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# Multiplication as groups of objects

Sal uses arrays to show different ways to multiply and get the same solution. Created by Sal Khan.

Video transcript

So, I have several groups of
these ball-looking things. And let's think about how
many balls are in each group. We have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11 12. And what I want to do is
think about the different ways of dividing these 12 balls into
different numbers of groups. So, for example, I could
view these 12 balls as one, so that's 1 group of 3, 2
groups of 3, 3 groups of 3, 4 groups of 3. So I could view 12 as
being 4 groups of 3. And the way that
we would write that is that 12 is equal
to 4 groups of 3. Or another way of reading this
is that 12 is equal to 4 times 3. If I have 1, 2, 3, 4 groups,
and in each of those groups I have 1, 2, 3,
objects, I'm going to have a total of 12 objects. But that's not the only
way we can get to 12. We could also view
it as 3 groups of 4. So, let's look at that. So we could have it as 1
group of 4, 2 groups of 4, 3 groups of 4. So now we could view 12
as being 3 groups of 4. Or we could say-- let me get the
right tool out-- that 3 times 4 is equal to 12. So, whether we're doing
4 times 3 or 3 times 4, they're both going
to be equal to 12. 4 groups of 3 is
12, 3 groups of 4. But we don't have to stop there. We could also view
12 as, well, we could view it as 2 groups of 6. Let's look at that. So this is 1 group
of 6 right over here. So, that's one 1 of 6. That's another group of 6. So, once again, we could
view this as 2 times 6. 2 times 6 will also get us a 12. Well, what about doing
it as 6 groups of 2? Well, we can draw that
out, too-- 6 groups of 2. So that's 1 group of 2. Let me do that in
a different color. Let me do it in
this purple color. We have 1 group of 2, 2
groups of 2, 3 groups of 2, 4 groups of 2, 5 groups
of 2, and 6 groups of 2. So once again, this
is all different ways of writing 12, something
equivalent to 12. We could write 6 times 2--
6 groups of 2-- 6 times 2 is also equal 12. But we don't have to stop there. We could also literally
view 12 as 1 group of 12. So how would that look? So 1 group of 12. So this whole thing is
just 1 group of 12 here. So we could literally say
1 times 12 is equal to 12. We have one entire group of 12. 1 times 12 is equal to 12. And we could think of
it the other way around. We could view this
as 12 groups of 1. Let me draw that. So 12 groups of 1. This is 1 group of
1, 2 groups of 1, 3, 4, 5, 6, 7, 8, 9, 10,
11, and 12-- 12 groups of 1. So we could also write 12. 12 groups, and in
each one, I have 1. Well, that's still
going to get me to 12.