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JOSSEY-BASS TEACHER GRADESGRADES K–123–5 Overand Problem-Solving Skills300 Puzzles 444that TeachGradeMaths 3–5 TTEERRRRYY SSTTIICCKKEELLSS
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Praise for Math Puzzles and Brainteasers Terry Stickels combines his masterful ability to create diverse, challenging and just plain fun puzzles with a wide range of math concepts, in a playful way that encourages the solver to discover their own unique methods of finding solutions. —David Kalvitis, author of The Greatest Dot-to-Dot Books in the World Logical, numerical, spatial/visual, and creative thinking problems can all be found within these covers, embracing a wide spectrum of thinking skills for developing minds. Terry Stickels also encourages indulgence in mathematical play, which for young students is an indispensable component of motivated and successful problem solving. —Barry R. Clarke, Mind Gym compiler, The Daily Telegraph (UK) Even kids who are not math nerds will enjoy this book. Stickels hits the perfect mix of brainteasers: They’re challenging while still managing to be great fun at the same time! —Casey Shaw, Creative Director, USA WEEKEND magazine Terry Stickels is clearly this country’s Puzzle Laureate. He has concocted a delightful and challenging volume of brainteasers that belong in every math teacher’s library. Focused specifically on grades 3–5 and grades 6–8, ThiNganHang.comthese puzzles both educate and sharpen children’s critical thinking skills. Asan award-winning puzzle constructor myself, I am always in awe of whatTerry comes up with. —Sam Bellotto Jr., Crossdown ThiNganHang.com
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Jossey-Bass Teacher Jossey-Bass Teacher provides educators with practical knowledge and tools to create a positive and lifelong impact on student learning. We offer classroom-tested and research-based teaching resources for a variety of grade levels and subject areas. Whether you are an aspiring, new, or veteran teacher, we want to help you make every teaching day your best. From ready-to-use classroom activities to the latest teaching framework, our value-packed books provide insightful, practical, and comprehensive materials on the topics that matter most to K–12 teachers. We hope to become your trusted source for the best ideas from the most experienced and respected experts in the field. ThiNganHang.com ThiNganHang.com
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MaTh Puzzles and BrainTeasers, grades 3–5 Over 300 Puzzles that Teach Math and Problem-Solving Skills Terry Stickels ThiNganHang.com ThiNganHang.com
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Published by Jossey-Bass A Wiley Imprint 989 Market Street, San Francisco, CA 94103-1741—www.josseybass.com No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the Web at www. copyright.com. Requests to the publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at www.wiley.com/go/permissions. Permission is given for individual classroom teachers to reproduce the pages and illustrations for classroom use. Reproduction of these materials for an entire school system is strictly forbidden. Readers should be aware that Internet Web sites offered as citations and/or sources for further information may have changed or disappeared between the time this was written and when it is read. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Jossey-Bass books and products are available through most bookstores. To contact Jossey-Bass directlycall our Customer Care Department within the U.S. at 800-956-7739, outside the U.S. at 317-572-3986, orfax 317-572-4002.ThiNganHang.com Jossey-Bass also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Printed in the United States of America ISBN: 978-0-4702-2719-0 first edition PB Printing 10 9 8 7 6 5 4 3 2 1 ThiNganHang.com
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Contents foreword vii ACKNowLedGMeNTS xi AbouT ThiS booK xiii The AuThor xv iNTroduCTioN 1 i. NuMberS and oPerATioNS 5 whole Numbers 6 rational Numbers 50 ii. GeoMeTrY and MeASureMeNT 67 Geometry 68 Measurement 90 iii. MATheMATiCAL reASoNiNG 101 ThiNganHang.comVisualother 102137 iV. ALGebrA, STATiSTiCS, and ProbAbiLiTY 169 ANSwerS 179 ThiNganHang.com
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Foreword One of the advantages of growing up in a large extended family is that on numerous occasions I have had the opportunity to observe a miraculous transformation: A young, helpless, and ignorant epsilon—as the mathematician Paul Erdos called young children— comes into the world, begins to eat, drink, cry, dirty his or her diapers, grow (and grow and grow), and, last but not least, assimilate and process information about the world around this new being. Somewhere around the age of three that child becomes a genius. The idea that all children for a short while in their lives are geniuses has been put forth in both humorous and serious ways by pundits and quipsters galore. The imaginations of children at this young age are unfettered, their preconceptions virtually nonexistent. The names “Plato” and “Socrates,” which appeared on ear tags of stuffed animals of a granddaughter of mine, immediately became “Playdough” and “Soccer-team.” After all, who were these strange Greeks who were impinging on her sovereign right to name her own animals? The animals still would recognize their names, wouldn’t they? ThiNganHang.comA daughter of mine once politely refused an offering of sauerkraut at a neighbor’s dinner table, saying that she didn’t like “sourcrap.” Another time I found one of my children blithely sitting in an empty laundry basket counting—or at least trying to count—the hundreds of square holes in its sides. The sole purpose of the endeavor apparently was to get some up close and personal information about the basket. vii ThiNganHang.com
