## ABSTRACT

*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 3
*a*+ 4*b*equal 7*ab*? - 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.

## TABLE OF CONTENTS

chapter 101|2 pages

#### Introduction

part |62 pages

11–14 years old

chapter 1|2 pages

#### Decimals and multiplication by 10, etc.

chapter 2|3 pages

#### Multiplying and dividing by decimals

chapter 3|4 pages

#### Adding fractions

chapter 4|6 pages

#### Multiplying and dividing by fractions; and by 0

chapter 6|7 pages

#### Use hundreds and thousands, not apples and bananas!

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 12|3 pages

#### Square of a sum and sum of squares, leading to Pythagoras’ theorem

chapter 13|3 pages

#### The difference of two squares

chapter 14|2 pages

#### Another look at (a − b)(a + b)

part |78 pages

14–16 years old

chapter 16|3 pages

#### The difference of two squares revisited

chapter 18|3 pages

#### Negative and fractional indices

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 22|5 pages

#### Straight line graphs and gradients

chapter 23|3 pages

#### Percentage changes

chapter 24|6 pages

#### Combining small percentage changes

chapter 25|4 pages

#### Angle properties of circles

chapter 26|5 pages

#### Trigonometry with general triangles

chapter 27|10 pages

#### Irrational numbers

chapter 28|4 pages

#### Minimising via reflection

chapter 29|4 pages

#### Maximum area with given perimeter

chapter 31|5 pages

#### Touching circles and Farey sequences again

part |73 pages

16–18 years old