Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson from our unit on multiplication and division of fractions.

Have you done lots of work on fractions recently? Do you like fractions? I hope you do.

I love fractions.

So let's have a think about how we're going to be able to multiply and divide using fractions.

Are you ready? Let's make a start.

In this lesson we're going to be explaining how to multiply two non-unit fractions.

So non-unit fractions are when the numerator is not equal to one.

So let's have a look at what's in our lesson today.

We've got one keyword and that's represent.

I'm sure it's a word you know, but let's go through and check it and then check what it means.

So I'll say it and then you can have a turn at saying it.

Are you ready? My turn, represent.

Your turn, excellent.

What does it mean to represent something? Well, it means to show something in a different way.

So we're going to be thinking carefully about how we can represent the multiplication of fractions.

It's really important that we understand what is happening when we multiply fractions together.

There is a rule, a generalisation that we can follow, but it's really helpful to know why that works, because then if we were ever to forget the rule or wonder whether we were right or not, we can go back to the representation to help us to understand what's really going on.

So there are two parts to our lesson.

In the first part, we're going to be multiplying non-unit fractions by a unit fraction, and in the second part we're going to be multiplying two-unit fractions.

So let's make a start on part one.

And we've got Andeep and Jacob with us in our lesson today.

Jacob ate half a bar of chocolate on Monday.

He ate 3/4 of what was left on Tuesday.

How much of the whole bar did he eat on Tuesday? So here's the whole bar of chocolate.

Can you imagine what the story is, written out, sort of on the bar of chocolate represented on this rectangle.

He had half a bar of chocolate on Monday and he had 3/4 of what was left on Tuesday.

Let's see what that looks like.

So he's eaten half of the bar, so he ate that on Monday, that's gone.

And then he had 3/4 of what was left on Tuesday, so that's gone as well.

But what we want to know is how much of the whole bar did he eat on Tuesday? So that's represented by those three parts, 3/4 of what was left after he had eaten half of it on Monday.

How are we going to work this out? So let's see that again.

Half a bar on Monday, and so this shows that he had half a bar left on Tuesday.

And then he ate 3/4 of what was left.

So the arrow represents the 3/4 he ate on Tuesday.

But what is that as a proportion of the whole? Well it's 3/4 of a half, isn't it? So 3/4 multiplied by a half, we know we can replace the word of with multiply.

So 3/4 of a half, 3/4 multiplied by a half or a half multiplied by 3/4.

If we split both of our halves into quarters, we've now got equal pieces, so we can work out what those three pieces are as a fraction of the whole.

So we want to know what 3/4 of one half is, and that is equal to 3/8.

We've now divided the whole into eight equal parts, and we can see that the part that Lucas ate on Tuesday is three of those parts, so 3/8.

So 3/4 of one half is equal to 3/8.

Lucas's sister had a third of a bar of chocolate left after she ate some on Monday, then she ate 4/5 of what was left on Tuesday.

How much of the whole bar did she eat on Tuesday? So let's have a think about that.

She had a third of a bar left, so she must have eaten 2/3 of the chocolate bar on Monday, and then on Tuesday she ate 4/5 of what was left.

So how much of the whole bar did she eat on Tuesday? So then we can see the 2/3 that she ate and then we can see the 4/5 of that remaining third that she left.

Aha, can you see something happening there? 4/5 of a third? Yes.

so this time we want to know what 4/5 of 1/3 is equal to.

So 4/5 of 1/3 we can write as 4/5 multiplied by 1/3, or 1/3 of 4/5.

So we know that that 1/3 was divided into five equal parts, so we can divide the other 2/3, that she's already eaten, into five equal parts.

So how many parts are there in the whole now? There are 15 and she ate four of them.

So we know that 4/5 of a third or 4/5 multiplied by 1/3 is equal to 4/15.

Who ate the most chocolate on Tuesday? Lucas ate 3/8 of the bar and his sister ate 4/15 of the bar.

Can you use your fraction sense to compare those? What do you think? Well, 3/8 is pretty close to a half, isn't it? And 4/15 is quite a lot less than a half.

So as Andeep says, three is a larger portion of eight than four is of 15.

So we can say that 3/8 is larger than 4/15.

Time to check your understanding.

Can you write an equation to represent this image? I'm going to show you a short sequence and then you are going to pause the video and write an equation to represent the yellow part that we're left with.

Are you ready? So here we are, what can you see now? And now? And now? Pause the video and write an equation to represent the calculation that we would use to work out the value of the yellow area with the question mark.

