Okay, for all you lactose
intolerant people, this may
not be a great lesson, but here we go.
If 16 people
try to share an 8 piece
pizza- so here we go.
Here's your 8 piece pizza, okay?
How many pieces
does each person get?
Assume everybody wants the
same amount of pizza. Okay,
so you have 16 people,
you only have 8 slices. What
are you going to have to
do with each slice? Well,
you're going to have cut each
slice. You're going to have to cut each
slice in half, so that you get 16 slices.
So, what we're dealing with
now, is we're dealing with
fractions. Ah, I knew, that's
why I didn't want to tell-
give it up before. Okay, so
we're dealing fractions.
So, let's take a look,
more closely at fractions.
Okay, so here we go. A
fraction is actually called a
rational number, okay? Does that mean
there are irrational numbers? Oh, maybe.
But, a fraction is actually called a
rational number. And, the definition is
"a fraction
is in the form a over b- that's what
makes it a fraction- where a and
b are integers." Let me explain, okay.
Remember the integers,
that's the z, the integers
are the negatives,
0, the positives.
So, you don't really have a fraction if
you have a decimal on the top. So, this
is not really a proper fraction. That's
not really a fraction yet. That's
not a rational number. You
have to get rid of the decimal
point. We'll talk more about
decimals later. Okay.
So, you can't have
decimals within fractions,
otherwise they're not
really fractions. Okay.
So, a over b, where a and b are integers,
and b cannot be 0. Let's talk about
what that says. First
of all, the top
number of a fraction
is called a numerator.
The bottom number of
the fraction is called
the denominator, okay?
So, this is saying
the denominator can't be 0. Why not? Well,
let's recall something we did earlier.
If my denominator is 0, it means we're
dividing by 0. And remember, we cannot
divide by 0, because we have something
that's undefined. Okay? Because,
nothing times 0 will give me that number.
So remember, all we have is a fraction,
that's a rational number.
And, rational numbers
are fractions. Or, any
number that can be
expressed as a fraction.
Any number that can be expressed
as a fraction. Its symbol is a q.
That means integers, if you think
about it, can be written like
fractions. Because, if you have
the number 2, I can write 2
as 2 over 1. So, every integer
can be written as a fraction.
Okay? Every whole number can
be written as a fraction.
Every natural number can be
written as a fraction. Let me
show you by taking a look over here,
and show you how they're all related.
Our numbers just keep
growing, and growing, and
growing. Again, remember,
we have natural numbers.
And, natural numbers were
the 1, the 2, the 3.
I'm going to get so annoying,
you're never going to
forget this. Have I achieved
that already? Okay.
Then we have the whole numbers.
And, I had to add a 0.
Then I had the integers. The integers
add all of the negatives. Okay? Not the
negative decimals or fractions,
but all of the negative
numbers. Okay, whole numbers
and their opposites, remember.
Okay, so let me just give you
1 integer in there, okay?
Now, we have a new set of numbers
we just talked about. Oh my gosh.
This is the big mama. The big mama.
Okay, these are the rational numbers.
The rational numbers are your fractions,
or any number that can be written as a
fraction. Notice, all of these
numbers, if I take all of these off,
all of the natural, all of the whole,
all of the integers, they're all
in the rational numbers,
because they can all be
written like a fraction.
Like, negative 2 over
1. 0 over 1. 2 over 1. 1 over 1.
Okay? So they can
all be written as a fraction.
Okay. But,
not all rational numbers can
be written as integers.
For instance, the number 3
fifths is not an integer.
The number 3 fifths is not a whole number.
The number 3 fifths
is not a natural number. but, if
you think about the number 3 over
3, that's a fraction. But 3 over 3 is
1, so that's a rational number that's
integers, whole, and natural.
Because it's
really 1, it's really
this number right here.
So any fraction that reduces to
something that's natural, whole,
or an integer, or in one
of these other sets
of numbers. And, that's
how they're related.
Let's take a closer look at fractions
as we look at your workbook.
Which will show you these colored parts.
Okay, so the first one you see is that
1 fourth. It says "1
fourth is shaded." Okay?
And, what they have is
a little square here.
