Lesson video

Hello.

Welcome to lesson four.

My name's Ms. Eaton and I'm going to be continuing with your journey, looking at equivalent fractions today.

So I know you might need to get a couple of things together, so make sure that you've got your work from yesterday.

You might want to refer back to that, and also any jot-ins or any drawings that you've done to represent your independent task, and you'll need some fresh paper and a pen for this lesson, too.

So if you just make sure you've got those and come back to me when you have.

Okay.

I like to start this session by thinking about the accurate mathematical language you've been using to explain your thinking.

And I know that you've met this generalisation here, which says, let's say it together, sometimes two fractions have the same value, and we call these equivalent fractions.

And in the last session we looked at equivalent fractions to a third, and your independent work has been to go away and see how many equivalent fractions to 1/5 you could find.

So I've also been having a go at them, and I got my blocks out.

So you can see here I've got some orange blocks and I've got some yellow blocks, and I tried to make some fractions that were equivalent to 1/5 with my blocks.

So if I look at the orange blocks, can you tell me how many equal parts the whole has been divided into? So we can say, can't we? But the whole is divided into five equal parts, which is what the denominator represents, and we have one of those equal parts that is expressed by the numerator.

Excellent.

What about my yellow blocks? How can we say that in a stem sentence? So now the whole has been divided into 10 equal parts, well done, which is represented by the denominator there, and we have how many of those equal parts? We have two of those equal parts, fantastic.

And those two equal parts are the same proportion of the whole, as this one part from 1/5.

Okay, well, if they're equivalent, what's the same and what is different? I'll just give you a moment to have a think about that.

What is the same and what is different? And maybe you can come back and tell me, 'cause I'm sure you've found 2/10 was equivalent to 1/5, as well, so I bet you've got some really good ideas about this.

All right, what do you think is the same? Yes, fantastic.

So the size of the whole is the same.

So the size of the orange blocks is the same as the size of the yellow blocks.

The size of the whole is the same.

Is there anything that's different? Ah, okay.

So here, the number of equal parts with the yellow blocks, I've got more of them, haven't I? I've got more equal parts.

But how have I got more equal parts if the whole is the same? Because the parts are smaller, fantastic.

So I've got more equal parts here, the pole has been divided into more equal parts, and I'm looking at two of those parts being the same proportion of a pole as one of the orange blocks.

Fantastic.

Okay, I'd like to dig a little deeper onto your independent task.

Now I know you had done it in those different ways.

Some of you might have gone pouring water.

I like to use rice in glasses so I make less mess and I can use after.

You might draw some number lines, and you might've folded paper or drawn some models on paper using your ruler.

I decided to use paper and fold paper, but I've drawn some diagrams to represent that.

So let's have a look.

I want to see if I've got the same as you got.

So here I've divided my paper into five equal parts.

So the whole has been divided into five equal parts and we have one of those parts, which is represented by this fraction, and we know this, don't we? The denominator represents the number of equal parts that the whole has been divided into.

Now I'm going to draw a line, a horizontal line, on my paper here.

And now do you think you can write down the fraction that is represented here? What is the equivalent fraction? Yes, we've just looked at it, haven't we? It's 2/10.

So again, we can see here, can't we, the whole's been divided into 10 equal parts, represented by my denominator, and we have two of those parts.

Fantastic.

What do you think I did next? Yes.

I bet you know.

I've got to put another line in.

Do you think you can write down the fraction that is represented there now and use the stem sentence to tell me how many equal parts I've got and how many of those parts I'm looking at, but shaded? I'll give you a moment to do that.

Have you written, brilliant, 3/15.

So the whole is divided into 15 equal parts, again represented by my denominator, and we have three of those equal parts.

Excellent.

And they're the same proportion of the shape as 1/5, aren't they? What do you think I did next? Okay, I put another vertical line in, horizontal line in, apologies, but another horizontal line in.

Do you think you can represent the fraction there? Okay, have you written 4/20? Fantastic.

So let's do the stem sentence together.

The whole is divided into 20 equal parts now, and we have four of those parts.

But again, we can see, can't we, that this shaded part is the same proportion of the whole as 1/5 was.

I put in another horizontal line.

Can you tell me now what fraction's represented and how many equal parts I have? Give you a moment to look at that.

Did you put 5/25? Fantastic.

