Trapezoids

20 Nov

The PARCC Evidence Statement Table for Geometry EOY, G-CO.3, says: A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

Usiskin (and I’m sure others – he was just the first one from whom I read it) call this the inclusive definition of a trapezoid.

I have been posing this definition to my students as a possibility for several years now. What would happen if we defined a trapezoid as a quadrilateral with at least one pair of parallel sides? Then a parallelogram is also a trapezoid. Our always/sometimes/never fill in the blank statements of “A parallelogram is ____ a trapezoid” or “A trapezoid is ____ a square” change from never to sometimes. But since our textbook defined a trapezoid as a quadrilateral with exactly one pair of parallel sides, we did too. This year, we defined a trapezoid as PARCC will. My colleagues and I have been thinking about the implications of this definition on our deductive system.

On a side note, I’ve been looking for real CCSS geometry resources. I usually first look at where the unit on transformations is located. If it’s not 1st or 2nd, then that is a clear sign to me that the text hasn’t really been edited for CCSS-M. I’ve been forgetting to look at how they talk about trapezoids. Not that every CCSS-M textbook resource has to define trapezoids like PARCC does, but are they even suggesting the inclusive definition as a possibility?

Our Venn Diagram arrangement of the quadrilaterals has been deleted.

Several of our practice problems have had to be rewritten:

Old: By definition, what is a quadrilateral with exactly one pair of opposite sides parallel?

New: By definition, what is a quadrilateral with at least one pair of opposite sides parallel?

Old: By definition, what is a quadrilateral in which both pairs of opposite sides are parallel?

New: By definition, what is a trapezoid in which both pairs of opposite sides are parallel?

Technically, we could have kept the old question, but we changed it since we are trying to emphasize the new definition.

Old: Which statement is NEVER true?

A. Square ABCD is a rhombus.

B. Parallelogram PQRS is a square.

C. Trapezoid GHJK is a parallelogram.

D. Square WXYZ is a parallelogram.

E. Trapezoid EFGH is an isosceles trapezoid.

New: We deleted this question completely. But now that I look at it more closely, I guess we could have changed parallelogram in choice C to kite.

Old: Two consecutive angles of a trapezoid are right angles. Four of the following statements about the trapezoid could be true. Which statement CANNOT be true?

A. The two right angles are base angles.

B. The diagonals are not congruent.

C. Two of the sides are congruent.

D. No two sides are congruent.

E. Exactly two sides are parallel.

New: We added the word Exactly at the beginning of the question stem.

Old: Write the number of each of the five figures in the appropriate region of the diagram.

New: We didn’t have to change the question…only the solution.

Other conversations we had were about how to define other quadrilaterals. If we define a trapezoid as a quadrilateral with at least one pair of parallel sides, then shouldn’t a parallelogram be a trapezoid with both pairs of opposite sides parallel? We have always defined a rectangle as a parallelogram with four congruent angles, a rhombus as a parallelogram with four congruent sides, and a square as a parallelogram that is both a rectangle and a rhombus. Those definitions still seem to work. But are they the best definitions? Do we change our definition of kite from a quadrilateral with two pairs of consecutive congruent sides to a quadrilateral with at least two pairs of consecutive congruent sides? And if so, then should we define a rhombus as a kite with four congruent sides? How do we define an isosceles trapezoid in this deductive system? A trapezoid with congruent legs no longer seems to work. What about a trapezoid with congruent diagonals? Or a trapezoid with congruent base angles? Do we even talk about bases and legs of a trapezoid anymore?

Another question I have had is whether K-8 teachers know about this inclusive definition of trapezoid. Does it bother students for a definition to change midway through their years of school? Or do we need to start a campaign to inform teachers about this definition who might not be reading the PARCC Evidence Statement Table for Geometry EOY? Just for the record, I think my students are okay with this definition. They have actually been more flexible in their thinking than the teachers.

Last year, my 2nd grader came home with this question on a worksheet, which I shared with my students.

Unfortunately, the intention was to mark one response even though my daughter marked more than one response.

Can you imagine the distress of having a 4th figure that is a non-special trapezoid with directions to “Mark the trapezoid”?

As far as exploring properties of a trapezoid, I really like the inclusive definition of a trapezoid. We built a trapezoid using our dynamic geometry software. We recognize that two pairs of consecutive angles are supplementary. But as we move the vertices to observe what stays the same and what changes, we recognize that there are times when all pairs of consecutive angles are supplementary. A trapezoid can be a parallelogram.

We also used our dynamic geometry software to observe what happens when we create a midsegment of the trapezoid. What is true about the midsegment of a trapezoid? It is parallel to the bases. It cuts the trapezoid into two smaller trapezoids. How do you know? How could we show that ABNM is also a trapezoid?

And what is true about the length of the midsegment compared to the bases? It is half of the sum of the bases. How can we show that the length of the midsegment is always half of the sum of the bases?

I am glad that this conversation was started elsewhere, and that I easily found it through Google. The graphic organizer on Mr. Chase’s blog posts has been most helpful in thinking through this change.

Why I hate the definition of trapezoids

Why I hate the definition of trapezoids (again)

Why I hate the definition of trapezoids (part 3)

What is a trapezoid? (Part 1)

What is a trapezoid? (Part 2)

It’s nice not to be alone as the journey continues ….

Posted by on November 20, 2013 in Geometry, PARCC, Polygons

7 responses to “Trapezoids”

1. November 20, 2013 at 1:08 am

Of course I absolutely love this post. Seriously!

I especially like the example of the proof of midsegments of the trapezoid. Interestingly, under the exclusive definition, a proof that ABNM is a trapezoid would have to include reasoning that shows AM and BN are *not* parallel. Because if they’re parallel, then ABNM is not actually a trapezoid, according to the old exclusive definition.

But that’s ridiculous!

You shouldn’t have to prove that AM and BN are not parallel. In fact, if they’re parallel, the result STILL holds and you have the proof of an even stronger statement that applies to all trapezoids, of which parallelograms are just a special case. That’s mathematically elegant and we should teach students to appreciate that!

• November 22, 2013 at 4:05 am

Thanks so much for helping me think through this. I’ve still been struggling with whether we still talk about legs and bases, base angles, etc. I do think that thinking about the median of a trapezoid whether it is a parallelogram or not is helpful for generalizing the area of a trapezoid.

2. November 20, 2013 at 3:01 pm

I teach at a private school and am not constrained by the CCSS, but I have to say that I am always dismayed by my 9th grade geometry students who come in with the idea (from elementary/middle school) that trapezoids have to be isosceles.

I do teach that trapezoids have exactly one pair of parallel sides – you are right that changing the definition (to the inclusive version) means changing the questions we ask. I am writing this as my students are taking the test on this unit, so will have to ponder this for next year. Thanks for giving me something to think about!

• November 22, 2013 at 4:07 am

Hi, Chrystal. It is interesting that your students think all trapezoids must be isosceles. I haven’t given much thought to my students having that misconception, but they might. Thanks for giving me something to think about as well!