G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Our content learning goal for the day: I can map a figure onto itself using transformations.
Our practice learning goal: I can attend to precision.
Combining those, we were working on: I can show my work.
Do your students know what you mean when you ask them to show your work?
Jill Gough has written a transformative leveled learning progression for showing your work. This was our first day in geometry this year to focus on it.
Level 4: I can show more than one way to find a solution to the problem.
Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.
Level 2: I can find a correct solution to the problem.
Level 1: I can ask questions to help me work toward a solution to the problem.
For this task, our focus was on describing clearly the transformations that would carry a rectangle or equilateral triangle onto itself so that a partner could follow the steps.
Which of the following is clear?
Reflect ABCD about a line through the middle of the rectangle.
Reflect ∆ABC about its center.
Rotate ∆ABC 60˚.
Reflect ABCD about the perpendicular bisector of segment AB.
Rotate ∆ABC 180˚ about point A.
Students set to work individually, paying attention to their language. I walked around to see what they were writing.
I noticed MR’s first, which said, Translate ∆ABC using vector AA. As I looked more closely, I realized that she was mapping the triangle on the left side of the page onto the triangle on the right side of the page, but even so, she had come up with a remarkably trivial solution, had she been mapping the triangle onto itself.
The next student that I saw had rotated ∆ABC 360˚ about point A.
And then the next student that I saw had dilated ∆ABC about point A using a scale factor of 1.
I decided at this point that perhaps a class discussion was in order to limit additional trivial solutions to this task. So we talked about transformations that will, of course, map the figure onto itself, such as rotating the image about one of its vertices 0˚ or 360˚, and also, really, are simple and not very interesting.
And then I let them work some more. The idea was for them to write a transformation or sequence of transformations and have their partner try it, following their directions exactly. The partner helped revise the directions as needed if the directions didn’t work the first time.
Instead of selecting particular students to share their work with the whole class, I asked students to write at least one set of their successful mappings in a shared Google Doc so that they could see multiple solutions to both the rectangle and the triangle.
Thanks to the leveled learning progression, I think we are off to a good start practicing “show your work”, as the journey continues …