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These personal recollections are not intended to impress the readers of this book with the cleverness or cuteness of my own progeny. Everyone who has raised a child or who is growing up will have story after story of their own. Some of these stories will be more humorous than those given here, some will show more intelligence on the part of the children involved, some will reveal unexpected turns of kindness, and some will parody the imperfections and mannerisms of the child’s parents. The point is that in the life of every child is a period when that child is highly creative, unassuming, and, in my opinion, highly intelligent. Children look at the world through a pair of magical glasses, wired to, and designed to program the most complicated computer in the world, the human brain. Then something goes terribly awry. Children gradually become larger physically, a bit more mature mentally, and we set about formally socializing and educating them. They lose their magical glasses and naiveté, and many of the educational processes to which we subject them seem to take on the form of a mass forced-feeding. But then, amazingly, a dozen or so years down the line, we begin to hear comments to the effect that the true geniuses among us, the truly creative people in our midst, are the ones who, for reasons that no one quite understands, have not lost their magical glasses, have not become fully educated in a sense. These elite thinkers are the ones who still see the world—even if it is an adult world they now see— through the eyes of a child. What has happened to most children in our educational systems, in modern parlance, is a failure to communicate. To be sure, part of this failure is necessary. No new educational theory, no new process or ThiNganHang.comprogram, no new technological process (at least presently available)will negate all of the negative aspects of having to introduce so muchinformation to so many children in what of necessity has to be a highly organized, almost regimented, manner. But we don’t have to throw all of the babies out with the bathwater. We can attempt to fight back. viii Foreword ThiNganHang.com
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Enter Terry Stickels and this book. It is a book intended to stem the loss of creativity in the educational process, grades 3 through 5, particularly in mathematics. Mr. Stickels is a highly successful and well-known creator of puzzles, one of the best we have in the country at this time. We need to only casually look at the quantity and variety of puzzles he has created to realize that he is one of those people who has a strong creative force permanently embedded and dispersed within them. He has for certain not lost that childlike ability to look at the world in new ways—and on a daily basis at that. His FRAME GAME puzzles, for example—some of which appear in this book—remind me of the way that children create words like “playdough” for “Plato” and “sourcrap” for “sauerkraut.” He is the only adult I know who I believe could compete with children in this regard. And that is a compliment. He has spent a lot of time and energy writing this book and has consulted with various knowledgeable experts concerning the mathematical content. Advice is given elsewhere on how to use this book, but I would like to throw in my two cents’ worth also. If you are a child reading this Foreword, send Mr. Stickels an e-mail and ask him to write a more advanced book for you. If you are a teacher, a parent, or a friend of a child in the appropriate age group, go ahead and browse, browse, browse. Pick problems that pique your curiosity, ones that turn you on. You will find many. Choose ones that concern the topic of interest at the moment. You will find several. Present these to the children you are concerned with as challenges—challenges to have fun with. Do not present many at once. Even one is sufficient sometimes. And finally, be patient, very patient. Don’t always expect success. ThiNganHang.comBased on my own experiences in mathematics, I can tell you withcertainty that an incorrect analysis of one puzzle, if only you will hang onto your thoughts, might well prove to be the key to solving another one, and actually might well make you appear to be a genius at some later time. No one has to know that most of your thinking came from an unsuccessful attempt with another puzzle! Foreword ix ThiNganHang.com
