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 how to compare pairs of non-related fractions by comparing them to a half.

So can you think about what fractions equal to a half have in common? I wonder.

We're going to be looking at that and how we can use that information to compare fractions during this lesson.

Let's have a look and see what's in our lesson.

So we've got a key phrase and a key word, proper fraction and order.

So there may be things that are familiar to you, but let's just practise them and then look at what they mean.

So I'll take my turn, then it'll be your turn.

So my turn, proper fraction, your turn.

My turn, order, your turn.

Well done.

Let's just remind ourselves what those words mean because they are going to be very useful to us in our lesson today.

So a proper fraction is a fraction where the numerator, the top number, is less than the denominator, the bottom number.

So we've got less than a whole.

And to order things means to put things in their correct place following a set rule.

So we might be ordering from the smallest to the largest or the largest to the smallest in this lesson.

There are two parts to our lesson.

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

And in the second part, we're going to be using that to order fractions.

So let's make a start on part one.

And we've got Andeep and Sofia helping us in our lesson today.

Andeep and Sofia are playing a game with a 10-sided dice.

Andeep says, "Let's roll the dice twice and make a proper fraction." He says, "The person who can make a fraction that is closest to a half gets a point." And Sofia says, "If you roll exactly a half, you get two points." Okay, so let's watch them play the game.

Are you ready? So Andeep rolls first and he rolls a three and an eight.

And remember, he's got to make a proper fraction.

Can you think what fraction he's going to make? Well, he says, "I could make three-eighths or eight-thirds." Oh, but he's gotta make a proper fraction, hasn't he? He says, "Eight-thirds is an improper fraction." The numerator is greater than the denominator.

"So I'll have to go with three eighths," he says.

So he's made his fraction from his two dice rolls.

Sofia's turn.

She's rolled a two and a 6.

"Two-sixths or six halves?" What do you think? She says, yes, "It'll have to be two-sixth." That's the proper fraction she can make.

So she's made two-sixths.

So who's closest to a half? Let's have a look and think about a number line.

Andeep says, "Let's try to put them on a number line to help us." So he's got three-eights and Sofia's got two-sixths.

So we've got two number lines going from zero to one whole, eight-eights and six-sixths.

Well, Sofia says, "Well this is one half of each hole." So four-eights is half of a hole if we're thinking about eights and three-sixths is half of the hole if we're thinking about six.

So those fractions are both equivalent to a half.

Where do their fractions sit in relation to those fractions equivalent to a half? Well, Andeep says, "We can place three-eights here" and Sofia says, "Two-sixths will be placed here." Anything you notice? Whose fraction is closer to one half? Well, Andeep says, "Three eights is closer to one half than two-sixths." The distance on the number line is greater from two sixths up to three sixths, which is a half, than it is from three eighths up to four eighths, which is a half.

Let's have a look at it on a bar model.

Again, we've marked half of our eighths and the half of our sixths and we've coloured in three eighths of the top bar and two six of the bottom bar.

And Andeep says, "We can also think of this as bar models." And he says, "Each fraction is one part away from half." Did you notice that? Three eighths is one-eighth away from half and two sixth is one-sixth away from a half.

And Sofia says, "Since one-eighth is smaller than one-sixth, the distance to one half is smaller." "So one point to you." Three-eighths is greater than two-six and it's closer to a half.

Both of them are smaller than a half.

Three-eighths is slightly closer to a half than two-sixes.

So one point to Andeep.

They're going to have another round.

This time Andeep rolls three and a four.

Can you think what proper fraction he's going to be able to make? He says, "I could make three-quarters or four-thirds." Three quarters is a proper fraction.

So there's his fraction.

What about Sofia? She says, "Okay, it's my turn." She's rolled a four and a five.

So what proper fraction can she make with four and five? "Four-fifths or five-quarters," she says It'll have to be four-fifths to be a proper fraction where the numerator is smaller than the denominator.

So there's her fraction of four-fifths.

