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

enter image description here

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.

The Chaz 2.0
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    You are right. You can construct many counterexamples by changing the angle between the two sides of length 9. – onetoinfinity May 19 '15 at 22:26
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    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 May 19 '15 at 22:33
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    @Daniel: Even if that were the case, do we expect students taking the test to answer the question as given, or do we expect them to somehow divine the intent of the question setters? ...On second thought, given the quality of standardized testing in many places, don't answer that. –  May 19 '15 at 22:54
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    Tell your student if this is for an exam, the answer can be a square (depends on the mindset of the teacher). If this is for his/her own learning purposes, the answer is a trapezoid. – achille hui May 19 '15 at 23:06
  • @Rahul The downside to answering the question as given is that the teacher/grader has an answer key ... – Daniel Fischer May 19 '15 at 23:08
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    Would it make sense to migrate this question to http://matheducators.stackexchange.com ? – Barry Cipra May 19 '15 at 23:19
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    As a tutor I've always felt that I have no standing in the classroom, however I've also felt that I'm really working on behalf of the parent, who certainly *does* have standing. I would suggest that a parent e-mail the teacher. I haven't run across a parent willing to pay for a tutor who is unwilling to go to bat with the teachers or administrators on behalf of their child. – Todd Wilcox May 19 '15 at 23:52
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    @daniel, that question is speculative at best! If the teacher didn't create the problem, then my immediate concern is still with the teacher (and not the creator). That they would insist on a wrong answer is the point of this post. – The Chaz 2.0 May 20 '15 at 00:09
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    Tell the truth. You based your answer on the fact that the angles of the right side are unspecified, and the fact that the angles on the left are explicitly noted as being right angles. This leaves you with a question the student *should* ask the teacher: if the angles on the left side are explicitly stated as right angles, why is the right side *not* explicitly stated if the intended answer is a square? In other words: the question is confusing and poorly presented (assuming the teacher is "correct"). Use this as an opportunity to teach the student to *think* and explain/defend their answer. – jpmc26 May 20 '15 at 00:21
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    @TheChaz2.0 They *see* a square. Probably, if you go and explain to them that the data does also fit non-squares, they'll see and understand. Maybe not if you try to do that in front of the class. But it will take an explanation for them to un-see the square. I don't expect that your student would have had the composure and self-assuredness to explain the geometry when the teacher said her answer was wrong, so the teacher's misconception probably still persists. Go and - calmly, it's an honest mistake on the teacher's part - explain the matter. – Daniel Fischer May 20 '15 at 00:22
  • My state's standardized test did something similar - asking for the area of what was clearly intended as a rectangle, but only had two opposite right angles marked (so the problem was unsolvable, if you actually did it properly). – Milo Brandt May 20 '15 at 03:43
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    Another issue with this poorly-constructed question is that if the figure is considered a square, it is then also a rectangle and a trapezoid, so only one answer could be considered wrong. – Kimball May 20 '15 at 03:55
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    Keep in mind that a trapeziod in England has _no_ parallel sides, so a square is the only possibility. However, given OP is from the US and presumably the school and test maker is in the US, trapezoid is the right answer. –  May 20 '15 at 06:43
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    @paxdiablo: According to [Wikipedia](http://en.wikipedia.org/wiki/Trapezoid) *the term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage. (The Oxford English Dictionary says "Often called by English writers in the 19th century".)* with a reference to the Oxford English Dictionary – WoJ May 20 '15 at 15:07
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    I wouldn't think of it so much as "humiliation" or "reputation", etc. I remember I once had a student tell a teacher that his computations were wrong. He said "Well, maybe, so either I made a mistake or you made a mistake, doesn't matter" – CaptainCodeman May 20 '15 at 15:55
