Definition of Trapezoid

Question

From one textbook we use in our High School -

Transcription:

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases of the trapezoid.

And from Wikipedia -

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium (/trəˈpiːziəm/) in English outside North America, but as a trapezoid[1][2] (/ˈtræpəzɔɪd/) in American and Canadian English.

One other textbook in my school follows the Wikipedia definition.

The former definition excludes parallelograms and rectangles. The latter, defines both to be a subset of trapezoids.

How do we address this with students? I'm starting to get objections to the textbook image I posted, with the student either recalling having learned it differently in a prior class, or searching and declaring another source as the contradiction to our textbook. I can offer a response of "This is one of those math naming properties that doesn't have 100% agreement." Although this doesn't seem satisfying.

For historical context on the inclusive and exclusive definitions of "trapezoid", see my answer at https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares/13766#13766 — mweiss, Jan 24 '21 at 14:45
An important thing for Math Educators is that if one of your students comes up with either of these definitions, even if only one of them is in the book, the student is not wrong. When I was a young student, I often got in trouble for answering questions that were beyond the lesson plan (most egregiously, I got sent to the principal for answering: "Jane, Bob and Bill each have three apples, how many apples do they have together?" by saying "3 people each with 2 apples, 3 x 2 = 6". The teacher argued "you haven't learned multiplication yet", with me answering "well, obviously, I have" — Flydog57, Jan 24 '21 at 22:57
Oops, "... each have two apples" (I can't edit a comment). In any case, the teacher insisted that I write the solution as "2 + 2 + 2 = 6" and that "2 x 3 = 6" was wrong because "you haven't learned multiplication yet". Ah, 3rd grade (back in the 1960s). — Flydog57, Jan 25 '21 at 00:46
And don't get started on "isosceles trapezoid", "acute vs. right vs. obtuse trapezoid", "tangental trapezoid", etc... — Darrel Hoffman, Jan 25 '21 at 14:57
Definition of Trapezoid: "Any of several mechanisms designed to catch zoids. Originally the brand name Trap-a-Zoid™ but as with Kleenex™ the term now refers to the category of devices." (I'll see myself out...) — T.J. Crowder, Jan 25 '21 at 15:08
As an American, I've only heard the term trapezoid used to refer to shapes with exactly one set of parallel sides. I think shapes that Americans would call trapezoids may be called trapeziums elsewhere, and the definition of trapezium allows for shapes with two parallel sides, but the term trapezoid does not. — supercat, Jan 25 '21 at 16:26
@supercat I think the general rule is to always use the most specific term that applies. You could call a parallelogram a "trapezoid", but it's a special case of trapezoids, so you'd use the more specific term. Just like you could call a square a rectangle or a rhombus (since it is in fact both, as well as a parallelogram, trapezoid, and kite by extension), but you use the more specific term since it's a special case. Just like "Snoopy is a beagle, a beagle is a dog, a dog is a mammal, a mammal is an animal, an animal is a living organism, etc." It applies in any field. — Darrel Hoffman, Jan 25 '21 at 17:11
@DarrelHoffman: In many cases, it is useful to have both exclusive and inclusive terms for things, such as "irrational numbers" versus "real numbers". If the definition of "trapezium" is as stated, then "trapezoid" and "trapezium" would have usefully distinct meanings, with the former being exclusive and the latter, inclusive, which seems more useful than trying to treat the words as synonymous and arguing about whether it should be exclusive or inclusive. — supercat, Jan 25 '21 at 17:27
I'm of a firm belief that there's a whole bunch of "knowledge" that only exists to fill up grade textbooks and not only is not found anywhere outside said textbooks, and is of no use but to punish students who find alternative sources of information. Personally, as a student 30 years ago, I'd refer to some well-regarded math encyclopedia. These days it could be https://mathworld.wolfram.com/Trapezoid.html, and go with what's there, and basically tell the teacher a polite equivalent of "put up or shut up". As far as references go, school textbooks come last. In some cases even after Wikipedia. — Kuba hasn't forgotten Monica, Jan 25 '21 at 23:00
And I do agree that there often isn't the one and true definition, as is the case here, but even then schoolbooks come last. If a teacher truly has to confront two opposing definitions from reputable sources, then it's a nice teachable moment, but I'd personally never consider a high school (or any other grade school) math text to be reputable unless there was nothing else available (some island cut off from the world, and even then I'd like to trust my memory first). School books are that bad. American ones, Polish ones, I'm sure the lot of them are really done with zero love of the craft. — Kuba hasn't forgotten Monica, Jan 25 '21 at 23:03
In elementary calculus, we teach something called the "trapezoidal rule". But in fact, some of those trapezoids could be rectangles or even squares. For that purpose, it is useful to think of rectangles (and squares) as special trapezoids. — Gerald Edgar, Jan 26 '21 at 12:28
I write questions for math tests for elementary schools. I have been told to stay away from trapezoids because of the conflicting definitions. — Amy B, Jan 26 '21 at 12:44

