### Video Transcript

Comparing and Ordering Unit
Fractions

In this video, we’re going to learn
how to use models such as tape diagrams and number lines to compare and order unit
fractions. Do you know what? Instead of watching this video,
shall we put the television on instead? Let’s see what’s on. Oh! We are just in time to start
watching the fantastic new game show: Which is the greatest? Round one: five or three. Which is the greatest? It’s an easy one to start with,
isn’t it? Why don’t you point to the number
you think is correct? Is that your final answer? And that’s right. Of course, five is greater than
three. Onto round two: one-fifth or
one-third. Which is the greatest? If you’re not sure, have a
guess. Point to the number you think is
the greatest.

Let’s pause the TV show at this
point and think carefully about the question that we’re being asked. We’ve already seen in round one
that five is greater than three, so surely one-fifth is greater than one-third. These two numbers are
fractions. They’re worth part of a whole. And in this video, we’re going to
learn how to compare fractions just like these. And although we’re going to go on
to use models to help us, the key to this is understanding what the numbers in a
fraction mean. So the fraction one-fifth is a one,
then a horizontal line, and then the number five underneath. The fraction one-third also has a
one above the line but has the digit three underneath.

Now, what do you notice about these
fractions? What’s different and what’s the
same? We can see that both of the bottom
numbers are different, aren’t they? The bottom number or the
denominator in a fraction shows us how many equal parts of the whole amount has been
split into. Let’s show this using some fraction
strips. Our first fraction has five as a
denominator. So we’re going to need to split the
whole strip into five equal parts, one, two, three, four, and five. We’ve split the whole strip into
fifths. If we look at the denominator in
our second fraction, we can see that it’s the number three. This tells us that we need to split
up the whole strip into three equal parts, one, two, and three. We’ve split up the whole strip into
thirds.

So that’s what’s different about
our fractions. But if we look at the top numbers
or what we call the numerators in these fractions, we can see that they’re both
one. Any fraction that has one as a
numerator is what we call a unit fraction. This is what we meant in the title
of this video when we said we were going to be comparing and ordering unit
fractions. It just means that all the
fractions we’re going to be looking at in this video are going to have one as their
numerator, one-half, one-quarter, one-ninth, and so on. Now what does this number one, this
numerator, tell us in a fraction? Well, it tells us the number of
parts that we’re talking about.

We divided our first fraction strip
into fifths and the fraction that we need to show on the strip is one-fifth. So let’s shade in one of our
fifths. There we go. We’ve shaded in one out of our
possible five equal parts. One out of five is one-fifth. Onto our second fraction. Now again, our numerator tells us
how many parts we’re talking about, and once again, it’s one, one out of a possible
three. So let’s shade in one out of
three. One out of three is the same as
one-third. Now, there’s something really
important about our fraction strips that we need to mention. They’re both exactly the same
length. And because the whole strip is the
same length, we can compare the fractions.

Which is the greatest, one-fifth or
one-third? Well, we could draw little arrows
to help us. But you should be able to see by
looking at these fraction strips that one-third is the greatest. We can say one-third is greater
than one-fifth. But wait; that doesn’t make
sense. How can one-third be greater than
one-fifth when three is less than five? Well, we know that it does make
sense, don’t we? Even though yes, three is less than
five, we need to keep remembering what these numbers mean in a fraction. Would you rather have part of a
cookie that’s been shared into three equal parts or part of a cookie that’s been
split into five equal parts?

A smaller denominator like three
doesn’t mean a smaller part. It just means that the whole amount
has been split into less parts, so they’re going to be bigger. And in the same way, a larger
denominator doesn’t mean a larger part. It means that the whole amount has
been split into more equal parts. Each part is gonna be smaller. Let’s try answering some questions
now where we have to put into practice what we’ve learned. And each time that we compare our
unit fractions, we’re not going to look for the largest number and just think
straight away that must be the greatest. We’re going to think very carefully
about what the numbers in a fraction mean. And it might be a good idea to
sketch some models too.

Complete one-ninth what one-third,
using the symbol for is less than, is equal to, or is greater than.

These symbols that we have at the
end of this question are those that we use when we compare two values together. And in this question, we need to
choose the correct symbol to write in between two fractions. So, in other words, we’re comparing
two fractions. Is one-ninth less than one-third,
is it equal to one-third, or is it greater than one-third? And you might’ve noticed we’ve got
some fraction strips underneath that can help us. And we could look at them straight
away and be able to compare one-ninth with one-third.

But before we do that, let’s look
at the fractions themselves in the question. Both fractions have the same top
number or numerator. But look at these bottom
numbers. One-ninth has a bottom number or
denominator of nine, but the other fraction’s denominator is only three. Nine is greater than three, isn’t
it? So don’t you think that’s going to
mean one-ninth is greater than one-third? Think it might be time to look at
our fraction strips, don’t you? The first thing that we can say
about our fraction strips is they’re both the same length. This is really important when we’re
comparing fractions. It shows that the whole amount is
the same.

Now, the first fraction that’s
labeled is one-ninth. And we know that the denominator or
the bottom number in a fraction shows how many equal parts the whole amount has been
split into. So although we can’t see them all,
this fraction strip has been split into nine equal parts. In fact, should we draw them
on? There we go. Now, the numerator or the top
number in this fraction shows us how many of these parts we’re talking about. And because it’s one, we’re only
talking about one out of these nine parts. That’s where this blue part comes
from. Now, if we look at our second strip
for a moment, we can see that the bottom number or the denominator is smaller. But this doesn’t mean it’s a
smaller fraction.

