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.

So in this lesson, we're going to be explaining how to subtract related unit fractions.

You might have done some thinking about related fractions already, looking at the denominators to see if there is a multiple or factor relationship between them, and how you can use that to create common denominators to make addition and subtraction of fractions easier to do.

So let's have a look at how that's going to work in our lesson today.

Ah, and there are our key words: related fractions and common denominator.

I'm sure you're familiar with them, but let's just practise them and think about what they mean before we start our lesson.

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

My turn, related fractions, your turn.

My turn, common denominator, your turn.

Excellent, as I say, I'm sure you're very familiar with those phrases, but let's just remind ourselves what they mean.

They're going to be really useful in our lesson 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.

Let's see how those phrases are going to help us today.

There are two parts to our lesson.

We're going to be subtracting related fractions with a model to begin with.

And then in the second part, we're going to be doing it without the model.

So let's make a start on part one.

And we've got Jacob and Jun helping us in our lesson today.

So what fraction of the shape is shaded purple? Well, Jacob says, "If I divided the whole into equal parts, we could see that 1/3 of the whole has been shaded." Can you see that 1/3? And Jun says, "I can show that by dividing the whole into 3 equal parts." There we go, you might already have been able to picture that there were 3 of those columns in the whole.

So 1/3 of the shape had been shaded.

So there it is, 1/3.

What do you notice now? And how can this be recorded as an equation? What happened? Let's look again.

Hmm, how could you describe that? Well, Jun says, "One part has been subtracted from 1/3." But what's the part that's been subtracted? Ah, he says, "The part that's been subtracted is 1/9 of the whole." If we imagine the whole divided into equal parts equal to that small part, there would be 9 of them in the whole.

And 1 of those 9 is now not shaded purple.

Jacob says, "So we can write an equation for this." 1/3 minus 1/9 is equal to something.

But Jun says, "I have another equation we can write for this too." Can you see what he's done there? He's divided the whole shape into those 9 equal parts.

So he has seen that we can rewrite the 1/3 as 3/9.

"Oh yes!" Says Jacob.

"Now subtract the 1/9." And Jacob says, "That makes it easier to see what is remaining.

It's 2/9." So what have we done there? Well, we had 1/3 subtract 1/9.

But that was harder to do because we didn't have a common denominator.

We didn't have a common unit for counting.

But what Jun showed us is that we can rewrite that 1/3 as 3/9.

Now we've got a common denominator and we can subtract the fractions more easily.

Let's have a look at how we'd represent this on a number line.

We know that 1/3 is equivalent to 3/9.

We could see that in our shape and we can see that on our number line.

1/3 and 3/9 sit at the same point on the number line.

There they are.

So we know that we need to subtract 1/9 which leaves us with 2/9.

3/9 subtract 1/9 is equal to 2/9, and 1/3 is equivalent to 3/9.

And as Jacob says, "We started with different units.

We had thirds and ninths." But they are related fractions.

One denominator is a multiple of the other.

9 is a multiple of 3.

And that means we can convert the 1/9 into a fraction which has ninths as its unit or its denominator.

1/3 is equivalent to 3/9.

We've scaled the numerator and denominator by the same factor of 3, multiplied them by 3.

Our fraction, it has the same value though as we could see in the shape and on the number line.

And then, we can substitute our 1/3 for 3/9 because we know they are equivalent.

Now we have the same unit or a common denominator and we can subtract.

3/9 minus 1/9 is equal to 2/9.

Oh, another shape here.

What fraction of the shape is shaded here? Well, that's an interesting one.

Have you seen something like that before? What can you say about those different shapes that we've created within our whole? Well, June says, "I think 1/4 of the shape is shaded here." He says, "I know that because the whole could be divided into 4 equal parts." Ah yes, so if he removes some of the extra divisions, he can see that he has 1 of 4 equal parts shaded, 1/4.

Ah, what fraction of the shape is shaded now? We've removed a part of our quarter.

And Jacob says, "I think the part that's been subtracted is 1/16 of the whole." How does he know that? He says, "If I do this, each quarter of the whole can be divided into 4 equal parts.

