FOURTH CANADIAN EDITION Elementary and Middle · PDF fileElementary and Middle School Mathematics ... Cover Designer: Anthony Leung ... an elementary school teacher in New York

Elementary and Middle School Mathematics Teaching Developmentally

F O U R T H C A N A D I A N E D I T I O N

John A. Van de Walle Late of Virginia Commonwealth University

Karen S. Karp University of Louisville

Jennifer M. Bay-Williams University of Louisville

Lynn M. McGarvey University of Alberta

Sandra Folk University of Toronto

With Contributions by

Jonathan Wray Howard County Public Schools

Toronto

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Library and Archives Canada Cataloguing in Publication

Van de Walle, John A., author Elementary and middle school mathematics: teaching developmentally / John A. Van de Walle, Late of Virginia Commonwealth University, Karen S. Karp, University of Louisville, Jennifer M. Bay-Williams, University of Louisville, Lynn M. McGarvey, University of Alberta, Sandra Folk, University of Toronto.—Fourth Canadian edition.

Revision of: Elementary and middle school mathematics: teaching developmentally / John A. Van de Walle ... [et al.].—3rd Canadian ed.—Toronto: Pearson Allyn & Bacon, c2011. Includes bibliographical references and index. ISBN 978-0-205-87195-7 (pbk.) 1. Mathematics—Study and teaching (Elementary). 2. Mathematics—Study and teaching (Middle school). I. Karp, Karen S., author II. Bay-Williams, Jennifer M., author III. McGarvey, Lynn M., author IV. Folk, Sandra, author V. Title.

QA135.6.V36 2014 372.7’044 C2013-906895-3

ISBN 978-0-20-587195-7

iii

About the Authors

John A. Van de Walle was a professor emeritus at Virginia Commonwealth University. He was a mathematics education consultant who regularly gave professional development workshops for K–8 teachers in the United States and Canada. He visited and taught in elemen-tary school classrooms and worked with teachers to implement student-centered math lessons. He co-authored the Scott Foresman-Addison Wesley Mathematics K–6 series and contributed to the Pearson School mathematics program enVisionMATH. Additionally, he wrote numerous chapters and articles for the National Council of Teachers of Mathematics (NCTM) books and journals and was very active in NCTM, including serving on the

board of directors, chairing the educational materials committee, and speaking at national and regional meetings.

Karen S. Karp is a professor of mathematics education at the University of Louisville (Kentucky). Prior to entering the field of teacher education she was an elementary school teacher in New York. Karen is a co-author of Feisty

Females: Inspiring Girls to Think Mathematically , which is aligned with her research interests on teaching mathematics to diverse populations. With Jennifer, Karen co-edited Growing Professionally: Readings from NCTM Publi-

cations for Grades K–8 and co-authored (along with Janet Caldwell) Developing

Essential Understanding of Addition and Subtraction for Teaching Mathematics

in Pre-K–Grade 2 . She is a former member of the board of directors of the National Council of Teachers of Mathematics (NCTM) and a former presi-dent of the Association of Mathematics Teacher Educators (AMTE). She con-

tinues to work in classrooms with elementary and middle school teachers and with teachers at all levels who work with students with disabilities.

Jennifer M. Bay-Williams is a professor of mathematics education at the University of Louisville (Kentucky). Jennifer has published many articles on teaching and learning in NCTM journals. She has also co-authored numerous books, including Developing Essential Understanding of Addition and Subtraction

for Teaching Mathematics in Pre-K–Grade 2 , Math and Literature: Grades 6–8, Math

and Nonfiction: Grades 6–8, and Navigating Through Connections in Grades 6–8 . Jennifer taught elementary, middle, and high school in Missouri and in Peru, and continues to work in classrooms at all levels with students and with teach-ers. Jennifer is on the board of directors for TODOS: Equity for All, is the editor for the 2012 NCTM Yearbook, and is a former president of the Associa-tion of Mathematics Teacher Educators (AMTE).

t

8ited and taught in elemen-to implement student-

iv

Jonathan Wray is the technology contributor to Elementary and

Middle School Mathematics: Teaching Developmentally . He is the instruc-tional facilitator for secondary mathematics curricular programs in the Howard County public school system. He is the president elect of the Association of Maryland Mathematics Teacher Educators (AMMTE) and past president of the Maryland Council of Teachers of Mathematics (MCTM). He has been recognized for his expertise in infusing technology in mathematics teaching, receiving the Out-standing Technology Leader in Education award for his school dis-trict from the Maryland Society for Educational Technology (MSET). Jon is also actively engaged in NCTM, serving on the editorial panels

of Teaching Children Mathematics and ON-Math. He has served as a primary and interme-diate grades classroom teacher, gifted/talented resource teacher, elementary mathematics specialist, curriculum and assessment developer, grant project manager, and educational consultant.

About the Contributor

About the Canadian Author

Lynn M. McGarvey is an associate professor of mathematics educa-tion at the University of Alberta. Lynn’s research focuses on the mathematical reasoning of young children. She has written many ar-ticles for research and professional audiences, particularly on the top-ics of algebraic thinking, spatial reasoning, and patterning in the early years. She is a long-time member of NCTM and has served on a number of task forces and committees, including as editorial panel chair for Teaching Children Mathematics . She is a former junior high school teacher who now spends a considerable amount of time work-ing with children and teachers in preschools and kindergartens. Lynn has taught mathematics curriculum and pedagogy courses to thou-sands of pre-service and in-service elementary teachers and has won multiple teaching awards for this work.

Brief Contents

v

SECTION I Teaching Mathematics: Foundations and Perspectives

Chapter 1 Teaching and Learning Mathematics in the Twenty-First Century 1

Chapter 2 Exploring What It Means to Know and Do Mathematics 11

Chapter 3 Mathematical Inquiry through Rich Tasks and Classroom Discourse 28

Chapter 4 Preparing to Teach and Planning for Mathematics Learning 45

Chapter 5 Blending Teaching and Assessment 64

Chapter 6 Teaching Mathematics for All Learners 84

Chapter 7 Tools for Learning Mathematics 103

SECTION II Development of Mathematical Concepts and Procedures

Chapter 8 Developing Early Number Concepts and Number Sense 119

Chapter 9 Developing Meanings for the Operations 139

Chapter 10 Helping Students Master the Basic Facts 162

Chapter 11 Developing Whole-Number Place Value Concepts 183

Chapter 12 Developing Strategies for Addition and Subtraction Computation 208

Chapter 13 Developing Strategies for Multiplication and Division Computation 231

Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions 253

Chapter 15 Developing Fraction Concepts 284

Chapter 16 Developing Strategies for Fraction Computation 310

Chapter 17 Developing Concepts of Decimals and Percents 333

Chapter 18 Proportional Reasoning 352

Chapter 19 Developing Measurement Concepts 369

Chapter 20 Geometric Thinking and Geometric Concepts 396

Chapter 21 Developing Concepts of Data Analysis 428

Chapter 22 Exploring Concepts of Probability 448

Chapter 23 Developing Concepts of Exponents, Integers, and Real Numbers 466

APPENDIX Guide to Blackline Masters 485

REFERENCES 497

INDEX 517

CHAPTER 1

Teaching and Learning Mathematics

in the Twenty-First Century 1

What Are Your Memories of School Mathematics? 1

Twenty-First Century Knowledge and Skills 2

Diversity in Today’s Classrooms 2

Mathematics Curriculum in Canada 3 Goals for Student Learning 4 Mathematics Content 4 Mathematical Processes 5

An Invitation to Learn and Grow 7 Becoming a Teacher of Mathematics 7

RESOURCES FOR CHAPTER 1

Recommended Readings 9 Online Resources 9

REFLECTIONS ON CHAPTER 1

Writing to Learn 10 For Discussion and Exploration 10

CHAPTER 2

Exploring What It Means to Know

and Do Mathematics 11

What Does It Mean to Do Mathematics? 11 A Classroom Environment for Doing Mathematics 12

An Invitation to Do Mathematics 12

Searching for Patterns 12 Analyzing a Situation 14 Generalizing Relationships 14 Experimenting and Explaining 16

Engaging in Mathematical Inquiry 16 Mathematics Proficiency 17

What Does It Mean to Learn Mathematics? 20

Mechanical Metaphors for Learning 20 Learning Mathematics as Acquisition 21 Learning Mathematics along a Linear Path 21

Ecological Metaphors for Learning 22 Learning Mathematics as Coherence 23 Learning Mathematics as Outward Growth 24 An Example of Mathematics Learning 24

Learning Theories and Implications for Teaching 25

Promoting Dispositions for Doing Mathematics 26





vii

Contents

Preface xvii

SECTION ITeaching Mathematics: Foundations and Perspectives

The fundamental core of effective teaching of mathematics combines an understanding of how children learn, how

to promote that learning by teaching through problem solving, and how to plan for and assess that learning on a

daily basis. Introductory chapters in this section provide perspectives on trends in mathematics education and the

process of doing mathematics. These chapters develop the core ideas of learning, teaching, planning, and assessment.

