Yes it’s true. I’m writing about trapezoids again (having written passionately about them here and here previously). I’ve been taking a break from blogging, as I usually do in the summer. For us, school starts in just two weeks. So I thought I’d come out of my shell and post something…and of course I always have something to say about trapezoids :-).
Let’s start with the following easy test question. Don’t peek. See if you can answer the question without any help.
Which of the following quadrilaterals are trapezoids?
Before giving the answer, let me first just remind you about my very strongly held position. I believe that instead of this typical textbook definition (the “exclusive definition” we’ll call it) that reads:
“A quadrilateral with one and only one pair of parallel sides.”
the definition should be made inclusive, and read:
“A quadrilateral with at least one pair of parallel sides.”
So the test question above was easy, right? Quadrilaterals (A) and (C) are trapezoids, I hear you say.
Not so fast!! If you’re using the inclusive definition, then the correct answers are actually (A), (B), (C), (D), and (E). But it gets better: If you were using the the exclusive definition, then NONE of these are trapezoids. In order for (A) and (C) to be trapezoids, under the exclusive definition, you must prove that two sides are parallel AND the two remaining sides are not parallel (and you can’t assume that from the picture…especially for (C)!).
Can you see the absurdity of the exclusive definition now?
I finish by offering the following list of reasons why the inclusive definition is better (can you suggest more reasons?):
- All other quadrilaterals are defined in the inclusive way, so that quadrilaterals “beneath” them inherit all the properties of their “parents.” A square is a rectangle because a square meets the definition of a rectangle. Likewise, parallelograms, rectangles, rhombuses, and squares should all be special cases of a trapezoid.
- The area formula for a trapezoid still works, even if the legs are parallel. It’s true! The area formula
works fine for a parallelogram, rectangle, rhombus, or square.
- No other definitions break when you use the inclusive definition. With the exception of the definition that some texts use for an isosceles trapezoid. Those texts define an isosceles trapezoid has having both legs congruent, which would make a parallelogram an isosceles trapezoid. Instead, define an isosceles trapezoid as having base angles congruent, or equivalently, having a line of symmetry.
- The trapezoidal approximation method in Calculus doesn’t fail when one of the trapezoids is actually a rectangle. But under the exclusive definition, you would have to change its name to the “trapezoidal and/or rectangular approximation method,” or perhaps ban people from doing the trapezoidal method on problems like this one: Approximate
using the trapezoidal method with 5 equal intervals. (Note here that the center trapezoid is actually a rectangle…God forbid!!)
- When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel (see the beginning of this post!).
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As I’ve chimed in before, I’m with you 100%. But just to keep the pot boiling, here goes.
First, see Wolfram on “trapezium” here: http://mathworld.wolfram.com/Trapezium.html .
The issue seems to be British vs. American usage of “trapezium” vs. “trapezoid” respectively. It seems as though “with 2 sides parallel” is agnostic to what is happening with the other 2 sides, regardless of which side of the pond you are on. In other words, the great, authoritative Mathworld agrees with you. 🙂
Second, orthopaedists refer to one bone in the wrist as the “trapezium” because it is shaped like a table, and the Greek word for “table” is τραπέζι /trapezi/. See
http://www.eorthopod.com/sites/default/files/images/adult_hand_fx_bones03.jpg .
Also the trapezius muscle is shaped like a table, one might suppose. See
I guess tables can be a variety of shapes. What shaped table did Euclid have in mind? Euclid’s definition of trapezia is the polar opposite of your (our) definition. He doesn’t even see squares as being rectangles (i.e. oblongs). See
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI22.html
I conclude that consistency demands that those who demand that squares be rectangles must also demand that the squares be trapezoids.
Trapezium doesn’t bother me, even though I can see why it would be confusing. I would, of course, insist that trapezoid (US) and trapezoid (UK) both be defined using the inclusive definition, and they mean exactly the same thing in their respective contexts.
As for the Greek discussion, and specifically Euclid’s interpretation, see my post here, along with the comment thread. Euclid defined all his quadrilaterals exclusively. But he also never really uses those definitions anywhere else ever, so they’re not so important.
For us, we use the inclusive definition for everything else. So I like what you said: consistency demands we use the inclusive definition for the trapezoid too.
