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Contents

0 General Advice 10.1 General Advice on Teaching Courses for Elementary Teachers . . . . 20.2 Comments and Advice on the Class Activities . . . . . . . . . . . . . 40.3 Comments and Advice on the Practice Exercises and Problems . . . . 60.4 Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60.5 Sample Syllabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80.6 Advice to Give Students Who Are Struggling . . . . . . . . . . . . . 100.7 What Should Be the Focus of Math Content Courses? . . . . . . . . . 120.8 How Is Teaching from this Book Different? . . . . . . . . . . . . . . . 13

1 Numbers and the Decimal System 151.1 The Counting Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 16

The Counting Numbers as a List . . . . . . . . . . . . . . . . . . . . 16Connecting Counting Numbers as a List with Cardinality . . . . . . . 16How Many Are There? . . . . . . . . . . . . . . . . . . . . . . . . . . 17Showing the Values of Places in the Decimal System . . . . . . . . . . 18

1.2 Decimals and Negative Numbers . . . . . . . . . . . . . . . . . . . . . 19Representing Decimals with Bundled Objects . . . . . . . . . . . . . 19Representing Decimals as Lengths . . . . . . . . . . . . . . . . . . . . 20Zooming In on Number Lines . . . . . . . . . . . . . . . . . . . . . . 20Numbers Plotted on Number Lines . . . . . . . . . . . . . . . . . . . 22Negative Numbers on Number Lines . . . . . . . . . . . . . . . . . . 23

1.3 Comparing Numbers in the Decimal System . . . . . . . . . . . . . . 23Places of Larger Value Count More than Lower Places Combined . . 23Misconceptions in Comparing Decimals . . . . . . . . . . . . . . . . . 24Finding Smaller and Smaller Decimals . . . . . . . . . . . . . . . . . 25Finding Decimals Between Decimals . . . . . . . . . . . . . . . . . . . 25Decimals Between Decimals on Number Lines . . . . . . . . . . . . . 25

iii

Copyright © 2011 Pearson Education Inc. Publishing as Addison-Wesley.


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“Greater Than” and “Less Than” with Negative Numbers . . . . . . 261.4 Rounding Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Why Do We Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Explaining Rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Rounding with Number Lines . . . . . . . . . . . . . . . . . . . . . . 28Can We Round This Way? . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Fractions 292.1 The Meaning of Fractions . . . . . . . . . . . . . . . . . . . . . . . . 30

Fractions of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 30The Whole Associated With a Fraction . . . . . . . . . . . . . . . . . 30Relating a Fraction to its Whole . . . . . . . . . . . . . . . . . . . . . 31Comparing Quantities with Fractions . . . . . . . . . . . . . . . . . . 32Fractions of Non-Contiguous Wholes . . . . . . . . . . . . . . . . . . 32Is the Meaning of Equal Parts Always Clear? . . . . . . . . . . . . . . 33Improper Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Interlude: Solving Problems and Explaining Solutions . . . . . . . . . 342.3 Fractions as Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Number Line Errors with Fractions . . . . . . . . . . . . . . . . . . . 34Fractions on Number Lines, Part 1 . . . . . . . . . . . . . . . . . . . 35

2.4 Equivalent Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Equivalent Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Misconceptions About Fraction Equivalence . . . . . . . . . . . . . . 36Common Denominators . . . . . . . . . . . . . . . . . . . . . . . . . . 36Solving Problems by Changing Denominators . . . . . . . . . . . . . 37Fractions on Number Lines, Part 2 . . . . . . . . . . . . . . . . . . . 38Simplifying Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Comparing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Can We Compare Fractions This Way? . . . . . . . . . . . . . . . . . 39What is another way to Compare these Fractions? . . . . . . . . . . . 39Comparing Fractions by Reasoning . . . . . . . . . . . . . . . . . . . 40Can We Reason this Way? . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6 Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Pictures, Percentages, and Fractions . . . . . . . . . . . . . . . . . . . 41Calculating Percents of Quantities by Using Benchmark Fractions and

Percent Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 41Calculating Percentages with Pictures and Percent Tables . . . . . . . 42Calculating Percentages with Equivalent Fractions . . . . . . . . . . . 43



CONTENTS v

Calculating a Quantity from a Percentage of It . . . . . . . . . . . . . 43Percent Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Addition and Subtraction 453.1 Interpretations of Addition and Subtraction . . . . . . . . . . . . . . 46

Relating Addition and Subtraction:The Shopkeepers’ Method of Mak-ing Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Types of Addition and Subtraction Story Problems . . . . . . . . . . 46Why Can’t We Rely on Keywords to Solve Story Problems? . . . . . 47Using Simple Diagrams to Decide Whether to Add or Subtract to Solve

a Story Problem . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Properties of Addition and Mental Math . . . . . . . . . . . . . . . . 49

Mental Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Children’s Learning Paths for Single-Digit Addition . . . . . . . . . . 50Children’s Learning Paths for Single-Digit Subtraction . . . . . . . . 50Solving Addition and Subtraction Story Problems: Easier and Harder

Sub-Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Using Properties of Addition in Mental Math . . . . . . . . . . . . . 52Writing Correct Equations . . . . . . . . . . . . . . . . . . . . . . . . 52Writing Equations that Correspond to a Method of Calculation . . . 53Other Ways to Add and Subtract . . . . . . . . . . . . . . . . . . . . 54

3.3 Algorithms for Addition and Subtraction . . . . . . . . . . . . . . . . 55Adding and Subtracting with Ten-Structured Pictures . . . . . . . . . 55Understanding the Common Addition Algorithm . . . . . . . . . . . . 56Understanding the Common Subtraction Algorithm . . . . . . . . . . 57A Third Grader’s Method of Subtraction . . . . . . . . . . . . . . . . 58Subtracting Across Zeros . . . . . . . . . . . . . . . . . . . . . . . . . 59Regrouping With Dozens and Dozens of Dozens . . . . . . . . . . . . 60Regrouping With Seconds, Minutes, and Hours . . . . . . . . . . . . . 61

3.4 Adding And Subtracting Fractions . . . . . . . . . . . . . . . . . . . 61Why Do We Add and Subtract Fractions the Way We Do? . . . . . . 61How Do We Find a Suitable Common Denominator for Adding or

Subtracting Fractions? . . . . . . . . . . . . . . . . . . . . . . 62Mixed Numbers and Improper Fractions . . . . . . . . . . . . . . . . 62Adding and Subtracting Mixed Numbers . . . . . . . . . . . . . . . . 63Addition with Whole Numbers, Decimals, Fractions, and Mixed Num-

bers: What Are Common Ideas? . . . . . . . . . . . . . . . . . 64Are These Story Problems for 1

2+ 1

3? . . . . . . . . . . . . . . . . . . 64



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Are These Story Problems for 12− 1

3? . . . . . . . . . . . . . . . . . . 65

What Fraction is Shaded? . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Adding and Subtracting Negative Numbers . . . . . . . . . . . . . . . 67

Story Problems and Rules for Adding and Subtracting with NegativeNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Multiplication 694.1 Interpretations of Multiplication . . . . . . . . . . . . . . . . . . . . . 70

Showing Multiplicative Structure . . . . . . . . . . . . . . . . . . . . 70Problems About Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 71Writing Multiplication Story Problems . . . . . . . . . . . . . . . . . 72

4.2 Why Multiplying Numbers by 10 Is Easy in the Decimal System . . . 72Multiplying by 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Multiplying by Powers of 10 Explains the Cycling of Decimal Repre-

sentations of Fractions . . . . . . . . . . . . . . . . . . . . . . 734.3 Commutative, Associative Properties, Area, Volume . . . . . . . . . . 74

Explaining the Commutative Property of Multiplication . . . . . . . . 74Multiplication, Areas of Rectangles, and the Commutative Property . 75Ways to Describe the Volume of a Box with Multiplication . . . . . . 75Explaining the Associative Property . . . . . . . . . . . . . . . . . . 76Using the Associative and Commutative Properties of Multiplication 77Different Ways to Calculate the Total Number of Objects . . . . . . . 78Using Multiplication to Estimate How Many . . . . . . . . . . . . . . 78How Many Gumdrops? . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . 80Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Explaining the Distributive Property . . . . . . . . . . . . . . . . . . 80Using the Distributive Property . . . . . . . . . . . . . . . . . . . . . 81Why Isn’t 23× 23 Equal to 20× 20 + 3× 3? . . . . . . . . . . . . . 82The Distributive Property and FOIL . . . . . . . . . . . . . . . . . . 82Squares and Products Near Squares . . . . . . . . . . . . . . . . . . . 83

4.5 Properties of Arithmetic, Mental Math, Basic Facts . . . . . . . . . . 83Using Properties of Arithmetic to Aid the Learning of Basic Multipli-

cation Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Solving Arithmetic Problems Mentally . . . . . . . . . . . . . . . . . 84Which Properties of Arithmetic do These Calculations Use? . . . . . 85Writing Equations that Correspond to a Method of Calculation . . . 86Showing the Algebra in Mental Math . . . . . . . . . . . . . . . . . . 87



CONTENTS vii

4.6 Why the Algorithm for Multiplying Works . . . . . . . . . . . . . . . 89The Standard Versus the Partial Products Multiplication Algorithm . 89Why the Multiplication Algorithms Give Correct Answers, Part I . . 90Why the Multiplication Algorithms Give Correct Answers, Part II . . 91The Standard Multiplication Algorithm Right Side Up and Upside Down 93

5 Multiplying Fractions, Decimals, Negatives 955.1 Multiplying Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

The Meaning of Multiplication for Fractions . . . . . . . . . . . . . . 96Misconceptions With Fraction Multiplication . . . . . . . . . . . . . . 99Explaining Why the Procedure for Multiplying Fractions Gives Correct

Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99When Do We Multiply Fractions? . . . . . . . . . . . . . . . . . . . . 100Multiplying Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . 100What Fraction is shaded? . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Explaining Why We Place the Decimal Point Where We Do When We

Multiply Decimals . . . . . . . . . . . . . . . . . . . . . . . . 103Decimal Multiplication And Areas of Rectangles . . . . . . . . . . . . 103

5.3 Multiplying Negative Numbers . . . . . . . . . . . . . . . . . . . . . . 104Patterns With Multiplication and Negative Numbers . . . . . . . . . 104Explaining Multiplication with Negative Numbers (and 0) . . . . . . 105Using Checks and Bills to Interpret Multiplication with Negative Num-

bers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Does Multiplication Always Make Larger? . . . . . . . . . . . . . . . 106

5.4 Powers and Scientific Notation . . . . . . . . . . . . . . . . . . . . . . 106Multiplying Powers of 10 . . . . . . . . . . . . . . . . . . . . . . . . . 106Scientific Notation Versus Ordinary Decimal Notation . . . . . . . . . 107How Many Digits are in a Product of Counting Numbers? . . . . . . 108Explaining the Pattern in the Number of Digits in Products . . . . . 108

6 Division 1116.1 Interpretations of Division . . . . . . . . . . . . . . . . . . . . . . . . 112

What Does Division Mean? . . . . . . . . . . . . . . . . . . . . . . . 112Division Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . 112Why Can’t We Divide by Zero? . . . . . . . . . . . . . . . . . . . . . 113

6.2 Division and Fractions and Division with Remainder . . . . . . . . . 114



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Relating Fractions and Division . . . . . . . . . . . . . . . . . . . . . 114Division with Remainder Notation . . . . . . . . . . . . . . . . . . . . 115What to Do With the Remainder? . . . . . . . . . . . . . . . . . . . 115

6.3 Why the Common Long Division Algorithm Works . . . . . . . . . . 116Can We Use Properties of Arithmetic to Divide? . . . . . . . . . . . . 117Dividing Without Using a Calculator or Long Division . . . . . . . . 117Why the Scaffold Method of Long Division Works . . . . . . . . . . . 118Using the Scaffold Method Flexibly . . . . . . . . . . . . . . . . . . . 119Interpreting the Standard Division Algorithm as Dividing Bundled

Toothpicks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Interpreting the Standard Division Algorithm in Terms of Area . . . 121Student Errors in Using the Division Algorithm . . . . . . . . . . . . 122Interpreting the Calculation of Decimal Answers to Whole Number

Division Problems in Terms of Money . . . . . . . . . . . . . . 123Errors in Decimal Answers to Division Problems . . . . . . . . . . . . 123Using Division to Calculate Decimal Representations of Fractions . . 123Rounding to Estimate Solutions to Division Problems . . . . . . . . . 124

6.4 Fraction Division, How Many Groups? . . . . . . . . . . . . . . . . . 124“How Many Groups?” Fraction Division Problems . . . . . . . . . . . 124Dividing Fractions by Dividing the Numerators and Dividing the De-

nominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.5 Fraction Division, How Many in One Group? . . . . . . . . . . . . . 127

“How Many in One Group?” Fraction Division Problems . . . . . . . 128Using “Double Number Lines” to solve “How Many in One Group?”

