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 thinking about how we divide a unit fraction by a whole number.

And by the end of the lesson, we should be able to explain how we can do that.

So let's see what's in this lesson.

We've got two key words.

We've got represent and dividend.

Let's just practise them.

They may well be very familiar too but always worth a practise.

And then, we'll have a look at what they mean.

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

So my turn, represent, your turn.

My turn, dividend, your turn.

Excellent, as I say, there may well be words you're familiar with but let's remind ourselves what they mean.

They are going to be useful to us in the lesson.

So to represent something means to show something in a different way and we're going to be looking at ways that we can represent the fractions that we're working with in order to help us to understand what's happening when we're dividing them by a whole number.

And the dividend is what we are dividing.

So it's the starting number of our division.

There are two parts in our lesson today.

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

And then in the second part, we're going to go beyond halving, but we're thinking about this as division.

And we've got Sofia and Alex in our lesson with us today.

So this rectangle represents the whole.

How can we find the value of the area marked with a question mark? So let's have a look at how this builds, and you think about what you'd say now? "The purple represents 1/4 of the whole," says Sofia, that's right.

What about this part? That's the bit we're interested in.

The value of the question marked area.

Well, the yellow part is 1/2 of 1/4.

We can say that 1/4 has been halved.

And you may know that you can represent this as multiplication: 1/2 of 1/4.

We know that the of can be represented with a multiplication symbol, 1/2 multiplied by 1/4 and we can read that as 1/2 of 1/4.

And that represents the yellow area with the question mark.

And because we know that multiplication is commutative, we can say 1/4 multiplied by 1/2 or 1/4 halved.

What about this representation though? Alex says, "I think you can also represent it as a division though." What do we know about multiplying by 1/2? Well, multiplying by 1/2 is the same as finding 1/2 of something which is dividing by 2.

So we can say that 1/4 has been divided into 2 equal parts and we can represent that with a division equation.

"We now need to find the size of the yellow part in relation to the whole." So we can divide all of those quarters in half so that we've got equal size parts in the whole.

And we can see that the yellow part represents 1/8 of the whole.

"So 1/4 divided by 2 is equal to 1/8." 1/4 divided into 2 equal parts or halved.

So let's have a look at the representation of these equations and have a think about what's the same and what's different.

You might want to pause the video and have a think yourself before Alex shares his thinking.

So we've got our three equations.

We've got 1/2 times 1/4 is equal to 1/8.

We can think of that as 1/2 of 1/4.

We've got 1/4 times 1/2 equals 1/8.

And we can think of that as 1/4 halved.

And we can think of 1/4 divided by 2.

So 1/2 times 1/4, 1/4 times 1/2, and 1/4 divided by 2 are all equivalent expressions.

They all represent the same piece of mathematics.

They represent that 1/4 of our shape divided into 2 equal parts, multiplied by 1/2, halved, or divided by 2.

If you multiply a number by 1/2, this is equivalent to dividing the number by 2.

Time to check your understanding.

Can you write a multiplication and a division equation to represent this image which we're going to build together now? So there's our whole, divided into hmm parts? So we need to write a multiplication and a division equation to represent the way that we would calculate the yellow shaded area of this shape, marked with a question mark.

Pause the video, have a go.

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

How did you get on? What did you think? Well, our whole was divided into 5 equal parts.

And then, we were finding 1/2 of that fifth.

So 1/2 of 1/5 or 1/5 multiplied by 1/2, 1/5 halved, and we know that that is 1/10.

We can see from the image that it's 1/10.

But we can also think of this as thinking of 1/5 divided by 2.

1/5 divided into 2 equal parts, each of those parts is worth 1/10.

So 1/5 divided by 2 is equal to 1/10.

It's the same as multiplying by 1/2.

So can we use what we know and understand to calculate 1/3 divided by 2? We've got a whole there divided into thirds.

So pause the video, have a think, what's the answer? And when you're ready for some feedback, press play.

How did you get on? Well, it's 1/6, isn't it? And "We could represent it like this," says Alex.

There's our 1/3 and we've divided it into 2 equal parts.

And we can see that one of those equal parts, our yellow with a question mark, is worth 1/6.

1/3 divided into 2 equal parts is equal to 1/6.

Time for you to do some practise with what you've learned about thinking about halving and the different ways we can think about it.

So can you complete each calculation? We've got 1/4 divided by 2, 1/5 divided by 2, 1/6 divided by 2, and 1/7 divided by 2.

And Alex says, "What do you notice about the solution?" So the answer, and the dividend, that number we're starting with.

And in question two, you're going to fill in the mixing boxes.

