I felt good about how I handled feedback today in class. The class has been studying transformations in 8th Grade, and I wanted kids to practice visualizing different transformations. The task was to match shapes with transformations.

The above work is something I saw a lot. My kids had a pretty good sense of the different motions that translations, rotations and reflections were describing, and overall were pretty accurate.

Rotations around centers besides (0,0), though? Kids didn’t seem to know how to handle that. Some treated a rotation around (2,0) as just another rotation around the origin. Others seemed to get the quadrant of the image correct, but the coordinates weren’t right.

I decided that this would be what I’d focus the start of my next lesson on. I’d start with a pre-feedback activity, give some quick comments on the work, and then ask kids to revise their work.

This was the diagram I started class with:

I was trying to figure out what prompt to start with. I thought about just asking kids, “Where would the image be?” but this idea had problems. First, I wanted to know a bit more about how they thought because I wasn’t sure exactly where the problem was with their thinking about this. Second, I knew that kids *would *have trouble with this based on their previous work.

Instead, I posted the image and asked kids to figure out as much as they could about the diagram. I figured that this vaguer prompt would be more revealing while also giving me a chance to respond to ideas and help kids learn how to think about this sort of rotation.

(I also thought about just telling kids, *here’s how you do this*. I think that would have been plenty effective in this case, but I also like giving kids chances to have interesting mathematical ideas, and this seemed like a chance for that.)

Here was the annotated picture from under the document camera.

The three ideas on the top were from kids. After hearing them I pointed out that some of these were correct, assuming that the transformation was around the origin.

Then I said, this is why I brought this image into class, because I noticed from your classwork that rotations not around the origin are hard. I have something that I think will help, I call it drawing a tether. And then I connected points A and B. Does this help you figure anything else out?

Very quickly a kid said, yeah, now I’ve got this and there was a lot of chatter about where the other points would go. I annotated those, and summarized that adding the tether would help. I also said that you might figure out where the other points are relative to B, and this led to more mathematical chatter and more asserting of where the points are. (Is that the sort of evidence I can use to know if a class is working out? Mathematical chatter?)

Then I said, I want you to try out these ideas while revising your classwork. I highlighted ones that could be revised, and I’d like you to draw a revision on the page. From past experience, I wanted to keep the timing of this quick, so I said you’ve got 5-10 minutes, and if you don’t start quickly then you probably won’t get a chance to learn much now.

Here was some of the revised work.

Not everyone got feedback about rotation around (2,0). These kids get feedback about reflections around y = x and y = -x (which is good, because we need to know those for the next activity we’re working on).

The questions that I’m left with are about how sure I can be that this lesson went as well as I felt it could be. What’s my evidence? Is it the way the kids were acting? Was it their engagement? Or their resulting work? Should I have done a quick exit assessment — would that have supported my claim that this worked?

And does “this lesson worked” mean that it just worked in the classroom? Or can it mean that the planning worked, that I made moves on purpose and with confidence?

How do you really know if a lesson worked?

My response to “did a lesson work?” is “if they remember the main idea next month, then it worked.” When someone asked me in August what went well last year, I said, “As me next month when I see what my students remembered from last year.” So, yes, an exit ticket gives good information. But a warmup in November would give different information. And I think that engagement, excitement, and math chatter are all good signs that you’re heading in the right direction.

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