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're 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 using our knowledge of adding and subtracting related unit fractions to solve problems. So related fractions are the ones where their denominators are related.

One fraction might have a denominator which is a multiple of the other fraction, and we can use that to create common denominators in order to solve problems. Let's have a look at what's in our lesson today.

Aha.

Well, there we are.

Keywords that we've just thought about, related fractions and common denominator.

Let's rehearse them anyway.

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

So my turn.

Related fractions.

Your turn.

My turn.

Common denominator, your turn.

I hope you're familiar with those.

Let's just remind ourselves, they're really key words in our learning today.

So when the denominator of one fraction is a multiple of the denominator of another fraction, we can say that these fractions are related fractions.

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

The denominator is common to both of them.

It's the same.

There are two parts in our lesson today.

In the first part, we're going to be working with three parts, and in the second part we're going to be reasoning with related fractions.

So let's have a look at part one.

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

So they're looking at this equation and wondering how they could solve it.

What do you think? Alex says, "We can draw this as a bar model to get a sense of the magnitude of the sum." The magnitude means the size of the sum.

It's about making an estimate, isn't it? And that's really useful to do whether you're calculating with whole numbers, large numbers, fractions or decimals.

So let's have a look at this bar model.

Well, we've got unit fractions here so we can compare them all.

So our largest fraction there is a third and adding to that is a quarter and then a one twelfth, which is the smallest fraction that we're adding.

So what do we think that's going to look like overall? Could we compare it maybe to a half? Well, two quarters is a half and we've got a quarter and a third, which is more than a quarter and then a bit more.

So it's going to be more than a half, but is it going to be equal to a whole? I don't think so.

A third is well under a half and a quarter and a twelfth are well under a half together.

So it's going to be between a half and a whole.

Andeep it says, "I've noticed that each fraction has a different denominator." But each fraction is a related fraction though.

When fractions don't have the same denominator, we can't just add them straight away.

We need to have them as the same unit of counting.

So they need to have the same denominator.

But they are related, as Andeep says.

We've got twelfths there and quarters and thirds and a quarter and a third are related because they are both factors of 12.

12 is a multiple of four and three.

And as Alex says, "Yes, the fractions with a denominator of four and three are related to the fraction with a denominator of 12." And to solve this, we'll need to convert each fraction so they have a common denominator.

So we can use twelfths as our common denominator because they're related fractions, as Alex says.

Let's start with one quarter.

So we've got to create a fraction equivalent to a quarter that has 12 as the denominator.

Well, we can see that the denominator has been scaled up by a factor of three.

So we know that we need to scale up the numerator by a factor of three as well for the fraction to be equivalent.

Four multiplied by three is equal to 12.

One multiplied by three is equal to three.

So one quarter is equivalent to three twelfths.

And we can substitute one quarter for three twelfths in our equation.

What about one third then? So again, we need to convert one third to twelfths so it has the same counting unit we're calling it, a common denominator of 12.

And we know that we can do this because 12 is a multiple of three.

It's four times three.

So if we scale the denominator up by a factor of four, we must scale the numerator up by the same factor so that the proportion of the whole stays the same.

One times four is equal to four, so one third is equal to four-twelfths.

And we can replace one third with four twelfths in our equation.

All of the fractions have a common denominator now, so we can add them easily, thinking about the numerators.

Three twelfths plus four twelfths plus one twelfth is equal to eight twelfths.

Now our estimate said it was going to be less than a whole, but greater than a half.

Well a half will be six twelfths and the whole will be 12 twelfths.

So eight twelfths is indeed bigger than a half, but smaller than a whole.

So I think our estimate was about right.

Andeep says he's noticed something else.

He says the numerator and the denominator of eight twelfths have a highest common factor of four.

Ah.

So we can simplify this answer.

We can convert our answer to a simpler form.

So just like we can scale up to create common denominators of equivalent fractions, we can scale a fraction down and represent it using smaller values in its simplest form.

