Connecting reflections and translations - proving relationships

Today did not progress as I hoped. Students did not make any progress on why the three segments connecting pre-image to double-reflected points were all the same length. There is a distinct lack of initiative and strong desire to be told what to do that can be disheartening.

I decided to try work through the problem as a proof. Unfortunately, not thinking this through, I tackled all three segments at once, which resulted in extensive repetition and loss of the central issue. Doing this again, I would prove that one segment was equal to twice the distance between the parallel lines and then say that the exact same argument would apply to the other two segments.

As part of this proof, we needed a working definition of what it means to reflect a point. One student suggested a reflection was a flip of 180^o over a given line. I drew an example that did not have the two flipped points equally distant from the reflection line. Students realized that the definition needed to include that the two points were the same distance from the reflection line. While the definition could be tightened up a bit, the critical aspect of equal distance apart was what was needed for the proof.

After plodding starting to plod through the proof, I could see that it was going to be too lengthy and confusing doing all three segments. I stopped along the way knowing that the pieces of thinking about what was given and how they relate had been communicated.

I then asked students to address how they knew if the three segments were parallel. Again, there was not much activity going on. Was it Monday doldrums or just a lack of interest? I told students to review angle relationships associated with parallel lines and transversals. At this point I had to ask the class about different angle relationships and their associated names.

The problem now made use of the reflection definition; a flip of 180^o equated with the lines being perpendicular. Now, all the angle pairs are 90^o and the converse relationships are in play.

The class ended with students tasked with determining how to position or to determine the location of the parallel lines given a specific reflection. As I walked around the room asking groups their thoughts before the end of class, I could see there was a lack of understanding about the concept of generalizing results. Yet something else that needs to be addressed.