Division Problems . . . . . . . . . . . . . . . . . . . . . . . . 129Are These Division Problems? . . . . . . . . . . . . . . . . . . . . . . 129

6.6 Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Quick Tricks for Some Decimal Division Problems . . . . . . . . . . . 130Decimal Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Proportional Reasoning 1337.1 The Meanings of Ratio, Rate, and Proportion . . . . . . . . . . . . . 134

Comparing Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Using Strip Diagrams, Ratio Tables, and Double Number Lines to

Describe Equivalent Ratios . . . . . . . . . . . . . . . . . . . . 135Using Ratio Tables to Compare Two Ratios . . . . . . . . . . . . . . 136

7.2 Reasoning with Multiplication and Division . . . . . . . . . . . . . . 137Using Strip Diagrams to Solve Ratio Problems . . . . . . . . . . . . . 137



CONTENTS ix

Solving Proportions by Reasoning About How Quantities Compare . 139

Going Through 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

More Ratio Problem Solving . . . . . . . . . . . . . . . . . . . . . . . 1407.3 Connecting Ratios and Fractions . . . . . . . . . . . . . . . . . . . . 141

Connecting Ratios, Fractions, and Division with Unit Rates . . . . . 141Solving Proportions by Cross-Multiplying Fractions . . . . . . . . . . 142

7.4 When You Can Use a Proportion and When You Cannot . . . . . . . 142

Can You Always Use a Proportion? . . . . . . . . . . . . . . . . . . . 142Who Says You Can’t Do Rocket Science? . . . . . . . . . . . . . . . . 143

7.5 Percent Revisited: Percent Increase and Decrease . . . . . . . . . . . 143

How Should We Describe the Change? . . . . . . . . . . . . . . . . . 144Calculating Percent Increase and Decrease . . . . . . . . . . . . . . . 144

Calculating Amounts from a Percent Increase or Decrease . . . . . . . 145

Percent Of Versus Percent Increase or Decrease . . . . . . . . . . . . 146Percent Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 146

Using the Commutative Property of Multiplication . . . . . . . . . . 147

8 Number Theory 1498.1 Factors and Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Factors, Multiples, and Rectangles . . . . . . . . . . . . . . . . . . . 150

Problems about Factors and Multiples . . . . . . . . . . . . . . . . . 150Finding All Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Do Factors Always Come in Pairs? . . . . . . . . . . . . . . . . . . . 151

8.2 Greatest Common Factor and Least Common Multiple . . . . . . . . 152Finding Commonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

The “Slide Method” . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Problems Involving Greatest Common Factors and Least Common-Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Flower Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Relationships between the GCF and the LCM and Explaining the

Flower Designs . . . . . . . . . . . . . . . . . . . . . . . . . . 155Using GCFs and LCMs With Fractions . . . . . . . . . . . . . . . . . 156

8.3 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . 157The Trial Division Method for Determining whether a Number Is Prime157

Factoring into Products of Primes . . . . . . . . . . . . . . . . . . . . 1588.4 Even and Odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158



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Why Can We Check the Ones Digit to Determine if a Number is Evenor Odd? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Questions About Even and Odd Numbers . . . . . . . . . . . . . . . 159Extending the Definitions of Even and Odd . . . . . . . . . . . . . . 160

8.5 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160The Divisibility Test for 3 . . . . . . . . . . . . . . . . . . . . . . . . 160

8.6 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . 161Decimal Representations of Fractions . . . . . . . . . . . . . . . . . . 161Writing Terminating and Repeating Decimals as Fractions . . . . . . 162What is 0.9999. . . ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163The Square Root of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Pattern Tiles and the Irrationality of the Square Root of 3 . . . . . . 164

8.7 Looking Back at the Number Systems . . . . . . . . . . . . . . . . . . 165

9 Algebra 1679.1 Mathematical Expressions and Formulas . . . . . . . . . . . . . . . . 168

Writing Expressions and a Formula for a Flower Pattern . . . . . . . 168Expressions for Dot and Star Designs . . . . . . . . . . . . . . . . . . 168Expressions with Fractions . . . . . . . . . . . . . . . . . . . . . . . . 169Evaluating Expressions With Fractions Efficiently and Correctly . . . 169Expressions for Story Problems . . . . . . . . . . . . . . . . . . . . . 170

9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171How Many Hi-Fives? . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Sums of Counting Numbers . . . . . . . . . . . . . . . . . . . . . . . 171Sums of Odd Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 172Equations Arising from Rectangular Designs . . . . . . . . . . . . . . 173Equations About Related Quantities . . . . . . . . . . . . . . . . . . 174Equations for Story Problems . . . . . . . . . . . . . . . . . . . . . . 174

9.3 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Solving Equations Using Number Sense . . . . . . . . . . . . . . . . . 175Solving Equations Algebraically and With a Pan Balance . . . . . . . 175

9.4 Solving Problems with Strip Diagrams . . . . . . . . . . . . . . . . . 175How Many Pencils Were There? . . . . . . . . . . . . . . . . . . . . . 176Solving Story Problems with Strip Diagrams and with Equations . . . 176Modifying Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Solving Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.5 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Arithmetic Sequences of Numbers Corresponding to Sequences of Figures179



CONTENTS xi

How Are Formulas for Arithmetic Sequences Related to the Way Se-quences Start and Grow? . . . . . . . . . . . . . . . . . . . . . 180

Explaining Formulas for Arithmetic Sequences . . . . . . . . . . . . . 181Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Repeating Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Sequences and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 184The Fibonacci Sequence in Nature and Art . . . . . . . . . . . . . . . 185What’s the Next Entry? . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.6 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Sums of Powers of Two . . . . . . . . . . . . . . . . . . . . . . . . . . 187An Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . 188Making Payments Into an Account . . . . . . . . . . . . . . . . . . . 188

9.7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Interpreting Graphs of Functions . . . . . . . . . . . . . . . . . . . . 189Are These Graphs Correct? . . . . . . . . . . . . . . . . . . . . . . . 190

9.8 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191A Function Arising from Proportions . . . . . . . . . . . . . . . . . . 191Arithmetic Sequences as Functions . . . . . . . . . . . . . . . . . . . 192Analyzing the Way Functions Change . . . . . . . . . . . . . . . . . . 193Story Problems for Linear Functions . . . . . . . . . . . . . . . . . . 193Deriving the Formula for Temperature in Degrees Fahrenheit in Terms

of Degrees Celsius . . . . . . . . . . . . . . . . . . . . . . . . . 194

10 Geometry 19710.1 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Visualizing Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . 198The Rotation of the Earth and Time Zones . . . . . . . . . . . . . . . 199Explaining the Phases of the Moon . . . . . . . . . . . . . . . . . . . 199

10.2 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Angle Explorers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Angles Formed by Two Lines . . . . . . . . . . . . . . . . . . . . . . 201Angles Formed when a Line Crosses Two Parallel Lines . . . . . . . . 202Seeing That the Angles in a Triangle Add to 180◦ . . . . . . . . . . . 202Using the Parallel Postulate to Prove that the Angles in a Triangle

Add to 180◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Describing Routes Using Distances and Angles . . . . . . . . . . . . . 203Explaining Why the Angles in a Triangle Add to 180◦ by Walking and

Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203



xii CONTENTS

Angle Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Students’ Ideas and Questions About Angles . . . . . . . . . . . . . . 204

10.3 Angles and Phenomena in the World . . . . . . . . . . . . . . . . . . 204Angles of Sun Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205How the Tilt of the Earth Causes Seasons . . . . . . . . . . . . . . . 205How Big Is the Reflection of Your Face in a Mirror? . . . . . . . . . . 205Why Do Spoons Reflect Upside Down? . . . . . . . . . . . . . . . . . 206The Special Shape of Satellite Dishes . . . . . . . . . . . . . . . . . . 206

10.4 Circles and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206Points That Are a Fixed Distance From a Given Point . . . . . . . . 206Using Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207The Global Positioning System (GPS) . . . . . . . . . . . . . . . . . 207Circle Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

10.5 Triangles, Quadrilaterals, and Other Polygons . . . . . . . . . . . . . 208Using a compass to draw Triangles and Quadrilaterals . . . . . . . . . 208Making Shapes by Folding Paper . . . . . . . . . . . . . . . . . . . . 209Constructing Quadrilaterals with Geometer’s Sketchpad . . . . . . . . 209Relating the Kinds of Quadrilaterals . . . . . . . . . . . . . . . . . . 210Venn Diagrams Relating Quadrilaterals . . . . . . . . . . . . . . . . . 211Investigating Diagonals of Quadrilaterals with Geometer’s Sketchpad 211Investigating Diagonals of Quadrilaterals(alternate) . . . . . . . . . . 212

10.6 Constructions With Straightedge and Compass . . . . . . . . . . . . . 212Relating the Constructions to Properties of Rhombuses . . . . . . . . 212Constructing a Square and an Octagon with Straightedge and Compass 213

11 Measurement 21511.1 Fundamentals of Measurement . . . . . . . . . . . . . . . . . . . . . . 216

The Biggest Tree in the World . . . . . . . . . . . . . . . . . . . . . . 216What Does “6 Square Inches” Mean? . . . . . . . . . . . . . . . . . . 216Using a Ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.2 Length, Area, Volume, and Dimension . . . . . . . . . . . . . . . . . 217Dimension and Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.3 Error and Precision in Measurements . . . . . . . . . . . . . . . . . . 217Reporting and Interpreting Measurements . . . . . . . . . . . . . . . 218

11.4 Converting from One Unit of Measurement to Another . . . . . . . . 218Conversions: When Do We Multiply? When Do We Divide? . . . . . 218Conversion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Using Dimensional Analysis to Convert Measurements . . . . . . . . 219



CONTENTS xiii

Area and Volume Conversions . . . . . . . . . . . . . . . . . . . . . . 220Area and Volume Conversions: Which Are Correct and Which Are Not?220Problem Solving with Conversions . . . . . . . . . . . . . . . . . . . . 221

12 Area of Shapes 22312.1 What Area Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Units of Length and Area in the Area Formula for Rectangles . . . . 22412.2 The Moving and Additivity Principles About Area . . . . . . . . . . 224

Different Shapes with the Same Area . . . . . . . . . . . . . . . . . . 225Using the Moving and Additivity Principles . . . . . . . . . . . . . . 225

12.3 Areas of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Determining Areas of Triangles in Progressively Sophisticated Ways . 226Choosing the Base and Height of Triangles . . . . . . . . . . . . . . . 228Explaining Why the Area Formula for Triangles Is Valid . . . . . . . 228Determining Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

12.4 Areas of Parallelograms and Other Polygons . . . . . . . . . . . . . . 230Do Side Lengths Determine the Area of a Parallelogram? . . . . . . . 230Explaining Why the Area Formula for Parallelograms Is Valid . . . . 231

