**Question: **Why did you decide to use the exclusive definition of a trapezoid?

As the question suggests, there is more than one definition of a trapezoid. Mathematicians define trapezoids in one of two ways:

Using the inclusive definition, all parallelograms (which include rectangles, squares, and rhombuses) are trapezoids. Using the exclusive definition, they are not.

In determining which definition to use, we thought about a couple of things:

- Most elementary textbooks use the exclusive definition, and the Common Core State Standards do not specify which definition should be used. A sidebar in the geometry progressions states that most colleges/universities use the inclusive definition, but the mathematical goals at that level are different.
- Using the exclusive definition makes mathematical sense in the elementary grades. It supports one of the main mathematical goals of the two-dimensional geometry work in
*Investigations 3*—getting students to think more deeply about the classification of 2-d shapes, and how changing an attribute or property can change how the shape is classified. Because this definition defines trapezoids as different from parallelograms, it complicates the hierarchy of 4-sided 2-d shapes in interesting ways—students need to identify not only parallel sides, but the number of parallel sides.

It often surprises people that there are two competing definitions. Isn’t mathematics supposed to be precise and have “one right answer”? This can be an interesting investigation to undertake with fourth or fifth grade students, who are often fascinated to learn of such mathematical disagreements. “What would happen if we change the rules, and use the definition of a trapezoid that says ‘*at least* one pair of parallel sides’? How would that change your categories and what shapes are in them”?

### The Investigations 3 Center Team

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We just had this EXACT conversation in our 5th grade math class! It allowed us to discover that there are two ways of looking at polygons in geometry, inclusive and exclusive. It first came up when one student said an equilateral triangle could be an isosceles triangle. He “constructed a viable argument” and the other students “critiqued” his reasoning (SMP 3). Fast forward a few days to when we were creating our quadrilateral relationship posters from Investigations 3, 5th grade Unit 8. There was a lively debate between groups on whether or not a parallelogram, rhombus, rectangle and square could be a trapezoid. One student said “If everything can be a trapezoid then why do we have the other categories?” This discussion opened the door to having a further discussion about inclusive and exclusive definitions. We decided that for now it was ok to leave the debate open. Thanks, TERC, for creating the opportunity for the conversation.

I wish we could listen in on these discussions! To be honest, I had a very similar moment to that 5th grade student as Keith and I were sorting out our answer to this question. Thanks for writing; we love hearing about Investigations in the classroom.