Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in our maths lesson today.

I hope you're ready to work hard and have lots of fun.

It's fractions, and I love fractions.

So let's make a start.

So in this lesson we're going to be using the relationship between the numerator and denominator in equivalent fractions to solve problems. And this lesson comes from our unit comparing fractions using equivalents in decimals.

So we're going to think about all that we've learned about equivalent fractions, and we're going to use that to solve problems. And we're gonna think in particular about the numerator and the denominator, and the relationship between them in our equivalent fractions.

So we've got some keywords today.

They may be quite familiar to you, but let's just practise them anyway 'cause they are going to be useful in the lesson.

So my turn, "numerator".

Your turn.

My turn, "denominator".

Your turn.

My turn, "equivalent fraction".

Your turn.

My turn, "proportion".

Your turn.

Let's just remind ourselves about what those words mean.

So the numerator is the top number in a fraction and shows how many parts we have.

The denominator is the bottom number in a fraction and shows how many equal parts the whole has been divided into.

Equivalent fractions are fractions which have the same value even though they may look different.

And the proportion of a number, shape, or group of objects is a part of the whole.

So there are two parts to our lesson today.

We're going to be creating and identifying equivalent fractions in the first part, and then comparing fractions and identifying equivalents in the second part.

So let's make a start on part one.

And Jun and Sofia are here to help us with our learning today.

So what's the missing number, and how do you know? We've got 1/3 is equal to 8.

Hmm.

8 somethings.

So we need to find the missing denominator.

And we've got some bars there to help us.

What do you notice about the bars? That's right, the same proportion of the whole is shaded, the same part of the bar.

The wholes are the same, the two bars are the same length, and the same proportion is shaded in each.

And in the top one we can see that 1 part out of 3 is shaded.

In the bottom part we can see 8 parts out of something, the rest of our bar is blank.

So what do we know about these fractions that can help us? Well 1 times something is equal to 3.

1 times 3 is equal to 3.

So we know that in an equivalent fraction that the numerator must be multiplied by the same number to create the denominator.

There's got to be that same relationship between the numerator and the denominator.

So 8 multiplied by 3 will give us our denominator.

And 8 times 3 is equal to 24.

That's right.

So if we think about our bar, we've got our 1 part out of 3 equal parts, and we knew that was divided into 8.

So now we need to repeat that two more times.

So we know that there are 24 parts in our whole, and that we have 8 of them.

And that 8 out of 24 is the same proportion of the whole as 1 part out of 3.

And there's our stem sentence that you may have seen before in another lesson.

1 out of 3 is the same proportion of the whole as 8 out of 24.

So, time to check your understanding.

Can you use that relationship between the numerator and the denominator and what you can see in the bars to fill in this missing numerator this time.

So 1/4 is equal to, hmm, 20ths.

And we've got the stem sentence there to complete.

So, what's the missing number and how do you know? Pause the video and have a go.

How did you get on? Did you spot that 1 times 4 is equal to 4? So we've got to have that multiplied by 4 relationship in our equivalent fraction.

So, what times 4 is equal to 20? 5 times 4 is equal to 20.

So we know that there must be 5 parts out of 20 that are shaded.

1 out of 4 is the same proportion of the whole as 5 out of 20.

And we can see that in the bars.

Did you notice this time we didn't know how many of the bottom bar were shaded in.

In the previous example, we didn't know how many parts there were in the whole.

This time we didn't know how many parts in the whole were shaded.

And we can think about that.

The numerator represents the number of parts we have.

The denominator represents the number of parts in the whole.

We knew that denominator this time.

We've got a missing denominator here.

What's the missing number, and how do you know? This time the missing denominator is in our unit fraction.

So how can we think about this without drawing the bars? 1 out of, hmm, is the same proportion of the whole as 3 out of 15.

Jun says, "3 times 5 is equal to 15.

So the denominator of the other fraction must be 1 multiplied by 5." So the other fraction must be 1/5.