12.5 Cavalieri’s Principle About Shearing and Area . . . . . . . . . . . . . 231Shearing a Toothpick Rectangle to Make a Parallelogram . . . . . . . 231Is This Shearing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Shearing Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . 232Shearing Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

12.6 Areas of Circles and the Number Pi . . . . . . . . . . . . . . . . . . . 233How Big Is the Number π? . . . . . . . . . . . . . . . . . . . . . . . . 233A Hands-On Method to Approximate the Value of π . . . . . . . . . 233Over- and Under- Estimates for the Area of a Circle . . . . . . . . . . 233Why the Area Formula for Circles Makes Sense . . . . . . . . . . . . 233Area Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

12.7 Approximating Areas of Irregular Shapes . . . . . . . . . . . . . . . . 235Determining the Area of an Irregular Shape . . . . . . . . . . . . . . 235

12.8 Perimeter Versus Area . . . . . . . . . . . . . . . . . . . . . . . . . . 236Calculating Perimeters . . . . . . . . . . . . . . . . . . . . . . . . . . 236Perimeter Misconceptions . . . . . . . . . . . . . . . . . . . . . . . . 237How Are Perimeter and Area Related? . . . . . . . . . . . . . . . . . 237Can We Determine Area by Measuring Perimeter? . . . . . . . . . . . 238

12.9 The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . 238Using the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . 238



xiv CONTENTS

Can We Prove the Pythagorean Theorem by Checking Examples? . . 239A Proof of the Pythagorean Theorem . . . . . . . . . . . . . . . . . . 239

13 Solid Shapes 24113.1 Polyhedra and Other Solid Shapes . . . . . . . . . . . . . . . . . . . . 242

Making Prisms and Pyramids . . . . . . . . . . . . . . . . . . . . . . 242Analyzing Prisms and Pyramids . . . . . . . . . . . . . . . . . . . . . 242What’s Inside the Magic 8 Ball? . . . . . . . . . . . . . . . . . . . . . 242Making Platonic Solids with Toothpicks and Gumdrops . . . . . . . . 243Why Are There No Other Platonic Solids? . . . . . . . . . . . . . . . 243Relating the Numbers of Faces, Edges, and Vertices of Polyhedra . . 243

13.2 Patterns and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . 244What Shapes Do These Patterns Make? . . . . . . . . . . . . . . . . 244Patterns for Prisms and Pyramids . . . . . . . . . . . . . . . . . . . . 244Patterns for Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 245Patterns for Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Cross-Sections of a Pyramid . . . . . . . . . . . . . . . . . . . . . . . 247Cross-Sections of a Long Rectangular Prism . . . . . . . . . . . . . . 247Shadows of Solid Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 247

13.3 Volumes of Solid Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 248Why the Volume Formula for Prisms and Cylinders Makes Sense . . . 248Filling Boxes and Jars . . . . . . . . . . . . . . . . . . . . . . . . . . 248Comparing the Volume of a Pyramid With the Volume of a Rectangular

Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250The 1

3in the Volume Formula for Pyramids andCones . . . . . . . . . 250

Volume Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 250The Volume of a Rhombic Dodecahedron . . . . . . . . . . . . . . . . 251Volume Versus Surface Area and Height . . . . . . . . . . . . . . . . 251

13.4 Volume Versus Weight . . . . . . . . . . . . . . . . . . . . . . . . . . 252Determining Volumes by Submersing in Water . . . . . . . . . . . . . 252Underwater Volume Problems . . . . . . . . . . . . . . . . . . . . . . 252Floating Versus Sinking: Archimedes’s Principle . . . . . . . . . . . . 253

14 Geometry of Motion and Change 25514.1 Reflections, Translations, and Rotations . . . . . . . . . . . . . . . . 256

Exploring Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Exploring Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Exploring Reflections with Geometer’s Sketchpad . . . . . . . . . . . 256



CONTENTS xv

Exploring Translations with Geometer’s Sketchpad . . . . . . . . . . 256

Exploring Rotations with Geometer’s Sketchpad . . . . . . . . . . . . 256Reflections, Rotations, and Translations in a Coordinate Plane . . . . 256

14.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Checking for Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Frieze Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Traditional Quilt Designs . . . . . . . . . . . . . . . . . . . . . . . . . 259

Creating Symmetrical Designs with Geometer’s Sketchpad . . . . . . 259Creating Symmetrical Designs (alternate) . . . . . . . . . . . . . . . . 260

Creating Escher-type designs with Geometer’s Sketchpad . . . . . . . 26014.3 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Triangles and Quadrilaterals of Specified Side Lengths . . . . . . . . . 260Describing a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Triangles with an Angle, a Side, and an Angle Specified . . . . . . . . 261Using Triangle Congruence Criteria . . . . . . . . . . . . . . . . . . . 261

14.4 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Mathematical Similarity Versus Similarity in Everyday Language . . 262A First Look at Solving Scaling Problems . . . . . . . . . . . . . . . 262

Using the “Scale Factor,” “Internal Factor,” and “Set up a Proportion”Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

A Common Misconception About Scaling . . . . . . . . . . . . . . . . 264

Using Scaling to Understand Astronomical Distances . . . . . . . . . 264More Scaling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Measuring Distances by “Sighting” . . . . . . . . . . . . . . . . . . . 265Using a Shadow or a Mirror to Determine the Height of a Tree . . . . 265

14.5 Areas, Volumes, and Scaling . . . . . . . . . . . . . . . . . . . . . . . 266

Areas and Volumes of Similar Boxes . . . . . . . . . . . . . . . . . . . 266Areas and Volumes of Similar Cylinders . . . . . . . . . . . . . . . . 267

Determining Areas and Volumes of Scaled Objects . . . . . . . . . . . 268A Scaling Proof of the Pythagorian Theorem . . . . . . . . . . . . . . 268

Area and Volume Problem Solving . . . . . . . . . . . . . . . . . . . 269

15 Statistics 271

15.1 Formulating, Designing, and Gathering Data . . . . . . . . . . . . . . 272Challenges in Formulating Survey Questions . . . . . . . . . . . . . . 272

Choosing a Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Using Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 272



xvi CONTENTS

Using Random Samples to Estimate Population Size by Marking (Capture-Recapture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

15.2 Displaying Data and Interpreting Data Displays . . . . . . . . . . . . 274What Is Wrong With These Displays? . . . . . . . . . . . . . . . . . 274What Is Wrong with the Interpretation of These Displays? . . . . . . 275Three Levels of Questions about Graphs . . . . . . . . . . . . . . . . 276Display These Data about Pets . . . . . . . . . . . . . . . . . . . . . 277Investigating Small Bags of Candies . . . . . . . . . . . . . . . . . . . 277The Length of a Pendulum and the Time It Takes to Swing . . . . . 277Balancing a Mobile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

15.3 The Center of Data: Mean, Median, and Mode . . . . . . . . . . . . . 278The Mean as “Making Even” or “Leveling Out” . . . . . . . . . . . . 278Solving Problems about the Mean . . . . . . . . . . . . . . . . . . . . 279The Average as “Balance Point” . . . . . . . . . . . . . . . . . . . . . 280Same Median, Different Average . . . . . . . . . . . . . . . . . . . . . 280Can More than Half Be Above Average? . . . . . . . . . . . . . . . . 280Errors with the Mean and the Median . . . . . . . . . . . . . . . . . 281

15.4 The Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . 281What Does the Shape of a Data Distribution Tell Us About the Data? 282Distributions of Random Samples . . . . . . . . . . . . . . . . . . . . 282Comparing Distributions: Mercury in Fish . . . . . . . . . . . . . . . 283Determining Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . 283Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Percentiles Versus Percent Correct . . . . . . . . . . . . . . . . . . . . 284

16 Probability 28716.1 Basic Principles of Probability . . . . . . . . . . . . . . . . . . . . . . 288

Probabilities with Spinners . . . . . . . . . . . . . . . . . . . . . . . . 288Some Probability Misconceptions . . . . . . . . . . . . . . . . . . . . 288Using Experimental Probability to Make Predictions . . . . . . . . . 289Experimental Versus Theoretical Probability: Picking Cubes From a

Bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289If You Flip 10 Pennies Should Half Come Up Heads? . . . . . . . . . 290

16.2 Counting the Number of Outcomes . . . . . . . . . . . . . . . . . . . 290How Many Keys Are There? . . . . . . . . . . . . . . . . . . . . . . . 290Counting Outcomes: Independent Versus Dependent . . . . . . . . . 291

16.3 Calculating Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 291Number Cube Rolling Game . . . . . . . . . . . . . . . . . . . . . . . 291



CONTENTS xvii

Picking Two Marbles From a Bag of 1 Black and 3 Red Marbles . . . 291More Probability Misconceptions . . . . . . . . . . . . . . . . . . . . 293Expected Earnings from the Fall Festival . . . . . . . . . . . . . . . . 293

16.4 Using Fraction Arithmetic to Calculate Probabilities . . . . . . . . . 294Using the Meaning of Fraction Multiplication to Calculate a Probability294Using Fraction Multiplication and Addition to Calculate a Probability 294

A Blackline Masters 297



Chapter 0

General Advice on Teaching, ClassActivities, Grading, and Syllabi

1



2 CHAPTER 0. GENERAL ADVICE

This manual contains solutions to the Class Activities as well as my advice andrecommendations on teaching the material. I hope the manual will help you teachsuccessful, enjoyable courses using my book. Teaching prospective elementary teach-ers can be extremely satisfying work. I look forward to going to class every day andhope that you will too!

0.1 General Advice on Teaching Courses for Ele-

mentary Teachers

It’s almost too obvious to say, but it helps me to keep the following key point inmind when developing and teaching mathematics courses for prospective teachers:These courses should help prepare teachers to teach mathematics. Therefore suchcourses should be designed and taught to “travel into the classroom.” In other words,teachers should be able to draw on what they have learned in their mathematicscourses (including mathematics content courses, not just methods courses) to teachmathematics effectively. Thus mathematics courses for prospective teachers shouldfocus on the mathematics that the teachers will teach, but should also help teachersunderstand how this mathematics will develop and be built upon in subsequent grades.

I wrote this book for use in courses that focus on explaining and making senseof the mathematics that teachers will teach, therefore I encourage this focus in yourcourses. Most prospective elementary teachers entering my courses do not know thatprocedures such as the standard longhand multiplication procedure can actually beexplained. They often seem to view the procedures and formulas of math as “given.”So expecting students to explain why these procedures and formulas work in termsof fundamental principles and concepts requires a shift in students’ thinking aboutmath. The payoff is that students will develop a more coherent understanding of mathbecause they will know how to link procedures and formulas to underlying concepts.

Students will need time and practice to help them learn what qualifies as a math-ematical explanation. If you ask students to explain why a formula or a procedureis valid, many will give mnemonic or other devices for remembering the formula, orthey will describe the procedure instead of explaining why it is valid. For example,to explain why we place the decimal point where we do when we multiply decimals, astudent might say that we “took out” some decimal places so we have to “put back”the same number of decimal places. Such a statement is not sufficiently specific orcomplete to qualify as a valid mathematical explanation. You will need to keep press-ing students to seek logical explanations that draw on central mathematical ideas.



0.1. GENERAL ADVICE ON TEACHING COURSES FOR ELEMENTARY TEACHERS3

In the decimal multiplication example you might ask what we do mathematically inorder to “take out” a decimal point (we multiply by 10 a suitable number of times tomove the decimal point or to shift the digits of the number).

