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Video: Comparing Fractions on a Number Line

Tim Burnham

Find how to use a number line to compare fractions with different denominators and how to use symbols, < less than and > greater than, to state which is larger.

13:30

Video Transcript

In this video, we’re going to use the number line to help us to compare fractions. We’ll see if two fractions are the same size or one is bigger and we’ll learn some symbols to help us to write this easily.

We will also use our knowledge of how to generate equivalent fractions, but remember there’s another video which covers that in more detail if you’re not too sure about them.

First then, let’s recap how to show fractions on the number line. Let’s say we want to show five-sixths. We zoom in on the number line between zero and one, and then we divide it into six equal parts.

We number them from zero to six, and then we use those numbers as the upstairs part of our fraction, the numerator. Because they’re out of six, we use six as the denominator, the downstairs part of the fraction. And then to represent five-sixths, we start our arrow at zero and we finish at five-sixths, like that.

Just to finish this off, we need to write each fraction on the number line between zero and one. Now zero-sixths is just zero and that’s already been written down there.

And if we thought about the sixth as being parts of a cake. If you had six, sixths of a cake, then you’d have the whole cake, and we use one to represent the whole thing. So one is already at the other end of the number line. Another way of thinking about that is that the fraction is a division, so six over six means six divided by six. How many six is going to six? One.

Now looking at the others, one-sixths means one divided by six. Well that’s just one over six. Now two over six well we can find an equivalent fraction with smaller numbers in it than that. Two and six are both divisible by two; two divided by two is equal to one and six divided by two is equal to three. So that is equivalent to a third, one over three.

Now three-sixths, well three and six can both be divided by three. Three is going to three once and three is going to six twice. So three over six is equivalent to one over two, a half.

Now four-sixths, well the upstairs and the downstairs can both be divided by two; four divided by two is two and six divided by two is three, so two-thirds is an equivalent fraction, the same position on the number line, to four-sixths, so we can write that as two-thirds and I’ve just got smaller numbers. Five-sixths, well five and six don’t have any common factors bigger than one, so there isn’t another number that we can divide them both by to make them smaller and that- this means that we just leave them as five over six, five-sixths.

So now we’ve remembered how to represent fractions on a number line and use arrows to represent those fractions. Let’s go ahead and do some comparing of fractions on the number line.

So let’s compare two-fifths and four-tenths. Well to represent two-fifths as a fraction on the number line, we draw the number line from zero to one, we split it into five, and then we draw an arrow from zero up to two.

And for four-tenths, again we’re gonna draw the number line from zero to one; this time I’m gonna split it into ten, but the arrow will go from zero up to the fourth division. So looking at two-fifths, it starts at zero and goes up to two-fifths on the line. Four-tenths starts at zero and it goes up to four-tenths on the line. Now provided we start our diagram in the same place as that they line up down here, the zeros are in line, and that the ones are in line at the other end, so these two things are at the same scale, we can then just compare the lengths of these arrows and we can see that two-fifths is exactly in line with four-tenths, so two-fifths and four-tenths must be equal.

So we can write that out like this with two-fifths then an equal sign and then four-tenths. Now let’s compare four-fifths and seven-tenths.

So for four-fifths, we’ve divided up the zero and one into five separate bits and drawn the arrow going from zero up to the fourth division. For seven-tenths, we’ve divided the zero to one into ten bits. We’ve started the arrow at zero and we’ve gone up to the seventh division, seven-tenths, so what we’ve gotta do before? Make sure that the zeros are lined up, as we said before, and they are, and then we’ve got to compare where the arrows end. And we can see that four-fifths actually lines up with eight-tenths, so that’s longer than the other arrow which doesn’t quite reach to the end of the first arrow. So four-fifths is- makes a longer arrow so four-fifths is bigger than seven-tenths.

So if I write four-fifths and seven-tenths next to each other, this symbol here tells us that the four-fifths is bigger than seven-tenths. So that’s our answer. Now this symbol here, how do you know which way round it is? Well let’s look at it. It’s-it’s big at one end so this end the symbol’s quite big, the lines are quite spaced out, and the other end okay the pointy end is quite small, so this end is the big end and this end is the small end. And they point to the bigger number and the smaller number, so it’s quite simple; it’s easy to work out which way around that sign has to go. The big end of sign- of the sign goes against the bigger number, and the small end of the sign goes against the smaller number.

Now that sign’s got a fancy name, greater than, so we’d say four-fifths is greater than seven-tenths, not too worried about the name for now, but do remember that bit about the big end of the sign goes against the big number, the small end of the sign goes against the smaller number.

