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## Multi-digit multiplication: place value and area models

Current time:0:00Total duration:8:47

# Multiplying with area model: 16 x 27

CCSS Math: 4.NBT.B.5

## Video transcript

So I'm going to
multiply 16 times 27. I'm going to do it using
something called an area model. And the whole point
of an area model is to really understand
what's going on in the multiplication process. So 16, you can represent
literally as 10 plus 6. This 1 is in the tens place. It represents one 10. So we can represent
that as 10 plus 6. Let me do the 6 in
that same green color. 10 plus 6. And let me mark
off 10 and 6 here. So this is 1, 2, 3,
4, 5, 6, 7, 8, 9, 10. So this part in
blue, that's the 10. So I'm representing--
I've gone 10 slashes, or I'm representing 10
boxes right over there. And then the 6,
which I want to do in that green color, the sixth,
let's mark off 6 boxes. 1, 2, 3, 4, 5, 6. So 16 is this whole length. The blue part is the 10. The green part is the 6. The 10 comes from
the one in the tens. This is literally 1 10 and
this is literally 6 1s. Now let's think about 27. Well, we already know that
the 2 in the 10s place is representing 20. So let's count off 20. So it's 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14 15, 16 17, 18, 19. 20. So up until that
point right over here, this line right over
here has a length of 20. But we're not just
talking about 20. We're talking about 27. So it's 20 plus 7. So let's count off 7
here, 1, 2, 3, 4, 5, 6, 7. Now if we look at
the total number. So if we have 16 times 27,
the product 16 times 27, what gives you the area of
a 16 by 27 rectangle. So let me draw that. So it's 16-- this
is a width of 16. Let's bring a little line down
right over here-- and then the height of my rectangle,
the way I've drawn it is 27. This is l, 2, 3, 4, 5, 6, 7. So the area of this rectangle
is going to be 16 times 27, the area of this
entire rectangle. Now, we could do is break it
up by parts, because it'll be easier to compute, and we
can see what part of the area those different
products represent. So, for example, we
can think about what 10-- so let me separate
out the sections. So let me draw it. So this section is 10 wide. And then let me draw
a section like this. So we could figure out the
areas of each of these sections, and then the area of
the entire rectangle, which is going to
be this product, is going to be the area of all
of these rectangles combined. So we could first
think about it-- well, let's think about
what 20 times 10 is. Well 20 times 10 is a fairly
straightforward thing. It's going to be 200. You could think of as 2 times
1, and you have two 0's there. Or you probably know
in your head 20 times 10 is just going to be--
you're going to add a 0 here. So it's going to be 200. So 20 times 10 is 200. And let me highlight that in. So you have-- that's
not the color I wanted to use-- let me use
this blue right here. So that's the blue from the 10. Let me put some orange
in there from the 20 to make it clear that this
is the product of both of those numbers. So 20 times 10 is 200. Now what's 20 times 6? Well 2 times 6 is 12,
plus you have this 0 here. So this right over here
is going to be 120. And it has the orange in it. So this is a 20. And then let me put
some green in for the 6. 20 times the 6. Now what's the area of this
section right over here? Well, it's 7 high
and it is 10 wide. So it's going to be
7 times 10 or 70. I'll have an area
of 70 square units. So let me do it in
this purple, and I'll thrown some blue in there, too. It's kind of a fun art project. I'll throw some
blue in there too. And then finally, what's the
area of this little section right over here? It's 7 high and it's 6 wide. So it's going to be 7 times
6 or 42 square units of area. Let me color it in. So I got some magenta
in there, and then I got some green in
there right over there. So what's the area of this
entire thing going to be? Well, it's going to be the
200 plus the 120 plus-- let me do it this way-- it's
going to be 200 plus 120 plus 70 plus 42. When you add that up, you get,
let's see, in the ones place, you get a 2. Then you get 2 tens plus 7
tens is 9 tens, plus 4 tens, is 13 tens, which is the same
thing as 3 tens and 1 hundred. And then this is a 4. Did I add that up right? Let's see, this will be 11. Yup, that looks right. So this is 432. So this is going
to be equal to 432. And you might be saying,
hey, Sal, why did we go through all of this business? I've seen before
that if I take 16-- I could take 16
times 27 like this. You probably learned
this type of a process. 16 times 27. Then you say, OK, let's start
with the 7 in the ones place and you do 7 times 6 is 42. And you carry the 4. You're really just putting
the 4 in the tens place because it's a 40. But right when you
did that 7 times 6, we essentially calculated
this right over there. And then when you multiply
the 7 times the 1, you really multiplying
7 times 10, and then you're adding
the 4 from the 42. So when you do the
7 times this 1. This is you're actually
calculating this area. And then when you
carry it when you add this carried 4, and
put it right down here. You're essentially taking the
sum of both of these things, because you're multiplying 7
times the 16 to get this area. So let's just do it. 7 times 1 is 7 plus 4 is 11. So 112, what you just
figured out right over here, is this area Right over here. So 7 times the 10 or the
70, plus 7 times 6, the 42. Now when you go
into the tens place you've probably always
said, oh, you know, I just throw a 0 down there. But why do you throw
a 0 down there? Because this 2 is
representing 10. So if you multiply 2 times
the 6, and you put a 12 and put a 2 down
and carry the 1. That wouldn't be right. This is a 20. So that's why you
put the 0 there. So let's scratch this out
so we don't get confused. 2 times 6 is 12. So we're used to carrying the
2 down here and carrying the 1. But notice we're really thinking
about 20 times 60 is 120. Just doing that we just
calculated that right over there. And then we do 20
times 10 is 200. But then we had that
one from the 120. So this is going to be 300. So what we just did when we
multiplied the 2 times the 16, we just calculated
this total area. We did the 20 times
the 6 to get that. And then we had the 20
times a 10 to get that. When we did the 20 times
the 6, we carried this 1 into the hundreds place. And so we added it all
together and got 320. And then this step
right over here, you're literally just
finding the combined area-- the area of this plus the
area of that the area. So that's going to get us to---
we deserve a drum roll now. 2 plus 0 is 2. 1 plus 2 is 3. 1 plus 3 is 4.