Main content

## Multiplication: place value and area models

Current time:0:00Total duration:6:58

# Multiplying with distributive property

CCSS Math: 4.NBT.B.5

## Video transcript

In this video, I'm going
to multiply 87 times 63. But I'm not going to do it
just by using some process, just showing you some steps. Instead, we're just going to
use the distributive property to actually try to
calculate this thing. So first, what I'm going
to do-- let me rewrite 87. So this is the same thing as 87. But instead of
writing 63 like that, I'm going to write
63 as 60 plus 3. Now, what is this
going to be equal to? Well, 87 times 60
plus 3, that's going to be the same thing
as-- and let me actually copy and paste this. So this is going to be the
same thing as 87 times 60 plus 87 times 3. You could say that we've
just distributed the 87. We're multiplying
87 times 60 plus 3. That's 87 times 60
plus 87 times 3. I could put parentheses here to
make it a little bit clearer. Well, fair enough. But then how do you
calculate what this is? Well, now we can
rewrite 87 as 80 plus 7. So let's rewrite that. So this is the same thing. Actually, let me
write it this way. I can swap them around. So this is the same
thing as 60 times 87. But I'll write that
as 60 times 80 plus 7. We could do it like this. 80 plus 7 plus 3 times
80 plus 7, or 3 times 87. Let me just copy
and paste that, so I don't have to keep
switching colors. Plus 3 times 80 plus 7. So copy and then
let me paste it. And then you have
it just like that. So all I did, just to
be clear-- all of what you see right over here,
87 times 60, well, that's the same thing as
60 times 87, which is the same thing as
60 times 80 plus 7. All that you see
here, 87 times 3, that's the same thing
as 3 times 87, which is the same thing as
3 times 80 plus 7. That's just that over here. But look, we can
distribute again. We can distribute the
60 times 80 plus 7. So this is going to be 60-- I'm
going to do that same color. Color changing is hard. This is 60 times 80 plus
60 times 7 plus 3 times 80 plus 3 times 7
right over here. So notice what we
really did is we thought about what each
of these digits represent. 8 represents 80. 7 represents 7. 6 represents 60, because
it's in the tens place. The 8 was in the
tens place, as well. This 3 is in the ones
place, so it's just 3. And we just multiplied
them all together. We multiplied the
80 times the 60. We multiplied the
80 times the 3. We multiplied the 7 times
the 60 right over here. We multiplied the 7 times the 3. And then we add them
all up together. And this will actually
give us our product. So for example, this right
over here, 6 times 8 is 48. But this isn't six 8's. This is 60 80's. So this is going to be 4,800. We've got two 0's
right over here, so 48 followed by the two 0's. This right over here,
60 times 7, is 420. 6 times 7 is 42. But it's going to be 10 times
as much, because this is a 60. And then 3 times 80--
well, same logic. 3 times 8 is 24. So this is going to be 240. And then, finally,
3 times 7 is 21. And then to get the product,
we can add these two together. And you might be
saying, hey, Sal, I know faster ways
of doing this. But the whole reason
I'm doing this is to show you that that fast
way you knew how to do it, it's not some magical formula
or some magical process you're doing. It just comes out of really
the distributive property and, hopefully, a little
bit of common sense. So what is this
going to be equal to? Well, we could add them all up. 4,800 plus 420 plus 240 plus 21. Well, you're going
to get a 1 here. Let's see. 20 plus 40 plus 20 is 80. Let's see. 800 plus 400 is 1,200
plus 200 more is 1,400. And so you get 5,481. It's equal to 5,481. And you might say, gee,
this was a bit of a pain to have to do the distributive
property over and over again. Is there a simpler way
to maybe visualize this? And there is. You could actually
write this as a grid. So we could say we're
multiplying 87 times 63. We could write it like this. We could say it's 80
plus 7 times 60 plus 3. And then you can set
up a grid like this. So let me set up
a little box here. It's 2 digits by 2 digits. So it's going to be a 2 by 2
grid, 2 rows and 2 columns. And then you just
have to calculate. Well, what's 60 times 80? Well, we already
calculated that. That's 4,800. What is 60 times 7? Well, that's going to be 420. What is 3 times 80? We already calculated that. That is 240. And I want to do that
same color-- 240. And finally, what is 3 times 7? 21. You add them all together. You get 5,481. And I encourage you to now just
do this same multiplication problem, the same
87 times 63, the way that you might have
traditionally learned it. And look at the different steps
and why they are making sense and why, at the end
of the day, you really are doing the same thing that
we just did in this video. You're just doing it
in a different way. And the whole point of
this whole exercise, this whole video, is so
you're not just blindly doing some type of steps to
find the product of two numbers. But you can actually
understand why those steps work and how those numbers
relate to each other.