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    I hope that you do send an email to the teacher with a polite, clear explanation of why the answer is what it is and why the answer key is wrong, perhaps with a link to this post (assuming none of the answers here have bad-mouthed the teacher). Sorry your student got caught up in this kind of thing. Please let us know how this resolves. – Reinstate Monica -- notmaynard May 20 '15 at 16:18
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    if [the creators of this test](http://www.mathmatuch.com/presentations/diagnostic_test.pdf) are ever tried, they will probably [plead incompetence](https://www.youtube.com/watch?v=G6D1YI-41ao) :-) – sds May 20 '15 at 20:51
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    "If" I was to give them (the makers of the test) the benefit of the doubt, I would say the length "9" might have been originally assigned to the top instead of the right side (so length of 9 for top and bottom), which would have made "J: square" the correct choice, and then during typesetting, the 9 was moved to the right since as a square, all sides would have to be length 9. Still an error on their part, but perhaps made by someone who was unaware of the ramifications of the change. – Kevin Fegan May 21 '15 at 00:49
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    @KevinFegan - surely the `9"` top-and-bottom only gets you *rectangle*, as the left-and-right sides are of unspecified length, and could be any length at all and still satisfy the abstract requirements of the diagram. – Dɑvïd May 21 '15 at 08:56
  • @WoJ No- US/UK differ on trapezoid/trapezium. Bizarre thing to do - but apparently we switch around which is one and which is no set of parallel sides. – OJFord May 21 '15 at 09:26
  • @OllieFord: I understand that there is a difference in the meaning of both words between countries (which is a nightmare on its own). What I am saying is that, in the UK, *trapezoid* in the sense *quadrilateral without any parallel sides* is deprecated (according to The Oxford English Dictionary). The same goes for France where the word *trapézoïde* mentioned in the Wikipedia article I reference is not used anymore. – WoJ May 21 '15 at 10:26
  • @TheChaz2.0: how old is the pupil? (I am in France so the "grade" per US classification may not be obvious, I will match the age with our school system here). – WoJ May 21 '15 at 10:30
  • @WoJ Ah okay. I certainly recall it's use to mean 'trapezium' being regarded an horrendous Americanism! Probably the fact that we didn't think "no, it has parallel sides" means you're correct; it is deprecated. Is 'trapezium' likewise deprecated in AE for parallel-less quads? – OJFord May 21 '15 at 10:31
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    @OllieFord: I cannot say, I am French :) I was curious as how it works in some other European languages and it is really messy. I have not, ever, heard the word *trapézoïde* being used for a geometric figure in France (the word is *trapèze*, which has two parallel opposite sides - for at least ~40 years) but there are indeed ancient references to *trapézoïde*. The depreciation is also [gaining traction in Germany](https://de.wikipedia.org/wiki/Trapez_(Geometrie)#Begriffsgeschichte). – WoJ May 21 '15 at 10:39
  • Looks can definitely deceive, doesn't mean the "elephant" is not an elephant! – MonK May 21 '15 at 11:26
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    @Kimball If the test stated that there is only one correct answer for each question then you could provide the correct answer by simply counting the sides of the figure to exclude that the figure is a triangle. After that, since there's only one correct answer and that square implies rectangle implies trapezoid the correct answer *must* be trapezoid. But in this case all information about the length of the sides and the angles would be superfluous... – Bakuriu May 21 '15 at 12:28
  • Out of curiosity, why didn't the student herself draw a (non-square) trapezoid which meets the given conditions ? Did the teacher not give her a chance to prove her point ? That would be really sad. – YoungFrog May 21 '15 at 13:18
  • @woj under 13 yrs – The Chaz 2.0 May 21 '15 at 15:25
  • @TheChaz2.0: thanks, this is equivalent to primary school in France, or the first year of *collège* (the next school after 5 years of primary school). I was wondering how this is aligned with what children learn here. For your information, the fact that, say, a square is also a rectangle is not particularly highlighted either (which does not say much about the reaction of the teacher when confronted to such a problem). – WoJ May 21 '15 at 15:29
  • @WoJ - indeed, I don't usually say "this shape **is** a square/rect/rhombus", but rather "this shape *has the properties of* such...". – The Chaz 2.0 May 21 '15 at 15:30
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    I know I am geometrically challenged, but I swear there's no way this is a trapezoid of any kind. A trapezoid has four straight lines. The figure clearly has more than four straight lines. – Git Gud May 21 '15 at 23:05
