# Month: May 2016

## Parallelograms generalise Pythagoras to any triangle

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 …

## The Ghosts of Departed Quantities and the difficulties of teaching and learning caculus

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus.  The infidel was probably Halley (of comet fame) or Newton. If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man. Quotes include: …

## Euler and the Properties of the Second Derivative

The summer examination season sees pupils searching for maximums and minimums on their text papers. This long-standing pursuit was initiated by the likes of Newton and Leibniz in their calculus. It can be all too difficult to think about for some: “our modern Analysts are not content to consider only the Differences of finite Quantities: …

## Yachting and similar triangles

A yacht has two parallel masts which are both perpendicular to a level deck.  One mast is 12 m high, the other is 8 m.  Stays are rigged from the head if each mast to the foot of the other.  Find the perpendicular height of the crossing point above the deck. Finding the height of …

## 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 …

## Exdexcel Formula Sheet lays Pooh Traps for Further Mathematics Pupils

FP3 integration standard forms could lead unwary pupils and teachers into a Pooh Trap. The following issue arises each year as the finer points of this course get sharpened. When consulting the Edexcel formula sheet to find the integral, two anti-derivatives are given, Some might be tempted to infer that these functions are equal, this …