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 comparing fractions.

Have you done lots of work on fractions, I wonder.

I expect you've heard of things like common denominators and maybe even common numerators.

Well let's have a look at all the different ways that we can think about comparing fractions.

If you're ready to make a start, let's get on with it.

So in this lesson, we're going to be looking at how we can use equivalent fractions with common denominators to compare non-related fractions.

So those are fractions that look as though they'd be really difficult to compare, but when we can find a common denominator to work with them, we can compare them more easily.

So if you're ready to make start, let's get going.

We've got three key words or key phrases in our lesson today.

We've got portion, common denominator, and related fractions.

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

So are you ready? My turn, portion, your turn.

My turn, common denominator, your turn.

My turn, related fractions, your turn.

Excellent.

Possibly words that you know already.

May be the first one, not necessarily in a mathematical sense.

We're not talking about portions of chips today I'm afraid.

Let's have a look at what they mean.

So a portion is a part of a whole.

And we'll hear the word portion and proportion perhaps in our lesson today.

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

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

And related fractions are where the denominator of one fraction is a multiple of the denominator of the other fraction.

Worth looking out for because they make comparing fractions and finding common denominators quite easy.

So look out for all of those words.

Think about what they mean.

They're gonna be really useful to us in our lesson today.

There are two parts to our lesson today.

In the first part, we're going to be comparing with common denominators.

And in the second part, we're going to be defining a comparison.

And we've got Laura, Jacob, and Jun in the lesson with us today helping us out.

Laura and Jacob have been making a tiling pattern in art.

How much of each pattern has been tiled? Laura says, I have used 15 tiles and you have used 14 tiles.

So I've used more and I've tiled more, she says.

Well, that's right, says Jacob, but I think we can reason this more precisely.

He says, we can say that I have tiled 14 out of 20 tiles.

And we can represent that with a fraction, can't we? 14/20.

So 14/20 of his patent has been tiled.

What about Laura's? Okay, she says.

So I can say that I've tiled 15 out of 20 tiles.

15/20 of her pattern is complete with tiles.

And Jacob says, we can say that 15/20 is greater than 14/20.

So when the denominators are the same, Laura says, the fraction with the largest numerator is the largest fraction.

And Jacob says, we can see that 15/20 is greater than 14/20.

So he can put that sign in to show the comparison.

15/20 is greater than 14/20 because the numerator is greater, the denominators are the same.

Jacob says, I think we can also say that you have tiled 3/4 of the pattern.

Can you see that? Can you see that her fraction of tiles completed is equal to 3/4? And Jacob says he has tiled 7/10 of his pattern.

Oh, let's have a look and see if we can see those.

Do you agree with them? Laura says, if we divide my whole into four equal parts, we can see that three of the parts are tiled.

So they're slightly odd looking parts, but can you see there's sort of five tiled bits in each of her quarters.

So one, two, three of them are tiled and one of them is still blank.

What about Jacob's shape then? Ah, so he's made 10 equal parts of his.

And we can see that three of them are not tiled.

So that means that seven of them must be 7/10.

So we know that those are now equivalent fractions, but when you look at the fractions, which set of fractions would you prefer to compare? Well, I think I agree with Laura.

It's a lot easier to compare the fractions when the denominators are the same.

Jacob says, I agree.

15/20 is easily seen to be greater than 14/20.

So 3/4 must be greater than 7/10.

That doesn't look as obvious, does it? But we know that 3/4 is equivalent to 15/20 and 7/10 is equivalent to 14/20.

So we know that those are equivalent fractions.

We can compare those ones easily.

So therefore, we can make the same deduction about 3/4 being greater than 7/10.

How would we place these fractions on a number line? Well, there's our 3/4 and there's our line of 10ths.

So we had 3/4 and 7/10 to begin with, didn't we? But we found a common denominator of 20ths.

And if you look up those lines, you can see that there are certain points where the 20ths and the quarters match up.

So 1/4 is equal to 5/20, 2/4 is equal to 10/20, and 3/4 is equal to 15/20.

And we can also see where the 10ths line up with 20ths as well.

So there's 3/4 and there's 7/10.

Where do they sit on that number line of 20ths? There we go.

If we move those number lines down, we could see that 7/10 would be equivalent to 14/20 and 3/4 would be equivalent to 15/20.

There we go.

Those two number lines have gone together.

Well, the quarters and the 10ths have.

They only really line up at the halfway point, don't they? 2/4 is equal to 5/10.

They're both equal to 1/2.

When we bring all three lines together, we can see that every 10th has an equivalent in 20ths, and every quarter has an equivalent in 20ths.

So we can see that we've now formed those common denominators.

