If you are a math teacher and you haven't yet seen James Tanton's "Exploding Dots" presentations, I encourage you to dive in. I've cloned a version like the one below to use in the latest release of the Long Division lesson app.

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In this version, I just let users circle their own groups, rather than handling this with code in some way (click the pencil icon to draw, and again to go back to dot manipulation). You can erase your groupings with the windshield wiper button, and you can clear the dots with the trash can button. Here's more of what you can do above:

- Click on an empty part of any column to insert a +1 dot (positive dot), right click on an empty part of a column to insert a –1 dot (negative dot), and double click on a dot or tod to remove it entirely from the division model.
- Click the "explode" button in the bottom right corner to create an explosion in a column, which, in the 1 ← 10 machine removes 10 dots (not tods) in a column and places a dot in the column to the left.
- Right click on a dot (or tod) in a column to "unexplode" it, which removes it from that column and inserts 10 of its kind in the column to the right.

So, for example, to model the subtraction 248 – 169, you can click on the appropriate columns to generate the number 248 with positive dots. Then, right click to insert "tods" in the columns representing –169. You can use the model to add 248 and –169 (the same as the subtraction 248 – 169). You will wind up with something like the model shown at the right:

You can simply be done here and read off the appropriate interpretation of the difference: 100 + –20 + –1. Or, as Tanton says, to make the answer more appropriate "for society," you can "unexplode" a positive dot in the hundreds column, giving you 0 hundreds 8 tens and –1 ones and then "unexplode" a positive tens dot to give you, finally, 0 hundreds, 7 tens, and 9 ones, or 79.

A division example is shown at the right. Here we see that 248 ÷ 11 = 22 (2 groups of 11 tens and 2 groups of 11 ones) with 6 (ones) left over, or 248 = 22 × 11 + 6.

What I find tremendously cool about this model is that, having played with it a lot, I find myself doing some forward thinking with the dividend. That is, I find that the model has me wondering **what numbers I can divide into the dividend**, rather than what the quotient to any particular problem is. So, I'll plop a number into the model and look for any divisor that will give me easy divisions (divisibility) or think about how I can slightly fudge the model (insert a positive-negative pair in a group, for example) to get cleaner divisions. All of this, for me, quickly transfers to just numbers, so it seems at least plausible that once a student is expert with the model, they don't have to live there. This goes for all the basic operations.

As Tanton goes on to explain in his presentations, the same model can be used for polynomial long division. So, for example, when I search for polynomial long division, the first example I get is \[\mathtt{2x^3 + 7x^2 + 2x + 9 \div 2x + 3}\] And one way to model that division is shown at the right. This time, from right to left, our columns represent 1s, \(\mathtt{x}\)s, \(\mathtt{x^2}\)s, \(\mathtt{x^3}\)s, and \(\mathtt{x^4}\)s. Interpreting the quotient, we see that it is \(\mathtt{x^2 + 2x – 2}\) with 15 left over. (You have to be careful with the 1 ← 10 machine above, though, when representing polynomial long division. Explosions and unexplosions are too difficult to represent on the model, since we don't know what x is (we just know, for example, that there are \(\mathtt{x}\) \(\mathtt{x}\)'s in \(\mathtt{x^2}\)). But the machine will let you do it anyway, because it's not built for algebraic polynomial long division.

What Curriculum Innovation Looks Like

When I hear the word "innovation" in education, something like the thinking that went into creating Exploding Dots is what I secretly wish everyone meant by the word all the time. The best part of that thinking in this case, for me, is how division is represented. It's fairly straightforward to see how the circling works when we're creating groups of 11. But why should I be able to circle 2 \(\mathtt{x^3}\) dots along with 3 \(\mathtt{x^2}\) dots to represent dividing by \(\mathtt{2x + 3}\)? The full answer to that question involves some factoring, but it connects the model with division all the way from elementary to high school. That's pretty powerful. \[\mathtt{10(10 + 1) = 100 + 10\,\,}\] \[\mathtt{x^2(2x + 3) = 2x^3 + 3x^2}\]

I see smaller versions of this kind of innovation occasionally in math ed, and my current opinion is that it's these kinds of small improvements that, over time, create real improvements in student learning: better explanations that allow more students to access and then follow through with the content, allow students to retain the information for longer, or allow students to make deeper connections to later material.