We've kept that relationship between the numerator and the denominator the same in both fractions.

You multiply the numerator by 5 to get the denominator.

Jun says, "I can imagine the bars and I can complete the stem sentence." 1 out of 5 is the same proportion of the whole as 3 out of 15.

And if you close your eyes, you can imagine those bars, two bars the same length, 1 divided into 5 equal parts and the other divided into 15 equal parts.

3 times as many parts, so we need 3 times as many of the parts to make the same proportion.

And Sofia's reminding us that 1/5 and 3/15 would sit at the same position on a number line.

So what's the missing number in this pair of equivalent fractions? We've got a missing numerator this time.

How can we think about it without drawing the bars? And can you complete the stem sentence? Pause the video and have a go.

How did you get on? So did you think about that relationship between the numerator and the denominator? 8 times what is equal to 56? Well, it's 8 times 7 is equal to 56.

So 1 times 7 is equal to 7.

So our missing numerator is 1.

1 out of 7 equal parts is the same proportion of the whole as 8 out of 56 equal parts.

And you might have imagined the bars to help you to make sense of that.

So, can we identify the pairs of equivalent fractions here? And can we then complete the stem sentence thinking about the proportions of the whole? So you might want to have a pause and have a think about that for a moment.

We've got lots of 8s and 5s in here, haven't we? Let's have a look and see what we can see with these fractions.

So we can see that in 1/8, 1 times 8 is equal to 8.

So if we're finding an equivalent fraction to 1/8, the denominator must be 8 times the numerator.

What about 1/5? Well that's 5 times.

So the denominator must be 5 times the numerator.

So let's have a look at the other fractions.

Well, 8 times 5 is equal to 40, and 5 times 8 is equal to 40.

So now we can see clearly that 1/8 is equal to 5/40.

1 out of 8 is the same proportion of the whole as 5 out of 40.

And we can see that 1/5 is equal to 8/40, 1 out of 5 is the same proportion of the whole as 8 out of 40.

There are other ways you might have been able to reason as to which with the equivalents, thinking about which fractions were greater and lesser in value.

But spotting that relationship between the numerator and the denominator in equivalent fractions is really important.

Time to check your understanding.

Can you create another equivalent fraction to go with each pair by filling in the missing denominator and the missing numerator, and use the stem sentences to check using that idea of proportion.

Pause the video and have a go.

How did you get on? So we've got a multiplied by 8 relationship here.

We've got fractions equal to 1/8.

1 times 8 equals 8, 5 times 8 equals 40, 11 times 8 equals 88.

That's right.

And then in the other set of fractions we've got 1/5.

So we've got a times 5 relationship.

1 times 5 is equal to 5, 8 times 5 is equal to 40.

And, hmm, times 5 is equal to 75.

What times 5 is equal to 75? 15 times 5.

So 15/75 is our equivalent fraction.

And if we think about those stem sentences, 11 out of 88 is the same proportion of the whole as 1 out of 8.

And 15 out of 75 is the same proportion of the whole as 1 part out of 5.

The new fractions we created had very large denominators, didn't they? Not the normal sorts of fractions that we come across.

But sometimes we will come across things like that, and it's useful to know that our rules for creating equivalent fractions, that relationship between the numerator and the denominator, will carry on however many parts we have in our whole.

That proportion will still stay the same.

Time for you to have a go.

So you are going to use the relationship between the numerator and the denominator to work out the missing numbers in these pairs of equivalent fractions.

And you've got five pairs to work on there.

And then in part two you're going to use the relationship between the numerator and the denominator to identify pairs of equivalent fractions involving one unit fraction and one non-unit fraction.

And you can see we've got the unit fractions on one side with one as the numerator, and the non-unit fractions, you're going to find those pairs.

And then can you add another fraction to each group to create three equivalent fractions? Pause the video, have a go, and we'll look through the answers together.

How did you get on? Did you spot that in A we had 1 times 6 equals 6.

