Explaining and Exploring Mathematics

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.

This practical book will help you to explain key mathematical concepts to your students. Written as a series of example lessons, it focuses on teaching for mathematical understanding and helping students to develop skills they can transfer to other subjects. The book covers all aspects of the secondary mathematics curriculum for ages 11-18 and reveals how you can build on students' existing knowledge to help them make sense of new concepts and avoid common misconceptions. It is essential reading for all trainee and practising teachers that 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