Golden Triangles

Megan Lonsdale asked on Twitter about some mathart ideas to decorate stairwell panels and fuel a neat first week for her learners. Sam Shah shared ideas and resources, including cellular automata which I've got to try with kids. In the course of the conversation, I realized I hadn't blogged about one of my favorite lessons of the past year. This was for a Festival of the Arts with Heather Minnebo, an art teacher who's always welcoming to me and my preservice teachers. (We've presented together, too.)

I have this standard framing for types of mathart lessons. They start from thinking about art as problem solving.

1. Art as the problem. (This lesson below.)
2. Art as inspiration. (Look at the effect this artist got... lets do math on it.)
3. Math as the problem. (Calder wanted his mobiles to balance perfectly...)
4. Math as inspiration. (Escher extending his tessellations to the hyperbolic plane.)

The inspiring math here is the golden triangle. It's such a great structure... The acute isosceles (1, $$\phi$$, $$\phi$$) decomposes into into a smaller acute isosceles and an obtuse isosceles. Or, equivalently, the (1, $$\phi$$, $$\phi$$) acute composes with the (1, 1 $$\phi$$) obtuse to make a larger acute. Here's artist Dusa Jesih playing with the structure. Here's some GeoGebra so you can play as well. 

For me these lessons can spend time as all the different types, depending on your objectives. Show the artist pictures, math as a problem, what makes these triangles fit together like that? What else do they do? (2) What angles do we need so that they fit together like this? (3) Show the triangles, what can we make with them? (4) But since this was a festival of the arts, I liked the idea of presenting an art problem. Look at these triangles, how they fit together, what can we do to make the different kinds of triangles show up distinctly when we put them together? (1)

With the fifth graders I brought a few cut out to show how they fit, and then together we drew a big one and started decomposing, counting up how many of each type as we cut it up, then did some noticing and wondering. They saw 1 & 1, 1 & 2, 2 & 3, then the surprising 3 & 5... maybe a prediction? 6! (4 was an anomaly.) 7! (We were every one but now we're skipping.) 8! (Are they, like, adding?) Who knows! (We were surprised once, now we know it's not a pattern.)

Digression about the Fibonacci numbers because what mathy person could resist.

Now the art problem: We're all going to make some of these triangles, and we know we need more of the acutes, but how can we decorate them to make them visually distinctive when put them together? Each of the classes debated different options, but each gravitated towards the same solution. Lines and curves, pictures and words, two different patterns, two different colors... but ultimately decided on warm and cool colors. (That had been a topic in art class in the past couple of months.) Some class discussion on what qualified as which. I had printed enough triangles for each learner to do two. (PDF).

When we had enough or time was running low, we gathered to try to put the flipped triangles together. Once they were taped, turn the whole thing for the dramatic reveal. Learners were curious about the reveal, happy of the results, and proud to point out their elements in the whole class mosaic. The assembly process is not automatic, and you can see that there was some difficulty making a perfect tiling. All in all, this one's a keeper, and I'll be looking for opportunities to try it or a variation.