So we can scale down the numerator and denominator by a factor of four, which gives us a fraction of two thirds.

We can leave our answer as eight twelfths, or we can simplify it to two thirds.

Sometimes, in a question it will ask you to simplify the answer.

Sometimes it's useful to simplify the answer because you've got to use it in another part of a question.

How could we solve this equation this time then? Again, Alex says we can draw this as a bar model to get a sense of the magnitude of the difference that we're looking for.

So this time, remember it's a subtraction.

So our whole is eight twelfths, the number we're starting with.

So that's our whole.

And from eight twelfths we're subtracting one third and one quarter, and we know that one third is a bit bigger than a quarter, and then we've got a small amount left.

Do you notice something about these values? Andeep says, "These are still related fractions." Alex says, "Yes, both fractions with a denominator of three and four are related to the fraction with a denominator of 12." So to solve this, we can convert them so they have a common denominator again.

Andeep says, "We could, but I don't think we need to." What's Andeep spotted? Maybe you've spotted it as well.

Let's revisit our bar model.

Andeep says, "I noticed the numbers used in this equation were used in our previous bar model." We had a quarter plus a third plus a twelfth, and we worked out that it was eight twelfths or two thirds.

Well, we've got a whole of eight twelfths here and we're subtracting a third and a quarter.

He says, "We know that the whole is eight twelfths and we know the size of all of the parts." "Oh, yes," says Alex.

"And if we subtract one quarter and one third from the whole, it leaves us with one twelfth." And we can see that clearly in our bar model.

"That was easy enough," he says.

Always worth looking at calculations to see if there's something you can use that you've done before.

So here our difference is one twelfth.

Over to you to check your understanding.

Can you tick the expressions where each fraction has a related value? So expressions where we could convert to common denominators very easily because there are related fractions in the expression.

Pause the video, have a go.

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

So which ones did you find? Well, the top three all had related fractions.

The smaller value denominators in these are all factors of the larger value denominator.

Or you could say that the largest value denominator is a multiple of the smaller value denominators.

For D, that wasn't the case though.

Seven isn't a multiple of three and it isn't a factor of 24.

Over to you for another check.

Can you solve the following equations? Have a look carefully at the denominators.

Do you need to do anything to make them easier to add? And what do you notice? Pause the video, have a go.

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

How did you get on? Well, the ones on the right hand side of the screen had common denominators, didn't they? Four twelfths plus two twelfths plus one twelfth.

Well that's four plus two plus one is equal to seven, and we know we're counting in twelfths.

And then for the second one, we've got nine twelfth, subtract four twelfth, subtract four twelfth.

Or we could combine the four twelfths and the four twelfths.

That's eight twelfths.

Nine twelfths subtract eight twelfths is equal to one twelfth.

What about the ones on the left hand side of the screen then? Anything you spotted? Ah yes.

Did you notice the connection between the first and the second equation in each row? They were equivalent fractions, weren't they? One third is equivalent to four twelfths.

One six is equivalent to two twelfths.

And then in the bottom one, one third is equivalent to four twelfths.

So the equations on the right hand side of the screen were the same as the equations on the left hand side, but with common denominators for the fractions so that we could add and subtract them easily.

I hope you spotted that.

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

You're going to solve the following equations.

You've got some additions and you've got some subtractions.

Have a think about related fractions and whether you can create common denominators so that you can add and subtract easily.

And for question two, you're going to fill in the missing numbers and we've got some additions and mixed additions and subtractions, but we've got missing elements.

We've got a missing numerator here and there and a missing denominator here and there.

Have a think through and see if you can reason your way to what the missing values are.

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

How did you get on? So for the first one in question one, we had a half plus an eighth plus a quarter.

So in question one, the fractions can all be represented as eighths.

So we can rewrite that equation as four eighths plus one eighth plus two eighths, and we can use our knowledge of equivalent fractions to do that.

So we've got four plus one plus two, which is equal to seven.

And so we've got seven eighths as our sum.

