Lesson video

Hi, my name is Mr. Peters, and welcome to today's lesson.

In this lesson we're gonna be thinking about how we can find a non-unit fraction of a quantity, using a range of both mental and written strategies.

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 use mental and written calculation strategies to find a non-unit fraction of a quantity.

In today's lesson we're gonna have two key words that we're gonna be referring to throughout.

The first word is denominator, and the second word is numerator.

Can you have a go at saying them? Well done.

So let's just recap what these mean.

A denominator is the bottom number of a fraction.

It tells you how many equal parts the whole has been divided into.

The numerator is the top number of a fraction, it tells you how many parts we have or are needed.

Today's lesson broken down into two cycles.

The first cycle is finding non-unit fractions of an amount, and the second cycle is solving problems in context.

Let's get started with the first cycle.

In this lesson today, you're gonna be joined by both Laura and Lucas, who will share their thinking, as well as any questions that they have throughout the lesson.

So I'm gonna start this lesson here.

I'm gonna ask you all, can you have a go at taking one-fifth of the oranges that you can see? I wonder how you would go about taking one-fifth of those oranges.

Let's see how Laura and Lucas go about it.

Lucas thinks he needs to take three oranges.

I wonder how he knows that.

Lucas is saying, "Well, to find one-fifth of 15, we need to take our whole of 15 oranges and we need to divide the whole into five equal parts." Now that we've divided the whole into five equal parts we can represent this as a division equation, and that would be equal to three, wouldn't it? There'd be three oranges in each part of the whole.

We know that each part represents one-fifth, so we can say that one-fifth or 15 or one-fifth multiplied by 15 is equal to three, or in this case, three oranges.

So what would happen if we took one more group of oranges now? This is one-fifth of 15, so we can say that one, one-fifth is equal to three.

(mouse clicking) And, again, we can write that as a multiplication equation.

However, what happens if we took two one-fifths? Well now we've got two one-fifths, and that would be equal to two-fifths, wouldn't it? So we can also record this as a multiplication equation as well.

We could say two-fifths multiplied by 15 or two-fifths of 15 is equal to.

Well, how many orange is that equal to? That's right, it's six.

We can say that two-fifths of 15 is equal to six, or in this case six oranges.

What if we take another fifth? Well, now we've taken three lots of one-fifth, haven't we, three one-fifths.

And we know that three one-fifths is the same as saying three-fifths, so we can write this as a multiplication equation as well.

three-fifths of 15 or three-fifths multiplied by 15, this time is equal to nine, isn't it? Nine oranges would be equivalent to three-fifths of 15.

So we can also represent this as a bar model, can't we? Let's have a look.

There's our 15 oranges now represented as a bar of 15, and we're gonna divide our 15 into five equal parts and each part has a value of three, or in the context that we're working with, three oranges.

And we know that one one-fifth is equal to three, so we can write that as one-fifth of 15 or one-fifth multiplied 15 is equal to three.

Now we have two one-fifths, don't we? We can write that as two-fifths of 15 or two-fifths multiplied by 15 is equal to six.

Now we've also got three one-fifths, which we can say is the same as three-fifths and we can say three-fifths multiplied by 15 or three-fifths of 15 is equivalent to nine altogether.

(mouse clicking) So if we were to think about this more deeply now, how could we go about finding the value of three-fifths of the whole? Well, we know that one-fifth of the whole is equal to three, don't we, so one-fifth multiplied by 15 is equal to three.

And as Lucas has pointed out, we know that three-fifths is three lots of one-fifth.

So if we know the value of one-fifth, we can multiply one-fifth by three, can't we, to find out the value of three-fifths? So we can say that three-fifths of 15 or three-fifths multiplied by 15 is equal to three lots of three oranges, or three lots of three, which was the value of each of the parts, wasn't it? (mouse clicking) We know that three lots of three is equal to nine, so we can say that three-fifths of 15 or three-fifths multiply by 15 is equal to nine.

Yeah, great thinking Lucas.

Well done.

So to calculate three-fifths, we can find the value of one-fifth, can't we, and then we multiply that by the number of fifths that we need, in this case, it's three.

Let's have a look at another example.

Can we find three-eighths of the whole? Well, there are 16 oranges altogether here, so let's get our 16 oranges and create them as our whole.

And we need to divide our whole into eight equal parts, as that is what the denominator requires of us.

So we now know that the whole has been divided into eight equal parts, and therefore the value of each one of those parts is equal to two, isn't it? So we can now say that one-eighth of 16 or one-eighth multiplied by 16 is equal to two.

