Congruence Proofs

If ABCD is a parallelogram…Prove that angle A is congruent to angle E

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These are from my classroom, and a little bit of context might be helpful:

  • This is from a very strong Honors class.
  • Proof, right now, means something less formal in our Geometry class than it might mean in yours’. Proof doesn’t mean “Statements/Reasons.” Proof means offering an explanation for why something is true.
  • We do this because it’s just as rigorous without crushing the souls of anyone in the classroom. Look at the array and variety of reasoning going on in these proofs. By keeping things less formal, we’ve got enough breathing room to actually do some Geometric thinking.
  • As the year has been going on, though, we’re getting more and more rigorous. These exercises help reveal some sloppiness in the kids reasoning. These proofs fail by their own standards of explanation. I’m thinking that I’ll be printing these out and handing them back for discussion. This is exactly what my English teachers friend does with essays.

Feel free to comment wildly here, either on my standards, some of my bulletted statements, or about any of the student work.

6 replies on “If ABCD is a parallelogram…Prove that angle A is congruent to angle E”

I think I know what the Z theorem means — what a descriptive name! (Is it a theorem or a postulate in your room right now?)

I’m really into the wordless proof at the bottom (yeah transitivity!), and the extending lines proof above that (what a great Geometric strategy — auxiliary lines! Solve a simpler problem!). The algebraic proof is one that always calls to me when working with angles too.

Especially intriguing to me is the idea of these proofs being in dialogue with each other. In what ways to proofs that look different have similar underlying architecture. In what ways do proofs that seem to start off similarly have very different leaps and gaps and bridges?

I’d be curious to post them just like this (anonymously and close together) and have students use highlighters or post-its to Notice, Wonder, and/or note Similarities, Differences, and things they Wonder if They’re the Same or Different.

I wonder whether we asked a different question – is angle A congruent to Angle E or not? Explain why – we might get a better answer? “prove that it is” implies to kids that the work has already been done, so they tend to put it in at the beginning.

I’m struggling a bit to follow this one. You made it open-ended and unstructured to keep it less soul-crushing, but upon discussion, these students are going to find out they were all effectively wrong. As a student, I found open-ended and loosely defined math problems to be soul crushing, because that’s not what math is.

I think I’m overthinking this. 🙂 Really, the bit that I catch on is the statement “prove that angle A ~= angle E.” I think I catch on it because the imperative tense conveys an expectation that the students already know how to prove a statement like this. Your write up makes it seem more like the class is still exploring what it means to prove something, so that’s an expectation that the students can’t fulfill, and that’s where I would have been crushed as a student.

So maybe, instead of the imperative “Prove this.” I would prefer a prompt that matches the open-ended intent you seem to have. “If …, how would you prove to me that angle A ~= angle E.” Or even, “how would you explain to me.” I’d want to replace the imperative command with an invitation to explore and discover the concept.

It’s subtle word choice, but I think it’s vital in setting the expectation appropriately, which prevents triggering mathphobias.

“Your write up makes it seem more like the class is still exploring what it means to prove something.”

I actually have a really good sense of what this group thinks a proof is. (See here.) And we’ve also been talking a lot about how proofs need to do more than justify, they also need to explain why something is true. And we’ve gone through the ringer with triangle proofs (SAS, etc.) and dissection proofs. We’ve seen things go , and we’ve figured out how they’ve gone horribly wrong.

So, no, proof is not open to anything for these kids.

But, yes, their concept of proof is evolving, as it should. These are mathematically immature students. They’re going to have relatively immature standards of evidence. They’re going to be sloppy.

What I haven’t done is impose the standards of incredibly precise mathematicians on these immature thinkers. I’m not caging them into formal statement/reason proofs, because doing that is imposing an abstraction and level of rigor on people who aren’t ready for it yet. As the year goes on, we’ll get more and more careful and more and more rigorous.

It’s the difference between asking kids to write an essay, that we’ll then critique and improve together through a series of revisions, and offering them the skeleton of an essay to work from.

(As a sidenote, I think a little bit of clarity about angle notation would actually make these proofs a whole lot clearer.)

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