We are now delving into some interesting coordinate geometry. We’re also beginning to use patty paper. And today the other geometry teacher and I had an awesomely fun conversation that revoles around something you might find it fun to think about.

*Here’s the impetus/setup.*

We have just finished talking about translations, and we’re moving on to reflections and rotations. When we introduce rotations to kids, we give them a backwards problem pretty early on: here is a figure and here is the rotation of the figure. Try to find the center of rotation. Use patty paper and guess and check.

This is an *awesome *exercise (inmyhumbleopinion) because it has kids use patty paper, it has them kinesthetically *see* the rotation, and it gives them immediate feedback on whether the point they thought was the center of rotation truly is the center of rotation. Simple, sweet, forces some thought.

This is our intro. We have a few more exercises in our problem packet similar to this. But then we move on. Le sigh. This felt wholly unsatisfying to us, because at its heart we left our kids hanging. We never get at the obvious question. How do I find the center of rotation *without* guessing and checking.

How do we take this introduction and make it mathematics?

**The Challenge**

[Baby challenge: Figure out how to find the center of rotation, given any figure and a rotated figure. This is not totally simple, okay? I'm calling it a baby challenge only because the next challenge is *sooooo* hard!]

Imagine you’re a geometry teacher, and you want students to **discover** a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to **understand (deeply)** and **articulate why** this method gives you *the* center of rotation.

What do you do to achieve these twin goals?

(This is what the other teacher and I were discussing today, and having loads of fun doing it!)

You *don’t* need to answer this question in the comments. (Though you’re welcome to throw down any ideas.) I’m actually not looking for advice. (We’re well on our way to coming up with an answer to this.) I just thought it would be a fun thought exercise for you, if you like thinking of lesson planning/curriculum design. This is the type of stuff I love thinking about — when I have time!

This is backwards planning at it’s most fun, in my opinion. I have a deep result that is abstract and hard to grasp. I have very concrete 9th graders who I want to get from knowing almost nothing to discovering, understanding, and marveling at this great mathematical insight. How do I get from Point A to Point B?