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At Right Angles November 2017

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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 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.

Enjoy your read and do send feedback to


V G Tikekar
Sums of Powers of Natural Number


Gaurav Bhatnagar
How to Discover the Exponential Function


Shailesh Shirali
The Exponential Series an Addendum

V Ramprashanth
Tonic Ambiguity


There are Infinitely many Prime Numbers

Shailesh Shirali
Nepoleon's Theorem


In the Classroom

Prithwijit De
A Triangle Problem and Three Variants

Dev Chandan and Adhiraj Patil
Happy Numbers


Shailesh Shirali
How to Prove it

Bodhideep Joardar
An Approximation of Π by using The Law of Cosines


Naveenkumar Yadav
On Scalene Triples

Ujjwal Rane
Fagnano's Theorem - Alternative Proof


Prithwijit De, Swati Sircar, Sneha Titus
LFHC - Area, Perimeter and Congruency (APC)


Haragopal R
G - Numbers


Course Correction - What to do when One Gets Two Different Answers to A Counting Problem


Shailesh Shirali
Inequalities in Algebra and Geometry - Part 3


Anant W Vyawahare
Divisibility By 7


Tech Space

Thomas Lingefjard and Djamshid Farahani
Teaching Concepts of Calculus


Sangeeta Gulati
Desmos Activity Builder Create Your Own Interactive Activity


Problem Corner

Shailesh Shirali
Harmonic Triangular Triples

A Triangle Problem


Sneha Titus and R Athmaraman
Problems for the Middle School

Prithwijit De and Shailesh Shirali
Problems for the Senior School

Shailesh Shirali
Adventures in Problem Solving



R Ramanujam
Review of G H Hardy's A Mathematician's Apology


Shashidhar Jagadeeshan
The Story of Maths A Brief Review


PadmaPriya Shirali
Teaching Integers


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