Lesson video

Hi there.

Welcome to today's lesson.

My name's Mr. Peters.

And in this lesson, we're gonna be building on our understanding of finding the whole when we know the size of a non-unit fraction.

If you're ready to get started, let's get going.

So by the end of this lesson today, you should be able to say that I can find the whole when the size of a non-unit fraction is known.

Throughout the lesson today, we've got five keywords.

I'll have a go at saying them first, and then you can repeat them after me.

The first one is represent.

Your turn.

The second one is unit fraction.

Your turn.

The third one is non-unit fraction.

Your turn.

The fourth one is numerator.

Your turn.

And the fifth one is denominator.

Your turn.

So to represent something means to show something in a different way.

The same thing can be shown using multiple representations.

A unit fraction is a fraction which has a numerator of one.

A non-unit fraction is a fraction which has a numerator that is greater than one.

The denominator is the bottom number in a fraction, and the numerator is the top number in a fraction.

Throughout this lesson today, we've got three cycles.

The first cycle is thinking about identifying the whole.

The second cycle will focus on finding the whole in different contexts.

And then the third cycle will be finding the whole from a known quantity.

Let's get started with the first cycle.

In this lesson today, we've got four students who'll be joining us.

Aisha, Sam, Sofia, and Jacob will all be sharing their thinking and any questions that they have along the way to help us with our learning.

So let's start our learning here for today then.

If this image here represents 3/5 of the whole, what does the whole look like? Take a moment.

Have a think for yourself.

Aisha starts us off by saying, "Well, if this is 3/5 of the whole, then it represents a non-unit fraction." If this represents a non-unit fraction, then we can say that each one of the parts represents a unit fraction of the whole.

In this case, it would be 1/5 of the whole.

There we go.

So each part represents 1/5 of the whole.

We can say that three lots of 1/5 is equal to 3/5.

So now we know that we've got 3/5 or three lots of 1/5, we now need to find a way to make the whole.

We know that the whole is made up of 5/5, don't we? So to make the whole, we need to find 5/5 all together.

And there we go.

We've got our whole.

Our whole is represented now by 5/5.

Sam thinks the whole could be represented slightly differently.

There we go.

This is how Sam has represented the whole, isn't it? And Sofia also thinks she could have represented the whole slightly differently.

Let's see how she did that.

There we go.

Sofia has represented the whole in a different way too.

Great question, Sam.

I wonder how many different ways we could have represented the whole each time.

So let's have a look at these three different wholes then.

What do you notice about them? What's the same and what's different about these examples? That's right, Aisha.

They're all composed of unit fractures, aren't they? They're composed of five 1/5, aren't they? Yep, and that's also right Sofia.

Each 1/5 can be placed in a different position within the whole, can't it? Okay, have a look now.

Jacob's come up with a different way of representing the whole.

What do you notice this time? What have you noticed, Sam? Ah, you think this can't be a whole because each of the parts is separated from each other.

They're not joined up together.

Ah, a good point there though, Jacob.

Just because they're all separated, it doesn't mean they can't represent one whole.

We could group them together like this to show that they represent one whole, couldn't we? There we go.

Yeah, well done.

And Jacob, you've really helped Sam's thinking there, so well done to you too.

Okay, time for you to check your understanding now.

Can you draw your own example of what the whole might look like? Take a moment to have a think.

There we go.

Here's what Jacob's come up with.

We started off with 3/5 of the whole, didn't we? And that was represented at the top by the three bars.

But we know to make the whole, we would need 5/5 all together.

So here we can see we've got 5/5.

And here Jacob's put them in the shape of almost an addition sign, hasn't he? Nice thinking, Jacob.

Okay, and time for you to practise now.

What I'd like you to do is draw the whole for each example given on the left-hand side.

And then for task two, what I'd like you to do is create three different wholes for the example given.

Good luck with those and I'll see you back here shortly.

Okay, let's have a look and see how you got on then.

