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Lesson 5.2 of Exploding Dots

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OK, let's explore division in a 1←10 machine. So let me start off with this problem. 3,906 ÷ 3. In fact, you can probably see what the answer is going to be right off the bat. It's going to be, what, 1, 3, 0, 2. 1, 302. But I wanted to illustrate how we could actually see that answer— I mean, literally see that answer— in a 1←10 machine.

So what I'm going to do is I'm going to draw a picture of 3,906 in a 1←10 machine. OK, fair enough. I can do that. So it would be three thousand, nine hundreds, no tens, and six ones. There's a picture of 3906. And I'm looking for, in that picture, three groups of three. And what does 3 look like? Well obviously, it looks like three dots.

All right, so the question is, can I see any groups of three in this picture of 3,906? And the answer is, yes, I can see lots of groups of three. For example, I see one group of three right here, at this one thousand level. Now that really is, actually, 1,000 groups of three. Because if I did all the unexplosion to be 1,000 groups of three there, they would all just like explode over here. So there's one group of three at the thousands level.

There's also, actually, three groups of three at the one hundred level. I just did tally them up. Maybe I should write 3. Maybe I should write 3, because look what's happening. 1, 3. Do I see any groups of three at that level? No. Do I see any groups of three at this level? Yep, I see two at that level, which I should guess right as 2.

And look at that. I see 1, 3, 0, 2. The answer to the division problem. I literally looked for groups of three at different levels in my 1←10 machine. I found one at the thousands level, three at the hundreds level, none at the tens level, two at the ones level. That is indeed the answer—1, 3, 0, 2. One at the thousands level, three at the hundreds level, zero at the tens, two at the ones.

Grand. So that's how division works in the 1←10 machine. Just draw a picture of the number you've got, draw a picture of the groups you're looking for, and go hunt for them—end of story. Kind of fun. Now, this was a nice one. That was a single digit division. Let's do a sort of more complicated one.

Let's do, say, 276 ÷ 12. This makes it a little more complicated. Now, 12 can be interesting for us. Let me show you. 276 is fine, we can definitely draw a picture of that. It is, what, two dots, seven dots, and six dots. And what does 12 look like? All right, 12 is going to be—think about it. 12 dots in a box would explode and make one dot one place over and two dots left behind.

So now this is more interesting. 3 was obviously all in one box, and I could find groups of three. But 12 is spilled over. So what I'm going to have to keep note of is that actually, all 12 dots are really sitting in there, the rightmost part of this picture. And then just the explosion must have occurred and then spilled one over into the next box over. So I'm going to have to keep that in mind. I'm really doing this, but 12 is a bit more complicated— a little explosion occurred.

So now the question is, 276 ÷ 12. Here's the picture of 276. Here's the picture of 12. Can I find any picture of 12 in this picture up here? Why, yes. I'm looking for one dot next to two dots. There's one dot next to two dots.

Now, be very careful. Where are the actual 12 dots in this loop? Well, they must all be here in this rightmost part. The explosion must have occurred. So you've actually got one group of 12 at that level there, the tens level. Just like I had one group of three at the hundreds level here, one group at the thousand level. That's where all the 12 dots actually are. The explosion must have happened on the rightmost part of the loop. That's where those 12 dots actually are.

Any more groups of 12? Why, yes. There's another one at that level. I'll just do a tally max right now. Uh, any more groups of 12 in this picture? Why, yes. Go one level over, see one dot next to two dots there. One dot next to two dots there. One dot next to two dots there. I see two at the tens level, three at the ones level.

I see the answer 23, which is indeed correct. Two groups of 12 at ten, three groups of 12 at one— 23 groups of 12 in total. Wow. There it is. I visually see it. In fact, I'm going to ask you just to keep this particular picture in mind. It's going to come back to us later on. Look at it for now—that's brilliant.

So let's do another example to show you how long division works with just pictures. I can really see what's going on here. All right, so I'll clean the board. I'll be back in just a moment.

All right, board is clean and I am back. So let's do another long division example. Let's make it a really big one. Let's do 38124 ÷ 102 this time. Good and juicy.

All right, here goes. I'm looking for groups of 102 and a picture of 38124. Well, let's draw a picture of 31824. In a 1←10 machine, it will be three ten thousands, one one thousand, eight one hundreds, two tens, and four ones. Beautiful. 102—what does that look like?

Now it's going to be a bit of a big picture, except I'm only actually physically drawing three dots. But really, it's a picture of 102 dots. They must all be there, and lots of explosions happen to spill them all the way over to two places to the left. All right, so there's a picture of 102. There's a picture of 31824.

I'm asking, how many groups of 102 can I find in this picture? I'm looking for one dot, zero dots, two dots—102 in this picture of 38124. Can I find some? All right, so I'm looking for one, blank, two. Now that blank is a little bit annoying. It's a little big of a hiccup for us. We've got to keep in mind we've got a blank to deal with.

Let me just get my blue pen. Can I see one dot, no dots, two dots anywhere in this picture? Yes, I can see one right here. What if I draw a great big long loop like that? One dot, no dots, two dots. All right, within that blue loop, where are the 102 dots, really?

Well, when all the explosions— they must be all right here in the midst— exploded this way, they're all at that level. There's one group of 102 at that level. Plus some explosions occurred to spill them over to the left.

All right, any other one blank two? All right, as you do this, you might find that drawing loops gets pretty messy. So I might just skip drawing loops. Can I do this? Can I do one dot, no dot, two dots? Is that OK? There's another one at that level then. But I can do it again—one dot, no dot, two dots. So I've got three at that level right now.

All right, how are we doing? How are we doing so far? All right, any more one, blank, two—one dot, no dot, two dots? Well, yes. One dot, no dot, two dots. One dot, no dot, two dots. Can I do this little split loop now, maybe, at that level? Remember, all 102 dots are really here, and there must have been some explosions spilling over yonder.

All right. So far so good. It's getting a bit tricky. Do you see any more? I do. One dot, no dot, two dots—those two there. And there's one at that level—one dot, no dot, two dots. Another one at that level. That's all the dots accounted for.

And look, we found a lot of groups—102 in this picture. In fact, we found three at the hundreds level, one at the tens level, and two at the ones level. The answer must be 312. We just did long division, purely with pictures. How gorgeous is that? Loads of fun.

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Duration: 7 minutes and 38 seconds
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Language: English
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Posted by: jamestanton on Jun 9, 2018

This is the 11th lesson in the Exploding Dots story for Global Math Week 2017 (www.theglobalmathproject.org).

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