### Video Transcript

Equivalent Fractions on a Number
Line

In this lesson, we’re going to
learn how to find equivalent fractions by modeling them on number lines. First things first. What is an equivalent fraction? When two things are equivalent, it
means they’re worth the same. They have the same value. And so equivalent fractions are
fractions that are worth the same. Let’s show what we mean.

These two children are learning
about fractions, and they’ve got strips of paper that are exactly the same
length. Now, the girl folds her paper once,
so her strip of paper is folded into two equal parts. Then the boy folds his strip
exactly the same way but then folds it a second time. So when he unfolds his strip of
paper, it’s been divided into four equal parts. Now the girl shades one part of her
strip orange, and then she challenges her friend. Can he color the same fraction on
his strip? Do you think this is gonna be
possible? Each of his parts is smaller.

Well, if we draw a dotted line from
the top strip, we can see that in order to color the same fraction as one part on
the first strip, we need to color two parts of the second strip. Let’s think about the two fractions
that we can see here. We could label them on number
lines. And we could use our fraction
strips or tape diagrams, as they’re sometimes called, to help us. To begin with, we could draw lines
that are just as long as our fraction strips. We could label one end zero and the
other end one to represent one whole.

Our first fraction strip has been
divided into two equal parts. So we can draw a little mark where
that division is. Because the whole strip has been
divided into two. We know the denominator for our
fraction is two. And we’ve only shaded one of these
parts. We’ve shaded one-half. And we can label this on our number
line. We divided our second strip of
paper into four equal parts. And by matching up our strip of
paper to our number line, we can draw the marks that separate our number line into
four equal parts too.

So what fractions can we see
here? Well, the whole amount has been
divided into four equal parts. So the denominator of our fraction
is four, and each part is one out of these four equal parts. It’s worth one-quarter. We can see that to shade the same
amount as the half that’s shaded in the first strip, we need to shade two of our
quarters. The fraction that we can see here
is two-quarters. We found two fractions that are
worth the same. One-half is equal to
two-quarters. Our fractions might be written
slightly differently, but they’re worth the same.

Look at how the numbers are linked
to each other, too. If we double one, we get two. And if we double two, we get
four. The numerator and the denominator
in our second fraction are both being doubled. But the fraction is still worth the
same. So one-half is the same as
two-quarters. Let’s try to find some more
fractions that are equivalent. So we’ll start by drawing two
number lines. And we’ll label them as we did
before; we’ll write zero at one end and one at the other, which represents one
whole. We’ll divide our first number line
into four equal parts. This means that we can label it
using quarters: one-quarter, two-quarters, three-quarters.

Now we can split our second number
line into eight equal parts, which means that each of these parts is worth
one-eighth. And we can label our number line
one-eighth, two-eighths, three-eighths, and so on. And because we’ve drawn our number
lines on top of each other and they’re exactly the same length, we can use these to
spot equivalent fractions. How many fractions that are worth
the same can you see? The first fraction on the top
number line is one-quarter. And if we draw a dotted line down
from this until we hit the bottom number line, we can see how many eighths this is
the same as. One-quarter equals two-eighths. And we can do the same when we get
to the next fraction on the top number line.

Can you see it’s exactly the same
as four-eighths along our bottom number line? Two-quarters and four-eighths are
equivalent fractions. And if we look at the last
fraction, we can see that three-quarters is the same as six-eighths. We found some fractions that are
eighths that are equivalent to fractions that are quarters, but can you see any
fractions that are eighths that aren’t equivalent to a number of quarters? Well, these are the ones in
between, aren’t they? We can’t write one-eighth,
three-eighths, five-eighths, or seven-eighths as a number of quarters. Now, when we look at the three
pairs of equivalent fractions that we’ve found here, I wonder, do you notice
anything? Each time, we’ve doubled the
numerator and done exactly the same to the denominator.

If you remember, this was the
pattern we saw with our first example, too. But, you know, we don’t have to
multiply by two every single time. It doesn’t matter what we multiply
or even divide by as long as we do the same to the numerator and the
denominator. Let’s show what we mean by
this. And this time, we could show both
of our equivalent fractions on one number line.

