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

274 Pages·2009·5.21 MB·English

by  Arthur Hyde

Unders t anding
Middle School
54 63 6
45 36 118
Math
14 41 181
27 66
71 17 35 72
12 53
22
21 50 34
46 8
5 43
64
25 52 33 57
75
92 29 38 83 18 74
76 81 47
85 73 65
67 56
Cool Problems
149 89 58 61
77 145 37 59
to Get Students
98 106
194 42 16 95
Thinking and
154 20 4
40
Connecting
24 2
224 62 26
11
Arthur Hyde 15 51

with Susan Friedlander, Cheryl Heck, and Lynn Pittner
DedicatedtoTeachers™ Foreword by Judith Zawojewski
Understanding

Middle School
Math
Heinemann_Hyde_Understanding Middle School Math.indd 1 12/17/2009 11:31:48 AM
Understanding
Middle School
Math 54 63 6
45 36 118
14 41 181
27 66
71 17 35 72
12 53
22
Cool Problems 21 50 34 46 8
5 43
64
to Get Students 25 52 33 57
75
Thinking and 92 29 38 83 18 74
76 81 47
85 73 65
Connecting
67 56
149 89 58 61
77 145 37 59
98 106
194 42 16 95
154 20 4
40
24 2
224 62 26
11
15 51
Arthur Hyde

with Susan Friedlander, Cheryl Heck, and Lynn Pittner
Foreword by Judith Zawojewski
HEINEMANN
Portsmouth, NH
Heinemann_Hyde_Understanding Middle School Math.indd 3 12/17/2009 11:31:48 AM
Heinemann
361 Hanover Street
Portsmouth, NH 03801−3912
www.heinemann.com
Offices and agents throughout the world
© 2009 by Arthur Hyde

All rights reserved. No part of this book may be reproduced in any form or by

any electronic or mechanical means, including information storage and retrieval

systems, without permission in writing from the publisher, except by a reviewer,

who may quote brief passages in a review.

“Dedicated to Teachers” is a trademark of Greenwood Publishing Group, Inc.

The author and publisher wish to thank those who have generously given permis­

sion to reprint borrowed material:

“A weighty matter” by Lauren Cabell and Phil Geib from Chicago Tribune, 2/1/2002.

Copyright © 2002 by Chicago Tribune. Used by permission and protected by the

Copyright Laws of the United States. The printing, copying, redistribution, or

retransmission of the Material without express written permission is prohibited.

All rights reserved.

Library of Congress Cataloging-in-Publication Data
Hyde, Arthur A.
Understanding middle school math : cool problems to get students thinking
and connecting / Arthur Hyde ; with Susan Friedlander, Cheryl Heck, and Lynn
Pittner.
p. cm.

Includes bibliographical references and index.

ISBN-13: 978-0-325-01386-2

ISBN-10: 0-325-01386-1

1. Mathematics—Study and teaching (Middle school). 2. Creative teaching.
I. Title.
QA135.6.H93 2009
510.71'2—dc22 2008047625
Editor: Emily Michie Birch
Production: Lynne Costa
Cover design: Night & Day Design
Typesetter: Val Levy, Drawing Board Studios
Manufacturing: Steve Bernier
Printed in the United States of America on acid-free paper
13 12 11 10 09 VP 1 2 3 4 5
Heinemann_Hyde_Understanding Middle School Math.indd 4 12/17/2009 11:31:48 AM
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

v
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vi contents 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

What Proportional 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

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CHAPTEr 6: gEOmETrY ANd mEASurEmENT 182 contents vii
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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Foreword

