Class 9-10

Pose these problems to the Senior School students...

The Pythagorean equation x^2 + y^2 = z^2 (to be solved over the positive integers N) is a much-studied one; many articles have appeared in this magazine alone, devoted to this equation. A close relative to this is the equation 1/x + 1/y = 1/z (which can be written as x^−1 + y^−1 = z^−1 ; in this form, its similarity to the Pythagorean equation is readily seen), and this too has been studied many times in At Right Angles.

On the Facebook page (AtRiUM: At Right Angles, Us and Math) linked to this magazine, one of our contributors, Arsalan Wares, has been astonishingly prolific in posting problems. A good many of these have had to do with regular hexagons; more specifically, with the areas of polygonal regions drawn within such hexagons. It is both astonishing and pleasing to see such a rich diversity of problems arising from this simple and familiar structure.

In this edition of ‘Adventures’ we study a few miscellaneous problems, mostly from the Pre-Regional Mathematics Olympiad (PRMO; this year’s PRMO was conducted on August 19 in centres all over the country). As usual, we pose the problems first and present solutions later.

For the middle school student, the transition from arithmetic to algebra is often quite daunting. In grade VI, the concept of a ‘variable’ is encountered for the first time. This is the stage where either a child embraces the newly introduced ‘Algebra’ or gets overwhelmed with the idea of numbers being replaced by letters of the alphabet. This is also the stage where the students learn to solve equations and find the value(s) of the unknown(s).

Websites and focus interest groups are a good source for interesting problems. But it’s rarely that one gets down to solving these; more often they go into a to-do list. We hope that the solution presented here will encourage you to try more of these. Look at the steps of the process: Visualization, definition of the problem, connection to known formulas and then good old mathematical processing. Problem solved!

Errors should not be viewed as a setback but as an opportunity to learn more about the student's thought processes. Correcting errors to get the right product instead of analysing the learning trajectory is a quick fix that does not nurture deep learning Very often, errors are caused by over-generalisation of rules which are transmitted to the student as short-cuts to getting the required answer.

The following diagram is a “proof without words” (also known as a ‘visual proof’) for the trigonometric identity.

In this article, we focus on investigations with graph paper. Pages 45 & 46 give guidelines for the facilitator, pages 43 & 44 are a worksheet for students. This time we explore quadrilaterals and triangles using lattice points.

In this article, student Satvik Kaushik investigates Armstrong numbers. Mathematical investigation is a powerful way for students to learn more about concepts that interest them. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation.


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