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.

So this lesson is part of the unit on comparing fractions using equivalents and decimals.

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

So by the end of this lesson, you should be able to do just that, use representations to describe and compare two fractions.

We've got two key words in our lesson today.

We've got divided and equal parts.

Now, I'm sure they are words that you know very well, but let's just practise them and think about what they mean 'cause they're going to be really useful today.

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

So my turn, divided.

Your turn.

My turn, equal parts.

Your turn.

As I say, I'm sure you're familiar with those words.

Let's have a look at what they mean, just remind ourselves.

So when something is divided, it is split into equal parts or equal groups.

So if we look at the conkers on the screen there, we can think about that in two ways.

They've been split into four equal parts with three in each part.

But they've also been split into equal groups with three in each group.

So two ways to think about it.

We're going to be thinking about dividing into equal parts today, in this lesson.

And equal parts can then be recombined to make a whole.

And you can see there that those circles are divided into four equal parts, and into 12 equal parts.

There are two parts to our lesson today.

Two equal parts, perhaps? Not sure they'll be exactly equal.

So in the first part of our lesson, we're going to be thinking about representing the same fraction in different ways.

And then we'll move on to represent and compare fractions.

So let's make a start on part one.

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

So Jun and Sofia have collected 12 conkers.

They're each going to take some and leave the rest for their friends.

Jun says, "I will take 1/4 of the conkers," and Sofia says, "I would like three of the conkers." Are you thinking ahead? Can you see what's going to happen here? Let's have a look.

So how many conkers will they each take? So Jun said he's going to have 1/4 of the conkers, and he says 1/4 means that the whole is divided into four equal parts, and I have one of the parts.

So he's dividing the conkers into four equal parts.

So let's have a look at that, okay? You can see on the screen that they're sort of divided into four equal parts of three.

And so he's put a ring around his conkers that he's going to take, 1/4 of the conkers.

And he's labelled it with 1/4, and he says, "these are my conkers." And Sofia says, "That's the same as my three conkers." She's put a box around her three conkers.

"What fraction of the conkers do I have," she says.

Hmm, I wonder, can we describe Sofia's conkers using a different fraction? Jun says, "the whole is divided into 12 equal parts, and you have three of the parts." So she's got three out of the 12 conkers.

So we can use the fraction 3/12 to describe Sofia's conkers.

And she says, "so I have 3/12 of the conkers." But Jun says, "how can we have the same number of conkers when we have a different fraction?" Hmm, I wonder.

Let's have a think about that, how can Jun and Sofia have the same number of conkers when they've described them with a different fraction? Let's have a think about how our keywords can help us to think about that.

So 1/4 means that the whole is divided into four equal parts and we have one of the parts.

So let's show that with these conkers.

So there are the four equal parts, and there's the one part that we have.

So one out of four equal parts is equal to 1/4 of the whole.

And 1/4 of the conkers, we can see, is equal to three conkers.

If we count the number of conkers we've got in that one equal part out of four, we've got three conkers.

Let's have a think about 3/12.

3/12 means that the whole is divided into 12 equal parts, and we have three of the parts.

So let's look at that.

Ooh, 12 equal parts, each conker is one part on its own, isn't it? And we have three of those parts.

So three out of 12 is the same number of conkers as 1/4.

And there we can see, our 1/4 and our 3/12 are the same number of conkers.

So we can say that the fraction 1/4 is equal to the fraction 3/12.

And we've shown that there using the conkers, 1/4 is equal to 3/12.

So time to check your understanding.

Have a look at these pairs of images and decide which of the pairs of images show that we can represent the shaded areas as 1/4 and 3/12, so we can say that 1/4 is equal to 3/12.

So which pairs of images show this? Pause the video and have a go.

How did you get on? Did you have some interesting discussions? Did you have a think about what fractions were shown by each of the images? And I hope you spotted that in A both of those bars show the same fraction.