Additional perspectives on mathematics for children with diverse backgrounds and the role of learning tools

(e.g., manipulatives, technology) are also discussed.

CHAPTER 3

Mathematical Inquiry through Rich Tasks

and Classroom Discourse 28

Why Mathematical Inquiry? 28

A Shift in the Role of Tasks in the Classroom 29

The Focus of Inquiry 30

What Is a Rich Task? 30

Features of a Rich Task 31 Variable Entry and Exit Points 31 Thinking Mathematically 32 High Levels of Cognitive Demand 34 Relevant Contexts 34

Evaluating and Adapting Tasks: What Do You Need to Do to Make Them Rich(er)? 36

Task Selection Guide 36

Developing Concepts and Procedures through Tasks 36

Concepts 36 Procedures 38

What about Drill or Practice? 38 New Definitions of Drill and Practice 38 What Practice Provides 39 What Drill Provides 39 When Is Drill Appropriate? 39 Drill and Student Errors 39

Creating a Culture of Mathematical Inquiry 40 Classroom Discourse 40 Questioning Considerations 42 How Much to Tell and Not to Tell 42

Preparing to Teach 43





CHAPTER 4

Preparing to Teach and Planning

for Mathematics Learning 45

A Three-Phase Lesson Format 45 Introduction to Inquiry: The Before Phase of a Lesson 45 Teacher Actions in the Before Phase 46 Development of Inquiry: The During Phase of a Lesson 48 Teacher Actions in the During Phase 48

Follow-Up to Inquiry: The After Phase of a Lesson 50 Teacher Actions in the After Phase 50

Process for Preparing a Lesson 51 1. Curriculum Learning Expectations 52 2. Student Experiences 52 3. Select, Design, or Adapt a Rich Task 53 4. Design the Lesson Assessments Tools 54 5. Plan the Before Phase of the Lesson 54 6: Plan the During Phase of the Lesson 55 7: Plan the After Phase of the Lesson 55 Final Revisions to Your Lesson Plan 56 Applying the Planning Process 56

Applying the Three-Phase Lesson Template to Other Lesson Structures 56

Applying the Three-Phase Model to Short Tasks 56 Applying the Three-Phase Model in Learning Centres 57

Homework and Parental Involvement 57 Effective Homework 58 Beyond Homework: Families Doing Math 58 Resources for Families 59

Frequently Asked Questions 59





EXPANDED LESSON Fixed Area 62

CHAPTER 5

Blending Teaching and Assessment 64

Purposes for Assessing Learning in Classrooms 64

What Should Be Assessed? 66 Mathematics Content 66 Mathematical Processes 67 Mathematical Disposition 67

What Tools Can Be Used to Gather Evidence of Mathematics Learning? 67

Performance-Based Tasks 68 Questions and Conversations 69 Interviews 69

Writing to Learn and Assess 70 Journals 72 Admit and Exit Slips 73

Inventories 74

Summative Assessment Tools 74 Cumulative Projects 74

viii Contents

Teacher-Made Tests 75

Provincial Assessment Programs 76

National and International Studies 76 The Pan-Canadian Assessment Program (PCAP) 77 Trends in International Mathematics

and Science Study (TIMSS) 77

Ways to Document and Communicate Evidence of Learning 77

Documenting Formative Assessment 78 Rubrics 79

Grades 81 What Gets Graded Gets Valued 81 From Assessment Tools to Grades 82

Using Assessments to Shape Instruction 82





CHAPTER 6

Teaching Mathematics for All Learners 84

Differentiated Learning and Teaching 84 Tiered Tasks 84 Learning Centres 86 Choice Boards, Menus, and Think-Tac-Toe 86 Think-Pair-Share 86 Graphic Organizers 86

Diversity in Our Classrooms 87 First Nations, Métis, and Inuit Students 87 Students Who Are Culturally and Linguistically Diverse 88 Culturally Diverse Students 88 Students Who Are English Language Learners (ELLs) 90

Providing for Students with Special Needs 93 Prevention Models and Interventions for All Students 93 Students with Mild Disabilities 95

Students with Moderate/Severe Disabilities 97 Students Who Are Mathematically Gifted and Talented 99

Final Thoughts 101





CHAPTER 7

Tools for Learning Mathematics 103

Tools for Learning 103

Types of Learning Tools 105 Concrete Materials and Physical Models 106 Visual and Graphic Representations 107 Technology-Based Tools 108

Calculators in Mathematics Instruction 110 When to Use a Calculator 110 Benefits of Calculator Use 111 Graphing Calculators 112 Portable Data-Collection Devices 113

Digital Tools for Mathematics Learning 113

Instructional Applications 114 Concept Instruction 114 Problem Solving 114 Drill and Reinforcement 115

Guidelines for Selecting and Using Digital Resources 115 Guidelines for Using Digital Content 115 How to Select Appropriate Digital Content 115

Final Thoughts 116





Contents ix

CHAPTER 8

Developing Early Number Concepts

and Number Sense 119

Promoting Good Beginnings 119

Number Concepts: Quantity, Counting, and Knowing How Many 120

Quantity and the Ability to Subitize 120 Early Counting 121 Numeral Writing and Recognition 123 Counting On and Counting Back 124

Comparing Sets: More Than, Less Than, and Equal To 125

Early Number Sense 126

Relationships for Numbers 1 through 10 127 One and Two More, One and Two Less 127 Anchoring Numbers to 5 and 10 128 Part-Part-Whole Relationships 130 Dot Cards as a Model for Teaching Number

Relationships 133

Relationships for Numbers 10 through 20 134 Early Place Value Concepts 134 Extending More-Than and Less-Than

Relationships 134

Number Sense in Their World 135 Estimation and Measurement 135 Data Collection and Analysis 136 Classroom Routines 136


Literature Connections 137 Recommended Readings 137 Online Resources 138



CHAPTER 9

Developing Meanings for the Operations 139

Addition and Subtraction Problem Structures 140 Join and Separate Problems 140 Part-Part-Whole Problems 141 Compare Problems 141 Problem Difficulty 142 Computational and Semantic Forms of Equations 142

Teaching Addition and Subtraction 142 Contextual Problems 142

MATH MAKES SENSE 144

Model-Based Problems 145 Properties of Addition and Subtraction 147

Multiplication and Division Problem Structures 148 Equal Groups Problems 148 Comparison Problems 149 Area and Array Problems 149 Combination Problems 151

Teaching Multiplication and Division 151 Contextual Problems 151 Remainders 152 Model-Based Problems 153 Properties of Multiplication and Division 155

Strategies for Solving Contextual Problems 156 Analyzing Context Problems 156 Two-Step Problems 158





x Contents

SECTION IIDevelopment of Mathematical Concepts and Procedures

This section serves as the application of the core ideas of Section I. Here you will find chapters on every major con-

tent area in the pre-K–8 mathematics curriculum. Numerous inquiry-based tasks and problems to engage students

are interwoven with a discussion of the mathematical content and how children develop their understanding of that

content. At the outset of each chapter, you will find a listing of “Big Ideas,” the mathematical umbrella for the chapter.

Also included are ideas for incorporating children’s literature, technology, and assessment. These chapters are de-

signed to help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future.

CHAPTER 10

Helping Students Master the Basic Facts 162

Developmental Nature of Basic Fact Mastery 162 Approaches to Fact Mastery 163 Guiding Strategy Development 164

Reasoning Strategies for Addition Facts 165 One More Than and Two More Than 166 Adding Zero 167 Using 5 as an Anchor 167 Make 10 167 Up Over 10 167 Doubles 168 Near-Doubles 169 Reinforcing Reasoning Strategies 169

Reasoning Strategies for Subtraction Facts 170 Subtraction as Think-Addition 170 Down Over 10 171 Take from the 10 171

Reasoning Strategies for Multiplication and Division Facts 172

Doubles 172 Fives 172 Zeros and Ones 172 Nifty Nines 173 Using Known Facts to Derive Other Facts 174 Division Facts 175

Mastering the Basic Facts 175 Effective Drill 175 Games to Support Basic Fact Mastery 176

Fact Remediation 177 What to Do When Teaching Basic Facts 179 What Not to Do When Teaching Basic Facts 179





CHAPTER 11

Developing Whole-Number

Place Value Concepts 183

Numeration System 183

Pre-Base-Ten Understandings 184 Counting by Ones 184

Basic Ideas of Place Value 184 Integration of Base-Ten Groupings with Counting by Ones 185 Role of Counting 185 Integration of Groupings with Words 186 Integration of Groupings with Place Value Notation 186

Base-Ten Models for Place Value 187 Groupable Models 187 Pregrouped or Trading Models 187 Nonproportional Models 188

Developing Base-Ten Concepts 188 Grouping Activities 188 The Strangeness of Ones, Tens, and Hundreds 189 Grouping Tens to Make 100 190 Equivalent Representations 191

