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 with and without an image how to add and subtract related non unit fractions.

Okay, so we're going to be using some images and then not using images.

We're going to be adding and subtracting related fractions, but this time they're non unit fractions.

So let's have a look at what that's going to be like.

We've got one key word and that's common denominator.

So I'll take my turn and then you can take your turn.

Are you ready? My turn, common denominator, your turn.

I'm sure you're familiar with that, but just let's remind ourself of the definition because we're going to be using this phrase a lot in our lesson today.

So when two or more fractions share the same denominator, you can say that they have a common denominator and we know that common denominators can really help us when we're adding and subtracting fractions.

There are two parts to our lesson today.

In the first part, we're going to be calculating with non-unit fractions, and in the second part we're going to be thinking about non-unit fractions beyond a whole.

So thinking about mixed numbers.

So if you're ready, let's start with part one and we've got Laura and Izzy helping us in our lesson today.

So what fraction of the whole is shaded red? What fraction of the whole is shaded blue and how much of the whole is shaded altogether? And we've got an image there.

You might want to spend a little time having a look at that before Izzy and Laura share their thinking.

So Laura's looking at the section that's got the three blue smaller squares in and she says, "If we make this section the same size as the others, then each red section would be one quarter of the whole." So she's ignoring the blue squares and just thinking about that area as a whole, so we can see that each red square is one out of four equal parts, one quarter of the whole.

And Izzy says, "There are three red parts shaded.

So this is three quarters." So the red part can be represented by the fraction three quarters.

Three quarters of the whole is shaded red.

Let's put those blue squares back in.

What fraction of the whole is shaded blue then? Well Laura says, "We can also divide up the whole into the same size parts as the blue squares." So can you see in each of those quarters we've got four smaller squares.

So if we repeat that across the whole shape, now we can see that each blue part is one 16th of the whole.

There are 16 small squares and one of the blue ones would represent one one 16th of the whole.

And Izzy says, again, "There are three blue parts shaded.

So this is three 16ths of the whole." But we want to know how much of the whole is shaded altogether.

So what do we need to do with these fractions? Ah, we know that we need to add the fractions together to find the total amount shaded.

So we've got three quarters, which is red and three 16ths, which is blue.

And we need to find the sum, the total, but we can't add them yet can we? Because they don't have a common denominator.

We've got quarters and we've got 16ths and we can't add those until we can count them in the same unit of counting, the same denominator.

Izzy says, "Let's go back to the image we had before." Oh well we have the whole divided up into the smaller squares, 16 smaller squares.

Now what can you see? Laura says, "If we look at the red squares now we can see that each one is also one 16th." So how many of those smaller red squares have we got altogether? Well, each one quarter is equivalent to four 16ths.

So we have three of the quarters.

So we have 12 16ths altogether that's a shaded red.

So how much of the whole is shaded altogether? Well, we need to do something with that three quarters to turn it into 16ths.

We could count the individual squares, but how does that work when we look at the fraction itself? So we're trying to create a fraction equivalent to 16ths.

Well, if we scale up the numerator and denominator both by a factor of four, we will do that.

Four times four is equal to 16 and four times three is equal to 12.

So three quarters is equal to 12 16ths.

And we could see that in the shape, we could have counted the red squares, either three lots of four or counted them individually.

Three quarters is equivalent to 12 16th.

So Izzy says, "Let's replace this in our equation." We know that those fractions are equivalent, so we can substitute three quarters for 12 16ths.

Now it's easy to add them together because we've got the same unit of count.

We've got 12 16ths, add three 16ths, and that is equal to 15 16ths.

And we can also see that that's right because we've only got one of those small squares which is not shaded, the whole would be 16 16ths.

So if one of them isn't, we must have 15 16ths shaded altogether.

And we can also represent this on a number line.

And we can see here really clearly that quarters and 16ths are related fractions, aren't they? 16 is a multiple of four.

So you can see that there are points where the quarters are marked and they line up exactly with an equivalent number of 16ths.

So we're starting on three quarters, which was our red squares shaded and that's equivalent to 12 16ths.

And then we were adding on the three 16ths that were blue.

So if we add that from the three quarters or from the 12 16ths, we can see that we can land on 15 16ths.

