How a tiny connection in geometry made me feel like a master teacher


This is my eleventh year teaching. And I’m starting to realize that I may be finally starting to develop, at least partially, into a “master teacher.”

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I cringe a little bit at that description – I mean, I still forget to take attendance pretty much every class, and sometimes there’s still just no lesson planned and things are a disaster, and every year I look at my stuff from last year and want to throw it all out and start over because I hate it and every year I know that a student leaves my class liking math less than when they came in no matter how hard I try.

But yesterday, something happened. I had a lesson planned. For geometry. A pretty simple lesson, outlineable in about five sentences.

Before you can understand this lesson, you should know that we are working toward congruence proofs by spending time on parallel lines and corresponding angles and their ilk. Students are newly inculcated in the “don’t assume anything we haven’t discussed/proven” idea. Students have learned a small set of theorems at this point, including the theorems of the form “If two lines are parallel and have a transversal, the corresponding angles are congruent.” (and AIAs, AEAs, SSIs are supplementary, etc – the family.) However, they have NOT learned the converses of those theorems. Nor have they proved that “if two lines are parallel to a third line, they are parallel to each other.” We actually, at this stage, have no way of deciding that two lines are parallel without it being given.

So. Here’s the lesson. 80 minutes.

  1. Do a warm-up problem where students find lots of missing angles on a big figure, using vertical angles, corresponding angles, linear pairs, etc.
  2. Talk about why the converses of all of the angle theorems with parallel lines hold (so we can use congruent CAs to decide lines are parallel, etc). In the process, talk a bit about contrapositives because we already did an in-class exploration and decided “If two lines are not parallel then the corresponding angles are not congruent” is probably true (a postulate!) so contrapositive equivalence lets us decide “if two CAs are congruent, the lines are parallel” is a postulate too.
  3. Do two simple proofs (four-five steps) in teams on white boards using these theorems – one where only reasons are filled in, one complete from scratch.
  4. Practice some questions just about the identification step: “If I tell you $\angle 1 \cong \angle 2$ what lines can you prove parallel?” with lots of highlighting. Basically in “Y’all do, we do” style.
  5. Look at a few corollaries and think about why they must be true with no formal proof, including “Through any point not on a line is exactly one line parallel to the line” and “if two lines are parallel to a third line they are parallel to each other”.

Okay, so why is this lesson, which honestly follows straight through the book, mostly, evidence of my teacher masteriness? It’s the little things that aren’t in those lines above.

It all started with the warm up problem.. The warm-up problem I used is from Discovering Geometry and it looks like this.

They love this problem. The sense of satisfaction is just great. It takes them a while, but most of them get there.

The hardest angle in this problem is q. It’s hard for many of them because the corresponding angle nature of it is hard to see since there’s a non-parallel line in the way. Because it’s so hard,  can usually jump almost straight to it during discussion. Somebody with a good eye always gets it and can explain that q and b are corresponding with parallel lines so q = 116.

To which I get to say “How do you know those lines are parallel?” and watch them flail a bit. Because it’s not given. This was really a happy accident, but what a happy one! Those two lines are not marked parallel. They may “know” it’s true, but at this point they could not prove that q is a corresponding angle to b.

At this point, I am merciful and say”I think your logic makes sense, but let’s see if we can find a provable way to get there. Is there another way to find q?” And there is! You can use the triangle with m and the supplement of 169 and the supplement of q to figure out q. Somebody always sees that one too!

To which I get to say “How do you know the angles in a triangle always add to 180?” and repeat the process again.

And suddenly, BAM!, I have some motivation for the rest of the lesson content. Because even though we “know” that q is probably 116 degrees, we can’t really, honestly, be sure in the most rigid of geometry senses. Yet. Time to explore some new theorems!

That connection is small. But it made this lesson feel so much more smooth, and vital, and interesting. I did some other good moves too – the two proofs were a nice touch, especially since they were just the right amount of same-but-different so that students can start to see common patterns in proofs (“Translate the given statements! Watch out for the transitive property!”) – but this moment of connection from the warmup was the moment of greatness. And that sort of tiny thing, so hard to quantify or explain, is the kind of master teaching moment I will continue striving for. Maybe next time I’ll choose the perfect warm-up problem on purpose!


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