A few weeks ago I got this question from Paula Beardell Krieg on Twitter:
Today I went through this problem with the boys – the difficulty of this exercise surprised me a bit. They really struggled to see how you could tell if an angle was 72 degrees.
Here’s the introduction. The boys noticed a few things about the picture and got some ideas with how to proceed:
Next we drew the two squares on a piece of paper and I let the boys explore the question. Here they struggled to make much progress beyond the things that they noticed in the first part of the project:
The thing giving them trouble was that they didn’t know any relationships between angles in a right triangle with a 72 degree angle. That left them completely stuck. Eventually they decided to measure the squares and found that they had something that looked like a triangle with side and .
Next we explored some of the ideas around triangles. After a little nudging from me they decided to measure the angles with a protractor.
Now I showed them my solution and let them see where the triangle comes up:
Finally, I let them play with two sets of triangles that I printed overnight. Two of these triangles are right triangles with 72 and 18 degree angles, and two of them are triangles. The question is -> are all 4 triangles the same?
Here are pictures (to scale) of the two triangles. You can see how similar they are.
First, the right triangle with a 72 degree angle:
Second, the triangle:
Tomorrow we’ll explore a second similarity between these two triangles. I found it playing around while I was making the triangles yesterday 🙂