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.

In this lesson, we are going to explain how to compare pairs of non-related fractions using our fraction sense.

So not always reaching for a common denominator, but thinking about what we know about fractions in order to help us to compare non-related fractions.

So those fractions where the denominators are not multiples or factors of each other.

Let's have a look at what's in this lesson then.

We've got one key word, and that's "magnitude." Can you say it? I'll take my turn, then it'll be your turn.

So my turn, magnitude.

Your turn.

Let's have a look at what it means.

It may be a new word to you.

So the magnitude of something refers to the size of something.

So we're going to be thinking about the size of fractions, and we can refer to that as their magnitude.

There are two parts in our lesson today.

In the first part, we're going to be comparing strategies.

And then in the second part, we're going to be reasoning proportionally.

So let's make a start on part one.

And we've got Sam, Lucas, and Alex helping us in our lesson today.

So how would you compare the following fractions? We've got one-third and three-quarters.

You might want to have a think before Alex and Lucas share their thinking.

Alex says, "Let's convert them to common denominators." Would you do that? Lucas says, "I don't think we need to.

We can just reason about their magnitude." It's our keyword, isn't it? Thinking about their size.

How could we position these on a number line then? Alex says, "I can visualise the whole being divided into three parts." So there's our whole divided into three equal parts.

And one-third would be about here," he says.

And Lucas says, "I can visualise the whole being divided into four equal parts." So there we go.

So three-quarters will be placed here.

So it's really clear, isn't it, which one has the greater magnitude? Alex says, "We can see that three-quarters is larger than one-third," and we could picture that quite easily, couldn't we? So one-third is less than three-quarters.

Can we think about reasoning around a half? Alex says, "Yes, we can compare each fraction to one half as another way of thinking about working out and using our fraction sense in order to compare the fractions." One-third is less than one-half.

We know that in fractions equivalent to one-half, the numerator is half the value of the denominator.

So in a fraction with an odd number denominator, that's a bit harder to think about.

But we know that half of three is one and a half, so it would be one and a half-thirds, so one-third must be less than a half.

And three-quarters is greater than a half.

We know that a half is equivalent to two-quarters, so three-quarters is greater.

So that's another way about reasoning to find out that one-third is smaller than three-quarters, and that doesn't rely on a sketching.

And that can be tricky when the fractions are closer in value to each other.

So, again, one-third is less than three-quarters.

We could draw a bar model.

We have to be sure that our bars are the same length because our whole has to be the same in each case.

But we could then look at the portion of each bar that is shaded.

And three is a bigger part of four than one is of three.

Oh, let's think about that.

That's thinking about the proportion.

So three-quarters is three out of four equal parts, and three out of four is a greater proportion of the whole.

A bigger portion of one than one is out of three.

So there's another way that we can think and reason about the size of those two fractions.

So there's three-quarters of one and one-third of one, and the three-quarters is a bigger portion.

Three is a bigger part of four than one is of three.

So three-quarters is larger than one-third.

And Lucas says, "You can also see that three out of four is nearer to the whole than one out of three." So how would you correct these bar models? What do you notice about them? You might want to have a think before Alex and Sam share their thinking.

Sam says, "Here are the bar models I drew to represent the fractions." So, remember, they were comparing one-third and three-quarters.

What's she done wrong? She says, "I think one third is larger than three quarters." Alex says, "I can see two problems here." Can you see the two problems that Alex has spotted? Firstly, Alex says, "The parts which are supposed to be one-third are not all equal in size." Can you see that? There are three parts in the whole, but they're not all equal.

And what else is wrong? Ah, that's right.

"And also," Alex says, "the wholes aren't the same size, so you cannot compare the areas of each fraction." When we are comparing one-third and three-quarters, we are comparing one-third and three-quarters of the same whole.

If we're just thinking about the numbers, we are comparing one-third of one and three-quarters of one.

But when we draw a bar model or another shape, we've got to make sure that the whole is the same before we start dividing it into three and four equal parts.

So Sam's models don't show us that one-third is greater than three-quarters.

We can't compare these.

The thirds aren't thirds because they're not equal parts and the wholes are not the same.

So we need to be really careful when we're drawing and sketching to help us to compare fractions.

Time to check your understanding.

We've told you here that two-thirds is greater than one-fifth.

Can you reason why? Can you find different ways to explain why that is true? You might want to draw or sketch something, but different ways to reason as to why that is correct.

