Algebra

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 …

Before the beginning – Euclid’s Common Notions

This preamble from Euclid’s Elements is where mathematics education still goes wrong, even after 2000 years or so. Sound notions of what equality is are required as a bedrock of algebra.  Dynamic approaches of teaching equation solving, in which terms and numbers move and change sign, seem to work at first but lack the simplicity …

Conic sections in the sand give shipwrecked philosophers good hope

Conic sections in the sand give shipwrecked Socratic philosopher, Aristippus, good hope. I wonder if this ancient tale, in which the mathematics of planetary motion some 2000 years prior to it’s application in space travel, is used to signify the presence of intelligent life and therefore hope of salvation has some echo in our …

Conic Sections Tangent/Normal Locus

A level further mathematics question animated.  Mid point of Normal x-intercept and Tangent y-intercept traces out another curve as original point moves round an ellipse.

Hyperbola – Focus Directrix

Hyperbola – locus of points in which ratio of distance from a focus point to the perpendicular distance to a line (directrix) is constant and greater than 1.  Staple diet of school Further Mathematics.

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.