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.

The Angle in a Semi-Circle, Euclid Book III, Proposition 31

Ancient theorem of Euclid (Book III, Proposition 31) still a fundamental part of school maths.  The angle in a semi-circle is a right angle.

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.

IGCSE Transformations – Rotation

Rotations in space make good Saturday morning maths at the King’s School.

Great Circles and Spherical Triangles

In the geometry of mariners, where straight lines are Great Circles, the angle sum of a triangle is more than 180 degrees.

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.

Have got absorbed by the National Cipher Challenge.  Not sure what this animation varying the multiplier in an ‘Affine Shift’ add 5.  Functional grid shows transition from Caesar Shift 5, multiply by 1, through to multiply by 13, which is a useless cipher.  In fact any multiplier not co-prime with 26 can be seen to be not one-one, and therefore useless too.  Modular arithmetic, straight line graphs, ciphers, dotty patterns, ….