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July 2015 Olume 4, No.2 V July 2015 olume 4, No.2 V Features Morley�s Miracle Paradoxes 3, 4, 5... And other Memorable Triples In the Classroom Keeping Things in Proportion (Low floor high ceiling) Tech Space Paper Folding and Geogebra Review �The Crest of the Peacock� T UT N LO E L EM PU R SU EA M THE CHARM OF PARADOXES Paradoxes have been around from the most ancient of times, and yet continue to be present “Correct spelling in our times, baffling and amusing us. At the same time, they often point to fundamental is esential.” All men are equal, but some are truths of life. (Perhaps the most hard‐hitting of more equal than others these is this one: .) You surely will have your own favourites. There are paradoxes of different kinds. Some lie strictly within the contours of mathematics. Everyone has seen the 'proof' that 1=2, or that every triangle is equilateral. Though these are labelled as paradoxes, they typically derive from a mathematically illegal step or an incorrectly drawn diagram. Precisely because of this, they have great value pedagogically. Then there are paradoxes in which the conclusion of a carefully constructed line of reasoning seems absurd, but this is only because our common sense is unable to accept the conclusion. Mathematically, there is nothing paradoxical about the conclusion. A well‐known such Paradox is Zeno's Paradox which derives from the notion that if you add an infinite number of quantities, the sum must be Don't read infinite (that's what common sense tells us). In more recent times, we have the Banach‐ this notice. Tarski Paradox, in which from a given object, we produce two copies of that same object by dividing it into a large Down with bumper stickers. number of pieces and rearranging the pieces, and the paradoxes concerning Infinity which derive from the work of Cantor. Note the feature that all these paradoxes share: the notion of Infinity. And then there are paradoxes which seem genuinely impossible to resolve. Such is certainly the case with paradoxes involving self‐reference. Well‐known examples are the Barber's Paradox, the Russell Paradox in set theory, and the Epimenides Paradox. The underlying logic here is of an utterly simple nature, yet the conclusion “I don't care frustrates us. Other such paradoxes which we find similarly frustrating are those involving time travel. (We do not possess the technology for time travel today, but to belong to any imagine that human beings of the future – say a century from now – master such club that will have technology. Would they be able to move backwards in time, and come and visit us in me as a member.” our time and teach us the technology for time travel? If so, it would lead to a most (Groucho Marx) delicious paradox!) All in all, paradoxes offer us a most There was a young lady of Crewe wonderful subject. Let us treat Whose limericks stopped at line two. them as gifts from the gods and delight in their study. From the Editor’s Desk . 'A popular mathematics magazine' seems to be a paradoxical phrase, but here is our tenth issue and with a growing subscriber list, we seem to be both popular and mathematical! Aptly enough, Punya Misra and Gaurav Bhatnagar wrap up their two part series on Paradoxes; selected visuals from these are featured on the cover too. Part III of Morley's Miracle by V.G. Tikekar and Features some stunning insights into the appearances of the 3‐4‐5 triangle by Shailesh Shirali completClassRoome the section. In the is packed with goodies for the practicing teacher. Majid Sheikh describes some Tests for Divisibility by Powers of 2 and CoMaC pitches in with an article on Prime Generation and an explanation for one of the patterns reported in a post from 'AtRiUM' – our FaceBook page. Swati Sircar and Sneha Titus continue the Low Floor, High Ceiling activity series and Swati Sircar also provides an addendum to the Pullout on Division. Shailesh Shirali pulls a family of circles out of his mathematician's hat and also continues the series on 'How To Prove It'. Tech Space Paper Folding and Dynamic Geometry software have blended seamlessly in where Swati Sircar begins a series on Conic sections. This section has a bonus this time – a GeoGebra investigation of the Rectangle Problem. It's time to say Open Sesame at your math lab sessions! Problem Corner Prithwijit De, R. Athmaraman and CoMaC give us more fodder for . This time we feature R. Ramanujam's review of the book by George Joseph Gheverghese, 'The Crest of the Peacock'. Padmapriya Shirali wraps up the issue with a pullout on Measurement. In the last issue, we had featured a challenge problem for readers and we are delighted to publish a solution sent in by reader Tejash Patel. A question on how we teach the rule of 'negative times negative is positive' which was thrown open to readers also sparked off several responses. This