Dilating a Line
CCSS-M G-SRT 1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
CCSS-M G-C 1. Prove that all circles are similar.
The Illustrative Mathematics task on Dilating a Line asks specifically about a dilating the given line by a scale factor of 2. I took out the specific scale factor when I gave it to my students. For one thing, I think we use a scale factor of 2 too often when we talk about dilations. I get it. It’s easy. It fits on the page. But things happen with the number 2 that don’t happen with every number. So I left the task more open-ended.
Students started their work on this task on paper. By themselves. One of our math practices is to use appropriate tools strategically. Some asked for straightedges, some used rulers, some used compasses. Eventually, we moved to our dynamic geometry software.
I watched. And took a few pictures.
And asked a few questions. But mostly I listened to the conversation once students talked to their groups about their work. And I selected which conversations needed to come out in our whole class discussion.
We talked about this picture (above). And what was correct about it. And what was incorrect about it.
We talked about this picture. What can we conclude about what happens when we dilate a line about a given point?
We talked about this picture. What does it mean for B to be between A and C? What does it mean for B’ to be between A’ and C’? I was glad for this picture to uncover a misconception about betweenness that several students still had a semester into our geometry course.
And we talked about this picture. What is different? Why?
Our technology gave us the opportunity to think about the dilation differently. How can we locate points A’, B’, and C’?
I also used a task from The Mathematics Common Core Toolbox on dilating a circle. Students worked on paper first, and then we moved to our technology to generalize the results.
How can we determine the location of R’?
If C corresponds to D on the circle, how can we determine the center of the dilation?
We keep hearing about the efforts of CCSS-M to do away with our mile wide inch deep curriculum. I am willing to testify to anyone needed that these questions about dilations dig far beneath the surface of what I have taught in the past. I have a much better understanding of what a dilation is because of my own studying, and of course, as a result, my students have a much better understanding of what a dilation buys us mathematically than any I have taught in the past. And so the journey of learning and teaching continues …