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 …
Once has been defined as the ratio circumference to diameter, the area of a circle must be . A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas. A circle is cut up into 6 sectors which are then rearranged into a near rectangle, What doesn’t look too close …
There’s always space on the inter-web for another proof of Pythagoras’s Theorem. Here’s one that uses the following equal areas property of parallelograms. This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom. The squares on the …
Parallelograms generalise Pythagoras to any triangle Read More »
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 …
First definition of triangles from Euclid’s Elements Book I (Fundamentals of Plane Geometry Involving Straight Lines), Definition 20.
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.
