Publications & Resources

Our faculty, students and researchers work together everyday to contribute to a better world by grappling with urgent problems we are facing in India. We conduct rigorous work to produce high quality learning resources and publications to contribute to public discourse and social change. Here, we feature a sample from our work for everyone to access. You can explore featured resources, policies, and the latest publications from the University. 

To explore all the work of our University, please visit our publications repository.

  • ARA March2021
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      Isn’t mathematics simply divine? Not an unexpected line from the At Right Angles team but one that we will back up with our Features article from the March 2021 issue: Geometric Constructions from the Sulvasutras. It’s an incisive dive from there to the Utilitarian Math World but your disillusionment will be healed with delightful articles on understanding some of the whys of procedures and rules in mathematics such as the divisibility rules, the square root algorithm, the formula for the median and the folk method to find the height of a tree.

      We bring you a new first with this issue: for the first time, AtRiA goes beyond its covers with three additional articles available only in the online mode. A big shout out to the numerous submissions which enable us to take this leap.

      Do revert with your feedback on AtRiA.​editor@​apu.​edu.​in

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    • RA Issue 8 Nov 2020 English Cover Page
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        Beyond Four Walls — Contain a mathematician in a room and you can be sure that some thinking on quadrilaterals will happen! Our second issue during the pandemic has much to keep you absorbed and happily engaged with problems- of the mathematical kind. From age digit reversals to an algorithm for evacuating a room — there is material for every age group. We begin with the all important question: Why Should You Study Mathematics? And we also carry a review of the Classes I‑III Sikkim Mathematics Textbooks, the development of which our URC-PDP played such a key role in. Do revert with your feedback on AtRiA.​editor@​apu.​edu.​in

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      • ARA July2020
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          As the world goes through lock down and social distancing, it is easy to feel less and less a part of the whole and more and more a small, isolated part. But that does not make us or anybody else, any less a part of the whole and we need to realise that every part of the whole matters. 

          But are all the parts of the whole equal? If not, what can we do to redress the balance?

          Think mathematically!

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        • RA Issue 6 March 2020 English Cover Page
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            Vision 2020 – As we move into the 9th year of AtRiA’s existence, we refocus for clarity. The March 2020 issue talks about Breaking Down Barriers: A Mathematical Approach. Launch right into the how and what with B. Sury’s article on Tiresome Paths, Water Gates and Euler’s Formula. But there’s more than networks and circuits here, starting with this issue, we will be focusing sharply on the challenges and difficulties of the average mathematics teacher and hoping to share and discuss strategies to help those who struggle with the nature and pedagogy of mathematics.

            We have some wonderful material to draw you in- Conversations on the Greatest Common Divisor, Trials with Triangles, Impossible Triangles on Dot Sheets and some great Search Problems for the Middle School to mention a few. We start a new section called ViewPoint and continue with our photo series Captured Mathematics. TechSpace features the free dynamic geometry software Robocompass and our PullOut is on Angles. Our review this time is on the Mathematics of India.

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          • RA Issue 5 Nov 2019 English Cover Page
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              AtRiA, November 2019 focuses on journeys ‑some are enjoyable, some tiresome, some endless. A mathematician’s journey is full of unexpected delight, simply because of unexpected detours, sudden discoveries and interesting alternatives. Have fun as you travel with us. Read all about the young Yatris and their journey of discovery in Features. This section also describes more abstract journeys from the Regular Pentagon to the Icosahedron and the Dodecahedron (Part 2) and from the familiar 2 and 3 dimensions to n dimensions in Extension of the Pythagorean Theorem. In ClassRoom, Simple Cryptography and Triangles with Integer Sides are articles sparked off by previous articles in AtRiA . An Unusual Proof of the Centroid Theorem, Modified Pascal Triangle and Orthocentre of a Triangle may motivate you to write similarly! You will find ideas in our regular columns – How to Prove It, Low Flow High Ceiling and TechSpace which features the simulation of a dice game this time. Our issue is short of 100% without the PullOut, enjoy the learning trajectory on Percentages, defined and illustrated with plenty of examples in this issue. How Craig Barton Wishes he’d Taught Maths is an excellent review by Sir Timothy William Gowers, reprinted from his Weblog.

