Publications & Resources
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Volume 6 Issue 3: The word ‘heuristics’ has always left me fumbling for the perfect definition. Rule of thumb seems adequate but too informal, as a teacher trainer, I’ve always preferred to illustrate rather than define this word. And the first article in the November issue does just that- Gaurav Bhatnagar, leads you gently into the Exponential Series- a heuristic definition! The relaxed tone continues with V. G. Tikekar setting the stage to arrive at formulae which are usually given and proved using mathematical induction. He, on the other hand, uses the narrative mode to set the context which explains and then proves the sum of squares and sum of cubes formulae. From story we move to song, V. Ramprashanth’s article Tonic Ambiguity is a delightful depart from our usual articles and it explains very simply, using just the concepts of ratios, how two hearers can place the same tones at different positions on the musical scale. CoMaC and then Shailesh Shirali, wrap up the Features section with a bouquet of proofs.
ClassRoom opens with Prithwijit De’s trademark investigative style – the title simply says A Triangle Problem – read on to find out more. Student Corner is our happy space and this time, we have Vinay Nair’s students talking about Happy Numbers and Bodhideep Joardar Investigating Pi. Next, we have Naveen Yadav combining Set Theory, Geometry and Combinatorics in Scalene Triples and Ujjwal Rane follows up on his previous article on Fagnano’s Theorem with a very visual, very practical alternative proof. A video of this result has also been uploaded by him on YouTube, it is available at https://youtu.be/5MrNM-VxXd8. Low Floor High Ceiling features a modification of an investigation into triangles having the same area and perimeter and conditions for congruency. Learn about G numbers in Hara Gopal’s article. We’re particularly delighted to bring in yet another new sub-section to ClassRoom – Course Correction features vignettes from pedagogical experience, ways in which pitfalls leading to misconceptions can be addressed and used to advantage to strengthen student understanding. This time, we feature the solution of a problem from a traditionally ambivalent topic- Combinatorics.
On to TechSpace, we have not one but two articles this time. Thomas Lingefjard describes the use of technology to introduce concepts in Calculus and Sangeeta Gulati describes changes in the Desmos software and how it can now be used to build activity kits. Problem Corner is a treasure trove of interesting problems in addition to the usual Middle School, Senior School and Adventures in Problem Solving articles.
Review features an old classic – R. Ramanujam takes us through a lovely guided tour of Hardy’s A Mathematician’s Apology. And there is an unexpected bonanza, with this issue, we will feature a mini- review, it may be of a video clip, a game, a story…………..all things mathematical in essence. This time, J. Shashidhar talks about Marcus du Sautoy’s Story of Mathematics. Read the facing page to know more about this fascinating mathematician.
Pullout has graduated to Upper Primary – with this issue, we bring you a compendium of information about teaching Integers- surely the point when students begin to have negative thoughts about mathematics. I’m sure you’ll enjoy reading how to combat those tendrils of fear.
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Volume 6 Issue 2: It is a pleasure to share with you, our readers, the collection of articles in this, the July 2017 issue. The (un)popular view of mathematics being a terrifying subject takes a completely new twist with the first article in which a mathematician takes on a terrorist threat! The hunt for answers to a mathematical problem is usually an absorbing one, at least to aficionados of the subject but Arun Vaidya’s fascinating story I M Code makes it a matter of life and death.