The ideas of elementary school math are surprisingly deep and subtle. It is there-fore likely that you and your students will encounter some mathematics in the bookthat you will find surprising, unfamiliar, and in fact, difficult. A natural reaction tosuch an encounter is to say “if this is hard for me, then it’s too hard for my students.”Such a reaction can be followed by the dreaded question, “why do we need to knowthis?”. This is a legitimate question, so I urge you to take it seriously and treat itrespectfully. (I actually rarely get this question any more.) It might also help to com-ment that young students are just as smart as we are, they are just less experienced.Therefore young students’ ways of thinking about mathematical ideas may actuallybe less rigid than ours.

To help your students see the relevance of what they are learning to their futurework, you may wish to point them to the mathematics standards in your state. Manystates are now expecting students to make sense of math and to discuss, reason about,and explain mathematical ideas rather than simply carry out procedures withoutunderstanding.

Some students may have their heart set on teaching a certain grade and mayresist thinking about material that is taught at other grade levels. If so, remind thestudents that they will be certified to teach a range of grades and that they must beprepared to teach the math at all of those grade levels. Also, to teach 5th grade, say,one must be thinking ahead to the material students will be learning at 6th and 7thgrades.

Even if you do not have experience with elementary school teaching, you can referto the carefully thought out recommendations of respected national organizations.The National Council of Teachers of Mathematics (NCTM) has developed principlesand standards for the teaching of mathematics in schools (Pre-K – 12) and CurriculumFocal Points for Prekindergarten through Grade 8 Mathematics. The ConferenceBoard of the Mathematical Sciences (CBMS) has developed recommendations for thepreparations of teachers. If we take these recommendations seriously, then teacherswill need to develop a much deeper understanding of elementary school mathematicsthan has been traditional.




0.2 Comments and Advice on the Class Activities

The Class Activities were developed in tandem with the text as a means to helpstudents engage actively with the ideas during class. They are intended to be usedflexibly to meet the needs of your students and to give you a choice of things to do inclass. I also hope they will inspire your own inventiveness and enthusiasm for teachingthe material.

It is unlikely that you will have time to do all the Class Activities in the sectionsyou cover, so most likely, you will have to pick and choose. I have labeled the activitiesthat I view as most central and important with the “core” symbol to help you. Evenso, please don’t feel that you must use all these core activities. It’s also fine not to doall the problems in a Class Activity. In many cases, there will be more problems inan activity than you have time for, especially if you have vigorous class discussions.Also, the Class Activities are not all of the same length. Some will be quick andwon’t require a lot of discussion, others will require more thought and discussion andtherefore more time.

It usually makes sense to precede a Class Activity with a short lecture on theunderlying principles or concepts that will be addressed in the activity. After a shortintroduction, have the students begin to work on one or several problems in the ClassActivity.

My favorite way to use the Class Activities is with a “think, pair, share” approach.First ask students to think individually about a problem in an activity (or perhapsthe activity as a whole), then, when the students are ready, or after a predeterminedamount of time, ask students to share their results or questions with a neighbor.Finally, have a whole-class discussion about the problem. I like to proceed throughthe activities one or two problems at a time rather than having students do the wholeactivity before we discuss it. By proceeding a problem or two at a time, students areless likely to get bogged down and are more likely to spend class time productively.At the end of the activity, summarize the findings and draw students’ attention tokey points.

How can you lead a discussion on a problem? You can begin such a discussion byasking for a volunteer to present a solution (either at the board or orally). Ratherthan stating whether the solution is correct or not, you can ask the other studentsif they have any questions or comments about the presentation. Or you can haveseveral students put their solutions on the board (especially if you believe they havesomewhat different approaches) and then ask the class to discuss the collection ofsolutions. In my experience, some of the most fruitful discussions occur when severaldifferent explanations can be examined, compared, and contrasted.



0.2. COMMENTS AND ADVICE ON THE CLASS ACTIVITIES 5

It will be important for you to set a classroom climate in which students areexpected and encouraged to discuss and critique each other’s work and feel comfort-able doing so. You will need to actively promote and encourage such discussion, forexample, by stating that in class, initial correctness is not important, and that theopportunity to discuss proposed solutions and to deliberate whether or not they arecorrect offers valuable learning opportunities. (It’s a good idea to warn students thatyou do expect to see correct arguments on tests.) Explain that teachers must be goodat discussing mathematics and at evaluating solutions proposed by their students, sothat for students who intend to become teachers, active participation in class discus-sions and careful listening to proposed solutions is especially important. Regularlyencourage your students to think carefully about whether proposed arguments makesense or not and give them time to think, decide, and comment.

Needless to say, it’s important never to belittle students’ understanding of math-ematics and to assure students that it’s ok to make errors during class discussions.When students have finished asking questions and making comments about a pro-posed solution, you can either ask your own questions and make your own comments,or you might first ask if anyone had a different way of solving the problem or explain-ing their solution. A major benefit of class discussions is that you will learn a lotabout how your students are thinking about the material and you will be better ableto gauge how you might push their thinking further.

A word of warning: even if your class has had an excellent discussion and youpulled the activity together at the end by summarizing the findings and drawingstudents’ attention to key points, do not assume that everything is crystal clear in allthe students’ minds. In my experience, follow-up reading, written homework, periodicquizzes or tests, and a comprehensive final exam are invaluable in helping students tosolidify their understanding. Should you assign the reading before or after you do therelated activities? I think that you can do it either way, but I prefer to have studentsdo the activities before I assign the related reading. For this reason I have generallyplaced the activities before the related portions of the text. I think the text is bestused to help students solidify their ideas after they have begun to explore the ideasin class.

Although my colleagues and I never took mathematics classes that used “think,pair, share” work on class activities, we have been successful in teaching this way byusing these Class Activities. If you are used to lecturing (as I had been), then it maytake some time to get used to “think, pair, share” and class discussions. But I thinkyou will agree with me that discussing mathematics helps students learn mathematics,and that class discussions and the opportunity for students to express their thinkingwill help you monitor your students’ progress.




One final comment about the Class Activities: please don’t be a slave to them!You are welcome to modify them, to use them as a starting point in your instruction,or as an inspiration for something different. The goal is to help the students learnthe material, not to “cover” activities. I think it’s important for instructors to beactively engaged in their teaching and to be thinking about how to make the coursematerial interesting for their students and themselves. No set of activities, no matterhow good, can substitute for your active engagement in and enthusiasm for helpingyour students learn!

0.3 Comments and Advice on the Practice Exer-

cises and Problems

Most students have little or no experience explaining why math problems are solvedthe way they are, and therefore require time, patience, and effort to learn to do this.As the instructor, you must be patient and perservere. I think it works well to givefrequent (e.g., daily) small assignments with prompt feedback. You may need toassign fewer problems to be turned in as homework than you are used to, otherwisethe grading can become overwhelming. I like to hold students responsible for doingthe practice problems by giving closed-book quizzes and tests with similar problems.Remind students that the answers to the practice problems provide them with manyexamples of good explanations of the sort they should learn to write, so that studyingthese explanations will help them improve their own.

0.4 Grading

Recently I’ve been grading students’ work on a 10 point scale. I give extra points fortruly exemplary work. I describe the scores to my students as follows:



0.4. GRADING 7

# of points description characteristics

10.5 exemplary work that could serve as a model for other students10 very good correct work that is carefully thought out and thorough9 work that contains only a minor flaw8 competent good, solid work that is largely correct7 work that has merit but also has some shortcomings6 basic work that has merit but also has significant shortcomings4 emerging work that shows effort but is seriously flawed0 no credit no work submitted or no serious effort shown

I determine the score on assignments and tests by the extent to which the workmeets the following criteria, which I give to students:

• The work is factually correct, or nearly so, with only minor, inconsequentialflaws.

• The work addresses the specific question or problem that was posed. It is fo-cused, detailed, and precise. Key points are emphasized. There are no irrelevantor distracting points.

• The work could be used to teach a student: either a child or another collegestudent, whichever is most appropriate. Thus points that are difficult or likelyto be confusing are attended to with sufficient care and elaboration.

• The work is clear, convincing, and logical. An explanation should be convincingto a skeptic and should not require the reader to make a leap of faith.

• Clear, complete sentences are used.

• Mathematical terms and symbols are used correctly and as needed.

• If applicable, supporting pictures, diagrams, or equations are used appropriatelyand as needed.

• The work is coherent.

I’m a firm believer in the importance of a comprehensive final exam, which I thinkis essential for helping students to solidify their understanding.




0.5 Sample Syllabi

A Three Semester Sequence

At the University of Georgia, we use this book for a 3 semester sequence:

Semester 1 A course on numbers and operations covering chapters 1 – 5 and sections6.1 – 6.3. This is a lot of material and I find I have to go more quickly than Iwould like to.

Semester 2 A course on geometry and geometric measurement covering chapters 10– 14.

Semester 3 A course on algebra, number theory, probability and statistics coveringsections 6.4 – 6.6, chapter 7, most of chapter 8 (usually omitting section 8.6,except for Class Activity 12U on Pattern Tiles and the Irrationality of theSquare Root of 3), chapter 9, and chapters 15, 16.

A Four Semester Sequence

If you have 4 semesters, you will have time to cover numbers and operations to thedepth they deserve and you will have more time for problem solving. I suggest thefollowing:

Semester 1 Course 1 on numbers and operations covering chapters 1 – 5.

Semester 2 Course 2 on numbers and operations covering chapters 6 – 8.

Semester 3 A course on geometry and geometric measurement covering chapters 10– 14.

Semester 4 A course on algebra, probability, and statistics covering chapters 9, 15,and 16.

A One Semester Course in Numbers and Operations

If you only have one semester in which you must cover the material of chapters 1 – 8,then the first thing I recommend is to lobby for an additional course! One semester isnot enough for all this important material, which elementary teachers need to knowwell. Here is what I recommend:



0.5. SAMPLE SYLLABI 9

• Chapter 1: Do sections 1.1 and 1.2 only.

• Chapter 2

• Chapter 3: Skip section 3.5 on adding and subtracting negative numbers.

• Chapter 4

• Chapter 5: Do section 5.1 only.

• Chapter 6: Do sections 6.1 – 6.3 only.

• Chapter 7: Do sections 7.1, 7.2 only.

• Skip chapter 8.

A Course on Proportional Reasoning

If you want to teach a course that focuses on ratios and proportions as they apply ina range of settings, consider using the following sections:

• Section 2.5, Percent

• If time: Section 5.1, Multiplying Fractions, preceded by a brief discussion of the(general) definition of multiplication in Section 4.1

• If time: Sections 6.4, 6.5 on Dividing Fractions, preceded by a brief discussionof the (general) definitions of division in Section 6.1.

• If there is not time for Sections 5.1 and 6.4, then discuss the definitions ofmultiplication and division in sections 4.1 and 6.1 briefly. It is essential tounderstand how to reason with these definitions since “proportional reasoning”is essentially reasoning about multiplication and division.

• Chapter 7

• Section 14.4 on Similarity.

• If time: Section 14.5, Areas, Volumes, and Scaling

• The part of Section 15.1 on using random samples.




Whole Number Arithmetic First, Then Fraction Arithmetic

If you prefer doing whole number arithmetic before you get into fractions, I suggestthe following sequence. It’s too much material for one semester, but can be turnedinto a two semester sequence by adding material from chapters 9, 15, and 16 or byspending more time on problem solving or on pedagogy.

1. Whole number arithmetic and number theory:

• Chapter 1, Numbers and the Decimal System: Section 1.1.

• Chapter 3, Addition and Subtraction: Sections 3.1 – 3.3.

• Chapter 4, Multiplication: All sections, 4.1 – 4.6.

• Chapter 6, Division: Sections 6.1 and 6.3.

• If time: Chapter 8, Number Theory: Sections 8.1 – 8.5.

2. Fraction and decimal arithmetic:

• Chapter 1, Numbers and the Decimal System: Sections 1.2 and 1.3. In-clude section 1.4 if there will be time.

• Chapter 2, Fractions: All sections.

