Active geometry

Formalised geometry can seem a meaningless set of hurdles. “To do geom” observes Geoffrey Willans’ schoolboy antihero, Nigel Molesworth (Down With Skool, 1958), “you hav to make a lot of things equal to each other when you can see perfectly well that they don’t”. Dynamic geometry software such as Cabrie-Géomètre II (CG2), a program developed in France and powered by the backing of calculator manufacturer Texas Instruments, offers the solution. It is an excellent platform for investigating detailed aspects of the Autograph models — as preliminary learning in advance, as subsequent consideration of observed phenomena, or as both in a refinement loop.

CG2 is a geometry processor adding to axiomatic Euclidean geometry the active, participatory element of transformational or analytic geometry. Here is an opportunity to discover for oneself, in a hands-on way, where the axioms came from. It allows fundamental components (points, lines, shapes) to be combined and moved in ways which obey geometric definitions. If a line is defined as a tangent to a circle at a particular point, for example, then the circle, line and point can all be freely moved around, the circle resized, and so on, but the line will remain tangential to the circle at that point. Additional constraints can be used for particular purposes, as can slider controls. A number of ready-made examples are provided, ready for instant classroom use. During our trial a physics teacher borrowed it and used two lines, a circle and an ellipse to demonstrate both the inverse square law and the cause of eclipses in a single pass.

At every stage, the software encouraged rapid explorative investigation whilst also pegging the mathematical representation back to a concrete reality comprehensible to the pupils.

[Contributed by AbsentCat]