Varignon’s Theorem

Yesterday Patrick Honner posted a nice illustration of Varignon’s Theorem by one of his students:

It is particularly fun to move the points around to form a non-convex quadrilateral and see that the midpoints still form a parallelogram.

As with many advanced concepts in geometry, my introduction to Varignon’s theorem came from Geometry Revisited by Coxeter and Greitzer. I remember the theorem partially because of the lovely introductory statement in the book:

“The following theorem is so simple that one is surprised to find its date of publication to be as late as 1731. It is due to Pierre Varignon (1654 – 1722).

Theorem 3.11. The figure formed when the midpoints of the sides of a quadrangle are joined in order is a parallelogram, and its area is half that of the quadrangle.

The chapter also presents three other wonderful theorems and some super problems including this one:

“1. [Show that] the perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrangle.”

So, this special parallelogram has some really interesting properties!

As a fun follow up this morning, my older son was working on some problems from the 2006 AMC 8. Looking over the test I noticed that problem #5 was a simple example of Varignon’s theorem:

Problem #5 from the 2006 AMC 8

I chose a different problem to go through with my older son, but thought my younger son would like this one. It was a challenge, but he eventually was able to work through it. It is kind of fun to think of this basic problem as one that opens the door to this beautiful theorem.

 

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A question from Tracy Johnston Zager that caught my eye

I saw an interesting question from Tracy Johnston Zager on twitter tonight. Unfortunately twitter isn’t cooperating with me right this second and I’m having trouble linking to all of the tweets (ARRGH!!!!), so I’ll have to summarize because I’m trying to get to be early tonight.

The conversation went along these lines:

Tracy was in a classroom working on a game similar to Nim. On each turn of the game a player can add either 1, 2, or 3 blocks to a pile. The player who adds the 10th block loses (or maybe wins, I forgot this detail, but luckily it doesn’t matter for purposes of this post).

The one tweet I can get to link are some of the “notice and wonder” questions that the students had:

The question that caught my eye was about the students’ questions about the actual pieces used to play the game. I’m paraphrasing, but the question from Tracy as I understood it was essentially – there’s not really that much math behind the questions about the game pieces, so is it productive to talk about those questions in a math class?

I thought that addressing those questions might be interesting, but I’m also a terrible judge as to what will be interesting, so I decided to talk about those ideas with my younger son (a 4th grader).

Here’s how the conversation went.

First I introduced the game and we played a few rounds (just 2 min here – the rounds go quick!):

 

Second – I asked him if he thought the game would change if we played with yellow blocks instead of orange blocks. He thinks that the game will not really change, but importantly for this blog post he does not appear to think this is a silly question:

 

Finally, I asked him what would change if we played with Lego mini-figures rather than blocks. Again, he thinks the game will not change.

Then I asked him what *would* change the game and something really cool happened!

 

It was an incredibly lucky break that the conversation about what would change the game led to my son figuring out how to solve a new game. But, even without that fun ending, I think talking about what would change the game and what wouldn’t change it was productive.

I can definitely believe that a kid would have questions about the game pieces and wonder if changing the pieces would change the game. In my mind it is similar to a kid learning algebra wondering if the way you solve an equation like 3x + 1 = 5 is the same way you solve an equation like \pi x + 1 = 5.

Anyway, I’m glad I tried out the question with my son – it was interesting to hear his thoughts 🙂