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.

So in this lesson, we're going to be explaining how to multiply two unit fractions.

Do you remember what a unit fraction is? It has a numerator of 1.

It's 1 part out of the whole.

So let's see what's in our lesson today.

We've got one keyword today and that's commutative.

Let's just take a chance to rehearse it.

I'll say it first, and then over to you.

My turn, commutative.

Your turn.

I'm sure you've been using the word commutative for a long time.

Let's just remind us though what it means.

It is going to be important in our lesson.

The commutative law states that you can write the values of a calculation in a different order without changing the calculation.

The result is still the same, and it applies for addition and multiplication.

We can swap the order of the factors, and the product remains the same when we're thinking about multiplication.

There are two parts to our lesson today.

In the first part, we're going to be thinking about halving fractions.

And in the second part, we're going to go beyond halving.

So let's make a start on halving fractions.

And we've got Laura and Lucas with us in the lesson today.

This rectangle represents 1 whole.

What about this then? How could we describe this image? Remember our rectangle represented 1 whole.

Laura says, "The rectangle represents 1 whole, so it is 1/4 of 1." 1/4 has been shaded.

Lucas says, "1/4 of what though?" What is it 1/4 of? Well Laura says, "We can say 1/4 of 1." Our rectangle was our whole, so our rectangle is 1.

So how could we record this as an equation? Well we could say 1/4 of 1 is equal to 1/4.

What does the of mean? How could we represent that with a mathematical symbol? Ah, she says, "We know that this can therefore be written as a multiplication equation." 1/4 of 1 is the same as saying 1/4 multiplied by 1.

So we can read the multiplication with the word of in this case, or we can think of of as multiplied by.

So 1/4 multiplied by 1 is equal to 1/4.

And we know that anything multiplied by 1 is equal to itself.

So 1/4 multiplied by 1 is 1/4, and 1/4 of 1 is 1/4.

What do you notice now, what's happened? Laura says, "The yellow part is half the size of 1/4." So we've taken that 1/4 and cut it in half.

So we're now showing 1/2 of 1/4.

And we know that we can represent the word of with a multiplication symbol.

So we've got 1/2 multiplied by 1/4.

How could we find the value of the yellow part? What would we do? Can you think of a way of annotating the image to help us find out the value of the yellow part? Lucas says, "To find the value of the yellow part, we need to find its size in comparison to the whole." It's 1 part out of how many equal parts in the whole? Ah, so it was half of a quarter.

So if we half all of the quarters, how many pieces do we have in our whole now? Right, the whole is now divided into 8 equal parts.

Each part is 1/8 of the whole.

So our yellow part is 1/8.

So we can now complete our calculation.

1/2 of 1/4 is equal to 1/8.

And we know that we can replace the word of with a multiplication sign.

And we can apply the commutative law to write an equation which represents this image as well.

We thought about 1/2 of 1/4, but we can also say 1/4 multiplied by 1/2 is equal to 1/8 as well.

Let's have a think about how that works.

1/2 of 1/4 is equal to 1/8.

And 1/4, halved, is equal to 1/8.

Let's have a look at another one.

So the rectangle represents 1 whole.

Again, so how could we describe the image this time? Laura says, "The rectangle represents 1 whole, so it is 1/3 of 1 whole." The shaded part is 1/3 of the whole.

Now what do you see? Now we have 1/2 of 1/3.

1/2 of 1/3 or 1/2 multiplied by 1/3.

"To find the value of the yellow part, we need to find its size in comparison to the whole," says Lucas.

So what could we do to the whole to find out the value of that yellow shaded area? Oh, we can divide all of the thirds in half.

So the whole is now divided into 6 equal parts.

Each part is 1/6 of the whole.

So 1/2 multiplied by 1/3 is equal to 1/6, 1/2 of 1/3 is equal to 1/6, as Lucas says.

And you can apply the commutative law to write an equation which represents this image.

We said it was 1/2 of 1/3 was equal to 1/6.

And we know we can replace the word of with a multiplication sign.

1/2 times 1/3 is equal to 1/6.

We know that multiplication is commutative, so we can also say that 1/3 times 1/2 is equal to 1/6.

So how would we read those? Well we know that we can replace the multiplication with of, so 1/2 of 1/3 is equal to 1/6.

But we started with a third, didn't we? So how could we read this one? So Lucas says, "1/3, halved, is equal to 1/6." We took 1/3 and we halved it, and the area we were left with was 1/6 of the whole.

So 1/3 halved is equal to 1/6.

