Lesson video

Hi, everyone.

Mr East here again.

I'm going to take you through today's lesson and we're going to begin by going through the practise activity from the previous lessons.

So, you were asked to look at these two fractions that are written on the screen.

We've got one 1/2 and 1/1000 and you were asked to write down five fractions in between these fractions so that they are in order.

So the first thing you would need to do is think, which is larger, 1/2 or 1/1000? 1/2 is larger, isn't it? We learned in other lessons that as the denominator grows, the size of the fraction decreases.

So the denominator of two is a lot smaller than the denominator 1,000, so that means one half is the largest fraction.

That then means that the next fraction and all the fractions after need to decrease in size each time until we get to 1/1000.

There will be loads of ways you could do this.

I might have 1/2, followed by 1/3, followed by a 1/4, followed by 1/5, followed by 1/6, followed by 1/7.

And then 1/1000, that would be fine.

But as long as the denominator each time increases, then that answer will be correct for that question.

Now on today's lesson.

We're going to be using lots of the maths that you've been learning in the last few lessons, so let's think carefully about that.

Let's start by looking at this first circle.

What fraction of the circle has been shaded? Shout it out.

Let's just make sure we've all got the same fraction.

How many parts is the whole being split into? That's going to be our denominator.

It's two.

How many of those parts have been shaded? One.

So the fraction is, how do we say that? 1/2.

What about our second circle? How many equal parts is this circle being split into? It's been split into four equal parts, so our denominator is four.

And how many of those equal parts have been shaded? Yeah, one of them.

So our fraction is 1/4.

Now in the last few lessons, we've been comparing fractions.

Just looking at those pictures, which fraction is larger, 1/2 or 1/4? Let's look at the amount of the shape, the fraction of the shape that has been shaded.

Which one's larger? The fraction that we've said is 1/2, or the fraction that we've said is 1/4? it's 1/2, isn't it? So, if I asked you to put these fractions in ascending order, so going from smallest to largest, we would have to move 1/2 over here.

Now these are in ascending order.

Now, if we're going to compare them, we could use this sign, that is-smaller-than sign.

So now we can say this statement.

1/4 is smaller than 1/2, or 1/4 is less than 1/2.

Can you say those with me? One quarter is less than a half, or one quarter is smaller than one half.

And something that we looked at together before is this statement.

When comparing unit fractions, remember that's fractions where the numerator is one, the greater the denominator, the smaller the fraction.

So if we look at our two denominators here, we have our denominator of four and our denominator of two.

Four is the larger number, so that means this fraction is smaller than in this fraction, where the denominator is two.

Now Emma has drawn these two diagrams. She's drawn a blue circle and split it into quarters and a red circle and she's split it in half and she thinks her diagram disproves what we've just said.

She thinks this shows that 1/4 is greater than 1/2.

What do you think? Do you agree with Emma, or do you disagree with Emma? Pause the video and see what you think.

Now, I can see Emma's point, because 1/4 of the blue circle is bigger than 1/2 of the red circle.

Let me put the red circle on top of the blue circle.

We can definitely see that for these examples, the quarter, the blue part of the blue circle is larger than the red part, the half of the red circle.

I'm not sure I agree with her though.

Do you, do you agree that 1/4 is greater than 1/2? No, we need to think and from the previous lesson, you should've thought about that when we compare unit fractions, the greater the denominator, the smaller the fraction.

So we need to look at this a little bit more.

So let's compare our first diagram, which I've shown here on the left and Emma's diagram on the right.

Let's look at those two diagrams carefully and think, what's the same and what is different? Pause the video and see how many similarities and differences you can find.

Okay, let's start off with the first similarity I've spotted.

We've got a circle that's been split into four parts on the left and we've got a circle from Emma that's been split into four parts on the right.

They've both been split into four parts.

And for both our diagram and Emma's diagram, one of those four equally sized parts has been shaded, so both of those are 1/4.

Let's look at the second circle.

Again, something that's the same.

They've both been split into two equal parts and again, one of those two equal parts has been shaded.

So that is the same.

So let's talk about the differences that you've spotted.

Again, if we start with our first diagram, can you spot that both of our wholes are the same size but for Emma's diagram, both of the wholes are not the same size? The blue circle is a lot larger than the red circle.

And because of this, that's why Emma has got an incorrect statement.

1/4 is not the same size as 1/2 and the reason for that is when we compare fractions, the whole has to be the same size.

So Emma thought her diagram proved that a quarter was larger than a half but we've just said that that's not the case.

We know that's not true.

And we said, the problem that Emma had is that her wholes are different sizes.

If we change the red circle to be the same size as the blue circle, we can clearly see that 1/4 is not greater than 1/2, because we can see here that the blue fraction is 1/4 and that's definitely smaller than the red part, 'cause it will come to here and this is larger in size.

So the correct statement is 1/4 is less than 1/2.

Uh oh, Zainab thinks that she has drawn two diagrams that prove a quarter is actually equal in size to a half.

Should we have a look at her diagram? Now I can see what she means because the blue part, which is 1/4 is equal in size to the red part, which is 1/2.

So does that mean those two fractions are equal in size? And what do we say needs to be the same when we compare fractions? the size of the whole needs to be the same and are our two wholes the same size? No, they're not.