A book of cool problems for middle school mathematics classrooms—
does it get any better? Yes, it does. Art Hyde and his colleagues, three
middle school math teachers, 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 all through the lens of the author-created Braid Model,
a coherent model of learning that links language, cognition, and math­
ematics through problem solving.
The middle school mathematics classroom has traditionally been a
place to review and polish some set of basic skills—primarily arithme­
tic—to prepare for the journey through high school algebra, geometry,
advanced algebra, and trigonometry. Although some innovative programs
have been designed that engage middle school students in the develop­
ment of concepts and intuitions about algebra, linear and exponential
growth, spatial relationship and geometric properties, statistics, and
probability, these programs are not found in the majority of American
classrooms.
I taught from traditional textbooks for nine years as a teacher of
sixth-, seventh-, and eighth-grade mathematics. Along with many of my
peers, I was always on the search for ideas, activities, lessons, and units
that would enrich the experience for all of my students—experiences that
would challenge, engage, stretch, and open my students’ minds to the
wonder, beauty, and application of mathematics. I ended up with two file
cabinets full of replacement lessons, ideas for units of study, collections
of problems, and various articles from the teaching journals. Every year,
I recreated my sixth-, seventh-, and eighth-grade plans to meet the needs
of the students I had that particular year, and assimilated new and inno­
vative ideas that I found during the year. What a resource Understanding
Middle School Math would have been for me!
Art Hyde and colleagues, in their quest for cool problems that teach
concepts in deep ways, help teachers make the connections between their
classroom needs and the resources provided in this book. Understanding ix
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x foreword Middle School Math acts as a lesson study, with its conversational style and
its vivid illustrations from teachers’ classrooms. As a result, teachers ob­
tain concrete ideas for differentiating challenges for students who bring
varied talents to the table. They help facilitate students’ evolution in the
use of various representations, from initial visual and verbal representa­
tions to abstract generalizations and symbols. Mathematics comes alive as
students engage in varying contexts, from applied situations, to games, to
explorations of mathematical environments for their own sake. Through­
out the book, the problems help teachers develop the problem-solving
aspect of a model curriculum—making connections within and across
problems, using rich mathematical problems as a means to enhance stu­
dents’ learning, and using problems as a basis for serious discussion and
to encourage conceptual dialogue.
The chapters in this book are connected by the KWC method for
scoping out a problem—What do I Know for sure? What do I Want to
find out? Are there any special Conditions?—and a view of mathemat­
ics learning and teaching that braids together language, cognition, and
the domain of mathematics. In particular, students are viewed as actively
developing their conceptual understanding in the context of mathemati­
cal problem solving. This approach is consistent with research (Lesh and
Zawojewski 2006) that reveals that higher problem-solving performance
is associated with breadth and depth of mathematical knowledge.
No compelling evidence exists that links direct instruction in isolated
problem-solving strategies (for example, draw a picture, make a table, con­
sider a similar problem) to improved problem-solving performance. Rather,
problem solving needs to be thought of as a means for learning math­
ematics. Simultaneously, mathematical knowledge is a launch for more
sophisticated problem solving, and students can use problem-solving
strategies as they engage in problem-solving activities. The authors of
Understanding Middle School Math show that students’ development of
solutions to mathematical problems goes hand in hand with students’
acquisition of deep conceptual understanding.
The role of representation and the development of representational flu­
ency is foundational to the cool problems in Understanding Middle School
Math. Most important in middle school mathematics education is that
instead of teaching representation simply as a skill for its own sake, teach­
ers must link representation to a function or purpose that is compelling
for students. For example, in Chapter 5, when reflecting back on the
Chocolate Algebra problem, a boy said, “I liked the tables the best for
figuring out the patterns and equations, but I liked the graphs the best
for making predictions” (158–59). Such insight on the part of students
can only be accomplished when teachers explicitly establish reasons for
engaging students in the production and analysis of a new representation.
Throughout Understanding Middle School Math, the authors traverse the
representational landscape from visual and verbal to general and sym­
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bolic by connecting each form to purposes and functions that are imme- foreword xi
diately apparent to the students.
With each set of cool problems, the authors provide vivid classroom
narratives, which help teachers make connections to significant math­
ematical content. Further, the narratives provide ideas for organizing stu­
dents for small-group work and clues for diagnosing and assessing what
relevant knowledge students possess. Organized around number, pro­
portional reasoning, algebra, geometry, data analysis, and probability, the
collection of problems in Understanding Middle School Math ranges from
reworked standards to new or unique problem situations, all the while
making connections across mathematics topics within each chapter.
Understanding Middle School Math provides teachers with sound
problems to augment any middle school mathematics textbook series, as
replacement lessons, enrichment activities, and replacement units. Most
importantly, the book is written in a way that helps middle school teach­
ers implement the problems with all students, using flexible grouping
that is responsive to students’ ongoing individual needs.
Understanding Middle School Math can also be used in professional
development settings, especially lesson study—which was also the origin
of the narratives for this book. In lesson study, teachers read activities,
plan common lessons, and then implement those activities while observ­
ing each other or making videos for later study. The follow-up reflection
session provides opportunities for teachers to discuss what happened, to
explore what modifications might be made to the planning and the im­
plementation, and to think about how the problems empower students’
problem-solving and learning abilities. Equally as important, Understand­
ing Middle School Math empowers middle school mathematics teachers to
become professionals who create challenging and exciting mathematical
environments for all students.
—Judith S. Zawojewski
Department of Mathematics
and Science Education
Illinois Institute of Technology, Chicago
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