Oral and Written Names for Numbers 192 Two-Digit Number Names 192 Three-Digit Number Names 193 Written Symbols 194 Assessing Place Value Concepts 195

Patterns and Relationships with Multidigit Numbers 195

The Hundreds Chart 195

Relationships with Benchmark Numbers 197 Connecting Place Value to Addition and Subtraction 198 Connections to Real-World Ideas 202

Numbers beyond 1000 203 Extending the Place Value System 203 Conceptualizing Large Numbers 204





CHAPTER 12

Developing Strategies for Addition

and Subtraction Computation 208

Toward Computational Fluency 209 Student-Generated (Personal) Strategies 211 Standard Algorithms 212

Development of Meaningful Strategies 214 Creating an Environment for Meaningful Strategies 214 Models to Support Meaningful Strategies 214

Student-Generated Strategies for Addition and Subtraction 217

Adding and Subtracting Single-Digit Numbers 217 Adding Two-Digit Numbers 218

Contents xi

Subtracting by Counting Up 219 Take-Away Subtraction 219 Extensions and Challenges 220

Algorithms for Addition and Subtraction 221 Standard Algorithm for Addition 221 Alternative Algorithms for Addition 223 Standard Algorithm for Subtraction 223 Alternative Algorithms for Subtraction 224

Introducing Computational Estimation 225 Understanding Computational Estimation 225 Suggestions for Teaching Computational Estimation 225

Computational Estimation Strategies 227 Front-End Methods 227 Rounding Methods 227 Compatible Numbers 228





CHAPTER 13

Developing Strategies for Multiplication

and Division Computation 231

Towards Computational Fluency with Multiplication and Division 232

Development of Meaningful Strategies for Multiplication and Division 232

Models to Support Meaningful Strategies 232

Student-Generated Strategies for Multiplication 235

Complete-Number Strategies (Including Doubling) 236 Multiplication of Larger Numbers 236

Standard Algorithms for Multiplication 238 One-Digit Multipliers 238 Two-Digit Multipliers 239 Alternative Algorithms for Multiplication 240

Student-Generated Strategies for Division 240

Missing-Factor Strategies 240 Cluster Problems 241

Standard Algorithms for Division 242 One-Digit Divisors 242 Alternative Division Algorithms 244

Computational Estimation in Multiplication and Division 245

Understanding Computational Estimation 245 Suggestions for Teaching Computational Estimation 245

Computational Estimation from Student-Generated Strategies 246

Stop before the Details 246 Use Related Problem Sets 246

Computational Estimation Strategies 247 Front-End Methods 247 Rounding Methods 247 Compatible Numbers 248 Using Tens and Hundreds 248

Estimation Experiences 249 Calculator Activities 249 Using Whole Numbers to Estimate Rational Numbers 251





CHAPTER 14

Algebraic Thinking: Generalizations,

Patterns, and Functions 253

Algebraic Thinking 254

Generalization from Arithmetic 254 Generalization with Operations 254 Generalization in the Hundreds Chart 255 Generalization through Exploring a Pattern 256

Meaningful Use of Symbols 257 The Meaning of the Equal Sign 257 The Meaning of Variables 262

Making Structure in the Number System Explicit 265

Making Conjectures about Properties 265 Justifying Conjectures 267 Odd and Even Relationships 267

Study of Patterns and Functions 267 Repeating Patterns 268 Growing Patterns 270 Linear Functions 274

Mathematical Modelling 276

Teaching Considerations 277 Emphasize Appropriate Algebra Vocabulary 277 Connecting Representations 278 Algebraic Thinking across the Curriculum 280



xii Contents



CHAPTER 15

Developing Fraction Concepts 284

Meanings of Fractions 284 Fraction Constructs 285 Why Fractions Are So Difficult 285

Models for Fractions 286 Area Models 286 Length Models 288 Set Models 289

Concept of Fractional Parts 289 Fraction Size Is Relative 289 Fraction Language 290 Partitioning 290 Sharing Tasks 292 Iterating 293 Fraction Notation 296 Fractions Greater Than One 296

Estimating with Fractions 297

Equivalent Fractions 298 Conceptual Focus on Equivalence 298 Equivalent-Fraction Models 299 Developing an Equivalent-Fraction Algorithm 301

Comparing Fractions 303 Comparing Fractions Using Number Sense 303 Using Equivalent Fractions to Compare 305

Teaching Considerations for Fraction Concepts 305

PROBLEM-BASED LESSON Equivalent Fraction

Challenge 306





CHAPTER 16

Developing Strategies for Fraction

Computation 310

Understanding Fraction Operations 311 Conceptual Development Takes Time 311 An Inquiry-Based Number-Sense Approach 311 Computational Estimation 312

Addition and Subtraction 314 Contextual Examples and Student-Generated (Personal)

Strategies 314 Models 315 Developing the Algorithms 317 Fractions Greater Than One 319 Addressing Errors and Misconceptions 319

Multiplication 321 Contextual Examples and Models 321

Developing the Algorithms 324 Factors Greater Than One 325 Addressing Errors and Misconceptions 326

Division 326 Contextual Examples and Models 326 Answers That Are Not Whole Numbers 328 Developing the Algorithms 329 Addressing Errors and Misconceptions 330





CHAPTER 17

Developing Concepts of Decimals

and Percents 333

Extending the Place Value System 334 A Two-Way Relationship 334 Regrouping 334 The Role of the Decimal Point 335

Connecting Fractions and Decimals 336 Base-Ten Fractions 336

Developing Decimal Number Sense 339 Familiar Fractions Connected to Decimals 339

Computation with Decimals 343 The Role of Estimation 343 Addition and Subtraction 344 Multiplication 344 Division 346

Introducing Percents 346 Models and Terminology 347 Percent Problems in Context 348 Estimation 349



Contents xiii



CHAPTER 18

Proportional Reasoning 352

Ratios 352 Types of Ratios 352 Ratios Compared to Fractions 353 Two Ways to Think about Ratios 353

Proportional Reasoning 354 Proportional and Nonproportional Situations 354 Additive and Multiplicative Comparisons in Problems 355 Covariation 356

Develop a Wide Variety of Strategies 360 Mental Strategies 360 Ratio Tables 362

CONNECTED MATHEMATICS Grade 7, Comparing

and Scaling 363

Double Line (Strip) Comparison 364 Percents 365 Cross-Products 365

Teaching Proportional Reasoning 366





CHAPTER 19

Developing Measurement Concepts 369

The Meaning and Process of Measuring 370 Concepts and Skills 370 Introducing Nonstandard Units 372 Developing Standard Units 372 Instructional Goals 372 A Brief History of the Metric System 373 Important Standard Units and Relationships 373 The Role of Estimation and Approximation 374 Strategies for Estimating Measurements 374 Tips for Teaching Estimation 375 Measurement Estimation Activities 375

Length 376 Comparison Activities 376 Using Models of Length Units 376 Making and Using Rulers 378

Area 379 Comparison Activities 379 Using Models of Area Units 380 The Relationship between Area and Perimeter 381

Developing Formulas for Area 382 Student Errors and Misconceptions 382 Areas of Rectangles, Parallelograms, Triangles, and Trapezoids 383 Circumference and Area of Circles 385

Volume and Capacity 386 Comparison Activities 386 Using Models of Volume and Capacity Units 386 Using Measuring Cups 387 Developing Formulas for Volumes of Common Solid Shapes 387 Connections between Formulas 388

Mass and Weight 388 Comparison Activities 389 Using Models of Mass Units 389

Angles 390 Comparison Activities 390 Using Models of Angular Measure Units 390 Using Protractors and Angle Rulers 390

Time 391 Comparison Activities 391 Reading Clocks 392 Elapsed Time 392

Money 393 Recognizing Coins and Identifying Their Values 393 Counting Sets of Coins 393 Making Change 394





CHAPTER 20

Geometric Thinking

and Geometric Concepts 396

Geometry Goals for Students 396 Spatial Sense and Geometric Reasoning 397 Geometric Content 397

Developing Geometric Thinking 397 The van Hiele Levels of Geometric Thought 397 Implications for Instruction 400

Learning about Solids, Plane Figures, and Their Properties 401

Solids, Plane Figures, and Properties for Level-0 Thinkers 401

xiv Contents

Shapes and Properties for Level-1 Thinkers 405 Shapes and Properties for Level-2 Thinkers 410

Learning about Transformations 413 Transformations for Level-0 Thinkers 413 Transformations for Level-1 Thinkers 415 Transformations for Level-2 Thinkers 417

Learning about Location 417 Location for Level-1 Thinkers 419 Location for Level-2 Thinkers 421

Learning about Visualization 421 Visualization for Level-0 Thinkers 422 Visualization for Level-1 Thinkers 423 Visualization for Level-2 Thinkers 424





CHAPTER 21

Developing Concepts of Data Analysis 428

What Does It Mean to Do Statistics? 429 Is It Statistics or Is It Mathematics? 429 The Shape of Data 429 The Process of Doing Statistics 430

