Intro to Math: Circles and Squares

School is back in session! One of the students I tutor apparently has no homework tonight, otherwise I don’t know when I would have had time to put this into writing – things got busy in a hurry.

I’m doing the whole #teach180 thing so if you’re desperate to see more frequent updates from the classroom you can follow me on Instagram, @multiplefactorsi. I’ll be posting every day, held accountable by a reminder on my phone. I hate tiny little red numbers on that screen, so you know I’ll be following the rules.

Anyway! The first week of math class I always do “intro to math.” We started with some talk about what topics in math we were going to do over the year, and then we spent some time on an arts integration project inspired by these tasks for small children. This would have been awesome if we had more time, or maybe they would have gotten old – who can say? A couple students finished them beautifully at home, and others have kept their unfinished projects in case of momentary boredom. I’ll attach that whole assignment.



PDF of the assignment: day-1-arts

The second day was all about seeing things from multiple perspectives and integrating multiple viewpoints into collaborative work. We started with a number talk, something I loved when I first saw it but just never used. I took the one from YouCubed’s “Week of Inspirational Math,” week 2 day 1. I had been very nervous that our conversation would be nothing like the thought-provoking and joyful example video, but this went amazingly. Here’s our board:


Inspired by Sarah at Math = Love, who was in turn inspired by other bloggers, we then launched Broken Circles. That was great! I love hearing students talk about math, but I also loved the no talking rule. It’s also an inspired touch that one circle completes itself. In one group, the person with the “A” pieces sat there, self-satisfied while the rest of the group struggled, and it was kind of glorious. It’s almost like the point of this task was to show that working together and paying attention is crucial.

Between that and our next collaborative task, we went over our group norms. I translated them into Spanish for an extra touch, and perhaps that will inspire me to do more group work in Spanish class? Time will tell.


The end of Wednesday we started the Pentomino task – I blogged about Pentominoes last time I did it, SO long ago! This year I didn’t leave as much time and consequently they didn’t come up with quite as many combinations, but I do still think it was a useful exercise in visualization and pattern recognition. I’m kind of in love with my independent reflection for that task, attached. Why indeed can you not build a 6×6 square?

Independent Reflection PDF: pentomino-ir

Things are shaping up, but I already know I won’t have nearly enough time to blog as I’d like : 0

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.

Math and Jazz

I have been hearing that math and music are strongly connected for quite some time. I’m always interested in integrating art into my classroom, so I have been listening. The only problem is that I have utterly no clue about music. I definitely cannot read music, have never played an instrument (except for the recorder in elementary school, but I accidentally snapped that in half…), and feel like Fall Out Boy is a genuinely good band. Luckily, the internet was there for me.

I looked at these lesson plans for some inspiration:
NCTM illuminations  “seeing music,” love you NCTM
PBS Ken Burns JAZZ, “Math and Jazz: The Beat Goes On

We just did the straw graphs, and then all the wrap-up needed for them to be extra familiar with the graphs of sine, cosine, and tangent. They know not just what y=sin(x) looks like, but also why it looks like that. Now they’re working on what happens when the graph is y=4sin(x), or y=sin(220x), or y=sin(x)+1 (which sidenote, I do not believe has a musical meaning).

The first introduction to sine shifts was watching this video. They agreed it was pretty cool. This also helped us to discover that the larger the wave’s amplitude, the louder the sound, and the higher the pitch, the greater the frequency of waves. This was a great class because most of them know more about music than I do, so we were all teaching each other.

Then we followed the NCTM illuminations plan pretty closely. They found the frequencies of the notes in two octaves on a piano. *SPOILERS* In addition to reviewing geometric series, they discovered that the frequency of an A note in the higher octave is twice the frequency of an A note in the lower octave.

Things got exciting when we graphed a lower A, a higher A, and then a B note. I asked them to observe what how these graphs compared. The two A notes are harmonious – they would sound nice played together, and their graphs intersect in even intervals:


An A and a B note are too close to each other. They would clash when played together, and their graphs reflect that, kind of hitting each other randomly, as a student put it “just off”:


For an excellent example of dissonance in jazz, listen to City of Glass by Stan Kenton. It’s on Spotify!

I’m really glad I finally did this. I feel like I understand music more, and the students have something concrete to tie the very abstract concepts of sine graph shifts to.