Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in our maths lesson today.

I hope you're ready to work hard and have lots of fun.

It's fractions and I love fractions.

So let's make a start.

This lesson comes from our unit on comparing fractions using equivalents and decimals.

And in this lesson, we're going to be learning about using representations to describe and compare fractions.

So, lots of different things to look at, lots of different things to discuss, and at the end of this lesson, you'll be able to use representations to describe and compare fractions.

We've got lots of key words in our lesson today.

They're words you may well know already, but they're useful for the learning today.

So let's have a quick practise of them.

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

So my turn divided.

Your turn.

My turn, equal parts.

Your turn.

My turn, numerator.

Your turn.

My turn, denominator.

Your turn.

So let's remind ourselves what these words mean, and we've sort of got them all modelled up together here.

So if we think about what the numerator is, it's the top number of a fraction and it shows how many equal parts we have selected.

And the denominator is the bottom number of the fraction.

And it shows how many equal parts the whole has been divided into.

So in our image here of our 12 conkers, we can see that we've divided the conkers into four equal parts.

So, our denominator is four.

And we've put a sort of shape around one of those equal parts.

So our numerator is one, showing us that we've selected one out of four equal parts of our whole.

We've got two parts to our lesson today.

In the first part, we're going to be identifying 1/2.

And in the second part we're going to be creating fractions with the same value.

So let's make a start on identifying half.

We've got Jun and Sofia.

They're going on a picnic today, I think.

Let's explore with them.

So here we are.

Jun and Sofia are sharing a picnic.

That's very good of them.

They share each item fairly.

So how much of the pizza, the sandwich, the cherries, and the strawberry lace will Sofia get? Hmm, I wonder.

Sofia says, "I'm hungry." But well, let's get started, so Sofia doesn't get too hungry.

Okay, so how much of each item will Sofia get? And how would you record it as a fraction? I wonder.

Sofia says, "We could use the stem sentences." The whole is divided into hmm equal parts.

Sofia has hmm of those parts.

So have a think about the pizza, the sandwich, the cherries, and the strawberry lace.

If we use those stem sentences, how much of each is Sofia going to get? Let's start thinking about the pizza.

How much of the pizza will Sofia get? How would you record it as a fraction? So have a look at the whole.

The whole has been divided into eight equal parts here, and Sofia is going to get an equal share.

She's going to share with Jun.

So she's going to get four of those parts.

So we've shaded out the bit that Jun's going to get.

We've left the bit that Sofia's going to get.

So what fraction of the pizza is that? And she says, "I have 4/8 of the pizza." How much of the strawberry lace will Sofia get? This is an interesting one, 'cause this is a bit more of a line.

We can't see individual bits of strawberry lace, but we do know that the strawberry lace is 30 centimetres long.

That's quite a long strawberry lace, isn't it? So what are we going to do this time? The whole is divided into hmm equal parts.

Sofia has hmm of those parts.

How can we think about this? We know that 30 centimetres is the whole.

So, how much strawberry lace will I get? So the whole is going to be divided into two equal parts, 'cause Sofia gets one part and Jun gets one part.

And Sofia has one of those parts.

And there we can see the strawberry lace being cut in the middle.

And Jun says, "You will get half of 30 centimetres, which is 15 centimetres, as we know that half of 30 is 15." And there we can see.

Sofia says, "I must have 15/30 of the lace." The whole is divided into 30 equal parts, 'cause it was 30 centimetres long.

And Sofia has 15 of those parts, 15 centimetres.

So, she could also say that she has 15/30 of the strawberry lace.

Okay, time to hand over to you.

So how much of the sandwich and the cherries will Sofia get? And how would you record this as a fraction? So, you've got the stem sentences there to help you.

So pause the video and work out how much of the sandwich and the cherries Sofia will get as a fraction.

How did you get on? Did you work it out? Okay, so with our sandwich, the whole is just divided into two equal parts.

Our sandwich is one.

It's one sandwich.

Sofia has one of those parts.

So Sofia has 1/2 of the sandwich.

What about the cherries? How many cherries are there in our whole? It was easy with the sandwich.

It was just one sandwich.

How many cherries are there in the hole? So this time we've got 10 cherries.

So the hole is divided into 10 equal parts, and Sofia's going to get five of those parts.

So the fraction we can say is Sofia has 5/10 of the cherries.

