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I think you've probably heard the word divide before, where

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someone tells you to divide something up.

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Divide the money between you and your brother or between

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you and your buddy.

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And it essentially means to cut up something.

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So let me write down the word divide.

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Let's say that I have 4 quarters.

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Do my best to draw the quarters.

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If I have 4 quarters just like that.

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That's my rendition of George Washington on the quarters.

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And let's say there's two of us and we're going to divide

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the quarters between us.

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So this is me right here.

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Let me try my best to draw me.

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So that's me right there.

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Let's see, I have a lot of hair.

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And then this is you right there.

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Do my best.

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So you're bald.

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You have side burns.

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Maybe you have a little bit of a beard.

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I want to get too focused on the drawing.

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So that's you, that's me, and we're going to divide these 4

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quarters between the 2 of us.

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So notice, we have 4 quarters and we're going to divide

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between the 2 of us.

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There are 2 of us.

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And I want to stress the number 2.

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So we're going to divide 4 quarters by 2.

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We're going to divide it between the 2 of us.

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And you've probably done something like this.

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What happens?

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Well, each of us are going to get 2 quarters.

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So let me divide it.

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We're going to divide it into 2.

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Essentially what I did do is I take the 4 quarters and I

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divide it into 2 equal groups.

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And that's what division is.

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We cut up this group of quarters into 2 equal groups.

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So when you divide 4 quarters into 2 groups, so this was

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4 quarters right there.

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And you want to divide it into 2 groups.

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This is group 1.

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Group 1 right here.

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And this is group 2 right here.

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How many numbers are in each group?

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Or how many quarters are in each group?

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Well, in each group I have 1, 2 quarters.

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Let me use a brighter color.

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I have 1, 2 quarters in each group.

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1 quarter and 2 quarters in each group.

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So to write this out mathematically, I think this is

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something that you've done, probably as long as you've been

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splitting money between you and your siblings and your buddies.

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Actually, let me scroll over a little bit, so you can

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see my entire picture.

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How do we write this mathematically?

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We can write that 4 divided by-- so this 4.

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Let me use the right colors.

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So this 4, which is this 4, divided by the 2 groups, these

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are the 2 groups: group 1 and this is group 2 right here.

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So divided into 2 groups or into 2 collections.

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4 divided by 2 is equal to-- when you divide 4 into 2

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groups, each group is going to have 2 quarters in it.

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It's going to be equal to 2.

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And I just wanted to use this example because I want to show

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you that division is something that you've been

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using all along.

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And another important, I guess, takeaway or thing to realize

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about this, is on some level this is the opposite

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of multiplication.

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If I said that I had 2 groups of 2 quarters I would multiply

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the 2 groups times the 2 quarters each and I would say

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I would then have 4 quarters.

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So on some level these are saying the same thing.

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But just to make it a little bit more concrete in our

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head, let's do a couple of more examples.

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Let's do a bunch of more examples.

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So let's write down, what is 6 divided by-- I'm trying to

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keep it nice and color coded.

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6 divided by 3, what is that equal to?

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Let's just draw 6 objects.

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They can be anything.

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Let's say I have 6 bell peppers.

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I won't take too much trouble to draw them.

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Well, that's not what a bell pepper looks like,

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but you get the idea.

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So 1, 2, 3, 4, 5, 6.

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And I'm going to divide it by 3.

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And one way that we can think about that is that means I want

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to divide my 6 bell peppers into 3 equal groups

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of bell peppers.

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You could kind of think of it if 3 people are going to share

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these bell peppers, how many do each of them get?

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So let's divide it into 3 groups.

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So that's our 6 bell peppers.

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I'm going to divide it into 3 groups.

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So the best way to divide it into 3 groups is I can have 1

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group right there, 2 groups, or the second group right there,

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and then, the third group.

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And then each group will have exactly how many bell peppers?

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They'll have 1, 2.

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1, 2.

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1, 2 bell peppers.

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So 6 divided by 3 is equal to 2.

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So the best way or one way to think about it is that you

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divided the 6 into 3 groups.

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Now you could view that a slightly different way,

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although it's not completely different, but it's a good

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way to think about it.

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You could also think of it as 6 divided by 3.

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And once again, let's say I have raspberries

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now-- easier to draw.

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1, 2, 3, 4, 5, 6.

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And here, instead of dividing it into 3

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groups like we did here.

