We’ve been working on transformations in the coordinate plane in two of my classes, and in each that work is leading us in different directions. The best part about writing my own curriculum is that I can orchestrate these mathematical story arcs with themes and threads and a plot progression.
In Math 2, we finally finished our work with probability, elections, and fairness. I say finally in spite of really enjoying the work we did. It was just a long long long unit. The first task I assigned was a flag enlargement project. Each student was given a very small flag, a scale factor, and a huge sheet of graph paper with a 1-inch grid. For some cross-curricular action, I asked them to look up the origins of their nation’s flag. It turns out that most of the time red is on a flag, it is symbolic of bloodshed. It also turns out that all of my red markers are completely dead by now.
This experience with flags leads perfectly into similar figures, which leads perfectly into proofs. See what I’m saying about a story arc?
Determining if two triangles are similar isn’t the most exciting thing, yet I’m very excited about it. Remember all of those rules, SSS, AAA, and SAS? These say that even though being similar means that every single angle is the same and every single side length is proportional, with triangles you can figure this out by testing just three things instead of everything. Because of their work with the flags, they had the exact prior knowledge to understand this concept. Most of them had not measured angles (nobody except for the Seychelles) and yet their angles were the same – that’s SSS! The case of the Seychelles verifies AAA. By the time we got to SAS I was looking pretty credible, plus they had fun with some SASsy puns, which I will consider an emotional connection with the mathematics.
I gave them a fairly boring worksheet of similar triangle practice where they had to determine if two triangles are similar. Examining angles and ratios (I used the language “is the scale factor the same for every pair of corresponding sides?” to connect to dilations) and using AAA, SSS, and SAS to justify their conclusions was secretly their very first exercise in proof.
And let’s talk about proof for a minute. The way that mathematical proof is taught in school is ridiculous and everybody knows it. Lucky for me, I have the leeway to not only create a plot progression in my class, but also completely abandon traditional proof pedagogy. I’m skipping these guys up to more college-level proof. It’s not more difficult but it is more logical, more interesting, and more like the way that mathematicians actually operate. Next week we’re going to start on proof by mathematical induction, a highly algebraic proof type. Our first exercise: prove that adding any 3 consecutive numbers results in a sum divisible by 3. I love this because you can actually test it out and convince yourself that it should be true. And then the proof. I’ll let everyone know how the rest of this goes.