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.