Formulating Questions 430 Classroom Questions 430 Beyond One Classroom 431

Data Collection 431 Collecting Data 431 Using Existing Data Sources 432

Data Analysis: Classification 432 Attribute Materials 433

Data Analysis: Graphical Representations 434 Bar Graphs and Tally Charts 435 Circle Graphs 436 Numerical Data Graphs 437 Scatter Plots 439

Data Analysis: Measures of Centre and Variability 440

Averages 440 Understanding the Mean: Two Interpretations 441 Variability 443 Box Plots 444

Interpreting Results 445





CHAPTER 22

Exploring Concepts of Probability 448

Introducing Probability 448 Likely or Not Likely 449 The Probability Continuum 450

Theoretical Probability and Experiments 453 Theoretical Probability 453 Experiments 454 Why Use Experiments? 456 Use of Technology in Experiments 456

Sample Spaces and Probability of Two Events 457 Independent Events 457 Area Models 459 Dependent Events 460

Simulations 461





CHAPTER 23

Developing Concepts of Exponents,

Integers, and Real Numbers 466

Exponents 466 Exponents in Expressions and Equations 466 Order of Operations 467 Negative Exponents 470 Scientific Notation 470

Integers 472 Contexts for Exploring Integers 472 Quantity Contexts 472 Linear Contexts 473 Meaning of Negative Numbers 474 Models for Teaching Integers 474

Contents xv

xvi Contents

Operations with Integers 475 Addition and Subtraction 475 Multiplication and Division 478

Real Numbers 478 Rational Numbers 479 Square Roots and Cube Roots 480





APPENDIXGuide to Blackline Masters 485

References 497

Index 517

xvii

New to This Edition The fourth Canadian edition has been revised to include the following changes to better prepare teachers to teach mathematics to Canadian learners:

● Section I ( Chapters 1 – 7 ) has been substantially revised to emphasize and describe math-ematics teaching, learning, and curriculum across Canada. In particular, twenty-first century knowledge and skills including problem solving, creativity, collaboration, com-munication, and social responsibility are introduced.

● In the third edition, Chapter 1 was based substantially on two aspects: NCTM Prin-ciples and Standards for School Mathematics (NCTM, 2000), and provincial, national, and international testing programs. Chapter 1 now examines the curriculum documents of the provinces and territories across Canada and notes their similarities and dif-ferences in terms of goals for student learning, mathematics content, and mathematical processes.

● Diverse learners are also introduced in Chapter 1 as diversity is the norm in most Canadian classrooms. Diverse learning needs are described here and throughout the text as opportunities to enhance learning for all students rather than as something to eliminate. The chapter explicitly addresses First Nations, Métis, and Inuit students, English language learners, and learners with special needs including those with cognitive, behavioural, and physical exceptionalities.

● Chapter 2 is substantially revised to expand on the nature of mathematics —from being the science of patterns or as problem solving to a more encompassing perspective that mathematics is a way of thinking and seeing . The tasks within all the chapters are revised to reflect this expanded viewpoint. Canadian examples are used throughout. The Chapter 2 section on learning theory is also revised. Rather than singling out specific learning theories, the emphasis is on helping teachers recognize widespread beliefs about learning and practices reflected in the metaphors we use (e.g., “I’m falling behind”).

● The first two chapters set the stage for revisions in Chapter 3 . Its previous focus on teaching through problem solving now takes a broader focus on mathematical in-quiry through rich tasks and classroom discourse . A more in-depth discussion is provided that outlines the features of a rich task using a multitude of examples. The discussion surrounding tasks such as drill and homework have been incorporated into Chapter 3 . Problem solving is considered one form of inquiry alongside other types of worthwhile tasks. Given this shift in focus, mathematical habits of mind replace prob-lem solving strategies to encompass broader perspectives of mathematical thinking and doing.

● In previous editions, lesson preparation and planning based on a three-phase lesson format was been spread across two chapters. In this fourth Canadian edition, preparation and planning are placed together in Chapter 4 , entitled “Preparing to Teach and Planning for Mathematics Learning.”

● Chapter 5 has been revised to use the framework of assessment for learning, as learning, and of learning throughout, using the questions: For what purpose? Who is responsible? What is assessed? What tools are used? and How is assessment data recorded? Substantial attention is paid to assessment of content, processes, and dispositions within Canadian

Preface

curriculum documents and in this chapter. Many more examples of children’s work and assessment tools have been added to this chapter.

● Chapter 6 , now Teaching Mathematics for All Learners, has also been substantially revised. In the previous edition, the chapter focused extensively on specific disabilities and general ways to address specific learning needs. The current chapter has been revised to focus on differentiated teaching and learning ; in particular, it builds on criteria for rich tasks from Chapter 3 to create tiered tasks, learning centres, choice boards, and so on. Diversity within the classroom is also addressed in more detail and takes into account different cultural ways of learning, culturally responsive teaching strategies, and linguistic diversity. Students with special needs are addressed in more detail through the framework of Response to Intervention, which offers a way to consider a tiered student support system and forms of intervention.

● The focus for Chapter 7 , which was previously on technology and school mathemat-ics, has been expanded to address Tools for Learning Mathematics , of which tech-nology is one type. The chapter is based on three types of tools used in Canadian curriculum, including (1) concrete materials and physical models (e.g., manipulatives); (2) visual and graphic representations (e.g., pictures, diagrams, schematic drawings, graphs); and (3) technology-based tools (e.g., calculators, dynamic geometry software). The chapter provides details on how tools can be used appropriately in the classroom to enhance learning.

Section II of the Textbook has been substantially updated with new research, along with a number of new features.

● New adaptations and accommodations for diverse learners, including English language learners and students with difficulties in mathematics, appear not only in the narrative in Section I but also in many activities through direct examples and descriptions for the various content areas in Section II. The increased emphasis on diversity will be obvious to those who have used the book in the past.

● New samples of authentic student work illustrate student thinking. Student responses present glimpses into how students think about problems and what students’ written work on mathematical tasks looks like, increasing teachers’ awareness of how rich students’ mathematical thinking can be—and how high our expectations should be.

● Increased early childhood coverage provides expanded emphasis on and reorganization of early numeracy in Chapters 8 and 9 . Based on learning for the areas of number, rela-tions, and operations, the work with early learners is seen as the essential foundation for number sense and mathematical ways of thinking.

● New Formative Assessment Notes in each chapter in Section II guide readers through ideas they can test with individual students or students in groups. Formative assessment is one of the key tools in finding out what students are thinking, and thereby identifying their areas of strength and weakness. Chapter 5 contains a more detailed description of forma-tive assessments organized in three major assessment areas: tasks, observation, and inter-views. To bring these ideas to life and to make them more directly linked to the content, these Formative Assessment Notes are included throughout the content area chapters to support teachers in the effective use of formative assessment, which is directly connected to increased student achievement.

● Extensively updated information on how to effectively integrate new techno-logical tools to support teaching and learning appears in select Activities throughout the text.

● A reorganization of Chapters 12 and 13 emphasizes both strategies for computa-tion and estimation for addition and subtraction in Chapter 12 , and the same for

xviii Preface

multiplication and division in Chapter 13 . This is a change from the third edition, which separated developing strategies for whole number computation and estimation for the four operations.

● A discussion on engaging families in meaningful ways to help students learn mathemat-ics appears in Chapter 4 .

● Additional attention to classroom discourse now appears in Chapter 3, Mathematical Inquiry through Rich Tasks and Classroom Discourse. The coverage includes how to conduct productive discussion sessions and develop effective questioning, and is illustrated with a vignette.

Other Changes of Note Much has changed on the landscape of mathematics education, and so many aspects of the book have been updated to reflect those changes. In addition to the changes listed above, the follow-ing substantive changes have been made:

● There is an increased focus on the research-based developmental model of developing basic facts, and new activities to support basic fact mastery appear in Chapter 10 .

● The content on algebraic thinking has been adapted to align with current research and changes to curricula across Canada. There is an increased emphasis on equivalence and variables, including adding the number-line representation of variables and making the properties more explicit.

● Chapter 15 (Developing Fraction Concepts) has greatly expanded sections on partitioning and on equivalence to reflect three recent research reviews that have indicated that this is essential to all advanced fraction work and success in algebra.

● Chapter 16 (Developing Strategies for Fraction Computation) now includes Activities—10 new ideas for developing understanding of fraction operations.

● Chapter 18 has been shortened, had new activities added, and been refocused to address understanding of ratios more deeply (with less focus on connecting to other content areas).

● The chapter on measurement, Chapter 19 , has been reorganized. Previously, the development of all measurement formulas was shared at the end of the chapter; now, the formulas are integrated with the corresponding measurement topic (e.g., area or volume). The content has also been revised to ensure the focus is on metric measurement.

● Chapter 21 gives more explicit attention to distinguishing between numerical data and categorical data.

● Chapter 23 includes a significantly revised section on order of operations and numerous new activities.