So we can see here that the first one is 4/6, isn't it? We know that 1/6 would be one of those triangles, so therefore 6/6 would be equal to six of them, which we can represent here.

For the second one, we can see that this bar represents 3/9 of the whole.

If this is 3/9, then we know we'd need 9/9 all together.

That means we'd need to add on another two lots of 3/9, wouldn't we? So we can find that here.

Next one represents 6/7.

We can find the unit fraction of this as each one would represent 1/7 of the whole.

But we need 7/7, so we're going to need seven of these cubes all together, which would look like this.

And then finally, the last one.

Well, here we've got four counters and they represent 2/5 of the whole.

The unit fraction would be two counters.

That would be 1/5 of the whole.

So therefore, we can now make 5/5 of the whole 'cause that would be five lots of two counters.

Well done if you got all of those.

And the next task then.

You are asked to have a go at creating different wholes that could represent what we'd given you so far.

We'd given you three pentagons to start you off with, haven't we? And each one of those pentagons represented 1/6 of the whole 'cause we had 3/6, and you had three of them.

So if one pentagon represents 1/6, then we need six pentagons to make 6/6 and these are the different ways you could have put your pentagons together.

Equally, you didn't even have to put them together.

You could have just grouped them together separately.

Well done if you managed to come up with those ideas for yourself as well.

Okay, onto cycle two now then.

Finding the whole in different contexts.

So we're gonna start here by looking at some pattern blocks.

Have a look here.

This block here represents 3/8 of the whole.

What could the whole look like? Well, we've given you a number of different pattern blocks here for us to think about how we could start creating these wholes.

So we've got these different shapes we can use.

And we know that this green trapezium here represents 3/8 of the whole.

So let's have a think then.

If the green trapezium represents 3/8 of the whole, which one of these shapes would represent 1/8 of the whole? That's right.

It would be the blue triangles, wouldn't it? We can fit three blue triangles into those 3/8, that each one of those blue triangles would represent 1/8 of the whole.

So we've now recorded this as 1/8 of the whole.

Can we find out the size of the other shapes? Well, that's right.

If we know that the blue triangle is 1/8 of the whole, we could then say that the parallelogram is 2/8 of the whole.

And we can see that because two of the blue triangles make the size of that parallelogram.

There we go.

And then what about the hexagon though? That's right.

Actually, we can fit six of our 1/8 into this hexagon, can't we? So the hexagon could represent 6/8 of the whole, wouldn't it? Okay, so now that we know the value of each one of the shapes, we can combine them in different ways to make the whole, can't we? We know that the whole needs 8/8 all together to make it.

So let's have a look.

Here's one way we could do it.

We could do it by adding 3/8 and 3/8 and another 2/8 together, which would be equal to 8/8.

Here's 3/8 and another 3/8 and another 2/8.

That would be equal to 8/8.

This is just one example of how it could have looked.

There are lots of different ways.

Here's Jacob.

He's come up with a different way.

Let's see what he's done.

He's used the hexagon, hasn't he? He said this represents 6/8.

And now he needs another 2/8 to make 8/8.

So he's gonna use, oh, he's gonna use two triangles for that.

So he's added 1/8 and another 1/8.

So we've got 6/8 plus 1/8 plus 1/8 is equal to 8/8 as well.

So this also represents the whole.

Okay, let's see how we can build on this understanding now, moving away from shapes into these figures here.

This set of figures represents 2/3 of Jacob's block figure collection.

So the question is, how big is his collection all together? Would you like to take a moment for yourself to have a quick think? We can say we've got four figures here, haven't we? And we know these four figures represent 2/3 of the whole collection.

That must mean that two figures represents 1/3 of the collection 'cause we can divide the collection into two groups because the numerator helps us identify how many groups we have, doesn't it? The numerator is two, that means we have two groups.

So the size of one group would be two block figures, wouldn't it? So 1/3 would be two block figures, and another 1/3 would be another two block figures.

