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 be ordering sets of non-related fractions using a range of strategies.

And by the end of the lesson, you are going to be able to decide on the most efficient strategy to compare and order fractions.

You may be getting really quite good with these strategies, so let's put them to the test and let's see if we can always find the most efficient way to sort, compare and order fractions.

We've got two phrases.

We've got improper fraction and proper fraction, so I'll take my turn to say them and then you could have a go at rehearsing them.

Are you ready? My turn, improper fraction, your turn.

My turn, proper fraction, your turn.

They may well be phrases you're familiar with.

Let's just double check their meaning though, because we are going to be using them in our lesson.

So an improper fraction is a fraction where the numerator is greater than the denominator.

It's a fraction greater than one.

And a proper fraction is a fraction where the numerator is less than the denominator.

And we know that when the numerator and the denominator are equal, fractions are equal to one.

There are two parts to our lesson.

In the first part, we're going to be thinking about increasing fractions, and in the second part, we're going to be using reasoning to order fractions.

So let's make a start on part one.

And we've got Sam and Sofia in our lesson with us today.

Sam and Sofia are playing a game.

They spin a one to 10 spinner twice to create a fraction with the digits it gives.

They then have to spin twice again.

However, this time they need to create a fraction which is larger than the last one.

The winner is the person who can make the longest sequence of increasing fractions.

If you haven't got a one to 10 spinner, maybe you could use some one to 10 digit cards as well.

Sam says, "I can't wait to get started." Sofia says, "In that case, you can go first." That's very kind of you, Sofia.

Okay, so Sam's going to create two fractions.

Here goes.

She spun the spinner and she's got one.

Okay, she says, "First number is one." So she knows she's going to create two fractions, so she's put one as the numerator for one fraction and as the denominator for the other fraction.

What sort of fraction is that going to create? Let's spin it again.

And she's got a three.

She says, "I could make one third or three over one." Three wholes, isn't it, three over one? So one third and three over one.

So now she's got to choose one to start the fraction sequence.

Sofia says, "One third is the smallest fraction here, you should probably go for that." That's a good strategy, I think, Sofia.

If we want a string of increasing fractions, starting with a small one is a good idea.

So she's gonna start with one third.

Okay, so Sofia says, "I will spin this time." So she's got a two.

It's her first number, so that could be the numerator or the denominator.

What's she gonna spin next, do you think? And she's got a four.

"And a four," she says, "I could make two quarters or four halves." "Well," says Sam, "I know two quarters is equivalent to one half and one half is larger than one third." Because that was Sam's fraction, wasn't it? "So let's go for that one," she says.

So we had our one third and now we've got two quarters.

And two quarters is greater than one third.

So we have got a fraction that's larger, so we are increasing.

"Let's see if we can get three in a row," says Sam.

So she's going to spin again.

She's spun an eight.

So that could be the numerator or the denominator of her fraction.

I wonder what her next one's gonna be.

What do you think she'd like to spin? And she spun a five.

"And the second number's five," she says.

"that could be eight fifths or five eighths." "What will you go for?" Says Sofia, What would you go for if you were Sam? Remember, we've got to keep the fractions increasing.

We've got one third and two quarters, which we know is equivalent to a half.

Does that give you a clue? She says, "Well, I know I could use eight fifths as it is an improper fraction, so it will be bigger." But five eighths is just larger than one half because four eighths is equal to one half.

"So I'll choose five eighths," she says.

So yeah, that is still right, the fractions are increasing.

Five eighths is greater than two quarters.

Time to check your understanding now.

Sofia spins a three and a six.

Can she make a larger fraction than the last example to keep the increasing pattern? Pause the video, have a think, and when you're ready for some feedback, press play.

What did you think? Sofia says, "Three six is equal to one half, so that will be smaller." So she can't use three-sixths.

"It would have to be the improper fraction of six thirds," she says.

So she has completed a fraction that continues that increasing sequence of fractions.

One third is smaller than two quarters, which is smaller than five eighths, which is smaller than six thirds.

And it's time for you to have a go, play the game with a partner.

So as I say, if you've got a spinner, that's fantastic.

