Video: Comparing and Ordering Unit Fractions

In this video, we will learn how to use models, such as tape diagrams and number lines, to compare and order unit fractions.

16:44

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.