Explaining and Exploring Mathematics is designed to help you teach key mathematical concepts in a fun and engaging way by developing the confidence that is vital for teachers. This practical guide focuses on improving students’ mathematical understanding, rather than just training them for exams. Covering many aspects of the secondary mathematics curriculum for ages 11-18, it explains how to build on students’ current knowledge to help them make sense of new concepts and avoid common misconceptions.

Focusing on two main principles to improve students’ understanding: spotting patterns and extending them to something new, and relating the topic being taught to something that the pupils already understand, this book helps you to explore mathematics with your class and establish a successful teacher-student relationship.

Structured into a series of lessons, Explaining and Exploring Mathematics is packed full of practical advice and examples of the best way to answer frequently asked questions such as:

  • Do two minuses really make a plus?
  • Why doesn’t 3a + 4b equal 7ab?
  • How do you get the area of a circle?
  • Why do the angles of a triangle add up to 180°?
  • How can you integrate 1/x and calculate the value of e?

This book will be essential reading for all trainee and practising teachers who want to make mathematics relevant and engaging for their students.

chapter 101|2 pages


part |62 pages

11–14 years old

chapter 3|4 pages

Adding fractions

chapter 6|7 pages

Use hundreds and thousands, not apples and bananas!

Helping beginners make sense of algebra

chapter 7|4 pages

Angles and polygons

chapter 8|4 pages

Special quadrilaterals

chapter 9|4 pages

Basic areas

chapter 10|3 pages

Circles and π

chapter 11|4 pages

Starting trigonometry

chapter 13|3 pages

The difference of two squares

chapter 15|6 pages

Number museum

How many factors?

part |78 pages

14–16 years old

chapter 17|6 pages

The m,d method

An alternative approach to quadratics

chapter 19|5 pages

A way to calculate π

chapter 20|4 pages

Pyramids and cones

chapter 21|4 pages

Volume and area of a sphere

chapter 23|3 pages

Percentage changes

chapter 25|4 pages

Angle properties of circles

chapter 27|10 pages

Irrational numbers

chapter 28|4 pages

Minimising via reflection

chapter 30|5 pages

Farey sequences

Fractions in order of size

part |73 pages

16–18 years old

chapter 33|4 pages

Adding arithmetic series

chapter 34|7 pages

d why? by dx

Or what is differentiation for?

chapter 35|3 pages

Integration without calculus

chapter 36|3 pages

Integration using calculus

chapter 37|5 pages

Summing series

Using differencing instead of induction

chapter 44|5 pages

Compound angle formulae

chapter 46|4 pages

Fermat centre of a triangle