Then that was our sum.

What if our calculation was this instead? Three quarters, subtract three 16ths.

Well, Laura says, "We can apply our understanding that we still need a common denominator for both fractions." Three quarters, subtract three 16ths.

We can't do that because we're not thinking in the same units of counting.

And we know from our previous calculation that three quarters is equivalent to 12 16ths.

We scale the numerator and denominator by a factor of four to create the equivalent fraction.

So we can substitute three quarters for 12 16ths in our equation.

Now we can subtract the three 16ths.

12 16ths subtract three 16ths is equal to nine 16ths.

Time to check your understanding.

Can you write an equation to represent the total amount shaded in red and blue in this image? Think about the fractions you're going to use.

Can you create equivalent fractions or can you represent them as fractions with different denominators? Pause the video, have a go.

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

(no audio) How did you get on? How did you think about the shape? Well, if we think about the blue parts, we can see that there would be four equal rows there and we've got three of them shaded.

So three quarters of the shape is blue.

But if we imagined each of those bars divided into three equal parts, like in the bar with the two reds, we would have three lots of four, which is equal to 12.

So those little squares represent one 12th of the whole and two of them are shaded.

So we can represent the total amount shaded as three quarters plus two 12ths.

Can you now calculate the total area shaded in red and blue in the shape? So you're going to need to think about your denominators here.

Pause the video, have a go.

And when you're ready for the answer and some feedback, press Play.

(no audio) How did you get on? Haha.

So there we go.

That's what we talked about in the first check.

If we add those extra lines to divide the whole shape into 12 pieces, we can see that our three quarters is equivalent to nine 12ths.

Let's just look at that again.

Three quarters, if we scale up the numerator and denominator by a factor of three, we've made each of those quarters into three times as many pieces.

We've got three times as many of them, three quarters is equivalent to nine 12ths.

And now we can add our two 12ths and we get a sum of 11 12ths.

And again, if we look at the whole shape, we've only got one of those 12ths that isn't shaded, we'd have 12 12ths in the whole, one of them isn't shaded.

So 11 12ths must be shaded.

Time for you to do some practise.

So in question one, you are just going to match the image to each equation, but you can see we've got six equations and only three images.

So each image must match to two equations and then you're going to give an answer to each of these equations.

So pause the video, have a go at questions one and two.

And when you're ready for the answer and some feedback, press Play.

(no audio) How did you get on? So how was your matching? Let's think about one third plus two 12ths.

So we're looking for a shape that we can see divided into three equal parts.

And then if we divided it into 12 equal parts, two of them will be shaded.

Well that matches that top image, doesn't it? So which other equation represents that image? Well, we've got 12ths there, haven't we? What is one third as 12ths? Well, if we look at that bottom row, we could see that those 12ths, so two red squares, there would be four of them in one third.

So the other equation that matches is four 12ths plus two 12ths.

Let's look at the middle image.

This time let's have a look what does that red one look like? Well, it's one column and if we use that top row to help us, we could see that there would be four columns.

So that must be one quarter.

So one quarter plus three 12ths.

So we can see three of the smaller parts shaded.

And again, those top two images would both have 12ths in them if we put all the lines in.

But what would our quarter be represented as 12ths? Well, we can sort of see that there are three rows there.

And so there will be three 12ths in each quarter.

So that's that top equation.

Three 12ths plus three 12ths.

Oh, there we go.

Three 12ths plus three 12ths.

And if we imagine rotating that red shape round, it would be the same size as the three blue squares, wouldn't it? So what about the bottom image? Well, I can clearly see half of it shaded.

So we must be looking at one half plus two 12ths.

So that must mean that in the other half there are six parts aren't there? So that would mean that our half is equal to six 12ths.

So that matches obviously our last equation, six 12ths plus two 12ths.

So now let's think about giving the answer to each calculation.

Can you see something that's the same here? We're starting with nine 12ths as our whole each time, aren't we? And all of the fractions that we're subtracting.

Two, three, four, and six are all factors of 12.

So we're going to be able to create equivalent fractions.

So one half in 12ths must be six 12ths because the numerator must be half the value of the denominator.

