Area and the distributive property
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 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 9, and we're going to multiply that times the width. But what's width here? Well, the width is going to be 8 plus 12. This entire distance right over here is 8 plus 12. So it's 9 times 8 plus 12. This is one way that we could figure out the area of this entire thing. 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-- is 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 going to be 9 times 8. And then what's the area of this the 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. 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'd 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 showed ourselves 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 9 times the sum of 8 plus 12 is equal to 9 times 8 plus 9 times 12. We essentially have distributed the 9-- 9 times 8 plus 9 times 12. And let's actually calculate it just to satisfy ourselves about the area. So 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 rectangle, that is 9 times 8. So that is going to be equal to 72. That's the purple rectangle. The area of the blue rectangle, 9 times 12, well, that's 108. And we're going to 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've verified. These, indeed, are equal to each other when we calculate them. And they make sense, because we are calculating the same exact area.