Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson from our unit on comparing fractions.

Have you done lots of work on fractions I wonder? I expect you've heard of things like common denominators and maybe even common numerators.

Well, let's have a look at all the different ways that we can think about comparing fractions.

If you're ready to make a start, let's get on with it.

So in this lesson we're going to be explaining which strategy for comparing non-related fractions is most efficient.

There's lots of different ways we can compare non-related fractions.

Those are fractions where the denominators are not factors and multiples of each other, and there's lots of different ways that we can compare them.

So we're going to think about being efficient in our choice of strategies in this lesson.

We've got some key words and a phrase there.

We've got common denominator, magnitude and efficient.

I'm sure they're words that are familiar to you, but let's just rehearse them and then check that we know what they mean.

So I'll say them first and then it'll be your turn.

So are you ready? My turn.

Common denominator.

Your turn.

My turn.

Magnitude.

Your turn.

My turn.

Efficient.

Your turn.

Well done! Okay, let's have a look at what those words mean.

They are all going to be really useful to us today.

So when two or more fractions share the same denominator, you can say that they have a common denominator and we can create common denominators to help us to work with fractions when we need to.

The magnitude of something refers to the size of something.

So we can think about the magnitude of a fraction, the size of a fraction relative to another fraction or relative to a whole and to be efficient means finding a way to solve a problem quickly whilst also maintaining accuracy.

We want to get the answer right and we want to be sure we're doing it in the best way.

So there are two parts to our lesson today.

In the first part, we're going to ask the question to convert or to reason? Are we going to use a common denominator or not? And in the second part, we're going to be thinking about a toolkit for reasoning.

All the different things that we have, all the things we know about fractions that can help us to make a decision as to which strategy is going to be the best.

So let's make a start on part one.

And we've got Izzy and Jun helping us in our lesson today.

So what would be the best strategy to compare these fractions? You might want to have a little look at them before Izzy and Jun share their thinking.

Izzy says, 'Shall we convert them to common denominators?' What do you think? Well, the answer to that is, well, we could, but is it the best strategy? And that's what Jun says.

'You could do that, but I don't think we need to.

Let's reason about their magnitude.

' So he's going to think about the size of the fractions.

And remember when we're just looking at numbers like this, fractions without a context, we are thinking about fractions of one whole of one.

So Jun says, 'One-tenth is only a small portion of the whole whereas seven-eighths is almost equal to the whole.

' If you pictured those fractions on a number line, one-tenth would be quite close to zero and seven-eighths will be really quite close to a whole.

So this is quite an easy one.

Seven eighths is larger than one-tenth, and we can use our understanding of fractions and our knowledge of where they would sit on a number line to be very certain that, that is the correct symbol to put there.

One-tenth is less than seven-eight.

Oh, we've got a Venn diagram here.

Izzy says, 'Let's record that.

' So we've got a Venn diagram here with one ring for common denominator and one ring for reasoning and one in the middle, well, some people might do it by reasoning and some people might use a common denominator.

We use reasoning for that one.

And Jun says, 'I think it was a lot more efficient to compare by just reasoning for these fractions.

' So let's put them in that part of our Venn diagram.

What about these fractions? What would be the best strategy for comparing them? Again, you might want to have a think before Izzy and June share their thoughts.

'How might you compare these ones then?' says Jun.

Izzy says, 'I think these are a little trickier to compare because they're both quite a large portion of the whole.

' We've got three quarters and we've got seven-ninths.

'It might be best to convert these ones to have common denominators.

' When fractions have common denominators, we know that we can just compare the numerators to decide which fraction is larger.

So Jun says, 'Let's multiply the denominators together to find the common denominator.

' And that's going to find us the lowest common denominator in this case.

If we wrote down the multiples of four and the multiples of nine, the first one that they would have in common would be 36.

So we're going to scale the denominator and the numerator by a factor of nine to turn three quarters into a number of 36.

So four times nine is 36, 3 times nine is 27.

So 27/36 is equivalent to three quarters.

What about the ninths? Well, we're going to scale everything up by a factor of four.

Four times as many pieces in the whole.

So four times as many of them to maintain the same proportion of the whole.

Nine times four is equal to 36 and seven times four is equal to 28.

So seven-ninths is equivalent to 28/36.

They were really close, weren't they? 27/36 and 28/36.

27 is less than 28.

And because we have a common denominator, we can now just compare the numerator.

So June says, 'I agree, those fractions were best to compare using common denominators.

' 'So let's put that on our Venn diagrams.' says Izzy.

So that's that pair of fractions.