So let's look at the fractions that Sofia gets, and what do you notice about them? So she gets 4/8 of the pizza.

She gets 1/2 of the sandwich.

She gets 5/10 of the cherries, and she gets 15/30 of the strawberry lace.

So Sofia gets 1/2 of each item, but the fractions look different.

Each whole has been divided into two equal parts, but the hole is not always one.

It's only the case in the sandwich where the whole is one, and possibly the way Jun looked at it with the strawberry lace as well.

So let's look at the fractions and the bar models and see what we notice about them.

So, here we're representing the pizza.

So there were eight pieces and Sofia got four of them, 4/8.

With the sandwich there was just one, and so Sofia got 1/2.

With the cherries, Sofia got 5/10 of the cherries, 'cause the whole was 10.

And with the strawberry lace, Sofia got 15/30.

Gosh, what do you notice about the bar models? Well, each whole has been divided into two equal parts so that Sofia gets one part and Jun gets one part, but the wholes are not the same, are they? Have a look at the fractions on the bar models this time and see what you notice about them.

There's 4/8, 1/2, 5/10 and 15/30.

What do you notice about the bars this time? Hmm, I wonder if you spotted this.

The whole is the same, but there's a different number of equal parts.

In the previous image, the equal parts were the same size, but the wholes were different.

This time we've made the wholes the same, but the equal parts are different.

But, we can still say that 1/2 of each bar is shaded.

Let's have a look at the fractions on their own.

So the 4/8 for the share of the pizza, 1/2 of the share of the sandwich, 5/10 for the share of the cherries, and 15/30 for the share of the strawberry lace.

What do you notice this time about the numerators and denominators in all of these fractions? And we know they all represent 1/2, because when two people share something, they get 1/2 each if they share it fairly.

Let's have a look at those numerator and denominators.

What do you notice? Jun's just reminding us or maybe reminding himself, "The numerator's the top number, and it tells us how many parts we have." So Sofia, "And the denominator is the bottom number, and it tells us how many parts there are in the whole." So they've just reminded themselves what the numerator and the denominator is all about.

So what do you notice about the numerators and the denominators in these fractions? What do you notice about four and eight? Hmm? Four is half of eight and eight is double four.

What about in the 1/2? What do you notice about one and two? Well, one is half of two and two is double one.

Have you looked ahead at the other fractions? What have you spotted? What do you notice about five and 10? Five is half of 10, and 10 is double five.

And what about 15 and 30? Can we do the same for 15 and 30? What do we notice? 15 is half of 30, and 30 is double 15.

Do you notice the word half coming in quite a lot there? Hmm.

I wonder what we can say? Do you think that's always true? So by looking at these fractions, we can say that when a fraction represents 1/2, the numerator is half the denominator or the denominator is double the numerator.

And we've shown that for those fractions that we've worked with today.

I wonder if you could try it with some other fractions to see if it's true for all fractions.

Sofia says, "This is useful to remember." It's going to help us in the rest of this lesson, and possibly some more lessons too.

So let's turn the bar models into number lines, see what we notice this time.

Let's have a look.

So there's our zero to one number line with 4/8 for the pizza.

Zero to one number line with 1/2 for the sandwich.

Zero to one number line with 5/10 for the cherries and zero to one number line with 15/30 for the strawberry lace.

What do you notice about them this time? Hmm.

So the fractions are all in the same place on the number line.

The fractions all represent 1/2.

So time to check your understanding.

Which fraction here is equal to 1/2? Can you use Sofia's generalisation to explain your thinking and explain why the others are not equal to 1/2, perhaps? And Sofia's reminding us, "When a fraction represents 1/2 the numerator is half the denominator." Pause the video and have a think about those three fractions.

How did you get on? Did you spot that 3/6 is the fraction that is equal to 1/2, because three is half of six, or six is double three? And in 3/4 and 3/7, three is not half of four, and three is not half of seven.

Three is more than half of four, and three is just less than half of seven.

So I wonder if you spotted that as well.

That's going to be useful in a future lesson as well.

And there's Sofia using her generalisation to explain what we've just talked about.

Time for you to do some practise.

So you are going to identify half of these items and explain how you know, using the stem sentences.

The whole is divided into hmm equal parts.

I have hmm, of those parts.

The fraction is, hmm.

So another sort of picnic situation here.