Again the whole rectangle represents 1 whole.

So what can we say about the image this time? And what can we say now? So Laura says, "The rectangle represents 1 whole, so it is 1/6 of 1 whole.

And now we have 1/2 of 1/6." So the purple area was 1/6 of the whole, and now the yellow area is 1/2 of 1/6, or 1/6 halved.

1/2 of 1/6 or 1/2 multiplied by 1/6.

To find the value of the yellow part, we need to find out its size in comparison to the whole.

So do you remember we halved 1/6.

So if we halve them all, we've now got the whole divided into 12 equal parts.

Each part is 1/12 of the whole.

So we can say that 1/2 of 1/6, or 1/2 times 1/6 is equal to 1/12.

1/2 of 1/6 is equivalent to 1/12.

And we can apply the commutative law again to write that equation in a different way.

We can say 1/2 times 1/6 is equal to 1/12, and because of the commutative law, 1/6 times 1/2 is equal to 1/12.

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

This is how we can read the first equation.

We can use the word of to replace our multiplication sign.

What about the other one? We started with 1/6, didn't we? And what did we do to it? We halved it.

So 1/6 halved is equal to 1/12.

Time to check your understanding.

So what will the value of this question mark be? Think carefully about the shape, and when you're ready for some feedback, press play.

What did you think? Well, it's a half of 1/5.

The purple area that was shaded to begin with was 1/5 of the whole, and the yellow area represents 1/2 of 1/5, 1/2 times 1/5, and it is 1/10 of the whole.

Time to check your understanding again.

Tick the equations that could represent this image.

We're interested in the equation that represents the yellow area with the question mark in it.

So pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? Did you spot that one of those columns? If it had all been shaded in purple would've been 1/8 of the whole.

And we have found 1/2 of 1/8.

So 1/2 multiplied by 1/8.

So a and c represent that.

1/2 multiplied by 1/8, 1/2 of 1/8 is equal to 1/16, or 1/8 halved is equal to 1/16.

Well done if you've got those two.

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

Can you draw an image to represent each equation, and give the answer.

Pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? So we had 1/2 of 1/5.

So we could draw a shape divided into 5 equal parts, and then find a half of that part.

And if we half all of those fifths, we find that we've divided our shape into 10 equal parts.

So 1/2 of 1/5 or 1/2 times 1/5 is equal to 1/10.

In the second one, we had 1/2 times 1/4, 1/2 of a quarter.

And if we drew our image, found 1/4 and then halved it, that half is equivalent to 1/8 equal parts in the whole.

1/2 of 1/4 is equal to 1/8.

And finally, we had 1/2 of 1/3, 1/2 times 1/3.

And for c, we've got 1/2 multiplied by 1/3 or 1/2 of 1/3.

But Lucas says this time, it shows the halves vertically, but it still represents the same thing.

So our third this time was a row rather than a column.

And half of that third is equal to 1/6.

The whole is now divided into 6 equal parts.

Well done if you were successful with those.

And now we're going to go onto the second part of our lesson, which is beyond halving.

How could we represent this image as an equation? So here we've got 1 whole divided into 3 equal parts.

The rectangle represents 1 whole.

So the green section is 1/3 of the whole.

What about now? What's happened to our third? And what would be the value of the yellow part with the question mark? Laura says, "The question mark represents 1/5 of 1/3." We've divided that third into 5 equal pieces.

And Lucas says, "I've got a cool way of describing what's happened." Go on then, tell us, Lucas.

He says, "We could say that the third has been fifthed." Do you remember he talked about the fractions being halved in the last cycle? Well now he's saying that our fraction has been fifthed, it's been cut into 5 equal parts.

And he says, "This is my way of saying divided into 5 parts," fifthed.

So the question mark represents 1/5 of 1/3.

How can we represent this as an equation then? So we can represent it as a multiplication equation.

1/3 of 1/5.

And we know that we can replace the word of with a multiplication sign, 1/3 times 1/5.

However, to calculate this, we need to find the value of the question mark in relation to the whole, just like we did when we were finding halves of fractions.

So what are we going to do this time? Yes, we can continue those lines.

So we've divided each of our thirds into 5 equal parts.

How many parts have we got now? There are 15 equal parts in the whole.

So our question mark area is 1/15 of the whole.

So 1/3 of 1/5 or 1/3 times 1/5 is equal to 1/15.

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

1/3 fifthed is equal to 1/15.

And because of commutativity, we can write the fractions in either order and our product still remains the same.

