-
- What we're gonna do in this video is try
-
to figure out what 259 times 35 is.
-
And there's a lot of
ways to approach this,
-
but one way to think about it is,
-
imagine we had a rectangle.
-
I'll make it a really wide rectangle.
-
So that's the width of our rectangle,
-
and then that's the
height of our rectangle.
-
And let me just draw the entire rectangle.
-
And let's imagine that this width
-
of this rectangle over here is 259 units,
-
whatever those units might
be, that's its width.
-
And then this height
over here is 35 units.
-
Well then the area of this
rectangle would be 259 times 35,
-
whatever that product is.
-
Now, why am I even talking
about this? Why is this useful?
-
Well, we can actually break
up 259 into 200 plus 50
-
plus nine and think about
those chunks of the area
-
and do the same thing for 35.
-
What do I mean by that? Well,
let's first imagine 200.
-
So if this whole thing
is 259, then maybe 200.
-
And I'm not going to get it exactly right,
-
but maybe 200 is going to be about,
-
about, let's see, am eyeballing it.
-
It's about that much
of it right over there.
-
And then the 50,
-
50 would be about that much of it.
-
And then the nine, I'll
do this in a new color,
-
might be that much of it.
-
That much of it.
-
So 259 is the same thing
as 200 plus 50 plus nine.
-
And I can divide this
rectangle into these areas.
-
So one way to think about it is
-
that this area over here is
going to be 200 times 35.
-
This area is gonna be 50 times 35,
-
and this area is nine times 35.
-
Well, as I said, I could
do the same thing with 35.
-
Let's say that, let's say that this is 30.
-
That is 30 right over there,
-
and then the rest of it is five.
-
So this right over here is five.
-
Well, I can further,
-
I can split up this bigger rectangle
-
into even more rectangles.
-
Now, why is this interesting?
-
Well, this top left area
is going to be what?
-
It's going to be 30 times 200.
-
And we might be able to
do that in our heads.
-
If I did three times two
-
it would be six, 30 times two is 60,
-
30 times 20 is 630 times 200 is 6,000.
-
Or some of you might say,
okay, three times two is six.
-
And then I have these three
zeros, which is the equivalent
-
to multiplying by ten three times
-
or multiplying by a thousand.
-
So it's 6,000.
-
And now what is the area of
this section right over here?
-
Well, that's going to be 30 times 50.
-
So 30 times 50 is going to be equal
-
to three times five is 15.
-
And then I have two more
zeros. So it's 1500.
-
And then this over here is
going to be 30 times nine,
-
which, three times nine is 27.
-
So 30 times nine is 270.
-
And now if I go down here,
-
this a little area is going to be,
-
I probably should have drawn
it a little bit bigger,
-
but it's going to be five times
200, which is equal to 1000.
-
This one over here is
gonna be five times 50,
-
five times 50, which is equal to 250.
-
And then last but not least
five times nine is 45.
-
Now why is this interesting?
-
Because each of these
little smaller areas,
-
if I add them all up,
I get the bigger area.
-
So the bigger area is going to be 6,000
-
plus 1500
-
plus 270.
-
And one way to think about it is each
-
of these are partial products.
-
They're partial products of
just 30 times 200 or 30 times 50
-
or 30 times nine.
-
But I'm gonna keep going.
-
And then I have this a thousand here,
-
which was the five times 200.
-
Then I have the 250 here,
and then I have the 45.
-
If I add up all of these partial products
-
or partial areas, I could say,
-
then I'm gonna get the total product
-
or the total area of this rectangle.
-
So let's see, in the
one's place, I just end up
-
with a five, in the tens
place, seven tens plus five,
-
that's 12 plus four is 16 10s.
-
So six tens and a hundred.
-
One plus five is six, plus two is eight,
-
plus two more is 10.
-
So that gets regrouped that one,
-
and we get one plus six is
seven, plus one is eight,
-
plus one more is nine.
-
So we get to 9,065 in total.