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 dividing a non-unit fraction by a whole number.

And by the end of the lesson we should be able to explain how we can do that, divide a non-unit fraction by a whole number.

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

We've got one key word and that's represent.

I'm sure it's a word you're familiar with, but let's just practise saying it and think about what it means.

I'll take my turn, then it's your turn.

My turn, represent, your turn.

Fantastic, as I say, I'm sure you know what this means, but let's just remind ourself because it's really useful.

Representing our mathematics is a really important way of making sure that we understand it.

So to represent something means to show something in a different way.

And that's going to be important today, thinking about how we can represent these divisions so that we really understand what's happening and it's not just about learning a rule.

There are two parts to our lesson today.

In the first part, we're going to be rewriting division, and in the second part, we are going to be dividing fractions in context.

So let's have a look at rewriting division.

And we've got Jun and Lucas in our lesson with us today.

So this rectangle represents the whole, and we are going to be thinking in these rectangles about the bit marked with a question mark and finding out how we can find the value of that part of the rectangle.

So let's have a look at what happens with this.

What can you see here? Well, Jun says, "The purple represents three-quarters of the whole." We've shaded three-quarters of the whole.

Now what have we done to those three-quarters? So three-quarters has been divided into five equal groups, and this is the bit we are interested in.

So our three-quarters has been divided into five equal groups and we've got one of those parts, one-fifth of three-quarters.

So we can rewrite the division, three-quarters divided by five, as three-quarters multiplied by one-fifth because dividing by five is the same as finding one-fifth.

And Jun says, "To find the value of the question mark area, we need to find the size of each part in relation to the whole." So we can divide our final quarter into five equal parts.

How many parts have we got in the whole now? "The whole is made up of 20 equal parts." "And we have three of these, so it's three-20th." So three-quarters divided by five is equal to three-20ths, and three-quarters multiplied by one-fifth, one-fifth of three-quarters is also equal to three-20ths.

Let's look at another example.

And remember, we are going to be interested in finding the value of the area marked with the question mark.

So what have we got here? Well, again, "The purple represents three-quarters of the whole," and each quarter this time is going to be divided into four equal parts.

So we've taken our three-quarters and we've divided it by four, and that's the value of the area we're interested in.

"So we can say that we have one-quarters of three-quarters." Three-quarters divided by four is the same as finding one-quarters of three-quarters.

So we've got three-quarters multiplied by a quarter, three-quarters quartered.

"Once again, we now need to know the value of each part in relation to the whole." So let's quarter our final quarter.

So, "The whole is made up of 16 equal parts." And, "We have three of them, which is three-16ths." So three-quarters divided by four or three-quarters multiplied by one-quarters is equal to three-16ths.

And if we remember about how we multiply fractions, we multiply the numerator and multiply the denominators.

Three times one is equal to three, four times four is equal to 16, and that is the same as dividing by four.

Finding one-quarters and dividing by four are equivalent calculations.

What do you notice about this example and how can we find the value of that question mark area again? So what have we got now? Well, the purple this time represents two-quarters of the whole, we could think of the half as well, but let's think of two-quarters for this example.

And now we're going to divide that into four equal parts.

So we've got two-quarters divided by four, and that's the bit we're interested in.

"So we can say we have one-quarters of two-quarters." Two-quarters multiplied by one-quarters, two-quarters has been quartered.

"So once again, we can find the size of each part in relation to the whole." So let's quarter our final two-quarters.

How many pieces have we got altogether? "The whole is made up of 16 equal parts." "But this time we have two of them, so it's two-16ths." So two-quarters divided by four or two-quarters multiplied by a quarter, two-quarters quartered is equal to two-16ths.

So these are the equations that we came up with based on those area models that we were looking at.

What do you notice? Well, Jun says, "Dividing a fraction by a whole number can be rewritten as multiplying the fraction by a unit fraction with the divisors as the denominator." Let's think about that.

"Dividing a fraction by a whole number can be rewritten." Okay, so three-quarters divided by five can be rewritten as three-quarters multiplied by one-fifth.

So that can be rewritten as a multiplication with a fraction.

And the number we were dividing by, the divisor, has become the denominator of our fraction and we've made it into a unit fraction.

