Don Steward’s Complete the Quadrilateral

This activity is from Don Steward’s post Complete the Quadrilateral. I’ve only learned of Don’s MEDIAN site more recently, and it’s a treasure trove because nearly every post can be an instant lesson in itself. Thank you!

I like this set of 24 problems a lot, but when I click on the image of the problem in his post, I get an image that is too small to fit on paper. It’s such a good task that I didn’t want the small size to affect the kids’ engagement with it in any way, so I recreated the whole handout from scratch using Geometer’s Sketchpad. (I’m too scared to contact him and ask for a regular size handout. You do it!)

I didn’t have room on the handout to fit his instruction, so I just told the kids what they needed to do, “Join the dots to complete these quadrilaterals — where there are options, try to find the one on the grid with the largest possible area.”


And I made an answer key too. (I wouldn’t trust it entirely. I’d be grateful if you found a mistake and let me know.)

My answers are not the same as Don’s because he felt it was better for this task that a quadrilateral not be a rectangle or a square, also a kite should not be a rhombus. I did not tell my kids this.[2/8/13: Our geometry textbook defines a kite as a non-rhombus anyway.]


My kids not only thought it was fun to form the requested quadrilateral given one segment and make it the largest possible area, they also found it challenging to find the area of each shape.

I was surprised that few of my students found the area the same way that I would. Most of them divided the drawn quadrilateral into smaller parts and tried to figure out the area of each part. Unfortunately a fair number of them made mistakes using this strategy. At least two students were using the Pythagorean Theorem to find the sides then plugged these into some fancy formulas. Oy.

I just found the area outside of the polygon and subtracted this from 16 square units. I didn’t share my strategy with them until after they’d turned their work in. One student said, “Geez. That is so much easier! It took me forever to find the area!” Live and learn, kid.

If appropriate for your class, I hope you’ll consider doing this lesson — it’ll make the hours I spent recreating the handout and key worth it! :)

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