We've got some cupcakes, a pizza, cherries and a, I dunno, a breadstick maybe perhaps to share there? So identify half of these items and use the stem sentence.

And then in part two, using Sofia's generalisation, can you spot which fractions are equal to 1/2? And there's Sofia's generalisation to help you.

So pause the video.

Have a go at your tasks, and then we'll look through them together.

How did you get on? So with the cupcakes, there were six of them.

So to find 1/2 the whole is divided into six equal parts, and I have three of those parts.

So the fraction is 3/6.

The pizza this time is divided into 12 equal parts, and we have six of those parts to have half of them.

So the fraction is 6/12.

For the cherries, this time, again, we had 10.

So the whole is divided into 10 equal parts, and I have five of those parts.

So the fraction is 5/10.

And for our breadstick, the whole is divided into two equal parts.

I have one of those parts, but we know this is 24 centimetres long.

So we've either got 12 out of 24 centimetres or one out of two.

So the fraction is 1/2 or 12/24.

And in part B, remembering Sofia's generalisation, when a fraction represents 1/2 the numerator is half the denominator.

So that's true in 4/8, 1/2, 5/10, 15/30, 3/6, and 50/100.

But in the other fractions, the numerator was not half the value of the denominator or the denominator was not double the value of the numerator.

So we come to the end of part one of our lesson.

Let's have a look at what's in part two.

We're going to be creating fractions with the same value.

So let's have a look at this.

What do you notice? Hmm? What did you notice happening there? I wonder what Jun and Sofia noticed.

Jun has spotted that, "The whole stays the same, and the shaded area stays the same." And Sofia spotted that, "The whole is divided into a different number of equal parts when we have those dotted lines in." Did you spot that as well? So Sofia says, "What happens to the shaded part?" Let's have a look.

Hmm.

What happened to the shaded part as we added those extra lines in? Jun says, "Nothing.

It stays the same size and shape." But Sofia says, "But it is divided into a different number of equal parts." So let's have a think about what fractions we can use to label this shaded part.

So here we at the start, how many equal parts have we got? We've got three equal parts in the whole and one of them is shaded.

So we can say that 1/3 of the shape is shaded.

So let's have a look as we add those extra lines in, what can we say, what fraction can we use to describe the shaded part? So now, how many equal parts have we divided the whole into? We've divided it into six equal parts, and we've got two of them.

So we can say it's 2/6.

How many equal parts now? Nine and three of them shaded, 3/9.

I'm gonna hand over to you to finish off the rest of this.

So for your check for understanding, you are going to decide what fractions can be used to label the shaded parts.

So A, B, and C, adding in those final extra dotted lines.

And you've got the stem sentence there.

It has been divided into hmm equal parts.

Hmm of the parts are shaded.

The fraction is, hmm.

Okay, so pause the video and see if you can label the fractions for A, B and C.

How did you get on? So for our stem sentence, for A, it has been divided into 12 equal parts.

Four of the parts are shaded, so the fraction shaded is 4/12.

What about B? It's been divided into 15 equal parts this time.

Five of the parts are shaded, so the fraction shaded is 5/15.

And for C, it's been divided into 18 equal parts.

Six of the parts are shaded.

So the fraction shaded is 6/18.

Well done if you've got all of those.

Well, what can we say about these fractions? Sofia says, "The fractions must be equal as they all represent 1/3 of this shape." So adding all those extra lines, we didn't change the fact that the whole was the same, and the shaded part was the same.

So all of those fractions are representing 1/3 of the shape.

So we can put equal signs in between them, because we know that they are all equal in value.

What do you notice about the fractions? We know that they're all equal, but what do you notice about them? Hmm, I wonder.

I wonder what Sofia and Jun are going to notice about them.

Jun says, "I can see the numerator increases by one each time." When we look along that row of fractions, we've got one for 1/3, two and 2/6, 3 and 3/9 and so on.

So the numerator increases by one each time.

And Sofia says the denominator is going up in threes, and we can see our three times table there.

Three, six, nine, 12, 15, and 18.

So, can we say then, for a fraction to represent 1/3, the denominator must be three times the numerator.

Should we look at that relationship? One times three is three.

Two times three is six.

Three times three is nine.

Four times three is 12.

Five times three is 15.

Six times three is equal to 18.