So, 6 multiply by 6 gives us 36.

So, 1/6 is equal to 6/36.

In B we had the completed fraction was 4/28.

And we can see that 4 times 7 is equal to 28.

So 1 times 7 is equal to 7.

Our missing numerator must be 1 to create our unit fraction of 1/7.

In C we had a missing denominator and we had a complete fraction of 11/99ths.

So 11 times 9 is equal to 99, 1 times 9 is equal to 9.

So our equivalent fraction was 1/9.

For D we had 1/12 is equal to, hmm, 36ths.

Or 1 times 12 is equal to 12.

So, something times 12 must be equal to 36.

And using our times table knowledge, we know that 3 times 12 is equal to 36.

So our missing numerator was 3, creating an equivalent fraction of 3/36.

Now in E there were different possibilities here, but we went for 1/3 is equal to 8/24.

1 times 3 is equal to 3, and 8 times 3 is equal to 24.

But you could have had 1/2 is equal to 12/24.

1 times 2 is 2, 12 times 2 is 24.

Or 1/4 is equal to 6/24.

One times 4 is 4, 6 times 4 is equal to 24.

Lots of different possibilities for E.

And so for part two, there were lots of possible extra equivalent fractions you could have created, but these are the groups we created.

1/7th is equal to 10/70, which is equal to 100/700.

1/8 is equal to 6/48, which is equal to 3/24.

1/9 is equal to 7/63, which is equal to 14/126.

And then we had 3 equivalences for this one.

1/6 is equal to 5/30 is equal to 8/48, is also equal to 4/24.

And 1/5 is equal to 6/30, which is equal to 3/15.

I wonder what fractions you came up with to add to each group, and what the biggest denominator you managed to find was.

700's quite high.

I wonder if you went any bigger than that for your denominator.

So into part two of our lesson, and we're going to be comparing fractions and identifying equivalents.

So, is this a pair of equivalent fractions? How could we decide? So have a look at those two fractions.

Are they equivalent? 1/6 and 3/12.

Hmm.

Jun says, "Let's look at the relationship between the numerator and the denominator." So 1 time something is equal to 6, and 3 times something is equal to 12.

And when those fractions are equivalent, that number that we multiply by, that factor, will be the same.

And Sofia spotted 1 times 6 is equal to 6, but 3 times 6 is equal to 18.

So that's not right.

Jun says, "3 times 4 is equal to 12.

So the relationship is not the same for both fractions." And Sofia says, "These are not equivalent fractions." That relationship between the numerator and the denominator is not the same.

So, it's not a pair of equivalent fractions, but which fraction is larger? How could we decide? I wonder how you would go about thinking about that.

Well, Jun says, "We could still look at the relationship between the numerator and the denominator." 1 times 6 is equal to 6, but 3 times 4 is equal to 12.

Ah, Jun's thinking about the proportion.

He says that 3 out of 12 is a larger proportion of the whole than 1 out of 6.

Oh, that's a lot to think about, isn't it? We know that the whole has to be the same when we are comparing.

So can you picture a whole divided into 12 equal parts with 3 of them, and a whole divided into 6 equal parts, and we have 1 of them.

Hmm, we might need to draw some pictures for this, mightn't we? Ah-ha, here we are.

Here's a bar.

Here's a bar model to help us.

Our bars are the same length, 'cause when we are comparing fractions, the whole has to be the same.

In our top bar we've got our whole divided into 6 equal parts and we've shaded 1.

And in the bottom one we've got our bar divided into 12 equal parts and 3 of them are shaded.

So let's look at Jun's sentence again.

"3 out of 12 is a larger proportion of the whole than 1 out of 6." And if we think about that, 3 times 4 equals 12, we'll need 4 of those 3s to equal 12.

But 1 times 6 equals 6, we'd need 6 of those 1s to equal 6.

The more of those smaller parts we need to make the whole, the smaller the proportion the whole we have.

