Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 8

The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the eighth-grade level through visualization, play, and investigation.

During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same messagethat they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designedMindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks inMindset Mathematics reflect the lessons from brain science that:

• There is no such thing as a math person - anyone can learn mathematics to high levels.
• Mistakes, struggle and challenge are the most important times for brain growth.
• Speed is unimportant in mathematics.
• Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics.

With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics,Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.

JO BOALER is a professor of mathematics education at Stanford University and co-founder and faculty director of youcubed. She serves as an advisor to several Silicon Valley companies and is a White House presenter on girls and STEM (Science, Technology, Engineering, and Math). The author of seven books, includingMathematical Mindsets, and numerous research articles, she is a regular contributor to news and radio in the United States and England.

JEN MUNSON is an assistant professor of learning sciences at Northwestern University, a professional developer, and a former classroom teacher. She received her PhD in teacher education and mathematics education from Stanford University. Her research focuses on responsive, equitable mathematics instruction and coaching.

CATHY WILLIAMS is the co-founder and the executive director of youcubed at Stanford University. Before working at youcubed, she was a high school math teacher and worked in mathematics curriculum and administration at the county and district levels in California.

Introduction 1

Low-Floor, High-Ceiling Tasks 2

Youcubed Summer Camp 3

Memorization versus Conceptual Engagement 4

Mathematical Thinking, Reasoning, and Convincing 5

Big Ideas 9

Structure of the Book 9

Note on Materials 17

Manipulatives and Materials Used in This Book 18

Activities for Building Norms 21

Encouraging Good Group Work 21

Paper Folding: Learning to Reason, Convince, and Be a Skeptic 22

Big Idea 1: Moving Shapes 27

Visualize: What Does It Mean to Be the Same? 29

Play: Pixel Puzzles 35

Investigate: Slide It, Flip It, Turn It 45

Big Idea 2: Zooming In and Out 53

Visualize: What is Similarity? 55

Play: Find the Fakes 60

Investigate: Growing on a Grid 65

Big Idea 3: Analyzing Proportional Relationships 71

Visualize: Dropping the Altitude 73

Play: Seeing Triangles under the Line 79

Investigate: Stairway to Eleven 85

Big Idea 4: Comparing Patterns of Growth 91

Visualize: Squared Squares 93

Play: Skip-Counting Arrays 100

Investigate: Stacking Pennies 107

Big Idea 5: The Ins and Outs of Functions 117

Visualize: Growing Functions 120

Play: Getting Warmer! 134

Investigate: The Functions of Near Squares 140

Big Idea 6: Finding Patterns in the Clouds 149

Visualize: Whats Going On in This Graph? 151

Play: Whats the Story Here? 158

Investigate: Creating Data Visualizations 166

Big Idea 7: Completing the Real Number System 177

Visualize: Square Sides 179

Play: Between 4 × 4 and 5 × 5 187

Investigate: The Hypotenuse Hypothesis 193

Big Idea 8: Discovering Pythagoras 201

Visualize: Unpacking Pythagoras 204

Play: Pythagorean Triples 209

Investigate: Approximating Square Roots 214

Big Idea 9: Exploring the Geometry of Ice Cream 221

Visualize: Building Intuition about Volume 223

Play: Comparing Cylinders 227

Investigate: Scooping Up Volume 233

Appendix 239

Centimeter Grid Paper 240

Grid Paper 241

Isometric Dot Paper 242

Dot Paper 243

About the Authors 245

Acknowledgments 247

Index 249