1st Edition

# Explaining and Exploring Mathematics Teaching 11- to 18-year-olds for understanding and enjoyment

226 Pages 75 B/W Illustrations
by Routledge

226 Pages 75 B/W Illustrations
by Routledge

226 Pages 75 B/W Illustrations
by Routledge

Also available as eBook on:

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.

Introduction  Part I: 11-14 years old  1. Decimals and multiplication by 10 etc.  2. Multiplying and dividing by decimals  3. Adding fractions  4. Multiplying and dividing by fractions; and by 0?  5. Using patterns with negative numbers  6. Use hundreds and thousands, not apples and bananas!  7. Angles and polygons  8. Special quadrilaterals  9. Basic areas  10. Circles and pi  11. Starting trigonometry  12. Square of a sum and sum of squares  13. The difference of two squares  14. Another look at (a-b)(a+b)  15. Number museum: how many factors?  Part II: 14-16 years old  1. The difference of two squares revisited  2. The m,d method: an alternative approach to quadratics  3. Negative and fractional indices  4. A way to calculate pi  5. Pyramids and cones  6. Volume and area of a sphere  7. Straight line graphs  8. Percentage changes  9. Combining small percentage changes  10. Trigonometry with general triangles  11. Irrational numbers  12. Minimising via reflection  13. Maximum area for given perimeter  14. Farey sequences  15. Touching circles & Farey sequences again  Part III: 16-18 years old  1. Remainder theorem…  2. Adding arithmetic series  3. D why? by dx; or What is differentiation for?  4. Integration without calculus  5. Integration using calculus  6. Summing series: using differencing instead of induction  7. Geometric series, perfect numbers and repaying a loan  8. Binomial expansion and counting  9. How to make your own logarithms  10. The mysterious integral of 1/x  11. Differentiating exponential functions  12. Why do the trig ratios have those names?  13. Compound angle formulae  14. Differentiating trig ratios  15. Fermat centre of a triangle

### Biography

Christian Puritz studied maths at Wadham College, the University of Oxford and completed his PhD at Glasgow University, UK. Following on from his studies, he taught mathematics at the Royal Grammar School, High Wycombe, UK for more than thirty years. He currently offers home tuition for children of all abilities.

Colin Foster, University of Nottingham

There is certainly a pressing need for something that would assist school teachers in presenting mathematics in more discursive and mathematically rigorous ways, and that seems to be very much the focus of this proposal. However, none of these is similar in style to the proposed book. The conversation format is a unique feature, and the focus on mathematics and quality of explanation, depth of thinking is particularly pronounced. I could imagine the book appealing to teachers who use the materials (and books) provided by the UK Mathematics Trust, but they don’t have anything like this. Many teachers would benefit considerably from the advice given (implicitly) here about how to introduce and build ideas and tackle misconceptions.

John Woolmer, Winchester College, UK

I like the proposed material which judging by the sample chapters will be well presented, at times unusual, and always stimulating and clear.

Juan Perez, Former Teacher, UK

This proposal aims to cover a genuine gap in the market of books for (primarily) Maths teachers offering a rationale and an explanation for many of the facts given to students. The proposed book brings together some beautiful mathematics, making strong connections between the foundations of the subject and the facts that are distilled and eventually presented to students. Maths teachers will find many of the ideas developed in this book of great value as "connectors" between topics, laying the foundations and the rationale of much of what they present to pupils. In summary, I am very happy to recommend the publication of this book.