For each of these the prompt is, “find as many missing things as you can.” And what I like about them is that it gave my students to use congruent triangles alongside other sorts of deductive moves. We have ways of finding missing angles in a diagram, we have other ways of finding missing lengths. A congruent triangle argument is just like that — it’s another way of using what you know to know a little bit more.

The other nice thing is that in each of these diagrams, there *is *information that we can’t know. I think it was good for my students to experience that too.

For each of these my routine was to show the image, state the prompt, ask for a raised thumb when you figured something out, assign partners, task them with finding more things out, quickly listing some of the lower-hanging fruit on the board, then discussing whatever I found most interesting in each diagram.

I thought it might be nice to make a set of practice problems to follow-up on these, but I haven’t gotten around to it yet.

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These look really fantastic for connecting constructions to work on congruent triangles. Was that your intention?

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Not at all! I just wanted to make connections between proof with SAS/SSS/etc. and the type of problems that students more typically associate with math.

I’ve always found that to be a tough bit of learning and a kind of jarring transition — you go from solving problems where there’s always a single number as the solution, and now you’re writing these arguments.

So I was trying to mush together some of the angle/circle work we had already done with proof, which we were just starting when I used these problems.

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You might like the Sharing Skepticism routine I created and then shared last week at a Geometry PD. It’s essentially trying to bridge the same gap between students being able to solve problems and generate proofs.

The idea is that students solve a problem and then create an argument why their solution is correct with a partner and then we study those arguments together, and then (and this is a critical step) we decide with our partner which of the arguments presented was most convincing to us. The routine, as usual for the routines I use, ends with a reflection.

I think your “find anything you can” kind of problems would fit nicely within this routine, since each thing found by students can be argued for.

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Sorry, I missed a critical step => As we decide which arguments we prefer, we offer suggestions for how to improve these arguments (which I usually annotate with a different color) so that students, collectively, get some feedback about how to improve their arguments. Unfortunately, their opportunity to act on the feedback is limited to their reflections but if you do the routine often enough, students get to act on the previous feedback during the next time they construct an argument (especially if you have the posters up from the previous class).

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I took the second problem, and drew a point X at the intercept of FG and the inner circle. I asked my maths students (mega high ability 12th grade) to find the arc length FX. One of them got it in three minutes but the others are still puzzling 😉

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Is there a way to do it without using trig?

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Doubt it!

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