Math Wonders to Inspire Teachers and Students
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to Inspire Teachers and Students Alfred S. Posamentier to Inspire Teachers and Students Alfred S. Posamentier Association for Supervision and Curriculum Development Alexandria, Virginia USA Association for Supervision and Curriculum Development 1703 N. Beauregard St. * Alexandria, VA 22311-1714 USA Telephone: 800-933-2723 or 703-578-9600 * Fax: 703-575-5400 Web site: http://www.ascd.org * E-mail: [email protected] Gene R. Carter, Executive Director; Nancy Modrak, Director of Publishing; Julie Houtz, Director of Book Editing & Production; Darcie Russell, Project Manager; Technical Typesetting, Inc., Typesetting; Tracey Smith, Production Copyright © 2003 by Alfred S. Posamentier. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission from ASCD. Readers who wish to duplicate material may do so for a small fee by contacting the Copyright Clearance Center (CCC), 222 Rosewood Dr., Danvers, MA 01923, USA (telephone: 978-750-8400; fax: 978-750-4470; Web: http://www.copyright.com). ASCD has authorized the CCC to collect such fees on its behalf. Requests to reprint rather than photocopy should be directed to ASCD’s permissions office at 703-578-9600. Cover art copyright © 2003 by ASCD. Cover design by Shelley Young. ASCD publications present a variety of viewpoints. The views expressed or implied in this book should not be interpreted as official positions of the Association. All Web links in this book are correct as of the publication date below but may have become inactive or otherwise modified since that time. If you notice a deactivated or changed link, please e-mail [email protected] with the words “Link Update” in the subject line. In your message, please specify the Web link, the book title, and the page number on which the link appears. netLibrary E-Book: ISBN 0-87120-852-0 Price: $27.95 Quality Paperback: ISBN 0-87120-775-3 ASCD Product No. 103010 ASCD Member Price: $22.95 nonmember Price: $27.95 LibraryofCongressCataloging-in-PublicationData (for paperback book) Posamentier, Alfred S. Mathwonderstoinspireteachersandstudents/[AlfredS. Posamentier]. p. cm. Includes bibliographical references and index. ISBN 0-87120-775-3 (alk. paper) 1. Mathematics–Study and teaching. 2. Mathematical recreations. I. Title. QA11.2 .P64 2003 510—dc21 2003000738 In memory of my beloved parents, who, after having faced monumental adversities, provided me with the guidance to develop a love for mathematics, and chiefly to Barbara, without whose support and encouragement this book would not have been possible. Math Wonders to Inspire Teachers and Students Foreword.................................. ix Preface ................................... xii Chapter 1 The Beauty in Numbers .................. 1 1.1 Surprising Number Patterns I ............... 2 1.2 Surprising Number Patterns II ............... 5 1.3 Surprising Number Patterns III .............. 6 1.4 Surprising Number Patterns IV .............. 7 1.5 Surprising Number Patterns V ............... 9 1.6 Surprising Number Patterns VI .............. 10 1.7 Amazing Power Relationships ............... 10 1.8 Beautiful Number Relationships .............. 12 1.9 Unusual Number Relationships .............. 13 1.10 Strange Equalities ...................... 14 1.11 The Amazing Number 1,089 ............... 15 1.12 The Irrepressible Number 1 ................ 20 1.13 Perfect Numbers ...................... 22 1.14 Friendly Numbers ...................... 24 1.15 Another Friendly Pair of Numbers ............ 26 1.16 Palindromic Numbers .................... 26 1.17 Fun with Figurate Numbers ................ 29 1.18 The Fabulous Fibonacci Numbers ............. 32 1.19 Getting into an Endless Loop ............... 35 1.20 A Power Loop ........................ 36 1.21 A Factorial Loop ......................√ 39 1.22 The Irrationality of 2................... 41 1.23 Sums of Consecutive Integers ............... 44 Chapter 2 Some Arithmetic Marvels ................. 47 2.1 Multiplying by 11 ...................... 48 2.2 When Is a Number Divisible by 11? ........... 49 v 2.3 When Is a Number Divisible by 3 or 9? ......... 51 2.4 Divisibility by Prime Numbers .............. 52 2.5 The Russian Peasant’s Method of Multiplication .... 57 2.6 Speed Multiplying by 21, 31, and 41 ........... 59 2.7 Clever Addition ....................... 60 2.8 Alphametics ......................... 61 2.9 Howlers ........................... 64 2.10 The Unusual Number 9 .................. 69 2.11 Successive Percentages ................... 72 2.12 Are Averages Averages? .................. 74 2.13 The Rule of 72 ....................... 75 2.14 Extracting a Square Root ................. 