So three-quarters divided by five is the same as three-quarters multiplied by one-fifth.

Finding one-fifth of something is the same as dividing by five.

So in the middle version, dividing by four is same as multiplying by a quarter.

Finding a quarter of something is the same as dividing by four, and that's the same for the third example as well.

So we can rewrite division as a multiplication by a unit fraction where the denominator is the divisors that we had in our division.

So can you write a division equation for this image? So write the division equation that would allow you to calculate the yellow area mark with the question marks.

Pause the video, have a go.

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

How did you get on? So our purple area represented two-fifths and we divided it into four equal pieces.

So to find that area, we'd be doing two-fifths divided by four.

And another one, write this as a multiplication equation and solve it.

So the same one, but let's think about a multiplication equation this time and see if we can solve it.

Pause the video, have a go.

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

How did you get on? So we knew that this was two-fifths divided into four equal parts, but we can rewrite that as two-fifths multiplied by one-quarters because calculating a quarter of something is the same as dividing by four, and we know that we can do that fraction multiplication.

So two-fifths multiplied by one-quarters is equal to two-20ths, two-fifths divided by four is equal to two-20ths.

And if we imagine completing that whole with all the fifths divided into quarters, we'd have 20 equal parts and we have shaded two of them in yellow.

Time for you to do some practise.

So in question one, you're going to write a multiplication and division equation for each bar model.

Let's have a look at these bar models.

So we're thinking about the yellow area each time.

So you're going to write a multiplication and division equation for each bar model.

And in question two, you're going to fill in the missing numbers, thinking about rewriting our divisions as multiplications or multiplications as divisions.

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

So can you think about the related multiplication of fractions that would help you to solve them? And for question four, you're going to fill in the missing numbers to make the statements true.

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

How did you get on? So for question one, you were asked to write a multiplication and division equation for each bar model or representation.

So in this first one, we could see that the whole had been divided into five equal parts and three of them were highlighted, so we had three-fifths, and then we had divided those three-fifths into three equal parts.

So we were finding three-fifths divided by three, and we can rewrite that as three-fifths multiplied by one third, three-fifths thirded, or one third of three-fifths.

And we can see that that is three-15ths.

If we imagine the final two-fifths divided into three equal parts, we'd have 15 parts in our whole, and three of them are yellow.

So three-fifths divided by three is equal to three-15ths.

And what about B? This time our whole was divided into six equal parts and we had four of them.

So we had four-sixth, and we divided those four-sixths into two equal parts.

So we had four sixths-divided by two, and we know that that can be rewritten as four-sixth multiplied by a half, four-sixths halved, and that's equal to four-12ths.

So four-sixths divided by two is equal to four-12ths.

In question two, you have lots of missing numbers to fill in, rewriting our divisions as multiplications and the other way round.

So for A, one-ninth divided by four is equal to one-ninth multiplied by a quarter.

In B, two-fifths divided by six is equal to two-fifths multiplied by one-sixth.

For C, four-seventh divided by three is equal to one-third times four-sevenths.

For D, one-sixth divided by five is equal to one-sixth multiplied by one-fifth, or you could have had it the other way round, one-fifth multiplied by one-sixth.

For E, six-eighths multiplied by one-fifth is equal to six-eighths divided by five.

In F, three-fifths multiplied by one-seventh is equal to three-fifths divided by seven.

For G, five-ninths multiplied by one-10th is equal to five-ninths divided by 10.

And in H, two-sixths multiplied by one-third is equal to two-sixths divided by three.

Well done if you got those right.

In question three, you are solving each equation.

Did you think of these as the related multiplications? One-ninth divided by four is equal to one-36ths, two-fifths divided by six is equal to two-30ths, four-sevenths divided by three is equal to four-21sts or 21s, in D, one-sixth divided by five is equal to one-30th.

In E, six-eighths divided by five is the same as six-eighths multiplied by one-fifth, which is the same as six-40th.

For F, three-fifths divided by one-seventh is the same as three-fifths multiplied by one-seventh, which is the same as three-35ths.

For G, five-ninths divided by 10 is equal to five-90ths.