We know that we can represent multiplying by 1/2 as dividing by 2.

So can you complete those equations, filling in the missing boxes? 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 1/4 divided by 2 is equal to 1/8.

1/5 divided by 2 is equal to 1/10.

1/6 divided by 2 is equal to 1/12.

And 1/7 divided by 2 is equal to 1/14.

How did you think about that? Did you picture a grid or an area model? Or did you think about multiplying by 1/2? Alex says, "When you divide a unit fraction by a whole number, the solution is smaller than the dividend, so the denominator increases." If we've taken 1/4 and divided it into 2 equal parts, each of those parts will be smaller than the 1/4 dividend that we started with.

Did you also notice what had happened to the denominators of our solutions, for the results of our divisions? They would double the denominator of the fraction that we started with, double the denominator of the dividend.

And then for question two, you were filling in the missing boxes to make the equations correct.

So we had 1/5 multiplied by 1/2.

1/5 halved is equal to 1/10.

And we can rewrite that as 1/5 divided by 2 because halving is the same as dividing by 2.

So this time, our completed equation was 1/10 divided by 2 is equal to 1/20.

So we must have been multiplying by 1/2.

1/10 multiplied by 1/2 is equal to 1/20.

And you may well have learned that when we multiply fractions, we multiply the numerator and the denominators.

We've got unit fractions, when we multiply unit fractions, the numerator is always 1 and we think about multiplying the denominators.

So in c, we had 1/6 multiplied by something is equal to 1/12, and 1/6 divided by something is equal to 1/12.

Well, thinking about that multiplication rule, 6 times 2 is equal to 12.

So 1/6 multiplied by 1/2, 1/6 halved is equal to 1/12.

So 1/6 divided by 2 is equal to 1/12.

And for d, we had 1/12 times something is equal to 1/24 and 1/12 divided by something is equal to 1/24.

Well, if we think about our multiplication, 12 times 2 is equal to 24.

So we were multiplying by 1/2 and we know that multiplying by 1/2 is the same as dividing by 2.

So 1/12 times 1/2 is equal to 1/24 and 1/12 divided by two is equal to 1/24.

And on into the second part of our lesson, beyond halving this time.

So we're going to be thinking about dividing unit fractions by something other than 2.

Let's think again, this rectangle represents the whole.

So let's think about what we could find the value of here.

So what have we done? We divided it into 4 equal parts.

So, "The green represents 1/4 of the whole," says Sofia.

Now how many parts have we divided our quarter into? We've divided it into 3 equal parts, haven't we? And we want to find the value of the yellow area with the question mark.

So the yellow area we can describe as 1/3 of 1/4.

And Alex says, "I've got a cool way of describing this.

We can say that 1/4 has been thirded." So how do we find 1/3 of something.

So, "We can represent this as two multiplication again." So there's our 1/4 divided into 3 equal parts and we've got one of them.

So 1/3 of 1/4, 1/3 of 1/4.

And we know that we can write that as a multiplication, we can replace the of with a multiplication sign, 1/3 multiplied by 1/4.

Or again, we can say 1/4 has been thirded and have 1/4 multiplied by 1/3.

But how else could we represent this? How else can we think about finding 1/3 of something Sofia says, "As you said before, we can also represent this as division." Alex says, "That's right, 1/4 has been divided into 3 equal parts." When we find 1/3 of something, we divide it into 3 equal parts, we divide it by three.

So we can also say 1/4 divided by 3.

And to find the value of one of those parts, we need to find the size of the part in relation to the whole.

So we can divide all of the quarters by 3 to find out how many of those equal size parts we have in our whole.

Each part is 1/12 of the whole.

So 1/4 divided by 3 is equal to 1/12.

So let's have a look at this.

What's the same and what's different? We've represented finding this yellow area as two multiplications and one division.

So have a think.

What's the same and what's different? Well we can describe this as 1/3 of 1/4, 1/4 thirded, or 1/4 divided by 3.

So those expressions are all equal, they're equivalent to each other.

If you multiply a number by 1/3, this is equivalent to dividing the number by 3.

So 1/3 of 1/4, 1/4 multiplied by 1/3, is the same as 1/4 divided by 3.

Okay, so what do you think the missing digit might be for this final one? We've looked at 1/2 of 1/4 or 1/4 halved is the same as dividing by 2.

1/3 of 1/4, 1/4 thirded, is the same as 1/4 divided by 3.

And Alex has got a sort of same sentence for us.

He says, "To divide a fraction by a whole number, we can change it to an equivalent multiplication.

To divide by hmm, we can multiply by hmm." So in the first one, to divide by 2 we can multiply by 1/2.

