# step maths

## t-Formulae and parameterisation of the circle

t-Formulae are used in integration to tackle rational expressions of tigonometric functions.  After a spell in the cold, when they were not included in some A level specifications, they are now back in sixth form lessons. It all starts with the subsitution, from which the following functions can be derived, ,  ,  . These derivations …

## 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 …

## 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 …

## Algebraic Form of the Modulus Function

Sneaky play with algebraic notation and function definition lays traps for scholarship students in school maths.

## Functional Transfromations – Core Maths into STEP

Linking core maths 3 functional transformations to Step algebra and solutions of equations.

The Natural Logarithm is an integration defined function. The shaded area under the graph is the value of the natural logarithm of the upper limit in the integration.

Pascal’s Triangle with binomial coefficients expressed modulo prime numbers in an ascending sequence.

## Ellipse, stuck at origin, rotates

Take ellipse: apply rotation to variables: to make new equations: then vary  to get rotating ellipse: note that addition of an  term only to the original equation, under certain limitations of coefficients, means a rotation only.

Summary of parabola facts, worthy of Victorian text book, attempts to exhaust obvious question contexts in Further Pure and STEP. Prove them!

Polynomial rule requires subtle proof by induction to establish it’s truth.  One for students of STEP or Further Pure mathematics.  Not sure this is the most elegant expression of this proof. Inspired by Carr’s Synopsis of Elementary Results in Mathematics.