# Patrick Honner’s rectangle dissection problem

Got a happy surprise last night when Patrick Honner sent me a draft of a geometry project to look through. I’d just finished up a chapter in our Geometry book yesterday and was looking for a project to do with my older son today, and, hey, a project just fell into my lap!

This one is a little longer than normal – I essentially filmed our entire 30 minutes this morning to provide a comprehensive review for Patrick. I think that my son enjoyed this project and learned some interesting ideas in geometry along the way. There’s a nice 3D printing project hiding in here, too, but as we are in the process of moving houses right now, I’m not sure that I have everything I would need for that project readily at hand.

Here’s the introduction to the activity along with step 1:

Step 2 is a basic construction. I’m just making sure that he understands the instructions at this point.

Step 3 is a second basic construction. In this part you have to remember how to measure a length with a compass.

Step 4 is the last construction required to set up the problem. Pretty similar to step 3.

Step 5 is the first problem in the project. We have to prove that two triangles are congruent. There’s a nice little “aha” moment around the 2:05 mark.

Step 6 is the first of two steps where we prove two line segments have the same length:

Step 7 is the second of the two steps where we prove that two line segments have the same length:

Step 8 is the doozy – we are going to try to show that the original rectangle that we’ve chopped up can be rearranged into a new rectangle with different side lengths. This part gave my son quite a bit of trouble. I had to chop up the conversation here into three pieces, and also had to give lots of help to get him to the end of this one. Around 1:50 In the third video in this sequence he sees how to slide the pieces into place:

The last part of this project is showing that this picture isn’t just geometry – there’s a neat algebraic identity hiding here, too. We looked at the geometry behind this algebraic identity in two different ways.

So, I really enjoyed the 30 minutes we spent on this project today. The basic ideas in geometry – some simple constructions and congruences – were things that my son was able to tackle without too much difficulty. The idea of rearranging the pieces to form a new rectangle was quite a challenge, though. But, not so challenging that he lost interest or got frustrated. It was also neat to be able to look at the connection between geometry and algebra at the end. Obviously it was especially nice to hear my son say that the project was fun.

As I said that the beginning of this post, it was a nice bit of luck that this project fell into place today.

## One thought on “Patrick Honner’s rectangle dissection problem”

1. Fascinating stuff, Mike. Thanks so much for trying this out, and sharing!

Really interesting to see how you and your son responded to the directives, and how you (as guide) worked with him. I love the “draw what you think the new rectangle would look like” move. And your explanation of the algebraic identity is so much clearer than mine!

Watching this certainly makes me think about how I will facilitate this activity. By the end, there’s so much going on in the diagram it may be hard to see the right moves. I also think I’ll put scissors into students’ hands at some point. And I agree–there are some great 3D printing possibilities here.

Really happy to know that you found this valuable. And that your son enjoyed it! Thanks for the feedback, and for sharing your son’s “Aha!” moment.