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FOURTH CANADIAN EDITION Elementary and Middle School

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Elementary and Middle School Mathematics Teaching Developmentally FOURTH CANADIAN EDITION 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. Editor ...



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10 9 8 7 6 5 4 3 2 1 CKV
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
About the Authors
John A Van de Walle was a professor emeritus att
Virginia Commonwealth University He was a
mathematics education consultant who regularly
gave professional development workshops for K 8 8
ited and taught in elemen
teachers in the United States and Canada He visited
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
About the Contributor
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 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
Teaching Mathematics Foundations and Perspectives
Chapter 1 Teaching and Learning Mathematics Chapter 5 Blending Teaching and Assessment 64
in the Twenty First Century 1
Chapter 6 Teaching Mathematics
Chapter 2 Exploring What It Means for All Learners 84
to Know and Do Mathematics 11
Chapter 7 Tools for Learning Mathematics 103
Chapter 3 Mathematical Inquiry through
Rich Tasks and Classroom Discourse 28
Chapter 4 Preparing to Teach and Planning
for Mathematics Learning 45
SECTION II
Development of Mathematical Concepts and Procedures
Chapter 8 Developing Early Number Chapter 16 Developing Strategies
Concepts and Number Sense 119 for Fraction Computation 310
Chapter 9 Developing Meanings Chapter 17 Developing Concepts
for the Operations 139 of Decimals and Percents 333
Chapter 10 Helping Students Master Chapter 18 Proportional Reasoning 352
the Basic Facts 162
Chapter 19 Developing Measurement
Chapter 11 Developing Whole Number Concepts 369
Place Value Concepts 183
Chapter 20 Geometric Thinking and
Chapter 12 Developing Strategies Geometric Concepts 396
for Addition and Subtraction Computation 208
Chapter 21 Developing Concepts
Chapter 13 Developing Strategies of Data Analysis 428
for Multiplication and Division Computation 231
Chapter 22 Exploring Concepts
Chapter 14 Algebraic Thinking Generalizations of Probability 448
Patterns and Functions 253
Chapter 23 Developing Concepts
Chapter 15 Developing Fraction Concepts 284 of Exponents Integers and Real Numbers 466
APPENDIX Guide to Blackline Masters 485
REFERENCES 497
Preface xvii
Teaching 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
An Invitation to Do Mathematics 12
CHAPTER 1 Searching for Patterns 12
Teaching and Learning Mathematics Analyzing a Situation 14
in the Twenty First Century 1 Generalizing Relationships 14
Experimenting and Explaining 16
What Are Your Memories of School Mathematics 1 Engaging in Mathematical Inquiry 16
Mathematics Proficiency 17
Twenty First Century Knowledge and Skills 2
What Does It Mean to Learn
Diversity in Today s Classrooms 2
Mathematics 20
Mathematics Curriculum in Canada 3
Mechanical Metaphors for Learning 20
Goals for Student Learning 4
Learning Mathematics as Acquisition 21
Mathematics Content 4
Learning Mathematics along a Linear Path 21
Mathematical Processes 5
Ecological Metaphors for Learning 22
An Invitation to Learn and Grow 7
Learning Mathematics as Coherence 23
Becoming a Teacher of Mathematics 7
Learning Mathematics as Outward Growth 24
RESOURCES FOR CHAPTER 1 An Example of Mathematics Learning 24
Recommended Readings 9 Learning Theories and Implications
Online Resources 9 for Teaching 25
REFLECTIONS ON CHAPTER 1 Promoting Dispositions for
Doing Mathematics 26
Writing to Learn 10
For Discussion and Exploration 10 RESOURCES FOR CHAPTER 2
Recommended Readings 26
Online Resources 27
REFLECTIONS ON CHAPTER 2
Exploring What It Means to Know
Writing to Learn 27
and Do Mathematics 11 For Discussion and Exploration 27
What Does It Mean to Do Mathematics 11
A Classroom Environment for Doing Mathematics 12
viii Contents
Follow Up to Inquiry The After Phase of a Lesson 50
CHAPTER 3 Teacher Actions in the After Phase 50
