# school maths

## Counting Pennies, Generating Functions and STEP

STEP I 1997, Question 1 Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins. In how many ways can you make up 20 pence using 20p,10p 5p, 2p and 1p coins? There is an interesting way of approaching this question using generating functions. A generating function …

## Differentiation From First Principles

The gradient of a smooth curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, . The idea behind differentiation is that as becomes very small, the gradient of tends …

## Carr’s Synopsis and Sixth Term Entrance Papers

Students in search of extension work in maths, particularly as far as Sixth Term Entrance Papers (STEP) are concerned, should peruse Carr’s Synopsis of results in Elementary Pure Mathematics. This book is famous for being the volume used by the autodidact Ramanujan to teach himself mathematics.  The book is merely a list of results and …

## Transformations of Sine

Functional notation and transformations is always tricky to teach and understand.  GCSE students will meet this in Year 11.   In general, transformations applied after the function are more easily understood:  or   Something very un-intuitive happens when the transformation is applied to the argument of the function:  or   with things stretching when they …

## The linear combination of a sine and a cosine is itself a sine wave

A linear combination of two functions, and is a sum involving constant multiples of the functions.  That is, where . So, in the case of and , we would have, . It is a slightly surprising fact that the linear combination of two sine waves is itself a sine wave.  The set of sine waves …

## third century wisdom of Diophantus gives us the essential truth about teaching and learning mathematics

The Google facsimile of the 1621 edition of Diophantus’s Arithmetica lays out, in original Greek and Latin translation, the essentials for success in teaching and learning mathematics.  Also included is the translation of the second half by Heath. When I first studied and learned to love mathematics I was beset by friends and family who stridently claimed that …

Edexcel’s January 2013 Mechanics 2 exam paper has massively intricate last part of last question to determine existence of third collision. Examiner’s report says not many candidates supplied an answer, which is not a surprise. This, and much more, is freely available on my exam bucket website: http://jped.co.uk/ExamBucket/exambucket.php.

Carr’s Synopsis – 19th century text for self taught geniuses – marks out this A-star topic for I/GCSE maths hopefuls.

Arithmetic Progressions, much loved by students of AS Core Mathematic, is an early result in my favourite Victorian textbook and has a good visual proof too.

## Theory of Equations

this fact about polynomials might be useful sometime …