The Edge

“I like to be surprised. The argument that follows a standard path, with few new features, is dull and unexciting. I like the unexpected, a new point of view, a link with other areas, a twist in the tail.” – Michael Atiyah

This page contains a few activities that are not normally covered on any traditional Maths syllabus, and that do not really fit anywhere else. They encourage free reign of a number of ideas, surreptitiously practice some skills that are universally applicable, pique the imagination and are fun.

Rational sided shapes

Golden Petals – Suman Vaze                               Construction of the Golden section

This activity exists in two forms, one more guided than the other. Either one works and you can choose your way in depending on the students. It brings in algebra, geometry and general investigation skills. Results such as generalising the observation that a 5/2 shape is the same as a 5/3 shape is quite neat and leads to a simple proof. Apparently it is likely that the attempted calculation of the Golden Ratio in the 5/2 was the first discovery of an irrational number. See “Trouble in Mathemagic Land

Vedic Mathematics

The rich history of Vedic Mathematics and its mathematical techniques is fascinating. It is so different from the Maths taught in western schools ad seems almost magical. I have never yet met a mathematician who fails to go “Wow” when they see the methods for generating recurring decimals. This sheet is a very short introduction, but is is a rich ground for exploration. How and why do these things work? Can you prove them? There are plenty of books (including this one) on the subject full of so many more techniques and linking them carefully to the Sutras. There is a bunch of resources and tutorials on the parent site here.

The Hat Problem

This is a tricky problem. It combines Maths and Logic and needs several steps to get to a solution. My solution is attached, and I think there is more than one answer. I would be grateful for any feedback from anyone at all as to whether I have made a mistake, and there really is a unique solution. However students working in groups can do this, if they are used to collaboration, systematic and disciplined. I have used this as a whole school Maths competition, but it really only got any traction with the older ones. A better whole school problem is Sum and Product. See below

The essence of mathematics lies in its freedom.” – Georg Cantor

Sum and Product (there is no linking document here)

Sharing a Pizza by Suman Vaze               How would you share a square pizza between any number of people (3 or 5 shown here) so that everyone gets an equal share of the crust?

The problem is simple. “When is the sum of two numbers a factor of the product?” If you like you can give and example of a pair where it is (3,6) and one where it isn’t (2,8).

It is a great problem because it can be approached on so many levels. Trail and error leads to hypotheses that can be tested and proved. A spreadsheet search leads to lots of data to be analysed. An algebraic attack leads to a variety of results, but no real final conclusion. As a whole school Maths competition it works really well. I also use it as the first lesson of an Advanced Maths course to see what their investigative skills are like.

Two Infinities and Beyond

This is a video presentation introducing Cantor’s ideas of transfinite numbers. It is done through the Infinite Hotel analogy. It starts very simply but gets quite interesting laster on.