Ram Kishore is a primary school teacher in a government school in Bidhoura, Sitarganj in Udhhamsingh Nagar, Uttarakhand For the 50 students of the school, he is the only teacher.

# Teacher Development

**Are there any reliable stewards of this blue planet, today?**

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Most students spend a big fraction of their study time preparing themselves for answering questions – for examinations. This becomes all the more important when a student writes a competitive examination such as the IIT entrance. In such a situation, the busy student tries to consult as many question banks from as many sources as possible, ensuring that (s)he has created his/her own database with (almost) all possible variations in questions from a certain theme!

- Read more about The Importance of Asking Questions – in Different Ways!
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We often find ourselves needing to predict the future value of something based on past trends. For instance, climate scientists have been trying to estimate the future rise in ocean temperatures based on (among other things) temperature data of the last 100 or more years. Or maybe you are a cricket enthusiast and want to predict how many centuries your favourite batsman will score this year based on his scoring statistics for past years. We all probably have our own ways of arriving at an estimate! In this article, let us explore one approach.

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In this edition of Adventures in Problem Solving, we study in detail a problem which first appeared in the 1987 USA Mathematics Olympiad. However, we adopt a different strategy this time. Faced with the given problem which looks quite challenging, we tweak it in different ways and obtain related problems which are simpler than the original one. Solving this collection of problems turns out to be a fun activity and demonstrates yet one more time the importance and utility of quadratic functions and quadratic equations.

Read on some interesting problems concerning rational numbers

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The following elegant geometric result concerning a triangle is based on a problem that appeared in the Regional Mathematics Olympiad (RMO) of 2016.

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Problem-posing and problem-solving are central to mathematics. As a student one solves a plethora of problems of varying levels of difficulty to learn the applications of theories taught in the mathematics curriculum. But rarely is one shown how problems are made. The importance of problem-posing is not emphasized as a part of learning mathematics. In this article, we show how new problems may be created from simple mathematical statements at the secondary school level.

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This article continues the investigation started by the author in the March 2017 issue of At Right Angles, available at: http://teachersofindia.org/en/ ebook/golden-rectangle-golden-quadrilaterals-and-beyond-1 The focus of the paper is on constructively defining various golden quadrilaterals analogously to the famous golden rectangle so that they exhibit some aspects of the golden ratio phi.

Elegant proofs of mathematical statements present the beauty of mathematics and enhance our learning pleasure. This is particularly so for ‘proofs without words’ which significantly improve the visual proof capability. In this article, we present three largely visual proofs which carry a great deal of elegance. In a strict sense they are not pure proofs without words because of the mathematical expressions and formulas that appear in them. Nevertheless, they carry a lot of appeal.

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This article is the second in the ‘Inequalities’ series. We prove a very important inequality, the Arithmetic Mean-Geometric Mean inequality (‘AM-GM inequality’) which has a vast number of applications and generalizations. We prove it using algebra as well as geometry

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