And, they're using
the whole square as
1. I'm sorry, they broke
up the whole square
into fourths. These
are each 1 fourth.
But they're saying 1
fourth is shaded. Okay.
So that's 1 fourth.
And what this means
is we have 4 equal parts.
4 equal
parts. And, 1 is how
many is shaded.
Okay. So the bottom, or the
denominator always talks about
how many equal parts your whole is broken
up into. Alright, this is kind of cool.
Here's my 1, okay? And again,
1 could be represented
by a big circle. 1
could be represented
by a big square, but
my 1 is just this bar.
When you break up the 1
into different pieces,
12 pieces, you're
actually going to get
all of the pieces. Think about this.
I think it's important
for you to see and visualize
fractions, because it's
a lot easier for me to talk
about them if you kind of get a
feeling for what they look like, okay?
If I break it up into
12 pieces, those pieces are going
to be really, really small. If
I break it up into 2 pieces,
those pieces are not going to be
that small, okay? Same thing, as we break
it up into more and more pieces, fourths,
that means 4 equal pieces.
Sixths means 6
equal pieces. Twelfths
means 12 equal pieces.
So, you break it up into more
and more pieces, and those
pieces actually get smaller,
which makes sense. Which is why
you don't want to split
a pie with 16 people.
You only want to split a
pie with 2 people, so
you get lots of pie. Okay.
Let's take a look at the next
thing in your work book. The
next thing in your work book
asks to compare the
following fractions to
1. So, it's 1 half,
2 thirds, 1 fourth,
5 sixths. So, I'm going to
do that by taking this 1,
and then I'm going to use the 1 half.
This is 1 half.
Well, how does that compare to 1.
Well, 1 half
is obviously less than 1.
Well, that makes
perfect sense. 1 half is less than 1.
Now, let's do 2
thirds. Oh, you're thinking I don't
have a thirds piece, but I do.
It's okay. I'll show you these
are third pieces by putting them
together. If these really
are third pieces that
means 3 of them, right,
would make a whole.
There's my thirds, but I want 2 thirds,
so I only want 2 of them. So, 2 thirds
is less than 1. You're
thinking "I know this."
I know, I know, just
watch the show for
a minute. Okay, now let's do 1 fourth.
Well these are 4
fourths. 1 fourth is
obviously smaller than 1.
It's less than 1. Now
let's do 5 sixths.
Ah, not worried are you,
because you see those sixths
pieces, right? Alright.
So, let's do 5 sixths.
Let's get this 1
fourth out of here.
3,
4, 5. There's your 5 sixths.
And, obviously 5
sixths is less than 1. Okay, so
that's how they compare to 1, okay?
Let's look at the top
number, let's look at the
bottom number. The bottom
number represents,
remember, how many equal many
pieces- it's okay, I'm good,
your whole is broken up into. Good thing
that's not a human, okay? How many equal
pieces. So, 1 half means you
broke it up into 2 equal
pieces. The top tells you
how many of those pieces
you want, want to shade, or just want.
Okay, 2 thirds means you're broken up into
3 equal pieces. And, this is what it says
in your work book, "the bottom is how many
equal pieces you're broken up
into." But, the top tells me
how many pieces I want.
Same thing all the way
down, okay? Alright, so
let's take a look at
another example (as soon
as I get my piece.)
Now I'm going to draw these.
Now, you should
know how good of a drawer I am by now.
I'm going to give
this my best shot. I'm just
going to move these over
to this side of the board.
Cranky, these sixths
pieces are cranky. Okay.
Here we go. The first
1 is 2 thirds.
Okay, so I'm going to draw 2 thirds.
Now, I could actually draw anything I
want, but you've seen me draw. So, I'm
probably just going to use a piece like
that, a little rectangle.
Now, please let me
explain this. This is not
going to look as good as
it should. When I draw this, let's think
about how many pieces I should draw it in.
I should draw it in 3 equal
pieces, because that's
what the bottom tells me.
But, here's the problem.
Did you hear that word equal? I have
to make these 3 equal pieces. Okay. So
I'm going to do my best, and
then we're going to pretend
that those are equal.
So, those are 3 equal
pieces. Okay, if you want
to see 3 equal pieces,
I'll show you 3 equal pieces.