We can express 7/10 as 14/20 and 3/4 as 15/20 and make that direct comparison.

Time to check your understanding.

Can you compare 4/12 and 3/12? And can you use a greater than, less than, or equal symbol to go in the circle? Pause the video, have a go.

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

How did you get on? Did you realise that we had common denominators already? We were thinking about 12ths.

So 4/12 has got to be greater than 3/12.

When the denominators are the same, the fraction with the largest numerator is the largest fraction.

And time for another check.

Can you place 2/3 on this number line? So what is 2/3 equivalent to in 15ths and can you place it on the number line? Pause the video, have a go.

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

How did you get on? Well, we were creating a fraction with a denominator of 15.

So we need to scale up our 2/3.

The numerator and the denominator by a factor of 5.

3 times 5 is equal to 15, so 2 times 5 is equal to 10.

So we've got 5 times as many parts in our whole, and we need 5 times as many of those parts for the fraction to have the same value.

So 2/3 is equivalent to 10/15 and they would sit at the same position on the number line.

Time for you to do some practise.

In question one, you're going to compare the following fractions and you're going to use a greater than, less than, or equal symbol to go in between those fractions to compare them.

In question two, you've got two fractions to compare and you're going to place them on the number line.

Why have we chosen that number line, do you think? And for B, again, you're going to compare those by using this number line.

And for C, comparing these two fractions using the number line.

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 let's look at these ones.

Did you notice that there was something common to all of them? So in A, we had a common denominator.

So we had 5/7 and 6/7.

When the denominators are the same, that the largest numerator gives us the largest fraction.

So 5/7 is less than 6/7.

In B, we had a common denominator of 14, so we were thinking about 14ths.

And 12/14 is less than 13/14.

In C, we had a common denominator of 6.

So we are thinking about 6.

This time though we had improper fractions, but the same thinking works.

The greater the numerator, the greater the size of the fraction when the denominators are the same.

So 8/6 is greater than 7/6.

And it still holds true when we've got our mixed numbers as well.

This time our one whole part was the same in each so we're just comparing the fractions.

And we have 3/5 and 4/5.

So 1 3/5 is less than 1 4/5.

Well done if you've got those right.

Now we've got two fractions with different denominators.

They're unrelated fractions, aren't they? 3/5 and 2/3.

5 is not a multiple of 3 and 3 is not a factor of 5.

In fact, 5 and 3 are both prime numbers, aren't they? So what could we do to compare these fractions? Well, we've got a clue there.

We've got a number line of 15ths.

So we can convert these into equivalent fractions.

3/5 is equivalent to 9/15.

Let's look at that again.

We scale up the denominator by a factor of 3, we have to scale the numerator by a factor of 3.

5 times 3 is 15, 3 times 3 is equal to 9, so the fraction is equivalent to 9/15.

What about 2/3 as 15ths? Well, this time we're scaling up by a factor of 5.

3 times 5 is equal to 15 and 2 times 5 is equal to 10, so 2/3 is equal to 10/15.

So there's 3/5 and there's 2/3.

They're close together, aren't they? But we can see that 3/5 is slightly smaller than 2/3.

9/15 is slightly smaller than 10/15.

And what's about the next two fractions? We've got 1/4 and 2/6.

Well, they're both even numbers, but they're not related are they? One is not a factor or a multiple of the other.

But we can see that if we've got quarters and 6ths, then both of them share a common multiple of 12 so we can convert them to 12ths.

So 1/4 is equal to 3/12 and 2/6 is equal to 4/12.

So there's 1/4 and there's 2/6.

So we can see that 1/4 or 3/12 is slightly less than 2/6 or 4/12.

And finally for C, we had 3/4 and 4/5.

Oh, that's interesting, isn't it? I can think of another way I might compare these ones, but let's have a look at those common denominators.

Again, they're unrelated fractions, but they do share a common multiple of 20.

So we can convert both fractions into 20ths.

4 times 5 is equal to 20.

So 3 times 5 is equal to 15, so 15/20.

And 5 times 4 is equal to 20, 4 times 4 is equal to 16, so 16/20.

So 3/4 is equivalent to 15/20, 4/5 is equivalent to 16/20, so we can see that 3/4 is slightly less than 4/5.

We might also have thought about the fact that they were both one of their parts away from a whole.

3/4 is 1/4 away from a whole and 4/5 is 1/5 away from a whole.

1/5 is smaller than 1/4, so therefore 4/5 must be closer to 1 and therefore greater.

Another way of thinking about it.

And onto part two of our lesson where we're thinking about defining a comparison.

So who has tiled the larger portion of the whole? So the larger part of the whole.

