For many years extra-mural classes have been held on Wednesday afternoons at the University of Birmingham for the benefit primarily of teachers of mathematics. Part of one of these courses for sixth form teachers involved taking A level questions and using them as starting points for further enquiries. The intention was to illustrate how almost any A level question can yield something fruitful, beyond its straightforward solution, if one perseveres with asking ancillary questions and exploring alternatives. The approach was presented as a possible means of encouraging students to adopt more divergent thinking than the solution of A level questions generally produces.

Decimal expansions

Every real number x has a decimal expansion of the form

x=±N.n1n2n3…,

where N is either zero or a positive integer and each ni is one of the digits from 0 to 9. The notation N.nln2n3… is an abbreviation for the infinite series

Most students in their undergraduate careers ensure that they have many examples of groups at their finger tips. These include not only whole systems such as cyclic groups and symmetric groups but also specific examples, in particular groups of small order. These latter groups are useful for trying out new ideas as soon as they are encountered, such as the bogeys of cosets, normal subgroups and factor groups. If we consider only groups of order 7 or less, then we already are furnished with nine examples of cyclic, non-cyclic, abelian and non-abelian groups.

The differential triangle of Leibniz for a real function f is found by taking an increment dx in the variable x, finding the corresponding increment dy in y = f(x) and drawing the ‘triangle’ in Fig. 1. Here ds is the increment in the length of the graph, where

and the derivative of f is