Here's 3 equal pieces.
Alright, there's 3 equal pieces.
Okay. Here's 3 equal pieces.
I want 2 of them.
2 thirds. So there's my
drawing of 2 thirds.
So we're kind of
pretending they're
equal. The next 1 is 5 fifths.
Well, let's talk about 5 fifths.
5 fifths means I have
to break this up into
5 equal pieces. Look,
I can't even draw the
rectangle straight. How could I
possibly draw it into equal pieces?
Let's try that again. 5 fifths.
Oh yeah, so much better.
Oh, we're in trouble.
5 fifths may be bigger than me.
5 fifths may be more than
I could really do. Here we go. I can
do it, I feel it. Alright, drum roll.
Okay, there's 5 equal pieces.
Yes, they are incredibly equal.
Alright, that's the
beauty of the pen. Okay.
And, you want 5 equal pieces, 5 fifths.
You want 5 of them, which is the whole
thing, okay. Which is why, think about
this. 5 fifths really equals 1.
Okay? So, we're all good.
Now we're going to
do 7 fourths. Oh,
that's just going to be
so much fun. I'm already sweating.
7 fourths.
Remember that math anxiety
in the first lesson
when you learned about that 0?
Okay, well
I'm having a little math anxiety
worrying about these 7 fourths.
But I can do it, I'm feeling good now.
I'm feeling good.
Let's do 7 fourths. Okay, so
I'm going to take this 1,
I'm going to break it up into fourths.
That's not bad, that's not bad.
Not bad. Okay, those are fourths,
but I only have 4 of them.
You know, don't make fun
unless you can do it.
Oh look. Some people
aren't blessed with the skill
of being an artist. Okay.
So, I need- those are fourths.
They're supposed to be all the same
size, but that's not bad. But, I need
how many more? If I already have
4, I need 5, 6, 7.
There is 7 fourths.
So, here's the point. Let's see what
the point is. This is greater than
1. This is less than 1.
This is equal to 1. So, we
should be able to tell
what fractions are
greater than 1, less
than 1, or equal to 1,
by looking at their
numerator and denominator.
Remember the top number
is the numerator,
and the bottom number is the denominator.
When the top number and the
bottom number are equal, that's when
they will be equal to 1, okay? When the
top number is bigger than the bottom
number, okay, that's when it's greater
than 1. And, when the top number,
which is numerator, is less than
the denominator, that's when it's less
than 1. Notice, numerator less than
than the denominator, less than 1.
Numerator
less than denominator,
less than 1.
That's how you know. And, this
is called an improper fraction,
when it's greater than 1. When
it's less than 1, it's called a
proper fraction.
When it's equal to 1, it
is also called an improper
fraction.
Okay, so that's what we know.
And all of that stuff
is in your workbook, so
don't worry about that.
Okay, in the next set of examples,
we're going to state the numerator and
denominator of each fraction, state
whether they're proper or improper, and
compare it to 1. Okay, so you
saw the little manipulatives
that I dropped all over the
floor, but that's cool.
Now, we're going to just
summarize this thing.
The numerator remembers
the top number,
(let's try this again).
The numerator is the top
number, the denominator
is the bottom number.
And, we want to know if it's proper,
or improper, and how it compares to 1.
Well, because the numerator is less
than the denominator, the numerator
is less than the denominator,
this is a proper fraction.
And, it's less
than 1. Okay? So now
let's look at 5 fifths.
Well, this is an easy question. The
numerator is 5, the denominator is 5.
And, the numerator equals the denominator.
The numerator equals the
denominator. So, that makes
that an improper fraction,
because it's equal to 1.
And, you knew it was equal to 1, right?
That whole division rule. Okay.
Numerator, denominator. The numerator
here is 7. The denominator is
4. And, I remember denominator, if you've
confused them, denominator, "d for down."
Denominator is down,
"d for down." And here
the numerator is greater than
the denominator, so this
fraction is greater than 1.
And, you
saw that, right? So this
1 is, the numerator
is greater than the
denominator, it's greater
than 1, so it is an
improper fraction.
So, remember
that improper fractions are
greater than 1. And, that
proper fractions are less than 1.