And for H, two-sixths divided by three is equal to two-sixths multiplied by one-third, which is equal to two-18ths.

And for question four, you were filling in the missing numbers to make the statements true.

So we had two-ninths divided by something is equal to two-36ths.

Well, nine times four is equal to 36.

So if we thought about a multiplication, we'd have been multiplying by a quarter, which is the same as dividing by four.

What about B? We had a number of 10ths divided by three is equal to a number of 30ths.

Well, we could have had one-10th divided by three is equal to one-30th.

But as Lucas points out, "You could have had any number as the numerator as long as they were the same number each time." For C, we've got a fraction divided by five is equal to three-35ths.

Well, we know that dividing by five is the same as multiplying by a fifth.

So that would've been three-sevenths.

Three-sevenths divided by five is equal to three-35ths.

Three-sevenths multiplied by one-fifth is equal to three-35ths.

And finally, we've got four-fifths divided by something is equal to one-10th.

Hmm, that's an interesting one.

Well, it's four-fifths divided by eight.

So it would be four-fifths multiplied by one-eighth, which would give us four-40th.

And we can simplify that to one-10th.

And Jun says, "If you convert the fractions so that the denominators are the same, it's easier to find the missing divisor." So we could have converted our four-fifths to be eight-10ths and then eight-10ths divided by eight, well, eight divided by eight is equal to one.

If we divided our eight-20ths into eight equal groups, each group will be equivalent to one-10th.

I wonder how you thought through D.

Lots of different ways to get to that answer.

And on into the second part of our lesson.

We're going to be dividing fractions in context.

So Sam has three-quarters of a litre of juice and she divides it equally between five glasses.

What fraction of a litre is in each glass? Jun says, "Let's represent this using an area model." Yes, I think I need a representation to help me make sense of this.

So there is our whole litre of juice and Sam has three-quarters of it, and she divides it equally between five glasses.

So the purple represents three-quarters of a litre of juice, and now it's been shared between five glasses.

So we've divided each of our quarters into five equal pieces, three-quarters divided by five.

And that's the bit we're interested in.

How much is in each glass? She had three-quarters and she divides it equally between five glasses.

So that row of three smaller parts represents how much is in a glass.

And Lucas says, "So we can say we have one-fifth of three-quarters," so we can write that as a multiplication as well.

And this will show us how much juice is in each glass.

So Jun says, "To find the amount of juice in each glass, we need to find the size of the part in relation to the whole." So we can divide that final quarter into five equal pieces.

And the whole is now made up of 20 equal parts.

And Lucas says, "Each glass has three-20ths of a litre of juice." Three-quarters divided by five is equal to three-20ths.

Three-20ths multiplied by one-fifth or three-quarters fifthed is equal to three-20ths.

So how much juice is in the glass? We know it's three-20ths of a litre, but how much of a litre is that? Can you see what we've done here? Well, Jun says, "One litre is equivalent to 1,000 millilitres.

So we can convert three-20ths into millilitres to find out exactly how much juice there is in each glass.

So let's look at the horizontal relationship between these fractions." Well, if we multiply 20 by 50, we'll get 1,000.

So we're going to have to scale up our numerator by the same factor.

And three times 50 is equal to 150.

So three-20ths of a litre is equivalent to 151-1,000ths of a litre.

So that means there's 150 millilitres of juice in each glass.

Let's have a look at another context.

Sofia has half a metre of ribbon and she uses this to make five identical bookmarks.

What fraction of the material is used to make one bookmark? So again, Jun says, "Let's represent this using an area model." A good representation will always help to understand the problem.

So there's our ribbon and she's got half a metre.

So that was our whole metre.

And now we can see the half metre.

"The purple represents one-half of a metre of ribbon." And she uses this to make five identical bookmarks, so she's divided it into five equal parts, one part for each bookmark.

So we've got one-half of the metre, one-half divided by five, divided into five equal groups.

So that is how much is used for one bookmark.

And Lucas says, "So we can say we have one-fifth of one-half." And we can represent that as a multiplication, one-half multiplied by one-fifth, one-half fifthed.

And this shows us how much ribbon is used for each bookmark.

So, "To find the fraction of the material used to make one bookmark, we need to find the size of each part in relation to the whole." So we can divide our other half into five equal pieces.

