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## Topic A: Foundations

## Video transcript

So we've got two
figures right over here, and I want to think about
how much space they take up on your screen. And this idea of how much
space something takes up on a surface, this idea is area. So right when you look at
it, it looks pretty clear that this purple figure
takes up more space on my screen than
this blue figure. But how do we
actually measure it? How do we actually know how much
more area this purple figure takes up than this blue one? Well, one way to do
it would be to define a unit amount of area. So, for example, I could create
a square right over here, and this square, whatever units
we're using, we could say it's a one unit. So if its width right
over here is one unit and its height right
over here is one unit, we could call this
a unit square. And so one way to measure
the area of these figures is to figure out how many
unit squares I could cover this thing with
without overlapping and while staying
in the boundaries. So let's try to do that. Let's try to cover each of
these with unit squares, and essentially we'll
have a measure of area. So I'll start with
this blue one. So we could put 1, 2, 3,
3, 4, 5, five unit squares. Let me write this down. So we got 1, 2, 3,
4, 5 unit squares, and I could draw the
boundary between those unit squares a little bit clearer. So we have 5 unit squares. And so we could say that
this figure right over here has an area. The area is 5. We could say 5 unit squares. The more typical
way of saying it is that you have 5 square units. That's the area over here. Now, let's do the same thing
with this purple figure. So with the purple figure, I
could put 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 of these unit squares. I can cover it. They're not overlapping,
or I'm trying pretty close to not make them overlap. You see, you can fit 10 of them. And let me draw the
boundary between them, so you can see a
little bit clearer. So that's the boundary
between my unit squares. So I think-- there you go. And we can count them. We have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10. So we could say the area
here-- and let me actually divide these with the
black boundary, too. It makes it a little bit
clearer than that blue. So the area here for the
purple figure, we could say, so the area here is equal to 10. 10 square, 10 square units. So what we have
here, we have an idea of how much space does
something take up on a surface. And you could eyeball
it, and say, hey, this takes up more space. But now we've come up with
a way of measuring it. We can define a unit square. Here it's a 1 unit by 1 unit. In the future we'll see that
it could be a unit centimeter. It could be a 1 centimeter
by 1 centimeter squared. It could be a 1 meter
by 1 meter squared. It could be a 1 foot
by 1 foot square, but then we can use
that to actually measure the area of things. This thing has an area
of 5 square units. This thing has an area
of 10 square units. So this one we can actually
say has twice the area. The purple figure
had twice the area-- it's 10 square units--
as the blue figure. It takes up twice the amount
of space on the screen.