Let's look at another example.

What does the green bar represent here? So the rectangle represents 1 whole.

So the green section is 1/4 of the whole.

So our green area is 1/4 of the whole.

Now what's happened? And what would that part be worth? "The question mark area represents 1/6 of 1/4," says Laura.

"Let's use my way of describing what's happened here," says Lucas.

"We could say that the quarter has been sixthed." It's been divided into 6 equal parts.

So how can we then change our image to work out the value of the yellow area? That's right, we could divide all of those quarters into 6 equal parts.

How many parts have we got now? We've got 24 parts in our whole.

So we can say that 1/6 of 1/4 is equal to 1/24, or 1/4 sixthed is equal to 1/24th.

There we go.

The two ways of reading our equations.

What do you notice about these equations? So we had 1/2 of 1/4 was 1/8.

1/2 of 1/6 was equal to 1/12.

1/5 of 1/3 was equal to 1/15.

And 1/6 of 1/4 was equal to 1/24.

And we can record all of those with multiplication, because we know that of can be replaced with a multiplication symbol in an equation.

What do you notice about the numerators? Well Laura spotted that the numerator stay the same throughout the calculation.

So when multiplying unit fractions, you just need to multiply the denominators.

2 times 4 is equal to 8.

And the denominator of the product is the product of the other two denominators.

2 times 4 is equal to 8.

2 times 6 is equal to 12.

5 times 3 is equal to 15.

And 6 times 4 is equal to 24.

So when we started with 1/4, and we wanted to find a half of 1/4, we divided each of our quarters into halves.

So we made twice as many pieces.

And 2 times 4 is equal to 8.

Let's think about 1/5 of 1/3.

When we had 1/3 shaded and we wanted to find out what 1/5 of 1/3 was, we divided each of our thirds into 5 equal pieces.

So instead of 3 parts in our whole, we had 5 times 3 parts in our whole.

5 times 3 is equal to 15.

There's a question for you here.

Is the product in these equations bigger or smaller than the fractions being multiplied? You might want to have a think about this before Laura and Lucas share their thoughts.

Laura says, "The product is always smaller than the fractions being multiplied." If we start with 1/4 and we want 1/2 of it, we're going to have a smaller area.

If we start with 1/3 and we want 1/5 of it, it's going to be a smaller part of the whole.

Lucas says, "So we can say that when you multiply two unit fractions together, the product is smaller than the fractions being multiplied." Remember, we are only looking at unit fractions in this lesson.

So what do you notice here? We've got 1/2 of 1/4.

And we've also got 1/4 of 1/2.

What do you notice? Laura says, "The factors have been swapped over, but the product is still the same." So this time, we are reading our equation slightly differently.

So for the first one, we could say 1/2 of 1/4 or 1/4 halved.

But this time, we're thinking about 1/4 of 1/2.

On the right-hand side, we had half of the shape shaded in green, and then we've identified 1/4 of it.

In the image on the left-hand side, we had 1/4 shaded in green, and now we've highlighted 1/2 of that.

The area is the same.

So we can think of 1/2 of 1/4 or 1/4 of 1/2.

And the answer for both of those will be 1/8.

And that's because multiplication is commutative.

So just like multiplication with whole numbers, multiplication with unit fractions is also commutative.

1/2 of 1/4 is equal to 1/4 of 1/2.

Time to check your understanding.

What would the value of the question mark be in this image? Pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? So we can think of this as finding 1/3 of 1/4, 1/3 multiplied by 1/4.

And we can see that each of the quarters has been divided into 3 equal parts.

So we've now got 12 equal parts in the whole.

And 1/3 of 1/4 is equal to 1/12.

So 1/3 times 1/4 is equal to 1/12.

Time for another check.

We've got no image here.

Can you either sketch it for yourself or picture it, and calculate 1/5 times 1/4? Are you going to think about 1/5 of 1/4 or 1/4 of 1/5? Pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? Did you remember what Lucas and Laura spotted in that when we're multiplying unit fractions, we just need to multiply the denominators.

If we thought about 1/5 of 1/4, we'd have divided each of our quarters into 5 equal parts.

So we'd have 20 parts in the whole.

And if we thought about 1/4 of 1/5, we'd have divided each of our fifths into 4 equal parts.

So we'd have had 20 parts in the whole.

Whichever way you thought about it.

1/5 multiplied by 1/4 is equal to 1/20.

1/5 of 1/4 is 1/20.

Time for you to do some practise.