So the whole is divided into 25 equal parts and we have five of those parts.

And once again, we can see, can't we, that the proportion of the shape is exactly the same as 1/5.

And one more horizontal line.

See if you can write down the equivalent fraction, tell me how many equal parts I have with them.

So the whole has been divided into 30 equal parts.

Well done.

So now I have 30 equal parts, and we have six of those parts, and we can see those six parts there.

And again, it's the same proportion of the shape as 1/5.

So I kept saying that, didn't I? I kept saying that these fractions all denote exactly the same proportion of the whole shape, and that means that they are taking up that same amount of space in the whole shape, aren't they? But they are equivalent.

I've listed all the fractions that we've found equivalent to 1/5 in the list now, and I'd like you to have a little look.

Did you notice the multiplicative relationship between the numerator and the denominator? I know you looked at the multiplicative relationship between the numerator and the denominator for 1/3.

Can you see what that is for fractions equivalent to 1/5? Pause me now and have a little look at that.

Okay, what did you find? Did you find that the denominator was five times the numerator? Let's check each one.

So for 1/5, the denominator is five times the numerator.

One multiplied by five is five.

Does that work for 2/10? Yes.

The denominator is five times the numerator.

Two multiplied by five is 10.

Tell me the next one.

3/15.

15 is five times 3.

15 is five times the numerator.

Well done.

Is the same true for 4/20? 20 is five times four.

The denominator is five times the numerator.

What about the relationship between five and 25, the vertical multiplicative relationship? 25 is five times five.

So the denominator is five times the numerator, again.

Do we think it's going to be true for this final one? Yes, of course.

So six multiplied by five equals 30.

The denominator is five times the numerator.

Say that with me now.

The denominator for all of these equivalent fractions is five times the numerator.

The denominator is five times the numerator.

And we can see, can't we, that these fractions are equivalent where this relationship is repeated.

So we have a vertical multiplicative relationship between the numerator and the denominator, which we can express with multiplication.

I've taken two of those equivalent fractions now.

1/5 is equivalent to 2/10 because the denominator is five times the numerator.

So quickly review the language we used then.

How do we express this? What operation do we use? Just pause, check if that works, and come back and tell me what operation we've used.

Okay, what did you find? What operation did you use to make the denominator five times the numerator? You used multiplication, fantastic.

So two multiplied by five is 10.

What do you notice about these pizzas? Take a moment to have a look at these two pizzas and tell me what you notice about them? Fantastic, yes.

So firstly, I really hope you noticed that the pizzas are exactly the same size, so we've got two identical pizzas in size.

What else did you notice about them? You noticed they're the equivalent fractions that we looked at in the last slide.

So we're looking at, we're comparing, 1/5 and 2/10, and these are equivalent fractions, aren't they? So we said this.

1/5 is equivalent to 2/10.

What else do you notice about the pizzas? The amount of slices.

So the pizza on the left has been cut into five equal slices.

The pizza on the right has been cut into 10 equal slices.

And two of the slices is equivalent to one of the slices where the pizza's been cut into five equal parts.

So 2/10 is the same proportion of the whole as 1/5.

And can we see that vertical multiplicative relationship? We can, can't we? So we've proven there that they are equivalent.

Well done.

I've taken another pair of our equivalent fractions, so 1/5 and 4/20.

Can you complete that stem sentence on your own now? Quickly pause me and use the language that we've been using to express the relationship between the denominator and the numerator.

Have you managed that? So 1/5 is equivalent to 4/20 because the denominator is five times the numerator.

Fantastic.

And did you use multiplication to make your denominator five times the numerator? So four multiplied by five equals 20.

I'd like you to take a look at the first pair of equivalent fractions.

So we know, don't we, that the denominator is five times the numerator for fractions equivalent to 1/5.

And can you use that fact to help you find the missing denominator? I bet you've already done it, haven't you? So 12 multiplied by five is 60.

Really well done.

I'm sure you did that in a variety of different ways mentally.

So 60 is five times 12.

The denominator is five times the numerator.

Let's have a look at that pictorial.

So for 1/5, the whole has been divided into five equal parts, and we can see one of those parts.

The five equal parts are represented by the denominator.

And for 12/60, the whole has been divided into 60 equal parts, represented by the denominator, and 12 of those parts are equivalent to the same proportion of the shape as 1/5.