If not, you could just use some one to 10 digit cards and turn them over to see what fractions you can make.

So play the game with a partner.

What is the longest sequence you can make? And Sofia says, "How could you adapt the game to provide a different challenge?" Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Did you enjoy playing the game? How long was your sequence? Sam made a sequence of seven fractions when she played.

Look at that.

So they started with one third and ended with nine halves.

Look at that, we've got eight halves and nine halves.

Six thirds, that's equivalent to two, isn't it? Five- eighths just bigger than a half and two quarters is equal to a half.

Fantastic.

Sam says, "This was the whole sequence I created before I ran out of options." She couldn't create another fraction from the numbers she had that was greater than nine halves.

She says, "I had to use three improper fractions, but they still increased in size." And Sofia says, "You could change the rules so that the fractions have to decrease in size each time." That's another way you could do it.

Or you could also say that you could only use proper fractions.

That would be a challenge, wouldn't it? I hope you had fun playing the game.

Maybe you could try one of those alternative rules that Sofia suggested.

So decreasing the size of the fraction each time, or only limiting yourself to being able to use proper fractions.

And on into the second part of our lesson, we are going to be reasoning in order to order fractions.

So Sam and Sofia decide to change their game.

"This time," Sam says, "we're going to spin the spinner eight times." She says, "I wonder if I can make fractions with the number it gives that can be ordered into this sequence." So she's going to generate her eight numbers.

So she says, "We've spun, these were the numbers we got when we spun the spinner." So they got four, five, one, one, two, nine, four, and five.

Can they create four fractions where the value increases each time? Let's see.

"I wonder what fractions we can make with these," says Sam.

She says, "I think we should try to make the smallest fraction we can now." That's probably good, if you're increasing, you want to start with a small value, don't you? So what's the smallest fraction they can make? You might want to have a think before they share their ideas.

Sofia says, "I think if we create a unit fraction with a large denominator, this would be quite a small fraction." I think she's right, isn't it? A unit fraction is just one of the whole, and the bigger the denominator, the smaller the part of the whole.

So what's the biggest denominator they could use? "Okay," says Sam, "should we say one ninth is the smallest fraction?" So they started with one ninth, so they've had to cross out a one and then nine.

"What should we do next," says Sofia.

Well Sam says, "What's the largest fraction we could make with what's left over?" Can you have a think? And Sofia says, "Well, the largest fraction we could make would be an improper fraction.

If we find the smallest denominator and the largest numerator, that would be the biggest improper fraction we can have." So what's their smallest denominator and the largest numerator? Oh, that would be five over one, five wholes.

"Okay," says Sam, "now just two fractions left to make." So they've got to make a fraction that's greater than a ninth, but leave themselves some numbers to make a fraction that is greater than that fraction, but less than five.

Sofia says, "They need to be greater than one ninth, but smaller than five ones.

I think we can do it." Now that we know the largest and the smallest fractions, we just need to make two fractions, one that is larger than the other.

'Cause they know they can't make anything smaller than a ninth, and they can't make anything bigger than five ones.

"Well, if we look at the proper fractions we can make," says Sofia, "they can make two quarters, two fifths, or four fifths," because they've only got two fours, a two and a five left.

So those are the three proper fractions that they can make.

Sam says, "I can see two quarters which is a half, and two fifths which is less than a half.

Let's use these," she says.

So two quarters is equal to a half, but it's certainly less than five whole ones.

And two fifths is less than a half, but greater than one ninth.

So they've done it, they've created four fractions from their eight spins of the spinner.

"I wonder if there was another way of doing it with the same numbers?" says, Sofia.

You might want to explore that.

Time to check your understanding.

Sofia spins the spinner six times and gets these numbers, and she puts them in the missing boxes like this.

What would she need to roll to fill in the remaining missing boxes? So she's created a quarter, four sevenths, and three halves.

What could she roll the spinner for her final two rolls that would give her the right numbers to make a fraction to fill in that box, which is greater than a quarter, but less than four sevenths? Pause the video, have a think, and when you're ready for some feedback, press play.

What did you think? Well Sofia says, "Here's one solution I found." She's rolled a three and a six and three sixths is equal to a half, and we know that's greater than a quarter.

