Paula Beardell Krieg’s 72 degree question

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 1, 3, and \sqrt{10}.

Next we explored some of the ideas around 1, 3, \sqrt{10} 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 1, 3, \sqrt{10} 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 1, 3, \sqrt{10} 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:

72 degree picture

Second, the 1, 3, \sqrt{10} triangle:

one three root 10 picture.jpg

Tomorrow we’ll explore a second similarity between these two triangles. I found it playing around while I was making the triangles yesterday 🙂

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Comments

One Comment so far. Leave a comment below.
  1. This is the most fun I’ve ever had watching my assumptions disproved! After having spent so much time staring at those angles, the surprise appearance of your 3D printed triangles was like experiencing a cameo from a movie star. Looking forward to the next posts.

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