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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, visual-spatial, 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. As an award-winning puzzle constructor myself, I am always in awe of what Terry 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 6–8 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.ThiNganHang.com Jossey-Bass books and products are available through most bookstores. To contact Jossey-Bass directly call our Customer Care Department within the U.S. at 800-956-7739, outside the U.S. at 317-572-3986, or fax 317-572-4002. 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: 978-0-4702-2720-6 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 xvii iNTroduCTioN 1 i. NuMberS and oPerATioNS 5 whole Numbers 6 rational Numbers 40 ii. GeoMeTrY and MeASureMeNT 63 Geometry 64 Measurement 75 ThiNganHang.comiii. VisualMATheMATiCAL reASoNiNG 8990 other 139 iV. ALGebrA, STATiSTiCS, and ProbAbiLiTY 153 ANSwerS 173 ThiNganHang.com
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Foreword In the 1950s we punished misbehaving students in middle school or junior high school by making them stay after school to do 100 multiplication or long-division math problems. Mathematics was taught in a highly ordered and mechanically repetitive manner with the objective being to master basic arithmetic skills. A decade later New Math emerged as a response to the Soviet Union’s early dominance in the race to outer space. The focus in the classroom shifted from drill to conceptual understanding to prepare students for early exposure to advanced mathematics. This theoretical approach failed miserably because middle school students entering high school did not possess the arithmetic skills necessary to do calculations in mathematics and science courses. Consequently, the 1970s saw a backlash and the Back-to-the-Basics movement emerged. During the last three decades we have witnessed the emergence of several more movements in mathematics education attempting to balance the mastery of computational skills and conceptual understanding. These movements included a problem- ThiNganHang.comsolving approach, a high-tech calculator/computer approach, and amathematical user-friendly approach. Why have all these movementsfallen short? The number-one complaint about mathematics from students is: Math is boring! Perhaps, in the 21st century there should be a Math for Fun movement that makes mathematics exciting, challenging, and rewarding. With this puzzle book, Terry Stickels vii ThiNganHang.com
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has pioneered an innovative approach to motivating students not only to learn and do mathematics but also to come away from the experience with a positive attitude. In this creative and stimulating book, students and teachers will find a potpourri of mind-expanding puzzles designed to enhance and enlighten, as well as to entertain. This book contains an incredible assortment of puzzles of various types including logical, geometrical, mathematical, and verbal. The diversity of the puzzles and the various approaches to solving them will improve the student’s problem-solving skills, as well as the general thinking skills required for subjects outside of mathematics. Using these puzzles as supplements to the traditional mathematics curriculum, the teacher can add a new dimension to students’ learning experience. For example, the puzzles can be used to introduce and motivate fundamental algebraic or geometrical concepts. The puzzles also can be used to apply these abstract concepts to concrete problems. Thus, these puzzles can supplement the traditional “story problems” that have been tormenting students for decades. In addition, the puzzles can be used to give bonus points or extra credit to students who finish their required daily assignments in a timely manner. An exciting and promising application of the puzzles would be in a math club where the students can compete and play games that challenge the mind and the creative spirit. Teaching and learning mathematics constitute a multidimensional ThiNganHang.comexperience. The first dimension consists of the rules and algorithms required to do calculations. Many students perceive mathematics as being one-dimensional—“number crunching.” To apply the potential power of mathematics effectively, however, they need a second dimension: a conceptual understanding and framework. To expand the potential power of mathematics, they must enter a third dimension: the intuitive and mind-expanding creative viii Foreword ThiNganHang.com
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process. In this dimension we are “thinking outside the box.” This puzzle book by Terry Stickels will take students and teachers on a multidimensional journey filled with multilevel thinking, creative and imaginative explorations, and exciting discoveries and solutions. One of the great challenges of teaching mathematics in the 21st century is how to reach the diverse student population and how to teach to a variety of individual learning styles. Some students are visual learners; others learn by studying concrete examples, and many students with strong verbal skills learn by translating the mathematical symbolism to words. This puzzle book offers the potential to be an effective alternative approach to solving this universal challenge. The flexibility and variety of these puzzles that span the whole spectrum of traditional mathematics in Grades 6–8, and the direct application of these puzzles to a wide range of learning styles, will make this Math for Fun approach a rewarding and positive experience for students and teachers alike. In conclusion, this puzzle book can be used to reward the motivated hard-working students but also to “punish” the misbehaving students by making them stay after school to do 100 Terry Stickels puzzles! Serendipitously, Terry Stickels has created an innovative way to reach out to troubled students that can change their negative attitudes to positive experiences and a positive perspective on thinking and creativity. February 2009 Dr. John Konvalina Department of Mathematics ThiNganHang.comUniversity of Nebraska at Omaha Foreword ix ThiNganHang.com