Pause the video, have a go.

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

How did you think about it? Well, you might have drawn some bar models, remembering, learning from Sam's mistake, that our bars have to be exactly the same length and our pieces have to be equal in size.

And that's quite tricky if you're just drawing them freehand for yourself.

So here we've got bar models that are the same length overall, so they represent the same whole and the parts are equal in size.

And so we can see here that two-thirds of this whole is greater than one-fifth of this whole.

And Alex says, "I know that two out of three parts is larger than one out of five parts." It's a larger proportion or portion of the whole.

How else could you have reasoned about this? Well, you could have thought about the fact that two-thirds is greater than a half and one-fifth is less than a half.

And that's another way to reason that two-thirds is greater than one-fifth.

Time for another check.

Can you draw a bar model to reason why eight-tenths is greater than two-fifths? Pause the video, have a go.

Think carefully about the size of the parts and the size of the whole, and when you're ready for some feedback, press play.

How did you get on? So there's a bar divided into 10 equal parts and we've shaded eight of them, so we're representing our eight-tenths.

Now our bar needs to be the same length, and this time it's divided into five equal parts.

And can you see that each of those fifths is worth the same as two of the tenths? Because two-tenths is equivalent to one-fifth, and we've shaded two of them.

So we can clearly see that eight-tenths is greater than two-fifths.

And we could also have reasoned about that thinking about a half.

Eight-tenths is greater than a half and two-fifths is less than a half.

Two and a half-fifths would be equal to a half.

And it's time for you to do some practise.

You are going to compare three-eighths and four-sixths.

How many different ways can you reason about your comparison? What could you draw and how else could you reason about those fractions and which is greater than the other? And question two, using the digits one to nine, at most once each, fill in the missing boxes.

There are eight boxes to fill in and nine numbers, so one of them won't be used each time.

How can you complete the fractions to make the inequality and the equality true? So one fraction is greater than another and one fraction is equal to another.

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

How did you get on? So how many different ways can you reason about your comparison of three-eighths and four-sixths? Well, three-eighths is less than four-sixths, and you might have drawn bar models making sure that your bars were the same length and your parts were equal sizes.

And you can see there that three parts out of eight is a smaller part of the whole than four parts out of six.

You might have put those numbers on a number line and compared them to a half.

So if we have a number line marked in eighths, four-eighths is our half, three-eighths is less than that, and a number line marked in sixths, three-sixths is our half and four-sixths is greater than that.

You might have been able to reason that without the number line as well.

And Alex says, "I know that four out of six parts is larger than three out of eight parts when we are thinking about the same whole." And on into question two, where you were using the digits one to nine.

You couldn't repeat them, but there would be one missing each time.

So, how did you fill in the missing boxes? So we've got seven-ninths and that's greater than three-sixths.

Well, seven-ninths is greater than a half, and three-sixths is equal to a half, so seven-ninths must be greater.

What would a half be in ninths? Well, half of nine is four and a half, so we'd have four and a half-ninths and seven is definitely greater than four and a half.

And what about the equivalent fractions? Well, we had one-quarter and two-eighths that we could make as our equivalent fractions.

And Lucas says, "Here is one solution I found.

I wonder how many different solutions you found." And on into the second part of our lesson where we're going to be reasoning proportionally.

Hmm, let's think about that word.

So how would you compare four-sevenths and five-eighths? And can these be compared without using common denominators? You might want to pause and have a think here before Alex and Sam share what they did.

Okay, well, Sam says, "Shall we start by drawing a bar model or a number line?" Well, Alex says, "These can take quite a while and we have to be really accurate with our drawings too." We've got sevenths and eighths here, and if we are just sketching, it would be really quite tricky to draw accurately enough and to mark a number line accurately enough to be able to make that comparison.

So I think we might have to think about some other reasoning here.

Alex says, "I've got an idea.

Let's start here," he says.

What about one-seventh and one-eighth? "Can we compare these two fractions?" he says.

"That's easy," says Sam.

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

So one-seventh is greater than one-eighth." I like your reasoning, Sam.

I hope you do too.

What about two-sevenths and two-eighths? "What do you notice now?" says Alex.

Sam says, "Okay, so the numerators are now both two.

We can still apply the same rule.

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

So two-sevenths are greater than two-eighths." If one-seventh was greater than one-eighth, then two-sevenths must be greater than two-eighths.

