Division 1

Introduction 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 four quarters.

Do my best to draw the quarters.

If I have four 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.

I'll do my best.

Let's say you're bald.

But you have side burns.

Maybe you have a little bit of a beard.

So that's you, that's me,

and we're going to divide these four quarters between the two of us.

So notice, we have four quarters

and we're going to divide between the two of us.

There are two of us.

And I want to stress the number two.

So we're going to divide four quarters by two.

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

And you've probably done something like this.

What happens?

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

So let me divide it.

We're going to divide it into two.

Essentially what I did do is I take the four quarters

and I divide it into two equal groups.

Two equal groups.

And that's what division is.

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

So when you divide four quarters into two groups,

so this was four quarters right there.

And you want to divide it into two groups.

This is group one.

Group one right here.

And this is group two right here.

How many numbers are in each group?

Or how many quarters are in each group?

Well, in each group I have one, two quarters.

I'll need to use a brighter color.

I have one, two quarters in each group.

One quarter and two 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 four divided by-- so this four.

Let me use the right colors.

So this four, which is this four, divided by the two groups,

these are the two groups: group one and this is group two right here.

So divided into two groups or into two collections.

Four divided by two is equal to--

when you divide four into two groups,

each group is going to have two quarters in it.

It's going to be equal to two.

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 two groups of two quarters,

I would multiply the two groups times the two quarters each

and I would say I would then have four 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 six divided by--

I'm trying to keep it nice and color coded.

Six divided by three, what is that equal to?

Let's just draw six objects.

They can be anything.

Let's say I have six 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 one, two, three, four, five, six.

And I'm going to divide it by three.

And one way that we can think about that

is that means I want to divide my six bell peppers

into three equal groups of bell peppers.

You could kind of think of it as if three people are going to share these bell peppers,

how many do each of them get?

So let's divide it into three groups.

So that's our six bell peppers.

I'm going to divide it into three groups.

So the best way to divide it into three groups is

I can have one group right there, two 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 one, two.

One, two.

One, two bell peppers.

So six divided by three is equal to two.

So the best way or one way to think about it

is that you divided the six into three 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 six divided by three.

And once again, let's say I have raspberries now-- easier to draw.

One, two, three, four, five, six.

And here, instead of dividing it into three groups like we did here.

This was one group, two group, three groups.

Instead of dividing into three groups,

what I want to do is say well,

if I'm dividing six divided by three, I want to divide it into groups of three.

Not into three groups.

I want to divide it into groups of three.

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

Well, let me draw some groups of three.

So that is one group of three.

And that is two groups of three.

So if I take six things and I divide them into groups of three,

I will end up with one, two 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 six divided by three and six divided by two.

Let me do that right here.

What is six divided by two,

when you think of it in this context right here?

Six divided by two, when you do it like that--

let me draw one, two, three, four, five, six.

When we think about six divided by two in terms of dividing it into two groups,

what we can end up is we could have one group like this

and then one group like this,

and each group will have three elements.

It'll have three things in it.

So six divided by two is three.

Or you could think of it the other way.

You could say that six divided by two is--

you're taking six objects: one, two, three, four, five, six.

And your dividing it into groups of two

where each group has two elements.

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

If each group has two elements, well, that's the one 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 two.

But how many groups do I have?

I have one, two, three.

I have three groups.

But notice something, it's no coincidence that six divided by three is two,

and six divided by two is three.

Let me write that down.

We get six divided by three is equal to two,

and six divided by two is equal to three.

And the reason why you see this relation where you can kind of swap this two and this three

is because two times three is equal to six.

Let's say I have two groups of three.

Let me draw two groups of three.

So that's one group of three and then here's another group of three.

So two groups of three is equal to six.

Two times three is equal to six.

Or you could think of it the other way,

if I have three groups of two.

So that's one group of two right there.

I have another group of two right there.

And then I have a third group of two right there.

What is that equal to?

Three groups of two-- three times two.

That's also equal to six.

So two times three is equal to six.