Now they're in the same unit effectively, they're all eights.

We can add them together very easily.

And then you could apply the same thinking to the other questions.

So one third plus one fifth plus one fifteenths.

We're looking at a common denominator of fifteenths there.

So five fifteenths plus three fifteenths plus one fifteenth is equal to nine fifteenths.

And we could simplify that because there's a common factor of three between nine and 15, isn't there? One third plus one twelfth plus one quarter.

Well that might look a bit familiar.

We've been looking at that already.

So we can have a common denominator of twelfths here.

And our answer is eight twelfths.

One third plus one ninth plus one 27th.

Well, our common denominator here will be 27ths.

So we've got 13 27ths.

A quarter, subtract one eighth, subtract one 24th.

Well, we're looking at a common denominator of 24s there, aren't we? Six 24ths, subtract three 24ths, subtract one 24th leaves us with two 24ths.

For our next set, the common denominator would be 20.

That's our related fraction.

We said we'd have five 20ths, subtract four 20ths, subtract one 20th.

Ah, that gives us an answer of zero.

We've subtracted all of our 20ths.

In our next one, we've got a third subtract a quarter, subtract a twelfth.

So 12 will be our common denominator.

Thinking about twelfths.

Four twelfths, subtract three twelfths, subtract one twelfths.

Do you notice that again? Zero again, isn't it? And then for our final one, we've got 24ths again.

one sixth, subtract one 12, subtract one 24th.

So that will be four 24ths.

Subtract two 24ths, subtract one 24th, which leaves us with one 24th.

I hope you were successful with those.

And onto question two where we had to fill in the missing numbers.

So in the first one we're looking for what that common denominator would be.

How can we use our knowledge of related fractions? Well, we'd want to convert all of those into twelfths, wouldn't we, to make the addition easy.

So our sum will be 10 twelfths.

The missing value was the 12 in that denominator.

So the next one, we've got a half subtract one 24th, subtract one sixth, and we've got seven of something left over.

Well, it would make sense to have 24 as our denominator there because we know that one half is equivalent to 12 24ths and one sixth is equivalent to four 24ths.

And 12 subtract one, subtract four is equal to seven.

For our next one, we're looking at 27ths.

So this time we've got to think about what our numerators must have been.

One third is equal to nine 27ths.

One ninth is equal to three 27ths.

So that's 12 27ths.

Subtract one 27th which leaves us with 11 27ths.

For the next one we were thinking in 24ths.

So one eighth is equal to three 24ths and one half is equal to 12 24ths.

So that's 15 and we needed 18.

So that missing numerator must be three.

We must be adding three 24ths.

So for the next one we're starting with a half, but we're ending up with a fraction with 32 as the denominator.

So we must be thinking about 32ths or 32nds.

Well, we know that a half must be 16 32ths and an eighth must be four 32ths.

So 16 subtract four is equal to 12, subtract another two is equal to 10.

So our missing numerator must be two.

So for our final one, we've got an answer of 38 20ths.

Our one 10th is worth two 20ths.

So we've had plus two 20ths and subtract four 20ths.

So we've sort of effectively taken away two 20ths.

So we need 40 20ths to be our whole And 40 20ths is equivalent to 10 fifths.

So 10 fifths plus one 10th subtract four 20ths is equal to 38 20ths.

That one took quite a lot of working out, didn't it? I hope you enjoyed wrestling with that one.

And on into the second part of our lesson, we're going to be reasoning with related fractions.

How can we compare these expressions? Alex says, "These fractions are related fractions.

Shall we convert them to a common denominator to find the value of each expression," he says.

We could do that first, couldn't we? Andeep says, "I don't think we need to.

The expressions are the same." Yeah, we've got one third minus one twelfth and one third minus one twelfth.

So we know that they will be equal.

No need to do any calculating there.

No need to convert anything to common denominators.

What about these two? What do you notice this time? Alex says, "The expression on the right is now subtracting two twelfths.

