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 to divide a fraction by a whole number efficiently.

You may have been doing some dividing of fractions by whole numbers, but let's have a think about whether we can make sure we choose the most efficient strategy for when we are completing those sorts of calculations.

Let's have a look at what's in the lesson.

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

So I'll take my turn and then you can say it back.

Are you ready? My turn, divisor.

Your turn.

You may well be familiar with what the divisor means, but let's just check because it's going to be really useful to us in our lesson.

So the divisor is the number that we divide by.

For example, in this equation, 6 divided by 3 is equal to 2, 3 is the divisor, the number we are dividing by.

There are two parts to our lesson.

In the first part, we're going to be deciding whether to divide or multiply.

And in the second part, we're going to be looking at dividing generalisations.

And we've got Laura and Izzy working with us in this lesson.

So how could you solve this equation? 6/8 divided by 2 is equal to something.

You might want to have a think before Laura and Izzy share their thoughts with us.

Laura says, "We know that when dividing a fraction by a whole number, this is equivalent to multiplying by a unit fraction where the divisor is the denominator of the unit fraction." Let's just think about that again.

Dividing a fraction by a whole number is equivalent to multiplying by a unit fraction where the divisor, that's the number we're dividing by, is the denominator of the unit fraction.

"So let's start with that," Laura says.

"Wait!" Izzy says.

"No, I don't think we need to do that.

Look at this," she says.

So Laura wants to turn our divide by two into multiply by a half, but Izzy says she doesn't think we need to do that.

Let's have a think about what Izzy says.

So she's drawn a representation, a bar model.

She's drawn a bar model with eight equal parts and six of them are shaded representing our six-eighths.

And we know that we've got to divide by two.

So she says here we have six-eighths.

And we can divide these six-eighths into two equal parts." There we can see it.

And each equal part is worth 3/8.

"Oh, of course!" says Laura.

"So six-eighths divided by two is equal to three-eighths." There we go.

Laura says, "We could also use our knowledge of multiplication facts to help us here." She says, "I know that 6 ones divided by two is equal to 3 ones." So can you see? We've subtly changed our bar model here.

So we've still got a bar model with eight equal parts and we've shaded in six of them.

But this time, we've said each of them is worth one whole rather than one-eighth.

So 6 divided by 2 is equal to 3.

So all we've done now is change the unit of our counting.

Instead of 6 ones, we've got 6 one-eighths.

And 6 divided by 2 is equal to 3.

So 6 one eighths divided by two is equal to 3 one-eighths.

And there we can see it.

6/8 divided by 2 is equal to 3/8.

Izzy says, "That is an efficient way to solve this." We didn't need to rewrite our equation this time.

We could use a known fact to help us.

What if our calculation was this instead? 5/8 divided by 2.

Can you see what's happened? What's changed? Laura says, "It's not as easy to do this time as the numerator is not a multiple of the divisor." Well, let's think about that.

The numerator is not a multiple of the divisor.

So the numerator is five in this case, and our divisors is two.

And no, five is not a multiple of two.

When we had six-eighths, six is a multiple of two.

So it was easier to do.

This time, we might have to think in a slightly different way.

Laura says, "That means when we divide the fraction, our numerator would not be a whole number." So if we had five-eighths divided by two, we wouldn't have a whole number of eighths.

"So it might be easier to apply multiplication here." So Izzy says, "Dividing by two is equivalent to multiplying by one-half." Finding a half of something is the same as dividing by two.

So we can rewrite this equation as 5/8 multiplied by 1/2.

"And so that will be equal, Izzy says, to five-sixteenths." Because we know we multiply the numerators and multiply the denominators.

So 5/8 divided by 2 is equal to 5/16.

Oh, can you see what's happened? When we divide by two, we make it half the size, don't we? 5/8 has become 5/16.

We've got double the number of parts, but the same number of them.

So we must have half the proportion of the whole.

That's an interesting way of thinking about it.

But let's think about that known fact.

Izzy says, "I think it might be helpful to think of this as an image as well." So we were able to use 6 divided by 2 to work out 6/8 divided by 2.

This time we had 5/8 divided by 2.

So let's have a look.

5 divided by 2.

Well, here's 5 ones divided by two.

And it's 2.

5 is our answer.

5 divided by 2 is equal to 2.

5.

So 5 ones divided by two is equal to 2.

5.

"5 eighths divided by two is equal to 2.

5 eighths." And Izzy says, "Whilst I get a sense of the magnitude of the result here, I sense of the size of the fraction we've created.