So we can say that we need three lots of 1/3 to make the whole, can't we? There we go.

We would need another lot of 1/3 to add on.

Therefore, we would have six block figures all together, wouldn't we? Which would make the whole collection.

Okay, and here's another example now.

Sofia likes to play the guitar.

This is one of her guitar strings that's been snapped.

So what you can see here is 5/8 of the guitar string that was left after it had been snapped.

So how long was the original guitar string? Well, we can say that this string here is 5/8 long of the whole string, can't we? So 1/8 of the guitar string would look like this.

We know that the guitar string at the top is 5/8 of the whole long.

And therefore, we know that the guitar string at the top is made up of 5/8.

The numerator is five.

That means it could be comprised of five parts and this would be one of those parts.

However, if we've already got 5/8, we know we need another 3/8 to make the whole, don't we? So another 3/8 would look like this.

And then we could apply this to the broken string of 5/8, which would now give us the whole length of the original string of 8/8.

Well done if you managed to get that too.

Okay, time for you to check your understanding now.

Can you tick the image that represents the whole collection? Sam is saying this represents 3/4 of her marble collection.

Take a moment to have a think.

Brilliant.

Yeah, that's right.

A would represent the whole collection, wouldn't it? And why is that? Well, the numerator is three, isn't it? That means it tells us how many groups we have.

We have three groups.

So if we divide the original image into three groups, we can see the size of each one of those groups.

That would represent 1/4 and we need 4/4, don't we? To make the whole.

So therefore, if 1/4 was made up of four marbles, then 4/4 would need four lots of four marbles, which would be 16 marbles.

Well done if you got that.

And another quick check.

I'm 2/3 of the way through my swimming lesson.

I've been swimming for 20 minutes.

How long is a swimming lesson all together? Take a moment to have a think.

That's right, it's B, isn't it? It's 30 minutes long.

And why is that? Well, if we've been swimming for 20 minutes so far, and that's 2/3 of the lesson, and we know that the numerator here is two, which represents two groups or two parts, so if the 20 minutes can be divided into two parts, each part would be worth 10 minutes.

So 1/3 of the lesson would be 10 minutes.

So 3/3 of the lesson would be 30 minutes all together as well.

Well done if you got that.

Okay, time for you to practise again now.

What I'd like to do is use the pattern blocks to recreate the whole.

For each example, you might like to come up with as many different solutions as you possibly can as well.

And for task two, what I'd like to do is to have a think about which of the wholes would be larger each time? And how do you know? Good luck with those two tasks and I'll see you back here shortly.

Let's have a look at some of the solutions that you could have come up with.

Okay, for this one here then, you can see that the parallelogram represents 2/5 of the whole.

That means the blue triangle would be 1/5 of the whole.

So we just need to add on another 3/5 of the whole to the 2/5.

That's equivalent to three blue triangles.

Or actually, we could have used the trapezium 'cause that's equivalent to three blue triangles, isn't it? And for the second one, we've got a hexagon, which represents 6/12 of the whole.

Once again, the blue triangle in this case would represent 1/12 of the whole.

At the moment, we've got 6/12, so we need to find another 6/12, don't we? So we could do it like this.

We can see here that I've got another three blue triangles that represents 3/12 and I've got a trapezium, which represents another 3/12.

So all together, we've got 12/12.

And for the last one there, this is a bit trickier, isn't it? 2/8 of the whole is represented by three blue triangles.

So what would 1/8 of the whole look like? That's right.

1/8 of the whole would be one and a half blue triangles, wouldn't it? So we now need how many lots of one and a half blue triangles? That's right.

That means we're going to need eight lots of one and a half blue triangles, doesn't it? And here we go.

Here's an example of which you could have come up with.

We equally could have said that we know that 2/8 of the whole is three triangles.

So 4/8 of the whole would be six triangles.

6/8 of the whole would be nine triangles and 8/8 of the whole would be 12 triangles.

So we could actually break down all these shapes here into 12 triangles, which would be equivalent to 8/8 of the whole as well.

