One of the tasks that we gave our students at the end of our unit on Rigid Motions came from Illustrative Mathematics: The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected. I took the task and made it dynamic in a TI-Nspire document. Instead of creating the line of reflection for two static triangles, they had to construct it so that it still worked when we moved one of the vertices of the original triangle. They were also required to show some measurements on their screen that justified their work.
On the next page, △ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.
Many students were successful on this task. Most of them constructed the perpendicular bisector of one of the segments with endpoints that were pre-image point and its image.
Then next task that we asked them to do was the following, which I first heard about from an instructor at the University Lab School in Honolulu.
On the next page, you are given segment AB. Construct a regular hexagon ABCDEF with segment AB as one of its sides.
-You may not use any Shapes tools.
-You many not use any Measurement tools.
When you are finished, we will use Measurement tools to justify your construction.
After proposing the task to the students, I made sure they knew what we meant by regular hexagon. Then I let them think and work for a little while before the class discussed a few questions. Someone wanted to know what one of the measures of the angles of the regular hexagon was. So I drew a triangle and asked for the sum of the measures. We ended up having a mini-lesson – and didn’t even get to the point of generalizing the sum of the measures of the triangle (that will come later) – just enough for them to have what they needed to make their hexagon using transformations.
Most students rotated the sides all of the way around to create their hexagon.
A few students rotated segment AB twice and then reflected the segments to get the rest of the hexagon.
All of the students learned something about hexagons that they had not previously considered.
And then one more…
I love the angles of rotation that the students used to rotate the pentagon onto itself…angles that are easy to use because of technology. My students are entering into the practice of use appropriate tools strategically.
And so the journey continues….