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 addition and subtraction of fractions.

Now, fractions are one of those things that some people absolutely love, and I hope you are in that camp.

I love fractions, I think they're really clever ways of expressing numbers and parts of wholes.

So if you're ready to make a start, let's do some work on adding and subtracting fractions.

In this lesson, we're going to be adding and subtracting non-related fractions with different denominators.

You may have been adding lots of related fractions where the denominators are related by being a multiple or a factor of the other one.

This time we're going to look at fractions where that's not the case.

We've got quite a few key words here, hopefully bringing together some learning that you might have been doing recently.

So we've got common denominator, related fractions, non-related fractions and common multiples.

So I'll take my turn to say them and then it'll be your turn.

Are you ready? Here we go.

My turn, common denominator, your turn.

My turn related fractions, your turn.

My turn, non-related fractions, your turn.

My turn, common multiple, your turn.

Okay, lots in there.

Common multiple you might not have been using so much recently, but let's have a look at what they all mean because they are all going to be really useful phrases in our learning today.

So when two or more fractions share the same denominator, you can say that they have a common denominator.

It's common to both of them or all of them.

Related fractions are when the denominator of one fraction is a multiple of the denominator of another fraction.

Non-related fractions are when the denominator of one fraction is not a multiple of the denominator of the other fraction.

And that's what we're going to be focusing on today.

And a multiple that is common between two or more numbers is known as a common multiple.

Okay, let's see how those words are gonna help us today.

There are two parts to our lesson.

In the first part, we're going to be applying our fractional sense.

It's not all about creating common denominators and finding related fractions.

It's sometimes about just pausing, thinking about what the equation is asking us to do or what the problem is asking us to do and making a reasonable estimate before we start getting into thinking about common denominators and so on.

And in the second part, we're going to be calculating with common denominators.

So let's make a start on part one.

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

Aisha is spending some of her birthday money.

She spends 1/3 of her money on a new bike helmet and she spends 1/4 of her money on a new computer game.

What fraction of her birthday money did she spend altogether? And we've represented this as a sort of pie chart here.

So the pie chart in its whole shows how much birthday money was spent.

We've represented the green 1/4 and we've represented 1/3 in purple for her bike helmet.

So 1/4 for the computer game and 1/3 for the bike helmet.

And then we've got some money remaining.

We want to find out what fraction of her money she spent altogether.

So Aisha's asking, what equation would we write to represent this? And Lucas says, to find the fraction of money Aisha has spent, we need to add together the fractional parts she spent.

So she spent 1/3 of her money and 1/4 of her money.

And Lucas says to solve this, we can just add the denominators together.

1/3 plus 1/4 is equal to 1/7.

Is he right? Hang on, says Aisha, that can't be right.

How does she know that can't be right? He says, what do you mean? Well, when the numerator are the same, the larger the denominator, the smaller the fraction.

So we're adding together 1/3 and 1/4.

So can our sum possibly be 1/7? 1/7 is smaller than both 1/3 and 1/4.

The sum should be larger than its parts in this case because we've got positive numbers that we're adding.

Also, she says if you look back at the pie chart, you can see that the amount shaded is much greater than 1/7 of the whole.

I think Lucas might have got confused.

Perhaps he knows that when we have something common in the fractions, we can just add the other bits.

But it's not a common numerator we're looking for, is it? It's common denominators.

Aisha says, what would your estimate for the answer be? And Lucas says, I'd said our estimate should be just over 1/2.

So looking at the pie chart, she has spent just over 1/2 of her birthday money.

Time to check your understanding.

So is this true or false? 1/5 plus 1/3 is equal to 1/8 true, false and why? Pause the video, have a go.

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

What did you think? It's false, isn't it? When the numerators are the same, the larger the denominator, the smaller the fraction because the ad ends are positive, the sum must be larger than both of the ad ends, not smaller.

But also we know that when we add fractions, we're not adding the denominators, are we? We need a common denominator and then we can add the numerator.

I think Lucas had got a bit confused about which was the numerator and which was the denominator.