What You Will Find in This Book When you look at the table of contents, you will see that the chapters are separated into two distinct sections. The first section, consisting of seven chapters, deals with important ideas that cross the boundaries of specific areas of content. The second section, consisting of 16 chapters, offers teaching suggestions and activities for every major mathematics topic in the pre-kindergarten to Grade 8 curriculum. Chapters in Section I offer perspectives on the challenging task of helping students learn mathematics. Having a feel for the disci-pline of mathematics—that is, to know what it means to “do mathematics”—is critical to learning how to teach mathematics well. In addition, understanding perspectives on learning

Preface xix

253

As Fosnot and Jacob (2010) write in their Constructing Algebra volume of Young Mathematicians at Work , “It is human to seek and build relations. The mind cannot pro-cess the multitude of stimuli in our surroundings and make meaning of them without developing a network of relations” (p. 12 ).

BIG IDEAS 1. Algebra is a useful tool for generalizing arithmetic and representing

patterns and regularities in our world.

2. Symbolism, especially involving equality and variables, must be well understood conceptually for students to be successful in mathemat-ics, particularly algebra.

3. Methods we use to compute and the structures in our number system can and should be generalized. For example, the generaliza-tion that a 1 b 5 b 1 a tells us that 83 1 27 5 27 1 83 without computing the sums on each side of the equal sign.

4. Patterns, both repeating and growing, can be recognized, extended, and generalized.

5. Functions in K–8 mathematics describe in concrete ways the notion that for every input, there is a unique output.

6. The understanding of functions is strengthened when functions are explored across representations, as each representation provides a different view of the same relationship.

Mathematics CONTENT CONNECTIONS

It is difficult to find an area of mathematics that does not involve generalizing and formalizing in some central way. In fact, this type of

Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions

Baloo -Rex May- / CartoonStock

F or many people, algebra is the topic that sent their mathematical careers “downhill.” Algebra is where

students begin to see letters in place of numbers, may expe-rience seemingly nonsensical manipulations of equations, and where everything they learned in elementary school arithmetic no longer seems applicable. When introduced in the middle years, algebra often becomes the fork in the road—the dividing point between those who “can” and those who “can’t” do mathematics.

The challenges of learning algebra have led to impor-tant research. Significant research over the past 30 years has focused on the difficult transition from arithmetic to alge-bra. This work has highlighted some of the obstacles to learning algebra due to the way arithmetic has been taught (Bednarz, Kieran, & Lee, 1996; Kieran, 2006; Mason, 2001). This earlier work has led to a transformation of curriculum in the early years (Cai & Knuth, 2011).

In Canada, algebraic thinking has become an important theme across the curriculum in the elementary and middle years in topics such as patterns or patterning, relations, number properties, variables and equations, and functions. The algebra envisioned for these grades is not the algebra that you may have experienced, including primarily symbol manipulation procedures and artificial applications with little connection to the real world. The focus now is on the type of thinking and reasoning that prepares students to think mathematically across all areas of mathematics.

Algebraic thinking (or algebraic reasoning) involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions. Far from a topic with little real-world use, algebraic thinking pervades all of mathematics and is essential for making mathematics useful in daily life.

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BIG IDEAS 1. Algebra is a useful tool for generalizing arithmetic and representing

patterns and regularities in our world.

2. Symbolism, especially involving equality and variables, must be well understood conceptually for students to be successful in mathemat-ics, particularly algebra.

3. Methods we use to compute and the structures in our number system can and should be generalized. For example, the generaliza-tion that a 1 b 5 b 1 a tells us that 83 1 27 5 27 1 83 without computing the sums on each side of the equal sign.

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Mathematics CONTENT CONNECTIONS

What students bring to the topic of fractions is an understanding of fair sharing. Other whole-number ideas can actually interfere in early frac-tion development, as discussed later in this chapter. Fraction concepts are intimately connected to other areas of the curriculum. In addition to the content connections listed below, fractions are used frequently in measurement ( Chapter 19 ) and in probability ( Chapter 22 ).

◆ Algebraic Thinking ( Chapter 14 ): As described in Chapter 14 , fractions are a part of algebra. Equations with variables often involve fractions or can be solved using fractions. For example, x4 5 5

16

is an equation involving equivalent fractions.

◆ Fraction Computation ( Chapter 16 ): Without a firm concep-tual understanding of fractions, computation with fractions is rel-egated to rules without reasons.

◆ Decimals and Percents ( Chapter 17 ): A key idea for students is that decimal notation and percent notation are simply two other representations of numbers. By making the connections between

mathematics and how different perspectives reflect different approaches to teaching provides a foundation and rationale for how to teach and assess pre-K–8 students.

Importantly, you will be teaching diverse students, including students who are English language learners, are gifted, and/or have difficulties learning. You will learn how to apply instructional strategies in ways that support and challenge all learners. Formative assess-ment strategies, strategies for diverse learners, and effective use of learning tools are addressed in specific chapters in Section I ( Chapters 5 , 6 , and 7 , respectively), and through-out the Section II chapters.

Each chapter of Section II focuses on one of the major content areas in pre-K–8 mathematics curriculum. It begins with identifying the big ideas for that content, then provides guidance on how students best learn that content, along with many worthwhile tasks to engage students in understanding mathematics. Reflecting on the activities as you read can help you think about the mathematics from the perspective of the student. As often as possible, take out pencil and paper and try the problems so that you actively engage in your learning about students learning mathematics. We hope this book will increase your own understanding of mathematics, of the students you teach, and of how to teach them well.

Some Special Features of this Text Upon flipping through the book, you will notice many sec-tion headings, a large number of figures, and various special features. All are designed to make the book more useful as a textbook and as a long-term resource. Here are a few things to look for.

Big Ideas ▼ Much of the current research and literature in mathematics education suggests that teachers plan their instruction around “big ideas” rather than isolated skills or concepts. At the begin-ning of each chapter in Section II, you will find a list of the key mathematical ideas associated with the chapter. Teachers find these lists help-ful for quickly getting a picture of the mathe-matics they are teaching.

Mathematics Content Connections ▼ Following the Big Ideas lists are brief descrip-tions of other content areas in mathematics that are related to the content of the current chap-ter. These lists are offered to help you be more aware of the potential interaction of content as you plan lessons, diagnose students’ difficulties, and learn more yourself about the mathematics you are teaching.

xx Preface

Data Analysis: Measures of Centre and Variability 443

Changes in the Mean. The balance approach to finding

the mean clearly illustrates that different data distributions

can have the same mean. Especially for small sets of data, the

mean is significantly affected by extreme values. For example,

suppose another toy with a price of $20 is added to the six we

have been using in the examples. How will the mean change?

If the $1 toy were removed, how would the mean be affected?

Suppose that one new toy is added that increases the mean

from $6 to $7. How much does the new toy cost? Students

should be challenged with questions such as these using small

sets of data and either the balance or the levelling concept.

The NCTM e-Examples, Applet 6.6, “Compar-

ing Properties of the Mean and the Median,”

shows seven data points that can be dragged

back and forth along a number line with the

mean and median updated instantly. The applet allows stu-

dents to see how stable the median is and how changing one

point can affect the mean.

Consider using a diagnostic interview

to assess whether students are able to

determine the best measure of centre to

use in a given situation, such as the

average height of students in the class. You can begin with

general questions such as these: “What is an average? What

does the mean represent? What does the median represent?

What is the difference between the mean and the median?

What is each useful for?” Then move to more analytical

questions: “Which should we use for this set of data? Might

we use a different measure of centre in another class? When

you’ve found the average height of the students in our class,

is it possible that no one is that height? Why?” ■

Variability

Measures of variability also need explicit attention (Franklin

& Garfield, 2006; Franklin et al., 2005; Rossman, Chance,

& Medina, 2006; Scheaffer, 2006). Students often do not

have a clear understanding of variability because the time

spent analyzing measure of centre dominates the data

analysis portion of the teacher’s long-range plan. Shaugh-

nessy (2006) summarized the findings on what students

should know about variability in the following list, starting

with basic notions and progressing to more sophisti-

cated ideas:

1. attending to outliers or extremes (but not necessarily

on the full distribution of the data)

2. considering change over time (which can lead into dis-

cussions of other types of variation)

3. examining variability as the full range of data (Range is

everything that occurs, but it doesn’t reveal the fre-

quency of different events within the range.)

4. considering variability as the likely range or expected

value 5. looking at how far data points are from the centre (e.g.,

the mean)

6. looking at how far off a set of data is from some fixed

value

In order to be prepared to teach students variability

beyond outliers and extremes, it is important to be aware

of the sources of variability that occur in statistics. Frank-

lin and colleagues (2005) suggest that students focus on

increasingly sophisticated sources of variability, starting

with variability within a group (e.g., the varying lengths

of students’ names, varying family sizes, and so on). When

students create a bar graph of class data and compare

the data collected, they are discussing the variability

within a group. Next is variability within and between groups. Students

might compare the variability of favourite music choices

of Grade 5 students compared to students in Grade 8.

