A triangle with mistakes. According to the student who made this triangle, the angles measure 101, 102, and 73 degrees.

You probably can’t read the writing on the triangle above, but the task was to cut out a triangle and measure all of the angles. It sounds fairly simple, right? What happened with it, I couldn’t have planned for.

This particular triangle was produced by a sophomore with reported interior angle measures of 73 degrees, 101 degrees, and 102 degrees. As the students meandered around the room recording triangle angle measures, several students stopped by this triangle to laugh, ridicule or otherwise comment on how silly it seems for a triangle to have interior angles that sum to 276 degrees.

But, the students also noticed that it wasn’t an isolated incident. Even so of the other triangles whose angle measures seemed a lot more reasonable still didn’t always add to 180 degrees. Some added up to 179 degrees, 181, 185, 178. What could be going on?

One student started the group conversation on accident when he asked: **“So, the one’s that don’t add up to 180 degrees, does that mean that they aren’t actually triangles?”**

As I turned the question back to the class, they weren’t exactly sure how to answer it. It looks like a triangle, but we know that all triangles have angles that add up to 180 degrees. This “triangle” doesn’t. So, is it a triangle-like thing, but not actually a triangle?

Which brought the class back to the red triangle shown above.

Student #1: “Well, this triangle is messed up.”

Student #2: “Messed up? Like how?”

Student #1: “Like, it’s a triangle. I mean, look at it. It’s a triangle. But it says that angle is 101 degrees, but it is an acute angle. So, the triangle could actually add up right, but the person who measured it just messed it up.”

The reason that I like this exchange is because it drives home the idea of math in theory and math in practice, which I made more explicit as a part of the big group conversation.

If I had “blah-blahed” about that, I would have reach 10% of them. Engagement in this activity was closer to 85% and I did way less talking.

That’s why I let the students explore and discuss.

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