Defining a trapezoid

This is the cue for my students to roll their eyes… Yesterday I got into a heated discussion with another math teacher about an important issue: how to define a trapezoid. He was arguing in favor of the position that a trapezoid has exactly one pair of parallel sides; I was arguing in favor of the position that a trapezoid has at least one pair of parallel sides. We both agree that it’s a quadrilateral.

My opponent made several good points:

  • Our current textbook defines the word his way.
  • So do some other textbooks.
  • The common image of a trapezoid has two non-parallel sides.
  • We don’t expect someone to look at a parallelogram and exclaim, “That’s a trapezoid!”

But I made, IMHO, several better points:

  • Nowhere else do we define a geometric object in this exclusionary way. We don’t say that a rectangle cannot have four congruent sides. We all agree that squares are rectangles, rhombuses are parallelograms, circles are ellipses, etc.
  • Many textbooks, including Moise’s and UCSMP, do define it my way.
  • In the software we use, the Geometer’s Sketchpad, it’s straightforward to construct a trapezoid with my definition but not with his.
  • Most importantly, the quadrilateral hierarchy should show parallelograms as a subset of trapezoids because theorems about trapezoids also apply to parallelograms.

Of course it all leads to a teachable moment — or more than a moment, actually. In my honors geometry class we devoted more than an hour of class and homework time to exploring the ramifications of the two definitions. Then the issue emerged in a question on the next quiz:

Always/Sometimes/Never: Under Mr. Davidson’s preferred definition of “trapezoid,” the diagonals of a trapezoid are congruent.

And then in a four-part question on the next test:

  1. Define “trapezoid” as the textbook does.
  2. Define “trapezoid” in the way that Mr. Davidson prefers.
  3. Former Harvard professor Edwin Moise has the following theorem (not definition) in his book: “A trapezoid is a parallelogram if its diagonals bisect each other.” Can you tell which of the two definitions of “trapezoid” must have preceded this theorem? Explain convincingly.
  4. Prove Moise’s theorem (using whichever definition you identified in part c).

Bonus: little did I realize that the embedded “if” in Moise’s theorem would confuse some students. Apparently they had never seen a theorem where the consequent preceded the antecedent. So that led to a worksheet for another assignment.

Thinking about what you’re learning is a good thing. And, as Humpty Dumpty said in Through the Looking Glass, “When I use a word, it means just what I choose it to mean — neither more nor less.”

Categories: Math, Teaching & Learning