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# Properties and patterns for multiplication

Sal changes the order of numbers or decomposes numbers to simplify multiplication problems. Created by Sal Khan.

Video transcript

We want to figure out how
many balloons we have here. And obviously, we
could just count these. But now, we have other
ways of thinking about it, especially because
they're arranged in this nice array, this
nice grid pattern here. And the reason why it's
useful to not just always have to count it, but to be able
to use little multiplication with the number of rows
and the number of columns is that you might
run into things and you will run
into things where it's very hard to count each
of the objects individually, but it might be a little
bit easier to count the rows and to count the columns. So for example,
right over here, we see that we have
1, 2, 3, 4 rows. And we have 1, 2, 3,
4, 5, 6, 7 columns. So you could view this
4 as an array of objects where we have 4 rows. Let me write that down. We have 4 rows and
we have 7 columns. And you might already
remember that we can calculate the
total number of objects by multiplying the rows
times the columns-- 4 rows times 7 columns. Now, why does this work? Why will this give us the
actual number of objects? Well, we could view
this-- we have 4 rows. So we have 4 groups of things. And how many things are
in each of those rows? Well, the number of columns. We have 7 things in each of
those 4 rows-- so 4 groups of 7. Or you could view it
the other way around. You could view that
each column is a group. So then you have 7 groups. And how many objects
do you have in each? Well, that's what
the rows tell you. You have 4 things in
each of those columns. And we already know that
both of these quantities are going to find the exact same
number, the number of things that we have right over here. So these two things
are equivalent. 4 times 7 is equal to 7 times 4. And there's a bunch
of ways that we can calculate
either one of these. We can skip count by 4. We say 4 times 4, 8,
12, 16, 20, 24, 28. And let's see. Let me make sure that that's 7. So 4 times 1, 2, 3, 4,
5, 6, 7-- so we get 28. We could just calculate that
there are 28 objects here. And likewise, we
could skip count by 7. So we could go 7-- 7 times 2
is 14, times 3 is 21, times 4 is 28, just adding
seven every time. And so we could get
28 the other way. Let's do that in
that same color. We could get 28 the other way. But if you had a situation
where you didn't know-- where you either didn't want
to do these techniques or it had been hard
to do these techniques or you didn't know what 4 times
7 was off the top of your head, which you should know at
some point in the very, very near future. Is there any way to break
this down to something that maybe you do know or maybe
that's a little bit easier to compute? Well, you could
realize that 7 columns is the same thing as 5
columns and then 2 columns. So you could view 7
columns as 5 columns-- so this is 5 columns right
over here-- plus 2 columns. So that's just like
saying that 4 times 7 is the same thing
as 4 times 5 plus 2. And all I do is I replace
the seven with a 5 plus 2. 7 has been replaced
with a 5 plus 2. Now, why is this interesting? Well, now, I can break this
up into two separate arrays. So I could say,
well, look, there's the array that has 4 rows and
2 columns right over here. And then there's the array that
has 4 rows and 5 columns right over here. So how many objects
are in this one, in the yellow one
right over here? Well, there's 4 times 5 objects. So there's 4 times 5
objects in the yellow grid or yellow array. And how many in this
orange-ish looking thing? Well, there's going
to be 4 times 2. And if we take the sum
of the 4 times the 5 and the 4 times the 2,
what are we going to get? Well, we're going to
get the 4 times the 7. We're going to get the
4 times the 5 plus 2. So if we take sum
of these things-- and we want to do the
multiplication first, so I'll just put a
parentheses around that to emphasize that--
this is going to be the same thing as
these things up here. And so you might say, oh,
well, I know what 4 times 5 is. 4 times 5 is 20. 4 times 2 is 8. 20 plus 8 is 28. And you might say,
OK, Sal, I get it. 4 times 7 is 28, which is the
same thing as 4 times 5 plus 2. And I see that that's the same
thing as 4 times 5 plus 4 times 2. And actually, this is called
the distributive property-- that 4 times 5 plus 2 is
the same thing as 4 times 5 plus 4 times 2. But I could just do one
of these first techniques you talked about. Why is this
distributive property that you just showed me
useful for computing or doing multiplication problems? Well, let me give you a
slightly more difficult one. Let's imagine you wanted
to multiply 6 times 36. Actually, I don't need to
write that parentheses. So how could you do this? Well, you could decompose
36 into two products or into two numbers
where it's easier to find the product
of that and 6. So for example 36, is the
same thing as 30 plus 6. So this is going to be
equal to 6 times 30 plus 6. And what's this going to be? Well, we just saw. 6 times these two things
added together first, this is going to be equal to
6 times 30 plus 6 times 6. Notice, we distributed the
6-- 6 times 30 plus 6 times 6. Now, why is this useful? Why was this useful at all? I'm going to put
parentheses to emphasize. We're going to do the
multiplication first. In general, when you see
multiplication and addition in a row like this,
or division, you want to do your multiplication
and division first, then do your addition
and subtraction. So what's 6 times 30? Well, this is
easier to calculate. 6 times 3, we know to be 18. So 6 times 30 is
going to be 180. And 6 times 6-- well, we
know that's going to be 36. So this is going
to be 180 plus 36. And what's that going to be? 180 plus 36-- well,
0 plus 6 is 6. 8 plus 3 is 11. 1 plus 1 is 2. So you just figured out that
6 times 36 is equal to 216. And what we just did with
the distributive property, this is actually
going to be how you're going to multiply all sorts
of larger numbers, way larger than what we just saw. So the distributive property,
which hopefully you're pretty convinced by, based
on how we broke things up, is a super useful
thing as you want to compute larger
and larger numbers. And you're going to find
it even more useful when you go even further in
your mathematical career.