Intro to division

I think you've probably heard the word divide before, where someone tells you to divide something up. Divide the money between you and your brother or between you and your buddy. And it essentially means to cut up something. So let me write down the word divide. Let's say that I have 4 quarters. Do my best to draw the quarters. If I have 4 quarters just like that. That's my rendition of George Washington on the quarters. And let's say there's two of us and we're going to divide the quarters between us. So this is me right here. Let me try my best to draw me. So that's me right there. Let's see, I have a lot of hair. And then this is you right there. Do my best. So you're bald. You have side burns. Maybe you have a little bit of a beard. I want to get too focused on the drawing. So that's you, that's me, and we're going to divide these 4 quarters between the 2 of us. So notice, we have 4 quarters and we're going to divide between the 2 of us. There are 2 of us. And I want to stress the number 2. So we're going to divide 4 quarters by 2. We're going to divide it between the 2 of us. And you've probably done something like this. What happens? Well, each of us are going to get 2 quarters. So let me divide it. We're going to divide it into 2. Essentially what I did do is I take the 4 quarters and I divide it into 2 equal groups. And that's what division is. We cut up this group of quarters into 2 equal groups. So when you divide 4 quarters into 2 groups, so this was 4 quarters right there. And you want to divide it into 2 groups. This is group 1. Group 1 right here. And this is group 2 right here. How many numbers are in each group? Or how many quarters are in each group? Well, in each group I have 1, 2 quarters. Let me use a brighter color. I have 1, 2 quarters in each group. 1 quarter and 2 quarters in each group. So to write this out mathematically, I think this is something that you've done, probably as long as you've been splitting money between you and your siblings and your buddies. Actually, let me scroll over a little bit, so you can see my entire picture. How do we write this mathematically? We can write that 4 divided by-- so this 4. Let me use the right colors. So this 4, which is this 4, divided by the 2 groups, these are the 2 groups: group 1 and this is group 2 right here. So divided into 2 groups or into 2 collections. 4 divided by 2 is equal to-- when you divide 4 into 2 groups, each group is going to have 2 quarters in it. It's going to be equal to 2. And I just wanted to use this example because I want to show you that division is something that you've been using all along. And another important, I guess, takeaway or thing to realize about this, is on some level this is the opposite of multiplication. If I said that I had 2 groups of 2 quarters I would multiply the 2 groups times the 2 quarters each and I would say I would then have 4 quarters. So on some level these are saying the same thing. But just to make it a little bit more concrete in our head, let's do a couple of more examples. Let's do a bunch of more examples. So let's write down, what is 6 divided by-- I'm trying to keep it nice and color coded. 6 divided by 3, what is that equal to? Let's just draw 6 objects. They can be anything. Let's say I have 6 bell peppers. I won't take too much trouble to draw them. Well, that's not what a bell pepper looks like, but you get the idea. So 1, 2, 3, 4, 5, 6. And I'm going to divide it by 3. And one way that we can think about that is that means I want to divide my 6 bell peppers into 3 equal groups of bell peppers. You could kind of think of it if 3 people are going to share these bell peppers, how many do each of them get? So let's divide it into 3 groups. So that's our 6 bell peppers. I'm going to divide it into 3 groups. So the best way to divide it into 3 groups is I can have 1 group right there, 2 groups, or the second group right there, and then, the third group. And then each group will have exactly how many bell peppers? They'll have 1, 2. 1, 2. 1, 2 bell peppers. So 6 divided by 3 is equal to 2. So the best way or one way to think about it is that you divided the 6 into 3 groups. Now you could view that a slightly different way, although it's not completely different, but it's a good way to think about it. You could also think of it as 6 divided by 3. And once again, let's say I have raspberries now-- easier to draw. 1, 2, 3, 4, 5, 6. And here, instead of dividing it into 3 groups like we did here. This was 1 group, 2 group, 3 groups. Instead of dividing into 3 groups, what I want to do is say well, if I'm dividing 6 divided by 3, I want to divide it into groups of 3. Not into 3 groups. I want to divide it into groups of 3. So how many groups of 3 am I going to have? Well, let me draw some groups of 3. So that is one group of 3. And that is two groups of 3. So if I take 6 things and I divide them into groups of 3, I will end up with 1, 2 groups. So that's another way to think about division. And this is an interesting thing. When you think about these two relations, you'll see a relationship between 6 divided by 3 and 6 divided by 3. Let me do that right here. What is 6 divided by 2 when you think of it in this context right here? 6 divided by 2, when you do it like that-- let me draw 1, 2, 3, 4, 5, 6. When we think about 6 divided by 2 in terms of dividing it into 2 groups, what we can end up is we could have 1 group like this and then 1 group like this, and each group will have 3 elements. It'll have 3 things in it. So 6 divided by 2 is 3. Or you could think of it the other way. You could say that 6 divided by 2 is-- you're taking 6 objects: 1, 2, 3, 4, 5, 6. And your dividing it into groups of 2 where each group has 2 elements. And that on some level is an easier thing to do. If each group has 2 elements, well, that's the 1 right there. They don't even have to be nicely ordered. This could be one group right there and that could be the other group right there. I don't have to draw them all stacked up. These are just groups of 2. But how many groups do I have? I have 1, 2, 3. I have 3 groups. But notice something, it's no coincidence that 6 divided by 3 is 2 and 6 divided by 2 is 3. Let me write that down. We get 6 divided by 3 is equal to 2 and 6 divided by 2 is equal to 3. And the reason why you see this relation where you can kind of swap this 2 and this 3 is because 2 times 3 is equal to 6. Let's say I have 2 groups of 3. Let me draw 2 groups of 3. So that's 1 group of 3 and then here's another group of 3. So 2 groups of 3 is equal to 6. 