Trapezoids

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Isosceles Trapezoidal Architecture Temple of Dendur, now housed at the MET in NYC

When it comes to math, I usually feel pretty good at it and fairly knowledgeable. Usually. The other day, this was not the case. Right before first period, one of my math colleagues huddled several of us up and asked, “What is the definition of a trapezoid?” This was more of a rhetorical question since we all knew the answer (or so we thought).

“A convex quadrilateral with exactly one pair of parallel sides,” I spouted.  Heads nodded in agreement. “Not anymore according to my nephew’s son’s third grade teacher in X district.” We were appalled. According to my colleague, a trapezoid was now defined as a “convex quadrilateral with at least one pair of parallel sides.” “But that makes a parallelogram a trapezoid,” gasped one teacher. “It makes a square a trapezoid,” I said flatly. Then we had to go to first period where I was teaching vectors to trig students.

Meanwhile, I googled “trapezoid” while my students were working the warm up problem. Stunned, I read that the definition depends on, well, your definition. Here are the first few paragraphs from Wikipedia’s entry:

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid inAmerican and Canadian English but as a trapezium in English outside North America. The parallel sides are called thebases of the trapezoid and the other two sides are called the legs or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to thespecial cases below.

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.[1] Others[2] define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition[3]), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.

Well, this made me mad.  How did I not know that there were two different definitions, one inclusive, and one exclusive?  I emailed my trusty cadre of math friends, and they replied back throughout the day.  And indeed, yes! The Common Core State Standards (CCSS) for mathematics strongly suggests using the inclusive definition of a trapezoid, starting in the lower grades. Honestly, I had not looked much at the lower grade CCSS until I started combing some of the inner documents, hoping to better understand this trapezoid issue. Parents nationwide have been complaining about CCSS and the ensuing PARCC assessments, but what I had seen to this point was mainly more rigor and more building understanding, both good. Now I understood the frustration. I wanted to scream and complain too.

But, by the end of the day, I was beginning to feel very happy about the inclusive definition since it was always little strange that the trapezoid sat off to the side under the quadrilateral tree. (Does anyone else remember the drawing from middle school — the family tree of parallelograms, but there’s that weird uncle Trapezoid sitting off under the tree?). Feast your eyes instead on this lovely Inclusive Venn diagram:

 

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I guess now the kite will be the only convex quadrilateral with interesting properties that is not a trapezoid.

Thinking back over my math lifetime, I could not remember a time when a definition changed. Sure theorems have been proved, like Fermat’s last theorem, but a definition changed? It is like having the mathematical rug pulled out from under your feet.

Scientists are used to this. Pluto’s not a planet anymore. Birds are avian dinosaurs. We can see further and further into organisms and into space with our improved technology. Some diseases are cured or held in very good check. The list goes on and on.

Mathematicians are not used to this. Well, maybe on a very advanced level where math is still unfolding, but not in Euclidian geometry or in algebra. Sure we teach it very differently than we were taught it, seeking to build conceptual understanding rather than to memorize procedure. We are able use technology such as spreadsheets and graphing calculators (which did not exist in the math classroom when I was in high school) towards this goal.

But 10 + 10 has always been 20. Right? Oh, I guess unless you are in base 2. Then it would be 100.

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