Mathematical Inquiry through Rich Tasks Process for Preparing a Lesson 51
and Classroom Discourse 28 1 Curriculum Learning Expectations 52
2 Student Experiences 52
Why Mathematical Inquiry 28 3 Select Design or Adapt a Rich Task 53
4 Design the Lesson Assessments Tools 54
A Shift in the Role of Tasks
5 Plan the Before Phase of the Lesson 54
in the Classroom 29
6 Plan the During Phase of the Lesson 55
The Focus of Inquiry 30
7 Plan the After Phase of the Lesson 55
What Is a Rich Task 30 Final Revisions to Your Lesson Plan 56
Features of a Rich Task 31 Applying the Planning Process 56
Variable Entry and Exit Points 31 Applying the Three Phase Lesson Template
Thinking Mathematically 32 to Other Lesson Structures 56
High Levels of Cognitive Demand 34 Applying the Three Phase Model to Short Tasks 56
Relevant Contexts 34 Applying the Three Phase Model in Learning Centres 57
Evaluating and Adapting Tasks What Do You Need Homework and Parental Involvement 57
to Do to Make Them Rich er 36 Effective Homework 58
Task Selection Guide 36 Beyond Homework Families Doing Math 58
Developing Concepts and Procedures Resources for Families 59
through Tasks 36 Frequently Asked Questions 59
Concepts 36
RESOURCES FOR CHAPTER 4
Procedures 38
Recommended Readings 60
What about Drill or Practice 38 Online Resources 61
New Definitions of Drill and Practice 38
REFLECTIONS ON CHAPTER 4
What Practice Provides 39
What Drill Provides 39 Writing to Learn 61
When Is Drill Appropriate 39 For Discussion and Exploration 61
Drill and Student Errors 39 EXPANDED LESSON Fixed Area 62
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 Blending Teaching and Assessment 64
RESOURCES FOR CHAPTER 3
Purposes for Assessing Learning
Recommended Readings 43 in Classrooms 64
Online Resources 44
What Should Be Assessed 66
REFLECTIONS ON CHAPTER 3
Mathematics Content 66
Writing to Learn 44 Mathematical Processes 67
For Discussion and Exploration 44
Mathematical Disposition 67
What Tools Can Be Used to Gather Evidence
of Mathematics Learning 67
CHAPTER 4 Performance Based Tasks 68
Questions and Conversations 69
Preparing to Teach and Planning
Interviews 69
for Mathematics Learning 45
Writing to Learn and Assess 70
Journals 72
A Three Phase Lesson Format 45
Admit and Exit Slips 73
Introduction to Inquiry The Before Phase of a Lesson 45
Teacher Actions in the Before Phase 46
Inventories 74
Development of Inquiry The During Phase of a Lesson 48 Summative Assessment Tools 74
Teacher Actions in the During Phase 48 Cumulative Projects 74
Contents ix
Teacher Made Tests 75 Students with Moderate Severe Disabilities 97
Provincial Assessment Programs 76 Students Who Are Mathematically Gifted and Talented 99
National and International Studies 76
Final Thoughts 101
The Pan Canadian Assessment Program PCAP 77 RESOURCES FOR CHAPTER 6
Trends in International Mathematics Recommended Readings 101
and Science Study TIMSS 77 Online Resources 101
Ways to Document and Communicate REFLECTIONS ON CHAPTER 6
Evidence of Learning 77
Writing to Learn 102
Documenting Formative Assessment 78 For Discussion and Exploration 102
Rubrics 79
What Gets Graded Gets Valued 81
From Assessment Tools to Grades 82
Using Assessments to Shape Instruction 82 Tools for Learning Mathematics 103
RESOURCES FOR CHAPTER 5
Tools for Learning 103
Recommended Readings 82
Online Resources 83 Types of Learning Tools 105
Concrete Materials and Physical Models 106
REFLECTIONS ON CHAPTER 5
Visual and Graphic Representations 107
Writing to Learn 83
Technology Based Tools 108
For Discussion and Exploration 83
Calculators in Mathematics Instruction 110
When to Use a Calculator 110
Benefits of Calculator Use 111
Graphing Calculators 112
Portable Data Collection Devices 113
Teaching Mathematics for All Learners 84 Digital Tools for Mathematics Learning 113
Instructional Applications 114
Differentiated Learning and Teaching 84
Concept Instruction 114
Tiered Tasks 84
Problem Solving 114
Learning Centres 86
Drill and Reinforcement 115
Choice Boards Menus and Think Tac Toe 86
Guidelines for Selecting and Using Digital Resources 115