The top one shows 3/12 and the bottom bar shows 1/4.

And we can see that those shaded areas are the same in both of the bars.

And we can say the same for D, with the circles.

The first circle is divided into four equal parts and one is shaded, and the second circle is divided in 12 equal parts and three are shaded, so showing that 1/4 is equal to 3/12.

In the others we've got quarters and twelfths, but the shaded areas don't show that 1/4 is equal to 3/12.

So can you represent another number of conkers with two fractions that have the same value? We've done it for three conkers.

I wonder if there's another number of conkers that you could choose and represent with two fractions that have the same value? Have a think about that.

I wonder what Sofia and Jun are going to come up with.

Jun says, "we could think about four conkers." Can you imagine what those four conkers might look like? Think about how we could split the conkers into equal parts and describe those four conkers in different ways, hmm.

Well, there are the four conkers, what can we say about them? Sofia says "we can use the stem sentences to describe the conkers." So the whole is divided into hmm equal parts.

We have hmm of the parts.

So what could we say here? We could say the whole is divided into 12 equal parts and we have four of the parts.

So what would that mean for our fraction? But Sofia says, "we could also do this." The whole is divided into three equal parts and we have one of the parts.

We can make three equal groups of four from our 12 conkers.

We can use our times tables to help us to think about that.

So we've completed the stem sentences in two different ways.

And what does that tell us about the fractions? Think about what those stem sentences mean if we represent them as a fraction.

So the whole is divided into 12 equal parts.

We have one of the parts, what does that mean as a fraction? And the whole is divided into three equal parts.

We have one of the parts, how would we write that as a fraction? Let's see what Jun thinks.

Jun says "we could describe four conkers as 1/3 or as 4/12." So 1/3 is equal to 4/12.

If we've got our conkers divided into 12 equal parts and we've got four of them, we've got four out of those 12, we've got 4/12.

And if we think of it as divided into three equal parts and we have one of those parts, we've got 1/3.

So we can say that 1/3 is equal to 4/12.

Time for you to do some practise.

So thinking about those 12 conkers again, can you represent two of the conkers, and then six of the conkers from the 12, with fractions that have the same value? So think about what two conkers would mean, how you could represent two of those conkers using different fractions.

And then how you could represent six of the conkers using different fractions.

Think about what we know for three and four conkers and use the stem sentences that are there for you to help your thinking.

You might decide to use counters or cubes to represent the conkers.

You may not have 12 conkers to hand.

And then Sofia's got an extra question for you.

Why is five missing from the numbers to try? We've looked at three and four, and we are now looking at two and six.

Why is five missing from the numbers to try? Have a think about that as well.

So for the second part of your task, you're going to use counters or draw a different representation to show that in A, 1/4 is equal to 2/8.

In B, 2/4 is equal to 4/8, and in C, 3/4 is equal to 6/8.

And we've got the stem sentences there to help you again.

So pause the video, have a go at your tasks and then we'll look at them together.

How did you get on? So let's have a think about those conkers and thinking about representing two of those conkers with two different fractions that have the same value.

So for two conkers we can say, that the whole is divided into 12 equal parts and we have two of the parts.

Or we could say, that the whole is divided into six equal parts and we have one of the parts.

So what would that mean for the fractions? So we could say that 2/12, two out of 12 equal parts is equal to 1/6, one out of six equal parts.

And you can see that with the purple shapes that we've drawn around the conkers there, and our two conkers.

So what about six conkers? So for six conkers we can say, the whole is divided into 12 equal parts and we have six of the parts.

Or the whole is divided into two equal parts and we have one of the parts.

So what fractions have we got then? That's right, we can say that 6/12 is equal to 1/2.

Those six conkers could be described as 6/12 of the conkers, or as 1/2 of the conkers.

Do you remember Sofia's question, why is five missing from the numbers to try? I wonder how did you describe this? How did you think about this? This is how we thought about it.