score 21 · Accepted Answer · 2021-01-25T22:40:02.117

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I would use this to help students understand three "meta" ideas:

(1) Math is not about memorizing lots of random trivia. In the real world, if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care.

(2) There is not always a consensus about definitions. Get over it, boys and girls! In STEM, it's very common that when you read something, you need to check which definitions they're using.

(3) In general, in math, we prefer to make our definitions in such a way that theorems come out tidy and with a minimum of special-casing. For this purpose, it's usually good to have the things that fit definition A be a subset of the things that fit definition B. By this rule of thumb, it's preferable to define a parallelogram as being a trapezoid. If not, then any time you want to prove a theorem whose conclusion is "X is a trapezoid," you will probably have to uglify it by making the conclusion "X is either a trapezoid or a parallelogram."

Often, a reason why books will sometimes choose exclusive definitions (so that a square is not a rectangle, and a parallelogram is not a trapezoid) is that they have a low estimate of their students' intelligence. Students operating at lower intellectual levels (as well as very young kids) have trouble understanding how these definitions can be inclusive.

In this particular example, there is a possible advantage of choosing the exclusive definition, which is that then we have two sides that we can pick out as the "bases." It's a trade-off.

edited Jan 25 '21 at 22:40

answered Jan 24 '21 at 14:48

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+1 "In STEM, it's very common that when you read something, you need to check which definitions they're using." As someone who works with safety-critical systems, I feel this is perhaps the most important thing the students can learn. Far more important than any of the properties of the trapezoid/trapezium. – Ben Hocking Jan 24 '21 at 22:05
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Counterpoint to (3): There are theorems that are true for trapezoids that are not parallelograms, but false for parallelograms. So if you use an inclusive definition and you wish to prove a theorem whose hypothesis is "X is a trapezoid", you would have to uglify it by specifying the exclusion "...but not a parallelogram". – mweiss Jan 24 '21 at 23:18
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@mweiss My go-to example of this is the computation of the area of a trapezoid by extending the non-parallel sides to form two similar triangles, which is impossible for a parallelogram without adding a point at infinity. But the conclusion is still true for parallelograms in this case. Do you have a stringer example on hand? – Kevin Carlson Jan 25 '21 at 13:42
@KevinArlin That result can be proved (more naturally?) for the inclusive definition of traps by drawing a diagonal and observing that forms two triangles with the same altitude relative to the parallel opposite sides we are considering. – Matthew Daly Jan 25 '21 at 14:21
@KevinArlin The Base Angles Theorem for trapezoids is false for parallelogrammatic (sp?) trapezoids. – mweiss Jan 25 '21 at 14:55
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Maybe rephrase or remove (1). If you say mathematicians don't care, students will not have a reason to care, and may just ignore the rest of your response, which has some good points. – Leland Hepworth Jan 25 '21 at 15:30
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(1) "if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care." -- That is very hard for me to believe as a general rule. – Daniel R. Collins Jan 25 '21 at 15:32
I agree with @DanielRCollins here. Plenty of mathematicians have strong opinions about which definition is better. – Kevin Carlson Jan 25 '21 at 21:44
I object to the idea, suggested by this answer, that an entire group of students is operating at a "lower intellectual level," and with the connection to "intelligence." Learners at a lower van Hiele level do not lack intellect, nor are they of low intelligence. They merely have less experience. Haphazard references to intellect and intelligence are problematic in a forum focused on education. Intellect is related to capacity to learn, not acquisition of content knowledge. Review the controversy generated by the book "The Bell Curve: Intelligence and Class Structure in American Life." – user52817 Jan 26 '21 at 15:41

