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 explain how to add related unit fractions without a representation or an image.

You may have been thinking about adding fractions recently, and you may have come across related fractions.

Let's see how we're going to use them today in our learning.

So we've got two keywords or phrases really this time.

We've got related fractions and common denominator.

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

So my turn related fractions.

Your turn.

My turn, common denominator.

Your turn.

Excellent.

Let's just double check what those mean so that we can use them accurately during our lesson today.

They're going to be really useful to us.

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

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

The denominator is common to both fractions.

Our lesson has two parts today.

In the first part, we're going to be identifying related fractions, and in the second part we're going to be using related fractions in equations.

So let's make a start on part one.

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

So what fraction of the vegetable patch is being used to grow carrots here? Hmm.

I wonder if you've looked at something similar to this before.

What fractions being used to grow carrots at the vegetable patch? Lucas says, "I think one-third of the patch is being used to grow carrots." Does it look like one out of three equal parts? Well, it might do if we sort of imagine some of those lines disappearing.

It would look like that, wouldn't it? He says, "I can show that by dividing the whole into three equal parts." So there we go.

Just like we said.

If we imagine some of those little lines disappearing, we can see that the carrots take up one of three equal columns in our image of our vegetable patch.

It's now not so easy to see what fraction is onions though, is it? But one-third is taken up by carrots.

What about the onions then? Well, we've sort of got to do the other way, haven't we? We've got to put some more lines in to work out how many equal parts there are in the whole.

Laura says, "I think one-ninth of the patch is used to grow onions." She says, "I can divide the whole into nine equal parts." So she's put in the extra parts, and we can see that the onions are one out of nine equal parts of the whole.

So one-ninth of the whole.

One-third and one-ninth are related fractions.

What do you notice? Do you remember how we described related fractions at the beginning of the lesson? Lucas says, "The denominators are both in the same times table." They're both in the three times table, aren't they? Laura says, "We can say that related fractions have denominators where one denominator is a multiple of the other." Nine is a multiple of three.

Or we could say that three is a factor of nine.

So can we use this information to help us to work out what fraction of the vegetable patch is being used to grow carrots and onions? Lucas says, "To find the total amount of the vegetable patch being used, we need to add the fractions.

But to add the fractions though, we need to rename them so that they're alike." We can't add a third and a ninth at the moment because they're not the same units, are they? Let's have a think about that idea of units being the same.

Here we've got the same unit.

We are counting cats.

We've got three cats plus two cats, which is equal to five cats.

The same unit is there, so we can add those easily.

But if we change two of the cats for dogs, then we've got a problem, haven't we? Well, we might have a problem anyway because cats and dogs don't always get on do they? But we've got a problem in adding them up together.

We haven't got a unit of counting, and we need to have the same unit of count to give a sum or a total.

So we need to change the equation so that we have a common unit.

So what could we call cats and dogs that would give them something in common? Well, we could call them pets, couldn't we? Three pets plus two pets is equal to five pets.

We still know it's three of something and two of something.

But we've changed the unit of count.

We've given them a common unit of count, a common name, a common denominator.

Sometimes you could have more than one unit that you could count with.

So what else could we call them? We could call them animals, couldn't we? So we could say three animals plus two animals is equal to five animals as well.

But whatever unit we choose, it's got to be something that can be common to both of our add-ins.

Both of the things we're adding together.

So let's think about how we can apply that understanding of common units or related fractions to adding our fractions with our vegetable patch.

One-third carrots and one-ninth onions.

And Lucas says, "In this example, we have two different counting units, thirds and ninths, but you can use your understanding of related fractions to create a common denominator." We can give these fractions the same counting unit because the denominators are related.

Nine is a multiple of three.

Three is a factor of nine.

So how can we change one-third plus one-ninth like we did with the cats and the dogs? Well we might know from our work on equivalent fractions that one-third is equivalent to three ninths.

We can rewrite our third in terms of ninths and give it the same unit of count, the common denominator.