But we can also see, can't we, that the vertical multiplicative relationship, the numerator is five times the denominator.

Brilliant.

Have a look at the second pair of equivalent fractions.

Now what do you notice? So we notice, don't we, that, actually, we're missing the numerator this time.

So how do you think we can find that? What is the relationship between the denominator and the numerator? So I'd like you to pause me now and have a think, if you can rewrite this stem sentence we used earlier to tell me what the relationship between the denominator and the numerator is.

What did you find? Did you find that, for fractions equivalent to 1/5, the numerator is 1/5 of the denominator.

Super.

So how do we find 1/5? How do we calculate 1/5? We can divide by five, fantastic.

So 1/5 of 40 is the same as dividing by five, 40 divided by five.

I'm sure you could do that mentally really quickly.

So 1/5 of 40 equals eight.

40 divided by five equals eight.

And we could see, actually, we've done the inverse, haven't we? So we multiplied by five to find the denominator and we're dividing by five, we're finding 1/5 of the denominator to find the numerator.

Let's look at that pictorial there.

So we can see, can't we, that the whole has been divided into 40 equal parts, and 1/5 of that whole is eight.

40 divided by five equals eight.

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

So we can find 1/5 of the denominator to identify the numerator.

Super.

Okay.

So we've really looked in detail there, at the relationship between the numerator and the denominator for 1/5 and we know that the denominator is five times the numerator and the numerator is 1/5 of the denominator.

Superb.

So do you think you can use that to tell me what the multiplicative relationship is between the numerator and the denominator in fractions equivalent to one quarter? I think what you're going to have to do here is start visualising that pizza again.

So have a think about that pizza.

Have a look at it, close your eyes, and think, oh, how many equal parts is my pizza divided into if I have quarters? Four equal parts, fantastic.

So we can also say a quarter is 1/4, can't we? So the four, the denominator, means that the whole has been divided into four equal parts.

So now we're really clear on that, can you tell me what the relationship between the numerator and the denominator is? So I'd like you to pause me again now and see if you can find the missing denominator and numerator using that relationship we've just spoken about.

How did you get on? What can you tell me about the relationship between the numerator and the denominator in one quarter? It's four times.

The denominator is four times the numerator.

Superb.

So does that help us find the missing denominator? Yes.

Because we can say three multiplied by four is, did you get 12? Fantastic.

So one quarter is equivalent to 3/12.

Super.

What about the second example? What can you tell me about the relationship from the denominator to the numerator? Did you find that the numerator is one quarter of the denominator? Fantastic.

How do we find one quarter? We can divide by four, super.

So 24 divided by four equals six.

Super, well done, if you got six.

And six multiplied by four is 24.

So we can use our knowledge of inverse relationships to help us, as well.

So I'm going to really remember, what we're really remembering is, that the denominator is four times the numerator and the numerator is one quarter of the denominator.

Super.

Can you now apply that to fractions equivalent to 1/10? So I bet you're already thinking now, aren't you, well, hang on.

Which relationship? Am I going from the numerator to the denominator, from the denominator to the numerator? And what operations do I use for that? I'm going to put those two up.

Pause me, have a go at those, and we'll come back and discuss the language.

How did you get on? Okay, what can you tell me about the relationship between the numerator and the denominator in fractions equivalent to 1/10? So the denominator is how many times the numerator? 10 times the numerator.

Fantastic.

So seven multiplied by 10 is 70.

The denominator is 10 times the numerator.

What about if we got a missing numerator? What did you notice? That the numerator was 1/10, 1/10 of the denominator.

Fantastic.

And what is 1/10 of 50? How do we find 1/10? We divide by 10.

Excellent, and I know you'll have a range of mental strategies to help you visualise multiplying and dividing by 10.

So 50 divided by 10 equals five.

1/5 of is five.

And we can see that, can't we, that we generated those equivalent fractions by using that vertical multiplicative relationship between the numerator and the denominator.

Okay.

I'd like us to think now about how we use that relationship to help us identify fractions are equivalent or not.

So I'm not telling you that these are equivalent this time.

They might be, might not.

You're going to work it out by using the relationship that we know exists between the numerator and the denominator.

So we can look at 1/6 and we can see how many times the numerator the denominator is.

Or we can find out if the numerator is 1/6 of the denominator.

That would work too, wouldn't it? And then we can do the same for 3/12.

