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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Far too often questions on New York State math exams demonstrate a disturbing lack of content knowledge.

Consider this multiple choice item from the June, 2017 Common Core Geometry exam.

Instead of testing a student’s understanding of triangle congruence, this question exposes a serious lack of mathematical understanding among the exam creators.

A superficial reading of the problem suggests that (3) is the correct answer.  In (1), the two triangles share only three pairs of congruent angles; in (2), two sides and a non-included angle are congruent in each triangle; and in (4), the triangles share only one pair of congruent sides and one pair of congruent angles.  None of these scenarios (AAA, SSA, SA) seems sufficient to guarantee that the triangles are congruent.  And in (3), the triangles have one pair of congruent sides and two pairs of congruent angles; this (ASA or SAA) is sufficient to conclude the triangles are congruent, so (3) is apparently the correct answer.

But closer inspection shows that, in fact, (1) and (2) are correct as well.

Consider choice (1).  While it’s not exactly clear what it means to map angle A onto angle D, it must require that point A gets mapped to point D.  Similarly, point C must be mapped to point F.  If a rigid motion maps A to D and C to F, then segment AC must be congruent to segment DF.  We now have one pair of congruent sides and three pairs of congruent angles: the triangles are congruent (by ASA or SAA), and choice (1) is a correct answer.

In (2), we are given that segment AC is mapped onto segment DF.  This means that point A gets mapped to point D and point C gets mapped to point F.  And since segment BC is mapped onto segment EF, we know that B is mapped onto E.  Therefore, the vertices of triangle ABC are mapped via rigid motion onto the vertices of triangle DEF.  This is sufficient to conclude that the triangles are congruent, and choice (2) is also a correct answer. (It’s also worth noting that, since the triangles are given as acute, SSA is actually sufficient to guarantee that the triangles are congruent.  This mathematical error turned up in a separate question on this exam.)

As it stands, the only option that is not a correct answer to this question is (4).

Within a few days, the NYS Education Department issued a directive to count all answers to this question as correct.  As is typical, no admission of an error was made:  the problem was blamed on discrepancies in wording.  Of course, there are no discrepancies in wording here:  this problem as written, reviewed, edited, and ultimately published is simply mathematically incorrect.  Its existence demonstrates a fundamental misunderstanding of the underlying concepts.

This isn’t the first time an erroneous question has made it onto one of these high-stakes Regents exams.  In fact, there were at least three mathematically invalid questions on this exam alone!  Over the past five years I’ve documented many others, and each time it happens, it raises serious questions:  Questions about the validity of these exams, how they are experienced by students, how they are scored, and the lack of accountability for those in charge.

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