It's not a convention to write a fraction without whole numbers for the numerator and the denominator." But let's have a look.

We can create equivalent fractions here, can't we? "5 eighths is equivalent to 10 sixteenths." "So 2.

5 eighths is equivalent to 5 sixteenths." And we know that that was the answer we got when we rewrote 5/8 divided by 2 as 5/8 multiplied by 1/2.

So we can still work it out that way, but we have to create an equivalent fraction in order to get a whole number as our numerator and our denominator.

So over to you to check your understanding.

Divide or multiply.

Can you calculate 6/12 divided by 2 using known facts? Pause the video, have a go, and when you're ready for some feedback, press Play.

What did you think? Well, we know 6 ones divided by 2 is equal to 3 ones.

So 6/12 divided by 2 must be equal to 3/12.

Well done.

So we were able to use a known fact.

And we could spot that the numerator of our fraction was a multiple of our divisor, so it made it easy to apply a known fact.

And can you calculate this one? 10/12 divided by 2.

Pause the video, have a go, and when you're ready for some feedback and to discuss the strategy you used, press Play.

How did you get on? Well, did you spot again that the numerator of our fraction was a multiple of our divisor, so we could use a known fact? 10 divided by 2 is equal to 5.

So 10/12 divided by 2 will be equal to 5/12.

Well done if you got that right.

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

So for question one, you are going to tick the equations that are best solved using known facts and explain how you know.

And in question two, you're going to solve each equation.

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

How did you get on? So are these the ones you ticked? So these are ones that we can solve using known facts.

The numerator of our fraction is a multiple of our divisor.

So 40/100 divided by 20, we can do 40 divided by 20, but we've just going to express our answer as a number of hundredths.

8/10 divided by 2.

Well, we can do 8 divided by 2, so we know that we can do 8/10 divided by 2.

9/15 divided by 3.

Again, 9 is a multiple of 3.

4/5 divided by 4.

Well, they're the same, aren't they? And we know 4 divided by 4 is equal to 1.

So 4/5 divided by 4 will be equal to 1/5.

And 24/25 divided by 6.

Again, 24 is a multiple of 6, so we can use a known fact to solve that equation.

And Izzy says, "The numerators are all multiples of the divisor." So in question two, we were going to solve each equation.

So for a, 8/10 divided by 2.

Well, 8 is a multiple of 2.

I know 8 divided by 2 is equal to 4, so 8/10 divided by 2 must be equal to 4/10.

The numerator is a multiple of the divisor 2, so division is a good strategy here.

What about b? 9/10 divided by 2.

Hmm, well, 9 isn't a multiple of 2 this time, is it? So 9/10 divided by 2 is equal to 9/20.

But this time we might have thought about rewriting this as 9/10 multiplied by 1/2.

And then 1 times 9 is 9 and 10 times 2 is equal to 20.

For c, 4/5 divided by 4.

Well, we know that 4 divided by 4 is 1, so 4/5 divided by 4 must be equal to 1/5.

What about 4/5 divided by 5? Ah, well, now we don't have that link between the numerator and the divisor, do we? So we maybe need to think about this as 4/5 multiplied by 1/5.

And that would give us 4/25.

The numerator is not a multiple of the divisor 5 so using multiplication is a good strategy here.

So for e, we've got 40/100 divided by 20.

Well, 40 is a multiple of 20.

So 40 divided by 20 is equal to 2.

So our answer will be 2/100.

For f, 9/15 divided by 3.

Well, 9 divided by 3 is equal to 3, so 9/15 divided by 3 must be equal to 3/15.

And for g, 24/25 divided by 6.

Well, 24 divided by 6 is equal to 4, so 24/25 divided by 6 will be equal to 4/25.

And finally, for h, 64 divided by 12.

Hmm, I don't think 64 is a multiple of 12, is it? 60 is but 64 isn't.

So we're going to need to think about this with multiplication.

So 64/120 multiplied by 1/12.

What's 12 times 120? Well, 12 times 12 is 144, so it must be 1,440.

So the answer to our equation is 61/1,440.

And on into the second part of our lesson.

This time we're going to be thinking about dividing generalisations.

So how would we solve this equation? What would you do? Laura says, "The numerator is a multiple of the divisor, so we can use division easily." 2/8 divided into two equal groups.

We can see it there in our bar model.

2/8 divided by 2 is equal to 1/8.

2 divided by 2 is equal to 1, so 2/8 divided by 2 is equal to 1/8.

What about this one? 3/8 divided by 3.

"The numerator is still a multiple of the divisor as they are the same." So there's our 3/8 divided into three equal groups.