You still have an issue with your calculus example, even the one you gave. At x = 0, for the function f(x) = 4x – x^2 (and also at x =4), you get y-values of 0. So, your first and last trapezoids aren’t trapezoids at all, even by your inclusive definition, they are triangles. I have heard the term degenerate trapezoid used in that case (where “degenerate” would refer to one of the parallel bases shrinking to a length of 0). However, the formula for area of a trapezoid still holds even here (just like it would hold for squares, rhombuses, and rectangles), because since one of the bases is 0, it just becomes (1/2)bh anyways. But that still doesn’t make the triangle a trapezoid, so we would still have to call it the trapezoid and triangle rule or (to quote you) we might have to “ban people from doing the trapezoidal method on problems like this one” where one or more of the shapes are triangles. Honestly, though, if a student taking Calculus isn’t smart enough to figure out that the trapezoid area formula can be applied to triangles, trapezoids, and rectangles (even if you don’t include rectangles in your definition of trapezoids) without renaming it the trapezoid/rectangular/triangular approximation method, then they probably shouldn’t be in the class anyways. Your other reasons are pretty good, but you also run into the problem of redefining the term “isosceles” as used in triangles when you get to trapezoids. Most (maybe all?) books define isosceles as having two or more congruent sides and then prove that an isosceles triangle also has to have two congruent angles. To fix your isosceles trapezoid vs. parallelogram problem, you have defined isosceles as having two congruent angles. Consistency there would be helpful (and is possible if books were to define isosceles triangles as having two congruent angles and then proving that they also have two congruent sides). However, that might be tough to do as well, because I believe (correct me if I am wrong) that the word isosceles comes from the Latin or Greek word scoles or skelos meaning leg and the prefix iso meaning equal. So, the root word itself focuses on the legs. The word isogonic though does refer to angles, so you might be better off calling yours an isogonic trapezoid, but the term isogonic doesn’t seem to be very widely used from my teaching experience.
Overall, good thoughts, but I’m still not convinced to abandon conventional thinking. Yet again, though I can understand either method and tell my students for consistency’s sake to go with what their book uses usually.
Wow, great contributions, Andy! You’re right that I’m letting triangles also be degenerate trapezoids… so perhaps my argument needs some work (or perhaps a more carefully chosen example :-).
And I love the Greek contribution…isosceles means legs are congruent.
That being said, we might have to take wikipedia and mathworld to task, since both define isosceles trapezoids without evoking the leg congruence (they use symmetry and base angles, respectively).
I think we can still call call them isosceles trapezoids (congruent-legged trapezoids), since the new definition implies leg congruence. It’s not like the new definition produces any non-congruent-legged trapezoids.
Thanks for the awesome comment!
Well, I don’t trust wikipedia all that much without corroborating websites/books/articles to back them up, and I honestly haven’t spent enough time on mathworld to know enough about them.
I like the fact though that mathematics isn’t finished and that there are still debatable topics and some things that are fluid or uncertain. Now, I wouldn’t like it if much of math was that way (like poetry for example), but I think it’s good to know that mankind doesn’t have it all figured out.
In a parallelogram, (under the inclusive definition) which are legs, and which are bases?
Whichever you like.
If you’ve got a parallelogram sitting in front of you, you can arbitrarily name one pair of sides “bases.” Now, if you want to apply any theorem or formula to your trapezoid, you may (try the area formula, for example).
Why waste time worrying about a definition and silly elementary school “hierarchies of quadrilaterals”? Cut straight to problem solving and definitions don’t matter:
http://fivetriangles.blogspot.com/2013/04/65-paper-cuts.html
http://fivetriangles.blogspot.com/2013/07/86-boomerang.html
Thanks for stopping by! I love all the problems you provide on your site. What a great resource for teachers and students of math.
On the importance of definitions, I would beg to differ. I think definitions are of first importance in math. Take the problem I gave in my post. What do you think the answer is? It truly depends on your definition. It’s either ALL or NONE of the quadrilaterals…big difference! 🙂
The other interesting thing about definitions in math is that they are arbitrarily chosen, and after we agree on them, we can see what consequences those definitions have. We hijack English words and make them more precise. Words like continuous, regular, equal, series, or product all have very particular meanings in various mathematical contexts. Why do we care about splitting hairs over orthogonal versus perpendicular? (There really is a slight difference here!) This is a huge part of mathematics. How can we tell that a polyhedron is regular? How can we tell that a function is continuous? How can we determine if the derivative exists? How can we prove two groups are homomorphic? We look to the definition. Always.
