Lesson Video: Equivalent Fractions on a Number Line Mathematics • 3rd Grade

In this video, we will learn how to find equivalent fractions by modeling fractions between 0 and 1 on number lines.

19:51

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.

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