And now we’re gonna compare two-thirds and two-fifths. So again we’ve drawn our diagrams, so we’ve lined them up. The zero ends line up together and the zero, but we can clearly see that two-thirds is a longer arrow because if we extend that down, we can see that there’s all this distance here between the length of the two arrows, so two-thirds is greater than, is bigger than, two-fifths.

So can you remember which way around we put the sign? Yup, that’s right. The big end of the sign goes against the big number, so we do this here. two-thirds is bigger than two-fifths.

But what if I wrote the numbers the other way around? Two-fifths and then two-thirds? Well we have to draw the sign the other way around; the big end of this sign goes against the big number and the small end goes against the small number.

And if we want to use the proper terminology, we could say two-thirds is greater than two-fifths or two-fifths is less than two-thirds.

Now if we think of fractions in terms of chunks of a cake, this all makes good sense, doesn’t it? Because let’s say our first cake we split into three and we plan to share that between three people, so we’ve created three slices of cake. There’s slice one, there’s slice two, and there’s slice three. Now another identical cake exactly the same size, we’ve split into five because we want to share that between five people. And here are those five pieces of cake being cut up now, and you can see that each, if we’re only sharing it between three people, each person is gonna get a bit bigger cake, so these bits of cake up here are slightly bigger than these bits of cake down here. So if I have two pieces of the cake that’s been chopped into three, then that’s gonna be bigger than two pieces of the cake which have been chopped into five.

Right, now you can have a go at a few questions. Simply pause the video for each question and work out if you think the fractions are the same or different and put the right sign between them. So pause now.

So we’ve labeled up our fractions so we’ve got a sixth, a third, a half, and so on and the same with the seventh, and now we’ve got to draw in our arrows. Four-sixths starts off at zero and goes up all the way to four-sixths. Five-sevenths starts at zero and goes all the way up to five-sevenths.

They both started in line at zero. The four-sixths only reached up to here. The five-sevenths reaches further than the four-sixths; it’s a longer arrow, so five-sevenths is bigger than four-sixths. So which way round do we think we need to put this sign?

Well the five-sevenths is the bigger number, so the bigger end of the sign needs to go against the five-sevenths. And if we wanted to say that, we would say four-sixths is less than five-sevenths.

Okay, next question, compare three-sevenths and two-fifths. Pause the video now. Okay we’ve labeled up the divisions now, so we’ve got our sevenths labeled and we’ve got our fifths labeled. so for three-sevenths, we’re gonna start at zero and we’re gonna draw a line going up to three-sevenths, the third division. For two-fifths, we’re gonna start at zero and we’re gonna go up to the second division, two-fifths. Now there’s not very much difference between these. These are quite tough ones to call, but if you look very, very carefully, I think you can see that the three-sevenths is slightly longer arrow than the two-fifths. So they both start at zero in line with each other here, but if we extend the three-sevenths downwards, I think you can see that it’s just a little bit longer than the two-fifths, so three-sevenths is bigger than two-fifths.

That means that the bigger end of the sign has to go against the three-sevenths, so three-sevenths is greater than two-fifths.

So in this last question, you’re comparing seven-tenths and two-thirds. This time, I want you to fill out both of those boxes at the end there. Okay, pause the video now.

So the first job was to fill in the names of the fractions, so we’ve done that for tenths and we’ve done that for thirds. And now we have to draw seven-tenths, so we’re gonna start from zero and we’re gonna go all the way up to the seventh division over here, seven-tenths. Then for two-thirds, we’re gonna start here from zero and we’re gonna go all the way up to two-thirds. And this is another really close one to call, but I think you’ll find that the seven-tenths arrow is slightly longer than the two-thirds arrow; let’s have a look at that.

They both start at zero and if I extend seven-tenths down, we can see that it comes just to the right of two-thirds, so seven-tenths is just a bit bigger than two-thirds. So in both of these cases, the big end of the sign is gonna go against the seven-tenths, so we’re gonna have to turn that sign round for the other one; the big end of the sign is against the seven-tenths and the small end is against the two-thirds.

And if we wanted to read that out, we would say that seven-tenths is greater than two-thirds, or we would say that two-thirds is less than seven-tenths.

So we’ve been representing fractions on the number line by drawing arrows. We divide that gap between zero and one into the relevant number of pieces, ten pieces if it’s tenths, three pieces if it’s third, and we can use the length of those arrows to compare fractions on a number line. We’ve also had a quick look at some equivalent fractions. If they’re the same length, if they’re equal to each other, then they’re equivalent fractions.