Top
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. Additional Problems ...................................................................... 132
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
7.6. Additional Problems....................................................................... 153
7.7. Pedagogical Explorations ............................................................... 156
7.8. Summary ....................................................................................... 157
Chapter 8: Quadratic Functions..................................................................... 158
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
8.7. Additional Problems ...................................................................... 178
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
9.9. Additional Problems....................................................................... 209
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
10.4. Additional Problems ...................................................................... 230
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
11.7. Additional Problems ...................................................................... 253
11.8. Pedagogical Explorations .............................................................. 254
11.9. Summary ....................................................................................... 255
