Edited by Doug French and Charlie Stripp

A Book of Ideas for Teachers of Upper Secondary School Students

The need to be sure that mathematical results are true makes the idea of proof vital to the activity of the mathematician. It is right therefore, as an important aspect of the school mathematics curriculum, to focus on proof, with its accompanying need to understand and be able to generate chains of logical reasoning. Concerns have been expressed that this important aspect of the subject has been relatively neglected in the school curriculum in recent years. As a consequence many students entering higher education have not developed either a sufficient appreciation of the importance of proof or the skills of understanding proofs and of generating their own proofs. However, the idea of proof has an important place in the mathematical education of all students. We hope that the book will make a small contribution to making proof more accessible and enlightening to a greater number of those who study mathematics.

The book has been written to discuss some of the issues related to learning about proof and to provide a source of ideas for teachers to use in the classroom. The level of the material is for the most part appropriate to students in the final three or four years of secondary education and includes ideas of varying levels of difficulty. The book has been designed to be dipped into: individual chapters and sections within chapters are largely independent of each other. Questions and exercises are provided throughout, with detailed commentaries in the final section of the book. In addition to providing ideas for teachers the book also provides a valuable resource to which abler students can refer, both to extend their understanding and appreciation of particular topics and to broaden their mathematical education. Permission is given by the Mathematical Association for purchasers to photocopy relevant sections of the book for use in their institutions.

For a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Pythagoras' Theorem is one of the most famous theorems in mathematics. Although it is commonly named after Pythagoras, it was known well before his time both in China and, at least in some particular cases, to the Babylonians.

PYTHAGORAS

Pythagoras came from the Greek island of Samos and lived in the sixth century BC. Little is known of the man, but he is thought to have founded a group known as the Pythagoreans whose ideas were part mystical and part mathematical, extending to number as well as geometry and the famous theorem which bears his name. Triangular numbers and the golden section are examples of familiar ideas that interested the Pythagoreans.

Proof 1

The diagram immediately suggests that a² + b² = c², by considering the equal areas of the squares. The right-hand diagram can be obtained from the left-hand one by translating the top right triangle to a position adjacent to the one at the bottom left. The other two triangles can then be translated to form the rectangle in the top right corner.

Alternatively, a more formal proof can be based on the left hand diagram:

Each triangle has area ¹/₂ab. The total area of the four triangles is 4(¹/₂ab) = 2ab.

Each square has area (a + b)².

Then, from the left hand diagram: (a + b)² = c² + 2 ab.

Multiplying out and simplifying gives a² + b² = c².

Proof 2

In this diagram four congruent right angled triangles are arranged inside a square with each hypotenuse, of length c, lying on one of the sides. The sides of the small square in the middle are of length a - b, where a and b are the lengths of the other two sides of the triangles.

Since the area of the large square is equal to that of the small square together with the four triangles:

(a - b)² + 2ab = c²

On multiplying out and simplifying Pythagoras' Theorem follows:

a² + b² = c².