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Regents Recap — June, 2017: More Mathematical Misunderstanding

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 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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6 Comments

  1. Willie Wong says:

    Being charitable, perhaps the original intent of the question was that for each choice there are TWO distinct rigid motions, the first of which effecting the mapping before the word “and”, the second of which after?

    This would still be a poor question (wrapping a simple question about the inadmissibility of SSA and AAA and mere SA as congruence tests in layers of obfuscation), but at least one that is mathematically correct.

    In fact, I can read this meaning into what they wrote as the question; of course, the wording used (“a sequence of rigid motions…”) is not clear about whether the tested conditions are during the sequence or after the entire sequence was applied.
    But perhaps therein lies the problem, and the official excuse is less inaccurate than you imagined.

    • MrHonner says:

      Willie-

      This is definitely not what was meant. This language is used consistently (in prior exams and in curricular materials) to refer to the original figure and the final image. I’m sorry to say your charity is in vain.

  2. Matthew Reardon says:

    Can you explain in answer (1) how the triangles are not equal? Sorry in advance if it is dumb question.

    • MrHonner says:

      According to answer choice (1), all three pairs of angles in the two triangles are congruent. This tells us the triangles are similar, but not necessarily congruent.

      However, the additional information regarding the mapping of point A to D and point C to F via rigid motion guarantees that segment AC is congruent to segment DF. Together with the congruent angles, this allows us to conclude the triangles are congruent.

  3. David Radcliffe says:

    Why do they say “sequence of rigid motions”? Are students not taught that the composition of two rigid motions is also a rigid motion?

    • MrHonner says:

      That the composition of rigid motions is a rigid motion is definitely part of the curriculum.

      The emphasis on a “sequence of rigid motions” (which comes up often in exams and curricular materials) may be because the curriculum emphasizes reflection, rotation, and translation as the basic rigid motions while mostly ignoring glide reflections (composition of translation and reflection).

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