And we can think about the numerator and the denominator being scaled up by a factor of three.

Three times three is equal to nine, one times three is equal to three.

We've got three times as many parts, so we need three times as many of them to have the same proportion of the whole.

And if we know that one-third is equivalent to three-ninths, we can substitute it in the equation, so now we have common denominators.

Three-ninths plus one-ninth is equal to four-ninths because now we have the same units of counting.

And we've got a stem sentence here to help us to think about those stages in the way that we worked.

Three-ninths and one-third are related fractions because the denominator nine is a multiple of the other denominator three.

Three-ninths plus one-ninth is equal to four-ninths.

So we can also say that one-third plus one-ninth is equal to four-ninths.

So let's have a check about related fractions.

Can you tick all the related fractions for one-third? Have a look carefully at A, B, C, and D and decide which are related.

Remember, think about the denominators.

Pause the video, have a go.

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

How did you get on? So five-sixths is related to one-third.

The denominator is six, which is a multiple of three.

What about three-eights? That isn't is it? We've got a three, but it's a numerator, and it's the denominator we're interested because that gives us sort of the name of the fraction, the unit of count, and eight is not a multiple of three.

What about six-ninths? Yes, that's right.

Nine is a multiple of three.

The six is as well, but it's not important when we're thinking about related fractions.

It's the denominators.

It's the number of parts in the whole that we're thinking about.

And what about seventeen-twelves? Yes, that is related as well, isn't it? Because 12 is a multiple of three.

Three is a factor of 12.

Well done if you've got all of those.

Another check.

What is it that related fractions have? Is it A, B, C, or D? Pause the video, have a look and when you're ready for some feedback, press play.

Which one was it? It was C, wasn't it? Related fractions have denominators where one denominator is a multiple of another.

The numerators aren't important.

And yes, eventually we'll give them the same denominators.

But related fractions don't have to have the same denominators.

Those are common denominators, aren't they? Things that we create perhaps by using our understanding of related fractions.

Time for you to do some practise and put this into action then.

In question one, you're going to circle the related fractions for one-fifth.

And in question two, you're going to tick the pairs of fractions that are related fractions and use that stem sentence to explain why.

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 which of these fractions are related to a fifth? Remember we're looking at the denominators.

So that is seven-fiftieths, three-tenths, four-twentieths, ten-tenths, three-fifths.

Well five is a multiple of five.

It's the first multiple of five.

So that is related.

They have the common denominator already.

One and seven-tenths is related.

It's a mixed number, but that doesn't matter.

It's the seven-tenths part, the fractional part that we are looking at.

And 10 is a multiple of five, so they are related.

And can you see one more? Yes, twenty four-fifteens.

Again, this is an improper fraction where the numerator is greater than the denominator, but it's still a number of fifteenths.

And fifteenths are related to fifths because 15 is a multiple of 5.

One-sixth and ten-twelfths are not related to one-fifth.

Six is not a multiple of five and 12 is not a multiple of five.

The fact that the numerator is doesn't matter, it's the denominator that's important for related fractions.

And for question two, you had to tick the pair of fractions that are related and explain using the stem sentence.

So again, we are looking for that relationship between the denominators.

So one quarter and four-eights are related.

One quarter and four-eights are related fractions because the denominator eight is a multiple of the other denominator four.

Can we apply that sentence to any other pairs of fractions here? Yes, we can to one-seventh and three-fourteenths.

They're related because 14 is a multiple of seven.

Four-twentieth and one-third are not related.

Three is not a factor of 20.

20 is not a multiple of three.

One-tenth and four-one thousandths are.

They are related fractions because the denominator 1000 is a multiple of the other denominator of 10.

Five-thirds and five-sixths are as well.

Five-thirds is an improper fraction, but that doesn't matter.

They are related fractions because the denominator six is a multiple of the denominator three.

And the same here with our mixed numbers.

They are related fractions because the denominator 12 is a multiple of the denominator four.

What about one and one-sixth and one and two-ninths? They're not related fractions.