What did you think? Did you notice that we had a quarter shaded to begin with and then we were working out 2/5 of that quarter? So we've got 2/5 of 1/4, and when we divided the shape up into equal parts, each of those quarters into five parts, we had 20 parts in the whole, and two of those parts are shaded in yellow.

So 2/5 of 1/4 is equal to 2/20.

Or 1/4 times 2/5 is equal to 2/20, because of commutativity, we can write those factors either way round and the product remains the same.

And another one to check your understanding with.

Again, I'm going to show you a short sequence and then you are going to pause and write the equation.

So what's been shaded now? We are interested in the yellow area.

So can you write the equation? Pause the video, when you're ready for some feedback, press play.

How did you get on? So we had 1/2 and then we divided our half into six equal parts, and we want to know what four of them is.

So we're looking at 4/6 of 1/2.

4/6 multiplied by 1/2, 4/6 of 1/2.

If we divide each of our halves into six equal pieces, we've got 12 pieces in the whole and we've got four of them.

So 4/6 of 1/2 is equal to 4/12.

Well done if you got that right.

Time for you to do some practise.

For question one, you're going to draw a model to represent each equation.

And for question two, you're going to write an equation to represent each model.

Pause the video, have a go at questions one and two, and when you're ready for some feedback, press play.

How did you get on? So you're going to draw an image to represent each equation.

So we were finding here 2/4 of 1/3.

So there is our 1/3, and that's been divided into four equal parts, and we've got 2/4 of that 1/3.

And if we imagined dividing each of our thirds into four equal pieces, we'd have 12 pieces in the whole and two of them are highlighted with our question marks.

So 2/4 of 1/3 is equal to 2/12.

What about 2/5 of 1/3? Oh, so again, we've divided into three equal parts, and then our third has been divided into five equal parts.

And if we imagine that happening to the other 2/3, we'd have 15 parts in our whole and two of them are highlighted.

So 2/5 of 1/3 is equal to 2/15.

And again, 2/6 of 1/3 this time, can you see that the part we are looking at is getting slightly smaller each time? So this time our thirds have been divided into six equal parts.

So that will give us 18 parts in the whole, and we've still got two of them highlighted.

So 2/6 of 1/3 is equal to 2/18.

Did you notice that you could always represent this as a portion of 1/3? So for two you were writing the equation to represent the image.

So this time we've divided our whole into thirds and then we've got 2/5 of 1/3.

So that's 2/15.

So for A, we were finding 2/5 of 1/3.

What about B? Well this time our whole is divided into quarters and then one of the quarters has been divided into fifths and we've got two of those fifths.

So this time we've got 2/5 of 1/4.

If we added in those extra lines to divide each of our quarters into five equal parts, we'd see that we had 20 parts in the whole, and two of them represented by our yellow question mark areas.

So this time we were finding 2/5 of 1/4.

And what about C? Well, this time we've got our whole divided into five equal parts and again, we're finding 2/5.

So this time we're finding 2/5 of 1/5.

And again, if we imagine each of those fifths divided into five equal parts, we'd have 25 parts in our whole, and so our yellow area would be 2/25.

So did you notice that each represented finding 2/5 of a different unit fraction? 2/5 of a third, 2/5 of a quarter and 2/5 of 1/5? Well done if you spotted that and got those correct.

So let's move on into the second part of our lesson.

This time we're multiplying two non-unit fractions.

So let's have a look at a problem here.

There were 4/5 of a pint of milk left in the fridge.

Jacob used 2/3 of this to make some porridge.

How much milk did he use? So let's have a think, there were 4/5 of a pint of milk left in the fridge and Jacob used 2/3 of the 4/5.

Oh, can you see something there? You know what we can replace of within an equation? Let's see how Andeep and Jacob are going to tackle this.

Andeep says, "Let's represent this with an area model." Good idea, Andeep.

Then we can really see what's happening.

So there is our pint of milk and we've represented 4/5.

So the blue represents the 4/5 of the pint of milk.

And we know that Jacob uses 2/3 of this to make some porridge.

So what are we going to need to do to this image to show that.

How that's right? So we can divide that 4/5 up into three equal parts and Jacob has used two of them.

So the yellow represents the 2/3 of the 4/5.

So 2/3 of 4/5 is equal to something, and we know that we can replace the of with multiplication.

So 2/3 multiplied by 4/5.