So we can still see one tile pattern with 3/4 covered and one tile pattern with 9/10 covered.

Jun says, I think you've tiled the largest portion of the whole because your tiles are larger.

Laura says, I disagree.

I think I've tiled a larger area, but not a larger portion of the whole.

Hmm.

Let's look at what our fractions mean in a bit of detail, she says.

So a fraction represents the relationship between a whole and a part.

The whole is made up of 4 parts in her fraction and 3 of them have been tiled.

And therefore he says, my whole has 10 parts even though they're smaller and 9 them have been tiled.

That means I've tiled a larger portion of my whole than you have of your whole.

And we might call that a larger proportion of the whole as well.

So a fraction might not help us to find out who's tiled the largest area if they represent different sized wholes, says Laura.

And that's something that's really important.

When we are comparing fractions and parts of wholes, we've got to think about is the whole the same? But they can tell us the portion of the whole that has been used or in this case tiled.

So 3/4 of Laura's shape is tiled and 9/10 of Jun's shape is tiled.

But that doesn't tell us who has tiled the largest area because the wholes are not the same size.

We can say though that Jun has tiled a larger portion or a larger proportion of his whole than Laura has.

Ah, do you see what's happened now? Laura says, however, if the wholes are the same size, we can convert to a common denominator to compare the area that has been tiled.

Now we're comparing two wholes of the same size.

And now we can convert to a common denominator.

Did you see what happened there? We put extra lines in.

So now each of those wholes is divided into the same number of small parts.

So there are 20 parts in each whole now.

So our common denominator of 20ths.

3/4 is equal to or we scaled up the denominator by a factor of 5, so we've got to scale the numerator up by a factor of 5.

So 3/4 is equal to 15/20.

So 15/20 of Laura's shape has been tiled.

What about Jun's shape? Well, he had 9/10.

And can you see that in each of those 10ths there are two little 20ths.

So we've scaled everything up by a factor of 2.

He's got 18/20, two times as many parts in the whole.

So he needs two times as many of the smaller parts for the same portion of the whole to be shaded or to be tiled in this case.

So 9/10 is equal to 18/20.

Now because we've made your whole the same size, she says, the fractions show us that you would've tiled a larger area because 18/20 is greater than 15/20.

But when the wholes were not the same size, we couldn't compare the areas because we weren't comparing the same wholes.

Two other pupils in the class have tiled the following proportion of their whole, 1/3 and 3/8.

Laura says, firstly, if we are to compare who has tiled the greatest area, we need to assume that we are comparing the same sized wholes.

So they started off with the same size board to put their tiles on.

And Jacob says, if that's the case, we can convert to a common denominator.

Jacob says, we can find a common denominator easily by multiplying the denominators together.

We can and that will always find a common denominator, but it might not always be the lowest common denominator that we can find.

3 times 8 is 24, so we can definitely use 24.

So if we scale up our numerator and denominator by a factor of 3, 3/8 is equivalent to 9/24.

And if we scale the numerator and denominator of 1/3 by 8, then 1/3 is equal to 8/24.

And now we can compare them.

8/24 and 9/24.

And we can see the 8/24 is less than 9/24.

So the 1/3 area was a smaller area than the 3/8.

How could you find out who tiled the greater area from these fractions? Do we need to do 6 times 12 this time? So once again, we'll assume that the wholes are the same size says Laura, otherwise we're not gonna be able to compare the areas these represent.

Ah, well done Jacob.

These are related fractions.

6 is a factor of 12, 12 is a multiple of 6, so we only need to convert one of the denominators to make them common.

We can convert 1/6 so it has 12 parts.

We couldn't simplify the 12ths because it's an odd number of 12ths and we'd have needed to scale down by a factor of 2.

We could have done, but it doesn't make the fraction look very ordinary, does it? So let's convert our 1/6 into 12ths.

So we're going to scale up by a factor of 2.

So 1/6 is equal to 2/12.

And then we can replace that.

So now we can see that 2/12 is less than 3/12.

So 3/12 is greater than 2/12 and greater therefore than 1/6.

Time to check your understanding.

True or false? Laura has shaded a greater proportion of her rectangle than Jacob has.

Do you think it's true or false and why? Pause the video, have a go.

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

How did you get on? Well, it's false, isn't it? A fraction describes the portion of the part in comparison to its whole.

Even though the area Laura shaded is greater, she has shaded a lesser proportion of the whole in comparison to Jacob.

She's got two parts out of hers that are unshaded.

Jacob's only got one part out of his.

Because we're only thinking about the portion or the proportion of the whole, we can compare these even though the wholes are not the same.

We couldn't make a comparison about the area though because the wholes are not the same.

