In math, what are “exploding dots?” I work with students and one of them is taking a course called Exploding Dots. I thought it sounded fun so I looked it up—I’m having trouble even understanding the concept of it, much less how it actually works.
In: 483
I hadn’t heard of this before and I just looked into it. It looks like a noble cause (although I dislike the name which seems to just be a byproduct of the creator’s enthusiasm rather than being a meaningful name).
Here’s how I imagine it came about: When doing arithmetic or in general when dealing with numbers, we make students learn certain rules. There’s a specific way to write numbers and then if you want to add two numbers there are specific rules to manipulate those numbers and get a new one. And similarly for subtraction, multiplication and division.
Let’s do addition in the purest way possible: To figure out 3+4, you just take 3 objects, then take 4 more objects, put them together and count how many objects you now have. This is the purest way and it is very good to not stray too far from it.
The problem with this way is that adding 123 and 322 will take ages. The cleverer way to do it is to group it into ones, tens and hundreds and add them in these groupings. But isn’t the way we are taught in school do exactly this? Yes it does, but that’s what’s happening behind the scenes. As a person carrying out the steps taught you’re not really interacting with the numbers. You miss out on the wonderful machinery that changes the numbers.
This is why I actually really like the exploding dots technique. It does the same job of grouping into ones, tens and hundreds. But it doesn’t act like there’s only way to write a number. **123 is 1 hundred, 2 tens and 3 ones, or just 123 ones, or 10 tens and 23 ones, or even 2 hundreds, -8 tens, and 3 ones. All of these are the same number and the basic idea behind exploding dots is to get you to embrace that**. Using this, you can do grouping and also addition in its pure form at the same time without relying on other steps.
And why should one embrace that? Because it makes playing with the numbers so much easier. There’s so much more you can do, like some mental tricks become visible tricks: you can add 8 by subtracting 2 and adding 10. Negative numbers are also numbers, and they visually cancel positive numbers when you use exploding dots. If you’re dividing 108 by 12, you can visually see 1 hundred and 8 ones change to 10 tens and 8 ones. You can get 4 twelves from that, leaving you with 6 tens. But that is 5 tens and 10 ones, which is equal to 5 more twelves. So that’s a total of 9 twelves. And this whole process used the basic idea that dividing by 12 is just the number of groups of 12 you can make. No need to do mechanical steps like a robot. (See pages 11 and 12 of [https://assets.ctfassets.net/p9j61e89vqti/5Y3wzZAy3uy0yKc0YAK0A8/3fe3bfcf4939f029b376e00b7e7b7fa2/Chapter_5_EXPLODING_DOTS_201709.pdf](https://assets.ctfassets.net/p9j61e89vqti/5Y3wzZAy3uy0yKc0YAK0A8/3fe3bfcf4939f029b376e00b7e7b7fa2/Chapter_5_EXPLODING_DOTS_201709.pdf) to see how this division is visually done.)
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