So what would be the best strategy to compare these fractions? Again, you might want to have a little discussion before Izzy and Jun share their thinking.

June says, 'I think maybe we should use a common denominator for these examples again.

' What do you think? Did you think a common denominator would be best? Izzy says, 'Actually, I think we can reason this quite easily.

' Have you spotted something about these fractions perhaps? 'Go on', says Jun, 'how would you do that?' Ah, she's drawn a bar model.

What do you notice about those two bar models? We'd have to be very accurate in drawing the bar models, but I think what they're going to do is show us how we could think about those fractions in a way that we could reason about them to compare them.

Izzy says, 'Well, four-tenths is one-tenth away from a half and five-twelfths is one twelfth away from a half as well.

' So if we look at those fractions, we know that a fraction equivalent to a half as the numerator is half the value of the denominator.

So that will be five-tenths and six-twelfths, and we've got one-tenth fewer at four-tenths and one-twelfth fewer with five-twelfths.

So there we can see that distance to a half.

And what do we know about twelfths compared to tenths? 'Oh, I see' says Jun, 'because one-twelfth is smaller than one-tenth, that means five-twelfths is a smaller distance from one-half.

' Five-twelfths is closer to one-half, so it's greater than four-tenths.

So four-tenths is less than five-twelfths.

Jun says, 'I think that's another example of one where we would reason instead.

' Izzy says, 'We could also have used a common denominator though, so maybe it's better off in the middle?' You have to think carefully about that common denominator because what is the first shared multiple of 10 and 12? I don't think we'd have to go all the way to a hundred and twentieth.

Time to check your understanding.

Which part of the diagram would you place these two fractions in to compare? We've got one-fifth and two-elevenths.

Would this be something you do to common denominator for? Could you do it by reasoning or is it well, we could do it either way? Does it go in the middle? Pause the video, have a think.

And when you're ready for some feedback, press play.

What did you reckon? Jun says, 'I know that one-fifth is equivalent to two-tenths and two-tenths is larger than two-elevenths.

So I think it can go in the reasoning section.

' That was really good thinking.

He didn't create a common denominator, but he did create a common numerator.

If we have numerator that is the same, then we can use the knowledge of the denominators to compare the fractions.

And in this case, the larger the denominator, the smaller the fraction.

So June would put that in the reasoning.

You may have thought a common denominator was a good way of doing it as well.

Both of the fractions have a very small magnitude, don't they? They're both very close to zero if we place them on a zero to one number line.

But Jun's reasoning, thinking about tenths I think was a really good way of thinking about those fractions.

Well done if you spotted that too! And it's time for you to do some practise.

You are going to sort these pairs of fractions according to the strategy you would use to compare them.

So would you use a common denominator? Would you use reasoning or could you do it either way and you couldn't quite decide which? And in question two, you're going to create your own sets of fractions that you would place into each part of the Venn diagram.

Izzy's made some up for you to start with.

So you are going to have a go as well.

Pause the video, have a go at the questions and when you're ready for some feedback, press play.

How did you get on? So Izzy says, 'Here's how I would've placed them.

' So we had one-third and one-quarter.

Well those are unit fractions, aren't they? They've got a common numerator, so we can definitely use reasoning to sort those ones out.

And we know that when the numerator are the same, the bigger the denominator, the smaller the fraction.

So one-third would be greater than one-quarter.

What about one-tenths and six-sevenths? Oh, what do you notice about that one? That's right, one-tenth is very close to zero on a number line and six-sevenths is almost one.

So again, that's one for reasoning.

What about three-eighths and six-fourteenths? Oh, that's a bit trickier, isn't it? Three-eighths is just less than a half and six-fourteenths is just less than a half.

That's only a fourteenth away from a half, isn't it? Whereas three-eighths is a whole eighth away from a half.

So we could do that by reasoning as well, thinking about how close they are to a half.

Six-ninths and six-tenths, we've got a common numerator there.

Again, so just like the unit fractions, the bigger the denominator, the smaller the fraction.

So we can use reasoning.

Gosh, we've used reasoning for all of them so far.

Did you expect that? What about four-twelfths and seven-ninths? Well, four-twelfths is less than a half, isn't it? And seven-ninths is greater than a half, so again, reasoning.

What about two-fifths and three-eighths? Oh, that's a bit trickier, isn't it? I think I might use a common denominator for that one, although there might be something around a half in there that we can think about, but it would be a little bit more complicated.

Common denominators might be the way to go for that one.

What about five-elevenths and three-fifths? We could probably compare to a half there.

