Extra teaching thought: will the real basic transformation please stand up?

I’m a pretty slow thinker. In an argument, I’m that guy who comes up with the perfectly witty comeback… about two hours after the conversation is over. This doesn’t just happen to me in social settings – it also happens to me in the classroom! Often I’ll teach through a skill or concept only to realise, after the lesson is over, that there was something else I should have said, or some other analogy that would have been immensely helpful, which would have added valuable insight or made things clearer.

Something like that happened when I taught this lesson on congruence transformations:

In geometry, a congruence transformation is – roughly speaking – a way that we can change (“transform”) a figure in such a way that it is still the same shape and size (“congruent”). There are three main kinds of transformation that we cover in year 7: translation (sliding the shape to a different position), reflection (flipping a shape over) and rotation (spinning the shape around).

After introducing the idea of transformations and how they work, we start to think about “composite transformations” – what happens when you combine more than one transformation and consider the total effect from original to final image. An interesting fact emerges: that not all transformations are created equal. In fact, one of the transformations is more simple and basic than the others. One transformation can be used to create each of the other two. Which do you think it is?

It turns out that the most basic transformation is reflection. You can make translation and rotation out of a series of reflections, but not the other way around; though it’s counter-intuitive, reflection is the most basic kind of geometric transformation. Perhaps that’s why reflectional symmetry is so deeply ingrained into our natural sense of beauty.