So nine 12ths subtract six 12ths is equal to three 12ths.

So what about the second one? Nine 12ths subtract one third.

Well a third is related to 12ths.

We'd need four times as many parts, four times as many in the whole.

So that would be four 12ths, nine 12ths subtract four 12ths is equal to five 12ths.

What about one quarter? How do we express that in 12ths? Well, we'd need three times as many pieces, so three times as many in the whole.

So one quarter is equivalent to three 12ths.

So nine 12ths subtract three 12ths is equal to six 12ths.

Oh, we know the six 12ths is equal to a half, don't we? Then we've got nine 12ths subtract one 6th well one 6th is equivalent to two 12ths, scale the numerator and denominator up by a factor of two.

Nine 12ths subtract two 12ths is equal to seven 12ths.

And then we've got two 6ths.

Well if we know that one 6th is two 12ths, then two 6ths must be four 12ths.

Nine 12ths subtract four 12ths is equal to five 12ths.

Ah, can you see another answer that's five 12ths? That's right, two 6ths is equivalent to one third as well.

So we could have used that answer too to help us.

And did you notice that your answers increased each time? Three 12ths, five 12ths, six 12ths and seven 12ths apart from the last one.

We just talked about the fact that the last one is equivalent to five 12ths because two 6ths is equal to one third.

But can you see that each time we were subtracting a smaller value unit fraction? So therefore our answer was increasing each time.

Well done if you spotted that.

And the second part of two, what about the answers to these calculations? Well, this time we're adding and this time we're always adding four 24ths.

So let's have a look at the fractions we're adding them to.

Quarters, thirds, 6ths, 8ths and 12ths.

They're all factors of 24.

So we're going to be able to create equivalent fractions.

Two quarters is equal to a half, isn't it? So that must be 12 24ths.

So 12 24ths plus four, 24ths is 16 24ths.

Two thirds, what's that equivalent to in 24ths? Well three multiplied by eight is equal to 24.

So two multiplied by eight is 16.

So it'll be 16 24ths plus four 24ths, which is equal to 20 24ths.

What's two 6ths going to be equivalent to as 24ths? Well, six times four is equal to 24, so it must be eight 24ths.

Eight 24ths plus four 24ths is equal to 12 24ths.

Oh, which is equal to a half.

12 is half of 24.

What about 8ths? Three times eight is equal to 24.

So six 24ths must be equal to two 8ths and six 24ths plus four 24ths is equal to 10 24ths.

And then finally 12ths.

Well, 12 is a factor of 24.

We multiply it by two to equal 24.

So we must need four 24ths to be equal to two 12ths.

Four 24ths plus four 24ths is equal to eight 24ths.

We already had two 12ths, so we could have simplified our four 24ths to two 12ths and our answer would've been four 12ths.

And on into the second part of our lesson.

This time we're going to be thinking about non-unit fractions bigger than a whole.

So mixed numbers.

So what fraction of the whole is shaded blue and what fraction of the whole is shaded red and how much is shaded altogether? Well we can see we've sort of got two wholes here.

We've got one whole that's blue, and then we've got another whole that's part blue and part red.

So we're thinking about fractions beyond a whole.

So we've got one and something haven't we? So let's have a think about how that's going to look.

Laura says, "The rectangle on the left represents one whole," and that's blue.

So we know we've got one whole and "There's another part shaded blue in the second whole." So our blue part is going to be represented by a mixed number, one and a fraction.

Izzy's thinking about breaking down that second whole.

And let's think about breaking it down into three equal parts.

Can you see three equal rows? And we've just removed those extra lines to show our three equal rows.

So she says if we think about it like that, "We would have one whole and two thirds shaded in blue." So the blue shaded area can be represented as one and two thirds.

What about the red area? Laura says, "How will we know how much is shaded in red?" And Izzy says, "Well, we could break the whole into equal parts that are the same as the smallest red part." Can you see the smallest red part on the right? And you can sort of see that we are going to get an extra line there, aren't we, to separate the biggest red part into two.

So let's put those lines in.

Aha, how many have we got in the whole now? She says, "All these parts are now equal in size," and we've got 12 of them.

"So we can say that three 12ths of the whole is red." "Oh yes.