• Chapter 3, Addition and Subtraction: Section 3.4.

• Chapter 5, Multiplying Fractions and Decimals: Sections 5.1 and 5.2.

• Chapter 6, Division: Sections 6.4 – 6.6.

• If time: Chapter 7 on Proportional Reasoning.

0.6 Advice to Give Students Who Are Struggling

Some students who are used to simply following and memorizing step by step processesin math have a hard time learning to give explanations. Here are some things youmight discuss with these students.

It’s important for prospective teachers to realize that today’s expectations formath in elementary school and for teaching to today’s standards call for a focus onreasoning and sense-making in math and for communicating that reasoning. Teacherswill be responsible not just for showing students how to carry out mathematicalprocedures but also for helping students understand the underlying mathematicalideas and why the procedures make sense. There is a much greater conceptual focus



0.6. ADVICE TO GIVE STUDENTS WHO ARE STRUGGLING 11

and teachers will need to be able to lead discussions about the ideas and reasoningthat is involved in math. It’s this same kind of conceptual focus on reasoning andsense making that we are after in math courses for teachers. Students need to realizethat their old approaches to math may no longer be adequate if they had previouslyonly focused on copying what the teacher did rather than trying to really understandthe ideas. Such students need the recognition, willingness, and patience to do the hardwork it takes to learn the reasoning behind the math and to focus on understanding.

Here are some specific steps I recommend for students:

• The student must take responsibility for his or her own learning. She has toconvincer herself that she’s the one who has to actually understand the ideas.There are no shortcuts to this. It takes work, time, commitment, and patience!

• The student must actively engage in the class discussions and class work andshould use class time to think about the ideas and the reasoning being discussed.Most instructors who use this book structure the class time so that studentswill spend time thinking through how to solve a problem and discussing theirideas with a small group or one other student. This provides a valuable learningopportunity—if students take it. Some students make the mistake of not takingadvantage of this time to really think, but instead they wait for someone to tellthem the answer. This will not lead to success in understanding the ideas. Andthe student then feels they haven’t been taught the material.

• The student should read the relevant sections of the textbook in depth forunderstanding and monitor comprehension. He might look into ”comprehensionmonitoring” strategies. One that might help is to repeatedly ask ”why?” whilereading.

• The student should do all the practice exercises at the end of the relevantsections. After working out the problem, the student should then look at thesolutions and compare critically.

• When writing up homework, the student should write a draft, read it critically,and revise. Repeat if necessary!




0.7 What Should Be the Focus of Mathematics

Content Courses for Prospective Elementary

School Teachers?

As mathematics teacher educators, we want teachers to develop a deep understandingof the mathematics they will teach. But what does this deep understanding consistof and how do we help teachers acquire it?

First, we should recognize that pre-service teacher education cannot hope to pro-vide teachers with a fully mature understanding of the mathematics they will teach orwith fully mature teaching abilities. But teacher education should provide a strongfoundation upon which this maturity can develop. Teacher education should takeinto account the current knowledge and views of prospective teachers and use theseto look for points of high leverage. In my experience, the more we focus on ideas,problems, and activities that are clearly linked and obviously applicable to teachers’future teaching, the greater the leverage. So it makes sense to focus squarely on themathematics prospective teachers will (or may) teach and related mathematics a fewyears beyond. But even if we settle on these topics, there are different ways to studythem. What should we focus on when we study elementary mathematics? I thinkwe should focus on examining why this mathematics works the way it does. Notonly can teachers use this in their own instruction, but it also emphasizes that rulesand formulas in mathematics have reasons behind them, and that sense-making is ofprime importance in mathematics.

What are the key mathematical topics on which mathematics courses for prospec-tive elementary teachers should focus? The National Council of Teachers of Mathe-matics (NCTM) has published Curriculum Focal Points for Prekindergarten throughGrade 8 Mathematics (Focal Points), which describes focal topics for PreK throughgrade 8. The Focal Points recommend a focus on numbers and operations and geo-metric measurement, and in the later grades, also on algebra. The Focal Points makeclear that school children should understand the meanings of numbers and operationsand the key concepts which underlie calculation methods, namely place value and theproperties of operations. Repeatedly, the Focal Points ask students to develop pro-cedures and formulas, understand why these procedures and formulas work (in termsof fundamental underlying ideas and concepts), develop fluency in using proceduresand formulas, and use procedures and formulas to solve a wide range of problems.

So I think it makes sense for mathematics content courses for prospective elemen-tary school teachers to focus on arithmetic, how arithmetic is connected with andleads into algebra, and on geometric measurement. I think prospective elementary



0.8. HOW IS TEACHING FROM THIS BOOK DIFFERENT? 13

school teachers should learn to explain why the standard procedures and formulas ofelementary mathematics are valid, why nonstandard methods can also be valid, andwhy other seemingly plausible ways of reasoning are not correct. In doing so, teachersshould draw on key concepts and principles, such as place value, what fractions standfor, the meanings of the operations, the properties of arithmetic, the meanings oflength, area, volume, and angle, and the additivity of these geometric quantities andtheir invariance under rigid transformations.

Although the “how” of of elementary mathematics is usually quite familiar toprospective teachers, the “why” is not, and the view that the familiar procedures andformulas can be explained in terms of more fundamental ideas is especially unfamiliar.I have found that “using key ideas and concepts to explain why” is productive terrainto work on with prospective elementary teachers. This focus on “explaining why”builds on and extends prospective teachers’ existing knowledge. Prospective teachersengage enthusiastically in learning this material since they see it as valuable to theirfuture teaching. This focus allows prospective teachers to discuss their mathematicalideas and to represent them in a variety of ways while engaging in logical reasoning.It helps prospective teachers connect ideas, such as connecting area to multiplicationand the distributive property in explaining multiplication algorithms. It also naturallyincorporates some problem solving because the meanings of the four operations ofarithmetic and the meanings of length, area, volume, and angle and what kinds ofproblems these operations and concepts are useful for are essential in explanations forwhy procedures and formulas work. Additional multi-step problem solving is easilyadded (which is a good idea).

I believe that these recommendations fit with those of the Conference Board ofthe Mathematical Sciences regarding the mathematical preparation of teachers. I alsobelieve that these recommendations are a sensible foundation for working toward thevision of NCTM’s Principles and Standards for School Mathematics.

0.8 How Is Teaching from this Book Different from

Teaching Standard Math Major Courses?

If you are used to teaching standard math major courses, teaching a math course forprospective elementary teachers will probably require some adjustments on your part.I think that it is more valuable for prospective teachers to learn to give clear “ex-planatory arguments” as opposed to fully rigorous, complete proofs that fit withinan axiomatic development. For example, whereas in more advanced mathematical




treatments we usually take the standard procedure for fraction multiplication as adefinition, and we check that this definition makes sense because it agrees with mul-tiplication for the integers and turns the rational numbers into a field, a differenttreatment of fraction multiplication will be more useful for prospective elementaryteachers. These teachers should be able to give examples of story problems for agiven fraction multiplication problem, and they should be able to explain, in thecontext of a story, why the answer provided by the standard fraction multiplicationprocedure agrees with what one expects. Thus the focus here is on giving legiti-mate mathematical arguments that focus on sense-making rather than on a rigorousaxiomatic development of the mathematics.

Another difference between the explanations that this book intends to encourageand proofs developed in more advanced math courses is that teachers should ideallyknow several different kinds of explanations. Teachers should know multiple expla-nations because they are charged with bringing diverse students to high levels oflearning. A teacher who only knows one way to think about the material will not bewell prepared for this difficult task.



Chapter 1

Numbers and the Decimal System

15



16 CHAPTER 1. NUMBERS AND THE DECIMAL SYSTEM

Please note that many of the solutions to the class activities given here are onlysketches and do not provide full, detailed solutions, which you should press studentsto develop.

1.1 The Counting Numbers

Class Activity 1A: The Counting Numbers as a List

1. Common errors that children make (and are shown here) are: putting numbersin the wrong order, skipping one or more numbers, repeating numbers.

In the correct list each counting number appears exactly once and in the correctorder.

2. You could use this question as a lead in to the next activity, which answers it.

Class Activity 1B: Connecting Counting Numbers as a List

with Cardinality

1. No, see the next two parts.

2. You can introduce the terminology “one-to-one correspondence” when you dis-cuss this part. When objects are counted correctly, a one-to-one correspondenceis made between the objects and the relevant part of the list of counting num-bers.

3. Although we don’t know for sure, Child 2 may not understand that the lastnumber word said also indicates how many objects there are. It’s usually sur-prising to adults that this is a separate piece of knowledge!

5. & 6. To be fully fluent, as demonstrated by Child 3, children must be able to go backand forth between the “list view” of the counting numbers and cardinality. Afterestablishing that there were 6 bears initially, the child must be able to thinkabout the number 6 in the list, realize that the next number in the list is 7, thengo back to the cardinality point of view to understand that there are 7 bears.



1.1. THE COUNTING NUMBERS 17

Class Activity 1C: How Many Are There?

Includes IMAP video opportunity

I recommend leading into this activity by raising the issue about how to representever larger numbers as discussed in the text.

Students should gradually come to see that we can determine how many toothpicksthere are by bundling them into tens, and then bundling bundles of 10 into hundreds(and perhaps even bundling bundles of hundreds into thousands). Most students findit interesting to “rediscover” the decimal system.

Instead of working through parts 1 through 4 step by step, you can just pass outa bunch of toothpicks and ask the students to come up with ways of determininghow many toothpicks there are without counting them one by one. You might letthem discuss their ideas in pairs first, then ask them to discuss as whole group. Letthem present all their ideas, then gradually steer the students toward grouping thetoothpicks. When the time seems right, give a short lecture on the decimal systemand place value.

Parts 5 and 6 might be best done after a brief lecture on the decimal system. For5, it’s better to say “2 tens” because that draws attention to a ten as a single unit andthat the digit 2 really does stand for 2 of something (just not 2 individual counters,but 2 bundles of ten counters). Then you might like to show and discuss IMAP videonumber 5, Zenaida, who answers questions on how many ones and tens there are invarious numbers. In particular, she notes, correctly, that there are 32 ones in 32. Soasking “how many ones are in 32?” is different from asking “what is in the ones placeof 32?”.

For part 6, students should show 1 bundle of 100, 3 bundles of 10, and 7 individualtoothpicks. Although their pictures may not show this detail, a bundle of 100 shouldbe thought of as 10 bundles of 10, in keeping with the structure of the decimal system.

At this point, you could watch a segment of the IMAP video number 7, Talecia,to emphasize the importance of understanding the value of each place as 10 of theprevious place’s value. The video shows a student trying to add two 3-digit numbers.The student is confused about what 10 hundred is. To save time, you might want tostart watching the video at time 2:24, when the student adds 600 and 400 and gets“10 hundred.”

For part 7, the problem is that there are more than 9 items in each “place”. Wemust bundle 10 of the individual toothpicks to make another bundle of 10 and thenbundle 10 of the bundles of 10 to make a bundle of 100, making a total of 1 bundleof 100, 3 bundles of 10, and 5 individual toothpicks, which is 135 toothpicks.

Follow this activity with a discussion of the place value drawings shown in the




text. Perhaps also show some drawings such as the ones in Figure 1.1.

1010100

For 2-digit

numbers:

OR: OR:

For numbers with 2 or more digits:

These two types of representations have

helped first graders learn to add and subtract

with regrouping.

This is too cumbersome, but

it is a natural starting place.

Figure 1.1: Simple Drawings to Represent Numbers in the Decimal System

Class Activity 1D: Showing the Values of Places in the Deci-mal System

This won’t take long since the calculations are straightforward. You might want tohave the whole class do it together.

