Midpoint Polygons
Rose Mary Zbiek asked her students to investigate midpoint pentagons, the figures obtained by joining the midpoints of a pentagon in the same order. What is the ratio of their perimeters? What about their areas? Computer experiments using Geometers' Sketchpad initially suggested that these ratios should be constant, but when attempts to prove this conjecture failed, further investigation led to the opposite conjecture, namely that this ratio depends on the shape of the pentagon. The object lesson was that technology can lead students to make conjectures, and it can also provide evidence to show that a conjecture is false. [RMZ]
This investigation formed the basis of an example in the Standards, in dialogue
form, showing how students might discover preliminary theorems and then go on to investigate more general cases, make conjectures, and accumulate evidence to see whether a conjecture is likely to be true or false. [NCTM]