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Getting this idea across to people in general, and to young people in particular, is difficult. But an old cliché in sports does a fairly good job of doing this: It’s not whether you win or lose that counts, but how you play the game. The follow-up, in intellectual matters especially, is that how you play the game determines how many games you win in the future. Again, it merits pointing out that, judging from my contacts with him and the impression he gives of being a workaholic, Terry Stickels has invested an enormous amount of time, physical labor, and highly skilled creative thinking in producing this book. More so than perhaps we realize. Based on numerous conversations with him, I can vouch for the fact he passionately cares about the American educational system. Let’s give the gentleman a chance to do what he can with the puzzles he presents here. Here’s hoping that this is not the last publication we see from him concerning the training of our young people in mathematics. February 2009 Dr. Harvy Baker Department of Mathematics University of Texas at Arlington ThiNganHang.com x Foreword ThiNganHang.com
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Acknowledgments This book would not have been possible without the work and sug- gestions of the following people: Mr. Sam Bellotto Jr. of CROSSDOWN.COM Ms. Terry Baughan of TALLROSE PRODUCTIONS Ms. Shelley Hazard of PUZZLERSPARADISE.COM Mr. Barry Finnen of PHYSICS247.COM Webmaster Mr. Roger Smith Mr. Robert Webb of SOFTWARE3D.COM Ms. Suzanne Alejandre of THE MATH FORUM@DREXEL Mr. Martin Gardner Mr. Casey Shaw of USA WEEKEND magazine Mr. Brendan Burford of KING FEATURES ThiNganHang.comMs. Kelsey FlowerMr. Alex Stickels Finally—a special thanks to my right hand and the person who makes all this happen, Ms. Christy Davis, owner of Executive Services, Arlington, Texas. xi ThiNganHang.com
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About This Book Puzzles and brainteasers are fun ways to get kids enjoying and thinking about math. The “thinking smart” puzzles in this book are designed to sharpen the creativity and problem-solving skills, as well as the mathematics content skills, of students in grades 3 through 5. The design for the book includes the following objectives: • Offer a panoramic approach to the thinking skills that kids need to excel in math • Incorporate a broad spectrum of different kinds of puzzles • Meet the grade-appropriate guidelines set forth by the National Council of Teachers of Mathematics • Venture into content areas where previous math/thinking skills books have not gone • Be challenging, but also offer lots of fun along the way The puzzles are easy, medium, and difficult, but none are so ThiNganHang.comdesignated. What one student will find easy, another may finddifficult, and vice versa. A difficulty rating also might be intimidatingto some students—and interpreted as a good reason for not solving a puzzle—the opposite of the book’s purpose. xiii ThiNganHang.com
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The range of puzzles incorporates multiple approaches to skill building, including numerical manipulation, spatial/visual problems, and language arts exercises. There is no one “best” pathway to solving each puzzle, and often there are numerous entry points to finding solutions. Students invariably will find the way, using a mix of intuition and thinking skills that are uniquely their own. ThiNganHang.com xiv About This Book ThiNganHang.com
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The Author Terry Stickels is dedicated to helping people improve their mental flexibility and creative problem-solving capabilities through puzzles— and making it fun. His books, calendars, card decks, and newspaper columns are filled with clever and challenging exercises that stretch the minds of even the best thinkers. And he especially enjoys creating puzzles for kids. Terry is well known for his internationally syndicated columns. FRAME GAMES, appearing in USA WEEKEND magazine, is read by more than 48 million people in six hundred newspapers weekly. STICKELERS, published daily by King Features, appears in several of the largest newspapers in America, including the Washington Post, the Chicago-Sun Times, and the Seattle Post-Intelligencer. Terry is also the featured puzzle columnist for The Guardian in London—the United Kingdom’s largest newspaper. As a highly popular public speaker, Terry’s keynote addresses are ThiNganHang.comfast-paced, humorous looks at the ability (and sometimes the lack thereof) to think clearly. Distinguished authorities such as the National Council of Teachers of Mathematics also praise his work as important in assisting students to learn how to think critically and sharpen their problem-solving skills. xv ThiNganHang.com