So even though the parts may look different, they all now represent 1/16 of the whole." Can you see the bit that's different? That bottom right quarter in a way has been divided in a different way.

It's been divided into 4 equal parts, but they don't look quite the same.

But they are equivalent in area to one of the small squares.

So they are each 1/16 of the whole.

"So now we can say that 1/16 has been subtracted." We started with a whole quarter shaded in and we've subtracted a sixteenth.

so we can record that as an equation.

"This image also helps us to convert the original quarter so that we have common denominators." Can you see what that quarter is equivalent to? It's equivalent to 4/16, isn't it? 1/4 is equal to 4/16.

We've got 4 times as many parts.

So we'll need 4 times as many of them.

We scale the numerator and the denominator by the same factor.

So 1/4 is equal to 4/16.

And now we can substitute our 1/4 for 4/16 in our equation.

We know they have the same value.

"And now we have a common denominator," a common unit of counting, "and we can find the difference," says Jun.

4/16 minus 1/16 is equal to 3/16.

How would we represent this on a number line? Here's another illustration of the fact that 4/16 and 1/4 are exactly the same in value.

They sit at the same point on the number line.

1 jump out of 4 is the same proportion of the whole as 4 jumps out of 16.

"So we know that 1/4 is equivalent to 4/16." "So we know that we need to subtract 1/16 which leaves us with 3/16." Time to check your understanding.

Can you write two equations to represent this image? So think about the two ways that we can represent the shaded area.

And then, we're going to subtract that part.

So there's our whole and that's the part we're subtracting.

So can you write two equations to represent this image? Pause the video, have a go.

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

How did you get on? What was shaded first? Well, it was 1/5, wasn't it? We can imagine 5 columns of the same size in that shape and 1 of them is shaded.

So our original shaded area was 1/5 and we've subtracted 1 of the small squares.

Well, again, if we imagine, we've got 4 in each column.

So that's 5 lots of 4 which will be 20.

So that's 1/20.

But what can we also say about our 1/5? If we put those extra divisions in, we can see that we've subtracted 1/20, but we can also see that our 1/5 is equivalent to 4/20.

So we can rewrite that equation as 4/20 minus 1/20.

And then, we've got a common denominator and we can work out that the difference is 3/20.

Well done if you spotted that.

Time for another check.

We're representing a subtraction on this number line as well.

Can you write two equations to represent this image of subtraction? Pause the video, have a go.

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

How did you get on? Did you spot that we were starting on 1/4 or 3/12 and we were jumping back 2/12? So we could write 1/4 minus 2/12 or 3/12 minus 2/12 because we can see that 1/4 is equivalent to 3/12.

And so we can substitute 1/4 for 3/12 in our equation, which gives us common denominators so they're much easier to subtract 'cause we've got the same unit of count.

And we can see that in both those cases, where we've landed on or the difference in our subtraction is 1/12.

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

You're going to write two equations for each image.

And a red cross represents a subtracted part.

So can you work out how much is shaded to begin with, how much we're subtracting, and what the difference is? And can you represent that with two different equations? Think about those related fractions and think about using common denominators.

And for question two, you're going to draw an image to represent each equation.

So think about what those equations would look like if you were writing them from an image.

See how creative you can be.

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 for the first one, you had to write the equations.

So oh, my goodness, that first one, let's have a look at what we've got.

Well, we can see that our original shape is 1 row and it's 1 row out of 1, 2, 3, 1 row out of 5.

So it's 1/5 of the whole.

So we're starting with 1/5, but what are we subtracting? Let's have a look.

Each of those rows is divided into 5 equal parts and we've got 5 of them.

So that must be 1/25 of the whole.

So we can say it's 1/5 subtract 1/25.

But we can also see that our 1/5 could be represented as 5/25.

And now we've got a common denominator.

5/25 minus 1/25 is equal to 4/25.

So 1/5 minus 1/25 must also be equal to 4/25.

Now for the second image, it's a slightly different looking image.

We've got sort of three hexagons there, haven't we? And we know that hexagons can divide into 6 equal parts.

So we would have 18 parts in our whole.

Now, originally we had 1/2 of each hexagon shaded in, so that's 1/2 of the whole, 9 out of 18 parts.

