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I’m a math coach and I’ve been working with one of my teachers on 8.G.2 and we are hitting a bit of a wall.
Here’s the standard.
Understand congruence and similarity using physical models, transparencies, or geometry software.CCSS.Math.Content.8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
So the standard asks the students to understand that if one figure is, for example, a rotation of another figure then the two figures are congruent, and then to show a sequence that exhibits how you get from one to the other.
This has been fine for reflection and translation. But for rotation it seems much harder. The problem we’ve been having is that it seems hard to demonstrate that one figure is a rotation of another in a mathematically rigorous way. Specifically, you can usually tell that a figure is a rotation by looking at it (literally drawing it on a piece of paper and turning the paper), but how can you spell out the conditions that prove that?
We thought about considering the distance of each point from the origin, since rotations will create images whose points are the same distance from the origin as the pre-image, but that is true of reflections also (or even of non-congruent figures, whose points are a rotation of the points in the pre-image, but the points of which have been rotated by differing amounts). We also thought about using the degree measures of each point relative to the x axis, i.e. something very similar to what you do in trigonometry with the unit circle, but that seems to require going far beyond the knowledge currently available to students in 8th grade.
So how do you demonstrate rigorously that one figure is a rotation of another?
If you have any insight on this, we could really use the help, as we are sort of stuck on this point.
Thanks for the help,
SilasIn my class I plan to keep the examples simple. This means rotations that are multiples of 90 degrees. Other rotations are more difficult to handle. How does one demonstrate, for instance, that one figure is the rotation of another 30 degrees about the origin? In general, that isn’t trivial, and it is of course beyond an introductory geometry class.
For my own part, I don’t think that analytic verification of a given transformation is of such great importance. A bit is good. But I do only so much as is necessary to motivate a set of results that follow: when polygons are congruent, sides which correspond and angles which correspond have equal measures; when sides and angles of polygons can be paired up in such a way that those which correspond have the same measure, then the polygons are congruent; SAS congruence and the rest of the triangle congruence principles.
So, when should rotations include all angles of rotation, in Geo-CO in high school? Or is it enough to stick to rotations that are multiples of 90 degrees there as well?
So, when do students need to work with any angle of rotation? In HS with GEO-CO standards? Or is it enough to stick with multiples of 90 degrees in high school as well?
It depends what you mean by “all angles of rotation.” Students can make arguments where the angle of rotation is specified by another angle in the diagram. So, you might prove SAS congruence by first translating so that the vertices with the A coincide, and then rotating so that two sides coincide, and then reasoning that the other vertices must coincide as well. This is an arbitrary rotation, but the argument doesn’t require you to ever specify the angle measure of the rotation: you would simply say “rotate by angle AOB” or something like that. So that is well within the standards.
But, giving a coordinate formula for an arbitrary rotation through a given angle measure is beyond the standards.
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