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More Crazy Looking (but easy to make) Origami… #made4math

Here’s another Origami Creation that I learned how to make at the Math Circle Summer Teacher Training Institute (I swear we did math too, I just like to have something to do while other people are talking!):

It’s a hyperbolic paraboloid encased in a tetrahedron. No glue was used and it stays together really well.

Hyperbolic paraboloid (inside) – 1 sheet of paper, lots of folding, about 20 minutes. Instructions.
Tetrahedron (outside) – 2 sheets of paper (same size as above) cut into thirds, about 30 minutes. Instructions (scroll down a bit – only make one of the tetrahedra in the model of course!).

Both units are pretty simple to make if you are patient and can follow instructions, and it is amazing to me that they fit together so well! The Tetrahedron is a bit harder because it’s tricky to fit together. Both units have really cool derivatives and variations that you can make, so they are worth learning how to make!

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Sonobe Unit Origami… #made4math

Every Monday, @druinok has been posting things that she has made for her math classroom on her blog. Tons of people have responded with really awesome ideas of things to make for their classroom. Though I consider myself really organized, I’m not good at the crafty type of things that people are posting. But the walls of my classroom this past year were a ridiculous expanse of nothingness, so one of my goals this summer is to brainstorm ways to make my classroom look less like a factory, which might require some #made4math creations.

At the Math Circle Summer Teacher Training Institute, I learned about a really cool, but simple, way to make awesome origami creations out of a single repeated unit. I want to string these together to create decorations, use them as balls for classroom activities, and just have them sitting around in the hopes that students will want to try to make them too! Here they are:

All of these structures (and many more) are made from this simple flappy foldy parallelogram with pockets called the Sonobe Unit:

If you Google “Sonobe Unit” you will find countless instructions on how to put these together. Here is one that I think is pretty good. Then, all you have to do is make a bunch of these and you can start putting them together in really cool ways. You can even invent your own variations of the unit, or how it is put together to get some really cool shapes.

Some tips:

  • Once you make the units, it’s really important to do the last step of folding them in half (so they should kind of look like W’s). This makes putting them together very intuitive.
  • In the picture above, the one on the left required 30 units and the one on the right required 12. The one in the middle is just a smaller version of the one on the right! If you used only 6 units you would get a cube (no need to fold in half, as mentioned above, for the cube).
  • It’s best to use three colors because of the way it is put together (so you can have three colors come together to make each one of those triangular peaks).
  • Once you start to experiment by putting it together, you should start to see how it works. Just slip one of the pointed ends of one unit into the small pocket on the middle section of another. On one of the peaks, each of the three units should connect to the middle section of the one to its right.
  • To make the creation with 12 units (the blue and green one on the right), just make sure that there are always 4 peaks around any given circle. To make the creation with 30 units, make sure that there are always 5 peaks around any given circle.

Math Circle Problem: Folding & Dragons

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.

So what’s going on with the pattern?
What can you say about the pattern created by the 100th fold?
What is the 112th digit in the 100th iteration of the pattern?

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