Table of Contents
Foreword ix
Introduction 1
Chapter 1 What You Teach and How You Teach It 7
The Power of KWC: An Alternative to Key Words 8
Using KWC to Tap Prior Knowledge 10
Using KWC to Structure Group Learning 12
Using KWC to Deepen Connections 13
Extensions 16
Chapter 2 Six Big Ideas 19
The Research on Mathematical Learning and Teaching 19
Principle 1 Engaging Prior Understanding 19
Principle 2 The Essential Role of Factual Knowledge and Conceptual Frameworks 20
Principle 3 The Importance of Self-Monitoring 21
Six Big Ideas: Building on Mathematical Research and Principles 22
Big Idea 1 Teachers Broaden Their View of Problem Solving 22
Big Idea 2 Making Connections Between the Problem and Their Lives 34
Big Idea 3 Creating Multiple Representations of Increasing Abstraction 43
Big Idea 4 Students Solving Problems: Same Concept, Multiple Contexts 51
Big Idea 5 Cognitively Based Planning for Language, Connections, Contexts, and Representations 55
Big Idea 6 Integrating Reading Comprehension Strategies and Math Processes via Cognitive Principles 56
Making Meaningful Connections Among Mathematical Concepts 61
The Connectedness of Strands 64
How Does This All Fit Together? 65
Chapter 3 Numbers and Early Algebra 68
Algebra in the Classroom, Then and Now 68
Partial Products Like You've Never Seen Them 69
Starting Out with Base Ten Blocks and Graph Paper 69
Moving on to More Abstract Representations and Mental Math 72
Red Dots 74
Algebra Tiles 76
Partial Quotients 80
Andy's Inheritance 83
Square the Digits and Sum the Squares 84
Summing the Cubes 87
'The Irrational Tangram 91
Chapter 4 Proportional Reasoning 95
WhatProportional Reasoning Looks and Sounds Like in the Classroom 95
Shampoo Bottle 95
Cats and Rats 96
Making Seismometers 99
Developing Students' Proportional Reasoning Skills 99
Understanding Differences Between Additive and Multiplicative Transformations 99
Understanding Ratios 100
Understanding Rates 105
Interesting Applications of Rate 110
Chapter 5 Algebraic Thinking and Modeling 127
Line of Best Fit and Linear Combinations 128
Positive Slope Situations 128
Inverse Linear Relations 138
Finite Differences: Quadratic, Cubic, and Beyond 168
Quadratic Equations 169
Cubic Equations 176
Conclusion 181
Chapter 6 Geometry and Measurement 182
Multiple Representations for Solving a Geometry Problem 182
Ordering Shapes by Two-Dimensional Size 182
Measuring the Area 191
Make My Polygon 193
A Great Extension: Making Dodecagons 196
What's Your Angle? 198
Tessellations: A Different Way 202
Pythagoras 'R' Us 209
Pythagoras and Similarity 214
Primitive Pythagorean Triples (PPT) 214
Geometry and the Metric System 216
Silent Snow, Secret Snow 216
Conclusion 219
Chapter 7 Data Analysis and Probability 220
Exploring Experimental Probability 220
Chevalier de Mere's Game of Chance 220
Inference and Prediction: Probability Bags 221
A Plethora of Pigs 225
Model Building with Montana Red Dog 228
Exploring Possible Outcomes in Theoretical Probability 235
Combination Pizzas and Permutation Locks 235
Product Versus Square 242
Montana Red Dog Follow-Up 245
De Mere's Bets Follow-Up 246
Concluding Thoughts 246
Appendix 249
References 253
Problem Index 255
Index 259
