Reflecting over the line y = 2x and connecting translations to reflections

Another short post. I had students try reflecting over y=2x. This was a struggle because students wanted the reflected points to behave nicely, i.e. reflect onto nice coordinates, which they weren't. I had to pass out tracing paper so that students could see that the image points they were drawing were not actually the reflection points. On the positive side, they stayed with the effort for over 30 minutes.

Students could see that the reflection points were not behaving well and that the nice symmetry of reflecting over a horizontal line, a vertical line, y=x, or y=-x was gone. Based on where they were, I determined that I needed to move on from this for now. Trying to investigate reflections over y=-2x or y=2x+3 would not be productive.

Since this was a block period, I did a quick brain break to re-energize the class and then moved on to translations. I had a triangle on a coordinate plane and asked what the translation of this figure by translating four points to the right and 3 points down would look like. I told students to guess at the meaning if they weren't sure what a translation was.

Students easily performed this task. We wrote out the notation (x, y) --> (x + 4, y - 3) as suggested by the class. I asked what the general expression would be for translating the x by a points and y by b points. One student suggested (x, y) --> ( x ± a, y ± b), which I hadn't expected. We tried a couple of more translations that I provided through expressions and then started to explore connections between translations and reflections.

I asked the class what connections they could think of between reflections and translations. No ideas were forth coming. I told the class we were going to explore this. The Slide Me Now activity from Navigating through Geometry provided the basis for this investigation. Students completed the first four questions during class. Their homework is to complete questions 5 & 6. I told the class to use what they know about parallel lines, transversals, and the angle relationships they learned to help them.

Next class I'll see what they come up with. I'm going to try to push this to begin looking a geometry proofs. I plan on completing the first two questions in the extension section of this investigation as well.

I'm thinking to revisit the reflection over y=2x and looking at why the segment connecting the pre-image and image points must be perpendicular to the line of reflection. This could lead into relationship of slope to perpendicular lines. I might also look at how to construct a segment bisector or perpendicular line as offshoots of this. I'm still debating which direction I want to take this piece.