So three 12ths have been shaded in red." That's right.

We've got three rows again, but we've got four in each row this time.

So we've got 12 parts altogether and three of them are red.

So how much of the whole is shaded altogether? Ah, Laura says, "Now we can add the parts together." So we've got to do one and two thirds plus three 12ths.

Izzy says that we need to convert the fractions to have a common denominator.

We can't add thirds and 12ths as they are.

And she says, "We can do this one of two ways." So let's have a look.

We could convert the mixed number into an improper fraction first.

So how would we do that? One whole is equivalent to three thirds and we've got another two thirds, so that would be five thirds.

So we could also represent the blue area as five thirds.

Now we can convert the improper fraction so that it has a common denominator with the three 12ths.

So we need to scale everything up by a factor of four.

Five thirds is equivalent to 20 12ths.

So we can then replace that in our equation.

And we know that our sum is now going to be 20 12ths plus three 12ths, which is equal to 23 12ths.

Now let's think about this in 12ths.

So 24 12ths would be two wholes.

So this must be one whole and 11 12ths.

And we can convert back to a mixed number.

But did you notice we didn't bridge through another whole, did we? We had one and two thirds to begin with and we've got one and 11 12ths.

Izzy says, "We could also just convert the fractional parts of the mixed number so it has a common denominator." So two thirds is going to be equivalent to eight 12ths.

We scale the numerator and denominator up by a factor of four.

So then we can replace one and two thirds with one and eight 12ths.

Eight 12ths plus three 12ths is equal to 11 12ths.

And we keep our one.

So our sum is now one and 11 12ths.

Because we didn't have to bridge through another whole that was a really efficient strategy this time.

But if we had to bridge through a whole then it's possible that converting to an improper fraction might have been the best approach to take.

So which strategy would you adopt for this calculation? Let's have a look at the numbers.

We've got one and two thirds and we're subtracting three 12ths.

Well, two thirds is bigger than a half, isn't it? And three 12ths is less than a half.

So we are not going to bridge back into that whole one, are we? I think I agree with Izzy.

"I prefer just to convert the fractional part of the mixed number to give it a common denominator," with our 12ths that we're subtracting.

So two thirds.

Well if we scale up the numerator and denominator by a factor of four, we create an equivalent fraction of eight 12ths.

So now we've got one and eight 12ths, subtract three 12ths, and that's going to be equal to one and five 12ths.

Over to you to check your understanding, can you solve one and five 7ths subtract eight 14ths? Which strategy will you use? Maybe you could do it both ways and see which you think is the most efficient.

Pause the video, have a go.

And when you're ready for the answer and some feedback, press Play.

(no audio) How did you get on? Well, five 7ths and eight 14ths are related fractions.

So let's think about just converting the five 7ths part of our mixed number into 14ths.

Well, we need to scale the numerator and denominator by a factor of two.

So five 7ths is equal to 10 14ths.

So one and five 7ths becomes one and 10 14ths.

So one and 10 14ths subtract eight 14ths is going to be one and two 14ths.

So we can also say that one and five 7ths subtract eight 14ths is equal to one and two 14ths.

Time for you to do some practise.

You are going to solve the following equations.

So you're going to do some subtractions here, thinking about how much of this are you going to convert, are you going to convert your mixed numbers into improper fractions and then common denominators or can you estimate that we are not going to bridge through the whole one and therefore you can just think about converting the fraction? And in question two you've got some more to solve as well.

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.

(no audio) How did you get on? So in A we had six 12ths subtract two 6ths, one and six 12th subtract two 6ths and two and six 12th subtract two 6ths.

Did you notice that? So let's have a think about six 12ths, subtract two 6ths.

How are we going to rate common denominators here? I can see two ways to do this.

We could have thought of two 6ths as 12ths and two 6ths is equivalent to four 12ths.

So six 12ths subtract four 12ths is equal to two 12ths, but we could have converted our six 12ths into 6ths and that would've been three 6ths.

The three 6ths subtract two 6ths is equal to one 6th and one 6th is equal to two 12ths.

We are thinking in 12ths though this time.

What about when we've got one and six 12ths subtract two 6ths? Well, we're going to end up with one and two 12ths, aren't we? Well our fractional parts have not changed.

