Making arguments in geometry and the need for vocabulary and notation

Today we continued looking at the connection between reflections and translations. Students struggled with explaining why the three segments connecting double-reflected images had to be parallel.

I had students talk things over in their groups. As I walked around asking what they were thinking, most groups expressed that they knew in their heads what they wanted to say but couldn't express it well. I told them to not worry about how rough or awkward the communication was, but to try to get their thoughts out.

The results were interesting. Most groups expressed conjectures or possible theorems about the relationship. For example, one group stated if the segments connecting the reflected points were parallel then the result would have to be a translation. Another group stated that if the segments were not parallel then the result could not be a translation. These indicated that the students were trying to think about the situation in a mathematical way.

One group said they were focused on the distance between the reflected figures and the given parallel lines. They weren't sure how to proceed with their thoughts but felt that the distances would be relevant to concluding the results were translations. Another student thought that the first reflection flipped the figure 180^o. The second reflection over a parallel line would then flip the figure back and therefore it would be a translation. This was a good intuitive way of thinking about what was happening.

I used this as an opportunity to discuss maths vocabulary and notation. The class understood how they were struggling to communicate what they were thinking and seeing. This is why vocabulary and notation was developed. Sometimes it took centuries to create but the need to express and communicate thoughts drove the vocabulary and notation. Students were able to better appreciate the value of learning maths vocabulary and notation.

I then asked students to focus on the idea of distance. They've been working on this aspect of reflections and know that a reflected point is the same distance away from the line of reflection as the original point. I want to see if students can use this result to determine that all three line segments must be equal. I'll see what they come up with next class.