This is one of those things that sneaks up on me. I never think it’s going to be as hard for students as it is.
Even though I don’t have a picture of it, some students also say that arc AB is 100 degrees. In other words, we know that something is double the central angle and that something is equal to it, but we don’t know what.
Every mistake points to something that students don’t yet understand as well as they could. In this case, what is it? It’s such a simple relationship.
For a lot of areas of math, we don’t have strong intuitions about the topic until we get our hands messy with it. That’s not really the case with a lot of geometry, where we can see the diagram itself and often have visceral reactions to it.
What does the above diagram suggest, to the untrained eye? Maybe it’s difficult to see that the inscribed angle must be smaller than the central angle it shares an arc with?
What I Did
This was one of our online days, so I asked students to find the error and respond to these prompts in the chat.
I asked the first question because I wanted to draw attention to the most important thing to notice — that the central angle must be larger than the inscribed angle. I figured, that would make it easier to remember that it’s exactly half the central angle. Then, I figured it would be good to draw attention to the arc and clarify that the inscribed angle is half of that as well.
During the discussion I remembered my favorite way of thinking about this: like a rubber band or a slingshot that is pulled back. If you pull it back really far, doesn’t the angle on your finger get tighter? That’s what’s going on here too.
After calling on a few students to explain their responses, I asked them to use these ideas to try a similar problem.
I guess the human eye isn’t really so sensitive to differences in angles! Makes me wonder whether I should incorporate some estimation activities into this unit. Would it be useful to estimate the sizes of the angles before learning any of the theorems, just to focus on the visual skill of seeing the inscribed angle as smaller? That sounds like it could help.
I don’t know much about research on geometric misconceptions. How does one train their eye to see things differently? The way I usually approach it in my teaching is to start by teaching the true relationship, then to clearly identify errors when they come up. Maybe I should identify those errors more systematically, earlier in the unit? Now that I have these little “mistake analysis” slides, I could use them soon after studying the inscribed angle relationship.