So now that we know that one-eighth of 16 is equal to two, we need three lots of one-eighth, don't we? There we go.

Now we've got ourselves three lots of one-eighth, haven't we? That means we need three lots of two.

So we know that three-eighths of 16 is equal to three lots of two, so as a result of that we can say three-eighths multiplied by 16 is equal to six, can't we? Hmm, have a think about what we did there again then.

That's right, we found the value of one-eighth to start off with, didn't we, and then once we found the value of one-eighth we need to identify how many lots of one eighth we need.

Well, it's the numerator that can help us with that, isn't it? If we look at the numerator, the numerator has a value of three and that tells us how many parts we need, doesn't it? So we can divide the whole into the number of equal parts by using the denominator that it tells us to do, and then we can use the numerator to tell us how many of those parts we actually need.

Once we found the value of one of those parts, we can multiply those parts by three because that's what the numerator is, and therefore we know that the value of three-eighths multiply by 16 would be equal to six.

Okay, moving on now to five-eighths of the whole, how could we calculate this? Well, we've still got 16 oranges as a whole, so there we go, and we can divide our whole into eight equal parts again.

And this time we know that the value of each one of those parts is two again, and we know that 'cause 16 divided by eight is equal to two.

We can now work out the size of one-eighth of 16, we know one-eighth of 16 is equivalent to two.

So now that we know the value of one-eighth of the whole, we can now start to think about five-eighths of the whole, can't we? At the moment we've got one-eighth but we actually need five-eighths, so we're going to need to multiply our one-eighth by five, aren't we? And we know that because the numerator is number five.

There we go, hopefully you can see that now.

We've now got five-eighths of our whole shaded, so we know that five lots of one-eighth in this case is the same as saying five multiplied by two, because the value of one-eighth is two.

So five lots of two is equal to 10, and then altogether we can say that five-eighths multiplied by 16 is equal to 10 altogether.

Well done if you managed to identify that for yourself as well.

(mouse clicking) So to summarise what we've been thinking about there, we know to find a non-unit fraction of an amount we need to find the value of the unit fraction of the whole, don't we, and then we need to multiply the value of that unit fraction by the amount of parts that we need, and we can identify the number of parts that we need by the numerator of the fraction that we're multiplying by.

Time for you to check your understanding now.

Which equations are represented by the bar model? Take a moment to have a think.

That's right.

It's A and B, isn't it? We can see that the whole is 24 and the whole has been divided into six equal parts, and we need two of those parts so we can represent that as two-sixths multiplied by 24, and that would be equal to eight altogether 'cause that would be two lots of four, or we could represent it as 24 multiplied by two-sixth because each one-sixth is a value of four, and therefore two lots of one-sixth would be equal to eight.

(mouse clicking) Another quick check.

Can you find three-fifths of 20? Take a moment to have a think, you might like to draw a bar model to help you.

Okay, let's represent it now.

Well, let's find the value of one-fifth to start us off with.

We can take our whole, which is 20, and divide it into five equal parts, which is what the denominator requires of us.

Each one of those parts has a value of four because 20 divided by five is equal to four.

So one-fifth of 20 is equal to four, therefore three-fifths of 20 would be equal to 12.

Okay, time for you to have a practise now then.

Use the representation to help fill in the missing numbers.

And then once you've done that, I'd like you to have a go at calculating each of the values as well.

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

Okay, let's see how you got on then.

Let's work through these then.

The first one, one-fifth of 15, we can see that's three, can't we? two-fifths of 15 would be equal to six, three-fifths of 15 would be equal to nine, four-fifths of 15 would be equal to 12, and five-fifths of 15 would be the whole, wouldn't it, that would be equal to 15.

Hmm, six-fifths of 15, wonder what that would look like? Well, if the value of one-fifth is three, we need to add on an extra one-fifth, don't we? We've got five-fifths, let's add on one more fifth which is a value of three, so that would be 18, wouldn't it? What about 10-fifths? Hmm, well if I know that five-fifths is equal to 15, then 10-fifths is double five-fifths, isn't it? So all we need to do is double the 15, that would give us a value of 30, wouldn't it? And what about the last one, 100-fifths of 15? Well, we know the value of one-fifth of 15 is three, the value of 10-fifths of 15 is 30, so the value of 100-fifths by 15 is 10 times the size the value of 10-fifths.

So 10-fifths multiplied by 15 is equal to 30, therefore 100-fifths multiplied by 15 would be 10 times the size, let's make the 30 10 times the size, that would be 300.

Well done if you've got all of those.

Okay, and to calculate the value of each of these then, I'm gonna run through these for you quickly.

