Clocks, origami, and sets

The first week in October, we did clock arithmetic. On a clock, if you count up to twelve you get back to the beginning, so 12 is basically the same thing as zero. Weird things happen in clock arithmetic. For example, it's possible to multiply two non-zero numbers (2 and 6, for example) and get zero. This doesn't happen with regular integers and real numbers!

The last couple of weeks we've been doing "magic tricks" with a compass and straightedge. We cut a line segment in half without even knowing how long it was, and we cut an angle in half without knowing the measure of the angle. These magic tricks are done in high school geometry. Magic trick 3, however, usually isn't done in school at all: while it's provably impossible to cut an angle into thirds using only a compass and straightedge, it can be done with origami paper!

Today we did some set theory. We created all whole numbers using only the symbols {, }, and a comma. Here's how:

0={} is the empty set

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

and so on. Since it's hard to make sense out of all those brackets, we can write the numbers they represent instead. Then we get something prettier:

0={}

1={0}

2={0,1}

3={0,1,2}

and so on, and so on...

Areas, volumes, and dimensions

This has been the week of finding areas. Renata's been working with the kids on finding the areas of funny shapes by decomposing them into rectangles and triangles and squares. I decided to approach areas from the point of view of units and dimension. If a square has side length 1 ft, what's the area of the square in square inches? Most of the kids said 144 very quickly, which made me happy (having had lots of college students who would have said 12). Then we went to volume, finding volumes in cubic feet and cubic inches.

Today we had a little time left at the end for more fun stuff, so we drew 4-dimensional cubes (and, in the case of one of the kids, a 3-dimensional pair of pants!).