  • So, just a question for the OP... did you see this question in isolation, or did you know it appeared on a test right next to "classify the [unlabeled] angle shown" and "name an obtuse angle in the [unlabeled triangle]"? – Random832 May 22 '15 at 02:40
  • @David Of course, you are right... never-mind. – Kevin Fegan May 22 '15 at 08:09
  • The way I view that diagram, it's saying that the right leg is 9 inches long (rather than the height), but that doesn't necessarily mean the left leg is 9 inches (but we can deduce that it is less than or equal to 9), so I'm in agreement that this is not definitively a square or a rectangle, but is definitely some sort of trapezoid. – GuitarPicker May 22 '15 at 13:58
  • @yes nice title change, – davidlowryduda May 22 '15 at 19:21
  • I would be curious to ask, (1) why and (2) how was she "publicly humiliated" in front of the class? (1) How did she present and argue for her answer? If she did that badly (e.g. arrogantly, confrontationally, or tactlessly), then this would be an important learning opportunity for her. (Not that it excuses the teacher.) I've been in that boat and had to learn the hard way. (2) Do you mean someone intentionally insulted and shamed her, or do you just mean she was embarrassed because people concluded she was wrong? The former is a real problem; the latter, more of a life lesson. – LarsH May 25 '15 at 21:10
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    If we're supposed to infer that the picture is a square based solely on its appearance, then why did the problem-setter bother to specify two of the angles and two of the sides? – WillO May 20 '15 at 01:40
  • @TravisJ Yes it does. It argues through the use of a rhetorical question that J is not the correct answer. – Will Sawin May 24 '15 at 07:31
  • @WillSawin, the questions were 1) who is right? (this post doesn't address that) and 2) what should I do? (the post doesn't address that either). In fact, the post only includes a question. – TravisJ May 24 '15 at 20:24
  • TravisJ: Suppose II had said "If we were supposed to infer that the picture is a square based solely on its appearance, then the problem-setter wouldn't have bothered to specify two of the angles and two of the sides", 1) Do you agree that this would have addressed the question? 2) Do you agree that it conveys exactly the same meaning as the rhetorical question I posted? – WillO May 24 '15 at 20:28
  • Perhaps more importantly, why did your student not produce a counterexample (e.g. by drawing on the board) after her answer had been questioned? Possibly shyness/anxiety? (apologies if someone asked this before, did not read all the comments) – Thomas May 27 '15 at 10:40
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    @TheChaz2.0 Can you give us an update? I want to know what happened! – Rushabh Mehta Nov 15 '18 at 17:39
  • i think i actually remember seeing this 5.5 years ago. wow over half a decade ago. good times. good times. – BCLC Dec 01 '20 at 00:46
  • related? https://math.stackexchange.com/questions/2249741/why-do-we-have-circles-for-ellipses-squares-for-rectangles-but-nothing-for-tria and https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares and https://matheducators.stackexchange.com/questions/13831/are-kindergartners-supposed-to-be-steered-from-squares-being-rectangles and https://korean.stackexchange.com/questions/3394 and https://chinese.stackexchange.com/questions/29214/ – BCLC Dec 01 '20 at 00:47
  • well good for you to the answerer Moby Disk so magnanimous and generous i don't see any moral obligation to be non-confrontational though. it's like henry mccord in this one episode of madam(e) secretary where his wife was being subjected to some senate hearing or something but then it was like the way it was done reeked of dishonesty. therefore, henry mccord, when subpoenaed refused to be honest and intended to commit perjury. this was despite that he was a religious, philosophy or ethics professor. – BCLC Dec 01 '20 at 01:12
  • but 1stly, the student was *humiliated*? i think BOTD is off the table. 2ndly, i mean, additionally, others even point out like the writer goes through the trouble of putting those 2 angles there. i mean obviously this isn't kindergarten teaching or anything (see a previous comment) so there's really some deduction to be done. it's not like you just look and that's that. 3rdly the whole 'OBVIOUS'. i mean come on. anyone who uses 'obvious' really deserves scrutiny in this case, where the case is like...it's like when people ask you a trick question but then they're the ones who are wrong. – BCLC Dec 01 '20 at 01:13
  • @DonThousand - I think I just expressed my concerns to the student and her mother, without pressing the issue too much. Sorry that there wasn't a more exciting end! (One time I confronted a teacher about her mixing up "point-slope" and "slope-intercept" forms of a line. She basically shrugged via email.) – The Chaz 2.0 Dec 01 '20 at 16:36
  • Ok the other angles can be $89^{\circ}$ and $91^{\circ}$. – Etemon May 20 '21 at 21:43