Anything you notice about these two fractions? Whose fraction is closer to one half? What do you think? What could they do to try and find out this time? Sofia says, "Why don't we find a common denominator this time to help us?" She says, "A common denominator for quarters and fifths would be twentieths." 20 is a common multiple for four and five.

If we listed the multiples of four and the multiples of five, the first one we get to that is shared by both four and five would be 20.

"So let's convert each fractions," says Andeep.

So to convert quarters into twentieths, we've got to scale everything up by a factor of five because four times five is equal to 20.

So we've got five times as many parts in the hole.

So we'll need five times as many of the parts to keep the fraction the same proportion of the whole.

So three quarters is equivalent to fifteen-twentieths.

What about four-fifths? Well this time we're scaling everything up by a factor of four because five times four is equal to 20.

And when we scale up the numerator as well to keep the proportion the same, we get a fraction of sixteen-twentieths, which is equivalent to four fifths.

"Now," says Andeep, "We can compare them to one-half." What do we know about a half represented as twentieths then? Well, let's have a look.

Here's a number line divided into 20 equal parts from zero to one.

So each part is one-twentieth, and ten-twentieths will be halfway along that number line.

"Ten-twentieths is equivalent to one-half," as Andeep says.

And Sofia says, "We can place fifteen-twentieths and sixteen-twentieths on the number line." So there's fifteen-twentieths and there's sixteen-twentieths.

So which one is closer to a half? Well, fifteen-twentieths is closest to one half.

That sixteen-twentieths is further away.

Fifteen-twentieths is closest to the half, so Andeep says, "Another point for me." He's doing well in this game, isn't he? Time to check your understanding.

On Andeep's next go he rolls a nine and a seven.

What proper fraction can he make? Pause the video, have a go, when you are ready for some feedback, press play.

How did you get on? He says, "Well, the only proper fraction I could make would be seven-ninths." Nine-sevenths is greater than one, so it's an improper fraction.

So seven-ninths is the proper fraction he can make.

And time for another check.

Sofia says, "Can you write down as many fractions equivalent to one-half as you can where the numerator and the denominator are only single digits?" So, as many fractions equivalent to a half as you can where the numerator and the denominator are only single digits.

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

How did you get on? Well, there's obviously one-half and then we can have two-quarters, three-sixths, four-eights.

If we had five as the numerator, what would the denominator be? It would be tenths, wouldn't it? And that's not a single digit.

So those are the four fractions equivalent to a half that we can write with single digit numerators and denominators.

And Sofia asks the question, "How would you reason what would represent one-half if the denominator was an odd number?" In all fractions equivalent to a half, the numerator is half the value of the denominator.

One is half of two, two is half of four, three is half of six, and four is half of eight.

So if we had an odd number as our denominator, we wouldn't end up with a whole number as our numerator.

And Andeep says, "One-half of three-thirds would be one and a half-thirds.

However, conventionally we don't represent fractions like this." But it can help us to think of what the half would be when we're comparing and ordering fractions.

One to watch out for in the rest of our lesson.

And a final check.

Can you tick the fractions that are less than one-half? Have a look at the numerators and denominators carefully, and remember the relationship between the numerator and the denominator when a fraction is equal to a half.

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

How did you get on? So one-tenth and eight-twentieths are both less than one half.

In a fraction equivalent to one half the numerator is half the value of the denominator.

So for fractions less than a half the numerator will be less than half the value of the denominator.

And one is certainly less than half of 10.

And eight is less than half of 20.

Five-thirds is an improper fraction, so it's greater than one.

And with four-fifths, we've got an odd number denominator.

So, half the value of the denominator would be two and a half-fifths.

And we wouldn't write that as a fraction, but we know that four is greater than two and a half.

So the fraction is greater than one half.

So one-tenth and eight-twentieths were our two fractions less than a half.

And as Sofia says, "When the numerator is less than half the denominator, the fraction is less than one-half." And it's time for some practise.