But we know that if the units are the same, then we can add our number of units together, 1/5 and 1/3 and not the same units.

And we certainly don't have a sum of 1/8.

Okay, time to check your understanding again.

Tick the correct symbol to compare these expressions.

We've got 1/4 plus 1/3 and we've got 1/2.

What symbol goes in the circle? Pause the video, have a go and when you're ready for some feedback, press play.

How did you get on? Did you remember to visualise the pie chart? It was B, wasn't it? And as Lucas says, if I visualise the pie chart, I can see that the first expression will be greater than 1/2.

And we could see that 1/4 plus 1/3 was greater than 1/2.

Just visualise it, you visualise 1/4 and then visualise adding 1/3 onto it.

The third is greater than 1/4, isn't it? We know two quarters are equal to 1/2.

So 1/4 plus 1/3 must be greater than 1/2.

Time for you to do some practise.

And remember, we're applying our fractional sense in here, so we're trying not to use common denominators here.

So can you put the right symbol into the circles, less than, greater than or equals to? And use your understanding of fractions to justify your response.

So pause the video, have a go.

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

How did you get on? So in A, we had 1/4 plus 1/3, and then we had 1/3 plus 1/3.

Well, we know that 1/3 is greater than 1/4.

So if we're adding 2/3 together, that's going to be greater than 1/4 plus 1/3.

What about the next one? 1/4 plus 1/3 and 2/3.

Well that's effectively the same as the first one, isn't it? It's just that we've given a sum to our thirds.

And you can see there our denominators were the same, so we could add our numerators.

1/3 plus 1/3 is equal to 2/3.

So this is just another way of expressing 2/3.

So that second expression is still greater.

And let's think about how we can carry on that thinking.

So in C, we had 1/4 plus 1/3, and then more quarter plus 1/4.

Well, we've replaced the third with 1/4, which is smaller.

So the second expression will be less than the first one.

And for D, again, we've just combined those two quarters.

So the same again, 1/4 plus 1/3 is greater than 2/4.

In E again, we've got that 1/4 plus 1/3 and we are comparing it to 1/2.

Well, we could visualise that, couldn't we? 1/3 is greater than 1/4 and we know that 2/4 is equal to 1/2, so that will be greater than 1/2.

And what do you notice about the next one? We've still got the quarter plus 1/3 and we've got 3/6.

Well, we know that 3/6 is equivalent to 1/2.

The numerator is half the value of the denominator.

So our addition expression will still be greater than 3/6.

1/4 plus 1/4 and 1/2.

Ah, well we know that 1/4 plus 1/4 is equal to two quarters, which is equal to 1/2.

So those have the same value.

And can you see in H We've replaced our 1/2 with 3/6 again.

We know that 3/6 is equivalent to 1/2.

So those expressions will be equal as well.

Well done if you used your reasoning to work your way through all of those.

It's really good to think about fractions before we dive into common denominators and making changes to them.

And onto part two, we're going to be calculating with common denominators.

So we're still thinking about Aisha's birthday money.

She spent 1/3 of it and 1/4 of it.

So exactly what fraction did she spend? Aisha says, can we find a common denominator? Are these related fractions, that might help.

Are thirds and quarters related fractions? Lucas says, no, they're not related, neither denominator is a multiple of the other denominator.

Three is not a multiple of four and four is not a multiple of three.

So we could call these non-related fractions.

So if we can't convert thirds into quarters or quarters to thirds, what are we going to do? Lucas says, we're going to need to find a new denominator that's common to both fractions.

Aha, so let's have a think.

We've been thinking about numbers that are multiples of another or factors of another, haven't we? So maybe we can think about those multiples.

So what will be a common denominator for both three and four, thirds and quarters? Aisha says we can use our times tables to help us by thinking of the multiples of both three and four.

So here are the multiples of three or some of them, three, six, nine and 12.

Wonder why we stopped at 12.

Can you see what's gonna happen here? Let's think about the multiples of four, four, eight and 12.

Ah, what have you spotted? That's right, 12 is a common multiple of both three and four.