Think Pair Share 86
Guidelines for Using Digital Content 115
Graphic Organizers 86
How to Select Appropriate Digital Content 115
Diversity in Our Classrooms 87
Final Thoughts 116
First Nations M tis and Inuit Students 87
Students Who Are Culturally and Linguistically Diverse 88 RESOURCES FOR CHAPTER 7
Culturally Diverse Students 88 Recommended Readings 116
Students Who Are English Language Learners ELLs 90 Online Resources 117
Providing for Students with Special Needs 93 REFLECTIONS ON CHAPTER 7
Prevention Models and Interventions for All Students 93 Writing to Learn 118
Students with Mild Disabilities 95 For Discussion and Exploration 118
x Contents
SECTION II
Development 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 8 CHAPTER 9
Developing Early Number Concepts Developing Meanings for the Operations 139
and Number Sense 119
Addition and Subtraction Problem Structures 140
Promoting Good Beginnings 119 Join and Separate Problems 140
Number Concepts Quantity Counting Part Part Whole Problems 141
and Knowing How Many 120 Compare Problems 141
Quantity and the Ability to Subitize 120 Problem Difficulty 142
Early Counting 121 Computational and Semantic Forms of Equations 142
Numeral Writing and Recognition 123 Teaching Addition and Subtraction 142
Counting On and Counting Back 124 Contextual Problems 142
Comparing Sets More Than Less Than MATH MAKES SENSE 144
and Equal To 125
Model Based Problems 145
Early Number Sense 126
Properties of Addition and Subtraction 147
Relationships for Numbers 1 through 10 127 Multiplication and Division Problem Structures 148
One and Two More One and Two Less 127 Equal Groups Problems 148
Anchoring Numbers to 5 and 10 128 Comparison Problems 149
Part Part Whole Relationships 130 Area and Array Problems 149
Dot Cards as a Model for Teaching Number Combination Problems 151
Relationships 133
Teaching Multiplication and Division 151
Relationships for Numbers 10 through 20 134
Contextual Problems 151
Early Place Value Concepts 134
Remainders 152
Extending More Than and Less Than
Model Based Problems 153
Relationships 134
Properties of Multiplication and Division 155
Number Sense in Their World 135
Strategies for Solving Contextual Problems 156
Estimation and Measurement 135
Analyzing Context Problems 156
Data Collection and Analysis 136
Two Step Problems 158
Classroom Routines 136
RESOURCES FOR CHAPTER 9
RESOURCES FOR CHAPTER 8
Literature Connections 159
Literature Connections 137
Recommended Readings 159
Recommended Readings 137
Online Resources 160
Online Resources 138
REFLECTIONS ON CHAPTER 9
REFLECTIONS ON CHAPTER 8
Writing to Learn 160
Writing to Learn 138
For Discussion and Exploration 160
For Discussion and Exploration 138
Contents xi
Basic Ideas of Place Value 184
CHAPTER 10 Integration of Base Ten Groupings with Counting by Ones 185
Helping Students Master the Basic Facts 162 Role of Counting 185
Integration of Groupings with Words 186
Developmental Nature of Basic Fact Mastery 162 Integration of Groupings with Place Value Notation 186
Approaches to Fact Mastery 163 Base Ten Models for Place Value 187
Guiding Strategy Development 164 Groupable Models 187
Reasoning Strategies for Addition Facts 165 Pregrouped or Trading Models 187
One More Than and Two More Than 166 Nonproportional Models 188
Adding Zero 167 Developing Base Ten Concepts 188
Using 5 as an Anchor 167 Grouping Activities 188
Make 10 167 The Strangeness of Ones Tens and Hundreds 189
Up Over 10 167 Grouping Tens to Make 100 190
Doubles 168 Equivalent Representations 191
Near Doubles 169 Oral and Written Names for Numbers 192
Reinforcing Reasoning Strategies 169 Two Digit Number Names 192
Reasoning Strategies for Subtraction Facts 170 Three Digit Number Names 193
Subtraction as Think Addition 170 Written Symbols 194
Down Over 10 171 Assessing Place Value Concepts 195
Take from the 10 171 Patterns and Relationships with Multidigit
Reasoning Strategies for Multiplication Numbers 195
and Division Facts 172 The Hundreds Chart 195
Doubles 172 Relationships with Benchmark Numbers 197
Fives 172 Connecting Place Value to Addition and Subtraction 198
