I have spent the last six weeks in my Geometry class developing definitions for rigid transformations (reflection, rotation, translation). Some might call me crazy for devoting that much time, but I would argue that in addition to learning the mathematical content, students have learned mathematical habits of mind that will carry them through the rest of the year, and hopefully, the rest of their lives. I'm a "big picture" kinda gal, and I want my students to learn skills that are transferrable to their experiences outside my classroom.
Toward the end of the unit, I gave my students a task that was written by Jade White (High Tech High) in order to sort of "bring it all together".
As you can probably guess, the last problem has students graph a letter "H", but it has been rotated 90 degrees and is therefore not oriented correctly. The task I gave my students was:
Identify a single transformation that will orient this letter correctly and send it to Quadrant 1.
I was impressed to see that no students were trying translations. I even heard them saying to each other, "We can't use a translation because that would not change the orientation." Pretty awesome stuff!
Students grabbed patty paper and were testing out their ideas. They were rotating, trying to figure out where the center of rotation should be. They were reflecting, trying to find a line of reflection that would work. I was impressed to see their level of commitment and engagement in this problem. In addition to that, I was excited to see that they were using their definitions of transformations to look for their, what we call, "key ingredients" (line of reflection, center and degree of rotation).
Once students had arrived at an answer, I asked them to share a transformation that satisfied our needs (proper orientation and located in Quadrant 1). One student had identified a line of reflection of y = -x. We tested it, and it worked. We connected it back to our definition of reflection and students saw the connections. Students presented other lines of reflection.
Another student had found a center of rotation at the point (-3, 3) and rotated it 90 degrees clockwise. This was also a viable transformation and again, students connected the fact that this "worked" back to their definition of rotation. Students presented other centers and degrees of rotation.
At this point in the discussion, we explored what constituted as "Quadrant 1". Students shared that they found lines of reflection or centers and degrees of rotation that had segments or points from "H" mapped onto the x- or y-axis. Is that allowed? Then a student asked, "Well, what's the definition of Quadrant 1?" After I picked my jaw up off the floor, I said that my understanding of Quadrant 1 was that the x-values and y-values had to be positive. Students decided for themselves if they wanted to "count" the x- and y-axis as Quadrant 1, but the fact that they even asked was exciting for me.
The question that inspired me to write this post came from a student who is always thoughtful in her learning. She does not take things at face-value, she wants to know and understand concepts deeply and as completely as possible.
"I noticed that the center of rotation of (-3, 3) is on the line of reflection y = -x, and both satisfied our requirement of sending H to Quadrant 1 with the correct orientation. Is it always the case that the center of rotation will be on a line of reflection that works?"
I didn't know the answer. I gave students some room to explore, which I guess was a way for me to buy myself some time, but honestly, not all of them were engaged or willing to persevere through the question, so we moved on. I have since explored this question using Geogebra and I have some ideas surrounding this question, but I still don't know the answer.
What are your thoughts? What if it's not an "H" we're talking about? Is this idea generalizable?
For me, this is exactly what I want to see in my classroom: students being curious about the mathematics they are learning and feeling safe enough to ask about it. This is the kind of magic that happens when we give students the time and space to explore.