Well, let's do something
we've already really done
before, just to review it.
Okay. Because I know you think
and eat math, you know,
think eat and sleep math,
so you remember everything.
But, just in case
you don't...okay.
Here we go, number 1. 3 over 1.
Remember a fraction line is really
just a division line. 3 divided by 1,
we have. We have negative
3 divided by negative 3.
We have 0 divided by
3. And, we have
finally, 3 divided by
0. 3 divided by 1 is 3.
I can check that if I want.
And also notice, the
numerator is greater than
the denominator, so that is
called an improper fraction.
The answer is 3. This is 1.
This is 0, and this of course
is undefined. 0 is under,
so it's undefined. And remember why, by
that's a bad trick I guess. It helps me
remember, but I have to
make sure you understand
why. There's no number
times 0 that will
give me 3. So, that's
why it's called
undefined. Okay? So
that's what we've already
done. Alright, now let's graph some
fractions. We're going to do this on
separate number lines.
Okay. The first one we're going to do-
now listen, you know what I've noticed
in the 150 years I've been
teaching (I know, I don't look
that old), but I've noticed
that people have a hard time
with a ruler. They don't
really know (yeah, that
was a ruler, those things
you measure with),
they don't really know how to measure
that, I mean how to read a ruler.
So, I know rulers don't come
in fifths, but I just want
to explain (they usually come
in sixteenths), but I just
want to explain this. I'm going
to start here at 0. Now,
I'm going to break this up into fifths.
So, I would have 1 fifth,
2 fifths, 3 fifths, I'm going to
keep going, I know I could be done,
4 fifths, 5 fifths- this is 5
fifths, but 5 fifths is really
1. So, 1, 2, 3, 4, 5, you need 5
spots before you get to
5, because you'll be at 5
fifths. So, this is 3
fifths right here. I
just want to make sure
you understand that,
that you'd have 1 fifth,
2 fifths, 3 fifths,
4 fifths, 5 fifths
broken up into 5 equal
places. Let's look at negative 2
thirds. And, notice again, what this
proves, is that's a proper fraction, it is
smaller than 1. Because, here's
1, here's 1. And, you're 3 fifths
is less than 1. Let's
do negative 2 thirds.
Well, negative 2 thirds is
obviously on the negative side,
or the left side. And, this
is broken up into thirds.
So, this is negative 1
third, negative 2 thirds,
and this would be negative 3 thirds.
Negative 3
thirds, or negative 1, okay? And
that's where you are right here.
Again, to show you there's 3 spots,
because you're dividing into 3 places.
Okay, and finally 1 half.
Here's 0. Here's 1 half.
Well, the next 1 would be 2
over 2, which would be 1.
Okay? Because, only 2
spots, you need, you get
to 1, because you break it up into 2.
So, there's your 1 half.
Okay. Before I actually
do this next problem that's in your notes,
I want to just ask you something. If
you have a whole orange,
throw me an orange.
If you have a whole
orange, okay? And you
have a half an orange,
okay, I can't really cut this.
So, if you
have a half an orange, I think
this might be a tangerine, but who
have a half an orange. If you have
a whole orange, and you have a
half an orange- I think this might
be a tangerine, but who cares?
Okay, you have a whole orange, you have a
half an orange, what do you really have?
Well, you would really have 1 and a half
oranges. Let's talk about that. So,
we have
1 plus a half is 1 and a half.
Okay?
Now notice, in algebra, if
we have 2x, I don't need
to put a symbol in
there, and I know it's
attached by multiplication.
However, There's no
symbol here, it's not
attached by multiplication,
it's actually attached by addition.
That
little rule only works with variables.
This is actually attached by addition.
1 plus a half is 1 and a half.
And, this is called
a mixed number. We're going to get
to this right now, as I do the
next problem. But, a mixed number has
2 parts. It has a whole number part,
right, and that's the 1.
And it's
got a fraction part.
And, that's the 1 half. So let's now
take a look at this next problem.
If I work 5 whole days, and
then 1 third of a day,
so I'm working 5 days.
Plus, I'm going to work
another 1 third of a day.
Okay? How many days have
I worked in total? Okay,
so let's take a look at this.
