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

1.10. Additional Problems ........................................................................ 23

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

2.6. Additional Problems ........................................................................ 44

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

3.4. Additional Problems......................................................................... 58

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

4.8. Additional Problems......................................................................... 80

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

5.5. Additional Problems....................................................................... 103

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

6.5. Additi