Lesson video

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

Fractions are great.

I love working with fractions.

And so I hope during this lesson you're going to see that fractions are really friendly and things that can really help us with our mathematical thinking.

So let's get started.

So in this lesson we're going to be explaining the relationship within families of equivalent fractions.

So these are fractions that are all related in some way.

And we're gonna have a look in this lesson at how those families of fractions are related and what makes them equivalent.

We've got lots of keywords in our lesson today.

We've got numerator, denominator, equivalent fraction, and scale up and down.

So I'll take my turn and then you have your turn.

So my turn, numerator.

Your turn.

My turn, denominator.

Your turn.

My turn, equivalent fraction.

Your turn.

My turn, scale up and down.

Your turn.

Now you may have come across those words before, but let's just check what they mean.

They're gonna be useful to us in our lesson today.

So the numerator is the top number in a fraction and it shows how many parts we have.

The denominator is the bottom number in a fraction and it shows how many equal parts the whole has been divided into.

Equivalent fractions are fractions that have the same value, even though they may look different.

And scaling is when a quantity has been made, hmm, times the size.

To scale up is to multiply by a factor and to scale down is to multiply by a fraction or to divide by a factor.

So look out for those words as we go through our lesson today.

We've got two parts in our lesson.

We're going to be explaining equivalents using scaling up and scaling down.

And we're going to be comparing fractions and identifying equivalence in the second part.

So let's make a start on part one.

And we've got Jun and Sophia working with us today.

So can you explain what's happening here? I'm going to run through a series of animations for you to watch, see if you can think what's happening, and then we'll talk about them in more detail.

So just watch this animation.

Can you explain what was happening? Well, Jun says, the whole and the shaded area stay the same.

And Sophia says, there are a different number of equal parts each time.

Did you spot that? So Jun says, the fraction stays the same.

Sophia says, well, in that case, they must be equivalent fractions.

So whilst the shaded area stays the same, we could describe it as a different number of parts of the whole.

So what fraction of the shape is shaded each time? Let's see if we can label it as we go through.

What fraction is shaded at the moment? Jun says, 3/5 of the shape is shaded at the moment.

The shape's been divided into five equal parts and three of them are shaded.

What about now? He says 6/10 of the shape are now shaded.

What about now? 9/15 of the shape have been shaded.

We've now got 15 equal parts and nine of them have been shaded.

Have we changed the area that's shaded? We haven't, have we? What about now? What fraction is shaded? 12/20 of the shape is now shaded.

And what about now? 15/25 of the shape is shaded.

Sophia asks: What do you notice about these fractions? Can you see anything about them? She spotted that the denominator is counting in 5s, but the numerator is counting in 3s.

So we've got the 3 times table in the numerator and the 5 times table in the denominators.

Let's look at the fractions on a number line as well, see what else we can spot.

So here's a number line from 0 to 1.

So our whole in our shape is our whole rectangle, and our whole here is 0 to 1.

So the whole is 1 on the number line.

So here is 3/5 on the number line.

What was the fraction we could see here? Well, it's 6/10, isn't it? But we can see that because our whole has stayed the same, 6/10 is in exactly the same position on the number line as 3/5.

3/5 is equivalent to 6/10.

They're equal in value.

What about now? What are we gonna have to do to the number line and what will the fractions tell us? Well, now we've got 15 parts in our whole.

So our number line has 15 equal parts and we're showing the fraction 9/15.

But it sits at exactly the same point on the number line as 3/5.

So 3/5 is equivalent to 9/15.

Time to check your understanding now.

Can you sketch the number line to represent this fraction which is equivalent to 3/5? So you've got the image there, we've got a number line in 5ths.

How can you change that number line to make it represent the fraction that we are displaying now, which is equivalent to 3/5? Pause the video, have a go and we'll come back some feedback.

How did you get on? Did you spot that this time our number line needs to be divided into 20 equal parts and we have shaded 12 of them.

So our fraction is 12/20, but it sits at exactly the same place on the number line as 3/5 because 3/5 is equivalent to 12/20.

Time to check again.

This time we've divided the number line up, pause the video and have a go at adding those extra lines to the shape so that it reflects the point on the number line as well.

How did you get on? Did you spot that this time our number line had 25 equal parts? Each of 5ths had been divided into five equal parts and we can show that by adding those extra lines onto our diagram.

So now the fraction we are representing is 15/25.

But 15/25 is equivalent to 3/5.

They have the same value.

So how can we prove that these fractions are equivalent? We sort of demonstrated it, haven't we? Is there another way we can prove it? Sophia says, the whole we think of must be the same.

So when we're comparing fractions, we are comparing fractions of the same whole.

She says, they sit in the same position on the number line.

So we've shown by dividing up our number line that 3/5 and 9/15 have the same position on the number line.

