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# Area and the distributive property

CCSS.Math:

## Video transcript

I have this rectangle here and I want to figure out its area I want to figure out how much space it is taking up on my screen right over here and I encourage you to pause this video and try to figure out the area of this entire rectangle and when you do it think about the two different ways you could do it you could either multiply the length of the rectangle times the entire width so just figure out the area of the entire thing or you could separately figure out the area of this red or this purple rectangle and then separately figure out this blue rectangle and realize that their combined areas is the exact same thing as the entire rectangle so I encourage you to pause the video and try both of those both of those strategies out so let's just try them out ourselves first let's look at the overall dimensions of the larger rectangle the length is nine and we're gonna multiply that times the width but what's the width here well the width is going to be 8 plus 12 that's this entire distance right over here is 8 plus 12 so it's 9 times 8 plus 12 8 plus 12 this is one way that we could figure out the area of this entire thing this is just this is just the length times the width 8 plus 12 is obviously going to be equal to 20 but the other way that we could do it and this must be equivalent because we're figuring out the area of the same thing it's to separate out the area of these two sub rectangles so let's do that and this must be equal to this thing so what's the area of this purple rectangle well it's going to be the length it's going to be 9 let me do it in that same color it's going to be 9 times the width which is 8 it's gonna be 9 times 8 and then what's the area of this blue rectangle well that's going to be 9 times so the height here is 9 still the height is 9 and what's the width well the width is 12 the width is 12 and what's the area of the combined if you wanted to combine the area of the purple rectangle and the blue one well you just add these two things together you would just add these two things together and of course when you add these two things together you get the area of the entire thing so these things must be equivalent they are calculating the same area now what's neat about this is we just kind of showed ourselves the distributive property the distributive property when we're dealing with these numbers you could try these out for any numbers they'll work for any numbers because the distributive property works for any numbers you see when nine times the sum of 8 plus 12 is equal to 9 times 8 plus 9 times 12 we essentially have distributed we have distributed the 9 9 times 8 plus 9 times 12 and let's actually calculate it just to just to satisfy ourselves about the area so if you multiply if you multiply the length times the entire width so that's 9 times 8 plus 12 that's the same thing as 9 times 20 which is 180 and over here if you calculate the area of this purple this purple rectangle that is 9 times 8 so that is going to be equal to 72 so that is going to be equal to 72 that's the purple rectangle area of the blue rectangle 9 times 12 well that's 108 108 and we're gonna take the sum of the two to find the area of the larger rectangle what is 72 plus 108 well 72 plus 108 is also equal to 180 so we verified these indeed are equal to each other when we calculate them and they make sense because we are calculating the same exact area