And that’s why we place a great importance on definitions.
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I came her to look for a definitive answer to the trapezium (I’m British) and Parallelogram question and see it is a can of worms. I have always gone for the inclusive definitions, but the plethora of diagrams that seem to imply exclusivity made me doubt myself. I have only skimmed over your ideas, and I fully agree with your thinking regarding properties being valid, but I am not too happy with the hierarchical diagrams as they don’t show the subset nature properly i.e. the intersection of rhombus and rectangle is square.
Now to my question. You seem very passionate about trapeziums/trapezoids, but less so about kites. Is a rhombus a special kind of kite? Is it true to say that the intersection of kites and parallelograms are the rhombi? And where does that leave your trapezoid?
Glad you found us, Dave!
I would absolutely say a rhombus is a kite. One way to define a rhombus, as you say, is that it is both a kite and a parallelogram. One of the diagrams labeled “Good” in this post shows it this way. I think this is elegant!
I am a math teacher, and back in college, and in every book I have used, the exclusive definition has been used. I have talked with my colleagues, and they are only familiar with the exclusive definition as well.
With the onset of the Common Core, we are seeing that we need to change things. I guess I don’t care too much, except that I have always viewed trapezoids as truncated triangles, and isosceles trapezoids as truncated isosceles triangles. Now, that is off the table.
It is also difficult to go against what you have always believed and taught.
Yes, with Common Core now in full swing, this discussion is more important than ever. But in my opinion, this particular change is a positive one (given my original posts, this shouldn’t be a surprise!).
Accepting the inclusive definition may mean jettisoning a few things, but I think you gain so much more, the most important things being consistency and elegance.
Thanks for weighing in, James!
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Hi Mr. chase! I saw your first post from working on a research to define trapezoids in my Geometry class. I then saw your second and third posts and checked them out. I thank you for all of your hard work on this. It is very intriguing! I think you are correct. In my class we have used Geometer’s sketch pad (which is very similar to geogebra). When you construct a certain shape, you can bend it but it stays the same shape. I went to GSP and constructed a Trapezoid and I observed the same thing you did on your first post. I think we need to change our text books and think more!
Thanks, Daniel.
From early years kids naturally like exclusive definitions, and have to be weaned off this. This would be easier if we were more careful with the word “is”. Even to me the statement ” a square is a rhombus” sounds weird, if not actually wrong. It would be better to be less brutal, and say “a square is also a rhombus” (and all the other such statements). Even better, and quite mathematical, is the phrasing “a square is a special case of a rhombus”, as the idea of special cases is very important, and usually overlooked. It is odd that the classification of triangles is done entirely with adjectives and the difficulty is thus avoided.
I appreciate all 3 blog posts. I got here because I thought a geometry app (Isosceles) was wrong when it told me a parallelogram is also a type of trapezoid. Now, with this fun discussion of the origins of definitions, the uses of formulas, and their applications in higher level math, I see the need for the changed definition. It is indeed exciting to see that there are schools of thought on this. I also recently found out that zero and one are both neither prime nor composite. I’m not sure if this has multiple schools of thought or if I was just wrong on the matter. I also believe in the Oxford comma! To some, all of these definitions might seem semantic or arbitrary, but I appreciate precision. Thanks again!
Thanks for stopping by and for your comment. I think you’re right that 0 and 1 are neither prime nor composite by most people’s reckoning. Definitions are arbitrary, yes, but some are more elegant than others.
By the way, I’m a huge fan of the Oxford comma, too! 🙂
The definition is very important. I am a middle school math teacher, and I would love to know which definition I should teach my students, so that they will be prepared for high school geometry. When I teach them that there are two definitions, they are told that is incorrect by their high school math teachers. They then go off to college, and depending on the college, they are incorrect again. *sigh*
Great practical question, Amanda.
If you must teach only one definition, go with the inclusive definition. Do this for two reasons:
That being said, it might be even better to “teach the controversy,” since you would be involving your students in a rich discussion about the nature of definitions in mathematics. Definitions are arbitrarily chosen, but the definitions we choose have different consequences–some of which are powerful and some of which are useless, some of which are beautiful and some of which are ugly. This would let your students see that “doing mathematics” is a creative human endeavor, not a formulaic endeavor.
Come back and let us know what you end up doing with your classes!