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CCSS Math: 4.NF.B.3

Write 5 and 1/4 as an
improper fraction. An improper fraction is just
a pure fraction where the numerator is greater than
the denominator. This right here, it's
not a pure fraction. We have a whole number mixed
with a fraction, so we call this a mixed number. So let's think about what 5 and
1/4 represents, and let me rewrite it. So if we're talking about 5 and
1/4, and you can literally think of this as 5 and 1/4 or 5
plus 1/4, that's what 5 and 1/4 represents. So let's think about 5. Five is 5 wholes, or if you're
thinking of pie, we could draw literally five pies. Let me just cut up the pies
from the get go into four pieces since we're dealing
with fourths. So let me just cut up the
pies right over here. So that's one pie right
over there. Let me copy and paste this. Copy and paste. So I have two pies, and then I
have three pies, and then I have four pies, and then
I have five pies. So this is what the
5 represents. 5 literally represents--
so let me circle all of this together. That is the 5 part
right there. That is what 5 literally
represents. It represents five whole pies. Now, I have cut up the pies into
four pieces, so you can imagine each piece represents
a fourth. Now, how many pieces do I
have in these five pies? Well, I have four
pieces per pie. Let me just right it here. 4 pieces per pie times 5 pies
is equal to 20 pieces. Or another way to think of it,
since each piece is a fourth, this is also equal to 20 times
1/4, or you could just write this as being equal to 20/4. So we have 5 whole pies is
equal to 20 fourths. Let me write it like that. 20 fourths. Or we could write it as 20/4. I've kind of done the
same thing twice. So that's what the five
pies represent. 20/4 or 20 pieces, where
each piece is 1/4. Now, the 1/4 right here
represents literally one more fourth of a pie or one more
piece of a pie, so let me draw another pie here. So that is another pie. Cut it into four pieces. But this 1/4 only represents
one of these pieces, right? This is one of the
four pieces. The denominator tells
us how many pieces. The 1 tells us how many of those
pieces we're dealing with, so it's just this
one piece over here. That right there is the 1/4. Now, if we write 5 and 1/4,
we just saw that the 5 right here is 20/4. So we could rewrite this. Let me write it like this. 5 and 1/4 can be rewritten as
the same thing as 5 plus 1/4, which is the same thing as--
we just saw that five whole pies is the same
thing as 20/4. And to see that these are the
same thing, you literally just divide 4 into 20. You get 5, and nothing
is left over. So 5 is the same thing as 20/4,
and then this plus 1/4 is the same thing as plus 1/4. So if I have 20 fourths and I
add one more fourth to it, how many fourths do I have? Well, I have 21. I have 21 fourths. Or another way of thinking about
it, this 5 is-- so this right here is 20
pieces of pie. You can even count it. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20. But a quicker way is to say,
well, we have five pies. Each of them have four pieces. 5 times 4 is 20. This 1/4 right here represents
one piece plus one piece, so total we're going to
have 21 pieces. So we have 21 pieces, where each
piece is 1/4, so we could say we have 21 times 1/4 or
21 fourths pieces of pie. However you want to
think of it, but we've solved the problem. We're at an improper fraction. We've written 5 and 1/4 as
an improper fraction. Now, I've gone through great
pains to give you the intuition of what 5 and 1/4
means, but there is a fairly straightforward process for
getting straight to the improper fraction. Let me color code it. So if you have 5 and 1 over 4,
to convert it into an improper fraction, you're going to keep
the same denominator, so you're going to have
the over 4 there. But your numerator is going to
be your numerator of the fraction part before. So it's going to be 1 plus your
whole number times your denominator. So 1 plus-- or actually, let
me do it the way I tend to think of it. What I do is I take 4 times 5. So let me write that down and
I want to color code it. 4 times 5, and then to that,
I add this numerator. So I literally do 4 times 5 plus
1, which is-- so this is equal to 4 times 5 is 20, plus
1 is 21, and then that's over 4, so it's 21/4. And all of this is kind of
a fast way to do it. We're literally doing the exact
same thing that we did here in kind of a slower way. We're saying, OK, 5 wholes is
the same thing as 20 fourths, so you take 5, and I figure that
out, 5 times 4, and then I have one more fourth there, so
4 times 5 plus 1 gives 21.