vii
About the Au thors xiii
Preface xv
1 I nstructional Activities:
T he Building Blocks for Effective Instruction 1
Instructional Activities 1
Developmental Activities 2
Exploratory Developmental Activities 2
Consolidating Developmental Activities 2
Practice Activities 2
Think-Time Practice Activities 3
Speed-Drill Practice Activities 3
Application Activities 3
Classroom Applications 3
Real-World Problems 4
Assessment Activities 4
Flexible Use of Activities and Materials 5
Exercises and Activities 5
References and Related Readings 6
Websites 7
2 Prior Student Knowledge:
T he Basis for Learning and Instruction 8
Building on Prior Knowledge 8
Assessing Prior Knowledge 9
Remediation of Prerequisite Knowledge 9
Additional Classroom Scenarios 10
Exercises and Activities 11
contents
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3 L ess on Design:
Planning for Effective Instruction and Maximum Learning 12
Combining Activities into a Lesson 12
What Is a Lesson? 12
A Traditional Lesson Plan 13
The Nature of Standard Traditional Lessons 15
Adapting Lessons for Diverse Learning Needs 15
A Lesson Adapted for Diverse Learners 18
The Planning Process and “Official” Lesson Plans 21
The Planning Process and Teaching Notes 21
Exercises and Activities 21
References and Related Readings 22
Websites 22
4 C ompu tation with Whole Nu mbers:
A Review of Addition and Subtraction 23
Three Important Considerations 23
Number Sense 23
Foundations of Algebra 24
Basic Concepts and Skills 24
An Overview of the Development of Computation 24
The Meaning of the Operation 24
The Basic Facts 24
The Algorithm(s) 25
Review and Remediation of Whole Number Addition 26
Relationships among the Addition Facts 26
Remediation of the Easy Basic Addition Facts 28
Remediation of the Hard Basic Addition Facts 29
Understanding the Addition Algorithm 31
The First Big Idea 33
The Second Big Idea 35
Use of the Developmental Sequence to Remediate Addition 35
Remediation of Student Difficulties with Whole Number Addition 36
Review and Remediation of Whole Number Subtraction 37
Developing the Meaning of Subtraction 37
Remediation of the Easy Basic Subtraction Facts 37
Remediation of the Hard Basic Subtraction Facts 38
Remediation of the Subtraction Algorithm 39
The Big Ideas for the Subtraction Algorithm 39
Use of the Developmental Sequence for Remediation of Subtraction 40
Remediation of Student Difficulties with Whole Number Subtraction 41
Teaching Problem Solving Using Addition and Subtraction 41
Translate Word Problems into Situations 41
Exercises and Activities 42
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References and Related Readings 44
Websites 44
5 C ompu tation with Whole Nu mbers:
D evelopment of Multiplication and Division 45
Teaching Multiplication of Whole Numbers 45
Developing the Meaning of Multiplication 45
Developing the Easy Basic Multiplication Facts 48
Developing the Hard Basic Multiplication Facts 53
Teaching the Multiplication Algorithm 58
Use of the Developmental Sequence for Remediation of Multiplication 70
Remediation of Multiplication 71
Teaching Division of Whole Numbers 77
The Meaning of Division 77
Developing the Easy Basic Division Facts 80
Developing the Hard Basic Division Facts 82
Remediating the Division Algorithm 84
Use of the Developmental Sequence for Remediation of Division 96
Remediation of Division 96
Adapting a Division Lesson 96
Teaching Problem Solving Using Multiplication and of Division 99
Exercises and Activities 99
References and Related Readings 101
Websites 101
6 Fractions:
W orking with Units Smaller Than One 102
Defining Fractions 102
Three Sides of Fractions 103
Fractional Units 104
Beyond Unit Fractions 106
Fractions of a Set 107
Equivalent Fractions 108
Using the Laboratory Approach 110
Comparison of Fractions 112
Adding Fractions 116
Subtracting Fractions 117
Least Common Denominators 118
Addition and Subtraction Activities 119
Improper Fractions and Mixed Numbers 121
Multiplying Fractions 123
Dividing Fractions 127
Multiplication and Division Activities 131
Adapting a Lesson on Fractions 135
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Solving Problems Using Fractions 137
Problem-Solving Strategies 138
Exercises and Activities 140
References and Related Readings 141
Websites 141
7 D ecimals and percents:
W orking with Base-Ten Units Smaller Than One and Using Hundredths
as a Common Denominator 142
Decimals 142
Place Value for Decimals 144
Comparing Decimals 149
Adding and Subtracting Decimals 152
Adding and Subtracting Activities 154
Multiplying Decimals 156
Dividing Decimals 161
Multiplication and Division Activities 163
Fraction Comparison Revisited 167
Defining Percent 168
Adapting a Lesson on Decimals 168
Using Decimals to Solve Problems 171
Exercises and Activities 173
References and Related Readings 174
Websites 174
8 M easu rement:
A ssigning a Number to a Quantity 175
Measurement and Geometry 175
Defining Measurement 175
Measuring Length 176
Measuring Area 178
The Big Idea for Developing Area Formulas 182
Measuring Volume 187
Measuring Angles 195
Measuring with Indirect Units 198
Measuring Weight 198
Measuring Temperature 198
Measuring Value 198
Adapting a Lesson on Volume 199
Using Measurement to Solve Problems 201
Exercises and Activities 203
References and Related Readings 204
Websites 204
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9 Geometry:
Using the Big Ideas to Study Geometric Shapes 205
The Big Ideas of Middle School Geometry 205
Straightness 206
Congruence 206
Similarity 207
Parallelism 208
Perpendicularity 209
Symmetry 210
Using the Big Ideas to Study Geometric Shapes 210
Rectangles in Middle School 210
Circles in Middle School 211
Angles in the Fifth through Eighth Grades 212
Prisms in Middle School 213
Adapting a Geometry Lesson 215
Using Geometry to Solve Problems 216
Exercises and Activities 218
References and Related Readings 219
Websites 219
10 D ata Analysis And Probability:
G etting Information from Data and Measuring Likelihood 220
Data Analysis and Probability: Two Distinct but Related
Areas of Mathematics 220
Data Analysis 221
Emphasizing the Big Ideas of Data Analysis 221
Adapting a Data Analysis Lesson 226
Using Data Analysis to Solve Problems 231
Probability 231
Emphasizing the Big Ideas of Probability 231
Adapting a Probability Lesson 237
Using Probability to Solve Problems 240
Exercises and Activities 241
References and Related Readings 242
Websites 242
11 Foundations of Algebra:
Properties of Operations on Integers, Rational Numbers,
and Irrational Numbers 243
Emphasizing the Big Ideas of Algebra 243
The Closure Property 244
The Associative Property 245
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The Commutative Property 246
The Distributive Property 246
The Identity Elements 247
The Inverse Elements 247
Computation with Integers 248
The Order of Operations 253
Evaluating Algebraic Expressions 255
Using Properties of Equality to Solve Equations 255
Exponents 257
Exercises and Activities 261
References and Related Readings 262
Websites 262
12 Eff ective Practice:
G ames and Activities for Practice and Fun 263
Matching Activities 264
Ordering Activities 271
Answer or Example Construction Activities 275
Summary 278
References and Related Readings 278
A ctivities to Take to Your Class room 279
I ndex 281