## Trying to find math inside everything else

### Counting Circles

Because I had different warm-up routines I wanted to try, I’m this week ending my second go at Counting Circles, and won’t be using them again until next year. But they’ve had a great run! I think the students got a lot out of them, and I experimented with them in lots of different ways, a few of which I captured as pictures, so I wanted to share them below.

I started, as with Sadie’s recommendation, with just a simple off-decade 10s, to practice the idea.

One of the earlier things I tried was do work with an open inequality – that can count by any amount they want, as long as they don’t go below 40.

Later on, we counted by monomials and, then, binomials. A fun thing that tricks them up is to swap the order of the binomial. (Commutative property!) Then see how starts adding the wrong thing together, just because they were going left to right.

Counting up with one term and down with another can take a few moments for some students.

Later, after we had done exponential functions, I tried out a geometric sequence. But I had to make sure I started low enough that we could get around the class!

Another geometric sequence was the powers of 10. I mostly wanted to make sure they could name them all! They weren’t allowed to just say digits for this one, they had to say the names.

Technically this one is still geometric, though it didn’t feel the same. But I also was a stickler here, too – if a kid said “2 x 26” that’s what I wrote, instead of “2x^26”

As my last thing, today we did a quadratic counting circle. Now, we haven’t done quadratic functions yet – that starts next week. So this was somewhat of a preview. They also weren’t expecting the perfect squares – only one students noticed that in time to help them on their turn. There was a lot more collaboration on this circle because they had to refer back explicitly to what the last person did. I’ll do two more of these (triangle numbers tomorrow), and then that’s it!

### What’s a routine?

At #TMC14, I made the following tweet,

while I was in a session about warm-ups as review. I was reminded of what Jessica posted about her warm-ups, and how she does something different each day of the week.

I was thinking of doing something similar, but I also wanted to include things like Counting Circles. But…I’m having trouble with the disparity. Something like Counting Circles is very different from Estimation180 which is very different from a Throwback Thursday review problem. How can we get used to the norms of a counting circle if we only do it once a week?

I could just commit to one of these things, but I feel they are all important, so I didn’t know what to do. But now I had the thought…what if I did these routines but, instead of once a week, I did them for, saying, a marking period, then switched to another?

This could work because many of the routines match up with certain units – Counting Circles would be very helpful for linear functions, while Visual Patterns would be useful for functions in general (or perhaps for when we do quadratics). Does this sound like a good idea?

### 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?