Reflecting over lines that are not vertical or horizontal

Things got back on track today. I had a brief discussion at the start of class. I explained that I expected them to struggle, that I expected them to discuss the problem, and that I expected them to ask questions of others or me. What I will not abide is intellectual laziness; the class sitting around waiting to be told exactly what steps to take. I related this to later in life when they are trained in a job. The training doesn't cover all situations. The expectation is that an employee will be able to apply their training to the situation at hand.

With that we explored reflections over the line y = x. I asked students to conjecture what would happen in this situation. I instructed students to close their eyes and picture the grid with the line y = x on it. I asked them to imagine what would happen as an object was reflected over this line. We discussed their thoughts, with many thinking about how the coordinates might go from positive to negative. One girl conjectured that this would be equivalent to reflecting over the x-axis and then over the y-axis. The students were ready to start exploring.

Some students did not know what the line y = x looked like and I had to assist them to get things started. The other thing that kept cropping up was that students would still end up reflecting as if it was over a vertical line or horizontal line. Once they got this down, they were able to see the result that the pre-image point (x, y) ends at the image point (y, x).

I asked the class to consider what if the line was y = -x. Students still felt that the x and y values would flip. A couple thought, in addition, that the coordinates would become negative. After exploring the results, the class saw that (x, y) --> (-y, -x).

We had time to start exploring further. I asked what would happen when reflecting over the line y=2x? Again, some students weren't sure how to graph the line. I reminded them how to create a table and then they were off exploring. The class ended at this point, so we'll pick up where we left off last time.

I really like this exploration as it is bringing back algebra review that students obviously need. Hopefully the energy level and effort will continue.