In other words, without any formal algebra, we have a locus that turns a bit for every step forward. You could model that with a velocity vector and an acceleration vector, if you like (or several other ways all relating to tangents as limits of slopes.)

I think that Alan Kay and Kim Rose’s car game is similar. I thought a rolling glass was a bit boring, but it is a surprisingly rich vein for investigation.

The cup is rolling, or to be more precise, rolling without slipping. Imagine that the rim of the cup is covered in paint so as it rolls, it paints an arc onto the floor. We only roll it a few inches for this particular thought experiment (less than the circumference of the rim).

Now imagine we roll it the same distance, but this time there is no paint on the cup, but there is paint on the floor. If you measure the arc on the floor in the first experiment, and the arc on the cup in the second, they should be the same length. If the cup were sliding as it went, one arc or the other would be longer.

Now, the bottom of the cup paints out a similar arc. Because the cup is solid, the bottom of the cup must rotate at the same speed as the top. And yet, by the time the cup has made a full rotation (around its own axis, not around the entire circle) the upper arc is equal to the upper circumference, whereas the lower arc is equal to the lower circumference. How do you have two circles rolling at the same rate, and yet one goes further in the same amount of time? Well they have the same *rotational* rate, but different linear rates. I think that basically proves it won’t roll straight. But why a circle? Why not an ellipse or a hyperbola? I guess because the problem doesn’t change once you’ve rolled the cup a few inches. The same thing applies, the top is still rolling faster than the bottom by the same factor, and so the curvature is constant.

And look! We have a new definition for a circle: It’s the shape whose curvature is the same everywhere. Of course that begs the question of how exactly we define curvature, potentially leading us to some circular logic. Sorry for the pun.

Another question that could come out of this lesson: Why don’t the wheels on a car slip when you make a turn? The answer is a clever little device ( http://en.wikipedia.org/wiki/Differential_(mechanical_device) ) your students may not have heard of, which might lead them to wonder how many other things are going on in their car that they simply never thought about before.

Your demonstration seems to be missing a crucial step. How did you connect the slant height of the cup’s cone (which you ably calculated using similar triangles) with the cone’s roll radius?