incentimetres andinches

0-322-07126-7

Scissors 1.32.38Task10.2, 10.3, 10.4

Square tiles 10.1

Stopwatches 5.6

String 5.6

Tape 10.2, 10.3, 10.4

l MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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

learning process. Some assessment data is used for the evaluation of students for the

purpose 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 to determine prior knowl-

edge, 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.

A s s e s s m e nt fo r Le a r n i n g

Assessment can determine where individual students will need support and will help

to determine where the classroom time needs to be spent.

Mathematics 8: Focus on Understanding provides the teacher with assessment

support at the start of the text and the beginning of every chapter.

• The Get Ready for Grade 8 section at the beginning of the student text (pages 2

to 7) provides an Assessment for teachers to assess student readiness for grade 8.

• Get Ready reviews at the beginning of each chapter provide coaching on

essential concepts and skills needed for the upcoming chapter.

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.




SuggestedQuantity

3-D objects (set of ) 9.3

Toilet paper roll 10.4

Toothpicks 8.110.1

Tracing paper 3.3, 3.4, 3R, 3PT10.3

Transparent mirrors 3.3, 3.4

Trays 7.2*

Water (or rice) 8.510.5

* = optional

Introduction • MHR li

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 8:

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.

• 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.

• Journal questions allow teachers insight into students’ thinking at key points.

• 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.

lii MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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’

mathematical thinking. 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 8: 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.

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 01Assessment Recording Sheet

This generic chart can be used to organize yourcomments for assessment of student observations,journals, portfolios, and presentations.

Assessment Master 02Attitudes Assessment Checklist

This checklist will allow you to assess a student’sattitude as he/she works on a task.

Assessment Master 03Portfolio Checklist

This checklist will allow you to assess students’portfolios.

Assessment Master 04Presentation Checklist

This checklist will allow you assess students’ oral andwritten presentations.

Assessment Master 05Problem Solving Checklist

This checklist will allow you to assess students’problem-solving skills.

Assessment Master 06Journal Assessment Rubric

This rubric will allow you to evaluate students’ journalentries.

Assessment Master 07Group Work Assessment Recording Sheet

This sheet will allow you to record comments asstudents work on group tasks.

Introduction • MHR liii

Generic Assessment BLM Purpose

Assessment Master 08Group Work Assessment General Scoring Rubric

This rubric will allow you to assess students’ group-related work.

Assessment Master 09How 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 theirunderstanding of 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 theirwork as part of a team.

Assessment Master 14My Progress as a Problem Solver

This checklist will allow students to self-assess theirown ability at solving problems.

Assessment Master 15Assessing Work in Progress

This sheet will allow student groups to assess theirprogress as they work to complete a task.

Assessment Master 16Learning Skills Checklist

This checklist will allow you to assess a student’s workhabits and learning skills.

A D A P TAT I O N S

Mathematics 8: Focus on Understanding considers a broad range of needs and learning

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.

• Communication Mathematically 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, scaffolding for selected questions and

activities, 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

reasons. 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.

liv MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

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

difficult 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.,

verbal 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

visually inputted information. Difficulties with near and far point copying, accu-

rately aligning numbers in columns, properly sequencing numbers, and illegible

hand 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 memory).

A weak working or active memory makes it difficult for students to hold informa-

tion 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

others, the retrieval of information from long term memory (e.g., remembering

number 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 students

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

student’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 headings:

• Presentation/Instructional adaptations refer to changes in teaching strategies

that allow the student to access the curriculum.

Introduction • MHR lv

Adaptations IPP

There are no changes to public schooloutcomes.

The teaching strategies are developed in oneor more of the following areas:• presentation• assessment/evaluation• motivation• environment• class organization• resources

Adaptations are not noted on the student’sreport card 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 outcome level than expected forthe grade level

• deleting specific curriculum outcomes when the deletedoutcomes are necessary to develop an understanding ofthe general curriculum outcome

• where needed, providing programming for outcomes thatare not part of Nova Scotia’s public school program (e.g.,behaviour programming, life skills)

• adding new outcomes where students require enrichment

An IPP is indicated on the student’s report card and/ortranscript. A copy of the IPP is filed in the student’scumulative record file.

• 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.

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

Presentation/Instructional Organizational/Environmental Assessment

• pre-teach vocabulary

• give concise, step-by-step directions

• teach students to look for clue words,highlight these words

• use visual models

• use visual representations toaccompany word problems

• encourage students to look forcommon patterns in word problems

• have students make useof a mathjournal

• 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)

lvi MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource


• 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 concepts relatedto daily living

• make available referencecharts with commonly usedfacts, formulae, and steps forproblem-solving

• allow use of a calculator

• use games and computerprograms for drill andrepetition

• allow to use multiplicationcharts

• allow to use other referencecharts as appropriate

• allow to use calculators


• present one concept/typeof question at a time

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 learning

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

student’s program for enrichment falls beyond the scope of this Teacher’s Resource.

Adaptations for Enrichment


• 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


• 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

Introduction • MHR lvii

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-

language is laden with vocabulary, comparative forms, figurative speech and com-

plex language structures that are not explained. By contrast, key words in math are

usually 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


• 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

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 physical

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.

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.

AdaptationsNeither 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.

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