So it helps to look at that relationship between the numerator and the denominator and not just the relationship going, adding one and adding three.

So we can say fractions have the same value when a part of the same whole is divided into a different number of equal parts.

So a part of the same whole is divided into a different number of equal parts.

So here we can see the whole stays the same, the part stays the same, but we've divided the whole up into a different number of equal parts.

So we can see 1/3, 2/6, and 3/9 there represented.

But they all represent the same shaded area of the shape.

But we can also say that fractions have the same value when different wholes are divided into the same number of equal parts.

So we've got four sets of green and white counters here.

We can see that same fraction represented.

So Sofia says, "What do you notice about these representations?" "All the rectangles," she says, "represent 1/3." And we talked about that as we saw them come up.

What about the sets of counters? What do you notice? Ah, all the dots or the counters represent half.

So we can see that in each of those sets, a half of the counters are green and half of the counters are white.

We've got a different whole each time, but each one represents half green, half white.

Time for you to do some practise.

So these fractions are not equal to 1/2.

Can you use Sofia's generalisation to change the numerator or the denominator of these fractions, so that they are equal to 1/2? So if you look at 3/4, how could you change the denominator to make it equal to 1/2? How could you change the numerator to make the fraction equal to 1/2? And can you do it in more than one way for each fraction? And remember that Sofia's generalisation says that, "When a fraction represents 1/2, the numerator is 1/2 the denominator." Or you can say the denominator is double the numerator.

Sofia's decided to use the half in it to help her remember the half bit.

And for part two, you are going to fill in the gaps to create two fractions that have the same value.

And then B, can you use Jun's generalisation to fill in the gaps again and create two fractions, which both represent 1/3? So for A, you can use any numbers to represent the same fraction.

And in B, you're going to try and make two fractions that are equal to 1/3.

Using Jun's generalisation that when a fraction represents 1/3, the denominator is three times the numerator.

Or we could say the numerator is 1/3 the value of the denominator.

So pause the video.

Have a go at your tasks, and then we'll look at them together.

How did you get on? So we've changed the fractions.

Let's have a look.

So 3/4, we could change the denominator and create a fraction 3/6, which is equal to 1/2.

or we could change the numerator and make a fraction of 2/4.

B was 3/7.

Can we change the num.

Oh, we can change the denominator to make a fraction of 3/6, but we can't leave the denominator as seven and create a fraction equal to 1/2.

And we've said there, when the denominator is an even number, you can change the fraction in two ways to make it represent half by changing either the numerator or the denominator.

But if the denominator is an odd number, you cannot change the numerator to create a fraction equal to 1/2, because half of an odd number is not a whole number.

So for B, we can only change 3/7.

We can change the denominator to make 3/6.

So for C, we had 4/9.

So again, we can't just change the numerator, 'cause half of nine is four and 1/2, and we can't have four and 1/2 as our numerator.

So we need to change the denominator to create 4/8.

6/18, we can change either the numerator or the denominator.

So we could have 6/12 or 9/18.

And again, for E, we could change the numerator or the denominator.

So we could have 24/48 or 25/50.

And all of those fractions underneath will be equal to 1/2, because they meet Sofia's generalisation that when a fraction is equal to 1/2, the numerator is half the value of the denominator.

So for the second part, you were filling in the gaps to create fractions that had the same value.

So we filled it in as 6/12 and 3/6.

And as Jun says, "Both those fractions represent 1/2." And Sofia says, "The numerator is half the denominator." Now in B, we were going to use Jun's generalisation to fill in the gaps and create two fractions which both represent 1/3.

So we needed a numerator that was 1/3 the value of the denominator, or a denominator that was three times the value of the numerator.

So what could we have done? Ah, so we could have had 6/18 and 2/6, both fractions, Jun says, "Now represent 1/3." The denominator is three times the value of the numerator.

Six times three is equal to 18, and two times three is equal to six.

I hope you enjoyed playing around with numerators and denominators to create fractions with the same value.

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

So what have we learned about today? We've learned that different fractions can represent the same part of a whole.

That when a fraction represents 1/2, the numerator is half the denominator in these fractions that we've got here.

And when a fraction represents 1/3, the denominator is three times the value of the numerator.

And you can see that in those fractions there.

This is really useful learning that's gonna help you a lot in future lessons on fractions.

And I hope I'll get to see you in some of those soon.

Bye-Bye.