All we've done is added in a one.

So can you predict what the answer to the third one is going to be? That's right, two and two 12ths.

We didn't actually need to worry about the fractions because we knew that we had the fractional answer from our first equation.

And so did you notice that the minuend increased by one each time? So that's our whole, our starting number.

The subtrahend, the number we were subtracting remained the same.

So the difference also had to increase by one.

If we make our whole one bigger and one of our parts is the same, then the other part must be one bigger.

And I wonder if you did notice that you could also convert the 12ths into 6ths to create a common denominator.

What about question two? This time we've got mixed number subtract a mixed number, but we've got two and six 12ths and we're subtracting one and two 6ths.

So two 6ths is going to be smaller than six 12ths.

Six 12ths is a half, isn't it? Two 6ths is less than a half.

So if we convert our two 6ths to 12ths, that's four 12ths.

So six 12ths, subtract four 12ths is two 12ths, and our two subtract one is one.

So the difference is one and two 12ths.

Can you have a look at what's changed in the second one? We've still got two and six 12ths.

This time we're subtracting two and two sixths, so we're subtracting an extra one.

So this time our answer will just be two 12ths.

Our whole is the same, but we've subtracted an extra one.

So one of our parts is one bigger.

So our other part must be one smaller.

Two and six 12ths subtract two and two 6ths is equal to two 12ths.

Can you think of some other equations that are equivalent to two 12ths? Lots of different ideas here.

So you could have had seven and six 12ths subtract seven and two 6ths is equal to two 12ths.

You could have followed the pattern of our second equation.

There are lots of ways you could have found a difference of two 12ths.

Did you ensure that the whole number was the same? Then you could have had something and six 12ths subtract the same something and two 6ths and we could have had an infinite number of answers.

Let's look at question two.

Can we spot anything here? Well, for A, we are subtracting two 16ths each time.

What's two 16ths the same as? It's the same as one 8th, isn't it? So we could create equivalent fractions with a denominator of 16 or we could create equivalent fractions with a denominator of eight.

We stuck with 16ths here, so we would've converted our five 8ths to 10 16ths.

So one and 10 16ths subtract two 16ths is one and eight 16ths.

Four 8ths is equivalent to eight 16ths, it's the same as a half.

So one and eight 16ths subtract two 16ths is one and six 16ths.

And one and three 8ths subtract two 16ths.

Three 8ths is equivalent to six 16ths.

So one and six 16ths subtract two 16ths is one and four 16ths.

But you could have thought about those as 8ths.

Two 16ths is one 8th, so one and five 8ths subtract one 8th is equal to one and four 8ths.

One and four 8ths subtract one 8th is equal to one and three 8ths and one and three 8ths subtract one 8th is equal to one and two 8ths.

And you can see that our numerators are decreasing by two each time.

And two 16ths is equivalent to one 8th.

So in B, we were starting with one and five 8ths this time and we were adding two 16ths, two 24ths and two 32ths this time.

So we can't just think about 8ths, although we could for the first one.

One and five 8ths plus one 8th would be equal to one and six 8ths and one in six 8ths is equivalent to one and 12 16ths.

Now we've got to think about changing our eights into 24ths here.

So we can see that we can multiply by three.

So that would be one and 15 24ths plus two 24ths, one and 17 24ths.

And this time we've gotta think about 8ths becoming 32ths.

Well that's multiplied by four.

So we can scale up by four one and 20 32ths plus two 32ths is one and 22 32ths.

And did you notice that you could have converted the 16ths into 8ths in A and the first one of B as well and created a common denominator that way? We went with 16ths, but you could have gone with 8ths.

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

We've been explaining with and without an image how to add and subtract related non-unit fractions.

We've learned that you can use knowledge of common multiples to convert fractions into an equivalent with a common denominator.

And also that you can sometimes scale a fraction down to create a common denominator rather than always scale a fraction up.

Fractions need to have the same denominator if you are to add and subtract them.

We have to have the same unit of count.

And when adding or subtracting fractions or mixed numbers with the same denominators, you just add or subtract the numerators.

Once we've got that constant unit of count, we can just think about how many of each of that unit we've got.

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.