The Trouble With Trapezoids

Did you know there are two possible definitions for a trapezoid? Take a look and see if you can spot the difference between them:

  • A trapezoid is a quadrilateral with at least one pair of parallel sides.
  • A trapezoid is a quadrilateral with exactly one pair of parallel sides.

They look almost identical, right? But there’s a slight difference in wording that has a profound impact on how we classify not just trapezoids, but all other quadrilaterals.

The first definition is known as the inclusive definition. Since it contains the phrase “at least one pair” that means we can include every quadrilateral that has at least one pair of parallel sides. A parallelogram, for example, meets this definition because it has at least one pair of parallel sides. In fact, parallelograms have two pairs of parallel sides! Here’s the impact the inclusive definition has on classifying quadrilaterals. Notice how parallelograms are a sub-group under trapezoids:

inclusive

The second definition is known as the exclusive definition. Since it contains the phrase “exactly one pair” that means we can only include quadrilaterals that have exactly one pair of parallel sides. A parallelogram, for example, does not meet this definition because it does not have exactly one pair of parallel sides. In fact, parallelograms have two pairs of parallel sides! Here’s the impact the exclusive definition has on classifying quadrilaterals. Notice how parallelograms and trapezoids are separate categories under quadrilaterals:

exclusive

So which definition are you supposed to use? The answer is you can use whichever one you want, so long as you are clear about which one you are using and you are correct in how you treat trapezoids (and other quadrilaterals) once you’ve made your choice.

Will STAAR ever expect students to use one definition or the other? We don’t know. So far the STAAR test has avoided making students address this choice head on. Here is a sample question from 2015 that skirts the issue by using the categories “Parallelograms” and “Non-parallelograms.” In this example, trapezoids fit in the “Non-parallelograms” category regardless of which trapezoid definition you choose to use.

image

So what should you do going forward? Let students know that both definitions exist! Let them explore classifying quadrilaterals using both definitions to see what impact it has. Show them completed graphic organizers and let them decide which definition was used to classify the shapes that way. Rather than thinking about this as the trouble with trapezoids, see it as an opportunity for some really interesting discussion in class about how this subtle difference in language can create large differences in how we treat shapes.

If you want to read more about this topic, check out this great Dr. Math post that dives more deeply into trapezoids. I also recommend another Dr. Math post about inclusive and exclusive definitions. This is what explains why squares are also rectangles and equilateral triangles are also isosceles triangles.┬áIn case you’re thinking to yourself right now, “Wait a minute! Equilateral triangles aren’t isosceles triangle,” then I definitely recommend reading the second Dr. Math post.

2 thoughts on “The Trouble With Trapezoids

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  1. Interesting because in Ontario we use the exclusive definition with students but we do usually have a discussion in the upper grades about the fact that there are two definitions out there. I love the quad study though and the students find it interesting as well.

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