Project work has become a major constituent of Applied Mathematics courses at most British polytechnics and universities. There is a general consensus among academics as to what constitutes a project, how to discover and vet suitable projects, what their value is as a teaching aid, and how a student’s project work should be assessed.

A question which often seems to interest and intrigue schoolteachers is to what extent A-level grades provide good predictions of final degree classification. It has been suspected by many university mathematics departments that the answer is ‘not very much’, and that other factors play a considerable rôle. This note provides some evidence in favour of this contention. The table below indicates what happened to 589 students (see Note 1) who entered a particular university mathematics department over a period of five years, 1971-75, and who were therefore expected to graduate in the years 1974-78.

The theory of projectiles moving under gravity is well known. The parabolic trajectories can be calculated and their envelope, as the direction but not the magnitude (u) of the initial velocity is varied, is the curve of safety, or bounding parabola. Where a given trajectory meets the envelope, the particle is moving at right-angles to its initial direction. In particular, the maximum range on a horizontal plane is u2/g and is attained by projection at 45° above the horizontal.

Of all mathematical problems, those concerned with probability seem to contain ambiguities more often than any others. In the majority of cases the reason is that too little thought has been given to the wording of the question, but occasionally the question sounds very sensible and it is only when a real attempt is made to solve the problem that difficulties in interpretation appear. I believe that this is mainly due to the loose way that the word ‘probability’ is used.

The two great monuments of Western science are Euclid’s Elements and Newton’s Principia. Both are severely mathematical works; both set forth cosmological theories; in their day both were considered daring and speculative; both achieved unprecedented accuracy in application, so much so that at one time they were thought to be based upon a priori truths of reason, and even today are preferred to experiment over a wide range of phenomena. These two theories remain indispensable as elements of modern scientific thought despite the fact that they were simultaneously superseded in 1915 when Einstein published his theory of gravitation, a theory even more mathematical and boldly speculative than its predecessors. (One must, of course, remember that the nineteenth century discovery of non-euclidean geometry, whilst destroying the belief that Euclid’s geometry was true a priori, did not shake the belief that it was true as a matter of fact.)

The rank size rule is an empirical law which describes part of the size distributions of towns within individual countries and regions. It is of particular interest to geographers; and it is one of many applications of mathematics which are now being introduced into school science and mathematics courses. Since a collection of topics does not necessarily add up to a satisfactory mathematics course, it is useful to consider their integration into more general courses. The rank size rule could be taught as part of a course in descriptive statistics and linked to a more general discussion of frequency distributions.

One card up the sleeve of many a teacher of mathematics involves the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,…, in which each number is the sum of the preceding two. These numbers and the closely related golden ratio (√5 − 1):2 have intriguing geometric and algebraic properties and appear mysteriously in nature ([1], 160-172). The main purpose of this article, which stems from a recent talk to prizewinners in the Scottish schools’ problem competition, Mathematical Challenge, is to advertise some less widely known occurrences. A reader interested in further references to any part of this article is invited to contact the author, who himself is grateful to several colleagues.