Teaching Mathematics in Diverse Classrooms for Grades 5–8: Practical Strategies and Activities That Promote Understanding and Problem Solving Ability, CourseSmart eTextbook

For elementary classroom teachers, special education teachers, and teachers of mathematics in inclusion classrooms

 

Practical help for connecting what students already know about mathematics to what they need to know–and improving learning and recall.

 

This new guide uses an approach that emphasizes the premise that effective mathematics teaching promotes understanding–and understanding provides sound bases for skill development and better retention of material. It gives teachers of mathematics a variety lesson plans and learning activities that include: a more complete development of mathematical ideas, visuals to support mental imagery, opportunities for kinesthetic learning activities, opportunities for children to communicate their understanding of mathematics, and continual opportunities for the monitoring of student learning.

 

With this guide, teaching mathematics is made more effective through straightforward, easy-to-use lesson plans and learning activities that illustrate specific mathematical concepts and skills, ideas for using visuals to develop mental imagery, and ways to promote active student learning using movement and student discussion. The development of the content is user-friendly and based on a small number of easy-to-understand and easy-to-teach big ideas, and the many activities help to emphasize and reinforce the development of the mathematical content.

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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