Similarity Transformation: A similarity transformation, or similarity, is a composition of a finite number of basic rigid motions or dilations. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.I argued that this level of rigor was inappropriate for eighth grade math, but that's only half the story. It is possible to make a case for teaching junior high kids using the language and structure of an abstract algebra class. Not a good case, in my opinion, but not a nonsensical one either.
If you can pull it off, there's something to be said for the axiomatic approach, having each statement emerge naturally from the postulates and theorems that came before. There is also something to be said for learning to formulate closely reasoned, exactly worded arguments. That's an experience probably best deferred until after puberty, but still...
What we have here, though, is something entirely different. The difficult technical language doesn't actually lead to anything. Students are dragged through these incredibly dense and confusing set-ups only to be have them followed by something like this (from the same lesson) [emphasis added]
No, even though we could say that the corresponding sides are in proportion, there exists no single rigid motion or sequence of rigid motions that would map a four-sided figure to a three-sided figure. Therefore, the figures do not fulfill the congruence part of the definition for similarity, and Figure A is not similar to Figure A′.
God, this is maddening. You can't say that "the corresponding sides are in proportion"; you can't say ANYTHING about the corresponding sides here. The term is meaningless when comparing a rectangle and a triangle.If you use this or any other kind of standard definition, that statement that has sides of a rectangle correspond to sides of a triangle is gibberish.
In geometry, the tests for congruence and similarity involve comparing corresponding sides of polygons. In these tests, each side in one polygon is paired with a side in the second polygon, taking care to preserve the order of adjacency.
I suspect that the authors were trying to model their problem after something like this:
I didn't bother to label these but you get the idea. It is easy to come up with examples where corresponding sides are in proportion but the figures are not similar as long as there are more than three sides to the polygon (and, of course, if the polygons have the same number of sides).
Regardless of what they were trying to do, the authors had a very weak grasp of the concept of corresponding sides and that's a pretty scary thought given that they were writing a lesson on similarity.
I need a current article (2018-2020) (preferably in a peer reviewed journal) that is critical of Eureka math. It seems to be taking over, but I have serious reservations about its efficacy in the classroom, especially in a Native American setting, where use of complex abstract language can be a barrier to learning. (I am a math guy; I love it and have a degree in abstract math from Cornell U).ReplyDelete