15 Answers15

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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.

enter image description here

David G. Stork
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    Right, and I showed this to my student yesterday. I'm wondering if the class is just at that place where, if you draw [what kids call a square], then it's a square. – The Chaz 2.0 May 19 '15 at 22:27
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    @TheChaz2.0 If that's the case, then it's pretty silly to give any angles or any side lengths, as it's just as "obviously" a square with or without that information. Or so I would think... – pjs36 May 19 '15 at 22:29
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    What software did you use to make the diagram? – GFauxPas May 20 '15 at 01:00
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    Top notch picture... is proof in of itself that the bottom and right can be the same length, the left angles can be 90(deg) and the figure not a square. Made peffect sense the second I saw it. – WernerCD May 20 '15 at 12:53
  • What grade level are we talking about? And though I agree with this answer I would word the email very carefully and avoid pedantic-sounding words like "clearly" and "of course". For one the problem isn't clear to a lot of people, including teachers, and should be explained as clearly and plainly as possible. For another, belittling the teacher will not help the student, the tutor or the class. – Paul Sasik May 20 '15 at 22:52
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    That picture itself is an amazing answer. – Barry May 21 '15 at 02:27
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    @Barry and GFauxPas: *Mathematica*. – David G. Stork May 23 '15 at 01:41
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    @DavidG.Stork: One day, Mathematica and I will be married. – Vatsal Manot May 23 '15 at 11:04
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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.enter image description here

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

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

Good luck!

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.

enter image description here

Adhvaitha
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    Note: "Trapezium" (UK) = "Trapezoid" (US). Another confusion. – GEdgar May 21 '15 at 12:04
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    I like your first picture. Your second will only really help if there is confusion about the definitions of the shapes involved. I don't think that's the case. – Teepeemm May 21 '15 at 15:04
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    @Teepeemm Judging from the way the instructor seemed to have answered the question, it does seems the case that (s)he is confused about the definitions of the shapes involved. – Adhvaitha May 21 '15 at 15:49
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    I disagree. I think the second picture only helps if someone thinks the first picture is a square. The fundamental problem the teacher and the test makers assumed that another side or angle were congruent just because they appeared to be. – Teepeemm May 21 '15 at 17:36
  • I think the first picture would be better if it had a 9.1 and an 8.9 or whatever it ends up being with an 89 degree angle or so, and the other angles marked. – Random832 May 22 '15 at 02:36
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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.

Moby Disk
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Barry Cipra
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    +1 Interesting, when I look up the publisher wiki's entry, it has been ranked the $10^{th}$ worst company to work for in the state. Not sure whether there has anything to do with the quality of its test. – achille hui May 19 '15 at 23:46
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    The sad thing is that lifes are destroyed by such tests set up by mathematical morons. – Christian Blatter May 20 '15 at 15:32
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    This is one of many problems with standardized tests, and I say this as someone who is very good at taking standardized tests. What's _really_ frustrating is when you're taking a standardized test and the correct answer isn't any of the ones presented. I remember this happening on a teacher certification test I took quite a while back. It was a simple physics problem but the correct answer wasn't one of the four listed. I re-read the question multiple times to make sure there wasn't some detail I missed, but, no, just a bad question. (I'll note that the test was heavily biology-centric.) – Ben Hocking May 20 '15 at 21:56
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    Q73 asks for the length of a frog, which is positioned against a ruler. But not positioned in any way that would facilitate determining the length. This would be a fun test to answer facetiously! EDIT: Oh, as @DavidK answers. – OJFord May 21 '15 at 09:28
  • Question 75 *before* the OP question asks to name an angle and I wondered why $\langle DBC$ was not among the suggestions, which shows that there is not *the* angle frmed by the dashed rays ... Nevertheless, I would certainly fail Q71 :) – Hagen von Eitzen May 24 '15 at 17:23
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    Ironically here is one of their glassdoor reviews from an employee: http://www.glassdoor.com/Reviews/Houghton-Mifflin-Harcourt-Reviews-E275.htm "Drastically undervalue people's experience and input. Middle Management has been promoted from various positions within education or sales and doesn't have solid foundation in product development and launch to manage teams well. Marketing leadership lacks insight, leadership, and notion of team development. Too much focus by middle managers to checking off boxes, and doing things because they think they should be done. " – don bright May 25 '15 at 13:08
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The point is that mathematically, you can't tell from the picture. It might be this:

enter image description here

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:

enter image description here

Who measures the lengths of frogs that way?

