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.
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Okay so I’m not sure if someone else has answered this yet, but I wrote my final seminar for my BS in Math on Exploding Dots, so I can explain this pretty well.
[Exploding Dots](https://www.explodingdots.org/) is a visual way to teach base systems in math. We work in a base 10 system (think place value!), but there are other base systems. Binary is base 2, time is base 60, etc. Practicing working in other bases improves number sense (which is your innate understanding of numbers and how they interact). James Tanton is an Australian Mathematician and teacher, and he created this concept.
In a practical sense, it works much like an abacus or counters. You use a “dot” to represent the value “+1.” Using the dots and the 1<-10 (read “ten one”) machine, you can visually replicate how various operations work. If you draw 2 dots and add 3 more dots, you can visually see there are 5 dots (2+3=5). If you draw 10 dots in the right-most box in the machine they “explode,” and this creates a single dot in the box one to the left. This is how we visually represent 10 dots, by having every dot in the second box be equivalent to 10 dots in the first box, every dot in the third box be 100 dots, every dot in the fourth box 1000 dots, and so on. Through slightly more complicated procedures, more operations can be represented using dots and machines.
Tanton took this idea and created a curriculum around it, with videos, worksheets, and an online course/game. The theory is that if Math is taught this way when students are young, they will have a better conceptual understanding as adults.
Hope this helps!
Did no one here go to a Montessori school and learn math from the stamp game?
A different way to conceptualize number bases.
Think of it like “carrying the 1” in the old way of teaching math. Once you have more than 10 (or whatever number base you’re in) in a given space, you have to add it to the next space.
In an exploding dot box, say you have 14 in the box. When you move a dot to the next box, 10 dots explode from the first box, leaving 4 behind and they move up to the next base as 1 dot, essentially “carrying the 1”.
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.)
Never heard of it until now, but basically I used a similar concept to introduce my daughter to binary numbers by reiterating what we do in normal decimal space and then applying the same rules to base 2…but I was lacking a good visual for that so I was a bit unsatisfied with my explanation even though she grasped the concept… Thanks for bringing this to my attention…
And to anser the question, it’s just a visualization of what you learned for basic arithmetic. Instead of a take over you got the exploding dot… It can help to make implicitly patterns explicit and show you what you are actually doing… The 2->1 machine is basically just binary and by getting to 10->1 machine you get normal decimal and by introducing it that way you basically showed the same patterns for all number spaces apply
Does anyone remember chisan bop(?)
This is the first I have heard of this, but to me it is a visual process of converting base-10 numbers to base-n systems.
If you ask a person what 2 is in binary, most will look at you cross-eyed. If you teach them this exploding dots, they definitely will be able to answer and understand.
Thanks for sharing!
As a maths teacher (primary), I’ve found it quite a useful intervention for explaining base 10 and how regrouping works to children who dont seem to understand manipulative like Dienes. Furthermore, been great way getting class to understand how base 3 would work for instance. Really refreshing, geeky challenge writing numbers in different bases
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[https://www.explodingdots.org/](https://www.explodingdots.org/)
It’s an interesting way to visualize the structure of numbers in various bases. It lends itself well to addition and subtraction but gets somewhat obscure when it comes to multiplication and division.
It also offers some interesting insight into infinite series, polynomials, and rational bases (base 3/2 for example)
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