Regents Recap — June, 2017: Trouble with Dilations (and Logic)
The emphasis on transformations in Common Core Geometry has proven to be a challenge for the creators of the New York State Regents. Here’s the latest example.
This is a tricky question. So tricky, in fact, that it tripped up those responsible for creating this exam.
Dilation is a similarity mapping (assuming, as we do, that the scale factor is non-zero), and translation is a congruence mapping. Thus, any composition of the two will be a similarity mapping, but not necessarily a congruence mapping. So in the above question, statement II will always be true, and statements I and IV are not always true.
Statement III requires closer attention. Under most circumstances, translations and dilations map lines to parallel lines, and so the same would be true of their compositions. However, if the center of dilation lies on a given line, or the translation is parallel to the given line, then that line will be mapped onto itself under the transformation.
This means that the answer to this test question hinges on the question, “Is a line parallel to itself?”
If the answer is yes, then statement III will always be true, and so (3) II and III will be the correct answer. If the answer is no, then statement III won’t always be true. and so (1) II only will be the correct answer.
So which is the correct answer? Well, that’s tricky, too. The answer key provided by New York state originally gave (3) as the correct answer. But several days later, the NYS Department of Education issued a memo instructing graders to accept both (1) and (3) as correct. Apparently, the state isn’t prepared to take a stance on this issue.
Their final decision is amusing, as these two answer choices are mutually exclusive: either statement III is always true or it isn’t always true. It can’t be both. Those responsible for this exam are trying to get away with quietly asserting that (P and not P) can be true!
Oddly enough, this wasn’t the only place on this very exam where this issue arose. Here’s question 6:Notice that this question directly acknowledges that the location of the center of dilation impacts whether or not a line is mapped to a parallel line. It’s not entirely correct (a center’s location on the line, not the segment, is what matters) but it demonstrates some of the knowledge that was lacking in question 14. How, then, did the problem with question 14 slip through?
As is typical, the state provided a meaningless and generic explanation for the error: this problem was a result of discrepancies in wording. But there are no discrepancies in wording here. This is simply a careless error, one that should have been caught early in the test production process, and one that would have been caught if production of these exams were taken more seriously.
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