We cannot divide 12 conkers into five equal parts or into an equal number of groups of five.

If you remember at the beginning, we talked about dividing being about making equal groups or groups of the same size, and we can't do that with five.

We can't find two fractions from 12 to represent five conkers.

We could only find 5/12, five conkers out of 12 conkers.

So that's why five was missing, we can't find another way of representing five conkers with a fraction that's equal to 5/12.

And then in B, you were using a representation, perhaps the counters here, to show that those fractions are equal to each other.

So let's have a look at that.

So for A, 1/4 of the counters is equal to 2/8 of the counters.

So we can see that one out of four equal parts is equal to two out of eight equal parts.

And for 2/4, well, 2/4 is equal to 4/8, two out of four equal parts is equal to four out of eight equal parts.

And for C, 3/4 is equal to 6/8.

So let's think about the stem sentences.

So for A, the whole is divided into four equal parts and we have one of those parts.

Or the whole is divided into eight equal parts and we have two of those parts.

And we could use that stem sentence to describe the other fractions of counters as we go through B and C.

Well done, we're onto part two of our lesson, thinking about representing and comparing fractions.

So in the first part of our lesson we started looking at one 1/4 and 3/12, and using representations to show that those fractions are equal to each other.

And we can see here, from that check for understanding in the first cycle, we've got those representations back.

Like this pair of circles show us that 1/4 is equal to 3/12, and the pair of bars shows us that 1/4 is equal to 3/12.

The same area of the circles is shaded and the same area of the bars is shaded.

So both fractions represent the same shaded area.

Let's just have a quick check because when we're comparing fractions, the whole is really important.

So what represents the whole in these images? Have a think for a moment, what represents the whole? Hmm.

Let's think about the circles.

One circle is the whole.

So when we are looking at that 1/4 and that 3/12, our circles are identical.

The whole has to be the same when we're comparing fractions.

What about the bars, what represents the whole? That's right, one bar is the whole.

So one bar we are showing 1/4, and in the other bar we're showing 3/12.

And we can say that they are the same because the bars are equal in size, the bars are the same.

What about for the conkers though? What represents the whole for the conkers? That's right, 12 conkers are the whole here.

So when we are thinking about our 1/4 and our 3/12, we're thinking of 1/4 of the whole group of conkers, and 3/12 of the whole group of conkers.

Again, we've got to have the same number of conkers if we're comparing one 1/4 and 3/12.

So think about that as we go on through this lesson, the whole is really important.

So when we compare fractions, the whole must be the same.

So make sure you spot what the whole is as we go through the rest of this lesson.

So we know that we can represent fractions on a number line.

And here, we've got a number line counting in twelfths.

So what's the whole this time? Hmm, I wonder.

That's right, this time the whole is one on the number line, and we're thinking of our fractions as a fraction of one.

So our whole is one.

Sofia says, "let's count in twelfths to and from one." So are you ready? Let's count from zero up to one in twelfths, and then backwards from one to zero.

So from zero we're gonna start at zero, we're gonna count, zero, 1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12, 11/12, 12/12, or one, and we know that 12/12 is equal to one.

That's right, well done.

Let's count backwards from one, back to zero, counting in twelfths.

So should we start at one or 12/12? Let's start at one, we know one is equal to 12/12.

So one, 11/12, 10/12, 9/12, 8/12, 7/12, 6/12, 5/12, 4/12, 3/12, 2/12, 1/12, zero.

So let's remember, this time our whole is one, when we are looking at fractions on the number line.

What do you notice about these fractions in the green rectangle? What do you notice? Well, I noticed that the numerator, the number at the top, is only a small part of the denominator, the number at the bottom, tells us how many equal parts.

So they're closer to zero on the number line.

One, two, three, and four are quite small parts out of 12.

So these fractions sit close to zero on the number line.

Let's have a look at another set.

