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- [Instructor] What we wanna do
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in this video is figure out the volume
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of this rectangular prism here.
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So pause this video and see
if you can do that on your own
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before we do this together.
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All right, now let's work
through this together
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and we're actually gonna think about it
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in three different ways.
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And to help us do that in
those three different ways,
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we'll visualize this volume
in these three different ways.
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So the first way to visualize
it is let's just think
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about it as if we think about
it as cubic centimeters,
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five going up, four going across,
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or four wide, and then three deep.
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And so that's what all of these show.
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And in this first diagram,
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what we first can think
about is how many cubes,
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how many orange cubes
are in this first layer.
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Well, this first layer
right over here is going
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to be four cubes in one dimension.
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Actually, let me scroll down a little bit.
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It's going to be four in this
dimension right over here,
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and there's going to be
three in this dimension.
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So if you think about
how many cubes are there,
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that is four times three
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on just one base or one layer.
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And then how many layers are there?
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Well, we can see
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that there are one, two, three,
four, five of those layers.
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So then you can multiply that times five.
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Or another way to think about it is,
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in that base layer there,
if you did four times three,
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you're going to have 12 of those cubes,
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and then you're gonna have
to multiply 12 times 5,
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which you might just know as being 60.
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Let me do that same color.
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As 60.
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Or you could do a repeated
addition if you like.
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You could say, "Well, that's going
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to be equal to 12 plus 12
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plus 12 plus 12
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plus 12
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plus 12,
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which also is equal to 60."
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And that's 60 of what?
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Well, each of these things
is a cubic centimeter.
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So that is 60,
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60 cubic centimeters.
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Now we could approach it a different way.
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We could think about one of these,
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I guess you could say these
vertical layers first.
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So this vertical layer in this example,
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it is one, two, three, four, five high
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and then it, in each of those five rows,
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we can see it has one,
two, three, four cubes.
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So how many cubes are just
in this orange part here,
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this orange layer?
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Well, it's five times four, which is 20.
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And then how many of
these layers are there?
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You can almost think of them as walls.
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Well, we could see that there's one,
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let me just see a different color.
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We could see that there are
one, two, and three of them.
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So then you can multiply that times three.
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Well, five times four is
60, sorry, five times four.
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My brain is cutting to the chase.
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Five times four is 20,
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and then 20 times e is going to be 60.
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Once again 60 cubic centimeters.
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And you can do that
with repeated addition.
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Five times four is 20, 20
plus, 20 plus 20 is 60.
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Now you can imagine where this last way
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of thinking about it could go.
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We could just think about how many cubes
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are in this right wall.
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We can see that it is one,
two, three, four, five high.
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So it is five high
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and it is one, two, three deep.
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So in this last diagram,
you have five times three,
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which is the same thing as 15 cubes.
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And then how many of
these walls do you have?
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Well, you could see you have
one, two, three, four of them.
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So times four.
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And so what is that going to get you?
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Probably not a surprise.
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Five times three is 15.
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15 times four.
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You might know that that is going
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to be 60 cubic centimeters.
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Or you could say that
that's the same thing as 15
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plus 15
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plus 15
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plus 15.
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15 plus 15 is 30.
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Plus 15 is 45 plus 15 is 60.
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It's 60 cubic centimeters.
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Why did I put cubic centimeters here?
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Well, that was what we were
counting the entire time.
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Every one of these layers we're counting
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in terms of cubic centimeters.
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Each of these little cubes is,
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each of these little cubes right over here
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is a cubic centimeter.
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Now you might be saying,
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"What was the whole point
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of me doing this three times like this?"
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Well, so just show you that
you can think of volume
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as multiplying the area of one side times
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the last or the area of one layer.
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Or you could think about
the area of the base.
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So in this first example,
if you think of the base
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as something like this,
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or if you think about the area
of this base right over here,
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so three times four, which we
did first or four times three,
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and then we multiplied
that times the height.
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So base times height,
that's what we did here.
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Or you could think of it the other way.
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You could think of it as,
well, we could multiply.
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We could multiply, we could
find the area of that side
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and then multiply it
times the other dimension.
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So five times four would be
the area of that backside.
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And then you could
multiply it times the three
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once again to get to 60.
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Or last but not least,
in this last example,
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you could say, "All right,
let's find the area."
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I could say the area of this
right side, right over here,
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which is gonna be five times three
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and then multiply it
times that fourth side.
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But in all the situations, I'm
just multiplying the sides.
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I have four times three times five,
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five times four times three,
five times three times four.
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I use the parentheses to do some
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of that multiplication first.
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But when you're multiplying
three numbers like this,
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it actually doesn't matter the order
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that you are multiplying them in.
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So you just have to think
about the three sides here
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and multiply them in some order.
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And any way you do it,
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you're going to get 60 cubic centimeters.