Well done if you managed to get that too.

Okay, well, if we have a look at those pizza slices carefully then, we can see that one of them represents 2/6 of the whole and one of them represents 3/6 of the whole.

We have to visualise here, can you visualise what 1/6 would look like for both of those parts, for both of those slices? Yeah, 1/6 of the slice on the left would be half the amount of that slice and 1/6 for the slice on the right would be 1/3 of that amount 'cause we're using the numerator to help us identify how many groups there are all together.

We know that 1/6 of the slice on the left-hand side is bigger than 1/6 of the slice on the right-hand side.

And as a result of that, when you make the wholes then, if 1/6 of the slice on the left is bigger than 1/6 on the slice on the right, then the whole on the left would be bigger.

Question B then.

We've got 3/5 and 3/8.

What did you notice this time? Okay, well, the numerators are the same, aren't they? But the denominators are different.

Well, if the top bar represents 3/5 of the whole and the bottom bar represents 3/8 of the whole, we can just look at these here, can't we? And see that this is three out of five equal parts and the bottom one is three out of eight equal parts.

So three out of eight equal parts is definitely going to be larger than three out five equal parts as it has more parts.

And then for the last one, we've got 2/3 and 3/4 of the whole.

Well, the numerators aren't the same or the denominators aren't the same here.

So we can't compare this very easily.

We're gonna have to find the unit fractions for both of these.

So we can see from the marbles on the left-hand side, this represents 2/3 of the whole.

So to find 1/3 of the whole, we'd need to halve the amount of marbles.

That would be equal to six marbles.

On the right-hand side, the numerator is three.

So this is 3/4 of the whole.

That means we need to divide this 3/4 into three parts to find the size of the unit fraction.

Therefore, on the left-hand side, the unit fraction represents six marbles.

And on the right-hand side, the unit fraction will represent four marbles.

If the unit fraction on the left is six marbles and the unit fraction on the right is four marbles, then each part with six marbles will obviously be larger than each part with four marbles.

So we can say that the marble collection on the left-hand side would have the larger whole.

Okay, onto our last cycle for today now then, finding the whole from a known quantity.

Okay, let's have a look here then.

If this shape was 1/3 of the whole, how many squares make up the whole shape? Take a moment to have a think for yourself.

Okay, so we can use the information from the fraction here to help us with this, can't we? We know that the whole has been divided into three equal parts.

This part here represents of those three parts.

So therefore, we need three parts all together, don't we? One part has four squares.

So three parts all together would have 12 squares, wouldn't it? We can represent this as a bar model as well.

If we walk through our parts again, we know that the whole was originally divided into three equal parts.

There we go.

And we had one of those parts and the value of that part was four squares, wasn't it? If we know that the value of one of those parts is four squares and each of the other parts are equal in size, then the other parts must also have a value of four.

And now we can find out the total amount, which would've been 12 squares all together, and we've recorded that in the top of the bar model.

Okay, quick check for understanding.

If this shape is 1/4 of the whole, how many squares is the whole shape made up of? Can you draw a bar model to represent this image as well? Okay, so we know this represents 1/4 of the whole.

So we know that the whole has been divided into four equal parts.

There are the four equal parts.

Let's draw on our bar model.

Here's the whole.

And we divided that into four equal parts.

We know that one of these parts has a value of five squares, doesn't it? And we can see that here.

Therefore, each of the other parts must also have a value of five squares.

That leads us to a total of 20 squares all together in the whole shape.

Okay, let's have a look at this example now then.

If this is 2/3 of a set, how many circles make up the whole set? Take a moment to have a think.

What would you do here? Well, once again, we can use the fraction to help us here, can't we? We know that the whole set has been divided into three equal parts.

Here we go.

Here's our bar model.

We're dividing that into three equal parts now.

But at the moment, we have two parts of this set and that's represented here by the numerator.

So I'm gonna draw another bar underneath two parts of the bar model to represent the two parts that we already have.

