My Geometry classes have just finished a few days considering different translations on the Cartesian plane. We are working toward being comfortable with rotations (almost always around the origin), reflections (horizontal and vertical lines and the lines y = x and y = – x), and vector translations. Last week I had a particularly unsuccessful lesson where I tried to help my students discover a pattern for 90 degree rotations around the origin. I want to try and outline my thinking here and I would love love love any insight into why it did not go well and how I can improve this in the future, or even how to go back to cycle and revisit this with my current team of scholars.

So, my idea was this – try to pull together what we know about perpendicular slopes, our developing ideas of vectors as a physical object similar in nature to a line segment, and developing an intuition about the fact that a 90 degree rotation should result in a move of one quadrant in a certain direction. I asked for three coordinates from my students and drew a triangle. I asked them to predict where this triangle would end up after we moved it 90 degrees clockwise. Two of the coordinates given were in quadrant I and one was in quadrant IV. It seemed that my students were happy/comfortable with the idea that the two quadrant I points would live now in quadrant IV and that the quadrant IV point would move to quadrant III. This may have been a tepid agreement in retrospect. Next, I focused our attention on one of the quadrant I coordinates and I drew a segment from the origin to that point. We talked about the slope of the segment, we compared this segment to a vector, we talked about the length of the segment. I then asked the students to imagine a wheel and I told them that when I think about rotations I think about a bicycle wheel. In my mind I saw this segment as a spoke and I thought about distance from the point to the center of the wheel. Here is one place where I know I failed my students. I did not explicitly stop at this point and discuss distance the way I could have/should have. I also have a handful of students who are still struggling terribly with the idea of calculating distance. We have been talking about it since day two, I have coached them to think about Pythagoras, we have practiced it repeatedly. The combination of squaring, of square roots, of subtraction in one piece of ‘the distance formula’ and addition in another piece of it, comfort with mental arithmetic, all of these factors are working against my students being unanimously comfortable with calculating distances. So, the next step in my plan was to ask them what they recalled about perpendicular slopes. They all should know this and most recalled it pretty quickly. We had a segment in front of us with a slope of 4/3 and my students quickly agreed, maybe passively maybe enthusiastically, that a segment perpendicular to this would have a slope of -3/4. So, the question at hand was now whether the fraction was in the form or -3 over +4 or in the form of +3 over -4. I was convinced in advance of this lesson that this string of conversations would be a positive path to take. I felt that the combination of recalling past slope ideas, looking at the physical Cartesian plane, tying in ideas of line equations, etc. would gel together to make a lasting learning experience. I was wrong. When I prodded them toward the conclusion that -3 over +4 was the conclusion we wanted I saw some uncomfortable faces. When I mentioned the idea of a spoke as a visual to hold on to, I saw blank faces at this point. I got a bit frustrated and asked for my students to describe to me what they were thinking of when I mentioned the word spoke. Nothing. I pulled up a google image of a bicycle wheel and asked them to tell me what the spoke was in the image. By this point their reluctance to engage in this conversation was building, my frustration was increasing, and any positive momentum in building this process was falling apart. My fault for showing my frustration. My fault for stacking up too many ideas at once, I think. When I spoke to my Geometry colleague she felt that adding on the layer of talking about perpendicular slopes was the tipping point of discomfort for my students. I trust her instincts on a number of levels in part due to her experience in teaching our Algebra I course. She knows these Algebra kiddos and knows not only what they know but how comfortable they are knowing it. So, at this point it was clear to me that this was slipping away. We limped to the end of the conversation. Most students were willing to agree that the point (4,3) would end up in quadrant IV. They were split on whether it would land on (4,-3) or on (3,-4) and it honestly felt like many were mentally tossing coins to make this call. I showed them the conclusion on GeoGebra and we sort of ran out of time by this point. We have since gone back and tried to reinforce the conclusion we reached and I think that most of my students can reliably answer this question, but I am completely uncomfortable with how we got there. I would love any insight/advice about how to best structure this info. You can certainly drop a comment her or over on twitter where I am @mrdardy

Thanks for sharing this. This is one of my favorite connections, b/c no student understands negative reciprocal for slopes. I usually try to get at it with squares. I wonder if you did draw squares, compare slopes, then could try rotating segments.

What about a miniproject where learners each draw a segment, then rotate it to a new group member, they have to rotate it +/- 90 deg, etc? The visual might help them confirm, then there’d be so much data on points and slopes.

It occurs to me that this fact about slopes is one of those things where I usually just tell which is something I hate to do. Nice idea, I’ll try this out.