What do you notice about these fractions and where they are on the number line? This time, the numerator is a large part of the denominator.

Eight, nine, 10, and 11 are large parts of 12, so they're much closer to one, on the number line.

Have you spotted some that we missed out? Let's have a think about those ones.

What about 5/12, 6/12 and 7/12, what do you notice about these fractions? Well, I notice that the numerator, the top number, is around half the value of the denominator.

So these fractions are close to or even equal to 1/2 on the number line.

So by looking at the relationship between the numerator and the denominator, we can start to think about ordering and comparing fractions and thinking about where they might sit on a number line.

So have a bit of a check here.

Where would you position 2/12 on the number line? Would you place it at A, B, or C? Pause the video, have a think and then we'll look at it together.

How did you get on, what did you reckon? Yes, B is the correct position.

We marked the twelfths on there for you to make it a bit easier, but B is the correct position.

2/12 is two steps along the number line, from zero to one, when we're counting in twelfths.

We've got two other positions there gone, what are the other positions? What fractions would go at A and C? Pause the video, have a think.

What did you think? Did you think about those numerators? So A is closer to zero, isn't it? So the numerator's got to be smaller than two.

And what about C? Well, that's closer to one, isn't it? It's bigger than 1/2, and we know that half of 12 is six, so it's got to be more than that.

It's one jump further along, so it must be seven.

So A shows 1/12, and C is the position of 7/12.

Well done if you got those right.

And I hope you thought about the value of the numerator to help you to position those numbers on the number line.

So we can represent fractions on a number line.

And here, we've got two number lines, one marked in quarters, and one marked in twelfths.

But as you notice, that zero to one distance is the same.

We are comparing those fractions on a zero to one number line, one is our whole in this case.

And we can use the stem sentence to describe the number lines.

The whole, which is one in this case, has been divided into hmm equal parts.

So in the top line, our whole has been divided into four equal parts.

And in our bottom line the whole has been divided into 12 equal parts.

And when we are looking at number lines with fractions, it's important to spend a bit of time thinking about what that number line is showing us.

So for this one, the whole has been divided into four equal parts.

And for our bottom number line, the whole has been divided into 12 equal parts.

Wonder if you can think about what would happen if we moved those number lines together so that the zero to ones sat on top of each other.

Let's have a look.

Hmm, so when we bring the number lines together, what do you notice? I wonder, hmm.

Well, there's 1/4, and there's 3/12, the fractions that we were looking at in the first part of our lesson today.

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

Which other pairs of fractions share the same place on the number line? Can you have a look and see? That's right, 2/4 and 6/12 are at the same place on the number line.

So we can say that 2/4 is equal to 6/12, they're in the same place on the number line, they represent the same fraction of one.

Did you spot the other ones as well? 3/4 and 9/12, 3/4 we can say, is equal to 9/12.

They are at the same place on the number line.

They represent the same fraction of our whole, which is one.

They look very different though, don't they? But if you think about the work we did in part one, with the counters and the conkers, and the bars and the circles, we could use those representations to show that 3/4 and 9/12 are equal, and that 2/4 and 6/12 are equal.

Time to check your understanding.

Have a think maybe about those representations and have a think which three fractions are missing from this number line? So pause the video and have a think.

How did you get on? Did you use the divisions on the number line to help you? Or did you think about those fractions with the same value and maybe think about cubes or conkers or counters or bars or circles? So let's have a look, which three fractions are missing? So 3/12 is missing from here.

3/12 sits at the same place as 1/4, on the number line.

And then 6/12 is missing here.

6/12 sits at the same position on the number line, as 2/4.

And 3/4 is missing here.

3/4 sits at the same position on the number line, as 9/12.

Can you think of another fraction that could go there? There's a big clue here, it's halfway between zero and one.

So yeah, we could put 1/2 at that position on the number line as well.

And we could say that 1/2 is equal to 2/4, which is equal to 6/12.