You're going to play the game that Andeep and Sofia were playing.

So roll a 10-sided dice or if you haven't got a 10-sided dice, you could use one to 10 digit cards or even just bits of paper with the numbers one to 10 written on them, and you could turn two over to generate your fraction.

So you've got to take your two numbers and turn them into a proper fraction.

Score one point for the fraction closest to one half and two points if you get one half exactly.

How will you convince each other whose fraction is closer to a half? Will you use a number line or will you be able to reason it using your knowledge of the properties of a fraction equivalent to a half? Who can be the first person to get to five points.

And have a think.

How could you adapt the game, change it in some way, to make it more challenging.

Have fun playing the game, and when you're ready for some feedback, press play.

How did you get on? Did you have fun? I hope you enjoyed playing the game.

I hope you used some reasoning, you maybe used some number lines, you might have used equivalent fractions, or you might have used your knowledge of fractions equivalent to a half where the numerator is half the value of the denominator.

And you might have been able to use that reasoning to find out whether your fraction was closest to a half or not.

And Andeep says, "To make it more challenging, we could roll the dice three times instead to give us more options to choose what the numerator and the denominator would be." And then you might be able to choose the numerator and denominator that gave you the best chance of being closest to a half.

And on into the second part of our lesson.

We're going to be ordering fractions.

So how can you order these fractions from the largest to the smallest? So that's our rule, largest to smallest.

So we're going to put them in order.

And we've got two-thirds, two-quarters and two-tenths.

Anything you notice? Andeep says, "I can see that one of the fractions is equivalent to one-half." Can you see which fraction that is? Remember when a fraction is equivalent to a half, the numerator is half the value of the denominator.

So it's two-quarters, isn't it? Andeep says, "We can now compare the other two fractions in comparison to that half." So we've got two-thirds and two-tenths.

So you might have been able to see, just by looking at the numerators and denominators, but let's have a look at those number lines.

So we can see here that two-quarters is equal to a half.

And as Sofia says, "We can see that two-thirds is greater than a half." And two-tenths is less than a half.

So remember, this time, we're just ordering them.

We're not scoring points on what's closest to a half, but we're ordering them.

So our largest fraction was two-thirds.

Our smallest fraction is two-tenths.

And two-quarters was equivalent to half, so that one goes in the middle and is in between the two fractions.

Did you notice anything else about the fractions? Sofia says, "We could have reasoned this differently." Have you spotted what she spotted? She noticed that all the numerator were the same.

They all had a numerator of two.

And she says, "When the numerator are the same.

." Oh, Andeep's worked it out.

"The fraction with the smallest denominator is the largest." There are fewer parts in the hole, the smaller the denominator.

So when the numerator are the same, we can look to our denominators to order our fractions.

So two-thirds is the largest, two-tenths is the smallest.

Of course the order's going to be the same.

We've just reasoned about it in a different way.

And two-quarters will go in the middle again.

So when we divide something into three parts, the parts are larger than when we divide it into four or 10 parts.

So therefore two of those larger parts will give us the largest fraction of the whole.

Well done if you spotted that.

Time to check your understanding.

Can you tick the fractions that are greater than one half? Have a look carefully.

Remember what it is that makes a fraction equivalent to a half.

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

How did you get on? Well, four-fifths, nine-tenths and seven-twelfths are all greater than one-half.

The numerator is greater than half the value of the denominator.

In four-fifths we have that odd number denominator.

So two and a half-fifths would be equivalent to a half, and four is greater than two and a half.

Half of 10 is five, so nine is greater than five.

And half of 12 is six, so seven-twelfths is just over a half.

And Sofia's reminding us, "When the numerator is more than half the denominator, the fraction is more than one-half." And another check.

Can you order these fractions from the largest to the smallest? Have a think about what you notice.

Are there two ways you could perhaps reason as to which is the largest and which is the smallest? Pause the video, have a go, and when you're ready for some feedback, press play.

