More Similar Figures

While I was at NCTM I left my Math 2 classes with this three-dimensional figures task. When I returned, I was greeted with a messy classroom full of ants (marshmallow bag left open : / ) , but also this:

IMG_1497

It’s pretty clear that these are mathematical shapes, but I could never get past the fact that Platonic solids, particularly stellated ones, are aesthetically pleasing. But I did some research and came up with a not totally visual task for my students to work on. Since I wasn’t going to be there, everything is extremely spelled out – on a non-sub day I may leave some things more open.

Platonic Solids task

This task first has them construct three out of cardboard. I had printed a bunch of nets, but I guess they couldn’t find them because nobody used them. I kind of like it without the nets better anyway, because that’s a nicer, richer visualization task. Then they were supposed to re-create two of them with toothpicks and marshmallows or with straws. Only a couple actually attempted straws. I made all of the shapes with straws just to try it out, and I’m still struggling with the icosahedron. I’m really excited to try to make a stellated one.

The question I wanted them to answer is why there are only a limited number of Platonic solids. I’ll admit the task only gets them most of the way there, but I think it’s a good start.

The coolest thing about this whole thing is that I gave nets for the stellated shapes to my advisee group, and now I’m seeing kids just folding un-assigned 3D figures. I’d say that’s an engaging task! I definitely want to continue in this vein of highly visual mathematics with rigorous tasks aside from just “see how cool math looks.”

Although it does look pretty cool!

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NCTM Conference Thoughts

My main thought at this point is just “!!!” but that’s not really a worthwhile blog post, is it?

More specifically: Boston is a really lovely city, with a great skyline, really good mass transit, and a Chinatown far less fake than the one in DC. It’s very obvious that the population is disproportionately intelligent, and nobody actually talks like Juliane Moore on 30 Rock, so that’s a relief. The NCTM conference itself was exactly as energizing and inspiring as I had hoped it would be.

The first talk I went to that I really loved was by David Peabody from the other Washington. He teaches a class entirely devoted to math in art, technology, and history, so I got some excellent ideas in my continuing quest to make my curriculum arts integrated, or at least arts enhanced. I just emailed him the link to my blog so there is at least a slight chance he could be reading this RIGHT NOW, hello if so!

Next I heard from Geoff Krall, who did an excellent talk on strategies for turning closed textbook problems into open, engaging tasks. I’m working on some ideas for “doing the dang thing” with three-dimensional design problems, possibly doing something with sketch-up. This talk made designing open problems seem entirely possible, and may inspire me to crack open a textbook for the first time in literally years. He also said it to me in a language I understand – Parks and Recreation clips.

That evening I went to Jo Boaler’s much-anticipated talk. I was so inspired that I cried, like actual tears. She showed clips too, but not funny ones, terrible ones of Disney Chanel shows implying that girls can’t be good at math. She shared that she is soon to meet with Disney to address this issue, and just continues doing everything she can to eliminate math anxiety in a fight-the-power, taking-on-the-patriarchy sort of way.

Friday I went to another great arts-integrated talk by Carole Desoe about frieze patterns and other tessellated/ grid-based designs, mainly in Islamic art and architecture. I want to go on a field study to the Freer/Sackler galleries to find some of these grid-based designs, then maybe to the Native American and African art museums to compare these intricate interlocking designs across cultures. I want students to be able to look at these tiles or paintings or cloth patterns and see the underlying grid structures. She gave us some great resources to be able to do that.

Next up was Dan Meyer’s talk, obviously great. I’m inspired to look for potential math modeling situations everywhere now. I’m envisioning a task involving mass transit payment options: here in DC it’s crazy complicated with a different rate depending on how many stops you go, and an additional $1/ride for paper cards, but a $10 up-front cost for SmartTrip cards. I can’t decide if he is more the math teacher Brad Pitt or Dalai Lama. I will admit that I soaked up his celebrity just like everybody else, and was just as starstruck as I went up to shake his hand. I asked him what to do for our new “learn what you missed week,” when kids will be back in class for 2 days AFTER final exams, so I’m eagerly awaiting a response because I’m basically one of those teens who got invited to Taylor Swift’s houses last year. As in “OMG MET DAN MEYER, I LITERALLY DIED!”

The last really exciting talk I attended was by Dan Anderson about the Mandelbrot set. I was tangentially aware of the Mandelbrot set and the Julia set because they are fractals, but I guess I assumed their generating functions were more complicated than they actually were? He has this great program that generates the set with increasing precision. The math is also approachable and involves all students in creating these stunning images. 100% using this program with my math 4 class next year, especially because all of them will have a background in complex number operations. I love being the only [non-senior] math teacher some days, it can have its advantages when it comes to prior knowledge.

So now I’m back in DC and should get to work on things like grocery shopping, writing exams, and just continuing to soak up the math that is all around me.

3D Figures

I remember in college we had this super big “unit plan” project where we had to write out lesson plans for every day of a unit about a particular topic. Some classmates felt this was unrealistic because nobody would create an entire unit from scratch. I hoped that it was unrealistic because I wanted to create all of my units from scratch, which I kind of am now, and I didn’t really think it was possible to devote all of this attention to every detail of everything. Our professors gave us some really good advice, that if you focus on making one unit each year really good, then you can save that, and the next year you can focus on making another one really good, and then eventually you can teach everything to the best of your abilities.

This is not to say that I have created a really good 3D figures unit, just that I’ve been improving it steadily. I had planned on revamping the similar figures stuff, but I ended up only slightly tweaking that. 3D figures definitely have so much potential.

I have 2 sections of this class, and each is focusing on something slightly different: one on the visual aspects, and one on the algebraic aspects. This is giving me a great chance to experiment. I have this exciting exploration with marshmallows and toothpicks that I’ve had kids do in the past. It’s designed to have them discover the Euler Characteristic, V – E + F = 2, for vertices, edges, and faces.

marshmallow

Cool Platonic solids, right? I had to look this up, but Platonic solids are polyhedra with regular shapes as faces. Because of reasons I’m having them investigate next week while I’m at the NCTM conference, there are only 5 of these. Making an icosahedron out of toothpicks and marshmallows is extremely difficult, FYI.

We’ve also been working on volume and surface area. We found the volume of this little house in our school’s back yard by measuring everything and splitting it into a rectangular prism and a triangular prism.

volume house

When I was looking for non-terrible proof stuff I came across this and needed us to do it. proof by clementine

See, the surface area of a sphere is 4 times the area of the great circle! I love this so much. We proved the surface area of a cone using party hats and sector area – some great review, and great fashion statements! This is going to allow us to compare the surface area of a cylindrical cup and a conical cup with the same volume.

There are so many more things we can cover in the 2 weeks (!!!) that I have left for content this year. Stay tuned for another post!