Let me change this diagram so our wholes are the same size.

Is it still true now? Is 1/4 equal to 1/2? No, that's not true.

We said that to compare fractions, the whole has to be the same size.

And when the wholes are the same size, what statement can we say for 1/4 and 1/2? 1/4 is less than 1/2.

Now Mrs. Forde wants to prove to her class that 1/3 is less than 1/2.

And to do this, she is going to use some liquid.

She has filled this container with 1/3 full of orange juice.

She's then taken another container and then she's filled it 1/2 full.

I can see that both those fractions are about right.

I can see that three of those parts would fit into the whole container for 1/3 and two of those parts would fit into the container for 1/2.

But does this help Mrs. Forde's class understand that 1/3 is less than 1/2? I'm not sure it does.

The containers look like 1/3 is very similar in size to 1/2.

I think that maybe this shows that 1/3 is a little bit bigger than 1/2.

What do you think Mrs. Forde could do to help convince her class that actually, 1/3 is less than 1/2? Did anyone say pour them into the same container? Because that's exactly what Mrs. Forde has done.

If they're in the same container, if the wholes are the same size, then we can compare the fractions and what's bigger, what takes up more of the container, 1/2 or 1/3? It's really obvious, isn't it? It's 1/2.

1/2 is a larger amount than 1/3.

So we can say a 1/3 is less than 1/2.

Now we're going to look at another example.

Here, I've got the journey from Sunny's house to school and Sunny moves this amount.

What fraction of the whole journey do you think is shown? Have a guess.

How are we going to work out what fraction that is of the whole journey from Sunny's house to school? We've got to use some visualisation.

Here, we've got to think how many of those arrows would fit in the whole.

I'm going to show you.

Did anyone say 1/4? The amount shown in the arrow is 1/4 of the whole journey, because it's one of those four equally sized parts.

Now Kofi lives a little bit closer to the school and I want to show you part of Kofi's journey.

What fraction do you think that is of the whole of Kofi's journey to school? Let me prove to you.

There's two arrows and at first, I showed one of those arrows.

So the fraction that I've shown there would be one half.

Now I've written those two fractions that we said on those two arrows and I want us to compare the fraction of Sunny's journey from his house to school, to the fraction we've shown of Kofi's journey from his house to school.

Which one looks bigger? Sunny's house looks bigger, right? Let me put it underneath.

Yeah, definitely.

That arrow for the fraction of Sunny's journey is bigger than the fraction for the arrow for Kofi's journey.

Why is that? We've said that when we compare unit fractions, the greater the denominator, the smaller the fraction.

So why is 1/4 here larger than 1/2? Did anyone say it? Are the wholes the same size? No, the whole, that is the whole of Sunny's journey from his house to school, is larger than the whole journey of Kofi's house to school.

So we cannot compare these two fractions because the wholes are different sizes.

Again, when we compare fractions, the whole has to be the same size.

What do you think about what Femi's written here? Femi said his diagram proves 1/4 is less than 1/2.

And then Femi's drawn a diagram to try and prove that.

If you a teacher, what would you do? Would you tick the picture and tick the statement? I'm not sure I'd tick both.

Let's look at the different parts.

Femi said his diagram proves 1/4 is less than 1/2.

Is it? Yeah, we know, again, when comparing unit fractions, the greater the denominator, the smaller the fraction.

four is greater than two, so that means 1/4 is smaller than 1/2.

So I agree with that.

But does his diagram show that? What do you think? Would you ask him to redraw part of his diagram? Why would you ask him to read his diagram? What have we said through this lesson? We've said the wholes must be the same size.

So I don't think in this diagram it proves the statement, because the wholes are not the same size.

So, time to explain the practise activities for you.

For the first question, I have drawn two diagrams and I want you to think which of the following statements is proven by this diagram.

Does this diagram prove that 1/4 is less than 1/8? Does it prove 1/4 is greater than 1/8? Does it prove 1/4 is equal to 1/8, or a none of these statements true? So have a think about what you think and why you think that.

Think about explaining your answer to an adult at home, or writing it down if you're sending your work to your teacher.

So the second thing I'd like you to do is very similar to what we just spoke about with Femi's diagram.

We said that we agreed with his statement that 1/4 is less than 1/2, because we know from all the work we've done that when we compare unit fractions, the greater the denominator, the smaller the fraction.

But Femi's teachers asked him to look at his diagram and change it to make it more accurate.

We said that his diagram doesn't prove that 1/4 is less than 1/2 because of the size of the wholes.

So can you think about different ways and a more accurate representation that does prove that 1/4 is less than 1/2, or the opposite, that 1/2 is greater than 1/4? Have a go at that in preparation for the next lesson.

How many different ways can you represent it? Think of all the different contexts we've used in the last few lessons.

The third thing I'd like you to do is to compare two fractions of your choice.

I want you to choose a fraction to write here, choose a fraction to write here and then use our greater than, or less than sign to compare the two fractions you have chosen.

Maybe you can even draw a really accurate diagram to prove the inequality that you have written.

The final thing I'd like you to do is to complete this sentence.

This should summarise the same thing that we have said all the way through this lesson.

"When we compare fractions, the has to be the." Can you write that sentence to summarise the whole lesson?.