Can you calculate the product of these equations? You might want to think about an image.

You might just be able to picture it now.

But think about why that rule of multiplying the denominators works.

Think carefully about that.

Make sure you can explain why it works.

In question two, you've got to decide whether these are true or false.

Are these equations correct or not? In question three, you're going to fill in the missing symbols.

Are these equal? Is one greater than or one less than the other? And in question four, which single digits could go in the boxes to make these statements true? And how many solutions can you find? Pause the video, have a go at those four questions, and when you're ready for the answers and some feedback, press play.

How did you get on? So in question one, you could just go down here and multiply all those denominators, and get the right answer.

But why does that work? 1/2 of 1/3.

So if we think about 1/3 of a shape being shaded in, we cut each of those thirds in half, we'll have 6 equal pieces, and 1 of those 6 will be shaded, 1/6.

And we can apply that same thinking.

1/8 of 1/3 is 1/24.

8 times 3 is equal to 24.

1/5 of 1/6.

If we imagine a whole divided into 6 equal pieces, and each of those pieces divided into 5, we'd have 30 pieces in the whole, and we'd have one of them highlighted.

So 1/5 times 1/6 is equal to 1/30.

And we could use the same images to think about 1/6 of 1/3, and we'd see that that would be equal to 1/18.

So with these ones, true or false.

1/4 of 1/7 is equal to 1/128.

But if we imagine, a shape divided into quarters, each of those quarters divided into 7 pieces, we'd have 1/28 shaded.

So yes, that's true.

1/9 of 1/3.

So if we imagine a whole divided into 3 equal pieces, each of those pieces divided into 9, we'd have 1/27.

So that's not correct.

1/9 of 1/3 would be equal to 1/27.

What about 1/5 of 1/5? Well that wouldn't be 1/10th, would it? It would be 1/25, because we'd have each of those fifths divided into 5 equal pieces.

So we'd have 25 pieces in our whole.

And finally, 1/7 times 1/3 is equal to 1/3 times 1/7.

Well, yes, it is, 'cause we've got the commutativity law working for us there.

1/7 of 1/3 or 1/3 seventhed would be equal to each other.

So can we fill in the missing symbols here? So we've got a mixture here of multiplying fractions by whole numbers and multiplying fractions by fractions.

So what would 1/3 times 6 be? Well, that would be 6 lots of 1/3.

And that's going to be greater than 1/3, isn't it? What about 1/3 times 1, 1 lot of 1/3? Well that's equal to 1/3, isn't it? 1/3 sixthed, so 1/6 of 1/3.

Well that's going to be a lot less than 1/3, isn't it? And again, 1/3 sixthed.

Is that gonna be greater or less than 1/6? We'd have a whole divided into thirds, each of those thirds divided into 6 equal parts.

That would be 18 equal parts, so that would be 1/18.

So much less than 1/6 as well.

I hope you were able to use some visualisation or maybe even some sketching to work out which of those last two especially were going to be greater or smaller than a third and a sixth.

Well done if you've got them right.

And for question four, which single digits could go in the boxes to make these statements true? So we want two unit fractions multiplied together to equal 1/18.

So we know that we multiply the denominators to get the product.

So we need two numbers that multiply together to equal 18.

So we could have had 1/2 of 1/9, but we could have had 1/3 of 1/6 as well.

And we could have had 1 whole of 1/18, if we really wanted to.

But there's one solution.

And here, we've got to find something, which gives us a product that is less than 1/10, but greater than 1/18.

So we want a product that is between 10 and 18.

So we could have had 2 and 6.

1/2 of 1/6 would be 1/12.

And 1/12 is smaller than a 10th, but greater than an 18th.

I wonder if there was anything else you found that could go in there as well.

I think we could have had 1/3 of 1/5 to make 1/15 in there as well.

1/15 is smaller than a 10th, but it's larger than 1/18.

And Laura says, "Here's an example of the solutions I found." I wonder how many other ones you found.

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

We've been explaining how to multiply two 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.

Multiplication can be represented by the word of, and that's really important to remember.

If you see the word of, we're thinking multiplication.

We've learned that just like whole numbers, you can apply the commutative law when multiplying unit fractions.

And multiplying by a unit fraction will give a product smaller than the value of the other factor.

The rules are really important to remember, but it's also really important that you can sketch those images and imagine what's happening, so that you understand why the product will be smaller than the value of another factor, and why it is that you can multiply the denominators of the two fractions you're multiplying to get the denominator of the product.

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

Bye-bye.