One of the high points of this last academic year has been volunteering in Katya's 2nd/3rd grade English and Math classes. I was suitably rewarded for this just a few weeks ago with a delicious treat of an elegant mathematical argument from a 2nd grader (Katya's in 3rd). Here's the puzzle and its solution (names of individuals have been changed).

It all started with the teacher, Judith, posing a seemingly simple problem to the kids. She was teaching fractions that Thursday. In the latter half of Math class, the kids were each given rectangular cards, and asked to imagine they were cookies (or pizzas). Since classes at Ohlone are always mixed between two grades, the problems are always posed in a range of difficulties. With this one, the kids had to divide up their cookies into 2, 4, 8 and 3 equal parts. The more different ways you could slice it up, the more points you get.

The puzzle I want to talk about concerns the division of the rectangle into 4 equal parts. Most kids did it the conventional way, by which I mean folding the rectangle in half, first vertically, then horizontally, and cutting along the creases. Edgar (name changed), however, adopted a different tack. He folded the rectangle along its two diagonals.

The cool thing about this is that it's not immediately obvious that this way of cutting the rectangle produces 4 equal cuts, especially not to 2nd or 3rd graders. See the figure below, for instance:

Can we tell immediately that Slice A is the same size as Slice B? Edgar claimed they were, but wasn't able to explain it - he just felt it in his gut. Judith, awesome teacher that she is, immediately recognized the opportunity to enrich and picked up on this issue, calling the attention of the entire class to the problem. She wrote it up on the white-board and named it "Edgar's challenge". Anyone who was done with the rest of the problems was welcome to give it a shot. Can they show that the four slices were the same size?

Continue reading "A proof from the mouth of a babe" »