www.mathcentre.ac.uk/.../Fractions%20-%20basic%20ideas.mp4

This tutorial is about the basic
concepts of fractions.

What they are, what they look
like, and why we have them.

A function is a way of writing
part of a whole.

And it's formed when we divide a
whole into an equal number of

pieces. Now let's have a look.
I've got a representation here.

Of a whole.

And let's say we want to divide
it into 4 equal pieces.

So there we've
taken 1 hole

and divided it
into 4 equal

pieces. So each
piece represents 1/4.

Wow.

I've now taken 1/4 away.

Now I've removed

two quarters. If
I take a third.

That's 3/4.
And if I take the false so I've

now got all four pieces.

I've taken all of them for
quarters, which is exactly the

same as taking the whole.

Let's just return for a moment
to the two quarters.

Now
two

quarters. Is exactly

the same. As if I'd started
with my whole and actually

divided it into.

2 pieces of equal size.

And you can see that that's

exactly the same. As two
quarters so I can write two

quarters. As one
half.

Let's have a look at
another illustration now.

Here I have a bar of chocolate.

It's been divided.

Into six pieces of equal size.
So we've taken a whole bar and

divide it into six pieces.

So each piece is
16.

Now, let's say I'm going to
share my bar of chocolate with

the camera man.

So I want to divide the bar
of chocolate into two pieces.

So if I do that.

Where each going to have one

236. So 1/2 is
exactly the same as 36.

But there's not just one
cameraman. We've got two

cameramen, so I need to share
it. Actually, between three of

us. So now if I put my bar back
together and I need to share it

between 3:00. Where
each going to get.

Two pieces.

So 1/3 is exactly the
same as 26.

But Let's say I
want to eat all my chocolate bar

myself, so I'm going to have all
six pieces, so they're all mine.

Not going to share them, so I
take all six pieces.

And I've taken away the whole

bar.

So.
Fractions we can look at.

In two ways.

We can look at it as the number

of pieces. That we've used.

Divided by the
number of pieces.

That make a whole.

Oh

As the whole.

Divided by. Number of pieces
or number of people that we've

divided it into.

So here we have a whole bar
divided into 6 pieces.

Here we have the number of
pieces that we've taken divided

note 5 the number of pieces that
make up the whole bar.

Let's have a look
at some other fractions.

Let's say
we have

3/8.

So we've divided a whole up into
8 pieces of equal size.

And we've taken three of them.

3/8

We could have 11 twelfths.

So we've divided a whole up into
12 pieces and taking eleven of

them. We could have
7/10.

Here we will have divided a hole
into 10 pieces of equal size and

taken Seven of them.

And we can have.

Any numbers in our fraction so
we could have 105 hundreds or

three 167th and so on.

Now we've looked at representing

fractions. Using piece of Cod
circular representation are

rectangle with our bar of

chocolate. Let's have a look at
one more before we move on and

let's let's see it on.

A section of number line.

So let's say we have zero here.

And one here.

So let's look at what 3/8 might

look like. While I need to
divide my section into 8 pieces

of equal size.

Now obviously this is an
illustration, so I'm not

actually getting my router
out to make sure I've got

equal size pieces.

But hopefully. That's about
right. So we've got 12345678

pieces of equal size and I'm
going to take three of them.

So if I take 1, two

3/8. That's where my 3
eights will be.

Let's have a look at
another one.

This time will look at

11 twelfths. So we need to
divide our line up into.

Pieces so we have 12 pieces
of equal size.

OK, so we wanted
eleven of them, so

we need to count
11 one 23456789 ten

11. So at 11

twelfths. Is represented there.

Let's look more
closely at our

fraction half.

Now we've already seen that half
is exactly the same as two

quarters. And it's exactly the
same as 36.

Well, it's also the same
as 4 eighths 5/10.

2040 deaths

9900 and 98th and so on. We
could go on.

And what we have
here is actually equivalent

fractions. Each one of these
fractions are equivalent at the

same as each other.

Now, this form of the fraction

half. Is our fraction in its
lowest form, and often we need

to write fractions in their

lowest form. It's much easier to
visualize them actually in this

lowest form than it is in any

other form. So we often want to
find the lowest form.

