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CCSS.Math:

so I have this rectangular prism here it's kind of the shape of a brick or a fish tank and it's made up of these unit cubes and each of these unit cubes we're saying is 1/4 of a foot by 1/4 of a foot by 1/4 of a foot so you could almost imagine that this is so let me write it this way this is 1/4 of a foot by 1/4 of a foot by 1/4 of a foot those are its that those are its length height and width or depth whatever you want to call it so given that what is the volume of this entire rectangular prism going to be so I'm assuming you've given a go at it so there's a couple of ways to think about it you could first think about the volume of each unit cube and then think about how many unit cubes there are so let's do that the unit cube its volume is going to be 1/4 of 1/4 of a foot times 1/4 of a foot times 1/4 of a foot or another way to think about it is it's going to be 1/4 times 1/4 times 1/4 cubic feet which is often written as feet to the 3rd power cubic feet so 1/4 times 1/4 is 1/16 times 1/4 is one 64th so this is going to be 1 over 64 cubic feet cubic feet or 1/64 of a cubic foot that's the volume of each of these that's the volume of each of these unit cubes now how many of the how many of them are there well you could view them as kind of these two layers the first layer has 1 2 3 4 5 6 7 8 that's this first layer right over here that's this first layer right over here and then we have the second layer down here which would be another 8 so it's going to be 8 plus 8 or 16 so the total volume here the total volume is going to be 16 times times 1/64 of a cubic foot which is going to be equal to 16 over 64 16 over 64 cubic feet cubic feet which is the same thing 1660 fourths is the same thing as 1/4 divide the numerator and the denominator by 16 this is the same thing as 1/4 of a cubic foot of a cubic foot and that's our volume now there's other ways that you could have done this you could have just thought about the dimensions of the length the width and the height the width right over here is going to be 2 times 1/4 feet which is equal to 1/2 of a foot the height here is the same thing it's 2 times so it's going to be 2 times 1/4 of a foot which is equal to 2/4 or 1/2 of a foot and then the length the length here is 4 times 1/4 of a foot 4 times 1/4 of a foot well that's equal to 4/4 of a foot which is equal to one foot so to figure out it's about the volume we could we could multiply we could multiply the length times the width the length times the width times the height times the height and I need these little dots here these aren't decimals I've written them a little higher these are another way it's a shorthand for multiplication instead of writing a this kind of X looking thing this cross looking thing so the length is 1 the width is 1/2 of a foot so times 1/2 and then the height is another half is another half let me do it this way the height is another half so what's 1 times 1/2 times 1/2 well that's going to be equal to 1/4 and this is a foot this is a foot this is a foot so foot times foot times foot that's going to be foot feet to the third power or cubic feet 1/4 of a cubic foot either way we got the same result which is good