So we can create equivalent fractions for thirds that have 12 as a denominator and for quarters that have 12 as a denominator.

So let's create those equivalent fractions.

Three times four is equal to 12.

So we've got to scale the numerator by the same factor.

One times four is equal to four, 1/3 is equal to 4/12 and four times three is equal to 12.

So we have to scale our numerator by the same factor to keep the fraction the same.

1/4 is equal to 3/12.

So now we've expressed both our fractions with the same denominator.

We've got the same unit of count, so we can rewrite our equation.

1/3 is equal to 4/12 and 1/4 is equal to 3/12.

So now we've got 4/12 plus 3/12.

And that meant that Aisha spent 7/12 of her money.

Now we can just think about the numerator.

Four somethings plus three somethings is equal to seven somethings.

The somethings in question here are 12ths 4/12 add 3/12 is equal to 7/12.

Ah, and Lucas says we were right.

It is slightly more than half as we estimated from the pie chart.

6/12 will be equal to 1/2 and we've got 7/12.

So let's look at these images.

We've got our pie charts back again here.

What's the same and what's different? Well let's think, our 1/3 could be represented as 4/12.

And if you see in the pie chart on the right with all those extra dotted lines, we can see that it's been divided into 12 equal parts.

And we've got four of those parts in the purple shaded area, which was labelled as 1/3.

Those are 4/12, let's look at the quarter, the green shaded area and 1/4 was equal to 3/12.

Three of those dotted line parts are making up the same as our 1/4.

So we can see there clearly the 3/12 plus 4/12.

So the first pie chart shows our original fractions, 1/3 plus 1/4.

And our second pie chart has had the whole divided into 12 parts.

So we can see that equality now of 4/12 and 3/12 with the same denominator equaling 7/12.

And we also know that 1/3 plus 1/4 therefore must also be equal to 7/12.

Time to check your understanding.

Tick the multiple that could be used as a common denominator for 1/4 and 1/5.

Pause the video, have a go.

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

What did you think? It was 20ths, wasn't it? Let's think about those multiples.

20 is a common multiple of both four and five.

Let's list the multiples of four, four, eight, 12, 16, 20.

and the multiples of five, five, 10, 15, 20, and 20 is the first common multiple that we come across.

So we could convert both quarters and fifths into 20ths and add them easily.

Let's have a look at another problem then.

Aisha's family eats cereal for breakfast most mornings.

Do you eat cereal? I do sometimes as well.

She says we bought two packets of cereal which have the same amount in each box.

So the boxes contain the same amount of cereal.

She says we at 2/3 of the Shred'ohs and 3/5 of the Ultra Flakes.

How many boxes of cereal do we eat in one week? Hmm, let's have a think.

So she said we at 2/3 of the Shred'ohs, so can we represent that.

The whole has been divided into three equal parts and you've eaten two of them.

So there we go.

Whole divided into three equal parts and they've eaten two of the parts.

What about the Ultra Flakes? She says we at 3/5 of the Ultra Flakes.

So Lucas says the whole has been divided into five equal parts and you've eaten three of them, five equal parts and they've eaten three of them.

Do you notice they've eaten just over half of both? So they can record this as an equation.

Remember we're trying to work out how much of the boxes they've eaten at the end of the whole week.

So we've got 2/3 plus 3/5.

Aisha says, oh, they're not related fractions, are they? We cannot convert thirds to fifths or fifths to thirds.

So we're gonna have to think about those common multiples, aren't we? She says we'll need to find a denominator that is common for both.

So let's think about the multiples of three, three, six, nine, 12, 15.

Why do you think we've stopped at 15 there? I wonder if you've spotted it already.

And the multiples of five, five, 10 and 15.

So the first multiple shared by three and five, so the first multiple common to both of them is 15.

So we will be able to use fifteenths as our common denominator.

So what's 2/3 equal to as a number of fifteenths? Well, three times five is equal to 15.

So we need to scale the numerator by the same factor of five.

So 2/3 is equal to 10/15.