The first one, three quarters of 16 is equal to 12, three-fifths of 25 is equal to 15, five-eighths of 24 is equal to 15 as well, four-ninths of 36 is equal to 16, and finally two-10ths of 50 is equal to 10.

Well done if you've got all of those.

Okay, onto our second cycle now, solving problems in different contexts.

It starts here with the class teacher estimating that they've read four-fifths of the class read or the class book that they're reading to the class.

The book has 200 pages, how many pages have been read so far? Let's think about how we could represent this then.

That's a good idea, Lucas, let's use a bar model to help us, shall we? Well, let's draw our two bars then.

We know the whole is going to be 200 pages so we'll write that in the top, and we need to divide that into five equal parts 'cause that's what the denominator of the fraction is asking of us, so there we go.

Now that we've done that we can write a division equation, 200 divided by five.

Well we know that's equal to 40, so each of the parts has a value of 40, doesn't it? Therefore we know one-fifth of 200 is equal to 40 or one-fifth multiplied by 200 is equal to 40, how many fifths do we need though? That's right, we need four-fifths, don't we? And what tells us that? That's right, the numerator tells us the number of parts that we have or that we need, doesn't it? So we need four-fifths of 200, don't we, so let's represent that by multiplying the number of fifths by four.

There we go.

So the numerator has helped us to identify that we needed four-fifths of the whole, and we can now hopefully calculate the total amount of pages that have been read so far.

We know that one-fifth had a value of 40, so four-fifths is 40 multiplied by four, and that gives us 160, doesn't it? So we can say, altogether, that four-fifths multiplied by 200 or four-fifths of 200 is equal to 160.

Well done if you've got that.

So that's right, Lucas, we've read 160 pages so far, haven't we? How many pages we've got left to read? Yep, that's right.

We've got 40 pages left to read, haven't we? Well done, if you've got all of those.

Let's have a look at another problem.

It takes three quarters of an hour to get to cricket club after school because of the traffic.

How many minutes does it take to get to cricket? Take a moment for yourself to have a think.

Let's see what Laura was thinking.

She knows that an hour is equivalent to 60 minutes.

So let's represent our bar model, so in the whole we're gonna write 60 at the top.

We know we're looking to find the value of three quarters, aren't we, so to find the value of three quarters we first of all need to divide the whole into four equal parts, don't we Laura? So if we look at the fraction, that will help us tell us exactly what we need to do.

The denominator tells us how many equal parts we need to divide the whole into, so we're gonna divide the whole into four equal parts as that is what the denominator is in our fraction.

We know the value of each of those parts would be 15, because 60 divided by four is equal to 15.

We know the value of one of those parts is equal to 15, so we can say that one quarter of 60 or one quarter multiplied by 60 is equal to 15.

And therefore if we know that one quarter of 15 is equal to 60, we can work out three quarters of 15 by multiplying it by the numerator, can't we? The numerator tells us that we need three parts.

So there we go.

We can take 15, which is the value of one of the parts, and we can multiply that by three because we need three parts, don't we, or three-fifths.

Or three quarters.

So now we can say we've got 15 multiplied by three and that's equal to 45.

So, as we know, three quarters of 60 is equal to 45, so it took 45 minutes to get to cricket, didn't it? That's quite a long time in the car, isn't it, to get to your cricket session, Laura.

Sorry you had to wait so long 'cause of the traffic.

Okay, time for you to check your understanding again now.

Can you draw a bar model for this problem? A pair of trainers usually costs £80, I managed to buy them for three quarters of the price.

How much did I pay? Take a moment to have a think.

Well, let's draw a bar model to represent this, here you go.

We know that the whole price of the trainers would've been £80, and if we divide that into four equal parts, we can find the value of one of those parts, which would be 20.

But we bought it for three quarters of the price, didn't we, We didn't buy it for one quarter of the price.

So three quarters of the price would be three lots of 20, and that would be equal to 60, wouldn't it? Well done if you've got that.

Okay, and another quick check as well.

Can you solve the following problem? On World Book Day, four-ninths of the class dressed as witches and wizards.

There are 27 children in the class.

How many pupils dressed as witches and wizards? Take a moment to have a think.

Well, that's right, so we know the whole class was 27 and we divided that 27 into nine equal parts, and each one of those parts gives us a value of three because we know that 27 divided by nine is equal to three.

We got the nine from the denominator of our fraction, which tells us how many equal parts we need to divide the whole into.

Then we can work out the size of one-ninth of the whole, can't we? We know that one of those parts has a value of three, so one-ninth of 27 is equal to three.

However, we don't need one-ninth, do we, we need four-ninths.

