Tuesday, November 24, 2015

Hyperboloids from Bamboo Skewers

I was fortunate to have a visit today from Dr. George Hart (sponsored by Math for America). George led students in an activity building hyperboloids from barbecue skewers and hair rubber bands. I combined my AP Calculus BC students with a colleague's AP Statistics students for this double-period workshop.

First George explained some of the basic geometry of hyperboloids. One interesting thing in particular is that, while the surface of the hyperboloid is what we would normally call "curved," each point on the surface is the intersection of two straight lines.

The plan was to model a subset of the surface using bamboo skewers. For the model to work, we had to keep a regular pattern of "in front" and "behind." George explained this.

Basic plan was to pair skewers in the middle with a rubber band. Then we would attach pairs of adjacent ends. All the while we must maintain the "in front" and "behind" orientation properly.

Students began work.

Once they reached 12 pairs, they brought the ends around to join in a cylinder.

At this point, each pair of skewers is joined by one rubber band in the middle, and each interior skewer is joined to a neighbor exterior skewer by rubber bands at the top and bottom ends. To make room for more joins, the "end" rubber bands are slid in towards the middle. We'll then pair the ends to the new next neighboring skewer with a rubber band

This process is repeated until there are seven or more rubber bands holding each skewer to seven (or more) neighbors.

The constructed hyperboloid can be spread, or collapsed into a bundle of sticks.

Combinations are possible.

A good time was had by all (for the cost of a pack of skewers and a pack of rubber bands)

Monday, November 16, 2015

Hands-On Activity Workshop

As a Master Teaching Fellow at Math for America I have the privilege of participating in a variety of meaningful professional development opportunities. One such opportunity began this evening, with the first of a 3 session mini-course led by Dr. George Hart of Stonybrook University.

George first led us through a few drawing exercises -- draw a cube, draw a cube in an isometric projection style, draw a truncated cube, draw an icosahedron, draw a truncated icosahedron. At each step he gave us some pointers on key tricks to make our sketch a bit better. For example, judicious selection of the "front" can make the rest of the sketch so much easier.

In the background, we explored the relationship between vertices, faces, and edges. The point was not to derive Euler's formula, but it was a nice aside. (faces + vertices = edges +2) The real point though was to prepare us for a physical construction.

With our truncated icosahedron sketches as a guide, we took CD-R media and zip ties and began building a physical model of the icosahedron. CD with shiny side out represented the vertices, CD with dull side out represented the edges.

A good time was had by all.