And what about 3/5 as a fraction representing fifteenths? Well we are going to scale up by three this time, aren't we? Five times three is 15.

So we'll need three times as many of those parts.

Three times three is equal to nine.

So 3/5 is equal to 9/15.

So now we can replace the fractions in our equation.

So let's rewrite it, 2/3 was equal to 10/15 and 2/5 was equal to 9/15, 10/15 plus 9/15 is equal to 19/15.

Ah, we've got an improper fraction, they've eaten over a box of cereal.

And Lucas says, we can convert it to a mixed number.

15/15 would be one whole and we'd have four extra fifteenths.

So that will be one and 4/15.

So Aisha says we eat one whole box and 4/15 of a box each week.

Let's look at those images again.

What's the same and what's different? So in the first image, we're representing the 2/3 and the 3/5.

But what have we done to the second image? Well, we've represented each as fifteenths.

So our thirds have been divided into five equal parts and our fifths have been divided into three equal parts.

And converting them into fifteenths allowed us to express the fractions with a common denominator, 10/15 plus 9/15.

And then we could add them to get our solution.

Over to you to check your understanding.

Can you solve 1/3 plus 2/7? Think about what you're going to use as a common denominator.

Think about those common multiples of three and seven.

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

How did you get on? Well, if we list the multiples of three and seven with multiples of seven, we've got seven, 14 and 21.

And if we carry on listing the multiples of three, we find that our first common multiple is 21.

So we're going to convert our fractions into 21ths.

So 1/3, we're going to scale everything up by a factor of seven is going to be equal to 7/21 and 2/7, we're gonna scale everything up by a factor of three to create 6/21.

So we can rewrite our equation as 7/21 plus 6/21.

And that will give us a sum of 13/21.

Well done if you got that right.

And if you spotted that that first common multiple was 21.

Back to the cereal now.

Aisha says we at 2/3 of the shreds last week and this week we at 3/5 of the packet.

What's the difference between how much we at last week and this week? So last week they ate 2/3 and this week they ate 3/5.

So now we're gonna find a difference.

Ooh, let's have a look at that again.

There are our 2/3 and there are our 3/5 of the packet.

And can you see there's a tiny bit that's the difference.

They ate slightly less this week than they ate last week.

How we calculate what that difference is? Lucas says you can record that as a subtraction equation to find out the difference.

And can we see that our 2/3 were slightly bigger? So that's going to be our hole that we're starting with, 2/3 subtract 3/5.

Again, because the denominators are thirds and fifths, we'll need to find a common denominator.

They're not related fractions.

So we need to think about those multiples.

We know that 15 is a common multiple of three and five.

It's the first common multiple that they share.

So 2/3 is equal to 10/15 and 3/5 is equal to 9/15.

And you can see the scaling up that we've done for both of those to create the equivalent fraction.

So now we rewrite the equation, 10/15, subtract 9/15.

Oh, that's equal to 1/15.

That's a tiny difference, isn't it? Not a lot of difference, just a few flakes or a couple of Shred'ohs I reckon.

So to add and subtract fractions with different denominators, first we need to convert to fractions with common denominators.

And we can do that even when we don't have related fractions by thinking about common multiples.

Over to you to check your understanding.

Can you solve 2/3, subtract 1/7, think about a common multiple that would allow you to create common denominators.

Pause the video, have a go.

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

How did you get on? Well, if we list those multiples again, we can see that there's a common multiple of 21.

So 2/3 is going to be equivalent to 14/21.

Scaling up by a factor of seven this time and 1/7 is going to be equivalent to 3/21.

We're scaling up by a factor of three.

So now we can rewrite our equation.

14/21 subtract 3/21.

It's the same equation 'cause we've used equivalent fractions.

14 subtract three is equal to 11.

So our difference is 11/21.

Well done if you got that right.

Time for you to do some more practise now.

In question one, you're going to solve the fraction equations and think about what you notice as you solve them and some more to solve in question two.

Question three, you're going to fill in the missing numbers.

So you're really going to think about what these fractions represent and what they would look like with common denominators as well.