So we can say one-ninth multiplied by four would be the same as taking the value of one-ninth, which is three, and then multiplying that by four.

So four-ninths of 27 would be equal to 12 altogether.

Well done if you managed to get that.

Okay, onto our last tasks for today then.

What'd like to do is represent each problem using a bar model, and solve them as well.

There are four questions to do, entitled A, B, C, and D.

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

Okay, let's see how you got on then.

The first problem says this, "To save money, a company make their chocolate bar four-fifths of the original size.

The bar was originally 20 centimetres long.

How long is the new bar?" Well, we could represent that as a bar model.

The whole was 20 centimetres and we divide the whole into five equal parts, and then we needed four of those parts, didn't we? The value of one of those parts was four, so four multiplied by four gives us 16; so we can say that four-fifths of 20 is equal to 16.

For question B, I'm aiming to raise £400 for charity.

So far I've raised five-eighths of the total.

How much money have I raised? Well, let's represent this again as a bar model.

The whole £400, that's what we're aiming for, and we can divide the whole into eight equal parts based on the denominator of the fraction.

And we need four of those parts, don't we? We know that one of those parts has a value of 50 because 400 divided by eight is equal to 50, so now we need to multiply 50 by the number of parts we need, which is five parts, so 50 multiply by five is 250.

That means we've raised £250 so far.

For question C, Jacob had £300.

He spends two-sixths of his money on a new computer game.

Then he spends three-fifths of the remaining money on a new gaming chair.

How much money does he have left? Well, let's start off by representing the first part.

Jacob had £300 to start off with, didn't he, and this is when he bought his new computer game.

So we know that the computer game cost two-sixths of the total amount of money he had, so we can divide the hold into six equal parts and we need two of those parts, don't we? The value of one of those parts is 50 because 300 divided by six is equal to 50, but we didn't just need one of those parts and we needed two of those parts.

So we can see that altogether 50 multiplied by two would be equal to 100, so he spent £100 on his new computer game.

Gosh, these computer games are going up in price, aren't they? If he spends £100 on his computer game and we have £300 to start off with, that means he has £200 left.

So that is our new whole.

He then spends three-fifths of the remaining amount of money.

We know the remaining amount of money is £200, and to find out how much he spends of that we can divide that into five equal parts 'cause the denominator is five, and that tells us how many equal parts we need.

200 divided by five is equal to 40, so we can see that each part has a value of 40.

But we don't just need one part, do we, we need three parts so we can multiply that 40 by three, which gives us 120.

So that's £120 that Jacob spent on the gaming chair.

How many parts does that leave us with then? That's right, that leaves us with two parts, doesn't it, or two-fifths of the whole, and that's the amount of money that Jacob has left.

two-fifths of the whole amount of money that's left is equal to 80, so we can say that Jacob has £80 left.

Well done if you managed to get that one.

Okay, and the last one then.

Laura completed one-12th of her sticker album.

She has 24 stickers so far.

How many stickers would she have if she had four-12ths of the album, or if she had completed the whole album? So here we don't know the amount of stickers that make up the whole, do we, but we do know the amount of stickers that make up one-12th of the whole.

We can represent this as an equation, can't we? We can say one-12th multiplied by something is equal to 24.

We know we can find the size of four-12ths, can't we, quite quickly, because we can multiply the value of one-12th by four.

So we can also say that four-12ths multiplied by something is equal to 96, because one-12th is 24 and we can multiply 24 by four, which gives us 96.

So if she had four-12ths of the whole, that would be 96 stickers, wouldn't it? And what about if she had the completed whole? We can say that 12-12ths multiplied by the whole is equal to 24, which is the value of one-12th multiplied by 12.

We want 12-12ths, and we can find 12-12ths by doing one-12th multiplied by 12.

So 24 multiplied by one-12th would give us a total of 288, so we know now that the whole album had 288 stickers.

There's a few more stickers to collect there, Laura, aren't there? Good luck with that, and try not to spend too much money doing so.

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

I've really enjoyed that lesson and, again, I'm feeling a lot more confident about finding the value of a non-unit fraction of a quantity.

Hopefully you are too.

To summarise what we've learned today, we can say you can use the fraction that you are multiplying by to help you represent and calculate the problem.

The denominator tells you the number of equal parts the whole needs to be divided into, and the numerator tells you the number of parts that we have or that we need.

To find a non-unit fraction of an amount you can find the value of a unit fraction, to start off with, and then multiply the value of the unit fraction by the number of parts that are needed; and that's represented by the numerator, isn't it? Thanks for joining me today.

Take care, I'll see you again soon.