In addition, middle school students study how the change

in one variable relates to change in another variable—yes,

algebra! Students analyze two variables to see whether

there is a relationship (as discussed earlier in the section on

scatter plots). Students also explore sampling variability

(Franklin et al., 2005). When students flip a coin 10 times

as a sample, they may get 5 heads and 5 tails, but they also

may get many other results (even 0 heads and 10 tails). This

is sampling variability. The larger the sample, the more the

data reflect the expected values (50 percent heads, 50 per-

cent tails). Lastly, students can examine natural and induced vari-

ability. For example, plants grow at different rates. When

one flower naturally grows taller than the one right next to

it in the garden, that is natural variability. If the two plants

were in two different gardens, then other variables come

into play: fertilization, amount of sunlight, amount of water,

and so on, which can “induce” different growth rates.

Knowing these variability terms is less important than

knowing that in designing an experiment, we must look at

one factor (e.g., sunlight) and all other factors should be

0 1 2 3 4 5 6 7 8 9 10 11 12

FIGURE 21.17 Move data points in toward the centre or

balance point without changing the balance around that point.

When you have all points at the same value, that is the balance

or the mean .

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Activities ▼ The numerous activities found in every chapter of Section II have always been rated by readers as one of the most valu-able parts of the book. Some activity ideas are described directly in the text and in the illustra-tions. Others are pre-sented in the numbered Activity boxes. Every activity is a problem-based task (as described in Chapter 3 ) and is designed to engage stu-dents in doing mathe-matics. New adaptations and accommodations for diverse learners and differentiation strategies for students with difficulties are included in many activities.

Technology Ideas ▼ Technology is an important tool for learning mathematics, as you will learn in Chapter 7 . We have infused technology ideas throughout Section II. An icon is used to identify those places within the text or an activity where a technol-ogy idea or resource is discussed. Descriptions include open-source (free) software, applets, and other Web-based resources, as well as cal-culator ideas.

Formative Assessment Notes ▼ Assessment should be an inte-gral part of instruction; so, it makes sense to think about what to be listening for (assessing) as you read about different areas of content devel-opment. Throughout the content chapters, you will see Formative Assessment Note icons indicating a short description of ways to assess the topic in that section. Reading these assessment notes as you read the text can help you understand how best to help your students.

Preface xxi

gforms of the fraction, including equivalent forms where appropriate.

Activity 17.2

Base-Ten Fractions to Decimals For this activity, have students use their paper place value strips and squares (Blackline Master 14). Agree that the large square represents 1. Have students cover a base-ten fractional amount of the square using their strips and tinies (remember to call the pieces “tenths” and “hun-dredths”). For example, have them cover 2 35

100 of the square. Whole numbers require additional squares. The task is to decide how to write and say this fraction as a decimal and demonstrate the connection using their physical models. For students with diff iculties, you may want to have the amount shaded rather than have the students try to cover the exact amount; then ask them to name and write the decimal fraction.

The calculator can also play a significant role in devel-oping decimal concepts.

Activity 17.3

Calculator Decimal Counting Recall how to make the calculator “count” by pressing 1 1 . . . . Now have

students press 0.1 . . . . When the dis-play shows 0.9, stop and discuss what this means and what the display will look like with the next count. Many students will predict 0.10 (thinking that 10 comes after 9). This prediction is even more interesting if, with each press, the students have been accumulating base-ten strips as models for tenths. One more press would mean one more strip, or 10 strips. Why doesn’t the calculator show 0.10? When the tenth press produces a display of 1 (calculators are not usually set to display trailing zeros to the right of the decimal), the discussion should revolve around trading 10 strips for a square. Continue to count to 4 or 5 by tenths. How many presses does it take to get from one whole number to the

DIVERSE LEARNERSDIFFERENTIATION

STRATEGIES

DIFFERENTIATION STRATEGIES

End-of-Chapter Resources ▼ The end of each chapter includes two major subsections: Resources , which includes “Litera-ture Connections” (found in all Section II chap-ters), “Recommended Readings,” and “Online Resources”; and Reflections , which includes “Writing to Learn” and “For Discussion and Exploration.”

Literature Connections Section II chapters contain examples of great children’s literature for launching into the mathematics concepts in the chapter just read.

Recommended Readings In this section, you will find an annotated list of articles and books to augment the information found in the chapter.

Online Resources At the end of each chapter, you will find an annotated list of some of the best Web-based resources along with their website addresses so that you can further explore how to infuse technological tools into instruction to support student learning.

Writing to Learn Questions are provided that help you reflect on the impor-tant pedagogical ideas related to the content in the chapter.

For Discussion and Exploration These questions ask you to explore an issue related to that chapter’s content, applying what you have learned.

xxii Preface

Reflections on Chapter 17 351

sneaker representing 0.50 or 50 percent of a pair of shoes. Real-world links such as one-seventh of a week and one- eleventh of a soccer team will resonate with students. Note that some decimals and percents are rounded.

Recommended Readings

Articles

Cramer, K., Monson, D., Wyberg, T., Leavitt, S., & Whitney, S. B. (2009). Models for initial decimal ideas. Teaching Chil-dren Mathematics, 16 (2), 106–117. This article describes ways of using 10 3 10 grids and decimal addition and subtraction boards to enhance students’ under-standing of decimals. Several diagnostic interviews and an emphasis on having students use words, pictures, and numbers are included.

Suh, J. M., Johnston, C., Jamieson, S., & Mills, M. (2008). Pro-moting decimal number sense and representational fluency. Mathematics Teaching in the Middle School, 14 (1), 44–50. A group of Grade 5 and Grade 6 teachers in a lesson study group explored a variety of representations to develop students’ profi-ciency with decimals. Ideas for games and strategies for ELLs and students with special needs are shared.

Online Resources

Base Blocks—Decimals http://nlvm.usu.edu/en/nav/frames_asid_187_g_2_t_1.html?open5instructions&from5topic_t_1.html

Base-ten blocks can be placed on a place value chart. The number of decimal places can be selected, thus designating any of the four blocks as the unit. Addition and subtraction problems can be created or can be generated randomly.

Circle 3 http://nlvm.usu.edu/en/nav/frames_asid_187_g_2_t_1.html?open5instructions&from5topic_t_1.html

This game challenges students to use logic as they combine decimals to add to 3. Not as easy as it sounds.

Concentration http://illuminations.nctm.org/ActivityDetail.aspx?ID573

This is an engaging matching game using representations of percents, fractions, and a regional model.

Fraction Model—Version 3 http://illuminations.nctm.org/activitydetail.aspx?ID511

Explore equivalence of fraction, decimal, and percent rep-resentations using length, area, region, and set models.

REFLECTIONS on Chapter 17

Writing to Learn 1. Describe three different base-ten models for fractions and

decimals, and use each to illustrate how base-ten fractions can be represented.

2. How can we help students think about very small place val-ues, such as thousandths and millionths, in the same way we get students to think about very large place values, such as millions and billions?

3. Use an example involving base-ten pieces to explain the role of the decimal point in separating the whole from the frac-tional units. Relate this idea to changing units of measure-ment as in money or metric measures.

4. Explain how the line-up-the-decimals rule for adding and sub-tracting can be developed through practice with estimation.

5. Give an example explaining how, in many problems, multi-plication and division with decimals can be replaced with estimation and whole-number methods.

For Discussion and Exploration 1. A way you may have learned to order a series of decimals

such as 0.34, 0.3004, and 0.059 is to annex zeros to each number so that all numbers have the same number of deci-mal places. For example, rewrite

0.34 as 0.3400 0.3004 as 0.3004 0.059 as 0.0590 Now, ignore the decimal points and any leading zeros, and

order the resulting whole numbers. This method was found to detract from students’ conceptual understanding (Roche & Clarke, 2004). Why do you think that was the case? What should you try instead?

Visit MyEducationLab to access an electronic version of the text, as well as a variety of topics that enhance the text material. The topics include the following to support your learning in the course:

● Assessment, including Building Teaching Skills and Dispositions and Video Assignments ● Discussion board questions ● Videos, simulations, activities, case studies, and other useful course resources

308 Chapter 15 Developing Fraction Concepts

RESOURCES for Chapter 15

Literature Connections

Context takes students away from rules and encourages them to explore ideas in a more open and meaningful manner. The way that students approach fraction concepts in these contexts may surprise you.

How Many Snails? A Counting Book (Giganti, 1988)

Each page of this book has a similar pattern of questions. For example, the narrator wonders how many clouds there are, how many of them are big and fluffy, and how many of them are big and fluffy and gray. Students can look at the pictures and find the fraction of the objects (e.g., clouds) that have the particular characteristic (big and fluffy). Whitin and Whitin (2006) describe how a class used this book to write their own stories in this pattern and record the fractions for each subset of the objects.

