A day s adventure in math wonderland

6864tp.indd 3/26/08 9:30:49 AM This page intentionally left blank Tokai University, Japan Ateneo de Manila University, Philippines World Scientific NEW JERSEY 6864tp.indd • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 3/26/08 9:30:49 AM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Akiyama, J A day's adventure in math wonderland / Jin Akiyama & Mari-Jo P Ruiz p cm Includes bibliographical references ISBN-13: 978-981-281-476-0 (pbk : alk paper) ISBN-10: 981-281-476-0 (pbk : alk paper) Mathematics Study and teaching Manipulatives (Education) Mathematical models Mathematics in literature I Ruiz, Mari-Jo P II Title QA19.M34A45 2008 510 dc22 2008009416 Book design by Irwin Cruz British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2008 by Jin Akiyama and Mari-Jo P Ruiz All rights reserved Printed in Singapore A bell signals recess time Boys rush to the playground from all directions Ichiro looks around for his best friends Jai and Kino He sees them with a group of boys gathered around Kentaro They seem so engrossed in what he is saying “I wonder what Kentaro is saying that is so interesting.” Ichiro moves to within hearing distance “It was really fun! I rode on a tricycle with square wheels , I ran down a musical staircase , I won a race on some huge slides ,” Kentaro goes on and on breathlessly “Where did he go?” Ichiro asks “Math Wonderland,” Kino answers “Oh, that place again.” Ichiro says “What you mean?” Jai wonders “My grandmother couldn’t stop talking about that place over breakfast She saw some TV footage and she said the kids looked like they were having fun with the mathematical models,” Ichiro answers “I can’t imagine what could be fun about math,” he adds “Math is not so bad,” Jai counters Ichiro is doing well in school without much ‡ơ‘”– „—– ƒ–Š †‘‡• ‘– ’ƒ”–‹…—Žƒ”Ž› ‹–‡”‡•– Š‹Ǥ‡ϐ‹†•Š‹•ƒ–Š…Žƒ••Œ—•–ƒ‘–…Šƒ„‘˜‡ boring At times, he can hardly keep awake Besides, his math homework takes much time away from computer games, TV, and his other favorite activity – taking apart and putting together his toy robots Kino, always eager and impulsive, taps him on the shoulder, “Let’s go there this weekend.” “I’d like to see it,” says Jai On his own, Ichiro would not have planned on going, but with his best friends, it might be an adventure “Did you notice that in order to align the edges, you had to bisect the angles?” Yuji asks The boys look back on their solutions “We bisected twice,” Ichiro notes “Bisection is usually part of the strategy,” Yuji states “There is another detail that you must take care of The boundary lines must align all the way across the place where you plan to make the cut.” With these hints, he asks them to try another exercise The guides hand out pieces of paper with a triangle 161 The boys wrestle with the problem It is not ƒ••‹’Ž‡ƒ•–Š‡ϐ‹”•–•‹…‡–Š‡–”‹ƒ‰Ž‡‹•‡‹–Š‡” equilateral nor isosceles “It’s not so easy to align the lines this time,” Kino comments “Maybe we should try bisecting angles like Yuji said,” Ichiro suggests He folds