Imagine handing students state-by-state data on the number of gallons of soft drinks sold per person in one year. Imagine using it to lead a vibrant problem-solving session in which students energetically pose and answer mathematical questions:

Arthur Hyde is the author or coauthor of the Heinemann titles Comprehending Math (2006), Best Practice, Third Edition (2005), and Mathwise (1991). A professor of mathematics education at National-Louis University, he received its Excellence in Teaching award. While teaching high school mathematics in Philadelphia, he developed a variety of creative methods for teaching math. He also obtained a doctorate in curriculum and instruction from the University of Pennsylvania, where he later was Associate Director of Teacher Preparation. He continues to work frequently in elementary and middle school classrooms and conducts extensive professional development programs on mathematics and problem solving in Chicago and its surrounding school districts.

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