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Acknowledgments This book would not have been possible without the work and suggestions 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 ThiNganHang.comMs. Kelsey FlowerMr. Brendan Burford of KING FEATURES Mr. 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 A good math puzzle, paradox, or magic trick can stimulate a child’s imagination much faster than a practical application . . . and if the game is chosen carefully, it can lead almost effortlessly into significant mathematical ideas. Martin Gardner America’s Mathemagician Mr. Gardner’s quote captures one of the main reasons for this book. My intention was (and has been with all my books) twofold: to provide challenging fun and to offer options to think differently— and maybe discover opportunities to become a better thinker. There are countless stories of great thinkers being puzzle-lovers, but have you ever wondered why that is so? What is the connection ThiNganHang.combetween creative, bright people and their insatiable thirst for puzzles? The father of modern-day puzzle writers, Henry Dudeney, gave us one clue when he said, “Puzzles, like virtue, are their own reward.” He also noted that “the fact is that our lives are largely spent solving puzzles; for what is a puzzle but a perplexing question? And from our childhood upwards we are perpetually asking questions or trying to answer them.” xiiixiii ThiNganHang.com
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A well-crafted puzzle seems to naturally encourage a nontraditional or more circuitous route to its solution. This emphasis on different approaches is like brain candy for thinkers. Thinkers begin by dissecting a puzzle and viewing it from different perspectives simultaneously. Great puzzle solvers enjoy twisting, bending, separating, and spinning a puzzle. They look at it backward, forward, upside down, and sideways. Is there a quick solution? More than one solution? What kinds of internal patterns do puzzles have? Can I get to the answer and then make it into a new puzzle? Does it have direct application to the real world? A quick comment about being a “world-class” thinker: You don’t have to be a genius to be one. You may have noticed with many of the stories about successful, bright people that they often are accompanied by a back story about how they weren’t the best students in class or weren’t initially successful in certain studies. Einstein and Edison come to mind. Maybe they already were employing their own ideas of arriving at solutions that weren’t considered acceptable practices at the time. Students are often inspired by these stories not to give up working to become better thinkers because they don’t fit mainstream profiles. Here’s one other important point, often lost in this day and age of timed standardized tests: The time it takes to solve any problem or puzzle has nothing to do with mental ability or intellectual level. Some of the greatest thinkers deliberately take their time—enjoying the play of a puzzle and savoring it like a good meal! The puzzles in this book are designed to sharpen the creativity and ThiNganHang.comproblem-solving skills, as well as the mathematics content skills, ofstudents in grades 6–8. This book is designed with the following objectives: • Offer a panoramic approach to the thinking skills that kids need to excel in math xiv About This Book ThiNganHang.com
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• 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 and thinking skills books have not gone • Be challenging, but also offer lots of fun along the way Although the puzzles are easy, medium, and difficult, none are so designated. What one student will find easy, another may see as difficult, and vice versa. A difficulty rating also might be intimidating to some students, and interpreted as a good reason for not solving a puzzle—the opposite of the book’s purpose. The range of puzzles incorporates multiple approaches to skill- building, including numerical manipulation, spatial and visual problems, and language arts exercises. There is no one “best” pathway to solve each puzzle, and often numerous entry points to find solutions. Students invariably will find the way, using a mix of intuition and thinking skills that are uniquely their own. Puzzles can offer an experience parallel to a new dive off a diving board—an exciting intuitive leap into the unknown, with possible scintillating results! If you look at some of the greatest discoveries of science and mathematics, they often are accompanied by intuitive leaps, supported in turn by clear thinking, logic, evidence, and repeated consistent results in trials. In this complex world, we need all the good thinkers we can get—and grow. Any opportunity we have to help uncover those talented young ThiNganHang.compeople should be welcomed and maximized. Fortunately for me, Ilive in a time when puzzles are being used increasingly in schools andbusinesses to promote critical thinking and mental flexibility. A stream of recent studies makes clear that puzzles and games indeed will help the capacity of your mind to grow and stay flexible. These puzzles also can contribute to the development of an expanding cadre of lifelong learners—a primary goal of every parent and educator. About This Book xv 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, such as the Washington Post, the Chicago Sun-Times, and the Seattle Post-Intelligencer. Terry also is the featured puzzle columnist for The Guardian in London—the United Kingdom’s largest newspaper. ThiNganHang.comAs a highly popular public speaker, Terry’s keynote addresses arefast-paced, humorous looks at the ability (and sometimes the lackthereof) to think clearly. Distinguished authorities such as the National Council of Teachers of Mathematics also praise his work as an important aid in assisting students to learn how to think critically and sharpen their problem-solving skills. xvii 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. 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. His first book, MINDSTRETCHING PUZZLES, became an immediate hit and continues to sell 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 xviii The Author ThiNganHang.com
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