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# Area and the distributive property

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 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.