math wonders to inspire teachers and students

to Inspire Teachers and Students Alfred S. Posamentier Association for Supervision and Curriculum Development Alexandria, Virginia USA to Inspire Teachers and Students Alfred S. Posamentier 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: member@ascd.org 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 books@ascd.org 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. [...]... letters and e-mail messages supporting this view and asking for ways and materials to have people see and appreciate the beauty of mathematics I hope to be able to respond to the vast outcry for ways to demonstrate the beauty of mathematics with this book Teachers are the best ambassadors to the beautiful realm of mathematics Therefore, it is my desire to merely open the door to this aspect of mathematics... for mathematics and make teachers more resourceful so that they bring these mathematics motivators into their classrooms Teaching time isn’t lost by bringing some of these motivational devices into the classroom; rather, teaching time is more effective since the students will be more motivated and therefore more receptive to new material So parent and teacher alike should use these mathematics motivators... student’s enjoyment (of course, presented appropriately), the 1 2 Math Wonders to Inspire Teachers and Students justifications or explanations will be kept simple and adequately intelligible By the same token, in some cases the explanation might lead the reader to further research into or inspection of the phenomenon The moment you can bring students to the point where they question why the property exhibited... + 5! (Remember that 0! = 1.) That appears to be all of this sort that exists, so don’t bother having students search for more ∗ The exclamation mark is called a factorial and represents the product of consecutive integers from 1 to the number before the factorial symbol That is, n! = 1 2 3 4 · · · · · n − 2 n − 1 n 14 Math Wonders to Inspire Teachers and Students 1.10 Strange Equalities There are... might experiment by adding digits to the first multiplicand or by multiplying by other multiples of 9 In any case, experimentation ought to be encouraged 6 Math Wonders to Inspire Teachers and Students 1.3 Surprising Number Patterns III Here are some more charmers of mathematics that depend on the surprising nature of its number system Again, not many words are needed to demonstrate the charm, for it... 125 365 831 764 483 9 = 731 9 = 653 9 = 523 9 = 685 469 812 194 129 285 479 876 347 10 Math Wonders to Inspire Teachers and Students 1.6 Surprising Number Patterns VI Here is another nice pattern to further motivate your students to search on their own for other patterns in mathematics Again, not many words are needed to demonstrate the beauty of this pattern, for it is obvious at first sight 1 1+2+1 1+2+3+2+1... that we all too often take for granted Students too often are taught mathematics as a dry and required course of instruction As teachers, we have an obligation to make it interesting To show some of the number oddities brings some new “life” to the subject It will evoke a “gee whiz” response from students That’s what you ought to strive for Make them curious about the subject Motivate them to “dig” further... then you made a calculation error Check it 16 Math Wonders to Inspire Teachers and Students Unlike other numerical curiosities, which depended on a peculiarity of the decimal system, this illustration of a mathematical oddity depends on the operations Before we explore (for the more motivated students) why this happens, we want you to be able to impress your students with a further property of this lovely... appropriate that the discussion be relegated to a unit of this wonderful book So if you wish to see the sort of thing that turned me on to mathematics in my youth, see Unit 8.13 Why do I relate these episodes now? You are about to embark on a lovely book that was carefully crafted to turn you, the reader, and ultimately your students, on to mathematics It is impossible to determine what an individual will... 194 447 042 584 001 = 20720 12 Math Wonders to Inspire Teachers and Students 1.8 Beautiful Number Relationships Who said numbers can’t form beautiful relationships! Showing your students some of these unique situations might give them the feeling that there is more to “numbers” than meets the eye They should be encouraged not only to verify these relationships, but also to find others that can be considered . to Inspire Teachers and Students Alfred S. Posamentier Association for Supervision and Curriculum Development Alexandria, Virginia USA to Inspire Teachers and Students 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. 500 letters and e-mail messages supporting this view and asking for ways and mate- rials to have people see and appreciate the beauty of mathematics. I hope to be able to respond to the vast