What did you spot there? Did you spot the common numerators again, that they all had a numerator of three? Or maybe you spotted that three-six is equal to one half because three is half of six.

So if we think about those common numerators, we know that when the numerator is the same, the fraction with the smallest denominator will be the largest.

So three-quarters will be the largest, three-six, which is equivalent to a half goes in the middle, and three eighths is the smallest.

But we could also compare to a half.

Three-six is our half.

Three-quarters is greater than a half because three is greater than half of four.

And three-eights is smaller than a half because three is smaller than half of eight.

So whichever way you reasoned, I hope you got them in the right order.

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

You're going to order these sets of fractions from the largest to the smallest.

See how many different ways you can reason about it without having to use a common denominator.

For question two, you're going to use the digits one to five.

How many different ways can you fill in the missing boxes? We've got a fraction of a half, and we've got to create a fraction that is less than a half, and a fraction that is greater than a half using just the digits one to five.

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 in our first set of fractions, we had a half, a quarter, and a sixth.

And you could see there we had common numerators.

They were all unit fractions.

So the fraction with the smallest denominator will be the largest fraction.

So one-half is the largest, then one-quarter, and then one-sixth.

Did you notice that the numerators were the same, so you could compare the denominators? The larger the denominator, in this case, the smaller the fraction.

What about B? We haven't got common numerators this time, but we have got common denominators.

So now we can just compare the numerators.

The smaller the numerator, the smaller the fraction.

So four-sixths will be our largest, three-sixths, which is equivalent to a half, in the middle, and two-sixths will be the smallest.

So we could have compared to a half as well.

But did you notice that the denominators were the same, so you could compare the numerators? And also three-six is equivalent to one half.

What about C? What did we notice here? Well, four-eighths is equivalent to a half, eight-seventeenths.

Well, eight sixteenths would be a half.

So eight-seventeenths is going to be just smaller than a half.

And four-sevenths is going to be just larger than a half.

So four-sevenths is the largest, four-eighths is the half, which is in the middle, and eight-seventeenths is the smallest.

So did you compare the numerators and denominators within the fraction? Four-eighths is equal to one half.

And even though we had odd numbered denominators, we could still see that it will be three and a half-sevenths that will be equivalent to a half.

So four sevenths is larger than a half.

And half of 17 is eight and a half.

So eight-seventeenths is just less than a half.

And what about for D? Got some really big numbers there, haven't we? But I wonder if we can spot anything that'll help us.

Well, 150/300ths is equal to a half, isn't it? 110/240th? Well, half would be 120/240ths, so that's got to be smaller than a half.

And if our denominator is 220, a half would be 110.

So 150/220th is greater than a half.

So that's our largest fraction.

150/300ths is equivalent to a half is in the middle.

And 110/240ths is the smallest fraction.

So one is equal to a half, one is less than a half, and one is greater than a half.

I hope you were able to use your reasoning for that one.

I wouldn't have wanted to create a common denominator for that set of fractions.

And for question two, you were using the digits one to five.

How many different ways can you fill in the missing boxes? And Sofia says, "Here's one solution I found." I wonder how many different ways you managed to find to solve the problem.

So she said that two-fifths is less than a half, half would be two and a half-fifths, so two-fifths is less than a half, and three-quarters is greater than half because we know that two quarters is equal to a half.

Well done for reasoning your way through those questions.

I hope you enjoyed it.

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

We've been explaining how to compare pairs of non-related fractions by comparing to one half.

What have we learned about and thought about today then? Well, we've learned that you can compare fractions to one half to help estimate the magnitude, the size of a fraction.

In fractions equivalent to a half, the numerator is half the value of the denominator.

When the numerator is more than half the value of the denominator, the fraction is greater than a half.

And when the numerator is less than half the value of the denominator, the fraction is less than a half.

And we've also found that we can use this reasoning even when our denominator is an odd number.

Although we don't have a right fractions with a fractional numerator, we can use it as part of our reasoning.

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

Bye-Bye.