Identification of a quadrilateral as a trapezoid, rectangle, or square

Question

Yesterday I was tutoring a student, and the following question arose (number 76):

My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel, and that the bottom and right sides are congruent. That's not enough to be "more" than a trapezoid, so it's a trapezoid.

Now fast-forward to today. She is publicly humiliated in front of the class, and my reputation is called into question once the student claims to have been guided by a tutor. The teacher insists that the answer is J: square ("obviously"... no further proof was given).

1. Who is right? Is there a chance that we're both right?

2. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

Answer 1

Of course, you are right. Send an email to the teacher with a concrete example, given that (s)he seems to be geometrically challenged. For instance, you could attach the following pictures with the email, which are both drawn to scale. You should also let him/her know that you need $5$ parameters to fix a quadrilateral uniquely. With just $4$ pieces of information as given in the question, there exists infinitely many possible quadrilaterals, even though all of them have to be trapezium, since the sum of adjacent angles being $180^{\circ}$ forces the pair of opposite sides to be parallel.

The first one is an exaggerated example where the trapezium satisfies all conditions but is nowhere close to a square, even visually.

The second one is an example where the trapezium visually looks like a square but is not a square.

Not only should you email the teacher, but you should also direct him/her to this math.stackexchange thread.

Good luck!

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EDIT

Also, one might also try and explain to the teacher using the picture below that for the question only the first criterion is met, i.e., only one pair of opposite sides have been made parallel.

Answer 2

Clearly the figure is a trapezoid because you can construct an infinite number of quadralaterals consistent with the given constraints so long as the vertical height $h$ obeys $0 < h \leq 9$ inches. Only one of those infinite number of figures is a square.

I would email the above statement to the teacher... but that's up to you.

As for the "politics" or "pedagogy" of drawing a square, but giving conditions that admit non-square quadralaterals... well, I'd take this as a learning opportunity. The solution teaches the students that of course any single drawing must be an example member of a solution set, but need not be every example of that set. In this case: a square is just a special case of a trapezoid.

The solution goes further and reveals that the vertexes (apparently) lie on a semi-circle... ("obvious" to a student). A good followup or "part b" question would be to prove this is the case.

Answer 3

I agree with @davidgstork re: the first question.

As for your second question, it's important to get the word out that it is a trapezoid, but you'll need to draw a few diagrams that actually show the trapezoids that conform to the conditions. (As they say on standardized tests all the time, just because it looks like a whatever, doesn't mean it automatically is a whatever unless specifically told.) I'd e-mail diplomatically, of course, but with clear diagrams.

Answer 4

FWIW, this question appears to come from a diagnostic test which can be perused at http://web.archive.org/web/20161228001122/http://www.mathmatuch.com/presentations/diagnostic_test.pdf -- where the official answer is given as J (the square). So it's not just the teacher who is wrong.

(Remark: I found the site by googling on "identify the figure shown" and "trapezoid" then looking for "76" and "J" in the results.)

Update Feb. 17, 2020: The original link (above) is now dead. By googling again on "identify the figure shown" and "trapezoid," I did find another version of the diagnostic test at https://blevinshornets.org/teachers/wp-content/uploads/sites/6/2018/12/diagnostic_test-7th.pdf with the same error. Interestingly, a similar test can be found at http://stpatrickschoolstoneham.org/wp-content/uploads/2018/06/Rising-6th-grade-math-packet.pdf but problem 76 there is somewhat different and has no error.
Update Dec 1, 2020: Replaced original link with Wayback machine link.

Answer 5

The right vertical side of 9 inch is free to rotate about any of two vertices 1. upper-right vertex & 2. lower-right vertex without changing any of the conditions provided. This rotation shows that the quadrilateral is a trapezoid (having two right angles & two parallel sides not necessarily equal in length).

Thus, the resulting figure (given here) is generally a trapezoid not a square. It will be a square only if the angle between the sides of 9 inch is $90^o$ as an additional condition for this question. Assuming this condition (although not given here) some answer it as a square that is absolutely wrong.

Answer 6

The point is that mathematically, you can't tell from the picture. It might be this:

It is easy enough to describe a construction of this with compass and straightedge, so it is definitely a legitimate geometric figure by any reasonable definition.

The same "diagnostic test" from which this came (thanks to Barry Cipra for finding it) has numerous other zingers like this where assumptions are made based on the fact that you can't tell from the picture whether two segments are equal or whether two angles are equal, so we assume they are equal.

But the question that really stood out to me was this:

Who measures the lengths of frogs that way?

Answer 7

In our 5th grade math group, our teacher came across this multiple choice problem that we thought about for a short period of time. We discussed the answer, which we concluded would be a trapezoid, square, or a rectangle. We also discussed what the tutor should do about this problem. We had varying answers, from the tutor having a conference with the teacher with/without the student to the tutor just letting it go. We also thought this was a very cool situation and we all could relate to having us being right and the teachers being wrong.We thought it could be a good idea to include the students when the tutor discusses the problem with the teacher because the students could state their thinking about the problem.

Answer 8

Of course the mathematically correct answer is as the OP and others have stated. However, if you showed this drawing to an architect, engineer or carpenter they would probably assume that all sides are 9 inches - you don't usually indicate all measures, but assume that the missing lengths are equal to the opposite sides.

Answer 9

One might also point out that a square is also a special case of trapezoid, rectangle, rhombus, quadrilateral and parallelogram, so any of those answers would technically be correct, assuming the object in question was, indeed, a square.