Imagine that we want to find out
how many sixths are worth the same as one-half. We could start by dividing our
number line into six equal parts: one, two, three, four, five, six. Now we can label our number line in
sixths. And we want to find out how many
sixths are the same as one-half. Let’s use a slightly thicker pen to
show where one-half is. It’s here, isn’t it, halfway
along. We can see that by drawing a mark
here, we’ve divided the number line into two equal parts. And so straightaway we can identify
the equivalent fraction just by reading the number line. Three-sixths is the same as
one-half. These two fractions are
equivalent. They’re written differently, but
they’re worth the same.

And like we said earlier, we can
spot that these are equivalent fractions because the numerator and the denominator
have been multiplied by the same amount. They haven’t been doubled this
time, though; they’ve been multiplied by three. One times three is three, and two
times three is six. Because the numerator and the
denominator have been multiplied by the same amount, we know that these two
fractions we’ve labeled on our number line are equivalent. Let’s have a go at answering some
questions now, where we have to find equivalent fractions by modeling them on number
lines.

Complete the following: Two-thirds
equals what sixths?

In this question, we’re given two
equivalent fractions. When two fractions are equivalent,
we know that this means they have the same value. They might look different, but
they’re worth the same. And we can tell in the statement
that we’re given that our two fractions are equivalent because there’s an equal sign
in between them. Our first fraction is complete. It’s two-thirds. But the top number, that’s the
numerator, is missing from our second fraction. We know we’re looking for a number
of sixths, but we don’t know how many. This is the part that we need to
complete. And we’re given two number lines to
help us do this.

What do you notice about these
number lines? Well, they both start with zero,
they both end with one, and they’re both exactly the same length. And these facts help us to be able
to compare them. If we look at the first number
line, we can see that it’s been divided into three equal parts. And there’s a red line that’s been
used to highlight two of these sections. Two out of our possible three is
two-thirds, and this is labeled. It’s the first fraction in our
statement, isn’t it? Now, if we look at our second
number line, we can see that it’s been divided into six equal parts or sixths.

Now, as we’ve said already, by
drawing these number lines exactly the same size and also on top of each other, it
means we’re able to compare the fractions on them. And importantly, we can find
two-thirds on our top number line and then draw a line downwards until we hit our
bottom number line and see how many sixths this is worth. Where do we end up? Well, straightaway we can see the
fraction that’s labeled. It’s four intervals along or
four-sixths. And another way we can see these
two fractions are equivalent is by comparing the length of the two red lines. They’re both the same.

If we look carefully at our number
lines, we can see that there are two-sixths for every third. And so if we’re looking for a
fraction that’s worth the same as two-thirds, we can find one that’s the same as two
lots of two-sixths or four-sixths. We’ve used number lines to help us
find a pair of equivalent fractions. Two-thirds is the same as
four-sixths. Our missing numerator is four.

Complete the following: Two-thirds
equals what.

In this question, we’ve been given
a pair of equivalent fractions. We can tell that they’re worth the
same because there’s an equal sign in between them. Our first fraction is two thirds,
but we don’t know what our second fraction is at all. And you know, the interesting thing
about this question is that if we weren’t given any more information to help us,
there’s actually more than one fraction we could give us an answer. There are several fractions that
are worth the same as two-thirds. But thankfully, in this question,
we are given some more information to help us. We’re shown two number lines. Let’s look at them one by one.

Our first number line is split up
into three equal parts. It shows thirds. And these are labeled for us:
one-third, two-thirds, and then three-thirds, which is the same as one whole. So why are we shown this number
line? Well, it links with our first
fraction. We can see two-thirds on it. Can you see that orange line that
goes all the way from zero, two-thirds of the way along the number line? Now, if we look at our second
number line, we can spot a few things. Firstly, it begins at zero, and can
you see both zeros are lined up? And it ends at twelve twelfths. And twelve twelfths is the same as
one whole. In fact, any fraction where the
numerator and the denominator are the same is equal to one whole.

So we can say that our second
number line goes from zero to one, and it’s completely lined up with the one above
it. And because it’s exactly the same
length, this means that we can compare one number line with the other. It’s very easy to do so just by
looking up and down. Another thing that we can spot is
that our second number line isn’t divided into three equal parts. There are much more. If we count them, we can see that
there are 12 equal parts. That’s why the denominator that’s
labeled here is 12. We’re talking about twelfths: one
twelfth, two twelfths, and so on.