So they must be the same number, the same value.

9 jumps out of 15 is equal to 3 jumps out of 5.

The fractions represent the same proportion of the whole, and we showed it with the diagrams as well.

There are 3 times as many parts in the whole when we're thinking about 15ths.

So there we go.

3 times as many as there were when there were 5ths.

So you need 3 times as many of the smaller parts to have the same proportion of the whole.

And you can see we've got 3 times as many.

The numerator and denominator have both been scaled up by a factor of 3.

3 times 3 is equal to 9 and 5 times 3 is equal to 15.

There are 3 times as many parts in the whole.

You need 3 times as many of the smaller parts to have the same proportion of the whole.

And that means that our fractions are equivalent.

They have the same value even though they look different.

So can you use that thinking or maybe draw a number line or sketch a rectangle and divide it up? Can you work out what the missing numerator is to make these fractions equivalent? Pause the video and have a go and we'll come back for some feedback.

How did you get on? Well, we can see with the denominator that it's been scaled up by a factor of 4.

5 multiplied by 4 is equal to 20.

5 times 4 is equal to 20.

So you need to scale up the numerator by the same factor of 4 for the fractions to be equivalent.

So 2 times 4 is equal to 8.

So our missing numerator was 8.

2/5 is equivalent to 8/20.

And you might have shown that on a number line.

So you might have noticed that in this lesson we've been looking at non-unit fractions, fractions which do not have a numerator of 1.

The relationship within a fraction, so between the numerator and the denominator of each fraction, is sometimes more difficult to describe when we've got non-unit fractions.

But it will be the same for both equivalent fractions.

So let's just have a look at that.

What is the multiplication we have to do to turn 2 into 5? Well, there are 2 and a half 2s in 5.

2 and a half times 2 is equal to 5.

So 2 and 2 and half of 2 which is 1 is equal to 5.

2 multiplied by 2 1/2 is equal to 5.

And the same must be true for 8 and 20.

Two 8 is 16 and half of 8 is 4.

16 and 4 is equal to 20.

So 2 1/2 times 8 is equal to 20.

So whilst it's not as easy to see that relationship, there will still be the same relationship between the numerator and the denominator in any pair of equivalent fractions.

It's just that sometimes it's easier to see the relationship between the two numerator and the two denominators.

But it's worth remembering that this is also the case as well.

Sometimes it's easier to see the relationship between the numerators and denominators across the fractions.

So between 2/5 and 8/20, which are equivalent fractions, we can see that the numerator and the denominator have both been multiplied by a scale factor of 4 to keep the fractions equivalent.

So 2/5 is equivalent to 8/20.

But sometimes it's easier to see the relationship between the numerator and denominator within a fraction.

It's harder to see what 3 has been multiplied to make it equal to 10 and what 6 has been multiplied to make it equal to 20.

But if we look within each fraction, we can see that the relationship is the same.

3 times 2 is equal to 6 and 10 times 2 is equal to 20.

And we know that when the denominator is double the numerator, the fractions are also equivalent to a half.

So those fractions are equivalent to each other and also to a half.

So what's the missing denominator in this pair of equivalent fractions? Time to check your understanding.

Can you use both relationships, the relationship across the numerators and denominators and the relationship within the fractions to find the missing denominator and to check that the fractions are equivalent? Pause the video, have a go and we'll come back for some feedback.

How did you get on? Did you spot that times 2 relationship between the numerators? So across the fractions, the numerators and denominators have both been scaled up by a factor of 2.

So we can see that 3 times 2 equals 6.

So something times 2 is equal to 18.

9 times 2 is equal to 18.

So that would make our missing denominator 9.

Let's check using the other relationship.

So we can see that 6 times 3 is equal to 18.

So the relationship between the numerator and the denominator in each fraction must be a multiply by 3 relationship.

So within the fractions, the denominators are 3 times the value of the numerator.

So 3 times 3 is equal to 9.

So yes, that makes 9 as our missing denominator.

3/9 is equivalent to 6/18.

Time for you to do some practise.

Fill in the missing numerators and denominators in these pairs of equivalent fractions.

Which relationship was easier to use across the fractions from numerator to numerator and denominator to denominator or within the fractions between numerator and denominator in each fraction? Pause the video, have a go and we'll come back for some feedback.

How did you get on? Which relationship did you use for each pair of equivalent fractions? Let's have a look at a.

So in a, it was easy to see a relationship across those numerator.

s There was a scale factor of 3.

2 multiplied by 3 is equal to 6.

So we had to apply that to the denominators.

3 times 3 is equal to 9.

So our missing denominator was 9.

2/3 is equivalent to 6/9.

What about b? Well, this time it was easier to see a relationship within that fraction.