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This was 1 group, 2 group, 3 groups.

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Instead of dividing into 3 groups, what I want to do is

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say well, if I'm dividing 6 divided by 3, I want to

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divide it into groups of 3.

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Not into 3 groups.

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I want to divide it into groups of 3.

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So how many groups of 3 am I going to have?

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Well, let me draw some groups of 3.

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So that is one group of 3.

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And that is two groups of 3.

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So if I take 6 things and I divide them into groups of 3, I

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will end up with 1, 2 groups.

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So that's another way to think about division.

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And this is an interesting thing.

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When you think about these two relations, you'll see a

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relationship between 6 divided by 3 and 6 divided by 3.

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Let me do that right here.

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What is 6 divided by 2 when you think of it in this

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context right here?

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6 divided by 2, when you do it like that-- let me

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draw 1, 2, 3, 4, 5, 6.

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When we think about 6 divided by 2 in terms of dividing it

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into 2 groups, what we can end up is we could have 1 group

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like this and then 1 group like this, and each group

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will have 3 elements.

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It'll have 3 things in it.

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So 6 divided by 2 is 3.

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Or you could think of it the other way.

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You could say that 6 divided by 2 is-- you're taking 6

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objects: 1, 2, 3, 4, 5, 6.

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And your dividing it into groups of 2 where each

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group has 2 elements.

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And that on some level is an easier thing to do.

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If each group has 2 elements, well, that's the 1 right there.

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They don't even have to be nicely ordered.

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This could be one group right there and that could be the

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other group right there.

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I don't have to draw them all stacked up.

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These are just groups of 2.

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But how many groups do I have?

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I have 1, 2, 3.

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I have 3 groups.

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But notice something, it's no coincidence that 6 divided by 3

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is 2 and 6 divided by 2 is 3.

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Let me write that down.

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We get 6 divided by 3 is equal to 2 and 6 divided

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by 2 is equal to 3.

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And the reason why you see this relation where you can kind of

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swap this 2 and this 3 is because 2 times 3

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is equal to 6.

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Let's say I have 2 groups of 3.

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Let me draw 2 groups of 3.

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So that's 1 group of 3 and then here's another group of 3.

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So 2 groups of 3 is equal to 6.

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2 times 3 is equal to 6.

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Or you could think of it the other way if I

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have 3 groups of 2.

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So that's 1 group of 2 right there.

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I have another group of 2 right there.

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And then I have a 3 group of 2 right there.

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What is that equal to?

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3 groups of 2-- 3 times 2.

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That's also equal to 6.

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So 2 times 3 is equal to 6.

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3 times 2 is equal to 6.

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We saw this in the multiplication video that

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the order doesn't matter.

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But that's the reason why if you want to divide it, if you

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want to go the other way-- if you have 6 things and you

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want to divide it into groups of 2, you get 3.

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If you have 6 and you want to divide into groups

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of 3, you get 2.

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Let's do a couple of more problems.

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I think it'll really make sense about what

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division is all about.

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Let's do an interesting one.

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Let's do 9 divided by 4.

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So if we think about 9 divided by 4, let me draw 9 objects.

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1, 2, 3, 4, 5, 6, 7, 8, 9.

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Now when you divide by 4, for this problem, I'm thinking

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about dividing it into groups of 4.

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So if I want to divide it into groups of 4, let

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me try doing that.

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So here is one group of 4.

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I just picked any of them right like that.

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That's one group of 4.

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Then here's another group of 4 right there.

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And then I have this left over thing.

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Maybe we could call it a remainder, where I can't put

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this one into a group of 4.

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When I'm dividing by 4 I can only cut up the

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9 into groups of 4.

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So the answer here, and this is a new concept for you maybe, 9

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divided by 4 is going to be 2 groups.

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I have one group here and another group here, and then

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I have a remainder of 1.

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I have 1 left over that I wasn't able to do with.

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Remainder-- that says remainder 1.

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9 divided by 4 is 2 remainder 1.

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If I asked you what 12 divided by 4 is, so let me do 12.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

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So let me write that down.

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12 divided by 4.

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So I want to divide these 12 objects-- maybe

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they're apples or plums.

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And divide them into groups of 4.

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So let me see if I can do that.

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So this is one group of 4 just like that.

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This is another group of 4 just like that.

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And this is pretty straightforward.

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And then I have a third group of 4 just like that.

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And there's nothing left over like I had before.

