>> I'm going to start with vocabulary.
What's a proper fraction?
A proper fraction is a fraction
where the numerator is smaller than the denominator.
So here's some examples: 3/5, 1/2, 5/8.
The numerator is smaller and each
of these have a value less than 1.
For instance, 3/5, if you took a pie and broke it
up into five pieces and shaded three of the pieces,
you could see it's less than a whole pie.
All right, what's an improper fraction?
It's just the opposite.
It's a fraction where the numerator is not smaller
than the denominator.
So that means either the numerator is bigger
or it could be the same.
So here's some examples of improper fractions:
5 over 3, 7 over 2, 4 over 4.
Notice that the numerator
and denominator can be exactly the same number.
Proper fractions have a greater --
have a value greater than or equal to 1.
All right, now, remember when we first started off
with fractions, 5 over 3,
there's two ways we could think about fractions.
One is, how many parts out of the whole,
so the numerator's how many parts,
the denominator says how many parts make up a whole.
So let's do it that way.
If three parts make up a whole, then let's say I'm making
up a pie into three parts, and I want five parts shaded --
1, 2, 3, 4, 5 -- then I can see what I really have here is one
whole pie from this pie where all three parts are shaded,
and then I've got two out of the three parts -- whoops --
two out of the three parts shaded, so this is 2/3,
so what I have is 1 and 2/3.
1 plus 2/3.
Now, the way we write a whole number added to a fraction --
usually it's going to be a proper fraction --
is you just put the numbers together and read it as 1
and 2/3, instead of 1 plus 2/3;
so the way we say this is "one and two-thirds."
5/3 is the improper fraction where the numerator happens
to be bigger than the denominator,
and written in this way, it's called a "mixed number,"
because you're mixing a whole number with a fraction.
So 5/3 is the improper fraction
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and 1 and 2/3 is the mixed number.
So that's an example of how
to change an improper fraction to a mixed number.
I could just draw a picture, right.
You can look at the denominator,
that tells you how many parts make up a whole,
then you shade however many of the numerator is.
So notice you might have to make a lot of pies to do that.
All right.
Now, the other way, remember,
to think about fractions is you could just take the numerator
and divide it by the denominator --
that's another definition for a fraction -- 5 divided by 3.
Now, what would that look like?
Well, I'd go 3 into 5 is 1, and that's 3, you subtract,
and then I have a remainder of 2.
So this is the way you would normally write this before you
ever learned anything about fractions.
But what we want to do is be a little bit more precise here.
If we have a remainder of 2, that means we have two parts
out of three still, so here's how I write my answer.
I do 5 divided 3, I write the whole number part down,
and then the remainder goes in the numerator,
and I write the same denominator.
So I could either draw a picture for 5/3 to get 1 and 2/3;
or 5 over 3 -- 5/3 means 5 divided by 3, which is 1,
and then I put the remainder over what I would divide it by,
which was the 3, okay.
So in both the improper fraction and the mixed number,
the denominator stays the same.
Any time you have an improper fraction, it could be written
as a whole number or as a mixed number.
So let's say you have 15/3, remember that's just 15 divided
by 3, so the answer is 5.
So I wrote this improper fraction as a whole number.
So if when you divide the numerator divided
by the denominator and there's no remainder,
you'll end up with just whole number; and, of course,
you could have also done this with a picture.
You could have said, well, I've got to shade 15/3,
so I'd have to break pies into thirds, right,
until I could shade 15 pieces, and if I shaded all of those --
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 --
I end up with five whole pies, another way to see why 15
over 3 is 5 in terms of fractions and what they mean.
How about if I were asked to change 5/3 --
I'm sorry, 5/2 to a mixed number.
If we want to think of it in terms of division,
and remember you don't have to write this down, 5 divided by 2,
but I'll just reminding you -- well, if you do 5 divided by 2,
how many times does 2 go into 5?
Twice, and the remainder is 1.
Now, if you need to get out
and do your division using a long division,
that's no problem, you could do that.
The remainder's 1, and then you --
whatever you were dividing by, which means whatever was
in the denominator in the first place stays
in the denominator, so that's 2 and 1/2.
And, again, this works with a picture as well.
What you're saying is, you're going
to have two parts to a whole, right?
Two pieces make up a whole, and if I were going to shade five
of those parts -- so I have 1, 2, 3, 4, 5 of them --
you can see that I've got two whole parts
and here is one half.
Now, remember, you read this 2 and 1/2
and it does not mean 2 times 1/2, it means 2 plus 1/2.
We just omit writing the plus sign
between the whole number and the fraction.
Here are two for you to try.
Here's some improper fractions.
Write those either as whole numbers or mixed numbers.
Put the video on pause.
Okay, let's do it.
25 divided by 7.
Well, 7 goes into 25 three times, right,
because if you went over here and needed
to do this, that's okay.
Goes in 3 times, and then I've got a remainder of 4.
The remainder is what goes in the numerator,
and the denominator stays the same, so there's our answer.
3 and 4/7, and you could do this with a picture if you'd like.
And 9/9, huh, that's 9 divided by 9, that's 1.
Hopefully you didn't have any trouble with either of those.
Next, how do you change a mixed number to an improper fraction?
So let's say you were given 2 and 1/3, how I'm going to write
that is just an improper fraction.
Well, it's going to be the same denominator, okay,
but let's look at the picture here.
I want 2 and 1/3.
All right, so I'm going to have two whole pies, right,
and 1/3 of the pie is shaded.
So what we want to do is take all these pies and put them
into thirds, so -- because we know this is 2 here,
and let's see, I know I'm shading all of this, right,
and then I also am only shading one piece
over here, that's my 1/3, okay.
So then how many thirds did I end up shading?
Well, count them.
1, 2, 3 for that one pie, and then three more
for the other pie -- 6.
That was 6/3 so far plus this extra piece.
You know an easy way to get that 6 that's pretty cool?
You know there's 3/3, right -- since the denominator's 3 --
three pieces for each of the whole pies,
so 3 for the first pie, 3 for the second, 3 times 2,
take the denominator times 2, that's 6 so far,
plus the extra piece here in the other --
that's shaded in the third pie, so 3 times 2 plus 1 is 7/3 --
so you can look at the picture --
the easy way to get it is you go the denominator 3 times 2
and add 1, that will be the new numerator.
So 2 and 1/3 is 7/3.
All right, let's write 5 and 3/4 as an improper fraction.
All right, the easy way is we know it's going to end up with 4
in the denominator, we need the numerator,
so we multiply the 4 times the 5, okay, 4 times 5,
and we're going to add the numerator.
So what's that going to be?
20 plus 3, so this is 23/4.
So that's the easy way to get the answer.
If you forget that, remember you could draw a little picture;
you've got five whole parts and then 3/4, all right.
So I'm shading 3/4 here, but I'm shading all of these, right,
the whole thing is going to be shaded,
but I want to see how many fourths in there, right.
Can you count them up?
We've got 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20.
You'd shade 20 of those fourths plus three more, so you're going
to have 23 if you counted them up; but the easy way, again,
you've got four in each pie, and you've got five pies,
so you know you really are just going to do 4 times 5 plus the 3
that are over there to get 23, and that's how
to change a mixed number to an improper fraction.
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