And four sevenths is just greater than a half.

So she could have picked two numbers that would've given her a fraction equivalent to a half.

And there are lots of other fractions between one quarter and four sevenths.

But being able to use that ability to reason to a half was good.

And time for you to do some practise, you are going to play this game with a partner.

So generate your eight numbers, either by spinning a spinner or by turning over some digit cards, who can create a sequence with the largest difference between the smallest and the largest fraction? What about the smallest difference between the smallest and the largest fraction? So you're really going to think about the smallest and largest fractions you can make.

See if you can create a really big difference between your four fractions.

And then can you complete the same task with increasing values? But this time with a really small difference between your smallest and largest fractions.

And in question two, the same digit can be used in each box so that the fractions are arranged in ascending order.

So that's getting larger each time.

What digit is this? So the same digit in each of those three boxes so that the fractions get each time.

I wonder what that is, can you work it out? Pause the video, have a go at those two questions, and when you are ready for the answers and some feedback, press play.

How did you get on? Did you enjoy playing the game? So we were asking you to think about creating a sequence of increasing fractions, but thinking about the difference between your smallest and largest fractions.

What was the biggest difference you could create, but could you also complete it by creating a small difference? So here are some numbers that we generated.

Sofia says, "Here's an example of one where the difference between the smallest and the largest fraction was the smallest it could be." So she created two tenths as her starting fraction, which is quite small, isn't it? Not the very smallest she could make.

And then she had two eighths.

So she used her common numerator.

So she knew that two eighths was going to be greater than two tenths.

And two eighths is equivalent to one quarter, so that is smaller than one third, and what's she got left? She's used two, eight, a 10, a three, a one, and so she's got two fifths left.

And two fifths, well, we know one third is equivalent to two sixths.

So two fifths must be bigger than one sixth.

So this time the difference between two tenths and two fifths is not that great.

All the fractions were less than a half.

I wonder if you managed to create any with a smallest difference as that.

And for question two, you were reasoning about which digit could go into the missing boxes, the same digit each time, so that the fractions continued in an ascending order, getting larger.

I wonder what your strategy was.

Did you maybe start thinking, well, was it one, could it be one third, one fifth? No, it couldn't be one, could it? Because one fifth is less than one third and we've got to have a fraction that's greater than one third.

I wonder if we could try two.

One third, two fifths.

Well, this, we know that two fifths is greater than the third, then we'd have one half.

Well, yes, one half is greater than two fifths.

Four sevenths is greater than a half, and then three halves, well, that's greater than one, so it must be larger than four sevenths.

So yes, that was right.

So two was the missing number.

And Sofia says, "The missing number could be two.

This would make the fractions go up from the smallest to the largest." I wonder what strategy you used.

Did you try with one and then try two? Sofia said, "It helped me to think of the fractions on a number line, making the middle fraction equivalent to one half." Oh, that's a really good idea.

So once she'd put that half in the middle, she could then test out whether two worked in the other two spaces.

A really good strategy, Sofia, well done.

And Sofia also says, "Can you make your own example for a partner?" Maybe you could create some fractions and have the same number crossed out from two or three of them, and challenge your partner to fill in the missing number.

I hope you've enjoyed reasoning about fractions in those tasks.

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

We've been ordering sets of non-related fractions using a range of strategies.

And hopefully you've been evaluating your strategies and deciding which is the most efficient to use.

So what have we been thinking about and learning about in this lesson? Well, we've learned that you can apply your knowledge of comparing fractions to help you to order fractions as well.

And when ordering fractions, it can be helpful to identify fractions that are close to or equivalent to a half, as well as fractions that are easily recognisable as the largest or the smallest of the group.

And those strategies that you may have come across before comparing to a half, comparing to one, finding fractions with common numerators and common denominators, and then thinking about the magnitude, the size of those fractions, imagining them on a number line.

Lots of strategies you can use that don't involve you having to use common denominators in order to compare and order fractions.

Thank you for all your hard work and your mathematical thinking in this lesson, and I hope I get to work with you again soon.

Bye-Bye.