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Born and raised in Omaha, Nebraska, Terry was given his first puzzle book at age eleven. Fascinated by the book’s mind-bending playfulness, he soon was inventing puzzles on his own—lots of them. He attended the University of Nebraska at Omaha on a football scholarship, and while he was at UNO tutoring students in math and physics, he saw the advantages of using puzzles to turbocharge understanding of several concepts within those disciplines. After several years as an occasionally published creator of puzzles, Terry was asked to produce a weekly column for a twelve-newspaper syndicate in Rochester, New York. Two years later, his puzzles caught the attention of Sterling Publishing in New York, and his first book, MINDSTRETCHING PUZZLES, became an immediate hit and is selling well to this day. Twenty-five more puzzle books have followed, three of them sponsored by the high-IQ society MENSA. Terry lives in Fort Worth, Texas, where he is working on his next generation of puzzles to once again captivate, challenge, and delight his worldwide readership. ThiNganHang.com xvi The Author ThiNganHang.com
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Introduction This book contains more than 300 puzzles, ranging from relatively easy logic challenges to more difficult math brainteasers, requiring math skills ranging from addition and subtraction to determining probability and algebraic thinking. Within these pages you will find these types of puzzles: Mathematical Frame Games Spatial/Visual Cryptograms Logical Analogies Analytical Reasoning Sequence Word Puzzles Sudoku By design I have included a large number and broad spectrum of puzzles, providing teachers and learners with multiple options. These are organized into parts devoted to numbers and operations; geometry and measurement; mathematical reasoning; and algebra, statistics, and probability. This arrangement will facilitate the ThiNganHang.cominstructor’s ability to enhance areas of the curriculum that are mostappropriate, adding richness, change of pace, and reinforcement tothe teaching/learning process. 1 ThiNganHang.com
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Some Puzzle-Solving Tips Puzzle solving is sometimes like mathematical problem solving, but sometimes you have to move away from the more standard approaches when working on puzzles. Think about the puzzles from different perspectives and with a sense of play. Consider some of the following: • Can the puzzle be solved by breaking it down into simpler components? • Are there patterns that repeat often enough to suggest a prediction for “what comes next”? • A puzzle may have one or more answers. • Try thinking of ways to “twist, bend, separate, or spin” the puzzle. What does it look like “backward, forward, upside down, and sideways”? • Does your answer make sense? Can you plug your answer back into the question to satisfy all the parameters? • If your answer seems strange or unlikely, it may well be correct. The answers to puzzles are often surprising! • Don’t worry about how you might be seen if you can’t solve the puzzle. We all make mistakes, and no one can answer every question. Just relax, have a good time, and never worry about other ThiNganHang.compeople’s opinions. Projects throughout the book marked with a symbol can be done using easy-to-find manipulatives, such as coins, blocks, and cut paper, to help learners who may have trouble visualizing some of the puzzles. 2 Math Puzzles and Brainteasers, Grades 3–5 ThiNganHang.com
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You may wonder why some language arts puzzles are included in a math puzzle book. Actually, puzzles and problems such as analogies and analytical reasoning that are more “language arts” in nature pro- mote and augment critical-thinking skills. Take the FRAME GAMES, for example. FRAME GAMES are words, letters, pictures, fonts, and the like, juxtaposed in a way to reveal a common idiom, famous person, athlete, movie, song title, and similar things. These include components of spatial/visual thinking, language, memory, vocabu- lary, and lighthearted fun. When people solve even one puzzle correctly—and find the fun in doing so—they are eager to jump to the next challenge, even if it is a puzzle of a different kind. Also, solving a type of puzzle in one area often triggers the mind into a flexible mode that makes it easier to solve problems/puzzles in other areas. Another appealing feature of the FRAME GAMES is that they don’t always follow the standard left-to-right or top-to-bottom pattern for their solutions. Mental flexibility from different perspectives is required. These puzzles can be used in a broad spectrum of classroom situations—from special education to warm-ups in calculus classes. They are placed periodically throughout the book, offering both a mental break and a different type of thinking challenge. There is no wrong way to use these puzzles. They’re meant to be treated like a good watch or pair of shoes: to be used over and over again. And they never wear out! Here are some application ideas: ThiNganHang.com••• As warm-ups to introduce a new element of math curriculumAs a focus for competition among teamsAs an inspiration for learners to create their own versions to share with classmates • Featured on posters or class newsletters as the puzzle of the day, week, month, or holiday Introduction 3 ThiNganHang.com
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