My students have always struggled to make sense of any sort of proof of the Pythagorean Theorem. It's hard math, but this year I have been trying to push myself to get clearer about why the hard things are hard.
I started making some progress on this when I started asking myself a series of questions during my planning time.
- Why am I good at reconstructing these proofs? What do I know that my students don't?
- If you were good at making sense of these Pythagorean Theorem proofs, what else would you be good at?
After thinking about these two questions, I realized that these visual proofs of the Pythagorean Theorem are part of an entire family of proofs. There are lots of proofs that require the same sort of analysis as these proofs, though they have nothing to do with the Pythagorean Theorem.
This quickly gave my lessons a new life. Here were some implications I drew from this realization:
- In these types of proof, we almost always make progress by describing the same area in two different ways.
- We usually get one of the ways of describing area from thinking about the shape as a whole and a second way by adding up the area of each of its parts.
(By the way, we were working on this activity.)
Figuring out where Pythagorean Theorem proofs exist in the mathematical family tree helped me clarify what I was trying to teach, and that in turn gave me ways to help kids along. I made these worksheets to draw out the connections between the Pythagorean Theorem proofs and other proofs in the "Visual Area Proofs" family.
If you want the files, they're here. I lifted the quadrilateral area problems from here and the dot problems from here.
The takeaway from all this, I think, is that it pays off to get specific about what mathematical knowledge we want our kids to have.
- Danielson has a similar moment where a student question prompts him to reconsider what's involved in determining the range of a function.
- One of the joys of teaching elementary math is that there's actually a fairly decent specification of what we mean by "fluency with arithmetic."
- This was another instance when directly telling came in helpful. Questioning was definitely important, but it was also important for me to emphasize the whole area/sum of partials parallel. And it was also important for me to encourage students to use this framework when they worked on other Pythagorean Theorem proofs.
- So, what do kids need to know to successfully make sense of these proofs? (1) a geometric interpretation of "a-squared" and so on; (2) a working understanding of the Pythagorean Theorem; (3) knowledge of how these "visual area proofs" work, as detailed above; (4) how to expand (a+b)^2 and do other binomial algebra; (5) how to find the area of triangles and various quadrilaterals. That's why it's hard math -- there's a lot of stuff you need to know, all converging in one problem.