Geometry brain warmup for high school students

The standard American high school math curriculum seems to regard geometry as something to be gulped down as quickly as possible, rather than savored and thought about in order to develop geometric intuition. Many bright high school students take geometry in 8th or 9th grade and their teachers then believe they are "done" with geometry.

Award-winning math teacher Pat Ballew, who taught in schools in Japan and England for 16 years, has a different perspective.

Mr. Ballew posted the following:

In the January, 1929, issue of the American Mathematical Monthly, there appears a problem submitted by J. Rosenbaum of Milford Connecticut. The problem begins, "It is well known that the radius of the inscribed circle of a right triangle is equal to half the difference between the sums of the legs and the hypotenuse." I ... suggest that the theorem suggested may be less well known now than it might have been in the past.

If it's been a while since you've taken geometry, and you're feeling uncomfortably rusty, see if you can prove this so-called "well-known theorem" by playing around with it, then look at Mr. Ballew's very nice presentation and discussion of several approaches to proving this theorem. Even if you came up with your own proof, you may find you get new insights by looking at his approaches.

If you liked working on this problem, there are more where that one came from.

Pentablocks and beautiful geometric puzzles

The NYC Math Circle facebook page asked:

What is the ratio of the blue area to the area of the whole regular star in this picture?

This problem has a very beautiful answer, but it's hard for most people to visualize, unless they have a lot of geometric intuition. You might want to try playing with it for a while before visiting the discussion in the NYC Math Circle Facebook page.

The Pentablocks I showed to the New York Math Circle Teachers summer workshop last week provide a very easy way to visually construct and/or confirm the solution to the problem above.



Pentablocks are a very nice way to develop your geometric intuition in understanding the relationships in pentagons, pentagrams, and related polygons, the golden ratio (phi), Penrose tilings (quasicrystals), and more. As their name suggests, Pentablocks are based on pentagons and pentagrams (five-pointed stars). All their angles are multiples of 18 degrees, and all their side-lengths are related to one another by the golden ratio, phi.

The more traditional (and easier to find) Pattern Blocks are based on hexagons. All their components have angles which are multiples of 15 degrees, and all their dimensions are related to one another by 1, 2, and the square root of 3.

You can use either set to tessellate, but the Pentablock tessellations are far more complex than the traditional pattern block tessellations.

I'll bring sets of both types of blocks to our Labor Day weekend picnic so you can explore more relationships with them.