Use this activity to make the point that powers of ten grow very quickly, so thedecimal system displays numbers compactly. Even though it’s easy to write numberssuch as one million, 1, 000, 000, and one billion,1, 000, 000, 000, showing that manythings is not so easy.



1.2. DECIMALS AND NEGATIVE NUMBERS 19

There are problems in later chapters that return to the issue of showing largenumbers.

1. To show 100,000 stars, 20 pieces of paper are needed.

To show 1,000,000 stars, 200 pieces of paper are needed.

2. To show one billion stars, 1000 times 200, or 200,000 pieces of paper are needed.This would require 200, 000÷ 500 = 400 packages of paper. It’s not realistic forMrs. Kubrick to show one billion stars.

1.2 Decimals and Negative Numbers

Class Activity 1E: Representing Decimals with Bundled Ob-jects

IMAP Video opportunityIn the second half of IMAP video 9, starting at around 1:04, Megan and Donna

are asked to let a “long” base ten block (ten cubes joined together lengthwise) standfor 1 and are asked to represent 1.8. They do so by using a small cube to representa decimal point and use 8 more longs to represent the 8 tenths. This representationdoes not fit with or show the structure of the decimal system, because tenths are notrepresented in a way that 10 of them make 1.

1. For 0.034 draw 4 individual paper clips and 3 bundles of 10.

For 0.134 add a bundle of 100.

For 0.13 drop the 4 individual paper clips.

2. For 0.0028 draw 8 individual beads and 2 bundles of 10.

For 0.012 drop the 8 individual beads and add a bundle of 100.

3. 137 if a single toothpick represents 1 (note that whole numbers are also “deci-mals”)

13.7 if a single toothpick represents 110

.

1.37 if a single toothpick represents 1100

.

1370 if a single toothpick represents 10. Other answers are possible, of course.




Class Activity 1F: Representing Decimals as Lengths

The idea behind this activity is that the number 1.234 can be physically representedby one 1 unit strip, two 0.1 unit strips, three 0.01 unit strips, and four 0.001 unitstrips. The remaining numbers are represented in a similiar manner. Emphasize thatthe value of each place in a decimal number is ten times the value of the place to itsright and that we can see this relationship in the lengths of the strips.

Instead of having students cut the strips out of the back of the manual, you mightprefer to make sets of strips out of card stock that students could use in pairs. Youcould also make a large set to stick up on the board (with tacky clay). Or you couldhave lengths of plastic tubing cut (and use washers for the thousandths) as describedin the reference given in the text.

Class Activity 1G: Zooming In on Number Lines

The issues on saying and writing decimals that are discussed next in the text will begood ones to bring up in the context of this and the next activity, or perhaps evenbefore starting these activities. In particular, it will be good to remind students thatthey can always put extra zeros after numbers to the right of the decimal point. Forexample, 3.2 = 3.20 = 3.200.

1. See Figure 1.2.

2. When observing the location of 3.2996 on each number line, point out that everynumber line has lots and lots of decimals on it, including those with many digitsdown to small decimal places, even if we don’t see the relevant tick marks onthe particular number line.

The number 3.2996 lies between the whole numbers 3 and 4, between the tenths3.2 and 3.3, between the hundredths 3.29 and 3.30, between the thousandths3.299 and 3.300. Discuss with the students that the decimal places fartherand farther to the right give more and more detail about where, specifically, anumber is located.

3. See Figure 1.3. If students get stuck, tell them to think about “zooming in”.Students may make the error shown in the next problem.

4. The problem with this labeling is that to fit with the structure of the decimalsystem, if the short tick marks are tenths, then the long tick marks should bewhole numbers, or if the long tick marks are tenths, then the short tick marks



1.2. DECIMALS AND NEGATIVE NUMBERS 21

3.2 3.3

3.29 3.30

3.1 3.2 3.3

3.21 3.22

3.9

3.293.23

3.291 3.292 3.293 3.299

3.2993.2991 3.2992

etc.

etc.

etc.

etc.3.2993 3.29993.300

3 4

Figure 1.2: Zooming In

4.9

4.9

4.9

4.84.81 4.82 4.83 4.89etc.

4.89 4.891 4.892 4.893 etc.

etc.

4.899

4.8994.8991 4.8992 4.8993 4.8999

Figure 1.3: Labeled Number Lines




should be hundredths. The long and short tick marks should fit with the waythe value of each decimal place is divided into 10 equal parts in the decimalsystem.

5. See Figure 1.4.

4.9 5.0

4.90

4.900

4.91 4.92 4.93 etc. 4.99

4.914.901 4.902 4.903 etc. 4.909

4.9014.9001 4.9002 4.9003 etc. 4.9009

Figure 1.4: Labeled Number Lines

Class Activity 1H: Numbers Plotted on Number Lines

1. The point labeled A was positioned by the computer to be at 1.4263. Of courseit’s not possible to tell by eye, or even by measuring that the point is exactlyat this location. Students should be able to tell the the decimal representationbegins 1.4 because it is between 1 and 2 and less than 1.5. They might even beable to tell that it’s roughly 1.42 or 1.43, but in any case less than 1.45. Thenumber 1.18 is too small to be A. The numbers 1.8, 1.861, and 1.6 are too largeto be A.

2. The first number line should be labeled 23 = 23.0, 23.1, 23.2, . . . 24 = 24.0.

The second should be labeled 0.03 = 0.030, 0.031, 0.032, . . . 0.039, 0.04 = 0.040.

The third should be labeled 0.0 = 0.00, 0.01, 0.02, . . . 0.09, 0.1 = 0.10.

The last should be labeled 7 = 7.0, 7.1, 7.2, . . . 7.9, 8 = 8.0. Note that on thelast number line, 7.0095 will be almost indistinguishable from 7.

3. First number line: the long tick marks are 9.6 and 9.7.

Second number line: the long tick marks are 9.616 and 9.617.



1.3. COMPARING NUMBERS IN THE DECIMAL SYSTEM 23

Third number line: the long tick marks are 9.61 and 9.62.

Fourth number line: the long tick marks are 9 and 10.

Fifth number line: the long tick marks are 9.6163 and 9.6164.

Class Activity 1I: Negative Numbers on Number Lines

1. Katie’s and Matt’s numbers are decreasing to the right, not increasing. Anegative number such as −7 is 7 units away from 0, but to the left of 0. So−8 and −7.2 have to be farther away from 0 than −7. It might help to reviewthe location of numbers such as −1 and −2 on a number line that also shows0 on it. Parna’s labeling does show −8 as farther from 0 than −7, but herlabeling doesn’t fit with the decimal structure that is indicated by the heavierand lighter tick marks. The left-most tick mark could be labeled as −8 or as−7.1 (or as −7.01 or −7.001 etc.). In the latter cases, it would make sense torewrite −7 as −7.0 (or as −7.00 or −7.000 etc.).

2. The heavy tick mark on the far left is −1. The numbers between 0 and 1 areall of the form 0. ∗ ∗ ∗ ∗ . . . and the numbers between 0 and −1 are all of theform −0. ∗ ∗ ∗ ∗ ∗ . . ..

1.3 Comparing Numbers in the Decimal System

Class Activity 1J: Places of Larger Value Count More than

Lower Places Combined

2. 99 is the largest. 100 is the smallest.

3. 999 is the largest. 1000 is the smallest.

4. If in the place of largest value one number has a larger entry than the other,then that number is greater no matter the entries in the lower place values. Thisis because the value of a decimal place is larger than any amount formed by allthe smaller places combined. We can see this when counting: we get a digit ina higher place value after we “fill up” all the lower (whole number) place valueswith 9s.




Class Activity 1K: Misconceptions in Comparing Decimals

IMAP Video opportunityIn IMAP video 9, up to about time 0:45 on the clip, Megan and Donna seem to

know how to compare decimals at first, saying that 4.7 and 4.70 are equal becauseyou can add a 0 to 4.7. But then, when they are told it’s “you can’t add a zero day,”they think 4.70 is bigger than 4.7.

See also IMAP video 8, in which a third grader and a fifth grader add decimalsand also compare decimals. Both children seem to have the “whole number thinking”misconception about comparing decimals.

1. correct order: 3.05, 3.25, 3.251, 3.3, 3.4

Whole number thinking: 3.3, 3.4, 3.05, 3.25, 3.251

Column overflow thinking: 3.05, 3.3, 3.4, 3.25, 3.251

Denominator focused: 3.251, 3.05, 3.25, 3.3, 3.4

Reciprocal thinking: 3.251, 3.25, 3.05, 3.4, 3.3

Money thinking: Either 3.05, 3.251, 3.25, 3.3, 3.4 or 3.05, 3.25, 3.251, 3.3, 3.4

2. Answers will vary. Notice that including pairs like the following can be useful:

3.25 and 3.3

3.05 and 3.3

3.05 and 3.25

3.251 and 3.05

3.26 and 3.251

Students who get the first and second one wrong may be using whole numberthinking.

Students who get the first one wrong and the second one right may be usingcolumn overflow thinking.

Students who get the first two right and the third and fourth one wrong maybe using reciprocal thinking.

Students who get the first three right and the fourth one wrong may be usingdenominator focused thinking.

Students who get the first four right and the fifth one wrong maybe be usingmoney thinking.



1.3. COMPARING NUMBERS IN THE DECIMAL SYSTEM 25

Class Activity 1L: Finding Smaller and Smaller Decimals

Students should come to see that no matter what decimal is listed that is greaterthan 2, there is always another decimal that is smaller than it. We can find such adecimal simply by putting 0s out beyond the first non-zero digit that occurs in thecandidate decimal and then putting a 1 in some place after that.

Expect some students to come up with the description given in part 4. This doesnot describe a decimal because there is no specific decimal place in which the digit 1occurs.

Class Activity 1M: Finding Decimals Between Decimals

If students are stumped by a pair such as 1.2 and 1.3 ask them to consider hundredths.Hopefully they will see that they can write these as 1.20 and 1.30. If not, remindthem of this.

Students should come to see that no matter what pair of decimals is given, thereis always another decimal in between.

Be prepared for the question: “What about 0.9999999. . . and 1?” The fact thatthese numbers are equal is explained in the chapter on number theory. You may wishto discuss that material if there is time.

Be prepared for someone to talk about a number that is “right next to” anothernumber (e.g., the number that is “right next to 1.1.” Ask what “right next to” wouldmean. The number would have to have some decimal representation. What wouldthat be?

Class Activity 1N: Decimals Between Decimals on NumberLines

You might want to remind students that they can always put extra zeros after numbersto the right of the decimal point. For example, 3.4 = 3.40 = 3.400.

1. See Figure 1.5.

1.6 1.71.65

Figure 1.5: A Labeled Number Line




We can think of 1.6 and 1.7 as represented by $1.60 and $1.70 respectively.$1.65 is in between.

2. See Figure 1.6.

12.540 12.54112.5405


3. See Figure 1.7.

2.78

2.781

2.772.762.752.74

2.7342


4. See Figure 1.8.

23.99 24 =

24.00

23.995


Class Activity 1O: “Greater Than” and “Less Than” withNegative Numbers

1. For one explanation you can use a number line, for another, view negativenumbers as amounts owed.

2. Use a number line or view the negative numbers as amounts owed.

1.4 Rounding Numbers



1.4. ROUNDING NUMBERS 27

Class Activity 1P: Why Do We Round

This is a quick activity to get students thinking about rounding.Often we round because we do not know the exact value. Other times we round

because the exact value is not important. Quite often both of these statements aretrue.

Class Activity 1Q: Explaining Rounding

1. The entries on the first number line: 34600, 34610, . . . , 34700.

The entries on the second number line: 34000, 34100, . . . , 35000

The entries on the third number line: 30000, 31000, . . . , 40000

2. Press students to explain why the number line they use to help them round isuseful. To explain rounding with the aid of a number line, students should lookfor a number line on which the number to be rounded is plotted in between tickmarks that are labeled with numbers that have consecutive digits in the placesthey are rounding to and only zeros in places of lower value. For example, ifthey are rounding a number to the nearest ten, they should focus on tick marksin the tens; if they are rounding to the nearest hundred, they should focus ontick marks in the hundreds. Then it’s just a matter of determining which tickmark the number is closest to.