Well, let's have a look 1st at
finding some other equivalent

fractions. So let's say I take

3/4. How do I find an
equivalent fraction? Well, what

I can do is multiply the top
number and the bottom number.

By the same number.

So let's say I multiply by two.

If I multiply the top number by
two, I must also multiply the

bottom number by two so that I'm
not changing the fraction.

3 * 2 six 4 * 2

is 8. So 6 eighths
is a fraction equivalent to 3/4.

Let's try another one.

This time, let's take our 3/4.

And multiply it by three. The
top numbers multiplied by three,

so most the bottom number B3
threes and 9 three force or 12,

so nine twelfths is equivalent
to 6 eighths, and they're both

equivalent to 3/4.

Let's do one more this time.
Let's multiply both the top

number on the bottom number by
10. So we have 3 * 10.

Giving us 30.

And 4 * 10 giving us
40. So another fraction

equivalent to 3/4 is 3040
deaths.

So it's very easy to find
equivalent fractions as long as

you multiply the top number on
the bottom number by the same

number. Now we have some
mathematical language here.

Instead of using the word top
number and write it down top

number. And bottom
number.

We have two words that
we use. The top number

is called the numerator.

On the bottom number the

denominator. Now let's have a
look at seeing how we go the

other way. When we have an
equivalent fraction, how do we

find this fraction in its lowest

form? Well, let's look at an

example. Let's say we've
got 8, one, hundreds.

Now we need to find the number
that the lowest form was

multiplied by. And that we ended
up with eight one hundredths.

Well, the opposite of
multiplying is dividing, so we

need to divide both the
numerator and the denominator by

the same number.

So that we get back to a
fraction in its lowest form.

Well, if we look at the numbers
we have here 8 and 100, the

first thing you should notice is
actually the both even numbers.

And if they're both even
numbers, then obviously we can

divide them both by two.

So let's start by dividing the
numerator by two and the

denominator by two.

8 / 2 is four 100
/ 2 is 50.

Now we need to look at our
fraction. Again. We found an

equivalent fraction, but is it
in its lowest form?

Well again, we can see that
they're both even numbers, both

4 and 50 even, and so we can
divide by two again.

4 / 4 gives us
2 and 50 / 2.

Gives us 25, so another
equivalent fraction, but is it

in its lowest form?

Well, we need to see if there is
any number that goes both into

the numerator and the
denominator. Well, the only

numbers that go into 2A one
which goes into all numbers, so

that's not going to help us. And
two now 2 doesn't go into 25. So

therefore we found the fraction
in its lowest form, so 8 one

hundreds. The lowest form is 220

fifths. So when a fraction is in
its lowest form, the only number

that will go into both the
numerator and the denominator is

one. Those numbers have no other

common factor. Now if we look
here, we can see that in fact.

We could have divided by 4.

Straight away, instead of
dividing by two twice, well,

that's fine. If you've notice
tthat for was a factor.

Of both the numerator and the
denominator, you could have gone

straight there doing 8 / 4 was
two and 100 / 4 was 25 and then

check to see if you were in the
lowest form. That's fine, but

often. With numbers, larger
numbers is not always easy to

see what the highest common
factor is of these two numbers,

the numerator and the
denominator. So often it's

easier to work down to some
smaller numbers, and then you

can be certain that there are no
other common factors.

Now.

If we take all the
pieces of a fraction

like I did with my
chocolate, I took all

six of them.

That's the same as 6 / 6.

And that was our whole.

And any whole number can be
written this way, so we could

have. 3 thirds if we take all
the pieces, we've got one.

8 eighths, if we take all the
pieces, we've got one.

Now I'm going to rewrite.

Mathematical words numerator.

Divided fight denominator.

Because we're now going to

look. Add fractions
where the numerator.

Smaller than the denominator.

And we have a name for these
type of fractions and they

called proper fractions.

And examples.

Half.

3/4
16

7/8 5/10 and
seeing all these cases, the

numerator is smaller number than

the denominator. And as long as
that is the case, then we have a

proper fraction so we can have
any numbers 100 hundred and 50th

for example. Now if

the numerator.
Is greater than

the denominator?

