“What is the definition of a trapezoid?” Previously in my CTSE 5040 class, the discussion livened when this question was presented. Many of us thought this was a trick question and many of us gave the question much thought. Hesitantly, we replied with bases are parallel and each lower base angle is supplementary to the upper base angles. There was a long pause because the question seemed too easy and we felt there had to be a catch. Our instructor grinned when he asked, “does a trapezoid have exactly one or at least one pair of parallel sides?” Some of us responded quickly with exactly one, but, again, the long pause caused some recoil of the statement. Everyone in the room began to question what they previously thought the properties of a trapezoid were. Then someone stated, “if it is at least one, then that means every parallelogram is a trapezoid”, this promoted a whole new level of controversy.

The most common definition of a trapezoid in U.S. textbooks is exactly one pair of parallel sides, but others use the inclusive definition of at least one pair. Our instructor provided us with an NCTM discussion link on this topic, so we could read various arguments and decide which definition we believed to be most valid. The discussion brought up many interesting points and opinions, but one stood out to me. The writer stated that they discussed this definition with NCTM around 2007-2010, and at that time the United States used exactly one pair to define a trapezoid. However, Canada and other parts of the world use at least one pair of parallel sides to define a trapezoid. I like the analogy used in this discussion post, the writer said that this dissimilarity can be compared to how the United States does not regularly use the metric system, but the rest of the world does. Other posts were from teachers asking what definition standardize tests use because they wanted to prepare their students. It seems like many simply want to be on the same page, but others are set in their ways and closed off to opposite opinions.

After reading the discussion posts, I still could not draw a conclusion on which definition I believed. Each side had valid arguments, so I turned to someone who’s opinion I respected, thinking it may help me clarify and draw an accurate conclusion. I asked my Euclidean Geometry teacher. After class, I cornered him and before I could finish the introduction of the question he replied confidently with at least one. I was shocked at his confidence and I am sure he saw the shock on my face. He reiterated, “A trapezoid has at least one pair of parallel sides”. Just to be sure, I said that with the inclusive definition, all parallelograms would be considered trapezoids. He nodded. With that nod, I had my answer.

When back in class, I told my instructor who presented the controversial question, what my Euclidean Geometry teacher proclaimed. My instructor, who was not surprised, said that many mathematicians prefer the inclusive definition because they enjoy connections. Having things relate back to something else is satisfying and convenient. I then asked my instructor what he believed to be the definition of a trapezoid to be, he grinned. This question of inclusive or exclusive would be interesting to ask students in a geometry class because it allows them to formulate their own opinions which results in high cognitive demand and feelings of independence. Then, from their decisions, a teacher may adapt their lesson to fit the definition chosen by students. I believe this is an effective way of involving students with the curriculum and assessing student thinking.

I also enjoy how this discussion topic can be related to real-life events. I had not thought about this before my instructor brought the idea up in class. In life, there will always be different opinions on different matters. However, having a different opinion does not make you wrong or right, it simply means you have a different viewpoint or way of thinking. This is an important lesson to educate students on. I appreciate how, on this discussion topic, mathematics can be compared with the real world.

References

National Council of Teachers of Mathematics. (2018, October 3). myNCTM. Retrieved from myNCTM: https://www.nctm.org/login/SSOReturnUrl=https%3a%2f%2fmy.nctm.org%2fcommunitis%2fcommunityhome%2fdigestviewer%2fviewthread%3fMessageKey%3d215ddaecc6ba4ede965e3a73ba5bda8b%26CommunityKey%3d0a10e70-ceb6-4514-ac1c413a77367143%26tab%3ddigestviewer#bm