We are working with isometries in our Geometry class right now. We are looking at vector translations of objects, rotations (primarily around the origin), and reflections over horizontal or vertical lines. I am only teaching one section of Geometry so sometimes I feel like if I don’t get it right, I’ve missed an important opportunity. There have already been a couple of instances where I explain something after school in a way that I wish I had done the first time around. I am trying to keep note and be a smarter Geometry teacher next year.
Today when we were wrestling with rotations about the origin (I am sticking mostly to 90 degree increments) there was a bit of a clamor with the students begging me for rules about how to transform a point in the general (a, b) form. I really want them to try and develop a better intuition so I have resisted the call for these shortcuts. One of my best students, a boy I’ll refer to as M, pointed out that these rotations always maintain distance (again, we are rotating about the origin) so I pointed out to my class that this really limits our choices. We were looking at rotating the point (4, 2) 90 degrees counter-clockwise. Everyone agreed that this would put us in the second quadrant and I used M’s idea to suggest that our destination was now restricted to either the point (-4, 2) or (-2, 4) since the distance was the same and the second quadrant has negative x values and positive y values. A quick sketch convinced us all that the target should be (-2, 4). I felt like we were in sync with each other and that my sketches convinced my students. A group of them came after school to work to get ready for tomorrow’s quiz and they all confessed to not being convinced. I pulled up GeoGebra and tried to show them what would happen. I referenced M’s idea about distance and one of the students reminded me that I had been emphasizing the Pythagorean Theorem more than the distance formula earlier when we talked about distance. This was the breakthrough that I should have had at 8:30 am instead of 2:45 pm. I drew a right triangle on GeoGebra, called up a slider to rotate the triangle and showed where the vertex originally located at (4 , 2) ended up. Then I reminded this gang of help-seekers that points on the x-axis move to the y-axis under a 90 degree rotation. So I convinced them (and this time I am pretty sure that I DID convince them!) that the side of the right triangle corresponding to the distance 4 represented by the x coordinate of our original point now HAD to represent a y quantity after the rotation. We tried a few other points as well and we were humming along. I tested their wits by asking them one more question before they headed off – I asked about a 270 degree clockwise rotation. I was thrilled that one of the guys quickly pounced on this and said it was simply our 90 degree counter clockwise example again. VICTORY!
At least it feels that way now. I’ll see (and report back) after our quiz tomorrow morning.