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In an article published in the November 2016 issue of At Right Angles we had seen how geometrical fractal constructions lead to algebraic thinking. The article had highlighted the iterative construction processes, which lead to the Sierpinski triangle and the Sierpinski Square carpet. Further the idea of self-similarity within these fractals was reinforced through the recursive and explicit relationships between various stages of the fractal constructions.

In this episode of “How To Prove It”, we consider two similar sounding terms which have great significance in higher mathematics: contradiction and contrapositive. Both of them arise in connection with proofs. We give several examples of proofs of both these kinds.

If two sides of a triangle have the same lengths as two sides of another triangle, and one angle of the first triangle has the same measure as one angle of the second triangle, what can be said about them? Under what circumstances will they be congruent to one another?

The ‘TearOut’ series is back, with perimeter and area. Pages 1 and 2 are a worksheet for students, while pages 3 and 4 give guidelines for the facilitator. This time we explore shapes with given perimeter or given area using the dots or the grid.

In our last Low Floor High Ceiling article, we had looked at Squaring the Dots... a series of questions on counting the dots inside squares of different sizes and orientations drawn on dotted paper with the dots as lattice points. The focus of the activity was to tilt squares and try to find a general formula for the number of dots inside the square of a particular tilt, as the side of the square changed.

O n observing the triple (25, 125, 225) in which 125 is a perfect cube, 25 and 225 are perfect squares, and the three numbers are in arithmetic progression (AP), I felt that 125 is a very special perfect cube which is guarded by two perfect squares on either side at equal distance.

A surprising discovery we make is that 125 is guarded by two perfect squares in another way, namely: (81, 125, 169); here, 81 and 169 are perfect squares, and the three numbers are in AP as earlier.

In this article, I have argued that it is important not to bunch all the students’ errors as “careless mistakes” or “over-generalisations”. We can classify the errors to understand – (a) what is their mathematical source and (b) what could be the student’s thinking underlying such responses. This will help us in designing appropriate interventions for handling these errors in classroom.

Consider the following situation. A regular polygon of n sides is placed symmetrically inside another regular polygon of n sides i.e. with corresponding sides parallel and with a constant distance between them. We now have a 'path of uniform width' running around the inner polygon. Now the following is asked: By how much does the perimeter of the outer polygon exceed that of the inner.

In this article, I provide a relation connecting the lengths of the tangents from the vertices of a right-angled triangle to its incircle and ex-circles, in terms of its inradius and ex-radii. I give a geometric proof as well as an analytic proof.

As I was going through Evan Chen’s Euclidean Geometry for Mathematical Olympiads, I came across in this remarkable problem.


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