What have we done to turn 5 into 2 is harder to spot than 2 multiplied by 3 is equal to 6.

So we then knew that in our other fraction to make it equivalent, we would have to have the denominator 3 times the numerator, 5 times 3 is equal to 15.

So 5/15 are equivalent to 2/6.

What about c? Well again, that relationship within the fraction was actually easier to spot, wasn't it? 3 multiplied by 4 is equal to 12.

So we knew we had to have a number multiplied by 4 equal to 28, or 28 divided by 4.

However you thought about it, our numerator was 7.

3/12 is equivalent to 7/28.

And what about d? Well, this time it was easier to see that relationship between the denominators of the two fractions.

8 times 5 is equal to 40.

So we had to have something multiplied by 5 equaling 15.

And we know that 3 times 5 is equal to 15.

So our numerators and denominators had both been scaled by a factor of 5.

3/8 is equivalent to 15/40.

So for e, f and g, there were lots of possible answers.

So here are some examples and the thinking that maybe you did.

So in e, we could only look at that relationship between the denominators and we could see that it was a times 4 relationship.

So as long as the numerator of our quarters multiplied by 4 equaled the numerator of our 16ths, our fractions would be equivalent.

So we put in 3, 12.

3 multiplied by 4 is equal to 12.

In f, the relationship was there.

In the numerators, there was a multiply by 5.

So as long as our denominators had that multiply by 5 relationship, the fractions would be equivalent.

So we went for 12 and 60.

'Cause 12 times 5 is equal to 60.

So 5/12 must be equivalent to 25/60.

What about g? Well, for g, the only relationship we could see was between the numerator and denominator in 4/20.

And we could see that that too was a times 5 relationship.

So we could create any fraction where the denominator was 5 times the numerator.

So we went for 3/15.

4/20 is equivalent to 3/15.

It's really useful to look carefully at the fractions to see where that easy relationship to spot is.

Is it across the fractions or is it within one of the fractions? And onto the second part of our lesson, we're going to be using all that knowledge that we've just gained to compare fractions and to identify equivalence.

So Jun and Sophia have a set of eight fraction cards and they think that they can be sorted into four pairs of equivalent fractions.

Are they correct? Well, let's have a look.

Jun's decided to start with 3/15.

And he says, I can see that 3 times 5 is equal to 15.

So he's seen that relationship between the numerator and the denominator, 3 times 5 is equal to 15.

So Sophia says, we need to find another fraction where the denominator is 5 times the numerator.

So have a look.

Can you find another fraction where the denominator is 5 times the numerator in that set? That's right, it's 4/20.

Jun says, this is true for 4/20.

I can see that 4 multiplied by 5 is equal to 20.

So those fractions must be equivalent.

And Sophia says, we can say that 3/15 is equal to 4/20.

So we found one of the pairs.

And Jun says, we used the relationship within the fractions to find an equivalent fraction.

In each fraction, the numerator multiplied by 5 was equal to the denominator.

Sophia says, let's find a fraction equivalent to 2/3.

Hmm.

Jun says, I think looking across the numerators and denominators will help.

Harder to see the relationship between the numerator and the denominator within the fraction this time.

Sophia says, what about 6/20? If I look at the numerator, 2 times 3 is equal to 6.

And Jun says, ah, but 3 times 3 is equal to 9, not 20 so they can't be equivalent.

Oh yes, says Sophia, the numerator and denominator must both be scaled up by the same factor.

So 6/20 is not equivalent to 2/3.

Over to you.

Can you find a fraction from the set that is equivalent to 2/3? Pause the video, have a go and we'll come back for some feedback.

How did you get on? Did you spot that it was 10/15? Jun says, these are equivalent as 2 times 5 is equal to 10 and 3 times 5 is equal to 15.

And Sophia says, the numerator and denominator have both been scaled up by a factor of 5.

And Jun says, 2 is the same proportion of 3 as 10 is of 15.

And you could show that on a number line.

So moving on from our pairs of fractions, if 2/3 is equal to 10/15, which of the following equations is correct? Remember, if a fraction is equivalent, we can substitute it for another fraction, so we can have two fractions that are the same, they have the same value.

So have a look at these equations and knowing that 2/3 is equivalent to 10/15, which of them is correct? Sophia says, the fractions are equivalent so they have the same value.

Jun says, well, a cannot be correct.

The fractions have the same value, and if we subtract the same value from itself, the answer will be zero.

So 10/15 subtract 2/3 must be equal to zero.

And Sophia says, you can replace the 10/15 with 2/3 and 2/3 subtract 2/3 is equal to zero.

So she's proving that both a is incorrect, but that b is correct.

So b is correct.

Jun says, I think c is correct.

He thinks that 10/15 plus 2/3 is equal to 4/3.

Sophia says, yes, you can replace 10/15 with 2/3.