Three times two is equal to six.

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 six things and you want to divide it into groups of two, you get three.

If you have six and you want to divide into groups of three, you get two.

Let's do a couple more problems.

I think it'll really make sense about what division is all about.

Let's do an interesting one.

Let's do nine divided by four.

So if we think about nine divided by four, let me draw nine objects.

One, two, three, four, five, six, seven, eight, nine.

Now when you divide by four, for this problem,

I'm thinking about dividing it into groups of four.

So if I want to divide it into groups of four--

Let me try doing that.

So here is one group of four.

I just picked any of them right like that.

That's one group of four.

Then here's another group of four, 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 four.

When I'm dividing by four,

I can only cut up the nine into groups of four.

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

nine divided by four is going to be two groups.

I have one group here, and another group here,

and then I have a remainder of one.

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

Remainder-- that says remainder one.

Nine divided by four is two remainder one.

If I asked you what twelve divided by four is-- so let me do twelve.

One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.

So let me write that down.

Twelve divided by four.

So I want to divide these twelve objects--

maybe they're apples or plums.

And divide them into groups of four.

So let me see if I can do that.

So this is one group of four just like that.

This is another group of four just like that.

And this is pretty straightforward.

And then I have a third group of four.

Just like that.

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

I can exactly divide twelve objects into three groups of four.

One, two, three groups of four.

So twelve divided by four is equal to three.

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

What is twelve divided by three?

Let me do a new color.

Twelve divided by three.

Now based on what we've learn so far,

we say, that should just be four, because three times four is twelve.

But let's prove it to ourselves.

So one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.

Let's divide it into groups of three.

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 three, right there.

Twelve divided by three.

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

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

And I'll take this group of three.

There was obviously a much easier way of dividing it up

than doing these weird l-shaped things,

but I want to show you it doesn't matter.

You're just dividing it into groups of three.

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 four 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 three,

I could have just done one, two, three, four groups of three.

Either of these, I'm dividing the twelve objects into packets of three.

You can imagine them that way.

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

Let's see.

What is fourteen divided by five?

So let's draw fourteen objects.

One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen.

Fourteen objects.

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

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

two groups right there.

But then this last one, I only have four left,

so I can't make another group of five.

So the answer here is I can make two groups of five,

and I'm going to have a remainder-- r for remainder-- of four.

Two remainder four.

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, fourteen divided by five, how do I figure that out?

Actually, another way of writing this,

and no harm in showing you :

I could say fourteen divided by five is the same thing as fourteen divided by--

this sign right here-- divided by five.

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

How many times does five go into fourteen?

Well, let's see.

Five times-- and you kind of do multiplication tables in your head--

Five times one is equal to five.

Five times two is equal to ten.

So that's still less than fourteen, so five goes at least two times.

Five times three is equal to fifteen.

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

So five only goes two times.

So it goes two times.

Two times five is ten.

And then you subtract.

You say fourteen minus ten is four.

And that's the same remainder as right here.

Well, I could divide five into fourteen exactly two times,

which would get us two groups of five.

Which is essentially just ten.

And we still have the four 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 eight divided by two.

And I could also write this as eight--

so I want to know what that is.

That's a question mark.

I could also write this as eight divided by two.

And the way I do either of these-- I'll draw the circles in a second--

but the way I do it without drawing the circles,

I say, well, two times one is equal to two.

So that definitely goes into eight,

but maybe I can think of a larger number that goes into--

that when I multiply it by two still goes into eight.

Two times two is equal to four.

That's still less than eight.

So two times three is equal to six.

Still less than eight.

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

Two times four is exactly equal to eight.

So two goes into eight four times.

So I could say two goes into eight four times.

Or eight divided by two is equal to four.

We can even draw our circles.

One, two, three, four, five, six, seven, eight.

I drew them messy on purpose.

Let's divide them into groups of two.

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

three groups of two, four groups of two.

So if I have eight objects, divide them into groups of two,

you have four groups.

So eight divided by two is four.

Hopefully you found that helpful!