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            • RA Issue 4 July 2019 English Cover Page
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                The July 2019 issue features Fractals on the cover — those distant and beautiful and highly mathematical creations have been brought a click away in our TechSpace section. Enjoy creating them and adding to your repertoire. We also take you through the magical tour of the beauty of mathematics starting with an article on Pentagons. Pi enters the ClassRoom section and the primary has plenty of learning — have fun joining the dots and exploring geometrical figures with different lenses. Students share their mathematical findings in two exciting articles. Our pedagogy article helps you to Understand Learners’ Thinking. And the Middle School problem corner is a Do_​It_​Yourself hands on problem solving section. The PullOut is on Ratio. 

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              • RA Issue 3 March 2019 English Cover Page
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                  The March 2019 issue is packed with articles for students, teachers and teacher educators of all classes. Striking results are not just reported, they are also explained and justified with proofs. And our sources vary from students to observations from colleagues in mathematics classrooms and mathematical tricks from websites- these are explored and explained.

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                • Magazine

                  At Right Angles | July 2018

                  in Azim Premji University

                  RA Vol 7 Issue 2 July 2018 English Cover Page
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                    Volume 7 Issue 2: And it’s out! Most of you would have already received your copy of the July issue of At Right Angles, here’s the official announcement and the link with the welcome news that AtRiA is finally a whole school math resource! The shift all the way to Primary is marked with several new sub-sections in ClassRoom: from Sense-Making in Mathematics – the cover says it all- to TearOut- where all a busy teacher has to do is to tear out a ready made worksheet – complete with facilitator notes- and use it to challenge students to think in fun mathematical ways. As always, PullOut shares strategies and tips on a specific theme, this time we continue our series on Algebra.

                    And to top it all, we have not one but two students contributing articles to the Features section, no less! Enjoy a great read!

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                  • Magazine

                    At Right Angles | March 2018

                    in Azim Premji University

                    RA Vol 7 Issue 1 March 2018 English Cover Page
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                      Volume 7 Issue 1: This issue of AtRiA welcomes you to the Math Village. Where house numbers, rangoli patterns, piles of fruit and even a child’s amusement device speaks to the instictive mathematician in each of us.

                      Read about these in Triangular Numbers in Features and the House Number Problem in ClassRoom. Appropriately, many of these problems were prompted by Ramanujan in his village. The Review is of a book which was used to teach a pre-course at the School of Arts and Sciences, Azim Premji University and it syncs perfectly with the theme of finding mathematics in the most unexpected places.

                      Hone up on how to create a Conjecturing ClassRoom or generate Elementary Cellular Automata! Misconceptions in Fractions and an analysis of common errors in Algebra give an insight into students’ struggles with mathematics and the PullOut, too, focuses on Algebra and how to make a seamless move from Arithmetic to Algebra.

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                    • RA Vol 6 No 3 March 2017 English Cover Page
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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/​5​M​r​N​M​-​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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                      • RA Vol 6 No 2 March 2017 English Cover Page
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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.…

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                        • Magazine

                          At Right Angles | March 2017

                          in Azim Premji University

                          RA Vol 6 No 1 March 2017 English Cover Page
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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 345 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.

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                          • RA Vol 4 No 3 Nov 2015 English Cover Page
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                              Volume 4 Issue 3: Two themes dominate this issue of AtRiA: Archimedes & Magic Squares — an unlikely combination! Both are exceedingly rich topics to write about, with histories that go far back in time. Who can’t be both charmed and thrilled by the story of Archimedes?

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                            • Magazine

                              At Right Angles | March 2015

                              in Azim Premji University

                              RA Vol 4 No 1 March 2015 English Cover Page
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                                Volume 4 Issue 1: The lead feature in the March 2015 issue of AtRiA is based on the theme Proof Without Words. In the Review section, Mark Kleiner discusses Edward Frenkel’s Love and Math — the Heart of Hidden Reality. Thomas Lingefjard, in his article Learning Math with a DGE system, addresses a pressing need of teachers using technology in the classroom. This issue features a new author Ali Hussen whose article weaves in algebra, geometry and arithmetic. It also introduces a new series on Low Floor High Ceiling activities.The pullout continues on the Teaching of Geometry (part ii).

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