Monthly Archives: July 2012
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!
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.
- 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.
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.
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?
Coolest thing about teaching math? You can use math to explore whatever the flip else you want (population, box office trends, the economy etc etc). Which is why I love love love data and try to start most units numerically with data either found or collected. Another math blogger posted looking for good data sources. I am also always looking for data sources, specifically ones that have interesting and extensive data that are easy to search through, and then easy to access in spreadsheet form. Here are some that I have found – please comment with any other sources you know (cough*stat teachers*cough).
UN data: A huge database of socioeconomic data broken down by country that could be really fun to explore, but might be a bit overwhelming if you don’t have an idea of what you are looking for. My favorite more specific UN data though is the…
UN World Population Prospects: Great population data for every country in the world (since 1950) and population predictions through 2100. Is is really easy to search for and then download the data in .csv form. There are also lots of great population factors that you can look at too. I actually do a whole week-long project with this database (I love it that much) .
Gapminder: Good, focused data about various human geography type topics.
Google Public Data Explorer: Like the UN data website, just a crap ton of random data that would be fun to sift through. They do have a rolling front page though that gives random data ideas, which is fun to look through.
Google Trends: This is a really cool Google service (surprise, something cool from Google) where you can get data on how popular search terms are over time. You can see the correlation between different terms and restrict data to different regions. I really want to try to do something with this next year!
Box Office Mojo: I love box office data! Different types of movies seem to have different models that fit their data (for example, the gross earnings for an indie movie turned popular fits a logistic function very well), and talking about weekly intake of money vs. gross earnings is a nice easy way to talk about derivatives. Sometimes it’s a little hard to find that data in a good form because people usually only care what’s going on this week, and also care more about rankings, but you can click on a movie and then get box office earnings by week for every movie ever, which I have found to be the most useful way to look at it.
Baby Name Voyager: I love this website, even though the data isn’t that accessible (just graphs and then you can see the numbers on the graphs). It shows trends of baby names over time (as in how many people are named Bowman by year) which is really fun. (Answer: not many).