## Trying to find math inside everything else

### 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?