Then the fraction is called
an improper fraction.

And some examples.

Three over two or three halfs.

7 fifths.
Eight quarters

We could have 12 bytes.

Or we could have 201 hundredths.

And in all these cases, the
numerator is larger than the

denominator. And it shows that
what we've got is actually more

than whole 1.

All these fractions, the
proper ones are smaller than a

whole one. We haven't taken
all of the pieces 3/4. We've

only taken 3 out of the four
161 out of the six, so that

all smaller than a whole one
with improper fractions.

They are all larger than one

whole 1. So if we take three
over 2 for example, what we've

actually got is 3 halfs.

Oh, improper fractions can be
written in this form.

All they can be written
as mixed fractions.

So let's have a look
at our three halfs.

And what we can do is put two
hearts together to make the

whole 1. And we've got 1/2 left
over, so that can be written as

one and a half.

So there are exactly the same,
but written in a different form

1 as a mixed fraction.

And one other top heavy
fraction, an improper fraction

where the numerator is larger
than the denominator.

Let's have a look at another

example. Let's say we
had 8 thirds.

This out the way.

Let's count

out 1234567.
8 thirds

How else can we write that?
How do we write that as a

mixed fraction?

Well, what we're looking
for is how many whole ones

we've got there.

Well, if something's been
divided into 3 pieces.

It takes 3 pieces to make the

whole 1. So that's one whole 1.

There we have another whole 12.

And we've got 2/3 left over,
so 8 thirds is exactly the

same as two and 2/3.

Let's look at one more.

Let's say we had Seven

quarters. Now we know that there
are four quarters in each hole,

one. So we see how many fours go
into Seven. Well, that's one.

And we've got 3 left over, so
we've got one and 3/4.

Let's have a look at one more.

37 tenths

Now we've split something up
into 10 pieces of equal size.

So we need 10 of those to make a
whole one, so we need to see how

many 10s, how many whole ones
there are in 37.

Well, three 10s makes 30, so
that's three whole ones, and

we've got 7 leftover, so we've
got 3 and 7/10.

Just move

those. Now let's have
a look at doing the reverse

process. So if we start with a
mixed fraction, how do we turn

it into an improper fraction?

Let's look at three and a

quarter. And if we look at this
visually, we've got.

3 hole once.

And one quarter.

And what we want to turn it

into. Is all
quarters.

So we have a whole 1.

And if we split it into
quarters, we know that a whole 1

needs four quarters.

So we have four there.

Another for their.

Another folder plus this one.

So we've got three force or 12.

Plus the one gives us 13
quarters, so 3 1/4 is exactly

the same as 13 quarters.

Well, let's have a look at how
you might do this.

If you haven't got the visual

aid. Well, what we've actually
got here is our whole number.

And the fraction.

We wanted in quarters.

So what we're doing
is right it again.

We're actually saying We want
four quarters for every hole

one, so we've got three lots of

four. And then what were
ranting on is our one, and

these are all quarters.

So it's the whole number
multiplied by the denominator.

We've added the extra that
we have here. Whatever this

number is, and those are the
number of quarters we've

got. So we've got our 3/4 of
12 + 1/4, so 13 quarters.

Let's have a look at one more

example. Let's say we've got
five and two ninths.

We want to turn it
into this format.

Ninths well, if we want to take
a whole one, we wouldn't need 9

ninths and we've got five whole
ones, so we're going to have 5 *

9 lots of 9th this time.

And then we need to add on the
two nights that we have here.

So 5 nines of 45 plus
the two and that all 9th.

So we have 47 ninths.

Any whole number can be written
as a fraction.

So for example, if we take
the number 2.

If we write it with the
denominator of one.

We've written it as a fraction.

And any equivalent form, so we
could have 4 over 2.

30 over

15. And so on.
So any whole number can be

written as a fraction with a
numerator and a denominator.

So fractions.
They can appear in a number

of different forms. You might
see proper fractions, improper

fractions, mixed fractions.

And you can see lots of
different equivalent fractions.

So that all different
ways that we see them.