Trying to find math inside everything else

Starting Over

One of the reasons I wanted to teach 9th grade when I first started was because I wanted to, eventually, know all the students in the school. (There were other reasons, but that’s one of them.)  So after four years, I’ve taught everyone math and really enjoy knowing all the students well. (Well, I didn’t teach everyone – the students who skipped me to go straight to Geometry, but I managed to get to know most of them in other ways.)

So, because of that, I’ve taught a new batch of students every year, and every year I can refine my routines, toss what didn’t work, keep what did, and try out new things.

But next year there’s a very good chance I’ll be teaching Algebra II – the first time I’m teaching the course (and any main math course besides Algebra I). And that means my students will be the current sophomores, who I taught last year. And I’m wondering, how does that work? What can I carry over easily? Will transferring routines and getting started be faster (not just because they know the routines but are also older)? Will it be harder to toss out routines they liked that I didn’t because they know them? Will the honeymoon period at the beginning of the year be shorter, or longer?

I don’t know, but I’m hoping some people will having some insight. What do you do when you teach the same kids again?

Counting Circles in Algebra II

So there’s a good chance I’ll be teaching Algebra II next year (will everyone leading a morning session in TMC14 change courses before it arrives?) and so I was thinking about my future routines. My students will be (mostly) students that I taught last year, plus the current freshmen who are advanced in Geometry. Last year and this year I was big on Estimation180 but, because I was so big on it, they’ve seen a lot of it. There’s still plenty they haven’t seen, but I wanted to expand. I remember reading someone who said they used Estimation180 one day, Visual Patterns another, Counting Circles another, and I think there was a fourth but I don’t remember what. I thought it sounded like a good idea.

I was hesitant about counting circles at first because, yeah, my students do need to boost their mathematical fluency, number sense, and mental math, and that is always helpful, but it’s a lot of time to spend on stuff that is technically not part of the curriculum. But then I started to think about all the things we could count that would specifically enhance the Algebra II curriculum, and I got excited.

Things We Can Count

• Monomials (2x, 4x, 6x, 8x…)
• Polynomials (a + 2x, 2a + 4x, 3a + 6x…)
• Fractions of $\pi (\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{5\pi}{6},$ etc)
• Sin/Cos/Tan values of those values above
• Imaginary numbers
• Complex numbers
• Geometric Sequences (1, 2, 4, 8….)
• Geometric Sequences with Negative Ratios (1, -2, 4, -8….)
• Monomials geometrically (x, x^2, x^3, x^4…)
• Irrational numbers ($\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}$…)

What else could we count?

How Many Subway Transfers Do I Have to Make?

I was talking with Sam Shah and had the following exchange:

Of course, after I said that…I had to find out if it was actually true. So I pulled up a map of the subway system and started analyzing.

I realized the best way to analyze the system would be to create a matrix of connections: if I can transfer directly between two lines (or the walking transfer from 59th St/Lex to 63rd/Lex, since you don’t need to pay again), then put a 1 and make the cell green. If not, put a 0 and leave the cell white. That’ll show a chart of all the places you can get to on a single transfer.

Most of the lines have direct transfers, with a few being tricky. Breakout stars are the A, which connects with all but the 6, and the F, N, and R, which only miss some or all of the shuttles. Particularly difficult train lines are the G, the J, and the 6.

So this answers the question of where you can get with only 1 transfer. But what about two transfers? For that, we can multiply this matrix by itself. This is the result:

What do these numbers mean? Well, to explain, let’s look at the G –> 6, which I have highlighted in blue. The number there is 8. This means that there are 8 ways to get from the G to the 6 with two transfers:

G –> 7 –> 6
G –> D –> 6
G –> E –> 6
G –> F –> 6
G –> L –> 6
G –> M –> 6
G –> N –> 6
G –> R –> 6

So this chart shows that you can get from any line to any other with at most two transfers*, with one exception: the Rockaway Shuttle to the 6. However! Those stops aren’t solely serviced by the S. (The only stop in the system solely serviced by an S train is Park Pl, on the Franklin Ave Shuttle.)

Because of that, I can amend my statement to the following, which I have proven true:

During rush hour, you can get from any stop on the subway to any other with a maximum of two transfers.

But then, that gets me wondering further…this chart was just made if the connections exist, but they weren’t time-sensitive. For example, the M does not run at my stop at nights or on weekends. How would that change this chart? Especially when you consider that the E, which does not normally go to my stop, DOES at night. I leave that problem open.

———————–

* Of course, fewer transfers doesn't always mean better. If I wanted to get
from Astoria to Greenpoint, sure, I could take the N to the G, but that
requires going all the way through Manhattan, way down into Brooklyn, and
then back up. Instead, a quick hop from the N to the 7 to the G is much
more sensible, even if it is an extra transfer.

Excel File - Subway Analysis