Well, I'm working 5 days, plus
1 third of a day, which
is just 5 and 1 third.
That's it. That's all you
have to do when you have
a whole number. It's a whole
number and a fraction
part. You just get back a mixed number.
So, let's just take a look at this
if a draw it. Draw it,
that sounded terrible, but
I'm going to say it anyway, draw it.
Here we go.
Alright, ready? I'm going
to draw it, because of this
third, I'm going to draw it into
thirds, watch. Remember they're equal.
Sorry. Okay.
Watch this. I know, it's going
to take a minute, hold on. 1, 2,
oh, that's supposed to
be the same size, 3.
It's supposed to be the
same size, bear with me.
4, supposed to be the
same size, bear with me.
5.
That's not even the same one, that's
really pushing that with you, isn't it?
Okay, so here's what I've
got for 5 and a third.
Alright, here we go, this is
incredible drawing. Okay.
So, 5 and a third. 1,
2, 3, 4, 5, I went
crazy. I went shading crazy.
Okay.
Now, let's count up the thirds. So, I have
1 third, 2 thirds, 3 thirds, 4 thirds, 5
thirds, 6 thirds, 7 thirds, 8 thirds, 9
thirds, 10 thirds, 11 thirds, 12 thirds,
13 thirds, 14 thirds, 15
thirds, 16 thirds. So, I know
5 and 1 third is the same as 16
thirds. Okay, so improper
fractions are greater than 1.
Okay, greater than
1, can be written
as
mixed number. So, let me just kind
of talk about that for a second.
And then, I'll show you how do to it.
A quick way, without
this big drawing, because
you know, you may not be as
artistic as I am. So, You ay not
want to do it. So, here's the deal.
The deal is, remember,
there are 2 types of
improper fractions. There
are improper fractions,
like this, 16 thirds,
where the numerator is
greater than the denominator.
And then, there are
improper fractions where the numerator is
equal to the denominator. Well, that's 1.
So, that's not a mixed number.
So, you cannot change
these improper fractions to
mixed numbers. They just
become who numbers. But,
you can change these,
okay? So, improper
fractions greater than 1.
So, this is the type
that's greater than 1.
We already know that
can be changed to make
these numbers. Alright,
let me show you how to
do it. It is so simple,
you're going to love it.
Alright. Now, changing
a improper fraction to a
mixed number is not the same
as reducing it. It is just changing its
form. Okay? So here's how it works. If
I have 5 and 1 third (make
that 5 bigger), and I want
to change that, to that
16 thirds, without doing
all of that unnecessary
drawing, because I am
awful at it, here's what we do.
We multiply the denominator and
the whole number, and
we add the numerator,
and we put that over
the denominator.
So, watch. It goes over
the denominator. So,
3 is the denominator.
And then you just
multiply. 3 times 5 is 15, plus 1 is 16.
There's your 16
thirds, which is the same
answer you got when I
drew the whole thing out.
So, that's all you do.
You multiply the denominator
by the whole number,
add the numerator,
then just put it over
the denominator. Pretty easy. So,
you're going to get to try some.
Give it your best shot, and then
we'll look at the answers together.
Alright, let's see how you did on these.
Remember we said that
any improper fraction greater
than 1 can be written
as a mixed number. So, if
these are mixed numbers,
then we're going to get back
an improper fraction greater
than 1. So, let's take a look at it.
Okay. So we say we
multiply, 3 times 7 is 21, plus 2 is 23.
And, we write
that over the denominator. So, 23 thirds
is your answer there. Okay, now here,
it might be easier to put the
denominator down first, okay? 5 times 3
is 15, plus 4 is 19. So, every
time I do these I'm going to write
the denominator down first. Because, that
never changes. So, write the 9 down,
9 times 5 is 45, plus 1, is 46.
So, it's 46
ninths. Really, really, really easy. Let's
talk about what happens if you have a
negative mixed number.
Okay, so let's take a
look at that. You might
think that's a little
harder, when seriously, it's not.
Okay. So, remember, we said
that a whole number
plus a fraction, that
is a mixed number. So,
let's take a look at
these 2 examples at the same time.
