CCSS-M G-SRT.A.1. Verify experimentally the properties of dilations given by a center and a scale factor:
- 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.
- The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
CCSS-M G-C.A.1. Prove that all circles are similar.
It probably shouldn’t surprise me that my students have a difficult time dilating lines and circles. Most of the objects that we transform in mathematics are polygons, and we often think about what happens to them in terms of their vertices, neglecting to note what’s happening to the other points on the polygons or the sides of the polygons.
Suppose we apply a dilation by a factor of 2, centered at the point P, to the figure below.
Can you anticipate misconceptions that students will have?
As I was monitoring student work, I saw several approaches that I wanted the class to see. I took some pictures and thought carefully about whether to start with one that was correct or one that was incorrect.
I decided to start with a student who had used her ruler as a measuring tool and not just as a straightedge. FS drew the line that contains PA past A, measured PA, and then marked off that same measurement for AA’. She did the same for PB and BB’ and the same for PC and CC’. She hadn’t yet drawn the line through A’, B’, and C’. What will be true about that line compared to the line through A, B, and C?
More than one student said the line would double. What does it mean for a line to double?
SC had a similar approach, except that she didn’t use her ruler to measure, she used her compass to measure. She demonstrated how she measured PA and marked AA’ the same length.
Then I showed BB’s work. What do you think?
What do you like about her work?
What do you wonder about her work?
BB notes that A’B’ is double AB in her answer to part (c), but her diagram isn’t convincing.
What do you think about BK’s work?
What do you like about his work?
What do you wonder about his work?
One of the other explorations was “How would you dilate circle C by a scale factor of 3?”
Here’s the work of one student.
Which made me realize I had not specified that A was the center of dilation.
And the last exploration was “Given any two circles, can you always find a dilation that maps one circle onto another?”
Which we didn’t get to that day.
I’ve written about Dilating a Line before, but the experience of the task changes each year, as the journey of teaching and learning continues with a different community of learners …