Quadrilateral Puzzle

I’ve been immersed in puzzle and lesson creation lately, and I thought I should take advantage and throw some of them out here on the blog. Please take, solve, use in your classrooms or at home, and let me know what you think. If people like the puzzles, I’ll make a point of putting them out here more often.

A Quadrilateral Question for today. This sub-questions goes from easier to harder.

The Big Question: Start with any quadrilateral (Quad 1), label its midpoints, and connect them to form another quadrilateral (Quad 2). When will Quad 2 take up exactly half the area of Quad 1?

Will it happen if Quad 1 is…

1. a square?

2. a rectangle?

3. a parallelogram?

4. a trapezoid?

5. a kite?

6. Nonconvex?

7. Can you find an example when Quad 1’s area isn’t double Quad 2’s? Or will it happen all the time?

You can post in the comments if you’ve got an argument to share…

(Another question is to show that Quad 2 is always a parallelogram. Here’s my proof of that if you get frustrated.)


Comments 6

  1. John Golden

    We did this in a teacher class recently (in the context of patty paper constructions, so really Michael Serra’s idea). I had them start with a rectangle, a rhombus and a parallelogram, which led to more conjecturing, and got them making other quadrilaterals on their own. It also brought up a nice area conjecture which they abandoned on concave quads, but then verified anyway on GeoGebra!

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  2. John Golden

    It was definitely a plus. They didn’t believe it for concave, so it allowed them to move past that, believe the conjecture and start worrying about proving it. They had the idea to prove the midpoint quadrilateral type with coordinates, but were still wondering how else you could prove it, and when would it be a more special type.

    Do you think you could do it just asking the students to look for a relationship instead of tipping your hand? Or could you start with a square and observe it’s half: a) true for any square? b) What is the ratio for other quads? Did they notice anything about the type of the midpoint quads while investigating the area?

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      The way you’re describing at the end is how I’d do it. It’s true for a square, so how much could we weaken the stipulation of squareness and still have it work? Then you’ve got a search for a proof in the cases where you think it works, and a search for a counterexample in the situations where you think it doesn’t, which would motivate dealing with the far out cases like nonconvex shapes. For me, this problem would be more alive if it felt like an open question until you could find a proof. The fact that it does work for such crazy quadrilaterals gives it a real sense of surprise. At least, it did for me…

  3. Stephanie O

    My 10 year old son and I had a great time with this problem. Thanks! Before we get started on stuff like this, I need to remember that I’m trying to *teach* him something, not just reach the answer. I get swept up in the problem (and I don’t know the answer) so when *I* make a breakthrough I share it quickly to see if he sees it too. Of course he’ll learn more if I hint more lightly rather than just telling him what I figured out…

    It is interesting to do this sort of problem with him. I was a “good student” in math at school, but more in the “I can learn how to quickly and accurately do problems of the type we’re studying right now” way. I would give up too easily with puzzles like these, and decided I was “bad” at them. With my son by my side, I recognize the point of frustration where I *would* have given up in the past, but now I don’t want to model giving up so easily, so we usually figure a few more things out. (My husband is scary good at this sort of thing, so dad is always the backup plan when we can’t get through something .)

  4. Raj Shah

    I’d never seen this before. What a great geometric exploration. I’m still surprised by the result! It seems almost impossible that this could be true for a concave quadrilateral.

    Agree with John that starting from a square and making conjectures is a great way to start this for students.

    Please keep posting problems like these.

    Stephanie – I think so many students would characterize themselves just as you did. I was a bit like that as well. I’m working hard to make sure my kids and my students see that problem solving and perseverance as where the beauty and fun are.

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