2 times 3 is equal to 6. Or you could think of it the other way if I have 3 groups of 2. So that's 1 group of 2 right there. I have another group of 2 right there. And then I have a 3 group of 2 right there. What is that equal to? 3 groups of 2-- 3 times 2. That's also equal to 6. So 2 times 3 is equal to 6. 3 times 2 is equal to 6. We saw this in the multiplication video that the order doesn't matter. But that's the reason why if you want to divide it, if you want to go the other way-- if you have 6 things and you want to divide it into groups of 2, you get 3. If you have 6 and you want to divide into groups of 3, you get 2. Let's do a couple of more problems. I think it'll really make sense about what division is all about. Let's do an interesting one. Let's do 9 divided by 4. So if we think about 9 divided by 4, let me draw 9 objects. 1, 2, 3, 4, 5, 6, 7, 8, 9. Now when you divide by 4, for this problem, I'm thinking about dividing it into groups of 4. So if I want to divide it into groups of 4, let me try doing that. So here is one group of 4. I just picked any of them right like that. That's one group of 4. Then here's another group of 4 right there. And then I have this left over thing. Maybe we could call it a remainder, where I can't put this one into a group of 4. When I'm dividing by 4 I can only cut up the 9 into groups of 4. So the answer here, and this is a new concept for you maybe, 9 divided by 4 is going to be 2 groups. I have one group here and another group here, and then I have a remainder of 1. I have 1 left over that I wasn't able to do with. Remainder-- that says remainder 1. 9 divided by 4 is 2 remainder 1. If I asked you what 12 divided by 4 is, so let me do 12. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So let me write that down. 12 divided by 4. So I want to divide these 12 objects-- maybe they're apples or plums. And divide them into groups of 4. So let me see if I can do that. So this is one group of 4 just like that. This is another group of 4 just like that. And this is pretty straightforward. And then I have a third group of 4 just like that. And there's nothing left over like I had before. I can exactly divide 12 objects into 3 groups of 4. 1, 2, 3 groups of 4. So 12 divided by 4 is equal to 3. And we can do the exercise that we saw on the previous video. What is 12 divided by 3? Let me do a new color. 12 divided by 3. Now based on what we've learn so far we say, that should just be 4 because 3 times 4 is 12. But let's prove it to ourselves. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 11, 12. Let's divide it into groups of 3. And I'm going to make them a little strange looking just so you see that you don't always have to do it into nice, clean columns. So that's a group of 3 right there. 12 divided by 3. Let's see, here is another group of 3 just like that. And then, maybe I'll take this group of 3 like that. And I'll take this group of 3. There was obviously a much easier way of dividing it up then doing these weird l-shaped things, but I want to show you it doesn't matter. You're just dividing it into groups of 3. And how many groups do we have? We have one group. Then we have our second group right here. And then we have our third group right there. And then we have-- let me do it in a new color. And then we have our fourth group right there. So we have exactly 4 groups. And when I say there was an easier way to divide it, the easier way was obviously, maybe not obviously-- if I want to divide these into groups of 3 I could have just done 1, 2, 3, 4 groups of 3. Either of these I'm dividing the 12 objects into packets of 3. You can imagine them that way. Let's do another one that maybe has a remainder. Let's see. What is 14 divided by 5? So let's draw 14 objects. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14. 14 objects. And I'm going to divide it into groups of 5. Well, the easiest thing is there's one group right there, two groups right there. But then this last one, I only have 4 left, so I can't make another group of 5. So the answer here is I can make 2 groups of 5 and I'm going to have a remainder-- r for remainder-- of 4. 2 remainder 4. Now, once you get enough practice, you're not always going to be wanting to draw these circles and dividing them up like that. Although that would not be incorrect. So another way to think about this type of problem is to say, well, 14 divided by 5, how do I figure that out? Actually, another way of writing this and no harm in showing you. I could say 14 divided by 5 is the same thing as 15 divided by-- this sign right-- divided by 5. And what you do is you say, well, let's see. How many times does 5 go into 14? Well, let's see. 5 times and you kind of do multiplication tables in your head. 5 times 1 is equal to 5. 5 times 2 is equal to 10. So that's still less than 14, so 5 goes at least two times. 5 times 3 is equal to 15. Well that's bigger than 14, so I have to go back here. So 5 only goes two times. So it goes 2 times. 2 times 5 is 10. And then you subtract. You say 14 minus 10 is 4. And that's the same remainder as right here. Well, I could divide 5 into 14 exactly two times, which would get us 2 groups of 5. Which is essentially just 10. And we still have the 4 left over. Let me do a couple of more just to really make sure you get this stuff really, really, really, really well. Let me write it in that notation. Let's say I do 8 divided by 2. And I could also write this as 8-- so I want to know what that is. That's a question mark. I could also write this as 8 divided by 2. And the way I do either of these, I'll draw the circles in the second, but the way I do it without drawing the circles, I say, well, 2 times 1 is equal to 2. So that definitely goes into 8, but maybe I can think of a larger number that goes into-- that when I multiply it by 2 still goes into 8. 2 times 2 is equal to 4. That's still less than 8. So 2 times 3 is equal to 6. Still less than 8. 2 times-- oh, something weird happened to my pen. 2 times 4 is exactly equal to 8. So 2 goes into 8 four times. So I could say 2 goes into 8 four times. Or 8 divided by 2 is equal to 4. We can even draw our circles. 1, 2, 3, 4, 5, 6, 7, 8. I drew them messy on purpose. Let's divide them into groups of 2. I have one group of 2, two groups of 2, three groups of 2, four groups of 2. So if I have 8 objects, divide them into groups of 2, you have four groups. So 8 divided by 2 is 4. Hopefully you found that helpful.