So pause me again now.

Find out what that multiplicative relationship is between the numerator and the denominator, and then we could come back and tell me if these are equal or not.

What did you find? What is the relationship between the numerator and the denominator in 1/6? So the denominator for 1/6 is how many times the numerator? Six times the numerator.

It's been scaled up by six.

So the numerator is 1/6 of the denominator.

Super.

Was that the same relationship in 3/12? Did you find that? No, it wasn't the same.

So the denominator for 3/12 is how many times the numerator? Four times the numerator, brilliant, because three multiplied by four is 12.

So actually we could say, if we were looking at the relationship the other way around, that the numerator is one quarter, one quarter of the denominator.

So it's been scaled up by four.

So it's not the same, is it? The relationship of the numerator to the denominator was six times for 1/6 and it's four times for 3/12.

Are those fractions equal? No.

No, they're not.

Well done.

And let's have a look at that pictorial, there.

So we can see, then.

This proves that they're not equal.

So for the bottom bar, the whole has been divided into six equal parts, represented by our denominator, and one of those parts, we can see there, or 3/12 is larger than 1/6.

Because the whole has been divided into 12 equal parts, and if we look at three of those parts, only two of those parts would be equivalent to 1/6.

So it's not the same proportion of the whole, is it? And it's not the same proportion of the whole because we didn't have that vertical multiplicative relationship.

Okay.

We've really gone through a lot of work there.

So I'm going to just introduce your independent task, and then you can carry on with the rest of this presentation on your own, carry on with the rest of the lesson on your own, and we can come back tomorrow and look at what you've done and move on from there.

So thinking back to the beginning of the lesson, when we were looking at 1/5, we looked at them in an area model, didn't we? So we're going to go back and have a look at that in an area model, and I want you to identify the equivalent fractions pictorial there, thinking about that stem sentence that we can always come back to, can't we, telling us how many equal parts the whole is divided into.

So we can see with the first one the whole is divided into three equal parts.

Really well done.

And we have one of those parts.

And the next one is 2/6.

The whole is divided into six equal parts and we have two of those parts is the same proportion as 1/3.

Next one, the whole is divided into nine equal parts and we have three of those parts is the same proportion as 2/6 and 1/3.

And finally, what fraction do we have? 4/12, really well done.

So the whole is divided into 12 equal parts and we have four of those parts.

And just thinking about that vertical relationship between the numerator and the denominator, so we know now, don't we, that the numerator should be 1/3 of the denominator and the denominator should be three times the numerator.

So for the rest of your independent task, I want you to check that that relationship works every time.

And then on this slide you'll see that I want you to find all the equivalent fractions for those images.

And then the second part is to match the equivalent fractions, but by thinking about that multiplicative relationship between the numerator and the denominator.

So is the denominator, how many times is the denominator to the numerator and what fraction of the denominator is the numerator? And you can always go back and have a look at those stem sentences we've used to help you frame your reasoning.

Now have a look at those, and then on the next slide you'll see that I've just put in a couple of little challenges for you.

So now that we've been doing a lot of thinking about pizzas, haven't we, and you might want to go and catch out an adult in your house and get them to see if they understand what equivalent fractions are.

So maybe you could say to them, ooh, would you prefer 1/5 of a pizza or 3/15? And the adult might say that they prefer 3/15, 'cause it sounds like more slices.

And then you can explain to them why they are the same amount.

But think of some of your own.

Think of some of your own questions to ask them.

You don't have to use pizzas.

You could use drinks or cakes.

And then the final challenge there.

I want you to think about, maybe you could draw this, and then you could share this with your teachers, as well.

There are four children in the playground and 2/8 of them are wearing a hat.

How many of them are wearing a hat, and how can you now use your knowledge of equivalent fractions to help you solve that problem more easily? And I just put some language here for you to think about, that can help you with all of those challenges.

So be keeping in your mind that sometimes two fractions have the same value, but a different appearance, and we call these equivalent fractions, and that these equivalent fractions all denote exactly the same proportion of the shape.

So they are all the same proportion of the shape, and we've that pictorial quite a lot today, and the denominator is however many times the numerator.

So the numerator is the fraction of the denominator.

Okay, I'm going to leave it there.

I hope you really enjoy your independent work and I look forward to seeing you at the next lesson.

Thanks, bye!.