along the angle bisectors and immediately the paper yields the solution naturally “Look what I got,” he tells his friends They examine his paper and notice that a fourth fold appears along the perpendicular from the point of intersection of the angle bisectors to one side of the triangle 162 “That’s cool,” Jai says “Would it work if the fourth fold is made along the perpendicular to another side?” They try this out using the other two sides, so they discover that the folding is not unique Yuji observing them says: “You guys have the right attitude for mathematics, you are asking good questions and pursuing the alternatives.” They are pleased with the compliment “Try this out,” Yuji says handing them three ’‹‡…‡•‘ˆ’ƒ’‡”™‹–Šƒϐ‹˜‡Ǧ’‘‹–‡†•–ƒ”Ǥ The boys decide to this exercise together “OK, anyone who has any ideas, speak up,” Ichiro tells his friends “I think we should fold it by bisecting the angle at the top so that the lines from both sides align,” Jai proposes They stare at the folded sheet for a while 163 “I know, I know,” Ichiro says excitedly “We can put the two points together by folding this way.” He gets hold of the paper and folds it “Then we can align the three lines together using two folds,” Jai continues Kino cuts their model and the star separates from the rest of the paper 164 “We’re done.” They show their work to Yuji “Good work!” he says The other kids look at them with a mixture of admiration and envy, since they are way ahead of the rest “Ready for another exercise?” Yuji asks “Yes!” they say 165 ‡‰‹˜‡•–Š‡ƒϐ‹‰—”‡™Š‹…ŠŠ‡•ƒ›•‹•ƒ‘Ž† Japanese crest Š‡›–ƒ‡‘”‡–‹‡™‹–Š–Š‹•„—––Š‡›ϐ‹ƒŽŽ› get it done and it gives them a great feeling of satisfaction 166 Here is their solution Yuji gives the boys a problem to take home He hands them a sheet of paper with a trapezoid When all the other kids are done with the star, Yuji says: “Now that you have some experience with this kind of problem, I will show you how to fold the 4x4 chessboard that I cut for you at the start.” 167 He goes through the process slowly step by step and when he is done, a single cut releases the black squares “Will that work for a bigger chessboard?” Jai asks “Only if the number of squares on each side is even,” Yuji answers “If it’s odd, you have to make some adjustment.” 168 “Great party trick!” Kino says ƒ‹ ϐ‹Ž‡• ƒ‘–Š‡” ‡–ƒŽ note: try a chessboard with three squares on each side They appreciate Yuji’s encouragement and go over to thank him 169 This page intentionally left blank 171 172 The boys sense a commotion in one of –Š‡”‘‘•ǤŠ‡›…‘‡ƒ…”‘••–Š‡‹”ϐ‹”•–‰—‹†‡ Keiko Dz‘— ƒ”‡ ‹ Ž—…Ǩdz  •Š‡ •ƒ›•ǡ Dz”‘ˆ‡••‘” Yamaaki is here for a visit He is the one who …‘ŽŽ‡…–‡† ƒŽŽ –Š‡ ‘†‡Ž•  ‹ ‘†‡”Žƒ†Ǥ  ‘— …ƒ™ƒ–…ŠŠ‹†‡‘•–”ƒ–‡‘†‡Ž•‹–Š‡‡š– ”‘‘Ǥdz Š‡›ƒ”‡ƒ„Ž‡–‘ϐ‹†ƒ’Žƒ…‡…Ž‘•‡–‘–Š‡ˆ”‘– ƒ‘‰–Š‡‰ƒ–Š‡”‹‰…”‘™†‘ˆ‹†•Ǥ˜‡”›‘‡ǯ• ƒ––‡–‹‘‹•ˆ‘…—•‡†‘ƒƒ™‹–Š—”—Ž›•ƒŽ– ƒ† ’‡’’‡” Šƒ‹”ǡ ƒ „‡ƒ”† Ž‹‡ ƒ Š‹‡•‡ •ƒ‰‡ǡ ‰Žƒ••‡•ƒ†ƒ…‘Ž‘”ˆ—ŽŠ‡ƒ†„ƒ†Ǥ ‡•‡‡•ˆ”‹‡†Ž›Ǥ‡‹•Œ‘‹‰ƒ”‘—†™‹–Š •‘‡‘ˆ–Š‡‹†•Ǥ Dz Šƒ˜‡ƒ›–Š‹‰•–‘•Š‘™›‘—–‘†ƒ›ǡdzŠ‡ says Dz‡ Ž‘‘• ˆƒ‹Ž‹ƒ”ǡdz …Š‹”‘ ™Š‹•’‡”• –‘  Š‹• ˆ”‹‡†•ǡDz