Three-fifths is bigger than a half.

Five-elevenths, I know eleven is an odd number, but we'd need five and a half elevenths to be equal to a half, so we could reason about that one as well.

And then seven-eighths and five-sixths.

Or we can think about how close they are to a whole, can't we? They're both one fraction away from a whole, one-eighth and one-sixth.

We know that one-eighth is a smaller fraction than one-sixth, so therefore it will be closer to one.

So again, we can reason.

So there was only really one in there where we might have thought about using common denominators.

I wonder if you did that.

If you put more of them in the common denominator section of our Venn diagram or in the middle, can you think now why reasoning might be a better and more efficient strategy? We're going to focus in on a couple.

There we go.

Seven-eighths and five-sixths.

So you may have said one-eighth is smaller than one-sixth.

So seven-eighths is closer to the whole, just as we said earlier.

And for this one, you may have said that these are harder to compare without a common denominator, although there is something about two and a half-fifths.

Well half of a fifth is a tenth , so that's a tenth away from a half.

And three-eighths is one-eighth away from a half, which is slightly bigger, isn't it? So we could have done that, but a common denominator might well have been the most efficient strategy there.

And for question two, you are adding your own examples.

So here were some of the examples that Izzy came up with, sixth-tenths and four-sevenths.

Well, they're both greater than a half and tenths and sevenths are quite difficult to visualise.

So yes, a common denominator might be a good one to use there.

Four-sevenths and sixth-ninths.

Well, we are three of everything away from a whole.

Three-sevenths away from a whole and three-ninths away from a whole.

So we could reason, or you might have decided to use common denominator for that one.

And then we've got seven-twelfths and eight-fourteenths.

Well those are both one away from a half, aren't they? Seven-twelfths is one-twelfth greater than a half and eight-fourteenths is one-fourteenth greater than a half.

A fourteenth is smaller than a twelfth.

So seven-twelfths will be larger than eight-fourteenths.

Head on into the second part of our lesson, we're going to think about that toolkit for reasoning, thinking about all those strategies we've been discussing so far.

So what do you notice about the placement of the fractions to compare? Izzy says, 'I've noticed that lots of fractions can be compared by reasoning rather than converting to a common denominator.

' She says, 'I have a little strategy to help me decide how best to compare sets of fractions.

' Ooh, let's have a look Izzy.

I'm interested to hear about this.

She says, 'Firstly, I tried to visualise the fractions on a number line.

Can I compare the magnitude of the fractions in relation to on- half perhaps?' So she's got here one-tenth and six-sevenths.

And one-tenth is very close to zero on a number line and six-sevenths is very close to one.

So she can compare the magnitude and she can see that one-tenth will be a long way from being a half, much smaller than a half and six-sevenths will be a long way in the other direction, much bigger than a half.

And she remembers that for a fraction to be equivalent to one half, the denominator should be twice the numerator or the numerator half the value of the denominator.

And June says, 'One-tenth is less than one-half because the numerator is less than half the denominator.

And six-sevenths is greater than one-half because the numerator is more than half the denominator.

' So we could think about where they would sit on a number line and also compare them to a half.

So six-sevenths is greater than one-tenth.

Izzy says, 'We could also consider the magnitude of the part in comparison to the whole.

Is each fraction a lot of the whole, or is it a little of the whole?' And Jun says, 'Six is a larger portion of seven than one is of 10.

So therefore, once again, we can reason that six-sevenths is larger than one-tenth or one-tenth is smaller than six sevenths.

' So two things in our toolkit, comparing to a half and thinking about the magnitude of the fraction, how many pieces in comparison to the whole.

Izzy says, 'We can also use our understanding of generalisation when the denominators or the numerator are the same.

' So this is another way of reasoning.

So here we've got six-ninths and six-tenths.

When the numerator are the same, the fraction with the largest denominator is the smallest.

So six-tenths is gonna be smaller than six-ninths or six-ninths is greater than six-tenths.

June says, 'Or when the denominators are the same, the fraction with the largest numerator is the largest.

So if I've got four parts out of seven and five parts out of seven, five parts out of seven is going to be larger.

So four-sevenths is smaller than five-sevenths or five-sevenths is greater than four-sevenths.

' Izzy says, 'Sometimes we can consider how far the fractions are away from completing a whole.

' So we've got seven-eighths here and five-sixths.

Seven-eighths and five-sixths are both just one part away from the whole.

We'd need one more eight to make eight-eighths and one more six to make six-sixth.

And June says, 'One-eighth is smaller than one-sixth.