So I suppose what the first thing
we could do here is to complete part of our missing fraction. We know that whatever we find here
is going to be a number of twelfths. And so the denominator in our
missing fraction is going to be 12. So we can ask ourselves, two-thirds
equals how many twelfths? Let’s draw a line along our second
number line until we get one that’s exactly the same as the first. And as we do so, we’re going to
count in twelfths: one twelfth, two twelfths, three twelfths, four twelfths. Let’s pause here to look at
something interesting. We can see that four twelfths on
our bottom number line is the same as one-third on the top.

Now this isn’t the answer to our
question, but we could use it to help, because if we know that one-third is the same
as four twelfths, surely two-thirds must be worth double this. What do you think? Do you think if one-third is worth
four twelfths, then two-thirds must be worth eight twelfths? Let’s carry on shading our line:
five twelfths, six twelfths, seven twelfths, eight twelfths. And we can see that this matches up
perfectly with two-thirds on our top number line. Our two fractions might look
different, but they’re worth exactly the same. Two-thirds is the same as eight
twelfths. We’ve used number lines to show
that these are equivalent fractions.

And another way that we know that
they have the same value is that we can see the numerator has been multiplied by
four, two times four is eight, and the denominator has also been multiplied by four,
three times four is 12. And if we multiply or divide a
numerator and denominator by the same amount, we get an equivalent fraction. So although our number lines show
it, we can also use this to check. There are lots of fractions that
are worth the same as two-thirds. But these number lines show us that
two-thirds is the same as eight twelfths.

What is the missing equivalent
fraction? One-third, two-fifths, five-tenths,
or one-half.

In maths, we use the word
“equivalent” to describe things that have the same value as each other. And this question asks us to find a
missing equivalent fraction. In other words, we’re looking for
some fractions that are the same as each other here. And we’re shown a number line to
help us. Number lines are a really useful
way to spot equivalent fractions. Let’s have a look more closely at
this one. We can see that this one begins
with zero and ends with one. And we can think of this as
representing one whole, and so each of the fractions that are labeled on our number
line are parts of the whole amount.

Now, if we look at all these
fractions, we can see that they have a denominator of 10. Our number line is labeled in
tenths: one-tenth, two-tenths, and so on. Above our number line, we can see
where the missing fraction belongs. What’s this fraction worth? Well, if we look underneath, we can
see how many tenths it’s the same as. It’s exactly at the point that’s
labeled five-tenths. So we know our missing fraction is
equivalent to or worth the same as five-tenths.

So which of our possible answers is
correct? Is one-third equivalent to
five-tenths or two-fifths? What about five tenths or even
one-half? Now, only one of these is the
correct answer. Two of them are completely wrong,
and one of them, well, we wouldn’t really write. Can you spot the one that we
wouldn’t write? It’s five-tenths. This point on a number line was
already labeled five-tenths. So although of course we could say
five-tenths equals five-tenths, it’s not what this question is asking us. Our equivalent fraction is going to
be one that’s worth the same as five-tenths but isn’t five-tenths.

This leaves us with three possible
answers, one-third, two-fifths, or one-half. Now this is tricky because our
number line isn’t split up into three parts to show thirds or five parts to show
fifths or even two parts to show one-half. We’re going to have to visualize or
picture in our heads what each fraction might look like. Let’s alter the top part of our
number line to help us imagine what they might look like. Firstly, one-third would be the
same as if we split the number line into three equal parts, and we labeled the first
one of them. One-third isn’t the same as
five-tenths, is it?

Secondly, two-fifths would be the
same as if we split our number line into five equal parts, and we labeled the second
one along. Interestingly, we can see that
two-fifths is equivalent to one of our fractions, but it’s the same as four-tenths
not five-tenths. This only leaves us with one
fraction. To show one-half, we need to divide
the whole number line into two equal parts and label where we know the mark
belongs. And we can tell by looking at our
number lines that we found the right answer. Five-tenths equals one-half.

Now we’ve used a number line here
to find this equivalent fraction. But perhaps you knew the answer
already. Perhaps you remembered that five is
half of 10. Maybe you spotted that five-tenths
was halfway along the line. Or perhaps you even noticed that if
we divide both the numerator and the denominator in five-tenths by the same number,
in this case, the number five, we’d find our answer. Our missing fraction that’s
equivalent to five-tenths is one-half.

So what have we learned in this
video? We’ve learned how to find
equivalent fractions by modeling fractions on number lines.