Zeros and Ones 172 Connections to Real World Ideas 202
Nifty Nines 173
Numbers beyond 1000 203
Using Known Facts to Derive Other Facts 174
Extending the Place Value System 203
Division Facts 175
Conceptualizing Large Numbers 204
Mastering the Basic Facts 175
RESOURCES FOR CHAPTER 11
Effective Drill 175
Games to Support Basic Fact Mastery 176 Literature Connections 205
Recommended Readings 206
Fact Remediation 177 Online Resources 206
What to Do When Teaching Basic Facts 179
REFLECTIONS ON CHAPTER 11
What Not to Do When Teaching Basic Facts 179
Writing to Learn 206
RESOURCES FOR CHAPTER 10 For Discussion and Exploration 207
Literature Connections 180
Recommended Readings 180
Online Resources 181
REFLECTIONS ON CHAPTER 10
CHAPTER 12
Writing to Learn 181 Developing Strategies for Addition
For Discussion and Exploration 182 and Subtraction Computation 208
Toward Computational Fluency 209
Student Generated Personal Strategies 211
CHAPTER 11 Standard Algorithms 212
Developing Whole Number Development of Meaningful Strategies 214
Creating an Environment for Meaningful Strategies 214
Place Value Concepts 183
Models to Support Meaningful Strategies 214
Numeration System 183
Student Generated Strategies
for Addition and Subtraction 217
Pre Base Ten Understandings 184
Adding and Subtracting Single Digit Numbers 217
Counting by Ones 184 Adding Two Digit Numbers 218
xii Contents
Subtracting by Counting Up 219 Computational Estimation from
Take Away Subtraction 219 Student Generated Strategies 246
Extensions and Challenges 220 Stop before the Details 246
Algorithms for Addition and Subtraction 221 Use Related Problem Sets 246
Standard Algorithm for Addition 221 Computational Estimation Strategies 247
Alternative Algorithms for Addition 223 Front End Methods 247
Standard Algorithm for Subtraction 223 Rounding Methods 247
Alternative Algorithms for Subtraction 224 Compatible Numbers 248
Introducing Computational Estimation 225 Using Tens and Hundreds 248
Understanding Computational Estimation 225 Estimation Experiences 249
Suggestions for Teaching Computational Estimation 225 Calculator Activities 249
Computational Estimation Strategies 227 Using Whole Numbers to Estimate Rational Numbers 251
Front End Methods 227 RESOURCES FOR CHAPTER 13
Rounding Methods 227 Literature Connections 251
Compatible Numbers 228 Recommended Readings 252
RESOURCES FOR CHAPTER 12 Online Resources 252
Literature Connections 229 REFLECTIONS ON CHAPTER 13
Recommended Readings 229 Writing to Learn 252
Online Resources 230 For Discussion and Exploration 252
REFLECTIONS ON CHAPTER 12
Writing to Learn 230
For Discussion and Exploration 230 CHAPTER 14
Algebraic Thinking Generalizations
Patterns and Functions 253
CHAPTER 13
Developing Strategies for Multiplication Algebraic Thinking 254
and Division Computation 231 Generalization from Arithmetic 254
Generalization with Operations 254
Towards Computational Fluency Generalization in the Hundreds Chart 255
with Multiplication and Division 232 Generalization through Exploring a Pattern 256
Development of Meaningful Strategies Meaningful Use of Symbols 257
for Multiplication and Division 232 The Meaning of the Equal Sign 257
Models to Support Meaningful Strategies 232 The Meaning of Variables 262
Student Generated Strategies Making Structure in the Number System
for Multiplication 235 Explicit 265
Complete Number Strategies Including Doubling 236 Making Conjectures about Properties 265
Multiplication of Larger Numbers 236 Justifying Conjectures 267
Standard Algorithms for Multiplication 238 Odd and Even Relationships 267
One Digit Multipliers 238 Study of Patterns and Functions 267
Two Digit Multipliers 239 Repeating Patterns 268
Alternative Algorithms for Multiplication 240 Growing Patterns 270
Student Generated Strategies for Division 240 Linear Functions 274
Missing Factor Strategies 240 Mathematical Modelling 276
Cluster Problems 241 Teaching Considerations 277
Standard Algorithms for Division 242 Emphasize Appropriate Algebra Vocabulary 277
One Digit Divisors 242 Connecting Representations 278
