Midpoint Quadrilaterals

05 Jan
Midpoint Quadrilaterals


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.

1 01-05-2015 Image009 2 01-05-2015 Image010

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.

3 Screen Shot 2014-11-05 at 8.55.19 AM

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.

4 Screen Shot 2014-11-05 at 8.57.12 AM

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

6 Screen Shot 2014-11-05 at 9.17.15 AM

(Yes, Connor your polygon can be concave.)

7 Screen Shot 2014-11-05 at 9.19.34 AM

What do you notice?

8 Screen Shot 2014-11-05 at 9.19.51 AM

It’s a parallelogram.

9 Screen Shot 2014-11-05 at 9.20.31 AM

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.

10 Screen Shot 2014-11-05 at 9.22.36 AM 11 Screen Shot 2014-11-05 at 9.22.57 AM

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

12 Screen Shot 2014-11-05 at 9.24.07 AM 13 Screen Shot 2014-11-05 at 9.24.34 AM

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

14 Screen Shot 2014-11-05 at 9.27.33 AM

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 …

1 Comment

Posted by on January 5, 2015 in Geometry, Polygons


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One response to “Midpoint Quadrilaterals

  1. 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 !


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