We know the value of those two parts is equal to eight, don't we? We can see there are eight dots, which make up 2/3 of the whole.

So can we now find out the size of one part? Yeah, that's right.

We can, can't we? If we wanna find out the size of 1/3 of the whole, we can now half the eights, can't we? So if we half the eight, we'll now find what 1/3 of the whole is.

Half of eight is equal to four.

That must mean that all the other equal parts are also equivalent to four.

And now we can find the total amount, which would be equal to 12, wouldn't it? So we can therefore say that 2/3 of the whole is eight.

So 1/3 of the whole is four, which means that 3/3 of the whole is 12.

Brilliant.

Did you notice the whole time there how we found the size of the unit fraction first and then multiplied that to find the size of the whole number? That's a really useful strategy to help us as we go through the remainder of the lesson.

3/5 of a number is 60.

What could the number be? How would you go about tackling this? Do you think you could use a bar model and the fraction to help you identify this? Let's see what Aisha's going to do.

She says we should use the fraction and we know from the fraction that the whole has been divided into five equal parts.

So there's our whole and we've now divided it into five equal parts.

And how many parts do we have? That's right, we have three parts, don't we? And we know that the value of three parts is equal to 60.

So we've drawn a bar underneath to show that the 60 is equal to those three parts.

Right, so how are we gonna find out the size then of the unit fraction, which we said before would be really helpful.

Well, that's right.

If we know 60 is made up of three equal parts, then we just need to find out what the size of 1/3 of 60 is.

We need 1/3 of 60 to find out the size of the unit fraction.

We know that 1/3 of 60 is equal to 20 because we can go 20, 40, 60.

Three lots of 20 are equal to 60.

And if that's the case, if the unit fraction is 20 and each part must also represent 20, and now we can find the size of the whole, can't we? We've got five parts, each of them represent 20.

So that would be 100.

So the whole originally was 100.

Well done if you managed to get that.

Okay, time for you to check your understanding now.

4/5 of a number is 24.

What is the number? Use the fraction to help you to draw a bar model.

Okay, let's work through this then.

So at first, we can divide the whole into five equal parts because that's what the denominator tells us to do.

And then we can have a look at the numerator, which is four.

And we know that four of those parts would be equal to 24, as it says in the question.

Can we find the size of the unit fraction now? Well, if 24 is equal to four parts, then we just need to find out what 1/4 of 24 is to find out the size of one of those parts.

So finding 1/4 of 24 would be equal to six.

And as Sam's pointed out, we could use our division facts to help us understand this.

We know that 1/4 of 24 is equivalent to saying 24 divided by four.

So 24 divided by four is equal to six.

So if each part is worth six all together, we can find out that five lots of six is equal to 30.

So the original number was 30.

Okay, and time for you to practise now then.

What I'd like you to do is draw a bar model for each of the problems and then have a go at solving them.

And then I'd like to do the same for question two as well, but look more carefully at the numbers.

Is there anything there that you can use to help you? And then finally, for question three, what I'd like you to do is have a go at solving these different problems in context here.

Good luck with that and I'll see you back here shortly.

Okay, let's have a look at number one then.

So 3/5 of a number is 30.

What is the number? We know that the whole would be divided into five equal parts and three of those parts would be equal to 30.

That must mean each part is worth 10 and all together, the whole would be 50.

For the second one, we need 4/7 of a number is 32.

Well, the whole would be divided into seven equal parts and four of those parts would be equal to 32.

Therefore, 32 divided by four would find the size of each one of those parts, which would be equal to eight.

If the value of each one of those parts is eight, then we've got seven dots of eight, which is equal to 56.

For the next one, 6/9 of a number is 42.

Well, the whole is divided into nine equal parts and six of those parts is equal to 42.

If we divide 42 by six, we can find the size of each of the unit fractions, which gives us seven and therefore, we've got nine lots of seven, which is equal to 63.