The Doorbell Rang (Hutchins, 1986)

Often used to investigate whole-number operations of multi-plication and division, this book is also an excellent early intro-duction to fractions. The story is a simple tale of two children preparing to share a plate of 12 cookies. Just as they have fig-ured out how to share the cookies, the doorbell rings and more children arrive. You can change the number of children to cre-ate a sharing situation that requires fractions (e.g., 5 children).

The Man Who Counted: A Collection of Mathematical Adventures (Tahan, 1993)

This book contains a story, “Beasts of Burden,” about a wise mathematician, Beremiz, and the narrator, who are travelling together on one camel. They are asked by three brothers to solve an argument. Their father has left them 35 camels to divide among them: half to one brother, one-third to another, and one-ninth to the third brother. The story provides an excellent context for discussing fractional parts of sets and how fractional parts change as the whole changes. However, if the whole is changed from 35 to, say, 36 or 34, the problem of the indicated shares remains unresolved. The sum of 1

2, 13, and 1

9 will never be one whole, no matter how many camels are involved. Bresser (1995) describes three days of activities he conducted with his Grade 5 students based on this book.

Recommended Readings

Articles

Clarke, D. M., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make sense. Math-ematics Teaching in the Middle School, 13 (7), 373–380. Each of the excellent tips is discussed, and favorite activities are shared. An excellent overview of teaching fractions.

Flores, A., & Klein, E. (2005). From students’ problem-solving strategies to connections in fractions. Teaching Children Mathematics, 11 (9), 452–457. This article offers a very realistic view (complete with photos of student work) of how children develop initial fraction concepts and an understanding of notation as they engage in sharing tasks like those described in this chapter.

Books

Burns, M. (2001). Teaching arithmetic: Lessons for introduc-ing fractions, grades 4–5. Sausalito, CA: Math Solutions Publications. This book offers well-designed lessons with lots of details, sample student dialogue, and Blackline Masters. These are introductory ideas for fraction concepts. Five lessons cover one-half as a bench-mark. Assessments are also included.

McNamara, J., & Shaughnessy, M. M. (2010). Beyond pizzas and pies: 10 essential strategies for supporting fraction sense (grades 3–5). Sausalito, CA: Math Solutions Publications. This book has it all—classroom vignettes, discussion of research on teaching fractions, and many activities, including student work.

Online Resources

Cyberchase (PBS) http://pbskids.org/cyberchase

Cyberchase is a very popular television series targeting important mathematics. The site offers videos that model fractions with real-world connections. Also offered are activities such as “Make a Match” ( http://pbskids.org/cyberchase/games/equivalentfractions ), in which students examine the concept of equivalent fractions and match a fraction with a graphic representation of that fraction. Another activity is “Thirteen Ways of Looking at a Half ” ( http://pbskids.org/cyberchase/math-games/thirteen-ways-looking-half/ ), in which students explore fractions of geometric shapes—in particular, the 13 ways half of an eight-piece square can be arranged.

Fraction Bars (Math Playground) http://mathplayground.com/Fraction_bars.html

The user sets the total parts and then the shaded parts for each bar. Explore fractional parts, the concepts of numera-tor and denominator, and equivalence. The user can turn the numbers on or off.

Fractions Model (Illuminations) http://illuminations.nctm.org/ActivityDetail.aspx?ID=11

Explore length, area, region, and set models of fractions including improper fractions, mixed numbers, decimals, and percentages.

Appendix ▼ The Appendix contains images of all of the Blackline Masters.

Expanded Lessons ▼ An example of an Expanded Lesson can be found at the end of Chapter 4 . In addition, eight similar Expanded Lessons can be found on MyEducationLab at www.myeducationlab.com .

Preface xxiii

Appendix

Guide to Blackline Masters

0.5-cm square grid 36

1-cm isometric dot grid 39

1-cm square/diagonal grid 40

1-cm square dot grid 37

1-cm square grid 35

2-cm isometric grid 38

2-cm square grid 34

2 more than 63

2 less than 64

10 3 10 grids 27

10 3 10 multiplication array 12

10,000 grid 29

Addition and subtraction recording charts 19

Assorted shapes 41–47

Assorted triangles 58

Base-ten grid paper 18

Base-ten materials 14

Blank hundreds chart (10 3 10 square) 21

Circular fraction pieces 24–26

Clock faces 33

Coordinate grid 48

Create a journey story 71

Crooked paths 72

Degrees and wedges 32

Design a bag 60

Dot cards 3–8

Double ten-frame 11

Five-frame 9

Fixed area recording sheet 74

Four small hundreds charts 23

Fraction names 66

Geoboard pattern 49

Geoboard recording sheets 50

How long? 65

Hundreds chart 22

It's a matter of rates 68

Little ten-frames 15–16

Look-alike rectangles 30

Look-alike rectangles recording sheet 31

Looking at collections 62

Missing-part worksheet 13

More-or-less cards 1

Motion man 52–53

Multiplication and division recording charts 20

Number cards 2

Place-value mat (with ten-frames) 17

Predict how many 69–70

Properties of quadrilateral diagonals 75

Property lists for quadrilaterals 54–57

Rational number wheel 28

Rectangles made with 36 tiles 73

Solving problems involving fractions 67

Tangrams and mosaic puzzle 51

Ten-frame 10

Toying with measures 77

Toy purchases 76

What are the chances? 61

Woozle cards 59

This Appendix contains images of all of the Blackline Masters (BLM) that are listed below. The full size masters can be found on the MyEducationLab website (www.myeducationlab.com).

62 Chapter 4 Preparing to Teach and Planning for Mathematics Learning

E X P A N D E D L E S S O N Fixed Area

GRADE LEVEL: 5

Curriculum Focus ● To design and construct different rectangles given a fixed area (whole numbers) and draw conclusions

● To develop an understanding of the relationship between the area and the perimeter of different shapes, particularly rect-angles, when the area stays the same

● To compare and contrast the units used to measure perim-eter and those used to measure area

Student Experiences Students have worked with the concepts of area and perimeter. Some, if not the majority of students, are able to find the area and perimeter of given figures. They may even be able to state the formulas for finding the perimeter and area of a rectangle. However, they may be confused about the terms and the types of units used for each form of measurement.

Materials and Preparation Each Student will Need

● 36 square tiles, such as colour tiles ● Two or three sheets of 0.75 cm grid paper or 1 cm grid paper on legal paper

Teacher will Need

● Document camera to show student records

Before Begin with a Simpler Version of the Task

● Have students create, at their desk, a rectangle using 12 tiles only and record the image on grid paper. After eliciting some ideas of what this might look like, invite a student to come to the document camera and show their drawing for one rectangle. Possible solutions include 1 3 12, 2 3 6, or 3 3 4 rectangles.

● Emphasize the need to record the dimensions of the rect-angle, for example, “2 units by 6 units.” Create a chart to model how you would like students to record the rectangles they find. Write the heading “Dimension” and record the rectangle as “2 units 3 6 units” below.

● Ask, “What do we mean by perimeter? How do we measure perimeter?” After helping students define perimeter and describe how it is measured, ask students for the perimeter of this rectangle. Ask a student to come to the document camera to measure the perimeter of the rectangle. Emphasize that the units used to measure perimeter are one-dimensional, or

linear, and that perimeter is just the distance around an object. Add the new heading “Perimeter” to your chart and record the perimeter as “12 units.”

● Ask: “How would we find the area? What do we mean by ‘area’?” After helping students define “area” and describe how it is measured, ask what the area of this rectangle would be. Here you want to make explicit that the units used to mea-sure area are two-dimensional (square units). That is the reason that they cover a region. After counting the tiles, add the heading “Area” to your chart and record, in this case, “12 square units.”

● Have students make, at their desks, a different rectangle using 12 tiles. They record the perimeter and area, as before. Together, decide what “different” means. Is a 2 3 6 rectangle different from a 6 3 2 rectangle? In this task, they are con-sidered different if the perimeters are different. Is that the case for these rectangles? (No.)

Present the Focus of Inquiry to the Class

● How many different rectangles can you make using 36 tiles? ● Note what is happening by recording the perimeter and area for each rectangle you create.

Provide Clear Expectations

Write the following directions on the board:

1. Find as many different rectangles as you can using all 36 tiles.

2. Sketch each rectangle on the grid paper. 3. Measure and record the perimeter and area of each rectangle.

● Students may work in pairs or small groups, but have each student draw her or his own sketches.

During Initially

● Question students to be sure they understand the task and the meaning of area and perimeter. Look for students who are confusing these terms.

● Be sure students are both drawing the rectangles and record-ing their dimensions, perimeter, and area appropriately in a chart.

Ongoing

● Observe students as they work and ask questions to assess their understanding. Pose one or two questions to each stu-dent as you move around the room (see “Assessment,” below).

62

Frequently Asked Questions 63

After ● Bring the class together to share and discuss the task. (They should identify 5 different rectangles: 1 3 36, 2 3 18, 3 3 12, 4 3 9, and 6 3 6. If some students disagree that the 6 3 6 is a rectangle, prompt discussion on the properties of rectangles.)