Alternative Division Algorithms 244 Algebraic Thinking across the Curriculum 280
Computational Estimation in Multiplication RESOURCES FOR CHAPTER 14
and Division 245 Literature Connections 281
Understanding Computational Estimation 245 Recommended Readings 282
Suggestions for Teaching Computational Estimation 245 Online Resources 282
Contents xiii
REFLECTIONS ON CHAPTER 14 Addition and Subtraction 314
Writing to Learn 283 Contextual Examples and Student Generated Personal
For Discussion and Exploration 283 Strategies 314
Models 315
Developing the Algorithms 317
CHAPTER 15 Fractions Greater Than One 319
Addressing Errors and Misconceptions 319
Developing Fraction Concepts 284
Multiplication 321
Contextual Examples and Models 321
Meanings of Fractions 284
Fraction Constructs 285
Developing the Algorithms 324
Why Fractions Are So Difficult 285 Factors Greater Than One 325
Addressing Errors and Misconceptions 326
Models for Fractions 286
Area Models 286
Division 326
Length Models 288 Contextual Examples and Models 326
Set Models 289 Answers That Are Not Whole Numbers 328
Developing the Algorithms 329
Concept of Fractional Parts 289
Addressing Errors and Misconceptions 330
Fraction Size Is Relative 289
Fraction Language 290 RESOURCES FOR CHAPTER 16
Partitioning 290 Literature Connections 331
Sharing Tasks 292 Recommended Readings 331
Online Resources 331
Iterating 293
Fraction Notation 296 REFLECTIONS ON CHAPTER 16
Fractions Greater Than One 296 Writing to Learn 331
Estimating with Fractions 297 For Discussion and Exploration 332
Equivalent Fractions 298
Conceptual Focus on Equivalence 298
Equivalent Fraction Models 299 CHAPTER 17
Developing an Equivalent Fraction Algorithm 301
Developing Concepts of Decimals
Comparing Fractions 303
and Percents 333
Comparing Fractions Using Number Sense 303
Using Equivalent Fractions to Compare 305
Extending the Place Value System 334
Teaching Considerations for Fraction Concepts 305
A Two Way Relationship 334
PROBLEM BASED LESSON Equivalent Fraction Regrouping 334
Challenge 306 The Role of the Decimal Point 335
RESOURCES FOR CHAPTER 15 Connecting Fractions and Decimals 336
Base Ten Fractions 336
Literature Connections 308
Recommended Readings 308 Developing Decimal Number Sense 339
Online Resources 308 Familiar Fractions Connected to Decimals 339
REFLECTIONS ON CHAPTER 15 Computation with Decimals 343
Writing to Learn 309 The Role of Estimation 343
For Discussion and Exploration 309 Addition and Subtraction 344
Multiplication 344
Division 346
CHAPTER 16 Introducing Percents 346
Developing Strategies for Fraction Models and Terminology 347
Percent Problems in Context 348
Computation 310
Estimation 349
Understanding Fraction Operations 311 RESOURCES FOR CHAPTER 17
Conceptual Development Takes Time 311 Literature Connections 350
An Inquiry Based Number Sense Approach 311 Recommended Readings 351
Online Resources 351
Computational Estimation 312
xiv Contents
REFLECTIONS ON CHAPTER 17 Area 379
Writing to Learn 351 Comparison Activities 379
For Discussion and Exploration 351 Using Models of Area Units 380
The Relationship between Area and Perimeter 381
Developing Formulas for Area 382
CHAPTER 18 Student Errors and Misconceptions 382
Proportional Reasoning 352 Areas of Rectangles Parallelograms Triangles and Trapezoids 383
Circumference and Area of Circles 385
Ratios 352 Volume and Capacity 386
Types of Ratios 352 Comparison Activities 386
Ratios Compared to Fractions 353 Using Models of Volume and Capacity Units 386
Two Ways to Think about Ratios 353 Using Measuring Cups 387
Proportional Reasoning 354 Developing Formulas for Volumes of Common Solid Shapes 387
Connections between Formulas 388
Proportional and Nonproportional Situations 354
Additive and Multiplicative Comparisons in Problems 355 Mass and Weight 388
Covariation 356 Comparison Activities 389
Develop a Wide Variety of Strategies 360 Using Models of Mass Units 389
Mental Strategies 360 Angles 390
Ratio Tables 362 Comparison Activities 390
Using Models of Angular Measure Units 390
CONNECTED MATHEMATICS Grade 7 Comparing Using Protractors and Angle Rulers 390
