What mistakes would you expect to see in the proofs of the problem below? Take a moment and make some predictions. You might find it helpful to know that this was part of an end-of-the-year exam, and that kids were able to use whatever proof representation they wanted. In other words, they could write two-column proofs, paragraph proofs or flowchart.

proof pic1


Here are the mistakes, pulled from the class set. Or maybe you feel uncomfy describing these proofs as “mistakes”? Maybe it’s better to say that they contain mistakes? Or that they are proofs that aren’t where we probably want them to be?

mistake2   mistake3


Do you feel comfy calling these “mistakes”? What would you call them?

What are the next steps for these kids? What would you recommend that their teacher do?

  • I’m surprised that neither of these students made any mention of angles at all. Especially given the vertical angles that should have jumped out at them. I’d applaud the recognition of congruent radii and I’d be pleased to see that there are reasonably smart decisions about which radii to compare. It would not have been tremendously helpful to point out that AE and BE are congruent, for example. The troubling thing to me is the complete lack of angle or arc relationships called upon here.

    • Hao

      Maybe it’s the fact that vertical angles jumps out is what caused students to leave it off. It’s *so* obvious from the picture that the triangles are congruent (once you establish the sides are the same) that these students fail to explicitly mention it?

      I think this kind of “mistake” (incomplete proofs) is actually fairly common. Perhaps if we trained students to constantly try to come up with counterexamples to “proofs”, they would be better at checking their own work for inconsistency/completeness?

      • Laura C.

        That sort of thing always messed with me, too. “Why do I have to prove it, when it’s obvious by looking at the picture?”

        There should be a huge neon sign in every math classroom: “GEOMETRIC DIAGRAMS ARE NOT TO SCALE.” “Because it looks like it” is a common non-proof.

  • I would initially characterize the first one as not having sufficient explanation: I want to understand what is meant by “take up maximum space”. It’s possible that what is to be learned is how to express more of the thinking represented by that phrase. This solution might reflect a complete proof but not adequately communicate it.

    For the second I would say that the proof is incomplete/interrupted. I am interested in this case to know what they would say next in their proof.

    I would not yet characterize either as mistakes. Which may be different than asserting that there were no mistakes made by the authors.

  • Perhaps the students did the best they could with an impossible proof. Unless we are also *given* that AD and BC are lines, we can’t possibly use vertical angles. (Never trust the drawing!) So really, the only thing we can possibly know is that the radii are equal.

    • Interesting! Do you think that was something that the kids were thinking about when they were working on the proof? I’m not sure, but I also know that just as teachers sometimes assume things about diagrams, so do kids, and maybe all parties involved (somewhat sloppily) saw these as lines.

      • I doubt the kids noticed that they weren’t “given” the lines. Because a student who really understood that would probably have made a more or less (depending on the kid) snarky comment and then continued with some form of “Assuming AD and BC are diameters, then we have vertical angles and SAS…” But these kids didn’t try to get any of the typical acronymic correspondence relationships.

  • Remind me of my school days, this post refreshed alot of memories.