So can you fill in those missing numerators and denominators? So for question four, the pie chart shows the time spans when the first try is scored in a rugby match in a local competition.

So a try is how you score points in rugby.

It's getting the ball over the line.

So the pie chart shows that in the first 20 minutes, 1/3 of the tries were scored, in the next 20 minutes, 21 to 40 minutes or the next 19 minutes, 2/9 of the tries were scored.

In the next section of the game, 41 to 60 minutes, 1/6 of the tries were scored and need to work out what fraction of the total number of tri scored happens between 61 and 80 minutes.

And finally question five, how could you solve this calculation without using 40/8 as a common denominator? So can you think about common multiples and find another one that is not 40/8? You've got lots to get on with there.

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

How did you get on with all of that? So we were solving the equations here, 1/3 plus 1/7.

So we were going to think about 21ths here, I should think.

7/21 plus 3/21 is equal to 10/21.

1/2 plus 1/5.

Well we have a common multiple of 10 here.

So 5/10 plus 2/10 is equal to 7/10.

What about 1/4 plus 1/10? So we could have had a number of different denominators here, but we've gone for 40ths, so that would be 10/40 plus 10/40, which would be 14/40.

For the next one, we've got some subtractions here, but we can find a common denominator of 28s here.

A common multiple is 28.

7/28, subtract 4/28, which is equal to 3/28.

Oh gosh, fifths and 12ths where we could have a common denominator here of 60ths.

So that would be 12/60 subtract 5/60, which is equal to 7/60.

And then for quarters and sixths we could have a common denominator of 24ths.

That's a common multiple.

So that would be 6/24, subtract 4/24, which is equal to 12/24.

Did you notice something here or have you noticed something in the denominators that we've got? You might have had different common denominators.

If you multiply the denominators together, you will always find a common denominator.

Seven times three and three times seven is going to give us our common denominator of 21.

Two times five, five times two will give us a common denominator of 10, four times 10, 10 times four will give us a common denominator of 40.

So we will always get a common denominator, but it might not be the most efficient one.

Let's have a look at a later question to think more about that.

So let's solve these equations.

So we've got 3/8 plus 2/3.

Well, our common multiple here would be 24ths, so that would be a nine plus 16 is 25/24, which would become one and 1/24 as a mixed number.

For sevenths and ninths we've got 63rds as our common multiple, goodness me.

So we'd have to scale up the 4/7 by a factor of nine.

So that would be 36/63.

The 2/9 will be scaled up by a factor of seven.

So that will be 14, 36 plus 14, which will give us our 50/63.

Wow, there's a lot of thinking to do here, isn't there? For 2/11 plus 4/3, our lowest common multiple will be 33rds.

So that will be 6/33 plus 44, 33, six plus 44 is equal to 50.

So that's one whole and 17/33.

I'm not sure that 80 is the lowest common multiple of eight and 10, but we know that we can just multiply the denominators to find a common denominator.

So that would be 50/80, subtract 24/80, and that will leave us with 26/80.

So now we've got 4/7 minus 2/8.

Well our first column multiple is going to be 7/8, and that's my favourite times table fact, 7/8 are 56.

So we're going to have eight times as many of our sevenths, which is 32, so 32/56, and we're going to be subtracting seven times as many of our eights, which will give us 14/56.

So 32/56 subtract 14/56 will give us 18/56.

And I'm quite looking forward to this next one because we've got thirds and halves.

So I think that's going to give us six, isn't it? We've got 7/3, which is an improper fraction.

So that will be 14/6, subtract 9/6, and that will leave us with 5/6.

Gosh, there was a lot of thinking to do there.

I'm still not convinced that we always need to multiply the denominator together to find the most efficient way of finding a common denominator.

Let's keep thinking.

Okay, so we've got some missing numbers to fill in here for question three.

So we've got 1/3 plus some fifths is equal to 8/15.

Well, we're thinking about 8/15, so that will be 5/15 plus 3/15.

And 3/15 is equivalent to 1/5.

We've got 44ths here.

