Just for fun you could give them the 2-dimensional version of the standing-in-a-train theorem. That theorem says that if you want to go to sleep while standing up in a train that goes along a perfectly straight track, there is some starting angle that will cause you not to fall over. (Proof: if the angle is almost all the way forwards, then you will fall forwards; if it is almost all the way backwards, then you will fall backwards; by the intermediate value theorem there must be an angle that leaves you still standing at the end.) Ian Stewart argues vigorously that the implicit continuity assumption (that your final position depends continuously on your initial position) is wrong, but you can just forget about that.

Now let's suppose that you are standing on a surface that can move horizontally in any direction. (Perhaps you are in a boat, say.) This time if you fall, your head will be somewhere in a circle centre your feet and radius your height. Assuming that the position you end up in depends continuously on your starting position, this defines for you a continuous map of the disc that preserves the boundary. By Brouwer it is not a retraction, so there must be a starting position that stops you falling over.

Even if the continuity assumption is not in the end justified, I think this is a good and amusing illustration of the theorem.