And the last one, we've got 2/6 of a number is equal to 200.

Well, the whole has been divided into six equal parts and we have two of those parts.

Those two parts represent 200.

To find out the size of one of those parts, we can half 200, that would give us 100.

Therefore, if each of the parts represents 100, all together the whole would've been 600.

Well done if you managed to draw those bar models and find the original number as well.

Okay, let's see how we get on this time now then.

1/4 of a number is 15.

So what is the number? Well, we know that 1/4 is 15, so we can represent it like this.

That means each of the parts is 15, which leaves us with a whole of 60.

For the second one, 2/4 of a number is 30.

So what is the number? Well, we can divide the whole again into four equal parts.

This time 2/4 of the whole would be equal to 30.

That leads us with 1/4 of the whole being 15 again.

So the original number again was also 60.

What about the next one then? 3/4 of a number is 45.

What is the original number? Well, the whole again has been divided into four equal parts.

Three of those parts is equal to 45.

So what would each part have a value of? That's right, each part would a value of 15, wouldn't it? So there we go.

And again, that gives us a total of 60.

60 was the original number.

And then for the last one, you could probably already guess, can't you? The whole has been divided into four equal parts and those four parts is equal to 60, wasn't it? So that means 60 divided by four gives us the size of each one of those parts, which is 15.

But of course, we already know that the whole would be equal to 60.

Well done if you managed to spot that as you went along.

Okay, and onto our final two questions for today then.

A gardener picks apples from three of her eight apple trees.

So far she had 180 apples.

How many apples will she roughly have when she's picked from all of the trees? Now, this problem assumes that she gets the same number of apples from each of the trees.

So assuming that she does do that, let's try and figure out roughly how many apples she would get.

Well, we know that a whole set of trees, there are eight of them.

So we can divide the whole into eight equal parts.

And after picking the apples off of three of those trees, that's equal to 180 apples.

So we've got 180 apples from three trees, haven't we? So we can find out how many apples we got from one tree by doing 1/3 of 180 or 180 divided by three.

That gives us 60.

So if we've got 60 apples from one tree, then we can get 60 apples from all of the other trees, assuming we got the same amount of apples from each tree.

We had eight trees, so that's eight lots of 60 all together.

I know eight lots of six is 48, so eight lots of 60 is 480.

So the gardener would've got 480 apples.

That's a lot of apples to eat.

She'd have had to have shared them around with the locals I expect.

And finally, a teacher hands out chocolate coins to her class.

So far, she's handed out 36 coins to 12 of the 30 children in her class.

How many more coins does she need to hand out? Well, if we were to draw the bar model for this one, well, the whole would've been how many chocolate coins she needs to hand out all together.

We know that there's 30 children in the class, so we need to divide the whole by 30.

And we know that between 12 of those children, 36 chocolate coins were received.

So if 12 children received 36 chocolate coins, that means they got three coins each.

And therefore, if 12 children got 36 coins, we can find out how many more coins she needed to hand out by multiplying 18, which is the remaining number of children by the amount of coins each child got.

If each child got three coins, then we need to hand out another 18 lots of three coins, and that in total would be equal to another 54 coins.

How many coins would she need to hand out all together then? Well, if she handed out 36 coins to 12 of the children and another 54 coins to the rest of the children, how many coins is all together? That's right.

90 coins all together.

Well done if you managed to get that too.

Okay, that's the end of our lesson for today now then.

Let's see if we can summarise what we've learned.

So to find the size of the whole when the size of the non-unit fraction is known, you can firstly, find the size of the unit fraction and then identify the size of the whole based on the number of unit fractions required to complete it.

You could also use the denominator to help you identify how many unit fractions make up the whole.

And then finally, you can apply your knowledge of times table and division facts to help you find the value of the whole.

Thanks for joining me again today.

Hopefully you're feeling a lot more confident about how you can apply your understanding of multiplication of whole numbers with unit and non-unit fractions to find different amounts.

Take care and I'll see you again soon.