Math L AB for Kids FUN, HANDS-ON ACTIVITIES FOR LEARNING WITH SHAPES, PUZZLES, AND GAMES REBECCA RAPOPORT and J.A. YODER 11 of of oo rr PP Math Lab For Kids 001-144_11616.indd 1 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:42 Math Lab For Kids 001-144_11616.indd 1 25/8/16 14:32 Text WWSSLL PPaaggee:: 11 © 2017 Quarto Publishing Group USA Inc. Text © 2017 Rebecca Rapoport and J.A. Yoder First published in the United States of America in 2017 by Quarry Books, an imprint of Quarto Publishing Group USA Inc. 100 Cummings Center, Suite 406-L Beverly, Massachusetts 01915-6101 Telephone: (978) 282-9590 Fax: (978) 283-2742 QuartoKnows.com Visit our blogs at QuartoKnows.com All rights reserved. No part of this book may be reproduced in any form without written permission of the copyright owners. All images in this book have been reproduced with the knowledge and prior consent of the artists concerned, and no responsibility is accepted by producer, publisher, or printer for any infringement of copyright or otherwise, arising from the contents of this publication. Every effort has been made to trace the copyright holders and ensure that credits accurately comply with information supplied. We apologize for any inaccuracies that may have occurred and will resolve inaccurate or missing information in a subsequent reprinting of the book. 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-63159-252-2 Digital edition: 978-1-63159-341-3 Softcover edition: 978-1-63159-252-2 Library of Congress Cataloging-in-Publication Data available Design and page layout: Laura Shaw Design, Inc. Photography: Glenn Scott Photography Illustration: J.A. Yoder & Rebecca Rapoport Printed in China The information in this book is for educational purposes only. PUBLISHER'S NOTE Quarry Books would like to thank the staff and students at Birches School in Lincoln, Massachusetts, which graciously agreed to host the kids' photography for this book. We are especially grateful to Cecily Wardell, Director of Admission and Placement, who generously gave our authors, art director, and photographer access to their facilities and helped us coordinate their students' participation to minimize disruption. 11 of of oo rr PP Math Lab For Kids 001-144_11616.indd 2 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:42 Math Lab For Kids 001-144_11616.indd 2 25/8/16 14:32 Text WWSSLL PPaaggee:: 22 FOR ALLANNA, ZACK, AND XANDER. May you always find joy in math and everything else you do. 11 of of oo rr PP Math Lab For Kids 001-144_11616.indd 3 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:42 Math Lab For Kids 001-144_11616.indd 3 25/8/16 14:32 Text WWSSLL PPaaggee:: 33 CONTENTS Introduction 7 How to Use This Book 8 Geometry: Learn Color Maps Like a 1 About Shapes 11 3 Mathematician 45 Prisms and pyramids and Using the fewest number of colors Platonic solids, oh my! possible, fill in a map so adjacent shapes are different Lab 1: Prisms 12 Lab 12: Map Coloring Basics 46 Lab 2: Pyramids 14 Lab 13: Efficient Map Coloring 50 Lab 3: Antiprisms 16 Lab 14: Squiggle Maps 54 Lab 4: Platonic Solids 18 Lab 5: Perfect Circles 24 Lab 6: Try a Triangle 26 Stitching Curves 57 Lab 7: Exact Ellipses 28 4 Learn to create curves using Lab 8: Draw Giant Circles and only straight lines Ellipses 30 Lab 15: Drawing Parabolas 58 Lab 16: Stitching Stars 62 Topology: Mind- Lab 17: Creative Curves 64 2 Bending Shapes 35 Learn about squeezable, squishable shapes and surfaces Fantastic Fractals 67 5 A fractal is a shape that is similar to Lab 9: Compare and itself no matter how far you zoom in Classify Shapes 36 on one particular part Lab 10: Möbius Strips 38 Lab 18: Draw a Sierpinski Triangle 68 Lab 11: Möbius Surprise 42 Lab 19: Build a Sierpinski Triangle 70 Lab 20: Draw a Koch Snowflake 74 Lab 21: Draw a Square Fractal Snowflake 78 11 Lab 22: Explore the Koch Snowflake’s of of Perimeter 80 oo rr PP Math Lab For Kids 001-144_11616.indd 4 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:43 Math Lab For Kids 001-144_11616.indd 4 25/8/16 14:32 Text WWSSLL PPaaggee:: 44 Terrific Tangrams 83 Graph Theory 111 6 9 Solve ancient Chinese puzzles Explore how points and edges by making different shapes from are interconnected the same seven pieces Lab 33: Eulerian Circuits 112 Lab 23: Tangram Basics 84 Lab 34: Secrets of Eulerian Circuits Lab 24: Teaser Tangrams 86 Revealed 114 Lab 25: Tougher Tangrams 88 Lab 35: Bridges of Königsberg 116 Lab 36: The Euler Characteristic 118 Lab 37: A Proof About the Euler Toothpick Puzzles 91 Characteristic 122 7 Create and solve brainteasers using patterns of toothpicks Lab 26: Starter Toothpick Puzzles 92 Pull-Outs 124 Hints and Solutions 131 Lab 27: Toothpick Puzzles: The Next Acknowledgments 142 Level 94 About the Authors 142 Lab 28: Challenging Toothpick Resources 142 Puzzles 96 Index 144 The Game of Nim 99 8 Learn the game of Nim and develop a strategy to win every time Lab 29: Learn to Play Nim 100 Lab 30: Win Nim: The Copycat Strategy 104 Lab 31: Copycat Nim as Player 1 106 Lab 32: Win Nim: 1 + 2 = 3 Strategy 108 11 of of oo rr PP Math Lab For Kids 001-144_11616.indd 5 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:43 