# a level maths

## Rabbits, Matrices and the Golden Section – nth term of the Fibonacci Sequence by diagonalising a matrix

Fibonacci Sequence Leonardo Pisano, or Leonardo Fibonacci, studied rabbit populations in 1202 in the following way. Rabbit couples (male and female) inhabit an island.  Each rabbit couple becomes fertile 2 months after being born and then begets a male-female pair every month thereafter.  If the population of the island starts with one couple, how many …

## Integration by Substitution

Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions. Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment. Try this, Let, then, Suppose that there exists a function g, of another variable , such that and …

## 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 …

## The Fundamental Theorem of Calculus

Integration is introduced as the reversal of differentiation i.e. in solving a differential equation, . The link between integration and area is often passed over and is the subject of the Fundamental Theorem of Calculus. [The following discussion can be adapted for a decreasing function or, piece-wise, a function which successively increases or decreases.] Consider …

## 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 …

## support for school mathematics exams

I have developed a database driven website which shares past papers, markschemes and my own ‘write outs’ for most Edexcel modules in mathematics and further mathematics: jped Exam Bucket I use this mostly as a resource for my own teaching; using it I can lay my hands of exam questions and solutions of different types …

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right  into the 21 century classroom.

The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle.  Later it is about infinite series or, even, differential equations.

Mechanics 2 angle of dangle top tip – don’t try and draw your shape at the dangle, instead, add an oblique vertical after. This example is from Edexcel Mechanics 2 June 2014. For this exam write out and others see: http://jped.co.uk/ExamBucket/exambucket.php.