WTF and LCM (AKA Bunk and Bonzo)

Back in September, Wendell Pierce (The Wire, Treme, Suits, Selma, Jack Ryan, …) was the guest on Marc Maron’s WTF podcast. In it, he describes what he calls “the American aesthetic”:

… Jazz is based on the emulation of human dialogue. When you’re trading fours, you’re having a conversation. And improvisation that it comes from, it’s really the American aesthetic on display. Freedom within form. You have to honour the form of the music but, as a soloist, you have the right as an individual to go as far as you want to go. We’re a nation of laws but as an American we celebrate individuality. Honour the form, honour the laws but be yourself, be free. It’s a finite amount of notes with an infinite amount of combinations. Improvisation. And then that taught me how to act. Because it’s a finite amount of words but with an infinite amount of ways of saying them and an infinite amount of ways of having those words affect you. And that’s jazz and that’s what the American aesthetic is, unique to our experience…

–Wendell Pierce

I love this! It reminds me of the relationship between freedom and constraints in mathematics. In fact, Pierce goes on to connect freedom within form to two different proofs that land on the same truth in Trigonometry class. He argues “the two can coexist” when talking about Shakespeare or stand-up. Or, more importantly, when talking about the Constitution and John Lewis or police reform. It’s quite the “coffee run” and starts at about 1:04:20 below:

But a less significant section of his riff grabbed my attention. I learned why Led Zeppelin’s “Kashmir” has always sounded “off” to me (1:06:30). John Bonham’s drums are in 4/4 time. Jimmy Page’s guitar is in 3/4 time. The effect is that the guitar (and strings) seem to be ahead and then behind the beat. Listen for yourself below:

They meet up on twelve, the lowest common multiples (LCM) of three and four! Play with the slider below:

Guitar: 3/4 time; Drums: 4/4 time

It’s like a variation of FizzBuzz. FizzBuzz is a child’s game — or drinking game — in which players count around a circle, replacing any number divisible by three with “Fizz” and any number divisible by five with “Buzz.” Numbers divisible by both (i.e., numbers divisible by 15) are replaced with “FizzBuzz.” If a player hesitates for too long or messes up, they’re out.

Top: Multiples of 3 (Fizz); Bottom: Multiples of 5 (Buzz); Centre: Multiples of 3 & 5 (FizzBuzz)

This podcast came along at a time when Marc and I had wrapped up a series of Math 6/7 videos for parents. Factors & Multiples was one of mine. A few activities, like the following, ended up on the cutting room floor:

The gist of this activity is that students are challenged with determining two mystery numbers as their factors (or non-factors) are gradually revealed. Like Wanted Parabola, with each new “clue,” students must assess their thinking. For example, suppose that after 9, 4, 3, and 7 are placed, a student believes that a and b are 18 and 12, respectively. The placement of 1 isn’t helpful; it’s a common factor of a and b, no matter their values. But the placement of 2 means that a cannot be an even number like 18; this student must revise their thinking. (Their choice of b can remain the same since any multiple of four is also a multiple of two.) Throughout, students can be asked “How confident are you?” Notice that with the placement of 5, b must be 60 (or some multiple of 60); the subsequent placement of numbers in this circle adds nothing new. (Maybe it’s worth tinkering with the order in which the clues are revealed?) After, students can be asked to reflect on their mathematical reasoning: “Which clues were most helpful? Which were unnecessary?”

I haven’t tried out this exact task with real students… yet. I’d welcome any feedback from anyone who’s able to test-drive the task for me during this time.

This activity was cut from the video for a couple of reasons. I questioned the task’s accessibility. And slow and patient disclosure of information just plays out better as a classroom activity. More on the design principles behind these videos in an upcoming post…


Woah! Joe Schwartz shared this video about Bonzo from Polyphonic:

Check out “Kashmir” at 5:00!

Uploaded these GCF and LCM GIFs, also cut from our Math 6 Factors & Multiples video. First area, then length:

gcf(20, 30) = 10
lcm(20, 30) = 60
gcf(20, 30) = 10
lcm(20, 30) = 60

Math Picture Book Post #3: Miss Lina’s Ballerinas

Miss Lina’s Ballerinas by Grace Maccarone is about “teamwork, making new friends, and the pleasures of ballet.”

It’s also about math.

In my previous post, I wrote about multiplication in terms of groups of and arrays. Both models can be explored in Miss Lina’s Ballerinas. Eight ballerinas–Christina, Edwina, Sabrina, Justina, Katrina, Bettina, Marina, and Nina–dance in four groups of two

Miss Lina's Ballerinas Groups

and four lines of two¹.

Miss Lina's Ballerinas Array

What happens when a new girl, Regina, arrives? Spoiler alert: three rows of three. What if there were ten dancers? Eleven? Twelve?

If you are playing alongMiss Lina’s Ballerinas falls into my third category; the math concept is between the pages but the author did not intend to write a math concept book.

¹ This bugs me. Should it?