"Three-eighths divided by three is equal to one-eighth." And what about this one? 4/8 divided by 4.

Are you spotting a pattern here? "Yeah, the numerator is still a multiple of the divisor because they are the same." So 4/8 divided into four equal groups gives us an answer of 1/8.

"Four-eighths divided by four is equal to one-eighth." What do you notice? Well, the answer was 1/8 each time, wasn't it? Whenever the divisor is the same as the numerator, the result will always be a unit fraction.

So 2/8 divided by 2.

We know that 2 divided by 2 is equal to 1, so 2/8 divided by 2 will be equal to 1/8.

When we divide a number by itself, the answer is always 1.

And that doesn't matter if we are counting in whole numbers or infractions.

"Can you come up with some of your own examples?" Izzy says.

Well, what could we have? We could have 17/28 divided by 17 is equal to 1/28.

Anything where the numerator of our fraction is equal to the divisor? So how would you solve this equation then? 3/9 divided by 3.

Ah, so we can see that the numerator and the divisor are the same.

"So the result will be a unit fraction," says Laura.

But what will the fraction be? Well, it must be ninths, mustn't it? "Three-ninths divided by three is equal to one-ninth." What do you notice this time? We've got 6/9 divided by 3 is equal to 2/9.

"This time, the numerator is a multiple of the divisor, so we can divide the numerator by the divisor." 6/9 divided by 3 is equal to 2/9.

The ninths doesn't change.

That's what we are counting in.

It's how many of them we've got, and we're dividing the number that we have of them by three.

"Six-ninths divided by three is equal to two-ninths." Oh, what do you notice this time? Well, 9/9 is equal to a whole, isn't it? "Again, the numerator is still a multiple of the divisor, so we can just divide the divisor by the numerator again." 9/9 divided by 3 is equal to 3.

And that doesn't matter whether we are counting in ones or in ninths as we are here.

"So nine-ninths divided by three is equal to three-ninths." So again, let's have a look at those three equations side by side.

What do you notice? Laura says, "This time the numerator of the result has increased by one each time." 1/9, 2/9, and 3/9.

And Laura says, "That's because there is another set of three-ninths to be shared each time." We started with 3/9 divided by 3, then 6/9 divided by 3, and then 9/9 divided by 3.

"When the numerator is a multiple of the divisor, you can divide the numerator by the divisor to find the numerator of the result." So in each of these, we could ignore the fact that we were thinking about ninths.

If the numerator of our fraction was a multiple of our divisor, then we knew that we could work out the answer.

And the answer would be that number of the fraction that we started with.

So 3/9 divided by 3 is equal to 1/9, 6/9 divided by 3 is equal to 2/9, and 3/9 divided by 3 is equal to 3/9 because 3 divided by 3 is equal to 1, 6 divided by 3 is equal to 2, and 9 divided by 3 is equal to 3.

Izzy says, "The denominator of the results stays the same." So what did you notice here? 9/9 divided by 3 was equal to 3/9.

Oh, Laura says, "Except if you decide to simplify an example." Ah, so we could simplify something here, couldn't we? We could just call this 1 divided by 3.

9/9 is equivalent to one whole, so we can rewrite our equation.

1 divided by 3 is equal to 1/3.

We now have one whole divided by 3, which we can write as 1/3.

When the numerator and the denominator are the same, of our fraction that we're dividing, so 9/9, that's equivalent to 1, the result of the division can be written as a unit fraction with the divisor, that's 3, as its denominator, because we've got 1 divided by 3.

9/9 divided by 3 is equal to 1 divided by 3, which is equal to 1/3.

And we also know that the 3/9 that we calculated before is also equivalent to 1/3.

So can you calculate 4/5 divided by 4? Pause the video, have a think, and when you're ready for some feedback, press Play.

What did you think? Well, it's equal to 1/5, isn't it? Why is it equal to 1/5? "Whenever the divisors is the same as the numerator, the result will always be a unit fraction." Our numerator of 4 is equal to the divisor.

4 divided by 4 is equal to 1, so we're always going to get our 1 as our numerator of our result, giving us a unit fraction.

And another one.

Can you remember what happens here? Have a think.

We've got 8/8 divided by 4.

Pause the video, have a think, and when you're ready for some feedback, press Play.

Did you remember this generalisation? 8/8 is equivalent to 1, isn't it? "When the numerator and denominator of the fraction we are dividing, our dividend, are the same, the result can be written as a unit fraction with the divisor as its denominator." We've got 1 divided by 4 here, and 1 divided by 4 is equal to 1/4.