So, "The whole is made up of 10 equal parts." "Each bookmark is one-10th of the material." One-half divided by five is equal to one-10th.

And we can rewrite that as one half multiplied by one-fifth is equal to one-10th.

So what is the width of each bookmark? We know it's one-10th of a metre.

Well, one metre is equivalent to 100 centimetres.

So we can convert one-10th of a metre into centimetres to find out the width of each bookmark.

Let's look at that horizontal relationship again.

Well, 10 times 10 is equal to 100.

So we need to scale the numerator by 10 as well.

So each bookmark has a width of 10 centimetres.

Time to check your understanding.

Can you write a multiplication and division equation to represent this problem? Jun has three-fifths of a packet of potatoes left.

He peels one-third of these for his family's roast dinner.

What fraction of the whole bag of potatoes did Jun peel? Pause the video, have a go.

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

How did you get on? So Jun had three-fifths of the packet and he peels one-third of these for the dinner.

So we had three-fifths of the bag and he divided it into three equal parts.

So three-fifths divided by three, but we know we can also write that as one-third of three-fifths, three-fifths multiplied by one-third.

So Jun used three-15ths of the bag of potatoes.

And time for you to do some more practise.

For question one, you're going to solve these worded problems and you've got three there to solve.

For question two, you're going to see if you can write your own worded problem for these equations.

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

How did you get on? So for 1A, Andeep swims for three-quarters of an hour.

He does breaststroke for half of this time.

What fraction of an hour does Andeep swim breaststroke for? So three-quarters of an hour and he does breaststroke for half of the time.

So three-quarters of an hour divided by two.

And we can rewrite that as three-quarters multiplied by a half, which is equal to three-eighths.

So he spent three-eighths of an hour swimming the breaststroke.

In question B, Jun has three-fifths of his birthday money left over.

He spends one-quarters of it on a new game.

What fraction of his birthday money does he spend on the game? So he is got three-fifths of his birthday money, and he spends a quarter of it.

So three-fifths divided into four groups, three-fifths divided by four.

And we know we can rewrite that as three-fifths multiplied by one-quarters.

So he spends three-20ths of his money on the game.

For C, Laura finds four-fifths of a pint of milk in the fridge.

She uses one-third of it to make a milkshake.

What fraction of the pint of milk is left? So we've got four-fifths and she uses a third.

So she's divided it into three parts.

So four-fifths divided by three, and we know that's the same as four-fifths multiplied by one-third.

So she uses four-15ths of the milk in her milkshake.

But we need to know what fraction of the pint of milk is left.

"Laura started with four-fifths of the pint, and then she used four-15ths of the pint." So we've got to calculate four-fifths minus four-15ths.

Ah, well, to do a subtraction like that, we need a common denominator, don't we? So let's convert our four-fifths into 15ths as well.

So that's 12-15ths.

12-15ths minus four-15ths is equal to eight-15ths.

So there's eight-15ths of the pint left, which as Lucas says, is just over half a pint left.

For question two, you are going to write your own worded problems for these equations.

So these are the ones that we came up with.

So for A, three-quarters multiplied by one-fifth.

Izzy travelled three-quarters of a mile to school.

She walked with her mum for one-fifth of the way.

What fraction of a mile did Izzy walk with her mum? That's three-20th.

And for B, we had six-10ths divided by four.

So we came up with Jacob reading a book.

Jacob had six-10ths of his book left to read.

He divides the remaining parts into four equal parts so he can read it to the next four nights.

What fraction of the whole book does he read each night? And six-10ths divided by four is equal to six-40ths.

So he reads six-10ths of the book each night.

And we might be able to simplify that, mightn't we? And we've come to the end of this lesson.

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

So what have we been thinking about learning about in this lesson? Well, we know that you can represent division of a non-unit fraction by a whole number using an area model.

And we've been drawing representations to help us to understand what the problem is asking us to calculate.

We know that dividing a fraction by a whole number is equivalent to multiplying the fraction by a unit fraction, where the denominator is the same as the divisor.

So if we had three-quarters divided by four, it's the same as saying three-quarters multiplied by one-quarters.

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