In this new resource, elementary classroom teachers, special education teachers, and teachers of mathematics in inclusion classrooms get practical help for connecting what students already know about mathematics to what they need to knowwhile improving learning and recall. Through a variety of straight-forward, easy-to-use lesson plans and learning activities that illustrate specific mathematical concepts and skills, Teaching Mathematics in Diverse Classrooms, K-4emphasizes the premise that effective mathematics teaching promotes understandingand understanding provides sound bases for skill development and better retention of material.

*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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*viii * *C o n t e n t s*

* * *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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*C * *o n t e n t s ix*

* * ** ** *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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*x * *C o n t e n t s*

* * ** ** *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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*C * *o n t e n t s xi*

* * *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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*xii * *C o n t e n t s*

* * *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*