Therefore, if we minus these parts from the whole, subtracting one-eighth leaves a larger fraction than if you subtracted one-sixth.

' We're only a jump of one-eighth away from the whole.

So we've got more of the whole than if we had five-sixths.

One jump of one-sixth is a bigger jump than a jump of one-eighth.

So seven-eighths is larger than five-sixths.

What about five-twelfths and six-fourteenths? Well, Izzy says, 'Sometimes we can also consider how far the fractions are away from one-half.

' So one-half would be six-twelfths.

So we are one-twelfth away and it would be seven-fourteenths.

So we are one-fourteenth away.

Five-twelfths and six-fourteenths are both just one part away from half.

But one-twelfth is larger than one-fourteenth.

So we can say that five-twelfths is further away from one-half than six-fourteenths.

That sounds as though we're doing it wrong, doesn't it? But these fractions are both less than a half and a twelfth less than a half is further away from a half than one-fourteenth less than a half.

So six-fourteenths is larger than five-twelfths, or five-twelfths is less than six-fourteenths.

And finally, some fractions may be more easily compared using a common denominator.

So we've got seven-twelfths here and five-ninths.

So they're both just a bit bigger than a half, but they're not very friendly denominators to think about.

They're both multiples of three, but that's not really gonna help us this time, I don't think.

A common multiple of both 12 and nine is 36.

So let's use this as a common denominator.

So seven-twelfth is going to be equivalent to 21/36 and five-ninths is going to be equivalent to 20/36.

They were very close, weren't they? Only 1/36 is apart.

But we can see that seven-twelfths is equal to 21/36.

Five-ninths is equal to 20/36.

So 21/36 is larger than 20/36.

So our seven-twelfth was larger than five-ninths.

'So to compare fractions by reasoning, you could ask yourself these questions,' says Izzy.

Are the fractions greater or smaller than one-half? Can we easily compare them by thinking about their relationship to a half? Does each fraction represent a large part of the whole or a small part of the whole? Thinking about that magnitude of the fractions, picturing them if they could be clearly seen at different points on a number line.

Are the numerator or the denominators the same? If they are, then it's easy to compare them.

Are the fractions an equal number of parts away from a whole or a half? And can we compare the missing parts perhaps because then we'll have a common numerator.

And if you cannot reason with any of these strategies, you might then like to use common denominators.

So lots of options before we get to common denominators.

Time to check your understanding.

Compare these fractions by reasoning.

What strategy are you going to use? Pause the video, have a go.

When you're ready for some feedback, press play.

What did you spot about these fractions then? Well, June says, 'Both fractions are four parts away from the whole.

One-eleventh is smaller than one-ninth.

So four-elevenths is smaller than four-ninths.

Therefore subtracting four-elevenths from the whole would leave a larger portion of the whole left.

So seven-elevenths is the larger fraction.

' Five-ninths is smaller than seven-elevenths.

And time for you to do some practise.

Can you compare the following fractions? Think about Izzy's toolkit and think about the best strategy you could use to compare these fractions.

See if you can get away without using any common denominators.

And for question two, you've got some problems to solve.

In a, Alex and Lucas are making pasta with tomato sauce and they're using a different amount of the jar each.

And in b, Izzy and Andeep are reading the same book and they've read a different fraction of the book.

So it's about comparing fractions to see who has the most sauce left over and who's read the most of the book? And then in question three, for each sequence of fractions, tick the correct box to show if the fractions in the pattern are increasing in size, decreasing in size or neither.

Pause the video, have a go at the three questions and when you're ready for some feedback, press play.

How did you get on? So let's have a look at these first ones.

So seven-eleventh and four-sevenths.

Well, seven-elevenths is greater than four-sevenths.

What strategy did you use? Oh Izzy says, 'I did use common denominators to calculate the first one.

' Seven-elevenths, well that's greater than a half and four-sevenths is greater than a half.

Seven and 11 are both prime numbers.

So we'd have had to have used some really quite clever reasoning.

So I think common denominator is probably the best way for that one.

Well done Izzy! What about eight-ninths and five-sixths? What did you notice there? Yes, we've got that one away from the whole, haven't we? One-ninth to make a whole and one-sixth to make a whole.

We know one-ninth is smaller than one-sixth.

So eight-ninths is closer to a whole than five-sixths.

So is therefore greater.

And as June says in the second example, they're both one part away from the whole.

What about twelve-fifteenths and thirteen-sixteenths? Did you notice that each one was three parts away from the whole? Yes.

And sixteenths are slightly smaller than fifteenths.