They look close to being related fractions.

They're both multiples of three, but nine is not a multiple of six.

So we can't say that by our definition they are related fractions.

Well done if you've got all those right.

And I hope the stem sentence was useful in allowing you to check your reasoning.

And on into the second part of our lesson.

We're going to be using related fractions in equations.

So can you calculate the sum of one-fifth and one-fifteenth? Lucas and Laura have got some ideas to us off.

Lucas says, "Let's convert one of the fractions so that we have a common denominator." Yep, we can do that.

But I think Laura's got some advice for us first.

She says, "Sometimes, it's useful to step back and estimate what we think the answer will be." And that's true whatever numbers we are working with: whole numbers, decimals, fractions or whatever.

Useful to have an estimate in our mind so that we can check to see if we've made a mistake.

So let's have a think about that.

Which fraction is the largest of the two? Well they're both unit fractions, aren't they? So the fraction with the smallest denominator will be the largest fraction.

So one-fifth is larger than one-fifteenth as Lucas says.

So Laura then says, "Is the total going to be bigger or smaller than a half?" What do you think? Well Lucas says, "You need more than two-fifths to make a half." Don't you? If we are thinking about the numerator being half the denominator, well it's difficult 'cause it's an odd number.

We'd need two and a half fifths, wouldn't we? But we're only adding one-fifteenth and that's much smaller than a fifth.

So our sum is going to be less than a half.

Let's have a think about this on a number line then.

We can see that those fractions are related, can't we? So we've got our two number lines here with our fifteenths and our fifths and that helps us again to show how related these fractions are.

So Laura says, "What would this look like on a number line?" "Let's have a look", says Lucas.

Here's one-fifth and here's one-fifteenth.

And we've shown them on the different number lines, but we can see a relationship there, can't we? We can see that those number lines line up the marks line up on them, don't they? So let's put them together and look at that again.

There's one-fifth.

Then what can we say about one-fifth? And there's one-fifteenth.

Laura says, "We can expect our answer to be four-fifteenths." But how are we going to prove that we've got there? So let's think again about calculating this sum.

"To get that four-fifteenths," Laura says, "We need to find a common denominator." But Lucas says, "One-fifth and one-fifteenth are related fractions.

15 is a multiple of five.

But Laura says, "We know we can make the unit of count into fifteens by multiplying our denominator of five by three." So we've got to turn one-fifth into an equivalent fraction representing fifteenths.

So we can see that we've got three times as many in the denominator.

So we're going to need three times as many in the numerator.

And if you remember the number line, the one-fifth and three-fifteenths marks were at the same point.

We've divided our whole into three times as many pieces.

So we need three times as many of them.

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

So one-fifth is equal to three-fifteenths.

And now we could substitute that into our equation.

We've now got three fifteenths plus one-fifteenths and that's equal to four-fifteenths.

Our unit of count is now the same.

Three plus one is equal to four.

This time we're counting in fifteenths.

Time to check your understanding.

Before we add, can we estimate? Will the value of this expression be larger or smaller than a half? Pause the video.

Have a think.

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

What did you think? Well Laura says, "One half is equivalent to two quarters and one-sixteenth is less than one quarter.

So it's going to be less than a half.

And another check.

The next step would be to give a common unit, wouldn't it? To find a common denominator.

So what would the common denominator be to add these fractions together? Pause the video, have a think.

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

What did you think? Well, they're related fractions, aren't they? One quarter and one-sixteenth are related.

16 is a multiple of four.

So we can find an equivalent fraction to one quarter with 16 as the denominator.

We are going to scale the denominator by four.

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

Four times four is 16, one times four is four.

So our common denominator would be sixteenths.

And we can then rewrite that equation as four-sixteenths plus one-sixteenth.

And our sum will be five-sixteenths.

Well done if you got that right.

Time for you to do some practise.

You're going to fill in the missing boxes here.

So we've given you the stages of thinking for these additions.

One half plus one quarter.