So that would make our equation 2/3 plus 2/3, and that is equal to 4/3.

So c is correct.

And Jun says, 2/3 and another 2/3 is equal to 4/3.

It's not equal to 12/18.

So d cannot be correct.

Okay, so let's then have another look.

2/3 is not equal to 6/20.

We found that out when we were looking for our fraction pairs.

So if 2/3 is not equal to 6/20, which of the following equations are correct? What do you think? Let's start looking at a.

Sophia says, which is greater, 2/3 or 6/20? That's what we need to find out to find out if a is correct or b is correct.

Jun says, it must be 2/3.

2/3 must be greater because it is greater than a half and 6/20 is less than a half.

We know that when a fraction is equal to a half, the denominator is 2 times the numerator.

6 times 2 is 12, not 20.

So 6/20 has to be less than a half.

So a must be correct and b must be incorrect.

Jun says, 2/3 is greater than 6/20.

So when you add them, the sum will be less than 4/3.

Oh, that's interesting.

2/3 plus 2/3 would equal 4/3.

But we know that 6/20 is less than 2/3, so it must be less than 4/3.

So d must be correct, says Sophia.

Because c has our two fractions adding to more than 4/3.

So by thinking about equivalence and then comparing the fractions that are not equivalent, we can find out that a and d are correct equations.

Time for you to have some practise.

Can you sort the cards into pairs of equivalent fractions? And for part two, if there are any cards left over, can you complete inequalities using the cards without a pair and explain how you know.

So which fraction is greater than the other, which fraction is less than the other if they're not equivalent? And then again, if the cards are left over, can you make up some cards with equivalent fractions for each card without a pair? So how many different ways can you find equivalent fractions so that they would have a pair? Pause the video, have a go at your tasks and we'll come back for some feedback.

How did you get on? So for part one, you found that there were three pairs and two cards which are not a pair.

And Jun said sometimes it was easier to see the relationship across the fractions.

So for 10/15 and 2/3, it was easy to see that 10 divided by 5 is equal to 2 and 15 divided by 5 is equal to 3.

So in this case, we've used the fact that they've been scaled down by the same factor.

And the same for the second pair.

It depends which way around you have them.

You might have had 4/5 is equal to 12/15 and been able to say multiplied by 3.

But again, we can see that there's a scale factor of 3 linking the numerators and the denominators.

And for the last pair, it was easier to see the relationship within the fractions.

So it was easier to see that in both these fractions, the denominator was 5 times the numerator.

I hope you found those pairs.

And then you also realised that 3/5 and 6/20 were not equivalent, so they didn't have a pair.

So can we compare them? So there are two cards which do not form a pair.

3/5 is greater than a half and 6/20 is less than a half.

So we could complete the inequalities like that.

And then we were going to look to see if we could create some equivalent fractions for those fractions that didn't have pairs.

And it's easier in this case to see the relationship across the fraction pairs rather than looking at the relationship between the numerator and the denominator in each fraction.

There will be that multiplication that we can apply, but it will involve fractions, and so that's harder to spot.

So let's look at the relationships across the fractions.

So we created 30/50.

3 times 10 is equal to 30, 5 times 10 is equal to 50.

So 3/5 must be equivalent to 30/50.

And you might be able to picture that number line from 0 to 1.

3/5 and each of those 5ths divided into 10 equal parts, so we had 50 parts along our number line.

We also found 9/15.

So the numerator and denominator had each been scaled by a factor of 3.

And 21/35.

And in this pair of fractions, you can see that the numerator and the denominator have both been scaled up by a factor of 7.

So for 6/20, we found 3/10, 24/80, and 36/120.

Can you see what the scale factors are in all of those? What do you notice about this one? Yes, the numerator and denominator have both been scaled down by a factor of 2.

So we don't always need to make the values of the numerator and denominator bigger.

We can make them smaller.

We can scale them down.

So 6/20 is equivalent to 3/10.

They've both been divided by 2.

6 divided by 2 is equal to 3, and 20 divided by 2 is equal to 10.

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

We've been explaining the relationship within families of equivalent fractions, fractions that we can link with scale factors.

In pairs of equivalent fractions, the numerators and denominators have been scaled up or down by the same factor.

The relationship between the numerator and the denominator in each fraction remains the same although this can be harder to see in non-unit fractions.

And in families of equivalent fractions, you can see times table patterns in the numerators and denominators.

So when we started looking at our 5ths, we could see that our denominators were all numbers within the 5 times table.

They were all multiples of 5.

And our numerators would all be multiples of whatever that initial numerator was.

So here you can see that our numerators are all multiples of 2 and our denominators are all multiples of 5.

I hope you've enjoyed exploring the relationships between numerators and denominators across and within fractions.

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

Bye-bye.