Math Lab For Kids 001-144_11616.indd 5 25/8/16 14:32 Text WWSSLL PPaaggee:: 55 11 of of oo rr PP Math Lab For Kids 001-144_11616.indd 6 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:43 Math Lab For Kids 001-144_11616.indd 6 25/8/16 14:32 WWSSLL PPaaggee:: 66 INTRODUCTION WELCOME TO THE SECRET WORLD OF MATHEMATICIANS. This is your introduction to the gorgeous, exciting, beautiful math that only professionals see. What’s truly astounding is that it’s accessible, even for kids ages six to ten. We think that if more kids had a chance to play with a wider world of math, there would be far more math enthusiasts in the world. Most people think you learn math by climbing a sort of ladder: first addi- tion, then subtraction, then multiplication, then fractions, and so on. In fact, math is much more like a tree. There are many different areas of math that require only a basic foundation. Plenty of this lovely and woefully ignored math doesn’t require any previous knowledge. It’s accessible to everyone, if they only knew it existed. Readers of this book may ask, “How is this math?” Kids cut and tape and sew and color. They imagine walking over bridges, reproducing the same problem that spawned an entire field of mathematics. They draw enormous shapes in parking lots. It may not look like math because there are whole chapters with no pencils or memorization or calculators—but we assure you, the math you’re about to encounter is much closer to what actual mathemati- cians do. Mathematicians play. They come up with interesting questions and investi- gate possible solutions. This results in a lot of dead ends, but mathematicians know that failure provides a great chance to learn. In this book, you’ll have a chance to think like a mathematician and experiment with a given idea to see what you can discover. That approach of just fiddling around with a problem and seeing what falls out is an extremely common and useful technique that mathematicians employ. If you take nothing else away from this book, learning to just try something—anything—and seeing what develops is a great skill for math, science, engineering, writing, and, well, life! This is your opportunity, your gateway, into little-known worlds of math. Turn the page and explore for yourself. 11 of of oo rr PP 7 Math Lab For Kids 001-144_11616.indd 7 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:43 Math Lab For Kids 001-144_11616.indd 7 25/8/16 14:32 Text WWSSLL PPaaggee:: 77 HOW TO USE THIS BOOK The chapters of this book can be worked in any order. Occasionally a chapter refers to a method learned in a different chapter, but there is always a work- around in case you haven’t done the other chapter yet. Within a chapter, we recommend working labs in the order given, as earlier labs often develop knowledge or tools you’ll use in later labs within the same chapter. All of the material in this book has been successfully play-tested by six- to ten- year-olds. We do assume elementary students will have a guide (parent/teacher/ older sibling) to help them work through the labs. Much of the material should be interesting to middle school, high school, and adult students. There are cases where older kids will be able to try a more advanced technique and younger kids will do something easier, or may need a little help. For example, in the Fantastic Fractals chapter, older kids will use a ruler to find the middle of a shape, whereas younger kids can just “eyeball” and estimate the middle. Their results will be surprisingly close. Younger kids may need assistance with certain labs (tying knots, threading needles, cutting with scissors, etc.). Each chapter introduction contains a Think About It question. The question is always related to the chapter’s content and is meant to be played with before reading the rest of the chapter. This gives you the opportunity to experiment with the topic before we’ve introduced any formal concepts. Sometimes we come back to the Think About It problem within the chapter and answer it directly. Some- times we don’t. (In that case, if you’re curious, check the Hints and Solutions section at the end of the book.) In general, we hope students will have time to experiment and not just race through each lab. Real math is so much more about curiosity and experimentation than most people realize. Some chapters have Try This! problems that cover additional or more advanced material relating to the chapter. We provide hints for most of the Try This! prob- lems either as the problem is stated or in the Hints and Solutions section at the end of the book. Some chapters contain a Math Meet, a group activity meant to add a collabora- tive aspect to the learning process. Each chapter just scratches the surface of a whole field of mathematics. If you’re interested in more on any given topic, we’ve included some good sources 11 of of oo rr PP 8 MATH LAB FOR KIDS Math Lab For Kids 001-144_11616.indd 8 JJoobb:: 1111661166 TTiittllee:: ##222200338800__MMaatthh LLaabb FFoorr KKiiddss ((RRoocckkppoorrtt)) 25/8/16 14:43 Math Lab For Kids 001-144_11616.indd 8 25/8/16 14:32 Text WWSSLL PPaaggee:: 88