We're going to think about how we can use our knowledge of related fractions to create common denominators.

And then we're going to add the fractions.

And then the same for one-third plus one-ninth and one-fifth plus one-twentieth.

So can you fill in the missing boxes to go through those stages of thinking? And then in question two, you are going to do the thinking for yourself.

So you're going to solve the following equations, thinking about related fractions and using common denominators.

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 for question one, we're thinking about the stages that we go through thinking about related fractions and creating common denominators.

So one half plus one quarter while they're related, aren't they? Four is a multiple of two.

So we can create an equivalent fraction to one half which has four as the denominator.

And we can see that we've scaled up the denominator by a factor of two.

So we do the same to the numerator.

One half is equal to two quarters, then we can substitute two quarters for one half because they are equivalent.

Two quarters plus one quarter is equal to three quarters.

What about one-third plus one-ninth? Well, thirds and ninths are related.

Nine is a multiple of three.

We need to scale the numerator and denominator by three.

So one-third is equal to three-ninths.

We can then substitute that into our equation.

Three-ninths plus one-ninth is to four-ninths.

What about one-fifth plus two-twentieth? Well they're related fractions.

20 is a multiple of five.

We multiply our numerator and denominator by four, scale them up by the same factor, and we create the equivalent fraction four-twentieth, and we can substitute that back in our equation.

Did you estimate as you were thinking about this one-fifth plus two-twentieth, they're both much less than a half, aren't they? And in fact, again, we're going to be under a half for our answer.

Four-twentieth plus two-twentieth is equal to six-twentieth, which is indeed under a half.

I hope you estimated the first two as well.

And onto question two, this time you had to go through those stages yourself and think about the relationship between the denominators and how we would use that to create equivalent fractions and common denominators.

Did you do the estimating as well? We're adding unit fractions for these top three, aren't we? So we've got one of everything and the denominators are quite large, so I think everything's going to be well under a half, isn't it? One-sixth plus one-eighteenth? Well sixth that we can convert to eighteenths.

So one-sixth will be three-eighteenths.

Three-eighteenths plus one-eighteenths would give us four-eighteenths.

One-seventh plus one-twenty once or twenty first you might have said it.

Again, 21 is a multiple of seven.

So we can convert three-sevenths into three-twenty ones and that will give us an answer of four-twenty ones again.

Oh, can you see a pattern here emerging? Is it the same for this one? What's the relationship between the denominators nine and 27? Again, it's times three, isn't it? So one-ninth we can rewrite as three- twenty sevenths added to one- twenty sevenths, gives us four-twenty sevenths again.

And let's have a look at the bottom row.

Well we're looking at ninths and twenty sevenths all the time here, aren't we? But this time we've got some mixed numbers.

Now the whole of our mixed numbers we don't need to worry about.

And because we've estimated, we know that our fractional parts are not going to bridge through another whole.

So we can sort of not ignore the whole numbers, but we know that in the first one there's going to be a two and and the second one, two and and in the last one 12 and.

And we can then just focus on the fractions.

But we can only do that because we've estimated, and we know that our fractions are going to give us a sum of less than one in all those cases.

So one and one-ninth we can rewrite as one and three-twenty sevenths.

So that will give us an answer of two and four-twenty sevenths.

One and two-ninths we can rewrite as one and six-twenty sevenths, so that will give us a total of two and eight-twenty sevenths.

And 10 and one-ninth we can rewrite as 10 and three-twenty sevenths.

So added to two and two-twenty sevenths will give us a total of 12 and five-twenty sevenths.

I hope you were successful with all of those and used your knowledge of related fractions to create those common denominators.

Well done.

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

We've been explaining how to add related unit fractions without a representation or image.

We've learned that when adding two fractions with different denominators, you need to find a common denominator.

And identifying related fractions can help with this.

And we've also learned that related fractions are fractions where one fraction has a denominator that is a multiple of the other fractions denominator.

Thank you for all your hard work and your mathematical thinking, and I hope I get to work with you again soon.

Bye-bye.