# Algebra for Elementary and Middle School Teachers: An Inquiry Approach, 2nd Edition

Published Date: Jun 28, 2012

Preface......................................................................................................................... ix

UNIT 1: LEARNING ALGEBRA THROUGH PROBLEM SOLVING ............................................................ 1

Chapter 1: Introduction to Problem Solving ..................................................... 1

1.1. Silver Coins ....................................................................................... 8

1.2. Crossing the River............................................................................ 10

1.3. Cutting Through the Layers.............................................................. 12

1.4. Bossy the Cow................................................................................. 14

1.5. Painted Cubes .................................................................................. 15

1.6. Chickens .......................................................................................... 16

1.7. Staircases ......................................................................................... 18

1.8. Counting Handshakes....................................................................... 19

1.9. Variables.......................................................................................... 20

1.11. Pedagogical Explorations ................................................................ 25

1.12. Summary ......................................................................................... 28

Chapter 2: Solving Equations........................................................................... 30

2.1. Relational Thinking.......................................................................... 34

2.2. Using Relational Thinking to Solve Equations ................................. 36

2.3 Working Backwards to Solve Equations........................................... 37

2.4. Operating with the Unknown ........................................................... 39

2.5. Farmer Brown.................................................................................. 41

2.7. Pedagogical Explorations ................................................................ 46

2.8. Summary ......................................................................................... 47

UNIT 2: LEARNING ALGEBRA THROUGH PATTERNS AND SEQUENCES .......................................... 49

Chapter 3: Introduction to Patterns......................................................................................49

3.1. Four Repeating Patterns ................................................................... 52

3.2. Predicting with Patterns.................................................................... 54

3.3. Number Array.................................................................................. 57

3.5. Pedagogical Explorations ................................................................. 59

3.6. Summary.......................................................................................... 60

Chapter 4: Growing Patterns ........................................................................... 62

4.1. Tile Patios........................................................................................ 66

4.2. Multiple Ways of Seeing ................................................................. 67

4.3. Rows and Columns .......................................................................... 69

4.4. Toothpick Patterns ........................................................................... 70

4.5. More Tile Patterns............................................................................ 73

4.6. Matching Rules to Patterns............................................................... 76

4.7. Pattern Block Patterns ...................................................................... 78

4.9. Pedagogical Explorations ................................................................. 84

4.10. Summary.......................................................................................... 88

Chapter 5: Sequences ....................................................................................... 90

5.1. Skip Counting .................................................................................. 95

5.2. Describe the Rule............................................................................. 97

5.3. Arithmetic and Geometric Sequences............................................. 100

5.4. Honeybees ..................................................................................... 101

5.6. Pedagogical Explorations ............................................................... 104

5.7. Summary........................................................................................ 105

UNIT 3: LEARNING ALGEBRA THROUGH FUNCTIONS AND MODELING......................................... 106

Chapter 6: Representing Functional Relationships ....................................... 106

6.1. Function Machine .......................................................................... 113

6.2. Jumping Jacks and Crackers........................................................... 119

6.3. From Graphs to Situations.............................................................. 122

6.4. From Situations to Graphs.............................................................. 130

6.6. Pedagogical Explorations ............................................................... 133

6.7. Summary ....................................................................................... 135

Chapter 7: Linear Functions........................................................................... 137

7.1. Skate Rental ................................................................................... 142

7.2. Matching Representations of Linear Functions............................... 144

7.3. Fahrenheit and Celsius ................................................................... 147

7.4. Identifying Linear Functions .......................................................... 148

7.5. Marbles in a Glass of Water ........................................................... 151

7.7. Pedagogical Explorations ............................................................... 156

7.8. Summary ....................................................................................... 157

8.1. Areas of Rectangles with Perimeter of 24....................................... 164

8.2. Graphs and Equations of Quadratic Functions................................ 166

8.3. Graphs, Tables, and Equations of Quadratic Functions................... 169

8.4. Tables with a Non-Unit Change in x............................................... 173

8.5. Slam Dunk!.................................................................................... 175

8.6 Covering Jar Lids with Cereal ........................................................ 176

8.8. Pedagogical Explorations .............................................................. 182

8.9. Summary ....................................................................................... 184

Chapter 9: Exponential Functions.................................................................. 185

9.1. Paper Folding................................................................................. 191

9.2. Cuisenaire Rod Trains.................................................................... 193

9.3. Exponential Functions Represented in Graphs................................ 194

9.4. Percentage Growth Rates ............................................................... 196

9.5. More Growth Factors and Decay Factors........................................ 198

9.6. Approximating Growth Factors...................................................... 200

9.7. Disappearing Ms ............................................................................ 202

9.8. Sorting Functions........................................................................... 204

9.10. Pedagogical Explorations .............................................................. 211

9.11. Summary ....................................................................................... 212

UNIT 4: LEARNING ALGEBRA THROUGH GENERALIZATION AND PROOF................................... 214

Chapter 10: Properties of Numbers and Operations.................................... 214

10.1. Which Does Not Belong?............................................................... 220

10.2. Building Rectangles with Algebra Tiles ......................................... 222

10.3. Properties of Sets and Operations................................................... 226

10.5. Pedagogical Explorations .............................................................. 232

10.6. Summary ....................................................................................... 234

Chapter 11: Algebraic Proof ......................................................................... 236

11.1. Analyzing Children’s Proofs .......................................................... 240

11.2. Patterns in a Multiplication Chart ................................................... 243

11.3. Department Store Discount ............................................................ 245

11.4. Changing Rectangles ..................................................................... 246

11.5. Numerical Relationships ................................................................ 248

11.6. Sums of Consecutive Numbers....................................................... 250