Ah, what about seven seven-sevenths and eight-eighths? "Okay, next one.

What do you notice this time?" says Alex.

What do you notice? Sam says, "Now the numerators are different! And you've jumped up quite a few parts.

When the numerators and denominators are the same, each fraction represents one whole." So what are we going to put into the circle here? Seven-sevenths is equal to eight-eighths.

So those are equal.

Six-sevenths and seven-eighths we've got.

"What about now?" says Alex.

"Look carefully." What do you notice? "Ooh, I've got it," says Sam.

They are both one part away from the whole now." Now six parts out of seven and seven parts out of eight.

So each of them are one part away from the whole.

One-seventh away for six-sevenths, and one-eighth away for seven-eighths.

"And because one-seventh is larger than one eighth, that means that one-eighth has had less taken away from the whole.

So seven-eighths is a larger fraction." Ah, let's just think about that reasoning.

Picture that, perhaps.

Seven-eighths is going to be closer to one because one-eighth is a smaller part of the whole than one-seventh.

So seven-eighths must be the larger fraction.

There's a smaller piece missing to make a whole.

And there we can see it.

And I think Alex was right, it would've been difficult to make that accurate unless we were drawing with rulers and then we'd have had to have done quite a lot of maths around the common denominators to get our bars the same length and to divide them accurately.

So, the reasoning that we've done has meant that we haven't had to rely on accurate drawing to help us.

And there we can see one-eighth is smaller, so a smaller part has been taken away.

So what can we think? Can we use all that reasoning to help us now? "Sneaky," says Sam.

"They're both now two parts away from the whole.

And because one-seventh is larger than one-eighth, that means that two-sevenths is a larger portion to take away from the whole than two-eighths.

So six-eighths is the larger fraction." And we can see that there.

If we accurately draw bar models, we can see that six-eighths is the larger fraction.

Two-eighths is a smaller part of the whole, so we are closer to one.

It's the larger fraction.

That was really good reasoning.

I hope you followed all that through.

And she's just pointing out again, "Two-eighths is smaller, so a smaller part has been taken away from the whole." "And finally," says Alex, "Back to where we started." So this is the fraction comparison that we started with.

"No way!" says Sam.

"That's so much easier to reason with now.

We don't need common denominators.

Also quicker than finding common denominators.

One-seventh is larger than one-eighth, that means three-sevenths," which is what we need to make four-sevenths up to a whole, "is a larger portion to take away from the whole than three-eighths." And we need to add three-eighths onto five-eighths to make our whole.

And three-eights is smaller than five-eighths.

So, therefore, five-eighths is the larger fraction.

Fantastic, we've reasoned our way right the way through.

Well done to Alex and Sam.

And I hope you followed on that and I hope you had Sam's reaction as well.

"No way, that's really easy now, now I've thought about it." And there are the bar models, just to prove it again.

But remember that the bar models we've created, we've created really accurately, we haven't just sketched them.

And sometimes if you sketch fractions that are very close together, we can make mistakes in the sketching.

Three-eighths are smaller, so a smaller part has been taken away from the whole again.

Time to check your understanding.

Can you compare these fractions and explain how you know we've got six-sevenths and five-sixths? Think about that reasoning that Alex and Sam were doing.

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

How did you get on? Did you spot something? Sam says, "One seventh is smaller than one-sixth." Why is that important? Our six-sevenths is one-seventh away from the whole, and five-sixths is one-sixth away from being a whole.

So one-seventh is smaller than one-sixth, therefore, taking one-seventh from the whole would leave a larger amount.

And there we've got an accurate representation with the bar models.

So six-sevenths is greater than five-sixths because that one-seventh is a smaller portion of the whole to take us up to the whole than the one-sixth would be.

I hope you got that reasoning right.

And time for you to do some practise.

Can you compare the following fractions? And remember we are using our fraction sense in this lesson, thinking about what we know about unit fractions, fractions with a common numerator, fractions with common denominators, fractions equivalent to half, trying not to use common denominators at this point.

So compare the following fractions and put the correct symbol into the circle.

And then can you create a pair of fractions that you could compare using this same sort of reasoning.

And for question two, you've got a couple of problems to solve.

One about bakery and some baguettes.

And in question B, Sofia and her sister with drinks bottles.

And it's worth remembering that in A, the two bakeries bake the same number of baguettes so our whole is the same.

And Sofia and her sister have the same drinks bottle.

