Sunday, April 3, 2011

What We're Doing with Proofs

Since some of you asked, here are some proofs I gave / plan on giving to my 9th-grade kiddies. We've been doing proofs using a top-down approach for a little over a week now. In terms of learning modes, I switch off between circulating and looking at their proofs while they work on them in groups during class, going over/highlighting key parts of some proofs on the board after they've all more or less tried / "completed" them, and collecting proofs to fakely grade them at home so that I could provide more detailed feedback.

I say "fakely grade" because I told the kids that these grades are not going into their grades but merely serve as feedback for them and for me, to see how they're doing on a "test- or quiz-level expectation." The 9th-graders are LOVING the fake grades, and take the fake grades way more seriously than I would have thought. I guess it's something along the lines of Shawn's observation that teenagers just want you to hear/see what they can do. They don't want you to actually judge them.

Anyway, since it was a bit of work to pull these proof exercises together, and I'm liking them all so far, I thought I'd share them.

The first thing I did was to give the kids a reading from Proofs for Dummies that gives them a super kid-friendly intro to what attitude they should take toward constructing proofs. (I'm not implying that my kids are dummies, obviously. This was the most kid-friendly article about proof-writing out there, period.) Following the reading, we looked at one example proof and highlighted reasons vs. statements to start to "see" what a proof structure looks like. Then they did one proof -- showing that when you draw two overlapping triangles with parallel sides, then the resulting triangles will be similar (by the AAA Similarity Theorem*). That stuff is all here: Proof Packet #1.

*The AAA Similarity Theorem is missing from Packet #2, which I handed out at the same time as Packet #1 in order to allow kids to reference the common theorems right from the start. I had the kids add it by hand.

I noticed while I was making the answer key to Proof Packet #1 that the proof for Midsegment Theorem is actually really algebraic and difficult, so I told the kids to skip it in Proof Packet #1 and to go on to Proof Packet #2 to try their hands at the easier triangle congruence proofs first. They worked on some of it in class, in groups, and the last triangle congruence proof they took home and did on their own. I collected that last proof of Proof Packet #2, gave it a fake grade, and went over it step-by-step upon returning it to the kids. (You should also note that the list of theorems in the front of the packet are not complete. Kids were able to use them to cover the majority of cases, but in some cases I still needed to have them add missing theorems to the list. That's why doing the answer key yourself beforehand is a good idea.)

After Proof Packet #2, I decided that all kids are now ready to attempt the algebraically hairy Midsegment Theorem proof. So we went back to Proof Packet #1, spent 2 days on that in class, and I had the kids bring me back fresh copies the next day to submit as a project. --Sorry but I won't get around to grading the projects till later today, so no student samples... yet. :)

When the algebraic proof "project" was all submitted, I handed out Proof #3 packet and had the kids read all of the circle definitions and basic circle theorems before we discussed each of them as a class. I gave them illustrated definitions on the board when I went over them, and made sure that every kid understood each part. I also had them add Thales' Theorem and Isosceles Triangle Theorem to their list of reference theorems, since I had forgotten to include those the first time and they're needed to complete some of the proofs. The kids then began working on the circle proofs in groups in class. That's where we left off last week, and I plan on picking up from there on Monday.

So far, I'm really pleased with how it's going. I think the kids are doing fabulously with all the increasingly difficult proofs. For both my regular and honors kids, after this third set of proofs (which is mostly circle proofs, plus one Triangle Angle Sum proof), I plan to switch gears and have kids work on some algebra problems to practice applying what they now know as common properties of chords and angles inside a circle. These easy circle proofs I pulled from here, by the way. Thank you Oswego School District!

Then, for the regular kids, we'll move on and do a series of simple algebraic proofs (Thanks, Guillermo!), and the honors kids will first do a series of advanced circle proofs as "borrowed" from Mr. Tytler of Rochester, NY, before moving on to those same algebraic proofs as the regular kids.

I'm excited. I can't prove it, but I think we're doing good stuff. I'm sure when I teach proofs again next year (or whenever) I'll want to revamp the whole thing again, but for now, I'm really loving the layers of logical thinking my kids are showing. It's miles beyond what I could pull out of my kids last year!