David K
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    I glanced though the test and had exactly the same thought. – WoJ May 20 '15 at 15:03
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    Yeah, it's a really poor test. – Reinstate Monica -- notmaynard May 20 '15 at 16:23
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    How the OP talks about this is a little worrying. The OP spotted a gotcha that even the test makers didn't spot but appears to be in some kind of conflict of wills with a child who's at the level of being asked to determine the length of the frog that is so serious that they're worried about their reputation. At that age if my teacher had brought up say, some example involving imaginary numbers technically allowed by a mistake in the phrasing of the question and actually got into a serious conflict with me I'd have only been put off math, not educated. – Murphy May 20 '15 at 16:54
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    If the tutor's answer delved into mathematics beyond what the question asked about, this would be a different question. The test clearly expects knowledge of what makes something a mere trapezoid rather than a rectangle, and clearly indicates _just enough_ information to establish a trapezoid (unless you choose to regard the right-angle symbols as superfluous). So we're left trying to guess what answer the test makers wanted, based on which rules of math they arbitrarily chose to think about that day. _That_ is what I think is likely to turn students off math. – David K May 20 '15 at 17:15
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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.

paul garrett
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    Though I understand where you are coming from, aren't there also mathematical situations where data genuinely look like something they aren't (for example, aliasing from undersampling)? –  May 21 '15 at 01:37
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    @Strants, oh, yes, indeed, sometimes _things_ misrepresent _themselves_, and it's good to be alert to that. For that matter, sometimes one deceives oneself via inadvertent semantic boo-boos. But, unlike a contrived testing situation, in those cases we behave as though there were a genuine underlying truth, rather than a truth accessible only to a capricious authority. – paul garrett May 21 '15 at 11:58
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    My issue is that three of the four options are correct answers. This isn't a case of a trick question or a kid (or tutor) over-thinking it. This is a bad question. – Kevin May 21 '15 at 12:02
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    @Kevin, yes, _also_ it is a bad question, but/and to analyze it as though there were underlying truth is a waste of energy. – paul garrett May 21 '15 at 12:17
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    If the "correct" answer was trapezoid then this would be an example of a "trick question". But since the test's answer key indicates that the answer should be square, it's not — only a very *bad* question. – hobbs May 21 '15 at 14:50
  • This is not a trick question. If you have a computer program, there is no common-sense anywhere in sight. Any ambiguity at all that makes it insufficient to claim 'square' means that you must settle for 'trapezoid' as an answer. – Rob May 21 '15 at 20:03
  • @Rob, there is a large range of opinion about whether mathematics should be written as though it had no context (that is, denied any notion of "common sense"). In practice, it mostly _does_ admit a huge "overhead" of prior experience and resulting sensibilities, contrary to what is often portrayed or claimed in school and in textbooks. Other people have the opposite opinion, etc. – paul garrett May 21 '15 at 20:38
  • For my fourth grader, I challenge answers that don't have units, or are missing algebraic steps (like "(y=x * x)/x, x!=0, (y/x = x * x/x), (y/x = x)"). The whole point of studying geometry and algebra is to study air-tight reasoning. The wrong reasoning of the diagram is surely responsible for expensive building mistakes where walls don't come out square. :-) – Rob May 21 '15 at 20:57
  • @Rob, by the idealized standard, there _are_ no square walls, etc. I'm not sure that airtight reasoning is better taught as adherence to rules, as opposed to exercise of judgement about what is _important_. If all details are equally important, then, in effect, _none_ is, and all that. It is very tricky to draw a useful line between pedantry and carefulness, etc. For my own kids, and undergrads, and grad students, I emphasize "sanity checks", and I tend to declare a long, line-by-line computation or proof "unpersuasive" if there's no over-arching _cause_ delineated. Tastes vary. – paul garrett May 21 '15 at 21:42
  • So given that the answer key is wrong, it proves the point. The teachers learned waffly "well obviously, it looks like..." reasoning, and can't even produce correct tests that children are supposed to pass. When you are using strings and stakes in the ground (ie: building a deck or doing minor construction), you have already gotten into an area where 'looks like' is not even close. – Rob May 22 '15 at 14:50
  • For geometric proofs, you can't assume anything. Is there a 90 degree-angle symbol? No? Then no matter how perpendicular it's looking, you can't assume it's a right angle. I've seen drawings that look like squares, but are rectangles labeled with side of 3 and 9. I remind students "not to scale." In this case the two symbols produce one set of parallel lines. That's it. The teacher is wrong, and if the book said "square" I'd mark it up with a sharpie. – JTP - Apologise to Monica May 22 '15 at 16:33
  • @Kevin No, only one of the choices is correct in this question: trapezoid. – Sildoreth May 22 '15 at 17:16
  • @Sildoreth, yes. I agree. I meant that, if you accept what Mr Garret is saying that it was obviously meant to be a square, THEN three of the four answers are correct. – Kevin May 22 '15 at 18:52
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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.