So, again, the whole is the same in both questions.

Pause the video, have a go at questions one and two.

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

How did you get on? So let's have a look.

So we had four-fifths and seven-eighths.

Did you notice that each one was one unit fraction away from the whole? So four-fifths is one-fifth away from the whole.

Seven-eighths is one-eighth away from the whole.

One-eighth is smaller than one-fifth, so, therefore, seven-eighths is going to be greater.

What about five-eighths and three-sixths? Well, three-sixths is equal to a half, isn't it? And five-eighths is just greater than a half.

But both of them, again, are three of their fractions away from the whole.

So we could either reason about that or we could reason about the comparison to a half.

But however we reason , five-eighths is greater than three-sixths.

What about nine-elevens and four-sixths? Well, I need two more elevenths to make the whole, and I need two more sixths to make the whole, but elevenths are much smaller than sixths.

So nine-elevenths is greater than four-sixths.

Oh, this is an interesting one.

99-hundredths and nine-tenths.

Well, I think we can picture that, can't we? You could picture a 100 square for this, couldn't you? 99 out of a 100 there'd be one tiny square not shaded in.

Nine-tenths we'd have nine of our 10 rows or columns coloured in and one whole row or column not shaded.

We can also see that that's one of our fraction to get away.

One one-hundredth or one-tenth.

And one one-hundredth is much smaller than one-tenth.

So 99 100-hundredths must be greater.

It's much closer to being a whole.

And finally, we've got one and two-thirds and one and twelve-thirteenths.

We've got mixed numbers here.

Both of our fractions are proper fractions and we've got one as our whole in each time.

So we've really only got to compare the fractions.

So we've got two-thirds and twelve-thirteenths.

So, I need one more third to make a whole, I need one more thirteenth to make a whole.

And a thirteenth is much smaller than a third, so, therefore, it's much closer to the whole.

So one and two-thirds is less than one and twelve-thirteenths.

And Alex said he chose twelve-sevenths and eighteen-tenths.

Ah, that's interesting.

He chose improper fractions.

But he says they're both two parts away from two wholes.

He'd need two more sevenths to make fourteen-sevenths, which is two.

And he'd need two more tenths to make twenty-tenths, which is equal to two as well.

Tenths are smaller than sevenths, so two-tenths is a much smaller gap to one than two-sevenths'.

So twelve-sevenths is less than eighteen-tenths.

So question 2A was about a bakery.

Two local bakeries bake the same number of baguettes.

One bakery sells four-fifths of their baguettes.

The other bakery sells six-sevenths of their baguettes.

Which bakery has the most baguettes left? So we're comparing four-fifths and six-sevenths.

Both of them, again, one fraction away from the whole.

Sevenths are smaller than fifths, so four-fifths is less than six-sevenths.

So, therefore, that bakery has most baguettes left.

So the second bakery sold most of their baguettes.

So the first bakery has more baguettes remaining.

And for B, Sofia and her sister have the same drinks bottle.

Sofia drinks three-fifths of her water and her sister drinks five-sevenths of her water.

Who has the most amount of water left to drink? So we're comparing three-fifths and five-sevenths.

So, again, they're two parts away from a whole, two-fifths and two-sevenths.

Two-sevenths are smaller than two-fifths, so, therefore, five-sevenths must be greater than three-fifths.

Or the other way round, three-fifths is less than five-sevenths.

So who has the most amount of water left to drink? So Sofia has drunk less of her water.

So Sofia has more water left.

Well done if you've got those correct and well done for reasoning proportionally and for using your fraction sense to think about comparing fractions.

And we've come to the end of our lesson doing exactly that, comparing pairs of non-related fractions using fraction sense.

What have we learned about and thought about today? Well, you can use multiple strategies to compare fractions by developing a sense of the magnitude of each fraction, that sense of size of the fraction.

You can compare fractions by thinking about their magnitude in relation to the whole.

So we've been thinking a lot about a fraction that is two parts away from being a whole comparing to another fraction that is two of its parts away from being a whole and thinking about the size of those parts.

And you can use your understanding of the magnitude of each unit fraction to help you reason about the size of each non-unit fraction.

That's been a really important part of our thinking today.

You've worked really hard in this lesson.

You've done a lot of mathematical thinking and I hope you've enjoyed it as much as I have.

Thank you for your hard work and I hope I get to work with you again soon.

Bye-bye.