So, can we use the fractions to complete these equations? Can we think about that relationship between the numerator and the denominator to help us to identify a pair of fractions that are equivalent, and then two pairs that aren't equal, one with one fraction greater than the other, and one with one fraction smaller than the other? So let's think about the stem sentence.

And we've increased the stem sentence now.

We've got, hmm, out of, hmm, is the same proportion, or a smaller proportion, or a larger proportion of the whole than, hmm, out of, hmm, when we are comparing our fractions.

Okay, where should we start here? So, Jun says he can see two fractions with the denominator of 24.

"4 and 6 are both factors of 24," he says.

So we can see two fractions with the denominator of 24.

So we could compare those two, couldn't we? But let's have a look and compare those to something else.

Sofia says "4 times 6 is equal to 24, so I think 4/24 is equivalent to 1/6." 1 times 6 is 6, 4 times 6 is 24.

So 1/6 and 4/24 are equivalent.

And we can complete the stem sentence.

1 out of 6 is the same proportion of the whole as 4 out of 24.

Okay, so if we know that, what else do we know? Jun says "If that is true, then 1/6 must be greater than 3/24." Okay, how does he know that, then? So 1/6 is greater than 3/24.

We know they're not equivalent because 1 times 6 is equal to 6 and 3 times 6 is equal to 18, and not 24.

But we do know that if 1/6 is equal to 4/24, 3/24 is smaller than 4/24, so therefore it must be smaller than 1/6.

1 out of 6 is a larger proportion of the whole than 3 out of 24.

And if you imagine that 1/6 would be 4/24, so if we coloured in the 24th for 3/24, we wouldn't have enough to be equivalent to 1/6.

And Sofia says, "I think 1/9 must be smaller than 4/24 because 1/9 is smaller than 1/6." Ah-ha.

That's interesting, isn't it? So this time she's compared the unit fractions.

1/9 is smaller than 1/6.

1 out of 9 equal parts is smaller than 1 out of 6 equal parts of a whole.

And if 1/6 is equal to 4/24, then 1/9 must be smaller than 4/24.

1 out of 9 is a smaller proportion of the whole than 4 out of 24.

So we've used lots of reasoning there to compare our fractions.

We've used reasoning to work out what is equivalent.

And then we've used our knowledge of fractions to be able to compare the fractions by thinking about parts of a whole, and thinking about numerators and denominators, and the relationship between them.

Time for you to have a go.

You've got the same set of fractions.

I wonder if you can complete those equations in a different way.

Think about the relationship between the numerator and the denominator and use that stem sentence to help you.

And there's an extra gap now.

Hmm, out of, hmm, is either the same, or a greater, or a smaller proportion of the whole than, hmm, out of, hmm.

Pause the video and have a go.

How did you get on? There were different ways of thinking about this.

We could say that 1/6 is greater than 3/24.

I think we had that one before, didn't we? We could say that 1/9 is equal to 4/36.

1 times 9 is equal to 9, 4 times 9 is equal to 36.

Can we use anything there to help us with the other fraction? Now this time we've decided to say that 1/8 is smaller than 4/24.

1 times 8 equals 8, 4 times 8 equals 32.

So no, they're not the same are they? And we can complete our stem sentences.

1 out of 9 is the same proportion of the whole as 4 out of 36.

1 out of 6 is a larger proportion of the whole than 3 out of 24.

And 1 out of 8 is a smaller proportion of the whole than 4 out of 24.

So Jun and Sofia are playing a game with fraction cards.

They turn over one grey unit fraction card and one white card to complete the equations.

If they can't use the fractions because that equation's already complete, they turn them over so they can use them again.

So Jun says, "I picked out 6/12 and 1/2.

Sofia says "6 is half of 12.

So 6/12 is equivalent to 1/2." So Jun can put his cards there And he says "I'll complete the stem sentence." 6 out of 12 is the same proportion of the whole as 1 out of 2.

