Today I wanted to start out Geometry with a challenge problem. I was anticipating the kids getting frustrated and after 5 minutes, and then I would show them how a simple theorem/idea they learned would help them with it. In essence, I was thinking of this as a 7 minute start to the class, ending up with me illustrating how a simple theorem can lead to beautiful results.
The question was to do what they could do decide whether the sum of the marked angles any star (like the one below) summed to a constant number. And then deductively prove what they found out.
I thought they would get frustrated after a few minutes, and I would show them a clever solution and move on. (I normally don’t “give” solutions, but I wanted to forge forward with the curriculum, and this was just a nice result I wanted them to see…) After a few minutes, I asked if they wanted to continue working on it, or if they wanted a hint, or if they wanted to see a solution and move on. They wanted to continue. And none of them wanted a hint.
What ended up happening was the kids were hooked. Two of them took out geogebra and started finding out that the sum is always a constant 180 degrees. I shared that result with the kids. The rest of the kids started labeling things in the diagrams they knew — especially using vertical angles.
After a few minutes, I gave them a hint: use the theorem they previously proved (an exterior angle of a triangle is equal to the sum of the two remote interior angles of a triangle). With varying additional nudging, three of the four groups got it, and the last one was very close to getting it.
What was interesting to me was that most students got to the very end, and had the right information on their diagrams, but needed a final nudge to see why their labeled diagram showed what they wanted it to show. (To nudge them, I simply asked them to remind me what precisely they were looking for… and then I said their diagram actually has enough information to show that… when they saw it, their faces lit up!)
What was going to be a brief start to the class turned into a rich almost full day of problem solving.