So we've got 1/2 or 9/18 as our whole that we're starting with.

And from that, we're subtracting 1 of those eighteenths.

So we can represent that as 1/2 minus 1/18 or 9/18 minus 1/18.

And what we've got left when we've completed the subtraction is a difference of 8/18 in each case.

An interesting one, it was an unusual shape to start with, wasn't it? So I hope you did some thinking about what the whole represented and what the parts represented.

So these are the images that we drew for question two.

I wonder what you came up with.

You may just have drawn some rectangles divided up, that's absolutely fine as well.

But for the first one, we decided that 1/2 could be represented by 4 shapes with 1/2 of each of them shaded in.

So we've got 4 shapes and 1/2 of each shaded in.

And we've subtracted from that 1 of those shaded parts, 1/8.

So if you imagine, our whole is our 4 shapes and we've cut each of them in 1/2.

So each of those parts is 1/8.

So we've got 1/2 subtract 1/8.

But we also know that 1/2 can be represented as 4/8.

So 4/8 subtract 1/8 would be equal to 3/8.

And in our second image, we've gone for something made from blocks.

So we've used 6 blocks to make our shape.

1 of those blocks represents 1/6 but we we're starting with 1/3 of our shape.

So that's 2 of those blocks.

So 1/3 can be rewritten as 2/6 and 2/6 minus 1/6 is equal to 1/6.

I hope you had fun coming up with some interesting images to represent those equations.

And on into the second part of our lesson, we're going to subtract related fractions but without a model to help us.

So we're gonna think about our related fractions and think about common denominators.

So how would you solve this equation? Jacob says, "In this example, we have two different units of count.

We've got thirds and fifteens." Are they related though? Ah yes, he says, "You can use your understanding of related fractions to find a common denominator." We can see that 15 is a multiple of 3.

So what do we need to do to create an equivalent fraction with 15 as the denominator? Well, we can see that we've scaled up the denominator by a factor of 5, multiplied it by 5.

So we need to do the same to the numerator to keep the proportion of the whole the same.

So 1/3 is equal to 5/15.

And now we can substitute our 1/3 for 5/15 because we know they are the same.

And now we have the same unit of counting, a common denominator, 5/15 minus 1/15 is equal to 4/15.

Could we prove this with a representation? Jacob says, "The whole could be divided into 3 equal parts or 15 equal parts." So there's a whole divided into 3 equal parts to represent our 1/3 and there's the same whole divided into 15 equal parts.

He says, "We have 1/3 shaded which is equivalent to 5/15." So if we put one shape on top of the other, we can see that 1/3 is equal to 5/15.

And now we can subtract our 1/15 and we can see that there are 4/15 remaining.

1/3 minus 1/15 is equal to 4/15.

And we can rewrite that as 5/15 minus 1/15 is equal to 4/15 'cause we know that 1/3 is equivalent to 5/15.

How would you solve the equation this time? Hmm, what can you see? Well, Jacob says, "This time, the whole is a mixed number.

However the fractional parts are the same." We've got 1 1/3 subtract 1/15 this time.

Does that make a difference? Well, we mustn't forget that we've got a whole number there as part of our mixed number.

But 1/3 subtracted 1/15 it's not going to bridge through into our 1 whole.

So we can sort of park the 1 whole and just think about the fractions at the moment.

They're still related.

We can still find a common denominator.

1/3 was equivalent to 5/15.

So we can rewrite our mixed number as 1 5/15 and then we can easily subtract our 1/15.

And 1/15 is far smaller than 1/3 or 5/15.

So we know that our whole is going to stay the same.

So 1 5/15 minus 1/15 is equal to 1 4/15.

Time to check your understanding.

Is this statement true or false? To convert a fraction to a related fraction, it must share the same denominator as the other fraction.

So is that true or false and how can you explain why? Pause the video, have a go.

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

What did you think? Well, it's false, isn't it? Why is it false? Well, the fractions need to be related, and related fractions have denominators where one denominator is a multiple of the other denominator.

They don't have to be the same.

So when they're the same, they're common denominators, aren't they? Aha and here we are, a check about common denominators.

