I love the geometry of this tower crane as it swings around, creating obtuse projections of its perpendicularity. I wonder how accurately I could estimate my altitude from this picture?

**Geometry**category.

## Varignon’s Theorem

Varignon’s Theorem is one of my favorite results in elementary geometry: connect the adjacent midpoints of the four sides of any quadrilateral, and a parallelogram is formed! It is a magical result that defies expectations, and it’s so much fun to play around with, explore, and extend.

Steven Strogatz shared his favorite proof of Varignon’s Theorem on Twitter yesterday, and so I felt compelled to share mine. This is a standard proof of Varignon, but it is so clean and elegant: it is an immediate consequence of the Triangle Midsegment Theorem and the transitivity of parallelism.

Strogatz’s vector proof is beautiful and efficient, but the power of transitivity really shines in this elementary geometric proof.

I created a a simple Desmos demonstration to explore Varignon’s Theorem. And like all compelling mathematical results, there are so many fascinating follow-up questions to ask!

**Related Posts**

## Regents Recap — June 2016: Are These Figures Congruent?

Given the congruent triangles below, is the statement “Triangle *ABC* can be proved congruent to triangle *ZYX*” true, or false?

I imagine most will say that the statement is false, and argue that the correspondence of the triangles is incorrect. That is, segment *AB* is not congruent to segment *ZY*, and so on. I think this is a reasonable response.

However, a substantial part of me believes the statement is true. “Triangle *ABC*” references an object, as does “triangle *ZYX*“. These two objects are indeed congruent. Thus, how can it be said they can’t be proved congruent?

In other words, I don’t believe the statement “Triangle *ABC* can be proved congruent to triangle *ZYX” *entails a binding correspondence in the way that the statement

does.

I was thinking about this because of this question from the June 2016 Common Core Geometry Regents exam.

According to the rubric, the correct answer is (3) *reflection over the x-axis*. The most common incorrect response, of course, was (1) *rotation. *But I’m not certain it’s really incorrect. I don’t think anyone would get this question wrong based on my objection, but since the question is designed to entice students to say *rotation*, I think it deserves some scrutiny.

**Related Posts**

- Regents Recaps
- Regents Recap — August 2015: Trouble with Transformations
- Regents Recap — January 2015: Questions with No Correct Answer

## Regents Recap — June 2016: What Do They Want to Hear?

I read this problem several times and still did not understand what it was asking for. It is the first part of problem 36 from the June 2016 Common Core Geometry Regents exam.

*“The base with a diameter of 2 inches must be parallel to the base with a diameter of 3 inches in order to find the height of the cone. Explain why.” *Explain why? What do they want to hear?

Is the expectation that students will say something like “Height is only well-defined when measured between two parallel objects”, or “If the bases aren’t parallel, the height will vary depending on where the measurement is taken, thus height is only a meaningful measurement when the bases are parallel”? As usual, the rubric was no help, simply awarding points if *A correct explanation is given*.

But the model student work says it all. Here are two examples of *complete and correct *solutions.

These are not explanations of why the two bases must be parallel. These are descriptions of how you might compute the height given that the two bases are parallel. This argument essentially says “The bases are parallel because in order answer this question I need to apply a technique that requires that the bases be parallel.”

Not only is this not an explanation, it’s a kind of argument we want to teach students *not* to make. Validating these responses works against what we should be trying to do as math teachers.

These high stakes exams shouldn’t encourage teachers to promote invalid mathematical thinking. Unfortunately, as the posts below suggest, it’s happening far too often.

**Related Posts**

- Regents Recaps
- Regents Recap — January 2015: It’s True Because It’s True
- Regents Recap — August 2015: Modeling Data
- Regents Recap — June 2014: Common Core Algebra, “Explain your answer”