Ken
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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.

Heath School 5th Grade
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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.

Stefan
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    Exactly! Such questions are perverse. We should not teach students to be paranoid about "gotchas", which are not at all the _primary_ issues and ideas in mathematics. Many pranks and traps are not even of much secondary interest, except to illustrate that the test maker(s) do not know any genuine mathematical questions to ask. Tsk! :) – paul garrett May 20 '15 at 18:25
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    If you want to be good at maths, you do need to be paranoid about "gotchas". They come up all the time in the real world by themselves! – Brice M. Dempsey May 21 '15 at 08:00
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    I don't think that's true at all. Admittedly I'm a mere 'engineer in training' - but if you showed that diagram to me I'd think "Right - these are 9in, these are 90deg, and I'm free to do whatever's most convenient to me with the others. Done: http://i.stack.imgur.com/rWjZM.png". This happens a lot. And it's usually good. When it isn't, it's the fault of the specification; not the work. – OJFord May 21 '15 at 09:36
  • Naturally, when you are doing something practical, you will make a lot of assumptions to get a conclusion. But, make "one too many" assumption and your project will break apart. This is why mathematics exists in a first place, to have a tool that allows us to investigate real life problems and find errors in our "natural thinking". This is an exercise in "seeing" beyond your senses, finding multitude of possibilities instead of one. This is a quality everyone should cultivate. – Ennar May 21 '15 at 18:55
  • @Ennar, I absolutely do not see any disconnect between "mathematical thinking" and "natural thinking". – paul garrett May 21 '15 at 22:01
  • @paulgarrett, really? You'd say that average human is inherently a mathematician? I'd quite disagree, I learned most of the "mathematical thinking" as a student, and lot's of it became "natural" only after a while. – Ennar May 21 '15 at 22:09
  • @Ennar, the average human has quite wonderful spatial visualization, avoiding walking in to walls, chewing gum and walking at the same time, hitting balls with bats, all kinds of stuff. Semantic things, especially _purely_ semantic, are not so good. Humans do not use language with any facility. So I argue against divorcing mathematics from physical reality and making it purely semantical. That kind of thing. I do tell all my students to trust their physical intuition (note, I don't say "semantic intuition"). – paul garrett May 21 '15 at 22:18
  • @paulgarrett, this a valid point about how our brains work and I agree it's quite amazing. I'm not advocating one should separate mathematics from reality, quite the opposite, but, mathematics is in part a language. Our physical intuition might tell us that the quadrilateral in question is square, but semantics tell us it might not be. Thus my distinction. – Ennar May 21 '15 at 22:37
  • @Ennar, a more important watershed, to my mind, is whether what we write is a _narrative_ of some trans-semantic events, or whether the written account _is_ the thing. The latter is brittle and fragile, the former elusive but adaptive. Thus, a flawed _account_ of events/phenomena that have their own reality is different from the failure of prose to conform to rule _for_prose_itself_. Information is always incomplete, context is always a little ambiguous, etc. Why draw a thing to look like a square if it did not look like a square "in life"? :) – paul garrett May 21 '15 at 22:50
  • @OllieFord If you're saying that these type of "gotchas" don't come up in the "real world", I've had the opposite experience in computer programming. Computers, users, and even other programmers, are fantastic at finding uses you nearly always won't dream of, and completely invalidate some assumptions you may have made during initial design. For the "what shape is this?" type questions, they don't seem relevant to "real life" because we're virtually standing in a room pointing at the same object. In my experience, I don't get to stand in the same (virtual) room and point at the same thing. – r12 May 22 '15 at 01:04
  • @r12 In CS, consider implementation-defined compiler behaviour. The standard is suitably vague as to allow the compiler designer some freedom - whether that's to best suit the eventual user, or just what makes for simpler compiler design, it's up to him. Also e.g. MIPS specification leaves room for different conforming designs. – OJFord May 22 '15 at 01:35
  • I shall confute that the sketch be acceptable by an engineer : it's so much undefined as for a mathematician , and even more, since tolerances on measures are unspecified !. – G Cab Jan 01 '20 at 12:49
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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.