In the second one, to divide by 3 we can multiply by 1/3.

What have we got in that bottom row with our missing number? So we've got 1/4 of 1/4, 1/4 quartered, or 1/4 divided by 4.

We can divide by the denominator of the fraction that we're multiplying by.

Time to check your understanding.

Can you convert each division expression to its equivalent multiplication expression? Pause the video, have a go.

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

How did you get on? So 1/5 divided by 2 is equal to 1/5 multiplied by 1/2, 1/5 halved.

1/5 divided by 3 is equal to 1/5 multiplied by 1/3, 1/5 thirded.

And 1/5 divided by 4 is equal to 1/5 multiplied by 1/4, 1/5 quartered.

And time for another check.

Can you calculate this? 1/8 divided by 3 is equal to? See if you can think about it using an image or maybe think about it using that known equivalence with a multiplication.

Pause the video, have a go.

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

What did you think? Well, 1/8 divided by 3 is the same to finding 1/3 of 1/8, 1/8 thirded.

So 1/8 multiplied by 1/3 is equal to 1/24.

So 1/8 divided by 3 is equal to 1/24 as well.

Well done if you got that right.

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

You are going to decide if these equations are correct or incorrect.

Give them a tick or a cross.

For question two, you're going to complete each calculation and Alex asks, "What do you notice?" For question three, how many different solutions can you find for this calculation? Pause the video, have a go at the three questions.

And when you're ready for the answers and some feedback, press play.

How did you get on? So for question one, you had to decide if these equations are correct or incorrect.

So we had 1/2 divided by 6.

1/6 of 1/2.

That's correct, isn't it? We know that when we're dividing by 6, we are finding 1/6 of something.

So 1/6 of 1/2 is the same as 1/2 divided by 6.

For b, we had 1/4 divided by 4 is equal to 1.

What if we had 1/4 of something and we divided it equally into 4 parts? We wouldn't have one whole as our answer, would we? No, 1/4 divided by 4 is the same as saying 1/4 of 1/4, and that would be 1/16.

For c, I think the symbols might be wrong here.

We know that 1/3 divided by 8 would be the same as 1/3 multiplied by 1/8.

So the symbols are the wrong way round.

Those expressions are not equivalent.

And what about d? We've got 1/5 divided by 5 is equal to 1/25.

Is that right? Yes, because we know that we are finding 1/5 of 1/5.

And 1/5 fifthed would give us 1/25.

So that one is correct.

Well done if you got those right.

So for question two, you were completing each calculation and Alex asked you what you noticed as you worked.

So a is 1/4 divided by 2.

So that's 1/2 of 1/4, which is an 1/8.

For b, it was 1/4 divided by 3.

It was 1/3 of 1/4 which is 1/12.

For c, we had 1/4 divided by 4 which is 1/4 of 1/4, which is 1/16.

And for d, we had 1/4 divided by 5, which is a 1/5 of 1/4, which is equal to 1/20.

What did you notice there? Alex says, "You can use your times table knowledge of the four times table to help you to calculate each of these." Well done if you spotted that times table link.

And for question three, how many different solutions can you find for this calculation? One something divided by something is equal to 1/24.

So what are we thinking about here? We are thinking about dividing something by a whole number to equal 1/24.

So what we're starting with will be a fraction that's greater in value than 1/24.

So we could have had 1/6 divided by 4.

1/6 divided by 4 is the same as saying 1/6 multiplied by 1/4 and that would give us 1/24.

Were there any other solutions you could have had? Well, Alex says, "This is one example.

The missing digits can be any factor pairs of 24." So we could have had 3 and 8.

We could have had 2 and 12.

We could have had 1 and 24.

I hope you found some different solutions to that one.

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

We've been explaining how to divide a unit fraction by a whole number.

So what have we been thinking about learning about in this lesson? Well, we know that you can think of dividing a fraction by a whole number as sharing.

So we've used that idea of dividing into equal groups to think about our divisions today.

When a unit fraction is divided by a whole number, the quotient is smaller than the dividend.

So the denominator increases.

The quotient is the result of our division and that is smaller than the dividend, the number we start with.

And we know that as a fraction gets smaller, the denominator increases, and that's true for our unit fractions.

And we've got a bit of a stem sentence here because we know that dividing a number by hmm is the same as finding one-hmm of the number, and this can be applied to other fractions as well.

So when we divided a number by 3, this was the same as finding 1/3 of the number.

When we divided by 4, it was the same as finding 1/4 of the number.

I hope you've enjoyed exploring dividing unit fractions by whole numbers.

Thank you for your hard work and I hope we get to work with you again soon, bye-bye!.