Shall we convert them this time so we can compare their values?" What do you think? Should we do some conversion and some calculating? Oh, Andeep says, "Again, I don't think we need to." He says, "The wholes are the same." One third is the starting number with each of these expressions, but one of them is subtracting more than the other, so we can compare them easily.

One third, subtract two twelfths is taking away more.

So it's going to be less than one third subtract one twelfth.

So again, we don't need to evaluate or calculate the value of these expressions.

We can use what we know to reason the way to the answer.

What about these ones? Alex says, "I noticed this time that one of the wholes has changed.

One is one quarter and the other is one third." But we're subtracting one 12th both times, aren't we? Andeep says, "Yes.

So if we're subtracting the same amount but one whole is larger than the other, then the whole that was the largest to start with will have the largest difference." Yes, if we start with more but subtract the same amount, we'll be left with more.

So which fraction is largest? It's a third, isn't it? A third is larger than a quarter.

So one quarter subtract one twelfth will be less than one third, subtract one twelfth.

And again, we didn't have to do any calculating or use common denominators.

We could reason our way through because of what we know about subtraction.

What about this one then? "Oh," says Alex.

"That's an easy one." Have you spotted what Alex has spotted? He says, "The fractions are the same, but this time one expression is adding and the other one is subtracting one twelfth." "Yes," says Andeep.

"So one third plus one 12 will be larger than one third minus one twelfth.

Again, no need to do any calculating.

what about these expressions? Alex says, "This is trickier." One of them is subtracting and one of them is adding.

But also the starting fractions are different sizes as well.

We've got one third subtract a twelfth and one quarter plus one twelfth.

Andeep says, "We have a larger fraction subtracting a small amount and a smaller fraction adding a small amount." Difficult to say, isn't it? He says, "I think it might be best to convert these expressions to common denominators and work out their values." So one third is equivalent to four twelfths.

They're related fractions, aren't they? We can scale up the numerator and denominator by four and create the fraction of four twelfths.

And Andeep says, "And one quarter is equivalent to three twelfths." Again, they're related fractions.

So if we scale up the denominator of a quarter by three, we get 12 and we have to scale the numerator as well.

So one quarter is equal to three twelfths.

Now we can do the calculations and compare.

And we see that four twelfths subtract one twelfth is equal to three twelfths and three twelfths plus one twelfth is equal to four twelfths.

So one quarter plus a twelfth was slightly greater than one third minus one twelfth.

Is that what you expected? Alex says, "Well, it's easy to compare the expressions," but he wasn't expecting that answer.

I wonder if you were? Time to check your understanding.

Use the correct inequality to compare these expressions.

Do we need the less than, equals or greater than sign between those expressions? Pause the video, have a go.

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

How did you get on? Well Alex says, "If we convert one fifth to two tenths, the expression on the left is equal to three tenths." Two tenths plus one 10th is equal to three-tenths.

And if we convert one half to five tenths, then the expression on the right is equal to four-tenths.

One half is equal to five tenths.

Subtract one tenth is equal to four tenths.

So it's slightly greater than the other value.

One fifth plus one 10th is less than one half subtract one 10th.

Well done if you got that right.

And time for another check.

Can you tick the expression with the largest value? Have a careful look, pause the video and when you're ready for some feedback, press play.

What did you think? Did you spot that while the fractions are the same, expression A had two addition signs, whereas B and C had mixed edition and subtraction? So A was going to be largest because we were combining all those fractions, whereas in B and C, some of them were being subtracted.

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

Can you use inequalities to compare these expressions? So greater than, equals or less than, to compare those expressions? Think about related fractions, but also think, do you need to use common denominators and create equivalent fractions or can you reason your way through? And for question two, you've got a problem to solve about Alex and his brother eating pizza.

Who had the most pizza left over? 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 let's have a look at the first one.

So the first one, we had one eighth plus one 24th, one 24th plus one eighth, well we know that addition is commutative, so those two expressions are equal.