And Sofia says, "I can visualise a number line with a 1/2 and 6/12 at the same point.

Now it's Sofia's go.

"I picked 1/4 and 4/12," she says.

Ooh, Jun's having a look at the relationship between the numerator and the denominators.

He says "4 times 3 is equal to 12, and 1 times 4 is equal to 4.

So 4/12 is not equivalent to 1/4." But which is larger? Sofia says, "I will use the stem sentence and imagine bars or a number line." So can we imagine those bars? Can we imagine a bar cut into 4 equal parts and we've shaded 1 of them, and the same length bar cut into 12 equal parts and we've got 4 of them.

So let's think, we'd need 3 of those 4s to make the 12 equal parts.

We'd need 4 of the 1/4s to make the whole.

So I think 1 out of 4 is a smaller proportion of the whole than 4 out of 12.

So therefore 1/4 is less than 4/12.

And that's what Sofia is saying.

"I can say 1/4 is less than 4/12." So she can use her cards.

How can we prove that that stem sentence is correct? We tried to visualise it.

Can we use a number line to help us? 1 out of 4 is a smaller proportion of the whole than 4 out of 12.

So let's have a think.

4 lots of 1/4 are needed to make the whole.

4 times 3 is equal to 12, so only 3 lots, 4/12 are needed to make the whole.

1 out of 4 is a smaller proportion of the whole than 4 out of 12.

So 1/4 is less than 4/12.

Time for you to have a go.

You're going to play Jun and Sofia's game.

So you've got some fraction cards, I hope.

You're going to turn over one unit fraction card, and one other card, and see if you can complete the equations.

If the equation's already been completed and you can't use your fraction cards, turn them back over, choose some others the next time.

And you're going to use the stem sentence to help you to decide.

And you might want to sketch some bars or sketch a number line.

So pause the video, have a go at playing the game, and then we'll have a talk about what we found out.

Did you enjoy playing Jun and Sofia's game? There were lots and lots of different ways that you could have completed the equations.

Here's one way you could have done it.

So 1/3 is greater than 2/8.

6/12 is equal to 1/2.

And 1/4 is less than 4/12.

Well, 6/12 is equal to 1/2.

That's quite easy to do, isn't it? Because we know that in all fractions equivalent to 1/2, the denominator is 2 times the numerator, or the numerator is 1/2 the denominator.

Double 6 is 12, 1/2 of 12 is 6.

So 6/12 and 1/2 are equivalent.

But Jun had to think about 2/8.

He said 2/8 is equal to 1/4, and 1/4 is less than 1/3.

So 1/3 must be greater than 2/8.

Let's think about that again.

2/8, he says, is equal to 1/4.

And if we think about 1/4 and 1/3, 1/4 is smaller than 1/3 'cause we've got 1 out of 4 equal parts, and not 1 out of 3 equal parts.

So he says, "If a 1/3 is larger than 1/4, then 1/3 must be greater than 2/8." I wonder how you decided which fractions would go in which positions.

Did you draw some pictures to help you? Did you use the stem sentence? And did you use that relationship between the numerator and the denominator? Whatever you did, I hope you enjoyed playing the game.

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

So, we've been thinking about using the relationship between the numerator and the denominator in equivalent fractions to solve some problems. We've learned that in equivalent fractions, the proportion of the whole represented by each fraction is the same.

We can use a bar model or a number line to represent the fractions.

And we can look at the relationship between the numerator and the denominator in fractions and decide if they are equivalent or not.

So here we've got 1/3 and 8/24.

1 times 3 is equal to 3.

8 times 3 is equal to 24.

So we know that 1/3 and 8/24 are equivalent, and we've got the bars there to show it.

And we've also learned that if fractions aren't equivalent, we can use our knowledge of equivalence to help decide whether one fraction is greater than or less than another.

Thank you for all your hard work today.

I've really enjoyed exploring equivalent fractions with you and I hope we get to work together again soon.

Bye-Bye.