score 8 · Answer 2 · answered Jan 24 '21 at 14:03

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Unfortunately, we don't have a set of universally agreed upon definitions in mathematics. It might seem like we do (or should), especially in Geometry with its long history and so much agreement, but the truth is that we use different definitions frequently. One proof of this is the differing definitions in your textbooks! That is just the nature of a subject that evolved over millennia across a world of cultures. This is why it is so important to rigorously define your meaning in whatever context in which you're working.

In the case of trapezoids, I have heard passionate arguments for both sides, but that debate really is not important. The important piece is: define your terms.

How to address this for students.

Two thoughts:

This is an excellent opportunity to demonstrate to students that they can use whichever definition they need as long as they clearly state which one they're using! It is also a great way to engage students in formalizing logical arguments. I understand it might not seem satisfying to just state that there is disagreement, but maybe you could turn that into a productive discussion.
Focus on whichever definition will best serve them. For example, use the definition that is used for SAT, ACT, AP Exams, etc. Not because those should be authoritative organizations in mathematics, but because it will simplify things for students by minimizing discrepancies in definitions when they sit for high-stakes exams.

answered Jan 24 '21 at 14:03

Carser

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Thank-you. You've given me my next task, to confirm which definition the US standardized tests use. I'd warn students they might see the 'other' definition, but be mindful of which one is needed for the exam. – JTP - Apologise to Monica Jan 24 '21 at 14:16
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Please please please don't give in to the pressure to make education into test preparation. Students do well on standardized tests if they have a good intellectual understanding of the subject. A heavy focus on test prep is a hallmark of the worst schools, because they're trying to substitute it for intellectual understanding. Every minute spent on test prep is a minute that could instead have been spent on education -- probably with superior results in terms of testing, although that's not the point. – Jan 24 '21 at 15:03
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I agree with @BenCrowell that test preparation is not the goal of education. I hope it is clear that was not my suggestion. The claim that "students do well on standardized tests if they have a good intellectual understanding of the subject" is not directly supported by evidence. There are several reasons to believe that students do better when they are prepared for tests, i.e., the testing effect. So while I likely share many views with Ben, I consider it educational malpractice to avoid preparing students. Here in Massachusetts, we have the MCAS, and my geo students need to know definitions. – Carser Jan 24 '21 at 16:08
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I am in MA, too, Carser. I'm ok with Ben's answer, and will embrace all three of his points, but if this (I need to check) is on MCAS, I think I should know how they treat it, same as when students come see me for tutoring and I ask "what book?" This - "I consider it educational malpractice to avoid preparing students." sounds like a great version of "First, do no harm." – JTP - Apologise to Monica Jan 24 '21 at 16:52
@JTP-ApologisetoMonica Yeah I wholly agree with Ben's answer. Just not the idea that curriculum and test prep are mutually exclusive. You've got me curious about MCAS now! I'll be looking in the "big blue book" of 2017 standards to see what's in there. – Carser Jan 24 '21 at 17:16
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Are there passionate arguments for "parallelograms are not trapezoids"? The only things I can think in its favor is "that's the way your parents learned it" (in the US at least) and the fact that it lets you say that every trapezoid is the "frustrum" of a triangle. I can't imagine anyone being passionate about either of those reasons. – Matthew Daly Jan 24 '21 at 17:30