and Scaling 363
Double Line Strip Comparison 364
Comparison Activities 391
Percents 365
Reading Clocks 392
Cross Products 365
Elapsed Time 392
Teaching Proportional Reasoning 366
RESOURCES FOR CHAPTER 18 Recognizing Coins and Identifying Their Values 393
Literature Connections 366 Counting Sets of Coins 393
Recommended Readings 367 Making Change 394
Online Resources 367
RESOURCES FOR CHAPTER 19
REFLECTIONS ON CHAPTER 18
Literature Connections 394
Writing to Learn 368 Recommended Readings 394
For Discussion and Exploration 368 Online Resources 394
REFLECTIONS ON CHAPTER 19
CHAPTER 19 Writing to Learn 395
For Discussion and Exploration 395
Developing Measurement Concepts 369
The Meaning and Process of Measuring 370
CHAPTER 20
Concepts and Skills 370
Introducing Nonstandard Units 372 Geometric Thinking
Developing Standard Units 372 and Geometric Concepts 396
Instructional Goals 372
A Brief History of the Metric System 373 Geometry Goals for Students 396
Important Standard Units and Relationships 373 Spatial Sense and Geometric Reasoning 397
The Role of Estimation and Approximation 374 Geometric Content 397
Strategies for Estimating Measurements 374
Developing Geometric Thinking 397
Tips for Teaching Estimation 375
The van Hiele Levels of Geometric Thought 397
Measurement Estimation Activities 375
Implications for Instruction 400
Length 376
Learning about Solids Plane Figures
Comparison Activities 376
and Their Properties 401
Using Models of Length Units 376
Solids Plane Figures and Properties for Level 0 Thinkers 401
Making and Using Rulers 378
Contents xv
Shapes and Properties for Level 1 Thinkers 405 RESOURCES FOR CHAPTER 21
Shapes and Properties for Level 2 Thinkers 410 Literature Connections 446
Learning about Transformations 413 Recommended Readings 446
Transformations for Level 0 Thinkers 413 Online Resources 446
Transformations for Level 1 Thinkers 415 REFLECTIONS ON CHAPTER 21
Transformations for Level 2 Thinkers 417 Writing to Learn 447
Learning about Location 417 For Discussion and Exploration 447
Location for Level 1 Thinkers 419
Location for Level 2 Thinkers 421
Learning about Visualization 421 CHAPTER 22
Visualization for Level 0 Thinkers 422
Visualization for Level 1 Thinkers 423
Exploring Concepts of Probability 448
Visualization for Level 2 Thinkers 424
Introducing Probability 448
RESOURCES FOR CHAPTER 20
Likely or Not Likely 449
Literature Connections 425 The Probability Continuum 450
Recommended Readings 426
Online Resources 426 Theoretical Probability and Experiments 453
Theoretical Probability 453
REFLECTIONS ON CHAPTER 20
Experiments 454
Writing to Learn 426 Why Use Experiments 456
For Discussion and Exploration 427
Use of Technology in Experiments 456
Sample Spaces and Probability of Two Events 457
Independent Events 457
CHAPTER 21 Area Models 459
Dependent Events 460
Developing Concepts of Data Analysis 428 Simulations 461
RESOURCES FOR CHAPTER 22
What Does It Mean to Do Statistics 429
Is It Statistics or Is It Mathematics 429 Literature Connections 463
Recommended Readings 463
The Shape of Data 429
Online Resources 464
The Process of Doing Statistics 430
REFLECTIONS ON CHAPTER 22
Formulating Questions 430
Classroom Questions 430 Writing to Learn 464
For Discussion and Exploration 464
Beyond One Classroom 431
Data Collection 431
Collecting Data 431
Using Existing Data Sources 432 CHAPTER 23
Data Analysis Classification 432 Developing Concepts of Exponents
Attribute Materials 433 Integers and Real Numbers 466
Data Analysis Graphical Representations 434
Bar Graphs and Tally Charts 435 Exponents 466
Circle Graphs 436 Exponents in Expressions and Equations 466
Numerical Data Graphs 437 Order of Operations 467
Scatter Plots 439 Negative Exponents 470
Data Analysis Measures of Centre Scientific Notation 470
and Variability 440 Integers 472
Averages 440 Contexts for Exploring Integers 472
Understanding the Mean Two Interpretations 441 Quantity Contexts 472
Variability 443 Linear Contexts 473
Box Plots 444 Meaning of Negative Numbers 474
Interpreting Results 445 Models for Teaching Integers 474