What about the next one? We had one eighth plus one 23rd and one 24th plus one eighth.

So our eighth is the same, but we're adding a 23rd on one side and a 24th on the other side.

And we know that one 23rd is slightly greater in size than one 24th.

So that expression on the left must be greater than the one on the right.

What's about one eighth plus one 24th and one 25th plus one eighth? Well again, the eighth is the same on both sides, so we're just comparing the 24th and the 25th.

And a 25th is slightly smaller than a 24th.

So one eighth plus one 24th is greater than one 25th plus one eighth.

Now we've got one eighth plus one 24th and two 24ths plus one eighth.

Ah, well we've got adding two 24ths on one side, so that side must be greater.

And then we've got two eighths plus one 24th and two 24ths plus one eighth.

Hmm, that's a bit trickier, but eighths are a lot larger in scale than 24ths, aren't they? So two eighths plus one 24th has got to be greater than two 24ths plus one eighth.

We've got some subtractions going on over here.

So we've got one eighth subtract one 24th, and then one 24th plus one eighth.

Well, we're combining on that side, so that side must be greater.

The addition side is greater.

Here we've got two subtractions.

We've got one eighth, subtract one 24th and one 24th subtract one eighth.

That's going to give us a negative number because an eighth is greater than a 24th, isn't it? So that value will be smaller.

And again, here we've got one eighth subtract one 24th and one eighth subtract one 24th.

So this time our eighth is the whole in both cases and the part we're subtracting is a 24th.

So those are equal.

Well now we've got some three part ones to have a think about.

So we've got one eighth, subtract one 24th, subtract one 48th.

So we're subtracting two parts from our eighths.

And in the other one we've got one eighth, subtract one 24th.

Well, that's the same, but then we're adding a half.

Well that's going to make it larger, isn't it? That's the largest value fraction in any of those expressions.

And then we've got one eighth, subtract one 48th, add one 24th, and then one eighth add one 24th, subtract a half.

Well again, that half is the greatest value fraction in either expression.

So if we're subtracting it, that one is going to be smaller.

So did you notice that we didn't actually have to convert any fractions to common denominators to work out which inequality symbol went into the gaps there? I hope you reasoned your way through as well.

And let's have a look at question two.

Alex and his brother were eating their own pizzas.

Alex ate a quarter of the pizza on the first night, then the next day he ate two twelfths for his lunch and another one third for tea.

Well, he made it last, didn't he? Good value there, Alex.

Can we represent that? He had a whole pizza, he ate one quarter, he ate two twelfths and then he ate one third.

What about his brother? He ate one sixth of the pizza on the first night, then he ate a third of it for lunch and a quarter of it for his tea.

So we've got to compare that to one whole pizza subtract one sixth, subtract one third, subtract one quarter.

Ooh, let's have a look.

Well, I can see that there's some common things there.

Obviously we're starting with a whole pizza.

They each ate a quarter at some point, and they each ate a third at some point.

And then Alex ate two twelfths and his brother ate one sixth.

Do you spot something there? Those are equivalent fractions, aren't they? Two twelfths is equal to one sixth.

So they actually have the same amount left over.

Again, a little bit of equivalent fraction knowledge, but lots of reasoning.

We don't always need to create common denominators to solve problems when we're adding and subtracting fractions.

It's always worth having a look first to see if you can reason the answer before you start doing lots of work to create common denominators.

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

So we've been using knowledge of adding and subtracting related unit fractions to solve problems. So what have we learned? We've learned that when adding and subtracting related fractions with three parts, two of the parts must relate to the third part.

So we must have related fractions across all three if we're going to be able to add and subtract them easily.

And when they are related, this will allow you to convert two of the fractions to have a common denominator and that common denominator will be shared with the third fraction.

And when comparing expressions with related fractions, you do not need to convert each expression straight away.

You can use your sense of fractional size to help you to compare.

I hope you've enjoyed reasoning about fractions today, and I hope I get to work with you in another lesson again soon.

Bye-Bye.