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@MatthewDaly I think that's exactly the source of confidence in an argument! When there is something that you were told was true that has always been "true" to you, it is upsetting to be presented with a contradiction. I think it is the same as "whole milk is better" or $0^0=1$. It is "obvious" to some people that it is true, while to other's it is obviously untrue. The passion I referred to was meant to be the passion of the arguer, not a description of the argument :) – Carser Jan 24 '21 at 17:39
I have no strong opinion in this one. My main role is as an in-house tutor, and I'm looking to be sure my answer to a student doesn't cost them on an exam. I do recall as a child, "all squares are rectangles, but all rectangles are not squares." And until now, thought this applied here, that a rectangle was, in fact a trapezoid. We have material that offers squares with diagonals, and various indication of lengths and angles. I tell the student "with no measures, this is a quadrilateral. Right angles, or congruent diagonals gives us rectangles.... "Looks like a square to me, Mr J" "I know" :( – JTP - Apologise to Monica Jan 24 '21 at 18:01
@JTP-ApologisetoMonica So why are there standardized tests, and why, being in this situation, aren't you on the picket line to drop the damn things? Wrote any lawmakers about it? If you don't, you're a part of the problem. Bystanding is not innocent in this case. Apologies if you're the rare activist in this area. – Kuba hasn't forgotten Monica Jan 25 '21 at 23:07
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@Kubahasn'tforgottenMonica - can't tell if your comment is tongue-in-cheek, but, if it's not obvious, this isn't an issue I'm that emotionally wrapped up in. Maybe Carser will answer, but I looked through 6 of the last MCAS exams and this issue doesn't come up. I see 2 text books contradicting, I ask. Ben and others set me straight "yes, it's not an agreed rule/definition." I'm sort of done, except for a bit more discussion. [Notation for inverse Trig functions, diff question, has me more riled up. Still, no picket line for me] – JTP - Apologise to Monica Jan 26 '21 at 12:45
@Kubahasn'tforgottenMonica - OTOH, if it's mocking my passion. I care about my students, and would feel awful if I pulled out a textbook, showed them something, only to find they lost even a point on an exam due to my missing something. The history of this is interesting to me. More interesting, say, than Tom Brady's under inflated balls. – JTP - Apologise to Monica Jan 26 '21 at 13:04

score 2 · Answer 3 · answered Jan 26 '21 at 03:27

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The wikipedia definition is the right one.

A square is a rectangle. A rectangle is a trapezoid.

Yeah, at times you can have specific/different definitions. But, the trapezoid one is pretty clear cut. You'd be hard pressed to find a (non contrived) theorem that applies to trapezoid that suddenly stops working because the shape is also a rectangle.

It seems to me like the "exactly one pair" definition is only there to be less confusing to students (but, is it a rectangle or a trapezoid?). To me, that's not a good reason. Rather using this weird definition is a missed opportunity to discuss interesting concept: A rectangle is a trapezoid, but a trapezoid is not a rectangle.

answered Jan 26 '21 at 03:27

Jeffrey

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I have no preference. I never had an issue with the 'all squares are rectangles', even to the point of taunting my teachers in grade school. "until you tell us the sides are equal and the angles are right, it's just a quadrilateral." But I do appreciate the interest this otherwise simple issue can stir up, in 2021. – JTP - Apologise to Monica Jan 26 '21 at 04:09
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@JTP-ApologisetoMonica ha! We must have all been locked up too long. – Carser Jan 26 '21 at 12:30
Jeffrey's definition and explanation are the ones I have always used. They are simple, concise, and sufficiently flexible to handle every situation. – End Anti-Semitic Hate Jan 26 '21 at 16:14

Definition of Trapezoid

3 Answers3

How to address this for students.