issue publishes an abbreviated version of some of these. It's been an interesting three years and we appreciate this journey with you, our loyal readers! Sneha Titus — Associate Editor A Publication of Azim Premji University together with Community Mathematics Centre, Rishi Valley and Sahyadri School, Pune Contents Features In the Classroom This section has articles dealing with This section gives you a 'fly on the wall' classroom mathematical content, in pure and applied experience. With articles that deal with issues of mathematics. The scope is wide: a look at a topic pedagogy, teaching methodology and classroom Chief Editor Associate Editor through history; the life‐story of some teaching, it takes you to the hot seat of mathematics mathematician; a fresh approach to some topic; education. 'In The Classroom' is meant for practising Shailesh Shirali Sneha Titus application of a topic in some area of science, teachers and teacher educators. Articles are Community Mathematics Centre, Azim Premji University engineering or medicine; an unsuspected sometimes anecdotal; or about how to teach a topic Sahyadri School (KFI) connection between topics; a new way of solving a or concept in a different way. They often take a new Editorial Committee known problem; and so on. Paper folding is a look at assessment or at projects; discuss how to Athmaraman R D D Karopady Shashidhar Jagadeeshan theme we will frequently feature, for its many anchor a math club or math expo; offer insights into Association of Mathematics Azim Premji Foundation, CentrSrirange For Learning, Bangavalli Kona alore mathematical, aesthetic and hands‐on aspects. remedial teaching etc. TGiridhar Seachers of India, Chennai BangPadmaprialore ya Shirali Written by practising mathematicians, the 20 Divisibility Tests by Powers of 2 Rishi VK. Subralleamaniamy School common thread is the joy of sharing discoveries Majid Shaikh Azim PrHriday Kant Deemji Universitywan SahPrithyadri School (KFI)wijit De and the investigative approaches leading to them. Homi Bhabha Centre for Science 24 Azim PrJonaki B Ghoshemji University Homi Bhabha Centre for Science Education, Tata Institute of 05 Morley's Miracle - Part III Shailesh Shir153 and so on and on and on . .ali Education, Tata Institute of FTundamental Ranuj Shah esearch, Mumbai V G Tikekar FRamgopal Vundamental Rallathesearch Lady Shri Ram College for Gautham Dayal 26 Women, University of Delhi, Rishi Valley School 09 Paradoxes - Part II CoMacA Flower with Four Petals Delhi. Azim Premji University Punya Mishra and Gaurav Bhatnagar Design & Print CERTAD, Srishti school of Art, Design and Technology 28 Generating the n-th Prime 15 Shailesh Shir3,4,5... And other Memorali able Triples - CoMac SCPL Please Note: Part I Bangalore ‐ 560 062 +91 80 2686 0585 30 Low Floor High Ceiling Tasks +91 98450 42233 All views and opinions expressed in this issue are those of the authors and Swati Sircar and Sneha Titus www.scpl.net Azim Premji Foundation bears no responsibility for the same. 36 Thoughts on the Division Operation Swati Sircar At Right Angles is a publication of Azim Premji University together with Community Mathematics Centre, Rishi Valley School 40 A Baby One Quarter the Size of its Parents and Sahyadri School (KFI). It aims to reach out to teachers, teacher educators, students & those who are passionate about CoMac mathematics. It provides a platform for the expression of varied opinions & perspectives and encourages new and informed positions, thought‐provoking points of view and stories of innovation. The approach is a balance between being an ‘academic’ and 'practitioner’ oriented magazine. 44 How To Prove It Shailesh Shirali A Publication of Azim Premji University together with Community Mathematics Centre, Rishi Valley and Sahyadri School, Pune Contents Features In the Classroom This section has articles dealing with This section gives you a 'fly on the wall' classroom mathematical content, in pure and applied experience. With articles that deal with issues of mathematics. The scope is wide: a look at a topic pedagogy, teaching methodology and classroom through history; the life‐story of some teaching, it takes you to the hot seat of mathematics mathematician; a fresh approach to some topic; education. 'In The Classroom' is meant for practising application of a topic in some area of science, teachers and teacher educators. Articles are engineering or medicine; an unsuspected sometimes anecdotal; or about how to teach a topic connection between topics; a new way of solving a or concept in a different way. They often take a new known problem; and so on. Paper folding is a look at assessment or at projects; discuss how to theme we will frequently feature, for its many anchor a math club or math expo; offer insights into mathematical, aesthetic and hands‐on aspects.
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