Here I'll summarize another excellent and accessible activity from Allan Kay. [The Real Computer Revolution Hasn't Happened Yet pp. 13-14] This also doesn't require a computer. On its own, it doesn't excite me as much as some of his other examples, but it does set up some important experience for other exercises.
Students are asked to measure the circumference of a bicycle tire and are surprised to get different answers from different materials. Most math activities set children up to expect exact answers (true for most teachers for that matter). This math exercise is really a science experiment in disguise. The students get real answers instead of ideal mathematical answers. In this case the teacher was also mislead to expect an exact answer by the measurements stamped on the tires by the manufacturer. Measuring the tire uninflated gave a different measurement than measuring when inflated. This stirred up questions about measuring under different pressures. The measurements will likely differ at different temperatures too. One of Kay's team contacted the manufacturer and eventually discovered that the engineers don't even know the exact circumference and diameter. "We extrude them and cut them to a length that is 159.6 cm ± 1 millimeter tolerance!"
Kay mentioned Mandelbrot and fractals almost in passing with this example, but that connection is worth describing in more detail, particularly for those who may be unfamiliar with Mandelbrot's work. Mandelbrot published a paper in Science in 1967 entitled "How long is the Coast of Britain?" Wikipedia's entry on Mandelbrot's article has a nice series of images which illustrate what's happening in this measurement exercise. If you measure the coast of Britain with a long ruler, you will get a shorter measurement than if you use a short ruler. Hopefully it's obvious that there's a connection between Mandelbrot's article and this exercise of measuring a bicycle tire. We'll come back to fractals in another post.