Moby Disk
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  • well good for you so magnanimous and generous i don't see any moral obligation to be non-confrontational though. it's like henry mccord in this one episode of madam(e) secretary where his wife was being subjected to some senate hearing or something but then it was like the way it was done reeked of dishonesty. therefore, henry mccord, when subpoenaed refused to be honest and intended to commit perjury. this was despite that he was a religious, philosophy or ethics professor. – BCLC Dec 01 '20 at 01:06
  • but 1stly, the student was *humiliated*? i think BOTD is off the table. 2ndly, i mean, additionally, others even point out like the writer goes through the trouble of putting those 2 angles there. i mean obviously this isn't kindergarten teaching or anything (see next comment) so there's really some deduction to be done. it's not like you just look and that's that. 3rdly the whole 'OBVIOUS'. i mean come on. anyone who uses 'obvious' really deserves scrutiny in this case, where the case is like...it's like when people ask you a trick question but then they're the ones who are wrong. – BCLC Dec 01 '20 at 01:08
  • related? https://math.stackexchange.com/q/2249741 and matheducators.stackexchange.com/questions/13700/ and matheducators.stackexchange.com/questions/13831/ and korean.stackexchange.com/questions/3394 and chinese.stackexchange.com/questions/29214 – BCLC Dec 01 '20 at 01:08
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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).

Tyler Durden
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    Apparently it also fooled the people who created the test... – TravisJ May 20 '15 at 19:54
  • @TravisJ ? The people who designed the problem know the correct answer. It is only this local teacher who got it wrong. – Tyler Durden May 20 '15 at 21:25
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    [Apparently not only the local teacher](http://math.stackexchange.com/a/1290350) – Daniel Fischer May 20 '15 at 21:36
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    "To fool the unwary"? Genuine mathematics is not created by pranksters to try to fool each other. Genuine problems are often difficult enough without nincompoops creating fake issues. This is why I am ever more disaffected with "school mathematics", despite having considerable enduring affection for real mathematics. I am not at all surprised that many kids grow up with a bad feeling about math, since it is too-often used as a way to prove to kids that the caprices of authority ... win every time... and that there's no sense to it, and that they're trying to prank you. Ugly. – paul garrett May 20 '15 at 22:39
  • @DanielFischer Hmmm, interesting. Maybe it is just a typo. They either might have meant to make the right angles katacorner or else the answer key just has a typo. – Tyler Durden May 20 '15 at 22:50
  • @TylerDurden "make the right angles katacorner" would yield a kite. – Random832 May 22 '15 at 02:33
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    @TylerDurden - That's a pretty significant typo, to match what's drawn rather than what's specified. In fact, an *answer key* having a typo invalidates the entire concept of multiple choice testing. – Bobson May 22 '15 at 11:23
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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.

Deepak
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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.

Harish Chandra Rajpoot
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  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.

  1. 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.

VividD
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    Why would she respect the teacher who is too narrow-minded to admit mistake or cannot comprehend it? Everyone can make mistake, there is no shame in that, but the teacher that can't appreciate being corrected by pupil should not be respected. I am a maths teacher myself. – Ennar May 21 '15 at 18:40
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    In my view, being a teacher is a sacred role, and being a pupil the same. Both teacher and pupil should have mutual respect, EVEN if one of them is just plain wrong, and EVEN if the opposite side shows disrespect. Don't forget that no man is perfect. On the other hand, role of the tutor is more practical. This means closer relation to pupil, and in a way more direct communication. @Ennar – VividD May 21 '15 at 21:32
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    I agree that there should be mutual respect between a teacher and a pupil, but I cannot and will not expect from my pupils to respect me if I can't show them the same. This actually extends to anything in my classroom, all rules apply to them as they apply to me. If I'd violate any rule, I cannot hold it against them if they follow my lead. A teacher must be a role model and has to accept that any action is copied by his/her pupils. But, not to to stray too much, should the pupil in question be courteous to her teacher? Yes. Should she respect him? No. – Ennar May 21 '15 at 21:53
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    In this situation, there is no reason whatsoever to respect the teacher. He publicly humiliated the pupil. Respect has to be earned. – Syd Kerckhove May 22 '15 at 08:55
  • I agree that this teacher does not deserve respect. However, it is not disrespectful to someone to point out their error. So even if they did deserve respect, they should know about their error. – Joshua Meyers Mar 01 '17 at 21:03
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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

sgrmshrsm7
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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...

Alan Amaral
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    The problem wasn't to have the teacher corrected in the first place but that the student with the right answer was humiliated in front of the class. – JFS May 20 '15 at 18:27