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flexible organization emphasizes an integrated view of macroeconomics, while enabling professors to focus on the theories, ... (fourth Canadian edition). Professor Johnson received his undergraduate degree from the University of Toronto, his Master’s degree from the University of Western Ontario, and his Ph.D. in 1983 from Harvard University, where Olivier Blanchard served as one of his ...

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The Price of Knowledge - York University

The Price of Knowledge - York University

The Price of Knowledge: Access and Student Finance in Canada — Fourth Edition Joseph Berger, Anne Motte and Andrew Parkin Includes bibliographical references. 1. College students—Canada—Economic conditions. 2. Student aid—Canada. 3. Student loan funds—Canada 4. Post-secondary education—Economic aspects—Canada I. Motte, Anne, 1974- II. Parkin, Andrew, 1966- III. Canada Millennium ...

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Nineteenth-Century Canadian School Mathematics

Nineteenth-Century Canadian School Mathematics

Nineteenth-Century Canadian School Mathematics Harold Don Allen Quantity is any thing that can be increa8ed, dimini8hed, or meas­ ured.Mathematics i8 the 8cience of quantity. - 1858 definition.1 The present generation of Canadian educators is sufficiently removed in time from the schools and programs of the nineteenth century that Victorian practices in curriculum and instruction are tending ...

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BLACK LETTER OUTLINES Torts - Westlaw

BLACK LETTER OUTLINES Torts - Westlaw

FOURTH EDITION Mat #40264726 BLACK LETTER OUTLINES. Thomson/West have created this publication to provide you with accurate and authoritative information concerning the subject matter covered. However, this publication was not necessarily prepared by persons licensed to practice law in a particular jurisdiction. Thomson/West are not engaged in rendering legal or other professional advice, and ...

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Adams-Gordon’s Spelling Power

Adams-Gordon’s Spelling Power

Adams-Gordon’s Spelling Power Fourth Edition First Edition Preface This book began five years ago (1989) when I left my teaching position at a private school to teach my two daughters Angelina (Angie) and Amelia at home. I soon discovered my younger daughter, then in sixth grade, could spell only on a second grade level. Always having been a hopeless speller, I did not want my daughter to ...

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WHO/HSE/WSH/10.01/13 English only

WHO/HSE/WSH/10.01/13 English only

fourth edition. Under the oversight of a group of coordinators, each of whom was responsible for a group of chemicals considered in the GDWQ, the draft health criteria documents were submitted to a number of scientific institutions and selected experts for peer review. Comments were taken into consideration by the coordinators and authors. The draft documents were also released to the public ...

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Placing Your Student Adams-Gordon’s Spelling Power

Placing Your Student Adams-Gordon’s Spelling Power

Adams-Gordon’s Spelling Power Fourth Edition First Edition Preface This book began five years ago (1989) when I left my teaching position at a private school to teach my two daughters Angelina (Angie) and Amelia at home. I soon discovered my younger daughter, then in sixth grade, could spell only on a second grade level. Always having been a ...

Continue Reading...