Why should a parallelogram also be a trapezoid? The answer may surprise you.

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Here is an excellent question from a user of Eureka Math:

I was reviewing the Grade 4 Module 4 lessons and videos and saw the definition of a trapezoid as a quadrilateral with at least one pair of parallel sides. I know this is correct, but my question is, how useful is this vs. a previous common understanding (or misunderstanding) that many of us had, that a trapezoid is a quadrilateral with exactly one pair of parallel sides? I was just thinking about this today and couldn’t think of how being able to classify all parallelograms also as trapezoids is useful, whereas it seemed very clean and clear to identify and recognize quads with only one pair of parallel sides as trapezoids and to be able to say “this is what a trapezoid is” rather than “this is one type of trapezoid”. Any thoughts on this?

First, in general, when it comes to whether an equilateral triangle is also an isosceles triangle, a square is a rhombus, and so on, it is very important to realize that the decision to include or rule-out one type of figure can be a purely human convention—that is, it is often a human decision rather than one based upon the structure of mathematics. Thus it is common to see some books include parallelograms as trapezoids (at least one pair) and others rule-out parallelograms as trapezoids (exactly one pair).

There are at least three very good reasons to include parallelograms and rectangles as trapezoids:

(1) To make categorizing figures easy to learn for students. In Eureka Math, it was decided that all relationships between figures should be defined by inclusion instead of relating some figures by inclusion while defining other figures by rule-out. Thus, a parallelogram is also a trapezoid, an equilateral triangle is also an isosceles triangle, a square is also a rhombus and a rectangle, and so on.

This conscious choice to make figures always related by inclusion rather than some inclusion and some rule-outs is a very important one for students. It means that learning which figures are also other figures always follows the same principle—students don’t have to learn a bunch of “special cases.” This makes categorizing figures enormously easier, more intuitive, and overall far more coherent than the old common understandings—students only have to remember one principle and apply it appropriately.

(2) To make the definitions consistent with how they are used in all future math courses. For example, calculus uses “trapezoidal approximations” for calculating integrals. This method (and its explanation) only makes sense if rectangles are also trapezoids. Since rectangles are also parallelograms by inclusion, then parallelograms should be trapezoids by inclusion as well.

(3) To make theorems and proofs more general. The use of relating figures by inclusion serves a mathematical purpose: If you prove an idea that holds for a rectangle, then it automatically holds for a square. Similarly, if you prove something about the parallel sides of a trapezoid, then you automatically prove it for both pairs of parallel sides of a parallelogram. The fact that you get “extra facts for free” is a mathematical reason for relating figures by inclusion. This “two birds with one stone” aspect helps students appreciate and learn the importance of precisely defined definitions, and to reason with those definitions.

It should be pointed out that the reason for the “old common understanding” may have had a lot more to do with testing than it did mathematics. States and assessment agencies liked using the notion that some figures were related by inclusion while other figures were related by rule-out because of the high p-scores they could get on such questions on high stakes exams. Basically, the state could ask a “math” question that could distinguish between students who memorized all the special cases versus those who didn’t. These questions are not really fair to ask since they are manufactured to be memorization questions that have little to do with mathematical structure.

When the Progressions document, “Progressions for the Common Core State Standards in Mathematics (draft). Grades K-5, Geometry,” correctly suggested in the diagram on page 18 (see below) that parallelograms should also be trapezoids, Eureka Math agreed and did the same.