So I threw the triangle congruency rules at my kids. Being smart kids, they dutifully memorized them, and some even figured out what they meant. Kind of. But they didn’t get it.
It’s the sort of thing you want to use construction for – to let them dig in and experiment and try for themselves to see what works.
The problem is that traditional construction (using only a straightedge and a compass) works because of these exact congruency rules. Trying to explore something while you’re standing on it is difficult, if not downright impossible.
Fortunately, I didn’t want purity (yet), I wanted understanding. There was no need to limit myself, or them.
I dug into that stack of cardboard backing that I get about a dozen of every time we do a district assessment, (which after about 5 years is now a stack over a foot tall) and cut a handful of them up to make angles.
So, with a little bit of introduction1, and a list up on the board, I set them to work on (a) creating a triangle, and (b) seeing whether duplicating certain aspects of that triangle resulted in only one possible new triangle. It went from “Duh, Mr K, of course the triangles are all the same – they’re congruent!” to “How do I copy a triangle using SAA?”. The easy one, of course, was AAA. But the kicker came when most of them got the two possibilities for SSA, but not everyone did. That kicked off a whole new discussion of when it might work, or not.
I think this was the first time I managed to get most of my geometry kids (several who’d had this class already in an accelerated summer program) to all have an AHA experience. Definitely a win.
1 The biggest deal? Realizing that the ruler is not for measuring, and the compass is not for drawing pretty circles. Rather, the ruler just draws lines (sometimes through one or two points) and all of the measuring is done with the compass.