Let's
take a look at 5 and 1 third
(back to him), and negative
5 and 1 third. Okay,
this actually came from
5 plus 1 third. Alright, we
talked about that earlier.
This actually comes from- watch-
negative 5, plus a negative
1 third, which is
actually negative 5
and 1 third. Because, think about
this, I owe you 5, and I owe-
but there's that double sign,
we can get rid of that
double sign, because we have been.
I will just do that
again. Okay, watch. I could write
it like this if I want to, right?
Sign, signs are different,
changes to a negative.
Because, this means I owe you 5.
I owe you
a third, so I owe you 5 and third. Okay,
so that's where you're going here.
So, it's very simple. Still
combining that whole number,
and that whole fraction, to get
your mixed number. What happens if
I want to go- well let's think about this.
What happens if I want to change
this. Just think about this
so we don't get confused
if I want to change this
to an improper fraction.
Well, all I have to do is
bring down the negative.
Don't disregard it,
just bring it down.
Don't do anything with it in problem.
Don't go 3 times
negative 5, I wouldn't do
that, just bring it down.
And, I go 3 times 5. Oh, hello.
I wouldn't write
that down either. In fact, let's
go back to writing you're 3 down.
3 times 5 is 15, plus 1 is 16. It's
just negative 16 thirds. Okay.
So, just bring it down. Don't
go 3 times negative 5, plus,
you're not going to get the
right answer there, okay?
Just bring down the negative, and deal
with this. So, 3 times 5 is 15, plus 1 is
16, over the 3. Okay, so
be careful there. It's
really very simple, but
unless it's explained to you
once, you may not understand how it goes.
Alright, what about going the other way.
Alright, let's
look at 13 fifths.
And, let's see if we draw
it, what that really means,
13 fifths. I'm drawing
again, we're in trouble.
Alright, oh, and it's fifths. You
saw how good I was with that
before, didn't you? You might be able to
realize what's already going to happen.
Oh, that's a beautiful
rectangle, isn't it? Okay.
Hey, that's not bad for me.
That's not bad. I could
be getting an award
soon. Okay, so we have 1 fifth,
2 fifths, 3 fifths, 4 fifths,
5 fifths, 6 fifths, 7
fifths, 8 fifths, 9 fifths, 10 fifths.
I need 3 more fifths, right?
But I need fifths, so they
have to be the same size.
Because, this size is a fifth.
So, that's
10, that's 11, 12, 13 fifths.
So that's
really 2 wholes (not holes like
hole in your shirt, you know).
2 wholes, and 3 fifths. So, that
really is equal to that.
So, 13 fifths is really
2 and 3 fifths. You might have
figured that out already,
because you might have said
"well, I need 2 rows of
5, and then there will
be 3 left over." So,
watch how you do this.
Let's see how you make
this change. Here's how
you make the change.
Remember, this is a division sign.
All you
do, and I like to do it just
like that, to make sure
you understand what you're dividing.
I'm doing this.
I'm putting the 5 into 13.
I'm going 5 goes into 13
twice. That would make 10,
with a remainder of 3.
So, now we're going to convert
this to my answer. It's 2
wholes, 3 left over, always over
your same denominator. So, it's
2 and 3 fifths. So, that's how you
convert from this improper fraction
to a mixed number. You just use division,
okay? So, let's look at some examples.
I'll do this next one with
you, and then you can do
the 2 that are left. So,
remember we're dividing.
Again, we might want to get
in the habit of putting down
a denominator first. That might make
it easier. So, let's take a look at 17
fifths. Let's make sure there are no
negatives here, creeping up on the board.
Okay.
Change it to a mixed number.
Okay.
5 goes into 17 3 times. That's
15, and we get a remainder of 2.
So, here's what I know. I know
my denominator is going to be 5.
My whole number part is 3. My
remainder is 2. So, 17 fifths is 3
and 2 fifths. I could check myself.
Watch me check myself. 5
times 3 is 15, plus 2 is 17.
17 fifths. So, it might be a
good idea to check. Okay,
so if you'd work on the other
2, and then we'll look
at it together.
Alright, let's see how you did. Here
we go, we're going to divide here.
Well, okay. 4 into 23.
4 goes into 23
5 times. 4 times 5 is 20.
