Understanding Middle School Math: Cool Problems to Get Students Thinking and Connecting

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A book of cool problems for middle school mathematics classrooms-does it get any better? Yes, it does. Art Hyde and his colleagues go far beyond providing a collection of problems. They address big ideas, make connections, nurture the use of varied representations, and provide vivid accounts of actual classroom implementation.

-Judith Zawojewski

Board of Directors, NCTM

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:

Why does it say sold instead of consumed?

What IS a soft drink? Is it the same as soda?

Who would collect this kind of data? Why would they collect it?

How was gallons per person calculated?

What was the total amount of soda sold in our state?

How many 12 ounce cans is that? 20 ounce bottles? How many of each per person?

Understanding Middle School Math gathers 50 cool problems like this that lead to deep thinking. Problems such as the Renovation Problem, in which students uncover ideas about how perimeter, area, length, and cost affect a construction project. Or Chocolate Algebra, where they discover linear relationships among the pocket money available to buy two differently priced chocolate candies.

Arthur Hyde combines the latest research and decades of classroom experience to braid language, cognition, and math. His approach can help any student, including underprepared ones, with the rigors of math in middle school and beyond. He has created and adapted problems that strongly connect math to the real world, to students' lives, and to prior knowledge. Problems that scaffold content and processes, and give students multiple entry points into learning. Every problem has been extensively field tested and refined by classroom teachers. And for each cool problem practicing middle school teachers describe how they used it to differentiate over a wide range of students and extend learning.

For fantastic problems your students won't soon forget and teaching solutions that are exciting, substantial, and transformative, turn to Art Hyde. Read and use Understanding Middle School Math and pass your love of math on as you meet your classroom goals.

Discover more resources for developing mathematical thinking at Heinemann.com/Math

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Product Details

  • ISBN-13: 9780325013862
  • Publisher: Heinemann
  • Publication date: 4/9/2009
  • Edition description: New Edition
  • Pages: 280
  • Sales rank: 964,394
  • Product dimensions: 7.40 (w) x 9.30 (h) x 0.60 (d)

Meet the Author

Arthur Hyde is the author or coauthor of the Heinemann titles Understanding Middle School Mathematics; Comprehending Math; Best Practice, Fourth Edition; and Mathwise. 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.
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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

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