# pure mathematics

## 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 …

## Area of a circle – proof by exhuastion

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 …

## 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 …

## 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: …

## depression near Greenland – logarithmic spirals in nature

Logarithmic spirals frequently occur in nature.  Is this such a manifestation? The general equation for a logarithmic spiral is as follows. Changing the variables and produces spirals of different qualities.  The is really an enlargement scale factor but the controls how the spiral grows per revolution. If is more representative of an ammonite then seems …

## 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.

Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams.  School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.

FP3 2014 gives students the chance to prove the surface area of a sphere is 4 pi r squared.  Interestingly four time the area of a cross section through the centre and the derivative of the volume.  Pupils with strong maths general knowledge from GCSE would know what is coming.  TOP TIP do as much …

Carr’s synopsis reveals much about cyclic quadrilaterals and the cosine rule.  This Victorian text book is packed with sparse facts.  Genius Ramanujan worked through these proving each one to himself.  Modern students would extend their skills by doing the same.  I have added some connections and associations to this sequence of facts to get you …