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Creating volume expressions

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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.
Title:
Creating volume expressions
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
05:52

English subtitles

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