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CCSS Math: 5.NF.B.3

When we were first exposed to
multiplication and division, we saw that they had an
inverse relationship. Or another way of
thinking about it is that they can
undo each other. So for example, if I had 2 times
4, one interpretation of this is I could have
four groups of 2. So that is one group of 2, two
groups of 2, three groups of 2, and four groups of 2. And we learned many, many
videos ago that this, of course, is going to be equal to 8. Well, we could express a very
similar idea with division. We could start with 8 things. So let's start with one, two,
three, four, five, six, seven, eight things. So now we're going
to start with the 8. And we could say, well,
let's try to divide that into four groups,
four equal groups. Well, that's one equal
group, two equal groups, three equal groups,
and four equal groups. And we see when we
start with 8 divide it into four equal
groups, each group is going to have
2 objects in it. So you probably see
the relationship. 2 times 4 is 8. 8 divided by 4 is 2. And actually, if we did 8
divided by 2, we would get 4. And this is generally true. If I have something
times something else is equal to whatever their product
is, if you take the product and divide by one of
those two numbers, you'll get the other one. And that idea
applies to fractions. It actually makes a lot
of sense with fractions. So for example, let's say
that we started off with 1/3 and we wanted to
multiply that times 3. Well, there's a couple of
ways we could visualize it. Actually, let me just
draw a diagram here. So let's say that this
block represents a whole, and let me shade in a 1/3 of it. So that's 1/3. We're going to multiply by 3. So we're going to
have 3 of these 1/3's. Or another way of
thinking about it, it's going to be 1/3 plus
another 1/3 plus another 1/3. That's our first 1/3, our
second 1/3, and our third 1/3. And we get the whole. This is 3/3, or 1. So this is going
to be equal to 1. So you use the exact same idea. If 1/3 times 3 is
equal to 1, then that means that 1 divided by
3 must be equal to 1/3. And this comes straight
out of how we first even thought about fractions. The first way that we ever
thought about fractions was, well, let's
start with a whole. And that whole would be our 1. And let's divide it into 3
equal sections, the same way that we divided this
8 into 4 equal groups. So if you divide this
into 3 equal sections, the size of each
of those sections is going to be exactly 1/3. Now, this leads to an
interesting question that might be popping
in your brain. Notice, we have 1
is the numerator, 3 is the denominator,
and we just said that this is equal
to the numerator divided by the denominator. 1 over 3 is the same
thing as 1 divided by 3. Is this always true
for a fraction? Well, let's just do the
same thought experiment, but let's do it with
a different fraction. Let's take 3/4 and
multiply it by 4. So multiply it by 4. So once again, let's see
if I could draw 1/4 here. Let me do this in a new color. So let's say that this block
right over here is a whole. We'll divide it into
four equal sections. So now I've divided
it into fourths. And let me copy and paste it
so I can use it multiple times. So copy. All right. Now, 3/4, that's
going to be-- we can assume-- I didn't
draw it perfectly. Actually, I could draw it a
little bit better than that just to make the four equal
sections actually look equal. So that looks like a
little bit better of a job. I'm trying to make them
four equal sections. Let me copy that one. So let me use it for later. Now, 3/4. This is four equal sections, and
3/4 represents three of them-- one, two, three. But now we're going
to multiply it by 4. So we're going to
have 3/4 four times. So we're going to need
some more wholes here. So let's throw in another whole. So this is one 3/4. Now let me do the next
3/4 in another color. So that's a 1/4, that's a
second 1/4, that's a third 1/4. That's another 3/4. And now let's do-- so we've
done two 3/4 just now. Let me make it clear. This is the first 3/4,
and then this plus this is the second 3/4. Now let's do a third 3/4. And we're going to have to use
another whole right over here. And I will do that
in this color. So my third 3/4,
so here's a 1/4, here's my second 1/4,
here's a third 1/4. So in green, I have another 3/4. And now we need four 3/4. So let's do that in a color I
have not used yet, maybe white. So that's a 1/4, that's two
1/4, and that is three 1/4. So notice, now I have now I have
one 3/4, two 3/4, three 3/4, and four 3/4. And what did I do when
I got those four 3/4? Well, it's pretty clear. This is turned into 3 wholes. So this is equal to 3 wholes. Well, if 3/4 times
4 is equal to 3, that means that 3 divided
by 4 is equal to 3/4. So the same idea again. 3 over 4 is the same
thing as 3 divided by 4. And in general, this is true. The fraction symbol here can
be interpreted as division. And looking at this
diagram right here, it made complete sense. If you started with
3 wholes, and you want to divide it into 4 equal
groups, one group, two groups, three groups, four
groups, each group is going to have 3/4 in it.