77 Chapter 3 Problems with Surprising Solutions ............ 79 3.1 Thoughtful Reasoning ................... 80 3.2 Surprising Solution ..................... 81 3.3 A Juicy Problem ...................... 82 3.4 Working Backward ..................... 84 3.5 Logical Thinking ...................... 85 3.6 It’s Just How You Organize the Data ........... 86 3.7 Focusing on the Right Information ............ 88 3.8 The Pigeonhole Principle ................. 89 3.9 The Flight of the Bumblebee ............... 90 3.10 Relating Concentric Circles ................ 92 3.11 Don’t Overlook the Obvious ................ 93 3.12 Deceptively Difficult (Easy) ................ 95 3.13 The Worst Case Scenario ................. 97 Chapter 4 Algebraic Entertainments .................. 98 4.1 Using Algebra to Establish Arithmetic Shortcuts .... 99 4.2 The Mysterious Number 22 ................ 100 4.3 Justifying an Oddity .................... 101 4.4 Using Algebra for Number Theory ............ 103 4.5 Finding Patterns Among Figurate Numbers ....... 104 4.6 Using a Pattern to Find the Sum of a Series ...... 108 4.7 Geometric View of Algebra ................ 109 4.8 Some Algebra of the Golden Section ........... 112 vi 4.9 When Algebra Is Not Helpful ............... 115 4.10 Rationalizing a Denominator ............... 116 4.11 Pythagorean Triples ..................... 117 Chapter 5 Geometric Wonders ..................... 123 5.1 Angle Sum of a Triangle .................. 124 5.2 Pentagram Angles ...................... 126 5.3 Some Mind-Bogglers on ................ 131 5.4 The Ever-Present Parallelogram .............. 133 5.5 Comparing Areas and Perimeters ............. 137 5.6 How Eratosthenes Measured the Earth .......... 139 5.7 Surprising Rope Around the Earth ............ 141 5.8 Lunes and Triangles .................... 143 5.9 The Ever-Present Equilateral Triangle .......... 146 5.10 Napoleon’s Theorem .................... 149 5.11 The Golden Rectangle ................... 153 5.12 The Golden Section Constructed by Paper Folding . 158 5.13 The Regular Pentagon That Isn’t ............. 161 5.14 Pappus’s Invariant ...................... 163 5.15 Pascal’s Invariant ...................... 165 5.16 Brianchon’s Ingenius Extension of Pascal’s Idea .... 168 5.17 A Simple Proof of the Pythagorean Theorem ...... 170 5.18 Folding the Pythagorean Theorem ............ 172 5.19 President Garfield’s Contribution to Mathematics .... 174 5.20 What Is the Area of a Circle? ............... 176 5.21 A Unique Placement of Two Triangles .......... 178 5.22 A Point of Invariant Distance in an Equilateral Triangle ................180 5.23 The Nine-Point Circle ................... 183 5.24 Simson’s Invariant ..................... 187 5.25 Ceva’s Very Helpful Relationship ............. 189 5.26 An Obvious Concurrency? ................. 193 5.27 Euler’s Polyhedra ...................... 195 Chapter 6 Mathematical Paradoxes .................. 198 6.1 Are All Numbers Equal? .................. 199 6.2 −1 Is Not Equal to +1................... 200 vii 6.3 Thou Shalt Not Divide by 0 ................ 201 6.4 All Triangles Are Isosceles ................ 202 6.5 An Infinite-Series Fallacy ................. 206 6.6 The Deceptive Border ................... 208 6.7 Puzzling Paradoxes ..................... 210 6.8 A Trigonometric Fallacy .................. 211 6.9 Limits with Understanding ................. 213 Chapter 7 Counting and Probability .................. 215 7.1 Friday the 13th! ....................... 216 7.2 Think Before Counting ................... 217 7.3 The Worthless Increase ................... 219 7.4 Birthday Matches ...................... 220 7.5 Calendar Peculiarities .................... 223 7.6 The Monty Hall Problem ................. 224 7.7 Anticipating Heads and Tails ............... 228 Chapter 8 Mathematical Potpourri ................... 229 8.1 Perfection in Mathematics ................. 230 8.2 The Beautiful Magic Square ................ 232 8.3 Unsolved Problems ..................... 236 8.4 An Unexpected Result ................... 239 8.5 Mathematics in Nature ................... 241 8.6 The Hands of a Clock ................... 247 8.7 Where in the World Are You? ............... 251 8.8 Crossing the Bridges .................... 253 8.9 The Most Misunderstood Average ............ 256 8.10 The Pascal Triangle ..................... 259 8.11 It’s All Relative ....................... 263 8.12 Generalizations Require Proof ............... 264 8.13 A Beautiful Curve ..................... 265 Epilogue .................................. 268 Acknowledgments ............................. 271 Index .................................... 272 About the Author ............................. 276 viii Foreword Bertrand Russell once wrote, “Mathematics