34617 rounded to the nearest ten equals 34620.

34617 rounded to the nearest hundred equals 34600.

34617 rounded to the nearest thousand equals 35000.

3. The entries on the first number line: 99.25, 99.251, . . . , 99.26

The entries on the second number line: 99.2, 99.21, . . . , 99.3

The entries on the third number line: 99, 99.1, . . . , 100

The entries on the fourth number line: 90, 91, . . . , 100

4. 99.253 rounded to the nearest hundredth equals 99.25.

99.253 rounded to the nearest tenth equals 99.3.

99.253 rounded to the nearest one equals 99.

99.253 rounded to the nearest ten equals 100.




Class Activity 1R: Rounding with Number Lines

1. The number 3872 is between 3800 and 3900 and closer to 3900, so rounds to3900.

2. The number 3872 is between 3870 and 3880 and closer to 3870, so rounds to3870.

3. The number 2.349 is between 2.34 and 2.35 and closer to 2.35, so rounds to2.35.

4. The number 2.349 is between 2.3 and 2.4 and closer to 2.3, so rounds to 2.3.

5. Label the tick marks 100, 200, 300, . . . , 54700, 54800, 54900.

6. Label the tick marks 0.01, 0.02, 0.03, . . . 16.92, 16.93, 16.94.

7. Label the tick marks 0.1, 0.2, 0.3, . . . , 16.8, 16.9, 17.0.

Class Activity 1S: Can We Round This Way?

Maureen’s method is not a valid way to round even though it does give the correctanswer in this case. If Maureen used her method to round 1.2487 to the nearest tenthshe would give the answer 1.3. But the correct answer is 1.2 since 1.2487 is closer to1.2 than to 1.3 because it is less than 1.25.

As an extension, you might ask students to determine when Maureen’s methodwill give the correct answer and when it won’t.



Chapter 2

Fractions

29



30 CHAPTER 2. FRACTIONS

Please note that many of the solutions to the class activities given here are onlysketches and do not provide full, detailed solutions, which you should press studentsto develop.

I have purposely put fractions and percents at the beginning of the book becauseit’s so easy to leave these important and difficult topics to the tail end of a courseand have to rush through them.

2.1 The Meaning of Fractions

You might like to begin to weave in a discussion of the next section on solving problemsand explaining solutions into this section.

Class Activity 2A: Fractions of Objects

1. Show 34

of the piece of paper because you want to show 3 parts, given that thepiece of paper is 4 parts.

2. Benton should use 13

of his dough, which he can get by cutting the dough into3 equal parts either horizontally or vertically. He needs 1

3of the dough because

the 34

cup of butter consists of 3 parts (each of which is 14

cup) but he onlywants 1 of those parts.

The issue of “what’s the whole?” will come up in this activity because a givenamount represents a different fraction, depending on what the whole is taken to be.This is a natural lead in to the next activity.

Class Activity 2B: The Whole Associated With a Fraction

IMAP Video opportunity

1. It’s true that 36

of the longer rectangle has a larger area than 23

of the shorterrectangle. But the two fractions each refer to different sized wholes.

2. Kayla might be confused because there are different wholes we can use here. Ifwe take the whole to be the full collection of 12 circles, then the 3 circles form14

of this whole. If we take the whole to be 1 circle, then it’s not appropriate todescribe these circles as representing 1

4.



2.1. THE MEANING OF FRACTIONS 31

3. 4 is the number of equal parts into which the whole is to be divided, or thatmake up the whole. The whole could consist of 24 objects, for example, inwhich case it is misleading to say that the “whole is 4”.

In your discussions, start encouraging students to talk about fractions as “of awhole” rather than “parts out of parts.” For example, encourage them to say “2

3of

the bar is shaded” rather than “2 out of 3 parts are shaded.” One problem with the“out of” language is that it doesn’t draw attention to the associated whole. Also,it seems to cause some students to focus on the remaining part of the whole whenthe fractional part is “taken out of the whole,” i.e., removed from the whole. Thismisconception is discussed in IMAP Video 12. In particular, the discussion at theend explains why the “out of” language may cause problems for students.

Identifying the whole that a fraction is “of” will be especially important in ana-lyzing fraction multiplication and division (in chapters 5 and 6) because in fractionmultiplication and division story problems, different fractions in a problem can referto different wholes. So you might want to tell students that they will be encounteringthe “what’s the whole?” question later on as well.

We also have to specify the whole in order to interpret unambiguously pictures thatrepresent fractional amounts. This is especially important when it comes to improperfractions and mixed numbers. In your class discussions, you might encourage studentsto draw a separate picture of the whole and to label the whole as such.

Class Activity 2C: Relating a Fraction to its Whole


1. The swing area consists of 112

of the park. The whole for this 112

and for the 13

is the area of the entire neighborhood park. The whole for 14

is the area of theplayground.

You don’t need to bring this up now, but the swing problem is a story problemfor 1

4· 1

3.

2. Ben should use 12

of the oil in the bottle. The whole for this 12

is the amount ofoil in the bottle. But the whole for the 1

3and the 2

3is a cup of oil.

You don’t need to bring this up now, but the oil problem is a story problem for13÷ 2

3.




Encourage students to talk about fractions as “of a whole” rather than “parts outof parts.” For example, encourage them to say “ 1

12of the park” rather than “1 out of

12 parts.” See the comments from the previous activity, and in particular, note thatthere is a relevant IMAP video for this discussion.

Class Activity 2D: Comparing Quantities with Fractions

1. Nate raised 23

as much as Tyler. The whole for this 23

is the amount Tyler raised.Tyler raised 3

2as much as Nate. The whole for this 3

2is the amount Nate raised.

It is also correct to say that Tyler raised 112

times as much as Nate, or eventhat Tyler raised 50% more than Nate (although the latter answer doesn’t fitwith the “as much as” wording).

2. The strip for Company A will consist of 4 equal pieces and the one for CompanyB will consist of 5 equal pieces of the same size. Company B sells 5

4as much as

Company A. It is also correct to say that Company B sells 114

times as muchas Company A and even that Company B sells 25 % more than Company A.

3. Students should notice that in each case, the two fractions we use to describe therelationship between two quantities are inverses of each other. This previewsthe relationship A

B· B

A= 1.

Class Activity 2E: Fractions of Non-Contiguous Wholes


2. Mariah’s method is not correct because the 10 parts are not equal in area. Ifthe two plots did have the same area, then her reasoning would be correct.

3. Yes, Aysah’s picture can be used. It shows that the shaded portion is 15

ofPeter’s garden because 5 copies of the shaded portion can be joined to makethe whole garden. Or in other words, picture shows that the garden is dividedinto 5 parts and one of those parts is shaded (note that a “part” can consist ofseveral distinct pieces).

4. Matt’s reasoning is not valid because the two 15

each refer to a different wholeand the question is about Peter’s entire garden.

If the two plots were each 1 acre, then the shaded pieces would in fact betwo parts, each of which is 1

5of an acre, and therefore the two parts together



2.1. THE MEANING OF FRACTIONS 33

would be 25

of an acre. So yes, Matt’s reasoning can be used to make a correctstatement. As always, attending to the whole is crucial when working withfractions!

At this point, you might like to show IMAP video 14, in which Felisha divides 2cookies equally among 5 people. There is some confusion about how to describeeach person’s share. Is it 2

10or 2

5? The answer depends on what we take the

whole to be. (This video is an even better fit for the discussion of the relationshipbetween division and fractions in chapter 6.)

Class Activity 2F: Is the Meaning of Equal Parts AlwaysClear?

1. In most situations, it’s clear what “equal parts” means. But when we havedifferent sized objects like these, it’s not clear if all the marbles can be consideredequal.

2. Arianna did divide the shape into 4 parts of equal area even though the 4 partsare not congruent.

3. See parts 1 and 2 of this activity and the discussion in the text.

4. 613

of the shapes used in the design are triangles. 14

of the area of the designis made of triangles. For (a) the equal parts were the 13 different shapes thatmake up the design. For (b) we can divide each rhombus into two triangles thatare congruent to the triangles in the outer ring of the design. The hexagon canbe divided into six more congruent triangles. The outer six triangles are now 1

4

of the 24 triangles in the design.

Class Activity 2G: Improper Fractions

1. Confusion about what exactly is the whole could arise. To interpret the shadedregion as 5

4we must state that the whole consists of one large rectangle. You

could suggest to students that they draw another rectangle and label it with “1whole” in order to make clear what the whole is.

2. You might tell Enrico that we can use copies of parts. You could show him thatit makes sense to have 5

4cups of water, or a string of length 5

4of an inch.




3. Meili might think that the 6 pieces together must make the whole since it islarger than the 5 pieces that make the strip that is labeled as the whole. Somequestions to ask Meili are why she changed from fifths to sixths and whetherthe whole has changed or not.

2.2 Interlude: Solving Problems and Explaining

Solutions

You might like to begin to weave in a discussion of this section into the previous sec-tion. Although the norms of mathematical reasoning and explanation are certainlybased in our natural capacity for logical thought, these norms are culturally trans-mitted, and students must learn what qualifies as valid mathematical reasoning. Thistakes many examples as well as guidance from instructors.

2.3 Fractions as Numbers

Class Activity 2H: Number Line Errors with Fractions

1. (a) Eric may understand that making fourths involves dividing a unit into 4pieces, but he may think he can accomplish that by inserting 4 tick marksbetween 0 and 1 instead of dividing the interval from 0 to 1 into 4 equalpieces. Eric has actually divided the interval from 0 to 1 into 5 equalpieces.

(b) Kristin may simply be counting tick marks. She may not understand thatthe interval between 0 and 1 must be divided into 4 equal pieces. Hernumber line doesn’t show 0. Although not incorrect, it probably meansshe is not attending to the distance from 0.

Students may notice that Kristin didn’t appear to be working on the num-ber line that was previously labeled with 0 and 1. Perhaps she drew a newnumber line to work on.

(c) See text for the idea of circling intervals.

2. Tyler may have thought of 34

as 3 intervals out of the 4 intervals drawn on thenumber line. Clearly he is confused about exactly what is the whole.



2.4. EQUIVALENT FRACTIONS 35

Class Activity 2I: Fractions on Number Lines, Part 1

1. Be sure students talk about dividing the interval between 0 and 1 into 4 equal-length segments, and in particular, focus on length. It will be worth noting thatdividing the interval from 0 to 1 does not involve making 4 new tick marks. Seetext for full discussion of where to locate fractions on number lines and see theanswer to the practice problem in this section.

2. The tick marks should be 14

apart.

3. As the tick marks should be 14

apart, draw 3 equally spaced tick marks betweenthe tick marks for 0 and 1 (so as to divide the interval between 0 and 1 into 4equal pieces) and then continue on.


apart, draw 2 equally spaced tick marks betweenthe tick marks for 0 and 3

2, so as to subdivide the interval between 0 and 3

2into

3 equal pieces, and then continue on. Note that students must recognize that32

consists of 3 pieces, each of length 12.



4and then continue on.



5and then continue on.

2.4 Equivalent Fractions

Class Activity 2J: Equivalent Fractions

1. See text for an explanation using different numbers. Press students to explainthe multiplication of the numerator and denominator and not just to say that23

= 812

.

2. The issue addressed in this part may already come up in your discussion of part1. Note that it makes sense that when we divide each piece into 4 equal piecesthere will be 4 times as many pieces (in all, and shaded). So the pieces becomesmaller and to compensate, there are more of them. The pieces themselves aredivided, but the number of pieces is multiplied. In the picture we are workingwith the pieces themselves, but in the numerical work we are working with thenumber of pieces.




