How many do you see? How do you see them?

This summer, as Gwyneth and I were packing up Othello, I started playing with different arrangements of discs – mostly arrays – and asked her “How many?” I remembered the following arrangement, taken from AIMS’ Cookie Combos activity.


3^2 + 4 * 4

“Sixteen plus nine, so nineteen plus six… twenty, twenty-five,” she said. (I don’t think that she actually said “twenty” aloud. That came after my clarifying question: “Wait. Huh?”)

There’s a lot happening in Gwyneth’s bridging through twenty strategy – partitioning of quantities, place value, commutative property, breaking apart to make (a multiple of) ten. All within a three count, standard algorithm be damned.

This invented strategy discussion was a happy accident. The goal of this problem when we pose it to teachers is to see different ways to visualize the group and represent these using expressions. It’s about valuing different methods; the solution – counting 25 cookies – is easy enough.

How many do you see? How do you see them? How many different ways can you find?


Some popular solutions:


7 + 2 * 5 + 2 * 3 + 2 * 1


3 * 5 + 2 * 4 + 2


4 * 5 + 5

If you look just right, you can see two arrays:


4 * 4 + 3 * 3

A creative solution that involves counting what’s not there:


7^2 – 4 * 6

And moving what is:



If you plan on using these images with your students, I recommend displaying the photo with just white discs. This leaves the problem open. Two colours were used above to illustrate various visualizations. This can steer student thinking. (See how the use of colour is intended to be helpful here.) If students miss one of the visualizations above, display that photo and ask for the expression (or vice versa).


3 thoughts on “How many do you see? How do you see them?

  1. Brilliant! I’ve got a few Othello boards, so I’ll give this a whirl.

    9, 16, 25 – makes me think of Pythagoras – is there a proof on an Othello board…??

  2. I also had a passing thought about Pythag. Funny how a 3-4-5 representation can do that to me (and apparently also to Simon).

    I could see using this post–specifically the pictures–to illustrate what the heck I mean when I say “using color for a purpose.” I think using technical writing tools (such as color-coding) is something I’m pretty clearly communicating to students that I like/want, but not doing so well to explain what exactly that looks like. Some kids get it, others haven’t a clue and wind up decorating their paper with different colors to make it “look pretty.” This could be a good first step in explicitly teaching what using color can do to communicate mathematical thinking.

    To the Evernote webclipper!

  3. Simon & Bree,

    You two aren’t alone. At a conference last week, I was in a session where we were given this task. One group presented this as their visualization:

    My guess is that these teachers were looking for Pythagoras because of the 3-4-5/9-16-25 connection. I’m skeptical that this was one of the first visualizations that came to mind; they went looking for it, I think. Still, fun.

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