It's time for you to put some of this into practise.

For question one, you're going to solve the following equations using an efficient method.

So can you spot that the numerator of the fraction is a multiple of the divisor? And use that.

Can you spot that the fraction is equivalent to 1 that we are dividing, or do you need to think about rewriting as a multiplication? So choose the most efficient method to solve each of these following equations.

And what do you notice when you've solved them? And in question two, what numbers could go in the empty boxes to make these statements true? And how many different answers can you find? Pause the video, have a go at your two questions, and when you're ready for the answers and some feedback, press Play.

How did you get on? Let's have a look at question 1.

So we had 1/10 divided by 2.

While the numerator isn't a multiple of the divisor is it this time, but what we can see is that we can multiply the denominator of our fraction by the divisor.

So when it's a unit fraction, we've got 1/10 divided by 2.

We're going to make 1/10 half the size, so we need one out of twice as many pieces.

So 1/10 divided by 2 is equal to 1/20.

In b, 2/10 divided by 2.

This time, the numerator of the fraction and our divisor are equal.

So we will have a unit fraction as our answer.

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

What about c? 3/10 divided by 2.

Well, 3 isn't a multiple of 2, is it? Our numerator of the fraction is not a multiple of our divisor.

If we divided 3 by 2, we'd get 1.

5/10.

Ooh, which is tricky.

We could though scale that up by a factor of 2 and get 3/20 as our answer.

So again, we've got 3/10, and we want to divide it by 2 to make it half the size.

So that means we've got three of twice as many pieces, 3/20.

You might though have thought about finding half of 3/10, so multiplying by a half, which would give us the answer 3/20 as well.

For d, 4/10.

4 is a multiple of 2, so 4 divided by 2 is 2.

So our answer is 2/10.

In e, we've got 5/10 and we're dividing by 2.

So we want to make it half the size.

So we've got 5/10, so half of that will be 5/20.

For 6, our numerator is a multiple of our divisor again.

6 divided by 2 is equal to 3, so our answer will be 3/10.

Because we are dividing by 2, have you noticed that this is going alternating between the ones that we can use a straightforward division for and ones where we have to think a little more carefully? So 7/10 divided by 2.

Well, we might want to think about this as being multiplied by a half, 7/20.

For h, we've got a multiple again.

8 divided by 2 is equal to 4, so our answer is 4/10.

9, no, that's not work, is it? 9/20, can you see a pattern here? Each time the numerator is a multiple of the divisor, we have an answer in tenths.

If the numerator is not a multiple of the divisor, we have an answer in twentieths and we have an odd number of twentieths.

So 1/20, 3/20, 5/20, 7/20, and 9/20.

And then we've had 1/10, 2/10, 3/10, 4/10.

So 10/10 divided by 2 must be equal to, oh, 1/2, which is 5/10, isn't it? So we've talked a little bit about this as we went through, but did you notice that alternate numbers of tenths could be divided by 2 with a whole number numerator? And that the result for odd numbers of tenths can be expressed as twentieths.

So we had 1/20, 1/10, 3/20, 2/10, 5/20, 4/10, and so on.

And for question 2, what numbers could go in the empty boxes to make these statements true? And how many different answers can you find? So we've got a number of thirtieths divided by 7 is equal to a number of thirtieths.

Well, if our dividend fraction is in thirtieths and our answer, the result of our division, is in thirtieths, then we want a numerator that is a multiple of seven.

So we could have had 7/30 equaling 1/7.

"So here's one example I found," says Laura.

But we could also have had 14/30 divided by 7 is equal to 2/30.

21/30 divided by 7 is equal to 3/30.

28/30 divided by 7 is equal to 4/30.

And we could have carried on with improper fractions.

35/38 divided by 7 is equal to 5/30 because 35 divided by 7 is equal to 5.

So as long as our numerator of our dividend was a multiple of seven, we would end up with an answer of thirtieths as our result of our division.

Well done if you spotted that pattern.

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

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

What have we been thinking about learning about today then? Well, we know that when the numerator is a multiple of the divisor, you can use known multiplication facts to help find the result or solution.

When the numerator is the same as the divisor, the result will be a unit fraction of the original fraction.

And when the numerator and the denominator are the same, so it equal to one, the result will be a unit fraction with the divisor as the denominator.

Lots to think about and to understand when we are dividing fractions by whole numbers.

And remember, those representations can really help you when you're working out what is happening and why these generalisations work.

Thank you for all your hard work and your mathematical thinking, and I hope I get to work with you in a lesson again soon, bye-bye.