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From one textbook we use in our High School -

Some text (see transcription below) with a drawing of a trapezoid to the right. The trapezoid is labeled "Trapezoid".

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.

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    $\begingroup$ For historical context on the inclusive and exclusive definitions of "trapezoid", see my answer at matheducators.stackexchange.com/questions/13700/… $\endgroup$
    – mweiss
    Jan 24 at 14:45
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    $\begingroup$ 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" $\endgroup$
    – Flydog57
    Jan 24 at 22:57
  • $\begingroup$ 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). $\endgroup$
    – Flydog57
    Jan 25 at 0:46
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    $\begingroup$ And don't get started on "isosceles trapezoid", "acute vs. right vs. obtuse trapezoid", "tangental trapezoid", etc... $\endgroup$ Jan 25 at 14:57
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    $\begingroup$ 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. $\endgroup$ Jan 26 at 12:28
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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.

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    $\begingroup$ +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. $\endgroup$ Jan 24 at 22:05
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    $\begingroup$ 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". $\endgroup$
    – mweiss
    Jan 24 at 23:18
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    $\begingroup$ @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? $\endgroup$ Jan 25 at 13:42
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    $\begingroup$ 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. $\endgroup$ Jan 25 at 15:30
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    $\begingroup$ (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. $\endgroup$ Jan 25 at 15:32
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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:

  1. 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.

  2. 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.

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    $\begingroup$ 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. $\endgroup$ Jan 24 at 14:16
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    $\begingroup$ 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. $\endgroup$
    – user507
    Jan 24 at 15:03
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    $\begingroup$ 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. $\endgroup$
    – Carser
    Jan 24 at 16:08
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    $\begingroup$ 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." $\endgroup$ Jan 24 at 16:52
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    $\begingroup$ @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 :) $\endgroup$
    – Carser
    Jan 24 at 17:39
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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.

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  • $\begingroup$ 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. $\endgroup$ Jan 26 at 4:09
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    $\begingroup$ @JTP-ApologisetoMonica ha! We must have all been locked up too long. $\endgroup$
    – Carser
    Jan 26 at 12:30
  • $\begingroup$ Jeffrey's definition and explanation are the ones I have always used. They are simple, concise, and sufficiently flexible to handle every situation. $\endgroup$ Jan 26 at 16:14

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