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I can exactly divide 12 objects into 3 groups of 4.

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1, 2, 3 groups of 4.

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So 12 divided by 4 is equal to 3.

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And we can do the exercise that we saw on the previous video.

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What is 12 divided by 3?

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Let me do a new color.

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12 divided by 3.

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Now based on what we've learn so far we say, that should just

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be 4 because 3 times 4 is 12.

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But let's prove it to ourselves.

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So 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 11, 12.

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Let's divide it into groups of 3.

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And I'm going to make them a little strange looking just

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so you see that you don't always have to do it into

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nice, clean columns.

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So that's a group of 3 right there.

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12 divided by 3.

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Let's see, here is another group of 3 just like that.

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And then, maybe I'll take this group of 3 like that.

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And I'll take this group of 3.

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There was obviously a much easier way of dividing it up

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then doing these weird l-shaped things, but I want to show

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you it doesn't matter.

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You're just dividing it into groups of 3.

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And how many groups do we have?

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We have one group.

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Then we have our second group right here.

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And then we have our third group right there.

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And then we have-- let me do it in a new color.

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And then we have our fourth group right there.

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So we have exactly 4 groups.

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And when I say there was an easier way to divide it, the

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easier way was obviously, maybe not obviously-- if I want to

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divide these into groups of 3 I could have just done 1,

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2, 3, 4 groups of 3.

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Either of these I'm dividing the 12 objects

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into packets of 3.

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You can imagine them that way.

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Let's do another one that maybe has a remainder.

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Let's see.

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What is 14 divided by 5?

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So let's draw 14 objects.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14.

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14 objects.

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And I'm going to divide it into groups of 5.

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Well, the easiest thing is there's one group right there,

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two groups right there.

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But then this last one, I only have 4 left, so I can't

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make another group of 5.

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So the answer here is I can make 2 groups of 5 and I'm

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going to have a remainder-- r for remainder-- of 4.

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2 remainder 4.

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Now, once you get enough practice, you're not always

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going to be wanting to draw these circles and dividing

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them up like that.

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Although that would not be incorrect.

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So another way to think about this type of problem is to

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say, well, 14 divided by 5, how do I figure that out?

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Actually, another way of writing this and no

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harm in showing you.

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I could say 14 divided by 5 is the same thing as 15 divided

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by-- this sign right-- divided by 5.

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And what you do is you say, well, let's see.

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How many times does 5 go into 14?

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Well, let's see.

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5 times and you kind of do multiplication

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tables in your head.

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5 times 1 is equal to 5.

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5 times 2 is equal to 10.

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So that's still less than 14, so 5 goes at least two times.

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5 times 3 is equal to 15.

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Well that's bigger than 14, so I have to go back here.

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So 5 only goes two times.

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So it goes 2 times.

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2 times 5 is 10.

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And then you subtract.

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You say 14 minus 10 is 4.

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And that's the same remainder as right here.

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Well, I could divide 5 into 14 exactly two times, which

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would get us 2 groups of 5.

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Which is essentially just 10.

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And we still have the 4 left over.

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Let me do a couple of more just to really make sure you get

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this stuff really, really, really, really well.

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Let me write it in that notation.

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Let's say I do 8 divided by 2.

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And I could also write this as 8-- so I want

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to know what that is.

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That's a question mark.

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I could also write this as 8 divided by 2.

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And the way I do either of these, I'll draw the circles in

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the second, but the way I do it without drawing the circles, I

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say, well, 2 times 1 is equal to 2.

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So that definitely goes into 8, but maybe I can think of a

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larger number that goes into-- that when I multiply it

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by 2 still goes into 8.

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2 times 2 is equal to 4.

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That's still less than 8.

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So 2 times 3 is equal to 6.

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Still less than 8.

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2 times-- oh, something weird happened to my pen.

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2 times 4 is exactly equal to 8.

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So 2 goes into 8 four times.

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So I could say 2 goes into 8 four times.

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Or 8 divided by 2 is equal to 4.

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We can even draw our circles.

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1, 2, 3, 4, 5, 6, 7, 8.

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I drew them messy on purpose.

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Let's divide them into groups of 2.

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I have one group of 2, two groups of 2, three groups

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of 2, four groups of 2.

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So if I have 8 objects, divide them into groups of 2,

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you have four groups.

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So 8 divided by 2 is 4.

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Hopefully you found that helpful.