Following this, we have an article on another application of mathematics: Interpolation by Sankaran Viswanath. You will see again how mathematics is a tool for prediction, and how data can be fitted into mathematical expressions which then provide a mathematical model. From here, we move on to card tricks; yes, fun and mathematics can go together — and At Right Angles shows you how in Suhas Saha’s Ternary Base Magic Trick. A quick peek behind the magic reveals patterns based on the ternary base, it’s not as complex as it sounds, read on to find out. Our Features section ends with Shailesh Shirali’s exposition on Quadrilaterals with Perpendicular Diagonals, a nice bouquet of Arithmetic, Algebra and Geometry for you.In ClassRoom, we have the second part of the Inequalities series started in the March 2017 issue, again, both Algebra and Geometry are used to first prove the arithmetic mean- geometric mean inequality and then apply it in several situations to illustrate the power of this relationship and also view its implications in graphs, geometric figures, functions􀀿􀀿􀀿.the list of connections seems endless! Moshe Stupel and David Ben-Chaim appear next with their article Three Elegant Proofs, the name says it all, we promise it lives up to its title. CoMaC, as usual, provides an indepth analysis of an often-asked question, now increasingly appearing even in WhatsApp forwards: What’s the next number? Is the answer really unique as the question implies it to be? More on numbers with Swati Sircar and Sneha Titus, writing on the Sums of Consecutive Natural Numbers; mental mathematics becomes visual all of a sudden, and this Low Floor High Ceiling activity is sure to appeal to a variety of learning styles. Vinay Nair takes up the theme of Divisibility by Primes and provides some powerful tests using an ‘osculator’. Students are sure to be intrigued. ClassRoom concludes with a Proof Without Words on a property of the Orthocentre of a triangle.
For some time now, we have been featuring articles by students and we are particularly happy when they write in with their own discoveries. So much so, that from this issue onwards, we have devoted space to Student Corner in the ClassRoom section. Featured this time are Bodhideep of class 6 and Parthiv of class 11, you are sure to be impressed with their discoveries.
Our cover this time, features Golden Quadrilaterals and the illustrations have been provided by Michael de Villiers, who continues his series on constructive defining. These beautiful quadrilaterals have been defined by investigation and are an interesting activity for students who believe that everything in mathematics is pre-defined and that there is nothing new in mathematics to be discovered.
Problem Corner has seen some changes over the last few issues. In a deliberate attempt to avoid a ‘camp’ approach to problem solving and to make this section more inclusive, we have a wide variety to interest our readers. Prithwijit De sets the ball rolling with his article on ProblemPosing. This is followed by Middle and Senior Problems addressed to different age groups. CoMaC presents a theorem about a triangle and a problem about a rational number; the titles are deliberately bland but these are as fun as Shailesh Shirali’s Adventures in Problem Solving.
The Review this time will certainly have you leaping to order this book: The Cartoon Guides to Calculus and Algebra, a series whose name says both all and nothing. Can such a serious subject be illustrated with cartoons? With mathematical rigour? Read the Review and I’m sure you’ll be convinced.
Our issue concludes with the PullOut — Padmapriya Shirali focuses on Large Numbers and how students can grasp this concept. I am sure that adults too will enjoy this refresher course and pick up tips on how to make this topic child- friendly and approachable. So it’s over to you now! Happy reading.…Links
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Volume 6 Issue 1: The Art of the Matter — can a picture be a powerful pedagogical tool? At Right Angles believes it can, and if you read the write up on the facing page, I’m sure you will agree with us. Going to the heart of the art is key to mathematics and there’s more of the same inside this issue. We begin with Ramya Ramalingam, a sixteen-year-old school girl, unravelling the mysteries of Knot Theory for us. And Haneet Gandhi picks up where her series on Tessellations stopped, with a fascinating article on tiling and the pictures we can create with different combinations and permutations of polygons.
In the Classroom section, Khushboo Awasthi opens up the Square Root Spiral with a series of investigative questions; Ujjwal Rane proves Fagnano’s Theorem in several innovative ways. CoMaC describes an unusual way to bisect an angle and also manages to pull yet another 3−4−5 triangle which has long connected math with art pops up in Kepler’s triangle — read more about it in Marcus Bizony’s article. And in How To Prove It, Shailesh Shirali uses Ptolemy’s theorem to reveal all kinds of fascinating relationships in cyclic quadrilaterals.Tech Space features the first part of a two-part series on constructive definitions; Michael de Villiers shows you how to do so with a GeoGebra activity centered on the golden rhombus. Truely cutting edge math pedagogy!
Our review this time is by Kamala Mukunda who shares her views on Liping Ma’s classic Knowing and Teaching Elementary Mathematics. This is a must-have for every school library and a must-read for mathematics teachers of all classes.
Finally, it’s Time — this Pull Out by Padmapriya Shirali will give you several new ideas to introduce this all important concept and help students quantify something which impinges on their consciousness long before they come to school.Links