However, I think what everyone is missing is that while in a classroom the "right" answer is the one the teacher is looking for, even if that answer isn't the "correct" one... I've found in my experience that teachers can be obstinate, and aren't likely to be pleased by being corrected by a student in front of the class. Better (for ones academic success) to do so after class and let the teacher correct him/herself in a future class, or not...

Answer 10

It's a trick question. The answer, as you concluded, is a trapezoid.

The figure is deliberately drawn to look like a square to fool the unwary.

The best procedure to teach your student how to determine and prove such things for herself. Once she can prove to herself the figure is a trapezoid, your "reputation" is irrelevant. Math does not depend on reputations, it depends on proof (thank god).

Answer 11

The answers are replying you your question #1, but not question #2.

1. Who is right? Is there a chance that we're both right?

2. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

1. As explained by other answers, you are right.

2. Rather than feel humiliated, the student could explain why it is a square, showing counterexamples, and feel like a genius. I would also call the teacher, and I will explain why:

First, as for the student: Part of being a tutor is making sure the student understands the "why." This is a great opportunity for them to learn to stand-up against peer pressure (and faculty!).

Second, if you approach the teacher carefully, they might give the student the opportunity to explain to the class the source of the misconception. That would boost their confidence and be a learning experience for all. If you think there is a chance for confrontation:

You are aware that whoever created the question most likely intended the answer to be "square" and just wasn't meticulous enough to notice that with too few angles specified it need not be one, I suppose? – Daniel Fischer♦ 22 hours ago

This angle allows to have the student say "Well, the author of the test intended... but..." which doesn't come-off as confrontational.

Answer 12

I can't help but say something... As noted in other answers, this is clearly a trick question, playing on deliberately misleading visuals, and potentially on delicate (non-universal!) semantic conventions. (I am disturbed by the idea that, for example, a "square" is not a "rectangle", because, supposedly, "rectangle" only refers to (actual) rectangles that are not squares, etc.)

The element(s) of "arbitrary/capricious authority" that enter in both the context and in responses is completely unsurprising, but also chronically upsetting to me. Such episodes advertise the apparent utility of mathematics for creating and enforcing arbitrary, unfathomable rules, as well as highlighting the specific irrationality of "external, uncommunicative, ineffable" authority. Really ugly.

Let's admit to the kids that the picture was drawn to look like the dang thing was a square. Seriously! It's not a klutz's drawing of a real thing, it's a test question. It's not that we have to wonder about the verisimilitude of an inadequate reporter, but, rather, to wonder about the ulterior motives of people at ETS in New Jersey, etc.

In particular, instead of the too-popular traditional rather sub-verbal responses to such questions (is it an X, or not?), there should always be sufficient room to explain/address the genuine issue, as opposed to merely-semantic, merely artifactual. That is, we should teach kids to write prose that says "well, the picture makes the figure look like a square... the given data wouldn't itself physically require that it be a square, but what bumpus would draw a thing to look like a square if it wasn't?..."

(Seriously, very many peoples' physical intuition is excellent, but then we consistently prank them so that they think that there's scant connection to mathematics, which is completely false. We should teach kids to trust their physical intuition at least as a first approximation! Math is not perversity!)

But, yes, for multiple-choice tests, ABSOLUTELY tell your kids to deconstruct the stupid things, and imagine what the test-maker was thinking. For that matter, we should admit to the kids that those test-makers have a streak of mean prankishness that they (the kids) should be aware of. Too bad.

Answer 13

1. Who is right? Is there a chance that we're both right?

I don't have anything to add to what is already said in other answers.

2. How should I handle this? I told my student that I would email the teacher, but I'm not sure that's a good idea.

Take the high road.

Unless your pupil is in a great danger (losing scholarship or similar), I think you should not contact the teacher.

Instead, talk with your pupil, explain honestly the situation, tell her/him that similar situations are frequent, and that she/he should be proud of seeing better/deeper than the teacher, but at the same time do not disrespect the teacher.

Answer 14

In Singapore math questions of this nature, they almost always preface the question statement with the phrase: "not drawn to scale".

That phrasing might seem redundant, but in cases like this, it becomes so very important. Even if the figure is printed as a perfect square, the disclaimer that the figure is not to scale means that no conclusions at all should be assumed from studying its general shape. Not even the acuteness/obtuseness of angles should be assumed from line segment orientation. Only angles, sides and relationships that are explicitly defined may be assumed in solving such questions.

If that phrase had been included, I would have absolutely no hesitation in stating that the only possible correct answer is "trapezoid" (or "trapezium" as we refer to it over here) and that "square" is totally wrong.

However, without that disclaimer, a case, however weak it may sound to a rigorous mathematician, may be made that the evidence of the senses (and actual measurement) indicate that it's a square, therefore the answer "square" is also acceptable! In fact, three of the answers now become admissible - trapezoid, rectangle and square, again invalidating the expected single choice format.

So either way you look at it this is a very poor question.

With regard to the other point about the student being humiliated for giving the "most correct" answer, tell her there's no shame in it, and she shouldn't feel bad about it. I know those are hollow words as I've been in the same position myself many years ago, having been embarrassed by my (completely correct) answer being dismissed by a Physics teacher who didn't know what he was talking about (he later sent out an erratum to correct his error, without apologising to me (or even acknowledging my correctness)). Did that hurt? Yes. Did I survive? Certainly.

We must remember that teachers are human, with very human foibles. They are certainly fallible.

Answer 15

It could not be only square because if it is square then it is a rectangle too and if it is rectangle then it is trapezium too. And I don't think so that it is multiple correct type. But it is sure that it cannot be a triangle as it is a four sided figure.
Yes you were right. By the given information it can't be proved that it is rectangle or square so the answer I think must be F.) trapezoid