3. In the picture, each piece should be divided into 5 equal pieces.

4. Repeat the above with 23

replaced by AB

and 4 (or 5) replaced by N . If thereisn’t time to work all the way through this, you could instead remark that therewasn’t anything special about the numbers 2, 3, 4, 5 used in the previous expla-nations and that the very same reasoning will apply no matter what countingnumbers are involved.

Class Activity 2K: Misconceptions About Fraction Equiva-lence

1. Even though Anna did “do the same thing to the top and bottom of the frac-tion,” she doesn’t get equivalent fractions this way, which we can see immedi-ately by drawing pictures. When we teach about equivalent fractions, we shouldbe more specific about how to get equivalent fractions than just saying “do thesame thing to the top and the bottom.”

2. Both fractions are “one part away from a whole” but 17ths and 12ths are dif-ferent size parts.

3. Both fractions are equal to 1. Peter may not realize that when the whole isdivided into more parts, each part becomes smaller.

A similar issue comes up again in the activity “Can We Reason this Way?” inthe section on comparing fractions.

Class Activity 2L: Common Denominators

1. See Figure 2.1. In terms of the pictures, giving the fraction common denomi-nators is making like parts.

2. The two smallest common denominators are 12 and 24. When giving thesefractions common denominators we are dividing the strips and number linesinto like parts.



2.4. EQUIVALENT FRACTIONS 37

4/12 9/12 8/24 18/24 1000440pt 166pt

Figure 2.1: Common Denominators

Class Activity 2M: Solving Problems by Changing Denomina-tors

1. 16

of the larger (imagined) piece ofpaper is formed by 14

of the given piece ofpaper. Since 2

3= 4

6, and since we want 1 sixth and we have 4 sixths, we must

show 1 out of 4 equal parts of the given piece of paper. In solving this problem,23

also appears as 46. We can make 4 equal parts horizontally or vertically, which

look different.

2. Using a picture, we can see that Jeremy uses 34

+ 13

= 912

+ 412

= 1312

= 1 112

cupsbutter. The fraction 3

4appears as 9

12and 1

3appears as 4

12. The whole associated

with each fraction is a cup of butter.

3. Using a picture, we see that Jean needs 12

= 36

cups of butter but only has 13

= 26

cups of butter, which means she has 2 of the 3 parts she needs, so she has 23

of what she needs. In solving the problem the fractions appear as equivalentfractions with denominator 6. The whole associated with 1

2and 1

3is a cup of

butter, but the whole associated with 23

is the amount of butter Jean needs,namely the 1

2cup of butter.

4. Using a picture, we see that Joey should eat 38

cups of cereal. In the picture,we turn 3

4into 6

8in order to solve the problem.The whole associated with the 3

4




in the problem is a cup of cereal. The whole associated with the answer, 38, is

also a cup of cereal. But the whole associated with 12

is a serving of cereal.

Class Activity 2N: Fractions on Number Lines, Part 2

In this activity, students will need to give fractions common denominators to solvethe problems and they will need to focus on the meaning of fractions in terms ofnumber lines.

1. Let the distance between adjacent tick marks equal 16

and let the leftmost tickmark represent 2

3= 4

6.


and let the leftmost tickmark represent 33

4= 99

12.


= 0.05 and let the left-most tick mark represent 0.7 = 14

20.

4. Draw seven equally spaced tick marks between the tick marks for 0 and 23. The

distance between adjacent tick marks equals 112

. If we extend these tick markspast 2

3the tick mark for 3

4is the first one past 2

3.

5. Draw five equally spaced tick marks between the tick marks for 0 and 12. The

distance between adjacent tick marks equals 110

. If we extend these tick markspast 1

2the tick mark for 3

5is the first one past 1

2

Class Activity 2O: Simplifying Fractions

1. See practice problem 9 and its solution for the same type of problem.

2. See Figure 3.38 in the text and the text discussion.

3. 1218

= 2·63·6 = 2

3

1218

= 6·29·2 = 6

9= 2·3

3·3 = 23

4. Use pictures like the ones shown at the beginning of the activity or in practiceproblem 9.



2.5. COMPARING FRACTIONS 39

2.5 Comparing Fractions

You could begin this section by putting two fractions on the board (such as 23

and35) and asking students what methods they know for determining which of those two

fractions is greater.

Class Activity 2P: Can We Compare Fractions This Way?

Use this activity to caution students about comparing fractions by converting todecimals.

1. The two fractions will appear to be equal when we use a calculator to representthem as decimal numbers.

2. The second fraction is bigger because when you divide an object into 999,999,999,999equal pieces, each piece is slightly bigger than if you divided the object into1,000,000,000,000 equal pieces.

Class Activity 2Q: What is another way to Compare theseFractions?

Use this activity to discuss the method of comparing fractions that have the samenumerator by considering the denominators. At the end of this activity, you couldmention that even if two fractions don’t have the same numerator to start with, wecan give the fractions common numerators to compare them. In some cases this iseasier than giving fractions common denominators.

1. The second fraction is bigger because when you divide an object into 39 equalpieces, each piece is slightly bigger than if you divided the object into 49 equalpieces.

2. Use the same reasoning.




Class Activity 2R: Comparing Fractions by Reasoning2743

> 2645

because 27 > 26 and forty-thirds are greater than forty-fifths.1325

> 3470

because the first fraction is greater than 12

but the second fraction is lessthan 1

2.

1718

< 1920

because each fraction is one piece less than a whole but 20ths are smallerthan 19ths, so the second fraction is closer to a whole than the first.

5153

< 6567

because both fractions are 2 pieces less than a whole but sixty-seventhsare smaller than fifty-thirds, so the second fraction is closer to a whole than the first.

940

< 1244

because 940

< 14

< 1244

.1325

< 58

because both fractions are just over 12

but 58

is one eighth more than 12

whereas 1325

is half of a twenty-fifth more than 12.

Class Activity 2S: Can We Reason this Way?


1. Claire reaches the correct conclusion but her reasoning is not valid because eventhough 4 pieces is more than 3 pieces, 9ths are smaller than 8ths.

2. Conrad’s conclusion is incorrect, in fact

4

11<

3

8

You may wish to show IMAP Video 11 in which Ally talks about changing digitsin order to determine which fraction is larger. It’s not necessarily the same reasoningthat Claire and Conrad are using in this activity, but is in the same style. You couldalso show Video 12 where prospective teachers discuss a misconception in comparingfractions that many noticed when they interviewed students.



2.6. PERCENT 41

2.6 Percent

You can introduce this section by asking students to jot down what they think “per-cent” means and why they think we have percents. You could then discuss thematerial at the beginning of the section. You could then ask students why we use100 as the denominator for percentages instead of 10, say, or 1000. The denominatorof 100 allows percentages to be expressed with 2 digits, which is short enough to bequickly grasped yet provides enough information to distinguish and separate manycases.

Class Activity 2T: Pictures, Percentages, and Fractions

This activity gives students a chance to see various percentages as arising from com-bining other common “benchmark” percentages. This will be useful in mental percentcalculations.

1. Diagram 1: 95%

Diagram 2: 80% (viewed as 75% plus 15

of 25%)

Diagram 3: 45%

Diagram 4: 12.5% (half of 25%)

Diagram 5: 87.5% (75% plus half of 25%)

Class Activity 2U: Calculating Percents of Quantities by UsingBenchmark Fractions and Percent Tables

Most students find the “percent tables” to be really helpful for developing a feel forpercentages. (If you used a previous edition of this book, note that I changed thename from “percent diagram” to “percent table” in order to make clearer later onthat a percent table is really a form of ratio table. And a percent table can also beviewed as a table for a function.)

1. 110

of 80,000 is 8000. So half of 8000, namely 4000, is 5% of 80,000. If we take4,000 away from 80,000 that will be 95% of 80,000. So the answer is 76,000.




Percent diagram:100% −→ 80, 00010% −→ 80005% −→ 4000

95% −→ 76, 000

Some students may also calculate 9 times 8000 and then add 4000 to that, soit’s a good idea to ask for their reasoning, not just their percent diagrams. Havestudents recognize that this is another valid way to compute 95% of 80,000.

2. 110

of 6500 is 650. So half of that, namely 325 is 5% of 6500. Since 10% + 5% =15%, we have that 650 + 325 = 975 is 15% of 6500.

3. 1% of $25 is $.25. Hence 7% of $25 is $1.75.

4. 12

of 810 is 405. 10% of 810 is 81. So 60% of 810 is 405 + 81 = 486.

5. 20% of 810 is 162. 40% of 810 is 162 + 162 = 324. Hence 60% of 162 is324 + 162 = 486.

6. 12

of 180 is 90. 10% of that is 9, which is also 5% of 180 (or you can find 5%of 180 by taking 10% of 180 first and then take half of that). So 55% of 180 is90 + 9 = 99.

Class Activity 2V: Calculating Percentages with Pictures andPercent Tables

1. As seen in Figure 2.2, last year, 3 out of the 5 parts of normal rainfall fell, so35

= 60% of normal rainfall fell.

normal

last year

1 inch of

rain

Figure 2.2: Rainfall



2.6. PERCENT 43

2. Since 14

cup gives 100% of your full daily value of clacium, 112

gives you 3313%

of your daily value. Hence 13

= 412

of a cup gives you 13313% of your daily value.

Class Activity 2W: Calculating Percentages with EquivalentFractions

1. 60400

= 640

= 320

= 15100

= 15%.

2. 225500

= 9·2520·25 = 9

20= 45

100= 45%.

3. 625

= 6·425·4 = 24

100= 24%

4. 1275

= 12÷375÷3

= 425

= 16100

= 16%

5. 4260

= 710

= 70%

6. You may need to discuss with students that just as we can have improperfractions, we can also have percentages that are greater than 100%. 24

16= 3

2=

150100

= 150%.

7. 1624

= 23

= 0.666...1

= 66.66...100

= 66.66 . . .%. Note that this used “going through 1.”

Class Activity 2X: Calculating a Quantity from a Percentage

of It

1. See Figure 2.3 for a picture.

2 boxes are in each of the 3 shaded rectangles,

so there are 20 boxes total

Figure 2.3: Boxes of Paper

2. 30%→ 6

10%→ 2

100%→ 20




3. The fraction of his order that the 6 boxes represent must equal the fraction that30% is of 100%. Since 30

100= 5·6

5·20 = 620

his order must have been 20.

4. 7%→ $2.10

1%→ $.30

100%→ $30

Class Activity 2Y: Percent Problem Solving

1. With a percent diagram:40% −→ 3020% −→ 15

100% −→ 75

With equivalent fractions: we want to solve

40

100=

30

?

40

100=

4

10=

2

5=

30

75

2. Draw two strips to represent the running distances, making Marcie’s 10 pieceslong and Andrew’s 4 pieces long. From the comparison, we see that Marcie’sdistance is 250% as long as Andrew’s.

3. The 8 extra female bugs are as much as 10% of the male bugs. So there are10 · 8 = 80 male bugs and 88 female bugs. This makes 168 bugs all together.Using a percent table we might say

10% −→ 8

100% −→ 80

but this 100% refers to 100% of the male bugs. It would be easy to thinkincorrectly that this 100% refers to all the bugs.

4. Draw two strips, one for the dogs, one for the cats. Make the dog strip 25%longer than the cat strip by making the cat strip 4 pieces long and the dog stripone piece longer, so 5 pieces long. Then the cats and dogs together are “madeof” a total of 9 pieces and the cats are 4 of those 9 pieces. So the percentage ofcats is 4

9, which is about 44%.


contents€¦ · full file at iv contents “greater than” and “less than” with negative...

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