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# Properties of multiplication

Sal uses pictures and practice problems to see commutativity and associativity in multiplication. Created by Sal Khan.

Video transcript

So if you look at each
of these 4 by 6 grids, it's pretty clear that there's
24 of these green circle things in each of them. But what I want to show
you is that you can get 24 as the product of three numbers
in multiple different ways. And it actually doesn't matter
which products you take first or what order you
actually do them in. So let's think about this first. So the way that
I've colored it in, I have these three groups of 4. If you look at the
blue highlighting, this is one group of 4, two
groups of 4, three groups of 4. Actually, let me make
it a little bit clearer. One group of 4, two groups
of 4, and three groups of 4. So these three columns you
could view as 3 times 4. Now, we have another 3
times 4 right over here. This is also 3 times 4. We have one group of
4, two groups of 4, and three groups of 4. So you could view these
combined as 2 times 3 times 4. We have one 3 times 4. And then we have
another 3 times 4. So the whole thing we could
view as-- let me give myself some more space-- as 2 times--
let me do that in blue-- 2 times 3 times 4. That's the total
number of balls here. And you could see it based
on how it was colored. And of course, if you did 3
times 4 first, you get 12. And then you multiply
that times 2, you get 24, which is the total
number of these green circle things. And I encourage you now to
look at these other two. Pause the video and
think about what these would be the
product of, first looking at the
blue grouping, then looking at the purple
grouping in the same way that we did right over here, and
verify that the product still equals 24. Well, I assume that
you've paused the video. So you see here in this first, I
guess you could call it a zone, we have two groups of 4. So this is 2 times
4 right over here. We have one group of
4, another group of 4. That's 2 times 4. We have one group of
4, another group of 4. So this is also 2 times 4 if
we look in this purple zone. One group of 4,
another group of 4. So this is also 2 times 4. So we have three 2 times 4's. So if we look at each of
these, or all together, this is 3 times 2 times
4, so 3 times 2 times 4. Notice I did a different order. And here I did 3 times 4 first. Here I'm doing 2 times 4 first. But just like before,
2 times 4 is 8. 8 times 3 is still equal
to 24, as it needs to, because we have exactly 24
of these green circle things. Once again, pause the video
and try to do the same here. Look at the groupings
in blue, then look at the groupings
in purple, and try to express these 24 as some
kind of product of 2, 3, and 4. Well, you see first we
have these groupings of 3. So we have one grouping
of 3 in this purple zone, two groupings of 3
in this purple zone. So you could do
that as 2 times 3. And we have one 3 and another 3. So in this purple zone,
this is another 2 times 3. We have another 2 times 3. Whoops. I wrote 2 times 2. 2 times 3. We have another 2 times 3. And then finally, we
have a fourth 2 times 3. So how many 2 times
3's do we have here? Well, we have one, two,
three, four 2 times 3's. So this whole thing could be
written as 4 times 2 times 3. Now, what's this
going to be equal to? Well, it needs to
be equal to 24. And we can verify 2 times 3
is 6 times 4 is, indeed, 24. So the whole idea of what
I'm trying to show here is that the order in which
you multiply does not matter. Let me make this very clear. Let me pick a different example,
a completely new example. So let's say that I
have 4 times 5 times 6. You can do this multiplication
in multiple ways. You could do 4 times 5 first. Or you could do 4
times 5 times 6 first. And you can verify that. I encourage you
to pause the video and verify that these two
things are equivalent. And this is actually called
the associative property. It doesn't matter how you
associate these things, which of these that you do first. Also, order does not matter. And we've seen that
multiple times. Whether you do this or
you do 5 times 4 times 6-- notice I swapped the 5
and 4-- this doesn't matter. Or whether you do this
or 6 times 5 times 4, it doesn't matter. Here I swapped the
6 and the 5 times 4. All of these are going to
get the exact same value. And I encourage you
to pause the video. So when we're talking
about which one we do first, whether we do the 4
times 5 first or the 5 times 6, that's called the
associative property. It's kind of fancy word for
a reasonably simple thing. And when we're saying
that order doesn't matter, when it doesn't matter whether
we do 4 times 5 or 5 times 4, that's called the
commutative property. And once again, fancy word
for a very simple thing. It's just saying it doesn't
matter what order I do it in.