So I think we might be thinking about quarters here.

Three quarters plus 1/11 is equal to 37/44.

three quarters is equal to 33/44 and 1/11 is 4/44.

So that gives us our 37/44.

Can you see what's happening with our denominators here? We've got 4/9 and then a missing denominator, but our common denominator is 45, so I think this must be 1/5 and that would give us a common denominator of 45ths, wouldn't it? What about the next one? This time we've got our sum at the beginning, one and 2/30.

Well, we've got 4/6.

So what's another factor of 30? Well, I think that's gonna be fifths, isn't it? 4/6 plus 2/5 is equal to one and 2/30.

We've got a missing numerator this time, which is slightly harder in some ways than finding the missing denominators.

I think I'm going to convert my mixed number into an improper fraction.

So that gives me 41/12.

36/12 will be three wholes, plus five is 41/12.

I know that 53 is going to be equal to 20/12, and I need 41/12, so I need another 21/12.

So 21, my 12ths and quarters are linked by a factor of three.

So 21 divided by three is seven.

I think it's 7/4.

That took a lot of thinking, didn't it? Let's see if we can think about that Same sort of thinking to solve our final one.

Again, we've got a missing numerator.

So my sum is one and 1/20.

Well, that's 21/20.

So let's remember that 21.

3/5 is equal to 12/20, 1/4 is equal to 5/20.

That's 17/20, I wanted 21/20.

So 17 at four is 21.

So that will be 4/20, which will be equal to 2/10.

Lots of thinking to do there.

I expect you made lots of jottings with those.

I have to admit, as I was thinking about this, I made some jottings as well, so I hope you did too.

Question four, this is the rugby match, wasn't it? So we're trying to work out the fraction of tries scored between 61 and 80 minutes.

So we are trying to look at the rest of our whole, we need to add together 2/9, 1/3 and 3/6.

So what could be our common denominator here? What we don't need to do nine times three times six.

We can think about those common multiples of nine, three and six.

While the nine times table we get nine and 18.

And I know that 18 is in the three and six times tables, so we can use 18ths here.

So that's 4/18 plus 6/18 plus 3/18 plus something is equal to 18/18.

Four plus six is 10, plus three is 13.

So we need another 5/18.

So that means that 5/18 of the tries were scored in the final 19 minutes of the matches.

And finally, question five, how could you solve this calculation without using 48ths? This I think is some thinking that we could maybe go back and think about question one again.

So let's think about multiples of six and eight.

So the multiples of six and eight, we've written them down there up to 48, but we don't want to use 48ths.

Is there a multiple of eight in there that you can see that isn't 48? Well, we've written them out again to 48, but can you see a common multiple? Yes, 24 is also a common multiple of six and eight.

You can just multiply the denominators together, but it often isn't the most efficient way of working.

So 2/6 is equal to, well, we've got to multiply, scale up by factor of four, 8/24.

3/8 are scaling up by a factor of three is equal to 9/24.

So 8/24 plus 9/24 is equal to 17/24.

Well done if you got that right.

And remember always to look for the common multiple that isn't necessarily the product of the two denominators.

What a lot of work you've done in this lesson.

I hope you're feeling really proud of yourselves.

We have come to the end of the lesson though.

We've been adding and subtracting non-related fractions with different denominators.

What have we been thinking about? Well, we know that fractions need to have the same denominator if we are to add and subtract them.

Common denominators for non-related fractions can be found using common multiples of the denominators in equivalent fractions.

The numerators and denominators have been scaled up or down by the same factor.

That's really important.

They may look different, but equivalent fractions have exactly the same value, and we found that you can multiply the denominators together to find a common denominator if they are non-related fractions, although there's sometimes more than one possible option, and you can find a lower common multiple, meaning that your numerators are smaller and you've got an easier calculation to do.

Always worth looking.

Your fraction sense is really important when you're calculating with fractions.

Thank you for all your hard work in this lesson.

I hope you're feeling proud and can see the progress you've made with adding and subtracting fractions.

I hope I get to work with you again soon, bye-bye.