Time for another check.

Can you compare these two fractions, 2/5 and 1/4? Pause the video, have a go.

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

What did you think? Laura says a common denominator of 5ths and quarters, with 5 and 4, would be 20.

That is the first common multiple we find when we list the multiples of 4 and the multiples of 5.

So we can convert both of those to fractions with a denominator of 20.

We're not talking about areas or anything here.

We're talking about these fractions as numbers so we can compare them.

They're effectively fractions of one.

So 2/5 is equal to 8/20.

And 1/4 is equal to 5/20.

So 8/20 is greater than 5/20.

So 2/5 is greater than 1/4.

Time for you to do some practise now.

So in question one, you're going to look at these and you're going to tick any pairs of fractions whose area can be compared.

So you've got A, B, C, and D.

Which ones can we compare the area of? In question two, you're going to compare the following fractions and put in a greater than, less than, or equals symbol.

Think carefully about whether you need to use a common denominator or is there another way you can think about comparing the fractions? And for question three, you are going to order these fractions from the largest to the smallest.

So pause the video, have a go at the three questions.

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

How did you get on? So in the first one, you were ticking any pairs of fractions whose area can be compared? Well in A, the two shapes have a different area, so they can't be can they? In B, those two shapes have the same area.

So we could compare the areas of the shapes in B.

What about C? Well, they might have the same area, but it's very difficult to tell.

One of them is a circle and one of them is a rectangle.

So not C.

And D, well, we're not quite sure what the area is, but if it's 3/5 of an area of 25 and 3/5 of an area of 20, then no we can't.

So we don't know what it's 25 of or what it's 20 of.

So we don't know enough there to be able to say that we could compare the area.

So it's only B where we would be able to compare the areas represented by those fractions.

In question two, how did you work these out? So for 5/7 and 2/3, then we would use a common denominator probably of 21s or 21st.

So we can convert them so they have a common denominator.

15/21 and 14/21.

And 15 is greater than 14.

So our original fraction of 5/7 is greater than 2/3.

Okay, and we can do this for the next pair as well, 3/4 and 7/9.

Well, they have a common denominator.

In fact, the lowest one we can use is 36.

So that will convert to 27/36 and 28/36.

So 7/9 is slightly greater than 3/4.

What about 2/3 and 7/10? Well, they're not related fractions either, are they? So we're actually gonna think about 30ths.

So we think about 20/30 and we think about 21/30 So 7/10 is slightly greater than 2/3.

Again for 3/11 and 1/3, we are gonna have to use a common denominator, aren't we? Because they're not related fractions.

And when we use the common denominator of 33rds, we can see that 3/11 is less than 1/3.

6/10 and 3/5, did you notice that those are related fractions? And in fact, 3/5 is equal to 6/10.

5/7 and 6/8.

Well, we could look at the fact that they are two of their fractions away from a whole and 8ths are smaller than 7ths.

So 6/8 is going to be closer to 1 than 5/7 is so it must be greater.

But you might also have used my favourite times table fact, 7 times 8 is 56, and used 56.

2/6 and 3/9.

6 and 9 look as though they might be related, but they're not actually a factor or a multiple of each other, but they are both related to 3rds.

And in fact in this, we could have simplified both of them to 1/3 and 1/3, and they are equal.

And 1/5 and 2/11.

Well, let me think.

2/10 is equal to 1/5.

2/11 are going to be smaller than 2/10.

So 2/11 must be smaller than 1/5.

So I've avoided using a common denominator of 55 there by thinking about what I know about fractions.

And in question three, did you convert into 20th in order to compare and order these? You might have done.

But if you did that, you'd have found that 2/5 was the largest, 3/10 was in the middle, and 1/4 was the smallest.

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

We've been explaining how to compare non-related fractions finding equivalent fractions with common denominators.

What have we been thinking about? Well, we've been realising that a fraction compares the size of a part in relation to its whole.

And a fraction can identify what portion or proportion of the whole has been used.

And we looked at tiling patterns for that and the areas, didn't we? The area of a pair of fractions can only be compared if the size of the wholes are the same.

That's something really important to look at when you're working with fractions.

If we are thinking about a defined whole, then we have to make sure that those wholes are the same if we're going to compare our fractions.

When we're just comparing fractions as numbers, those fractions are fractions of 1.

So we can compare them.

The whole is just 1.

And it's easier to compare fractions when they share a common denominator, or it can be.

In this lesson certainly, we had fractions that it was much easier to compare them when they shared a common denominator.

And we can use our knowledge of multiples in order to work out what the common denominator will be.

Thank you for all your hard work in this lesson and I hope I get to work with you again soon.

Bye-bye.