Category Archives: Math Circle
What can you do with this?
Today, we did what I thought was one of the coolest explorations I have seen in a while. It is called the “One Cut” or “Fold and Cut” problem, with inspiration from Patrick Honner (@MrHonner, this part of his website). The premise is simple: you have a shape drawn on a piece of paper. How can you fold the paper so that you can make one straight snip with the scissors and cut the shape out?
Weirdly enough, there is a theorem that says that no matter what shape, this is possible (concave, convex, numerous closed figures, as long as the sides are all straight) . Which totally blows my mind. Because it’s really friggin’ hard in practice (though, the star above is pretty easy). Check out some patterns for some other shapes here.
With one group today, we started doing this problem as a way to see what vocabulary they knew with polygons and to talk a bit about symmetry. I printed out little shapes on 1/4 sheets of paper and the levels went as follows, getting a bit harder as the levels go up:
Equilateral Triangle –> Square –> Isosceles Triangle –> Rectangle –> Regular Pentagon –> Regular 5 pointed Star –> Scalene Triangle –> Arbitrary Quadrilateral
They were TOTALLY hooked. Every single kid was working on their own and kept either having the “YAYY, TEACHER LOOK!” reaction or laughing hilariously at the silly shapes that they made by accident. The kid in the picture with his hand over his face kept yelling out “AGHHH, TRICKY TRIANGLE, TRICKY TRIANGLE” because he couldn’t do the scalene triangle. But he wouldn’t accept a hint from me because he wanted to find out on his own!
45 minutes later, the classroom was a total mess, and I was wondering where all the time had gone…
Now, working with the regular polygons you might get duped into thinking it’s pretty easy. But try a scalene triangle. Or a non-special quadrilateral (i.e. most quadrilaterals). It’s actually VERY difficult. And I think the solution is pretty fascinating, because my solution to it (which I got after about 10 triangles and an hour) heavily involved the triangle’s incenter (the intersection of the three angle bisectors). Which made me think that this would be a super cool thing to do in a geometry class when talking about angle bisectors. Are there multiple “incenters” on shapes with more sides, even if you have to define the idea in a different way?
I won’t share the solution but just post a picture to show you that YEAHHH it’s possible.
(If you haven’t heard me bemoaning how much energy it takes to do math for 2.5 hours with 11 and 12 year olds at a summer camp, well then FYI: I am currently teaching at a summer camp for 6th and 7th graders from China who are here doing an academic program that is half ESL, half math. It’s getting more and more fun, but certainly has been an adjustment from teaching seniors in high school for only 45 minutes at a time).
This problem was originally posed by Sue on her blog Math Mama Writes (and was presented by her at the Match Circle Institute). There’s a kids’ game called Spot It, where there are cards with pictures all over them in a pile. If you have a match with the pile in the middle, you call out the name of that icon and grab the card. The person who collects the most cards win.
But here’s the interesting part – despite there being 57 different pictures and 55 different cards, every card has one and only one match with every single other card.
How did they make this game?
Would it work for every number of pictures?
Is there an algorithm for every number of pictures?
The best way to see why this is such an interesting question is by trying to make your own deck with 3, 4, 5, 6 etc different pictures on each card. If you do that, my “solution” below might make some sense!
This past week I attended the Math Circle Summer Teacher Training Institute. The idea of a Math Circle is that students learn math best by constructing it themselves. In Math Circles, you pose interesting problems that could probably lead to deep mathematical insight, and then let students discover those insights through conversation and collaboration. The leader is there to ask questions, moderate and guide, but should not steal the opportunity from any child to discover something for themselves. It’s a very cool way of learning math, which I personally thoroughly enjoyed. Is it the future of education in schools? I’m not 100% sure, but that’s a topic for a future post.
My favorite part of the week was struggling through some awesome problems with other math teachers in the training. I wanted to share a few of these problems because I think they are really fascinating, and could be used in traditional classroom environments too. For the next few posts, I’ll pose the problem, and then discuss some of the solutions a little lower in a “read more” tab (so if you don’t want any spoilers before you solve it, don’t click on that!).
Folding Paper and the Dragon Curve
The first problem is courtesy of James Tanton, who poses a ton of rich mathematical problems on both Twitter (@jamestanton) and his website (jamestanton.com). Here’s the problem:
Take a strip of paper and fold the right side over to the left. Unfold. Notice that if you hold it as you originally held it and then look at it from the side, the fold makes a little valley instead of a peak. We are going to number valleys with 1s and peaks with 0s. So to convert the fold pattern into a sequence, after the first fold the sequence would be just 1.
Now take the same paper, and always holding it in the same orientation, fold your original fold again from right to left, and then fold one more time. So you are bringing the fold from the right to the edge on the left. After you crease and unfold, and then hold it in its original orientation, you notice that there are two valleys and then one peak, so the sequence for this fold is 110. If you fold one more time (again being careful to always fold right to left), the next sequence should be 1101100.