Can you rewrite this equation so that the fractions share a common denominator and then solve it? From our previous check, we can see that these fractions are related.

12 is a multiple of 4.

So pause the video, have a go.

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

So what's our common denominator going to be here? Well, it's going to be twelfths, isn't it? As we've said, 12 is a multiple of 4.

So if we scale our denominator by a factor of 3, we must scale our numerator by a factor of 3.

1/4 is equal to 3/12.

And then, we could substitute 3/12 into our equation, and 3/12 minus 1/12 is equal to 2/12.

Well done if you spotted that.

And time for you to do some practise.

Again, you've got some missing boxes to fill in here.

So in question one, you've got three equations to complete and we've given you the sort of steps in thinking.

So for 1/4 minus 1/8, you're going to think about creating an equivalent fraction with a common denominator substituted in.

And then, complete the equation.

And you're going to do that for 1/3 subtract 1/9 and 1/4 subtract 2/16 as well.

And then for question two, you're going to solve these equations.

But this time, we haven't given you the different stages to think through.

So you've got to do that bit for yourself.

So 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 here we were thinking about the different stages that we go through as we create a common denominator so that we can solve a subtraction equation with fractions.

So we've got related fractions.

8 is a multiple of 4 so we know that we can convert 1/4 into an equivalent number of eighths.

We've got to scale the numerator in denominator by 2.

So 1/4 is equal to 2/8.

We can then substitute 2/8 into our equation.

2/8 minus 1/8 is equal to 1/8.

What about 1/3 minus 1/9? They're related fractions again, 9 is a multiple of 3.

So we know we can create an equivalent fraction to 1/3 with 9 as the denominator.

We're going to scale the numerator and denominator by 3.

So 1/3 is equivalent to 3/9.

We can then substitute that into our equation: 3/9 minus 1/9 is equal to 2/9.

And what about 1/4 minus 2/16? Well, again, they're related fraction.

16 is a multiple of 4.

So this time, we're going to scale our numerator and denominator by 4.

So 4 times 4 is 16.

1 times 4 is 4.

We've got 4 times as many pieces, so we need 4 times as many of them.

And we can substitute 4/16 for 1/4.

4/16 minus 2/16 is equal to 2/16.

Now in question two, you were doing all those stages but you were going to go through them yourself.

So let's have a look: a was 1/4 minus 1/32.

I like saying thirty-twoth because I like the fact that after fifths, we can just say the number of the denominator with the -th on the end.

So one thirty-twoth is the way I like to say it because it gives us that consistency with all fractions.

Unfortunately, not with 1/4.

1/4 minus 1/32, well, 32 is a multiple of 4.

So we'd have 8/32 minus 1/32, which is 7/32.

We've got more thirty-twoth, haven't we? You're gonna have to get used to me saying this.

So 1/8 minus 1/32.

This time, 1/8 is equal to 4/32.

So our difference will be 3/32.

And can you see again what's happening here? We've got another related fraction here.

This time, 1/16 is equal to 2/32.

So our difference will be 1/32.

And look at the next row.

We've still got thirty-twoths.

This time, we've got all eighths.

So we know that 1/8 is equal to 4/32.

This time, we've got mixed numbers as our whole, but that doesn't matter.

We can still think about the related fractional parts and create common denominators.

So we'll have 1 4/32 minus 3/32 which will be 1 1/32.

2 4/32 minus 3/32 which will be 2 1/32.

And then, 2 2/8 this time.

So we know that 1/8 is 4/32.

So 2/8 must be 8/32.

So our difference will be 2 5/32.

8 minus 3 is equal to 5.

What's a lot of thirty-twoths or thirty-secondths you might have called them.

I hope you are successful with all of those.

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

We've been explaining how to subtract related unit fractions.

So what have we discovered? Well, we've learned that when subtracting two fractions with different denominators, you need to find a common denominator.

And identifying related fractions can help with this, looking at those denominators and seeing if one is a multiple of the other.

Oh, and related fractions are fractions where one fraction has a denominator that is a multiple of the other fraction's denominator and spotting those helps us to create those common denominators so that we can subtract and add fractions easily.

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!.