● Ask students what they have found out about perimeter and area. Ask: “Did the perimeter stay the same? Is that what you expected? When is the perimeter big and when is it small?”

● Ask students how they can be sure they have all the possible rectangles.

● Ask students to describe what happens to the perimeter as the length and width change. (The perimeter gets shorter as the sides become similar in length. The square has the short-est perimeter.) Provide time to share ideas.

ASSESSMENT Observe

● Are students confusing perimeter and area? ● As students form new rectangles, are they aware that the area is not changing because they are using the same number of tiles each time? These students may not know what area is, or they may be confusing it with perimeter.

● Are students looking for patterns for how to find the perimeter? ● Are students discussing important concepts or patterns with their partners?

Ask ● What is the area of the rectangle you just made? ● What is the perimeter of the rectangle you just made? ● How is area different from perimeter? ● How do you measure the area of a rectangle? The perimeter?

63

Supplements MyEducationLab MyEducationLab for Curriculum and Instruction ( www.myeducationlab.com ) is an online learning solution that provides interactive exercises designed to help teacher can-didates develop the knowledge and skills that teachers need. Using classroom video, authentic student and teacher arti-facts, and other resources and assessments, the learning experiences in MyEducationLab offer you a unique and valuable education tool.

For each mathematics topic covered, you will find the following features and resources:

● Expanded Lesson Activities: these expanded activities will help you explore how to use the lessons in the classroom.

● Artifact Analysis Activities: activities contain audio and visual records of students interacting with mathe-matics, accompanied by questions that focus on the analysis of student thinking.

● Activities for Developing Mathematics Content Knowledge: designed to assess and develop your math-ematics content understanding, activities focus on areas where people typically have misconceptions.

● Building Teaching Skills and Dispositions: learning units that help teacher candidates practise and strengthen skills that are essential to effective teaching.

● Video Assignments: a robust selection of classroom videos accompanied by a set of questions.

● IRIS Modules and Case Studies: course-enhancement materials from the IRIS Center, designed to better prepare school personnel to provide an appropriate education to students.

● Lesson Plan Builder: an easy-to-use tool you can use to create, update, and share quality lesson plans.

Pearson eText Pearson eText gives students access to the text whenever and wherever they have access to the Internet. eText pages look exactly like the printed text, offering powerful new functionality for students and instructors. Users can create notes, highlight text in different colours, create bookmarks, zoom, click hyperlinked words and phrases to view defini-tions, and view in single-page or two-page view. Pearson eText allows for quick navigation to key parts of the eText using a table of contents, and provides full-text search. The eText may also offer links to associated media files, enabling users to access videos, animations, or other activities as they read the text.

CourseSmart for Students CourseSmart goes beyond traditional expectations—providing instant, online access to the textbooks and course materials you need at an average savings of 60%. With instant access from any computer and the ability to search your text, you’ll find the content you need quickly, no mat-ter where you are. And with online tools like highlighting and note-taking, you can save time and study efficiently. See all the benefits at www.coursesmart.com/students .

For Instructors The following instructor’s supplements are available for downloading from a password-protected section of Pearson Education Canada’s online catalogue: www.pearsoncanada.ca/highered . Navigate to your book’s catalogue page to view a list of those supplements that are available. See your local sales representative for details and access.

Instructor’s Manual The Instructor’s Manual includes a wealth of resources designed to help instructors teach the course, including chapter notes, activity suggestions, sug-gested assessments, and test questions.

PowerPoint Presentation Ideal for instructors to use for lecture presentations or student handouts, the Power-Point presentation provides dozens of ready-to-use graphic and text images tied to the text.

CourseSmart for Instructors CourseSmart goes beyond traditional expectations—providing instant, online access to the textbooks and course materials you need at a lower cost for students. And even as students save money, you can save time and hassle with a digital eTextbook that allows you to search for the most relevant content at the very moment you need it. Whether it’s evaluating textbooks or creating lecture notes to help students with difficult concepts, CourseSmart can make life a little easier. See how when you visit www.coursesmart.com/instructors .

Pearson Custom Library For enrollments of at least 25 students, you can create your own textbook by choosing the chapters that best suit your own course needs. To begin building your custom text, visit www.pearsoncustomlibrary.com . You may also work with a dedicated Pearson Custom editor to create your ideal text—publishing your own original content or mixing and matching Pearson content. Contact your local Pearson Representative to get started.

Learning Solutions Managers Pearson’s Learning Solutions Managers work with faculty and campus course designers to ensure that Pearson

xxiv Preface

Preface xxv

technology products, assessment tools, and online course materials are tailored to meet your specific needs. This highly qualified team is dedicated to helping schools take full advantage of a wide range of educational resources, by assisting in the integration of a variety of instructional materials and media formats. Your local Pearson Education sales representative can provide you with more details on this service program.

Acknowledgments from the US Edition Many talented people have contributed to the success of this book, and we are deeply grateful to all those who have assisted over the years. Without the success of the first edi-tion, there would certainly not have been a second, much less eight editions. John worked closely with Warren Crown, John Dossey, Bob Gilbert, and Steven Willoughby, who gave time and great care in offering detailed comments on the original manuscript.

In preparing the eighth US edition, we have received thoughtful input from the following educators who offered comments on the seventh edition or on the manuscript for the eighth:

Margaret Adams, University of North Carolina at Charlotte

Joohi Lee, University of Texas at Arlington Sandra J. Phifer, Metropolitan State College of

Denver Diana Piccolo, Missouri State University Janet Lynne Tassell, Western Kentucky University

Each reviewer challenged us to think through important issues. Many specific suggestions have found their way into this book, and their feedback helped us focus on important ideas. We are indebted to these committed professionals.

We also would each like to thank our families for their many contributions and support. On behalf of John, we thank his wife of more than 40 years, Sharon. Sharon was John’s biggest supporter in this process and remained a sounding board for his many decisions as he wrote the first six editions of this book. We also thank his daughters, Bridget (a fifth-grade teacher in Chesterfield County, Virginia) and Gretchen (an associate professor of psychology at Rutgers University–Newark). They were John’s first students and he tested many ideas that are in this book by their sides. We can’t forget those who called John “Math Grandpa”: his granddaughters, Maggie, Aidan, and Gracie.

From Karen Karp : I would like to express thanks to my husband, Bob Ronau, who as a mathematics educator gra-ciously helped me think about decisions while offering insights and encouragement. In addition, I thank my chil-dren, Matthew, Tammy, Joshua, Misty, Matt, Christine, Jeffrey, and Pamela for their kind support and inspiration.

I also am grateful for my wonderful grandchildren, Jessica, Zane, and Madeline, who have helped deepen my under-standing about how children think.

From Jennifer Bay-Williams : I am so grateful to my hus-band, Mitch, who offers support, guidance, and wisdom to my writing, and my children, MacKenna (8 years) and Nicolas (6 years), who enjoy doing a little extra math from time to time. My parents, siblings, and nieces and nephews have all contributed ideas and support to the writing of this edition. Finally, I want to thank Brandy Jones, who has been invaluable in helping me find research to inform my writing for this edition.

Most importantly, we thank all the teachers and students who gave of themselves by assessing what worked and what didn’t work in the many iterations of this book. In particular for the eighth edition, we thank teachers who generously tested activities and provided student work for us: Kyle Pat-terson, Kim George, Kelly Eaton, Sarah Bush, and Elizabeth Popelka. If future teachers learn how to teach mathematics from this book, it is because teachers and children before them shared their best ideas and thinking with the authors.

Acknowledgements from the Canadian Edition From Lynn McGarvey: I am indebted to my husband, Don, and my son, Matthew, who have been incredibly patient and supportive throughout the many phases of this project. I need to thank the creative and inspiring instructors who have been part of the Mathematics Subject Area Committee at the University of Alberta. These educators are constantly expanding my understanding of mathematics teaching and learning: Jane Gannon, Audrey Hodgson-Ward, Bev Kula, Julie Long, Janelle McFeetors, Gladys Sterenberg, and Car-rie Watt. Finally, my contribution to the textbook has been with my students in mind. They have been instrumental in challenging me to connect theory, practice, and their per-sonal histories to what it means to think mathematically.

In preparing this fourth Canadian edition, the follow-ing educators were among those who offered helpful and insightful comments on the previous edition and the draft manuscript:

Jerry Ameis, University of Winnipeg Jayne Asselin, Fleming College David Bale, University of Regina Norma Fraser, Thompson Rivers University Nahid Golafshani, Laurentian University Audrey Hodgson-Ward, University of Alberta Harpreet Kaur, Simon Fraser University Wendy Klassen, UBC Okanagan Jo-Anne Lake, Queen’s University Annie Savard, McGill University Gladys Sterenberg, University of Alberta David Watson, Western University

Documents

FOURTH CANADIAN EDITION Elementary and Middle · PDF fileElementary and Middle School Mathematics ... Cover Designer: Anthony Leung ... an elementary school teacher in New York