Creating volume expressions

- [Instructor] What we wanna do

in this video is figure out the volume

of this rectangular prism here.

So pause this video and see
if you can do that on your own

before we do this together.

All right, now let's work
through this together

and we're actually gonna think about it

in three different ways.

And to help us do that in
those three different ways,

we'll visualize this volume
in these three different ways.

So the first way to visualize
it is let's just think

about it as if we think about
it as cubic centimeters,

five going up, four going across,

or four wide, and then three deep.

And so that's what all of these show.

And in this first diagram,

what we first can think
about is how many cubes,

how many orange cubes
are in this first layer.

Well, this first layer
right over here is going

to be four cubes in one dimension.

Actually, let me scroll down a little bit.

It's going to be four in this
dimension right over here,

and there's going to be
three in this dimension.

So if you think about
how many cubes are there,

that is four times three

on just one base or one layer.

And then how many layers are there?

Well, we can see

that there are one, two, three,
four, five of those layers.

So then you can multiply that times five.

Or another way to think about it is,

in that base layer there,
if you did four times three,

you're going to have 12 of those cubes,

and then you're gonna have
to multiply 12 times 5,

which you might just know as being 60.

Let me do that same color.

As 60.

Or you could do a repeated
addition if you like.

You could say, "Well, that's going

to be equal to 12 plus 12

plus 12 plus 12

plus 12

plus 12,

which also is equal to 60."

And that's 60 of what?

Well, each of these things
is a cubic centimeter.

So that is 60,

60 cubic centimeters.

Now we could approach it a different way.

We could think about one of these,

I guess you could say these
vertical layers first.

So this vertical layer in this example,

it is one, two, three, four, five high

and then it, in each of those five rows,

we can see it has one,
two, three, four cubes.

So how many cubes are just
in this orange part here,

this orange layer?

Well, it's five times four, which is 20.

And then how many of
these layers are there?

You can almost think of them as walls.

Well, we could see that there's one,

let me just see a different color.

We could see that there are
one, two, and three of them.

So then you can multiply that times three.

Well, five times four is
60, sorry, five times four.

My brain is cutting to the chase.

Five times four is 20,

and then 20 times e is going to be 60.

Once again 60 cubic centimeters.

And you can do that
with repeated addition.

Five times four is 20, 20
plus, 20 plus 20 is 60.

Now you can imagine where this last way

of thinking about it could go.

We could just think about how many cubes

are in this right wall.

We can see that it is one,
two, three, four, five high.

So it is five high

and it is one, two, three deep.

So in this last diagram,
you have five times three,

which is the same thing as 15 cubes.

And then how many of
these walls do you have?

Well, you could see you have
one, two, three, four of them.

So times four.

And so what is that going to get you?

Probably not a surprise.

Five times three is 15.

15 times four.

You might know that that is going

to be 60 cubic centimeters.

Or you could say that
that's the same thing as 15

plus 15

plus 15

plus 15.

15 plus 15 is 30.

Plus 15 is 45 plus 15 is 60.

It's 60 cubic centimeters.

Why did I put cubic centimeters here?

Well, that was what we were
counting the entire time.

Every one of these layers we're counting

in terms of cubic centimeters.

Each of these little cubes is,

each of these little cubes right over here

is a cubic centimeter.

Now you might be saying,

"What was the whole point

of me doing this three times like this?"

Well, so just show you that
you can think of volume

as multiplying the area of one side times

the last or the area of one layer.

Or you could think about
the area of the base.

So in this first example,
if you think of the base

as something like this,

or if you think about the area
of this base right over here,

so three times four, which we
did first or four times three,

and then we multiplied
that times the height.

So base times height,
that's what we did here.

Or you could think of it the other way.

You could think of it as,
well, we could multiply.

We could multiply, we could
find the area of that side

and then multiply it
times the other dimension.

So five times four would be
the area of that backside.

And then you could
multiply it times the three

once again to get to 60.

Or last but not least,
in this last example,

you could say, "All right,
let's find the area."

I could say the area of this
right side, right over here,

which is gonna be five times three

and then multiply it
times that fourth side.

But in all the situations, I'm
just multiplying the sides.

I have four times three times five,

five times four times three,
five times three times four.

I use the parentheses to do some

of that multiplication first.

But when you're multiplying
three numbers like this,

it actually doesn't matter the order

that you are multiplying them in.

So you just have to think
about the three sides here

and multiply them in some order.

And any way you do it,

you're going to get 60 cubic centimeters.