Remember, the denominator shows us
how many equal parts we’re splitting the whole amount into. And if we’re splitting it into
three parts rather than nine, each of our three parts is going to be larger. And because our numerator is one
again, we’re only thinking about one out of these three parts. So that’s where our fraction strips
come from. Now we understand them; we can use
them to help. We can see that one out of nine
equal parts is a lot smaller than one out of three equal parts. One-ninth is less than
one-third. The symbol that we need to use in
between these two fractions to compare them is the one that means is less than.

Which fraction is greater,
one-quarter or one-fifth?

In this question, we’re being asked
to compare two fractions together, and they are one-quarter and one-fifth. This is quite an easy question,
isn’t it? Five is a bigger number than four,
so the answer must be one-fifth. Maybe we’d better not be so quick
at coming up with an answer. It might be a good idea to model
these two fractions and think about what the numbers that make them up mean. Let’s use number lines to help
us. Now, often with a number line, we
might start at zero and then carry on counting one, two, three, and so on, up to
maybe some larger numbers. But the number lines that we need
to draw to help us here are going to start at zero and end at one. In other words, we’re thinking
about the parts that are less than one.

We can think of one as being the
whole amount, and these fractions are part of a whole. So we know they’re going to appear
somewhere on this number line between zero and one. Let’s think about our first
fraction, one-quarter. Do you remember what the
denominator or the bottom number in a fraction tells us? It tells us the number of equal
parts that the whole amount is split into. And in one-quarter, we need to
split the whole amount into four equal parts. Not sure if you knew this, but a
quick way to do this is to split the whole amount into half and then split each of
those halves into half. Can you see we’ve now split our
number line into four equal parts? And because the numerator or the
top number in our fraction is one, we only need to think about one of those
parts. One-quarter is here.

Now let’s think about
one-fifth. Rather than getting muddled up with
lots and lots of notches, why don’t we draw a new number line? And because we want to compare our
number lines, let’s make sure it’s exactly the same length and it’s lined up. So again, this number line
represents one whole. Now, the denominator or the bottom
number in this second fraction is greater than the first. It’s five. But remember what this five
means? It’s the number of equal parts we
need to split the whole amount into. So we need to split our number line
into more equal parts this time. They’re going to be smaller, aren’t
they? We’ve split the number line into
fifths, five equal parts. And because the numerator or the
top number in our fraction is one again, we only have to think of one of these five
equal parts. One-fifth is here.

Now, hopefully you can see by
comparing these number lines, especially if we put in some dotted lines to help us,
that one-quarter is greater than one-fifth. Even though four is actually less
than five, we know that a lower denominator in a fraction tells us that the whole
amount needs to be split into less equal parts, so the parts are going to be
bigger. Would you rather have part of a
chocolate bar that’s been split into two equal pieces or 200 equal pieces? Well if you like chocolate, you’d
probably prefer one out of two equal pieces. One out of 200 would be tiny. And in the same way, because our
numerators are both one, we can see that the greater fraction is the one with the
smaller denominator. Out of one-quarter and one-fifth,
the greater fraction is one-quarter.

Arrange one-fifth, one-third,
one-seventh, and one-ninth in ascending order.

In this question, we’re given four
fractions and we’re told to arrange them or put them in a certain order, ascending
order. Now, when a balloon ascends in the
sky, it goes up and up. And so if we want to put these
fractions in ascending order, we need to arrange them from lower to higher. In other words, we need to start
with the smaller fraction and get greater and greater as we go along. Now what can we use to help us
compare these fractions? Perhaps you can put them in order
without using a model to help. But fraction strips can be
helpful. Let’s use some of those. We have four fractions and we’re
going to need four fraction strips. Our strips need to all be the same
length for us to compare them.

If we look at our fractions, we can
see that they’re all what we call unit fractions. They all have a numerator of
one. This means that although we’re
going to split the whole strip into lots of different parts, we’re only thinking
about one of those parts each time. Now we know that the bottom number
in a fraction tells us how many equal parts to split the whole amount into. To show one-fifth, we need to split
the whole strip into five equal parts and then shade one of those five. Now, three is a lower number than
five. But this doesn’t mean that the
fraction is smaller. We know that it means that we split
the whole amount into less equal pieces, three equal pieces. And because we’re thinking of
one-third, we simply need to shade one of these parts again.

Now we’ve modeled two fractions; we
can start to compare them. We can see that one-fifth is less
than one-third. So if we want to arrange our
fractions starting with the smallest, let’s write one-fifth before one-third. Of course, we haven’t looked at the
other fractions yet. So we don’t know where we need to
put those. One-seventh means one out of a
possible seven equal parts. This is interesting. Although it’s a larger denominator
than the other two fractions, we can see that one-seventh is actually a smaller
fraction. Of course it is. A larger denominator means more
parts, and more parts means smaller parts. We can see that one-seventh is our
smallest fraction so far, so we’re going to have to shuffle our fractions along a
bit. Our final fraction has the largest
denominator of the lot.

Let’s make a prediction. Because it has the largest
denominator, this means that the whole of the fraction strip is going to be split
into more parts. And so each of those parts is going
to be smaller. Do you think one-ninth might be our
smallest fraction? Nine equal parts and we’ll shade
one of them. We were right; the more parts we
split the whole amount into, the smaller they’ll be. Because all of the numerators in
our fractions are the same, if we want our fractions to go from the smallest to the
largest, our denominators need to go from the largest to the smallest. Look at how they do, nine, seven,
five, three. But by drawing these fraction
strips, we know why this is. And it’s all to do with what each
number in a fraction means. These fractions in ascending order
are one-ninth, one-seventh, one-fifth, and one-third.

So what’ve we learned in this
video? We’ve learned how to use models
like tape diagrams or fraction strips and number lines to compare and order unit
fractions.