Transformations

[Kathleen]

I have seen both “transformations in the plane” and “transformations of the plane” used when talking about the standards. To me they are not interchangeable. Please clarify for me whether the standards address only transformations in the plane, both in and of the plane, or if indeed the two are interchangeable. Thank you.

[Dr. M]

I take it that the core definitions – of ‘rigid motion’, say – can be given either in terms of transformation of the plane or transformation in the plane. In the first, we map the whole of the plane to itself. In the second, we map one subset of the plane to another. I’m tempted to say that it’s just a matter of taste.

[Bill McCallum]

I’m not sure there is a mathematical difference here, but rather a difference in point of view. A transformation takes points in the plane as inputs, and outputs other points in the plane. We conceptualize this as motion: the transformation moves the input point to the output point. This conceptualization is useful but not strictly part of the mathematical definition. Whether you think of this as moving points in the plane (a transformation in the plane) or whether you think of it as moving the entire plane all together (a transformation of the plane) strikes me as a matter of point of view, or taste as Dr. M. says.

[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.

[Bill McCallum]

It might help to make a distinction between the plane and the coordinate grid imposed on it. The plane in which geometric figures live does not come automatically equipped with a coordinate grid; indeed, there was no such thing as a coordinate grid in Euclid’s day. You could think of the plane as a featureless plane in which geometric figures exist, and the coordinate grid as an overlay which can be used to measure positions in the plane. So when I perform a transformation, I move points in the plane to other points, but the grid stays where it is so I can use it to measure the new position of a point. This might be where the in/of confusion comes from: I can think of this as a transformation of the plane, but I can also think of points moving in the coordinate grid.

By the way, you don’t really need the coordinate grid at all to talk about transformations. You can talk about a rotation about a certain point through a certain angle without necessary giving coordinates to the point. It could just be some point in a geometric figure (say, the vertex of a triangle). It seems to me that the coordinates are almost getting in the way of things with your teachers.

As for dilations, I think the phrase “dilation of a point” is awkward. A dilation with center $O$ takes all the points in the plane and moves them along rays from $O$, scaling the distance from $O$ by a certain scale factor. Again, there is no need for coordinates, and no need for vectors.

[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

[Bill McCallum]

It might be worth looking at these materials. (I haven’t had a close look myself.)

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