I recently read Kate Nowak’s post “Progress on the Wanted Board”, where she references an exploration from Paul Lockhart’s book Measurement. The exploration is of the quadrilateral formed by connecting the midpoints of any given quadrilateral. The post reminded me of a Geometry Nspired activity, Creating a Midpoint Quadrilateral.
So last week during our polygons unit, students created a midpoint quadrilateral on a Geometry page of a TI-Nspire document. We never made it to the question of why the midpoint quadrilateral is a parallelogram. Instead, we got stuck answering the question of how we know the midpoint quadrilateral is a parallelogram. It was actually exciting, because the day before, I had sent students a few Quick Polls to begin thinking about the sufficient conditions to know that a quadrilateral is a parallelogram.
They were pathetic questions. Certainly questions that have been asked plenty of times…in fact, most textbooks have a section of the Special Quadrilaterals chapter devoted to proving when a quadrilateral is a parallelogram. But most of our students answered “yes” simply because the given diagram looked like a parallelogram.
With the Midpoint Quadrilaterals exploration, I was able to ask similar questions in a way that had more students interested. Students decided pretty quickly into the exploration that the midpoint quadrilateral was a parallelogram. But I asked them to justify on their screen that it was. I happened to start with someone who had measured both pairs of opposite sides. Is this enough to show that the midpoint quadrilateral is a parallelogram?
Our definition of parallelogram is a quadrilateral with both pairs of opposite sides parallel. Can we determine whether the figure is a quadrilateral when we know both pairs of opposite sides are congruent? Most students was no. Partly because I asked a second time, which always makes them second guess their initial reaction. So we moved away from the technology, entered the practice of “look for and make use of structure” by drawing an auxiliary line (diagonal), and went from there to prove the two triangles formed by a diagonal congruent by SSS, corresponding angles congruent since “corresponding parts of congruent triangles are congruent”, and then the lines parallel because alternate interior angles are congruent.
In the meantime, some students went to town measuring (and adding color to their construction) … just in case having opposite sides congruent wasn’t enough.
I will say that rethinking geometry using CCSS has made me think of the sufficient conditions to make a parallelogram in a different way. I stay as far away from things to memorize as I can, so I’ve not been asking my students to memorize which conditions are sufficient to make a parallelogram, but neither have I emphasized how we can just take our given information, draw in a diagonal or two, prove some triangles congruent, and ultimately determine whether the opposite sides are parallel.
And so the journey continues ….