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Since 8th grade is meant to be an informal and intuitive exploration into the world of transformations, should we be reflecting over lines other than the axes when exploring reflections on the coordinate plane? And….should we be rotating around points other than the origin when rotating on the coordinate plane? I understand that we should be exploring rigid motions outside of the coordinate plane which allows for lines of reflection other than the axes and points of rotation other than the other than the origin, but I’m not sure about what the limitations are when on the coordinate plane.
If you want students to give coordinates of reflected or rotated points then you have to restrict to reflections and rotations where that is possible given what they know, so yes, that limits what you can do. You could have rotations about points other than the origin in multiples of 90°, and you could probably dream up other situations where special placement of the points or symmetry would make it possible, but basically you are right.
Hello Bill,
I am currently working with a group of 8th grade teachers on 8.C.1. We noticed in the progression document it states “They should encounter and experience transformations which do not preserve lengths, do not preserve angles, or do not preserve either.”
We are all stumped. When we are thinking of ‘transformations’ we are thinking of reflections, rotations, translations, and dilations. A dilation does not preserve length, but what is an example of a transformation that does not preserve angle measure OR either length or angles? Besides saying it is not a transformation, we are not sure how to show that it is a transformation yet does not preserve angle measure.Any help in clarification would be much, much, much appreciated!
Sincerely,
Mindy SingerMindy, a transformation doesn’t have to be made up of reflections, rotations, translations, and dilations, although those are certainly the main ones under study. For example a vertical stretch is a transformation (that is, stretch every vertical line by, say, a factor of 2). Or a shear transformation, which slides every point along a horizontal line a distance proportional to its height above the x-axis (think of shearing a square into a parallelogram of the same height). Here is an activity in the Illustrative Mathematics middle school curriculum which illustrates some of these transformations. It’s in Grade 7, in the section on scaled copies, but it paves the way for the work on rigid transformations in Grade 8.
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