Introducing rigid transformations in geometry

Today I introduced rigid transformations. We started by discussing transformations and students were able to identify translations, reflections, rotations, and dilations (although they only described them and didn't name them).

I had mini-whiteboards with axes on them and I started the students off by plotting some points. This helped me see who might be struggling with coordinates and helped students refresh their memory on coordinates and plotting points.

Satisfied that students could correctly plot points, I proceeded to introduce reflections. I decided to use NCTM's Navigating through Geometry sequence for transformations. The first activity deals with reflections over the y-axis.

The beginning prompts dealt with drawing and tracing to create a reflection. Students began to have issues when they were asked to drawn a segment between two points they had selected. The notation seemed to baffle them, which was disheartening given how much time we had spent on notation and understanding what structure and information is given in prompts.

After moving beyond this hurdle, students were to identify how the two points (pre-image and image) were related to the line of reflection. Students readily recognized that the two points were the same distance from the reflection line, lying on opposite sides of the line.

The next piece became a problem. Students were to draw a triangle and then draw its reflection based upon what they had just discussed. Students wanted to fold the paper and trace out again or seemed totally lost. I realized that using the whiteboards and grids could help. I moved students to using these and they were able to proceed without much incident.

We discussed what was happening with the coordinates and students could more readily see that the pre-image point (x , y) became the image point ( -x, y) when reflecting over the y-axis (line x = 0).

At this point I deviated from the activity slightly by focusing on reflections over vertical lines. I asked students to pick a different vertical line, such as x = 2, and reflect their pre-image over this line. I told students to focus on what is happening to the x coordinate now. How does reflecting over a different line change things.

The class struggled mightily on this task. Many wanted to give up and several asked me to just tell them what the result was. I finally told the class that I expected them to make reasonable conjectures based upon what they were seeing. I told them if it takes the class two weeks to work through this issue then it will take two weeks.

Their homework is to come up with some conjectures. I'll see what they bring to class.

My hope is that we can understand the mathematics of reflections over vertical lines and the resulting expressions that describe these reflections. It should then be fairly easy to conjecture about the results we would see reflecting over horizontal lines and verifying the results.

My goal is for students to tackle lines that are not horizontal or vertical. I'll start with y = x and y = -x and then have them tackle a line such as y = x + 2. From there, I hope to look at more general lines such as y = 2x - 3.

After these explorations, I'll revert back to using more of the Navigating through Geometry activities.