What do we need to do now? Well Jacob says, "To find out how much I used, we need to find the value of each part in relation to the whole." So we've divided four of those fifths up into three equal parts.

So we need to do the same for the final one.

So now we've got our whole divided into equal parts, and we know that we must have three times five of those equal parts, so we must have 15 equal parts in the whole, and Jacob used eight of them.

So 2/3 of 4/5 is equal to 8/15.

He says that, "We can see that I used 8/15 of the whole." So 2/3 of 4/5 is equal to 8/15, 2/3 multiplied by 4/5 is equal to 8/15.

Let's have a look at another one.

We're thinking about growing vegetables in the garden here.

3/8 of the garden is used for growing vegetables and 3/4 of that space will be used to grow potatoes.

How much of the whole garden is used to grow potatoes? What do you think? You might want to pause here, have a think about it, before Andeep and Jacob share their thinking.

Andeep says, "Let's represent this with an area model as well." Good idea, Andeep.

So there's our whole garden and there's the 3/8 that's going to be used for growing vegetables.

The blue represents the eighths that's used for growing vegetables, but we want to know what 3/4 of that 3/8 would look like.

So can you picture what's gonna happen to the image? That's right, we need to divide it into four equal parts and highlight three of those parts.

So the yellow represents the 3/4 of 3/8.

3/4 of 3/8, and we know that we can replace that of with a multiplication symbol.

So 3/4 multiplied by 3/8.

We've got to work out what that is worth.

So what are we gonna have to do? Well, Jacob says, to find out how much is used, we need to find the value of each part in relation to the whole.

So three of those eighths, we've divided into four equal parts.

So we need to do the same for the whole garden, don't we? So we've got eight parts divided into four equal parts, so we've got 32 parts altogether, and we've used nine of them for growing the potatoes.

So we can see that 9/32 or 9/32, however you like to say it, of the whole are used to grow potatoes.

3/4 of 3/8 is equal to 9/32.

3/4 multiplied by 3/8 is equal to 9/32 or 9/32.

So what do you notice about that? We found 2/3 of 4/5 and 3/4 of 3/8, and we wrote them as multiplication, and those are the multiplication with their answers.

Andeep says, "I noticed that the numerator of the product is the result of multiplying the numerators of the fractions being multiplied." So let's think about that.

The numerator of the product is the result of multiplying the numerators of the fractions being multiplied.

So let's look at that.

So we've got 2/3 multiplied by 4/5.

So we've got two and four as our numerators, and yes, two times four is equal to eight.

And then we have 3/4 of 3/8, 3/4 multiplied by 3/8.

Three times three is equal to nine, that's right.

"And the same can be said for the denominators," says Jacob.

If you multiply the denominators together, you get the denominator of the product.

So in our first pair of fractions, we had 2/3 multiplied by 4/5, so three times five is equal to 15, and then we have 3/4 multiplied by 3/8, four times eight is equal to 32.

So to multiply two non-unit fractions, multiply the numerators together and multiply the denominators together.

It's a really good rule to remember, but it's always useful to be able to go back to the area models that Andeep was drawing to see why that works.

We know that if we've got 2/3 of 4/5, we need to split each of our fifths into three equal parts, that will give us 15 parts in the whole.

And that those 4/5 will each have been split into double the number of parts, eight equal parts.

So our result will be 8/15, which we can get by multiplying the numerators and the denominators of the fractions that we're multiplying, but if we can relate it back to the area model, we can understand why that works.

Time to check your understanding.

Can you write an equation to represent this image? So look carefully as the image builds.

So there's our first part, our second part, our third part.

Can you write the equation that represents the image or that would allow us to calculate the yellow shaded area? Pause the video, have a go, when you're ready for some feedback, press play.

How did you get on? So initially, we had our whole divided into four equal parts and three of them were shaded, and then we divided each of those into five equal parts and two of them were shaded in each case.

So we had 3/4 and then we were finding 2/5 of 3/4, so 2/5 of 3/4, and we can see on the image and we can understand why the multiplications work, that that's going to be equal to 6/20.

Each of our quarters has been divided into five equal parts, and we know that we've got 2/5 in each of those parts, and we need three of them.

So the yellow shaded area is 6/20 of the whole.

And here's another check.

Can you write an equation to represent this image? So look carefully, what can you see? And then there.

So pause the video and write the equation that represents this image to calculate the area of the yellow part of the shape.

How did you get on? So this time we had fifths and we had 3/5 shaded, and we were finding 2/5 of our 3/5.