Got some slightly different representations to have a look at here.

What do you notice about these representations? What's the same and what's different? So here we've got four counters and one of them is green.

Eight counters and two of them are green, we could say.

12 counters and three of them are green.

16 counters and four of them are green.

So what do you notice? What's the same and what's different? Let's see what Jun's got to say about it.

Jun says, "I can see the same row in each set of counters." So let's put a box around those.

So he can see this row repeated in all of the patterns.

And in fact, all the patterns are made up from different numbers of that same row, aren't they? What do you notice about the row? Jun says, "one out of four equal parts is shaded and three parts are not." So one out of four equal parts, what can we say then? We could label that, we could say, 1/4 of each of those rows is shaded, couldn't we? But Jun says he spotted something else.

He says, "I can see one equal part out of four, shaded in each set." So we can see it in each of those rows.

He says he can see it in each set as well, let's have a look.

So one out of four equal parts, one out of four equal parts, one out of four equal parts, and one out of four equal parts.

So if we think about the columns, we can see that one out of four of the columns is shaded in green as well.

So that also shows us that it's one out of four equal parts, it's 1/4.

And Jun says, "1/4 of each set is shaded." You noticed anything this time? What about the whole? That's right, the whole is different in each case.

We can say that 1/4 of each set is shaded, but in this case, the whole is not the same each time so the quarters look slightly different.

Time for you to do some practise.

You're going to write in the fractions on the marks on the number line, so to complete those number lines.

And then find pairs of fractions that have the same value, and record them underneath.

So you've got some boxes there with equal signs between them.

So mark all the fractions on and then find pairs that have the same value.

And part B says, can you label A in more than one way to complete this, and have three fractions that are equal to each other.

And for the second part of your task, you're going to collect some counters or cubes so that you have six of one colour and 18 of another colour.

It's quite a lot of counters and cubes, I know.

I'm sure you can find them.

And you're going to arrange the objects into sets, where 1/4 of them are one colour in each set, and the rest are the other colour.

So you're going to think about finding 1/4 of a set, and then representing it with another fraction.

So arrange the counters, and then write another fraction that represents 1/4 as well.

Pause the video, have a go at your tasks and then we'll look through them together.

How did you get on? Did you complete the number line with the fractions? We'd looked at this one before, so hopefully, that wasn't too big a task for you to do.

And then finding pairs of fractions that sit at the same position on the number line so that we can say they are equal.

So 1/4 is equal to 3/12, 2/4 is equal to 6/12, and 3/4 is equal to 9/12.

And then A, mark the position of 2/4 and 6/12.

So we had to find another fraction.

So there we go, A could also be labelled as 1/2.

It's halfway between zero and one, it's one out of two equal jumps along the number line.

So 6/12 is equal to 2/4, which is equal to 1/2.

Ooh, so for B there were lots of different possibilities.

So your sets might have looked like this, each time, and we've arranged them like the ones that Jun was looking at, so you can see that one out of four equal parts is always shaded in green, in our case.

So how could we describe those fractions? They're all 1/4, but how else can we describe them? So we can say 1/4 is equal to 4/16.

1/4 is equal to 3/12, 1/4 is equal to 2/8.

Think about the last ones though.

1/4 is equal to 5/20, and 1/4 is equal to 6/24.

Exciting, I wonder which fractions you found.

Did you find all of those ones? Ooh, we've come to the end of our lesson.

You've worked really hard today.

I hope you've enjoyed finding fractions that have equal value, or describing part of a set using two different fractions.

So we've learned that the same whole can be divided into different numbers of equal parts, and that the same number of parts can be described using a different fraction.

We've also seen that different representations can be used to show these fractions.

And can you spot the missing representation? We haven't got a number line on here, have we? But we've used circles, we've used the fractions themselves, and we've used objects.

Thank you for your hard work today, and I look forward, I hope to see you in another lesson soon.

Bye-bye.