We started out our unit on rigid motions with a thinking routine called “Zoom In”, described in Making Thinking Visible. I showed part of a piece of fabric and asked students to write down what they saw, keeping in mind that we were in a mathematics class. Then I uncovered more of the picture and gave students time to write down what they saw. Eventually I showed them the entire piece of fabric and asked them to discuss what they saw in their groups. I had no idea if students would see reflections, rotations, and translations (and in fact, some saw flips, turns, and slides), but they did. It actually worked…they saw transformations…the perfect lead-in to our unit on rigid motions. And so we began to think about how we could show that one of the images was congruent to another image on the fabric. We began to develop the idea that two images are congruent if there is a rigid motion that maps one image onto the other.
This quote by Nobel Prize winner Albert Szent-Gyorgyi hangs on one of the walls of my geometry classroom: “Discovery consists of looking at the same thing as everyone else and thinking something different”. While we were all looking at the same piece of fabric, we didn’t all see the same transformations. We opened each other’s eyes to many types of transformations as we shared with each other and listened to one another.
We moved next to a MARS (Mathematics Assessment Resource Service) MAP (Mathematics Assessment Project) Classroom Challenge, Representing and Combining Transformations. This classroom challenge had some good ideas for talking about transformations, and it turned out to be a great way to start the unit. Students were given a coordinate plane with an L-shape drawn in (pre-image) and a cut-out L-shape (image) to transform as directed.
We began with having them translate the L-shape, then moved to some reflections (about the x-axis, the y-axis, the line x=-2, the line y=x), and then moved to some rotations (90 degrees clockwise about the origin, 180 degrees counterclockwise about the origin). Physically moving the L-shape helped students differently than drawing the transformation or recognizing the transformation – even distinguishing between a reflection about the x-axis and about the y-axis. We asked students why they had transformed their shape as they did, getting them to start using the language of transformations: “The image is the same distance from the x-axis as the pre-image.”
Then we showed students a transformation and asked them to describe it in a Quick Poll. At this point, we had them enter into the CCSS Mathematical Practice of “attending to precision”.
Notice that the class decided not to give “a translation” credit. They wanted students to be more specific about the type of translation.
There was a matching game in the MARS lesson that we didn’t get to. We should still be able to use it at some point in the unit. During this lesson, though, we eased the hurry syndrome by giving students time to understand the big picture of transformations before we dig deeper, spending a day on each type of transformation, making conjectures about what each type of transformation “buys us” mathematically.
And so, the journey continues…