So we were calculating 2/5 of 3/5, and we know we can just multiply the numerators and then multiply the denominators, but we can see why that works by looking at the image.

So the yellow shaded area represents 6/25 of the whole shape.

Well done if you got those right.

And it's time for you to do some practise.

Can you, for question one, complete the calculations, and you may well just be able to apply the rule, but think in your mind, what would the area model look like? Why does the rule work? And some more to completing question two.

And then for question three, can you fill in the missing numbers? So you might need to use that rule that you know about multiplying the numerators and denominators, to work out what the missing fraction is in each case.

And now here we've got three fractions to multiply together.

Just like whole numbers, we can multiply more than two values together.

So what three fractions could you multiply to get a product of 3/20? Pause the video, have a go at your four questions, and when you're ready for some feedback, press play.

How did you get on? So here were our calculations.

So we had 4/5 of 1/3 or 4/5 multiplied by 1/3, which is equal to 4/15.

For B, 1/7 multiplied by 5/8 is equal to 5/56.

One times five is five, seven times eight, my favourite times table fact, is equal to 56, so 5/56.

For C, we had 2/9 multiplied by 7/8.

So two times seven is equal to 14, nine times eight is equal to 72.

So 14/72 or 14/72.

For D, 3/4 multiplied by 5/6.

So 3/4 of 5/6, we could picture a whole divided into six equal parts, five of them shaded, 3/4 of those 5/6, and we would see that that was representing 15/24 of the whole.

4/5 of 5/12 will give us the fraction 20/60.

And for F, we were finding 3/10 of 5/6, and that gives us a fraction of 15/60 of the whole.

And Andeep says, "You may have simplified some of your answers as well." Can you see that quite a lot of them have common factors shared by the numerator and denominator? And so we could simplify them.

Not all of them, but some of them.

And some more to do for question two.

So 1/5 of 2/3 is equal to 2/15.

2/5 of 2/3.

Ooh, can you think what that's going to be? 2/5 of 2/3? 1/5 of 2/3 was 2/15.

So 2/5 of 2/3 must be 4/15.

What about 3/5 of 2/3? Well that must be 6/15.

4/5 of 2/3, that must be 8/15.

So for E, 5/5 of 2/3 must be equal to 10/15.

Or we could have said just 2/3 there because 5/5 is the same as one, isn't it? And then for F, 6/5 of 2/3, it's an improper fraction, but the multiplication still works the same.

And in fact, we can think about that pattern, can't we? So 6/5 of 2/3, if 5/5 of 2/3 was equal to 10/15, then 6/5 of 2/3 must be equal to 12/15.

Well done if you spotted a pattern there.

We were multiplying by an extra fifth each time.

So all of our answers were increasing by 2/15, which was that 1/5 of 2/3 that we started with.

In question three, you're filling in the missing numbers to create that missing fraction factor.

So 1/2 of something is equal to 5/16, so whatever it is must be twice the size of 5/16 and that's 5/8.

But you might have thought, well, one times something is equal to five, one times five is equal to five, two times something is equal to 16 and two times eight is equal to 16.

But when we think about halving something, our answer must be half the size of what we started with, and 5/8 is double 5/16.

So then we've got 3/4 multiplied by something is equal to 3/20.

Well, if we know the rule about multiplying the numerators and denominators, we know that three times one is equal to three and four times five is equal to 20.

So 3/4 of 1/5 is equal to 3/20.

And for question four, how many solutions can you find to make the statement true? So here's one example Andeep says that he came up with.

So we've got to have a product of three from our numerators, and we can really only do that with one times one times three.

But what three numbers can we multiply together to equal 20? Well, we could have two times two, which is four, and four times five, which is equal to 20.

So one half multiplied by one half multiplied by 3/5 is equal to 3/20.

Well done if you got that, I wonder if you found any other solutions as well.

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

We've been explaining how to multiply two non-unit fractions.

So what have we learned about today? Well, we've learned that when you multiply by a unit fraction, you find that fraction of the other factor.

We know that multiplication can be represented by the word of, so we can replace that multiplication symbol with the word of when we are reading our equations.

And when multiplying fractions together, you can multiply the numerators together and the denominators together.

That's our shortcut, isn't it? To getting the answer.

But it's really important to understand why that works.

So it's useful to go back and use those area models to convince yourself that strategy is going to be correct, and is going to give you the right answer, the right product.

Thank you for your hard work and your mathematical thinking today, and I hope I get to work with you again soon.

Bye-Bye.