[Kathleen]

Thank you for the insight. I have never taken the time to help them grasp that the plane and the grid should be thought of as “independent.” How simple and clean. Most of these teachers have only taught transformations without ever thinking of coordinates on a plane, unless it was specifically on the Cartesian plane and in Algebra, not Geometry. They are having trouble with the big picture. The whole problem arose when we were looking at task 602 on IM, Dilating a Line. So it started with one of them saying, “You can’t dilate a line because it is already infinitely long.” I responded, “What about the distance between specific points on the line?” Then we started the task. In the commentary it states: “The points A’, B’, and C’ appear to be collinear. If we choose more points on line l and dilate those points about point P, we will see that the dilations of those points also appear to lie on the line through A’, B’, and C’. It appears that the dilations of the points on the line l form a new line l’ that is parallel to line l”. That’s when they all agreed, “You can’t dilate a point!” But, in fact, GeoGebra allows you to do just that. It was not a good day for me. I was not ready for the discussion. I agree with you that the expression “dilation of a point” is awkward, but it is out there and I don’t think we can take it back. Older teachers do not want to make dilations “more difficult” than just finding ratios, and young teachers who have just finished college courses still don’t remember doing dilations in high school geometry. I have found that some teachers do not even understand that a dilation needs a center. They think of “similar” and “dilation” as the same thing, and the pictures in the texts do not have “centers” of similarity. We have a lot of work to do on this specific idea. Can you suggest some good resources I might suggest to them?

Thank you again. I feel like a real pain about this, but I do want to help these teachers be less anxious. Kathleen

[Kathleen]

Thank you for this reply. Your description of moving in the plane, or moving the entire plane is exactly the problem the teachers with whom I am working find troubling. Moving the points in the plane (not the entire plane) is much easier for them to grasp and does not seem to concern them at all. Picking up the origin and moving it seems to cause them great distress. I think this goes back to how they view “families of functions” where say the vertex of the parabola is “translated” rather than the origin of the plane.

So, if I understand you, I can just tell them to think “in the plane” when they see something written as “of the plane” and no harm will be done? This appears in some of the tasks on IM.

Also, is there an easy way I can discuss “dilation of a point” with them? They are focused on dilation as “change in size” and contend a point has not “size”. If we discuss “point has location” and if we change it’s location from the origin, then we change it’s “distance”, they say we are talking about vectors which are not “points” and hold that vectors are not addressed in the CCSM before we begin dilations. I am having a very hard time with this. I need guidance. Thank you very much for answering my questions.

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