Today we introduced ourselves to rigid motions for the first time in geometry. Here’s the run-down:

- Give our definition of congruence.
- Give students one minute of think time and four minutes of write time to brainstorm all their possible answers to “How can I show that these two objects are congruent?” (I should have called them “figures”. I probably should have at least rotated one of them.)
- Have
*every*student write their favorite method up on the whiteboard. - Ask students to respond to the various methods on the board. Lead a pretty good discussion.
- Settle on, and define, superposition as our method of choice for the time being.
- Give students one minute of think time and four minutes of write time to brainstorm all their possible answers to “What moves are valid to use for superposition?”
- Have
*every*student write their favorite move on the board. - Ask students to respond to the various moves on the board. Lead another pretty good discussion.

Here’s a pdf of the (unedited) responses my students put on the board. Some thoughts:

- Students struggled to differentiate between the
*definition*of congruence and the method by which figures could be*shown*to be congruent. Many students said we could show the two figures congruent by showing they had the same size and shape. This is true, but also just the definition. - The Greeks’ obsession with compass and straightedge geometry stands very far afield from my own students’ relationship to geometry. Some form of “just measure it” was common to almost every student.
- My students understand that figures have properties, but struggle to identify which of those properties are sufficient conditions for determining other things. Last year, I kind of assumed that students came with both abilities.
- My students are more comfortable with translation and rotation than reflection. Most students are aware of all three concepts, if not the terms.
- Two of the three classes came up with superposition on their own. Cool.

I’ve been stressing the idea of informal proof and making a case for *why* a thing is true (or false) this year. Today we went through each item on the board and formulated arguments for the truthiness of them. We came up with counterexamples for things they claimed were false (“draw two figures with the same area that aren’t the same shape”, “draw two figures with six squares that aren’t the same size or shape”). We made arguments to explain why certain pairs couldn’t exist (“why can’t we draw two congruent triangles, one with a right angle and one with no right angles?”). We talked about how it’s important to actually be clear in what you write, so people understand what you mean.

The conversations were good. Not great, but good. I am a much better facilitator of discussion than I was when I first started all this. I still need to be better. But students were engaged and thinking, and, most important when compared to last year, students were developing their own counterexamples and arguments to develop mathematical understanding.

Tomorrow we will practice our rigid transformations by using them to show that various pairs of figures are congruent and begin to think about how the coordinates change under each operation so we can develop algebraic rules. If the future, we will hopefully utilize these experiences to move toward more formal proofs.

This year is going a lot slower than last year, but I think it is paying off in a big way, because my students are actually thinking and justifying their own thoughts.