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## Arithmetic (all content)

### Unit 5: Lesson 21

Multiplying fractions- Intro to multiplying 2 fractions
- Multiplying 2 fractions: fraction model
- Multiplying 2 fractions: number line
- Multiplying fractions with visuals
- Multiplying 2 fractions: 5/6 x 2/3
- Multiplying fractions
- Finding area with fractional sides 1
- Finding area with fractional sides 2
- Area of rectangles with fraction side lengths
- Multiplying fractions review

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# Finding area with fractional sides 2

CCSS.Math:

Sal discusses intuition for area of a rectangle with fractional side lengths.

## Video transcript

- [Voiceover] So I have a square here and let's say that its
height is one meter. So this is height right over here. That is one meter. And let's say its width is also one meter. So I'm talking about the
dimensions of the entire square, not just the shaded region. So this is also, that right
over there is also one meter. What's the area of the
entire square going to be, not just the shaded, the entire square? Well, the total area, total area is going to be equal to the height times the width. So one meter times one meter. One times one is of course one and meters times meters we could write that as a square meter or meter squared, however
you want to think about it. Now with that out of the way now let's focus on the shaded area. Let's think about what that is. So the shaded, shaded, shaded area is equal to what, and I encourage you to pause the video and try to figure that out. Well the one thing that
might jump out of you is that our entire area, our entire square is divided into these equally
equal, equal rectangles. So one way to think about it is well, what is the area of each
of these equal rectangles? For example what is the area of that rectangle right over there? And to figure it out we can say well, what fraction is that of the whole? And to figure that out
we have to figure out how many of these rectangles has our whole been divided into? We could try to count them out or we could say let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 columns, and each columns has
one, two, three, four, five, six, seven. So we have 10 columns of seven or we have 70 of these rectangles, that our entire whole is
divided into 70 equal sections that we see these
rectangles right over here. This character, this
character right over there that is 1/70 of the entire area. So 1/70 of one square meter which is of course just going
to be 1/70 of a square meter. That's just one of these. That's just one of these rectangles. Now if we cared about the shaded area we can just count how many
of these rectangles there are and we see that there are
one, two, three, four, five, six, seven, eight, nine columns of one, two, three. So there's 27 of these rectangles, of these equal rectangles
in the shaded area. The shaded area is going to be, we have 27 of these
rectangles and each of them, and then each of them have an area of 1/70 of a square meter. 1/70 of a square meter and what does that give us? Well that gives us the area of the shaded or the shaded area is going to be 27/70. 27 times 1/70 is going to be 27/70. 27/70 square meters and we're done. But what I want to appreciate now is that there's multiple ways that we could have tackled this. Another way we could have tackled it is to figure out what the dimensions, what the dimensions
are of the shaded area. So for example. For example, what is the height of just the shaded area? So just that height right over there and I encourage you to pause the video and try to think about what it is and it's going to be a fraction. Well, we see if we're going
in the vertical direction we've divided this one meter. We've divided it into one,
two, three, four, five. So let me do it a little bit differently. We've divided it into one, two, three, four, five, six, seven equal sections. That might have been
a little bit confusing the way I just drew it. So you can see it when
you look at the actual... Actually let me do it
in a more vibrant color. We have... I'm having trouble picking colors. All right, here we go. We have one, that's that right over there. Two, three, four, five, six, seven equal sections that we've
divided this one meter in. And the height of the shaded
area is three of them. So this height right over here, this height right over
here is 3/7 of the whole and the whole is a meter. So it's 3/7 of a meter. Now by that same logic what
is the width going to be? What is the width going to be? Well we can see that the entire meter has been divided into one,
two, three, four, five, six, seven, eight, nine, 10 equal sections so going from here to here
is going to be a tenth. So this distance right over
here is going to be a tenth. Let me do that in a
color that's different. So this distance right over
here is going to be a tenth and so how many tenths represent the width of the green area? Let's see, we get, have 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10. So this width is 9/10 of this whole length which is a meter so it's 9/10 of a meter. Now to find the area we can multiply the width times the height or the height times the width. So we could say, so I'll write this again. The shaded area. Shaded area instead of doing it this way we could say I have a
height of 3/7 of a meter. So 3/7 meters and then I can multiply
that, times our width for just the shaded area which is 9/10 of a meter. 9/10 of a meter. And now what is this going to get us? Well, this is going to be equal to the meters times the
meters is going to get square meters which is what we want and then we could multiple the numerators and multiply the denominators. Three times nine is going to give us 27 and seven times 10 is going to give us 70. Exactly what we had before. 27/70 square. Let me write that a little bit neater. 27. 27/70 of a square meter. And you could think about
why did this work out regardless of how we did it? Notice, three and nine are the numerators. That was how many rows and columns we had of these little rectangles. And then the seven and 10 that's to figure out how many
rectangles we actually had. So this is saying okay, the three times nine is how
many rectangles we have? And then the seven times 10 is what fraction of the whole each of those rectangles represent and that's essentially
what we did up here. So either way, you're going
to get the right answer but I really want you to
think about why this was.