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!
James Cleveland-Tran
The Roots of the Equation 
Comments on: "Counting Circles" (2)
Hey James – I’m such a fan of this idea, but don’t know how to execute it in a high school classroom – did you do it as a warm up? Go around the room? I’d love to hear details. – WM
I did do it as a warm-up and I went around the room. Class size matters there, of course – last semester I had 20 students per class, so it worked out well. This semester I have more, but because they already mostly knew the routine it was still fine. If you have very large classes, I floated the idea of running two smaller circles, maybe with a student running one of them, to save on time.
I was definitely more comfortable with the pace of the circle when I first introduced it in September. When it came back now in February it felt like I was using up more time – but we did interesting things with it, so I don’t think it was wasted.