We subtract.
We get a remainder of 3. So, remember
how this goes. The denominator is
always going to stay the
same, no matter which way we
can, improper fraction to
mixed number, or vice versa.
And now, the whole number is 5,
remainder 3. So, it's 5 and 3 fourths.
We can check ourselves. 4
times is 20, plus 3 is 23
fourths. So, that works out.
62 over 3. Okay.
We get 20. 3 times 20- we did a
little short division there. 3 times
20 is 60. We get a
remainder of 2.
Okay? So remember, the
denominator stays the same.
20 is our whole number,
the remainder goes
into the numerator. So,
it's 20 and 2 thirds.
And, we can check ourselves. 3
times 20 is 60 plus 2 is 62
over 3. Awesome, but remember, this is
just changing its form. That's what
it's called, "changing its form" from
an improper fraction to a mixed number.
Alright. Let's take a look at how we can
possibly write a negative fraction.
So, let's take a look at
this fraction, this fraction,
this fraction-
I think I skipped 1, but
I'll put it here. You know
what, let me put it in the
same order you have it in.
Okay, I'm going to put this one,
this fraction, they almost look
alike. But,
okay, the biggest problem
have with negative fractions
is they don't know where to put the
negative. They're like, "alright,
I'll write the fraction. You want
me to make it negative? Uh...
okay, I'll put the negative
right here, because I don't
really know where to put it."
They line it up. Now, you can
do that. But, you need to understand that
it doesn’t matter. Let's take a look.
This fraction is clearly positive, okay?
That's 1 half. That's
clearly positive, I can see it as
positive. This fraction is clearly
negative, because that's a negative
right in front of it. Right by
the line. Let me line it up.
Okay. Now,
the other three issues. Let's take a
look at this one. I don't actually have
to divide, but I know
that if I did divide,
a negative divided by
a negative would be a
positive. So, I should
never see a fraction that
looks like this. Because,
when I simplify it,
that's going to be a
positive, because a negative
divided by a negative is a positive.
I don't
have to actually divide.
And, if I did divide
this, remember, it's a
number smaller than 1.
So, I'm going to get a decimal
number out of it. Okay, so these
are both positive. That one is positive,
this one is positive, and this
one is positive. Now, let's go to the
negative sides. We know this is negative.
Now let's look at this one.
Again, we're not going to
divide any of these. We don't
the decimal, we want the
fraction. Okay, so this one has a
negative in the numerator, but no
sign in the denominator. If
there's no sign, it means
it's positive, so that's
a negative divided by
a positive, which makes it a negative. So
again, I don't have to actually divide.
I just know that
fraction is negative.
Here, a positive divided by a
negative, that's also going to be
negative. So, here's
what you have to know.
If you have a negative sign, if
you have a negative sign, in only
1 place, either in the middle, or only on
the top, or only on the bottom, that's
still a negative fraction.
These are
equivalent fractions. I
should never see that,
that should always be changed to positive
1 half. Okay? Actually, but when I
write it in my answer, they
wanted me to group my like
fractions together. So, these
are both positive, but I
never should see this. Okay,
these are all negative,
and it doesn't matter what you see.
Okay?
Sometimes this is a little
easy to work with, we'll
see later on. But, it doesn't
matter what you see.
Alright, well just like
before, we talked absolute
value with whole numbers
and natural numbers.
We can talk about absolute
value with fractions.
And, it's the same kind of thing.
Absolute value is
the distance you are from 0 on the
number line, the number of units.
So, let's take a look at these 2
next questions. It says evaluate,
that means we're going
to get a number answer,
remember. So, let's
look at number 1.
Okay, 7 eighths.
The absolute value of 7 eighths.
So, what
they're asking is if I'm
on the number line,
how many units is
7 eighths from 0.
Well, 7 eighths, that's it.
It's 7 eighths units.
Here, absolute value,
distance from 0?
If I'm at negative 2 thirds,
how many units am I from
0? 2 thirds, I'm just in
the negative direction.
So, I'm 2 thirds units in
the negative direction.
You do the same thing we
did with whole numbers.
Okay? And, that takes care of that. The
next time we see you, we're going to be
reducing fractions.
