Getting to grips with geometry
Geometry has been described as the branch of mathematics where the real world is studied. The problem in the past was that all too often physical reality was the last thing to intrude on the teaching of geometry which seemed concerned with an over-formalised system of proof using pencil and paper (geometry, of course, is described in national curriculum-newspeak as Shape and Space in case anyone is put off by thoughts of Euclid).
There is today no longer any excuse for not making the study of shapes, angles, lines, distances and the transformations they can undergo, as interesting as possible by using physical apparatus and computer programs. The British company Polydron introduced its geometric building system 12 years ago and has established a well-deserved reputation for the quality of its interlocking components made from high-quality ABS plastic. The pieces join together by means of a unique clip hinge which provides both strength and flexibility, and all sides are based on a 70mm unit length.
Most primary schools have some of the Polydron apparatus, but secondary schools have been relatively slow to take it up. The pyramid kit and the book should help to convince secondary mathematics teachers that here is a resource with many possibilities.
The pyramid kit contains a selection of equilateral and isosceles triangles and a base on which to build, along with a booklet written by Bob Ansell. It is possible to construct small pyramids or a large one in several different ways, including one from four small pyramids, four tetrahedra and an octahedron which is especially satisfying. The pieces lock together well and the base gives stability to the models.
The booklet suggests many ways in which the shapes may be used, including an exploration of Pythagoras and the relationship between scale factors, area and volume. There are also explorations of the volumes of tetrahedra and pyramids, and puzzles involving isosceles triangles and tetrahedra. Final-ly, the booklet discusses convex deltahedra, which are solids made entirely from equilateral triangles, and the stellated octahedron.
My one minor criticism is that pupils are asked at one point not to look at the facing page which shows the remaining deltahedra which they have to find. I think very few pupils, if they are struggling as I did with both visualising and constructing the models, will be able to resist the temptation. The approach of the booklet will encourage pupils to explore the situations, because of the many interesting questions posed, and there is work here to cover all of key stages 3 and 4. Of course because making all the models suggested in the booklet involves several of the senses, understanding of the relationships and patterns is more likely to occur.
Mathematics with Polydron contains 35 photocopiable worksheets with accompanying teachers' notes. It is based on the School Geometry Set (Pounds 116.50 plus VAT) which comprises large quantities of equilateral, isosceles and right-angled triangles, squares, hexagons, pentagons and two solid protractors, packed in a solid case with a storage tray.
A few of the activities in the book are the same as those in the Pyramid booklet, but there are many others which explore both two- and three-dimensional geometry. The general approach is again that of exploring situations by encouraging the pupil to answer "can you?" and "what if?" type questions. It is pleasing to see that the authors have taken the opportunity to link geometry with algebra by using shapes in investigations such as Euler's Law and building staircases. There is also a good section on developing imagery, where pupils are encouraged to conjure up images and work in a mathematical way with them.
As with the pyramids kit, there is only one minor criticism I would make. I am not quite convinced that using the Polydron pieces to explore tessellations is sensible: the hinge projections on the sides of the pieces mean they do not fit together exactly at a vertex without gaps.
This book provides an excellent illustration of the power and versatility of the Polydron pieces and should prompt pupils and teachers to explore even further.