So twelve-fifteenths will be less than thirteen-sixteenths because it's closer to the whole.

For the next one, one-third and 998/1000.

I think we can think about the magnitude of the fractions here.

One third is less than a half and 998/1000 is very, very close to one whole, isn't it? So one-third must be less than 998/1000.

What about nine-sevenths and ten-eighths? Well they're improper fractions.

Both of them are two of their unit fractions away from a whole, but greater in this case.

Nine-sevenths is two-sevenths more than a whole.

And ten-eighths is two-eighths more than a whole.

Sevenths are larger than eighths.

So nine-sevenths must be greater than ten-eighths.

And then we've got one and four-fifths and one and two-thirds.

Well, the ones are the same and both of our fractions are proper fractions less than one.

So let's compare the fractions.

Four-fifths is really quite close to a whole, isn't it? It's only one-fifth away and two-thirds is a third away from a whole.

So one and four fifths is going to be greater than one and two-thirds.

And on in to B where we were solving some problems. So Alex and Lucas are making pasta.

Alex uses eight-tenths of the jar, and Lucas uses four-sixths of the jar.

Who has the most source left over? So we're comparing eight-tenths and four-sixths.

Well, what did you notice? Well, they're both greater than a half, but they're both two of their unit fractions away from a whole.

Two more tenths would give us a whole, and two more sixth would give us a whole.

And two-tenths is smaller than two-sixth.

So therefore the eight-tenths is greater than four-sixth.

Alex has used more pasta sauce, so he has less left in his jar.

So Lucas has the most source left in the jar.

And what about Izzy and Andeep? Izzy has three-sevenths of the book remaining and Andeep has two-sixth of the book remaining.

So that's how much they've got left to read.

So who has read the most so far? So the person who's read the most will be the person with the smallest fraction left, won't they? So we're looking for the smallest fraction to win.

So we've got three-sevenths and we've got two-sixths.

So what can we think about these fractions? Well, they're four of each away from the whole, aren't they? But we could also think about, we could think about comparing them to a half here perhaps.

Three-sevenths, I know that's an odd denominator.

We'd need three and a half sevenths to be equal to a half.

So we're half a seventh, less than a half.

And two-sixth is one whole six less than a half.

So three-sevenths must be greater than two-sixths.

So Andeep has two-sixth of the book remaining, and that's the smaller fraction.

So he must have read most of the book because he has less left to read.

He had to think really carefully about what the fraction was representing and what the question was asking you there, didn't you? I had to think twice about that one as well.

And for question three, we were deciding whether the fractions in each sequence were increasing in value, decreasing in value, or staying the same.

So let's have a look.

In the first one, we've got all our unit fractions, and we know that when the numerator is the same, the bigger the denominator, the smaller the value of the fraction.

So these fractions are decreasing in size.

What about the next one? One-half, two-quarters, three-sixth, four-eighths.

Did you notice there all those fractions are equivalent to a half? So they're neither increasing nor decreasing.

What about the next one? Three-halves, four-thirds, five-quarters, six-fifths.

Well, they're all improper fractions and they're all one of their unit fraction above a whole.

So again, we can think about the denominators.

The smaller the denominator, the bigger the value of the fraction.

So three halves is going to be greater than four-thirds, which is greater than five-quarters, which is greater than six-fifths.

So those are decreasing in size.

And what about the last one? Six-thirteenth, seven-twelfths, eight-elevenths, and nine-tenths.

Well, our numerator are increasing by one each time and our denominators are decreasing by one each time.

So we're getting a larger number of parts of a larger part of the whole.

So that is an increasing sequence.

Well done if you spotted all of those.

And we've come to the end of our lesson.

So we've been explaining which strategy for comparing non-related fractions is most efficient and deciding how to compare fractions.

So what have we been thinking about today? While I hope we've learned that it's often more efficient to compare fractions using your sense of each fraction's magnitude, thinking about what that fraction means, what it would look like, where it would sit on a number line maybe in comparison to a half or a whole.

And to help you compare fractions without converting both to a common denominator, you could ask yourself Izzy's questions.

Do you remember? Are the fractions greater or smaller than one-half? Are the numerator or the denominators the same? Does each fraction represent a large part of the whole or a small part of the whole? And are the fractions an equal number of parts away from a whole or a half? Because then you can compare fractions with the same numerator.

Thank you for all your hard work and your mathematical thinking, and I hope this will help you, next time you are comparing fractions, not just to rush for a common denominator, but to use your fraction sense and your understanding of fractions to help to compare them.

I hope I get to work with you in a lesson again soon.

Bye-Bye!.