Finding unusual topics for teaching geometry is a welcome challenge for Peter Ransom.
Part of the buzz I get from teaching maths at The Mountbatten School, an 11-16 comprehensive in Hampshire, comes from not knowing exactly what the day has in store. I believe that pupils need to make many connections between maths, its history and everyday life, and I try to incorporate some practical geometry with the theoretical to make it as interesting as possible.
There is much useful geometry, for example, in the type of sundial where the shadow cast by a person standing on a marked-out area indicates the time, such as the one in Lancaster. With a graphical calculator it is also possible to model this type of sundial on a calculator screen by using the mathematical functions to calculate the positions of the hour markers and using the statistical plotting functions to plot them.
Students can produce a working sundial with the aid of a staple to provide a shadow. The positions of the hour markers depend on the latitude of the observer. With a bit of geometry and simple trigonometry these can be calculated in the statistical lists of the calculator. Then, by plotting these postions on the calculator screen, the layout of the sundial is shown. To operate the sundial a gnomon (the rod that casts the shadow) has to be placed on its face. A staple fixed to the calculator with a bit of Blu-tak forms a gnomon of a suitable size and can be easily stored on a corner of the screen.
For key stage 3 pupils, the sundial work involves constructing models from a net, measuring and drawing angles, symmetry, Pythagoras' Theorem and interpreting a graph. At KS4, the sundial work involves ruler and compass construction as well as trigonometry and the aforementioned topics.
A second useful idea is based on teaching geometry through the use of old instruments. One such is the cross staff. This was an early navigational instrument, used primarily for determining latitude. It measures the angle subtended at the eye by a bar that slides along a staff. KS3 pupils make such an instrument, drawing angles to a scale before they are enarged to the working model. Pupils then use this model to measure angles outdoors - recording their results and working on scale drawings.
The instrument can also be used to measure the width of an inaccessible object. By lining up the bar with the object, then moving it along the staff through a distance equal to the bar and measuring how far you have to move to line up the bar again, the width of the object is ascertained. Results may be accurate to within a 1 per cent error. High-attaining pupils find it possible to follow the proof by similar triangles.
Additionally, the principle behind the cross staff gives rise to the following rule of thumb (the saying "rule of thumb" comes possibly from the fact that the end joint of the thumb is one and a half inches long, and was used as a rule in medieval times). If you line up your little finger with the horizon, then each finger's width represents about 10 minutes to sunset for the setting sun. As KS4 pupils investigate why this is true, they learn much geometrical modelling. They measure the width of the hand and length of the arm and use trigonometry or arc lengths to model the situation.
I find this of great use for revision purposes as it also links in with data handling. Data from males and females is collected and by plotting the data as box and whisker plots on the graphical calculator comparisons can be made.
Ensuing discussion on accuracy brings in other areas of maths, and helps make exam revision less tedious.
* Full working notes on both these activities can be obtained from Peter Ransom, 29 Rufus Close, Rownhmas, Southampton SO16 8LR, for pound;2 each or pound;3 for both, including postage.
Write-ups of the work can be read in Mathematics in School, a magazine published by the Mathematical Association. "Time for some mathematics" (November 98 issue) covers sundial work, and "Body Functions" (November 00) deals with the angle subtended at the eye. In October 93 an account of the classroom use of old instruments appeared under "The ACD of old instruments".
Peter Ransom teaches maths at The Mountbatten School, Romsey, Hampshire