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:
- Define “trapezoid” as the textbook does.
- Define “trapezoid” in the way that Mr. Davidson prefers.
- 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.
- 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.”