**CCSS-M G-CO.C.11**

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

While I am not exactly certain what “and conversely” modifies in this standard, I do want my students to think about not only the necessary conditions for naming a figure a parallelogram but also the sufficient conditions.

Our learning goals for the unit on Polygons include the following I can statement:

I can determine sufficient conditions for naming special quadrilaterals.

I’ve sent Quick Polls before asking students to determine whether the given information is sufficient for naming the figure a parallelogram.

Luckily the teachers with whom I work have kindly let me know how pathetic the questions are, and so I no longer send them. So how can we get students to determine the sufficient information for naming a figure a parallelogram without giving them the list from their textbook to use and memorize?

I started this lesson by showing three (pathetically drawn) figures with some given information and sending a poll for them to mark each figure that gives sufficient information for a parallelogram (more than one, if needed). Granted it’s only a bit better than the Yes/No Quick Polls, but it is better, and it did give students more opportunity to **construct a viable argument and critique the reasoning of others** than the one-at-a-time polls.

For an item like this, I especially like showing students the results without showing the correct answer, as that leaves room for even more conversation about math.

Next I asked them to construct a non-special quadrilateral and then its midpoint quadrilateral.

(Yes, Connor your polygon can be concave.)

What do you notice?

It’s a parallelogram.

How do you know?

I blog to reflect on my practice in the classroom. And so what I know now is that I should have asked students to measure and/or construct auxiliary lines using a sufficient amount of information to show that their midpoint quadrilateral was a parallelogram. Everyone wouldn’t have measured the exact same parts, and I could have used Class Capture to select students to present their information to the class. But I didn’t think of that during the lesson. The students played with their construction, some recognizing that the midpoint quadrilateral is a parallelogram no matter how they arranged their original vertices.

Others recognizing that every successive midpoint quadrilateral would also be a parallelogram.

And none connecting what we had done at the beginning of the lesson with what we were doing now.

And none proving why the figure had to be a parallelogram. I feel like the proof of why should come after we study dilations. But I like students figuring out that the figure is a parallelogram during our unit on polygons.

So maybe, eventually, we will move dilations earlier in the course.

Or maybe we can revisit the why-they-are-parallelograms after or during the dilations unit.

Either way, I’m grateful for a do-over next year as the journey continues …

Or maybe we can revisit the why after or during the dilations unit.

Either way, I’m grateful for a do-over next year as the journey continues …

howardat58

January 6, 2015 at 5:19 am

When I came across this midpoint thing for quadrilaterals recently, having completely forgotten about it, I thought Wow!

When I reached the last row of the last of your pics, number 2 from the left I suddenly thought “Damn ! It’s obvious !”

If you want to leave them with a hint then “Didn’t we join some midpoints once before?” is the least likely to hand them the answer on a plate !