Lesson video

Hello there.

My name is Mr. Tilston.

I'm a teacher.

My favourite subject to teach is definitely maths, and my favourite part of maths is probably fractions, and that's what we're going to be learning about today.

Today we're going to be comparing fractions.

You may have had some recent experience of comparing fractions where the denominator's the same.

Today we're going to look at fractions where the numerator's the same.

So if you are ready to begin, let's start.

The outcome of today's lesson is this.

I can compare and order mixed numbers when the whole and the numerator of the fractional part is the same.

And our key words, my turn, mixed number, your turn.

I think you know what a mixed number is, but let's have a little recap.

A mixed number is a whole number and a fraction combined.

For example, this is one and a half.

That's a mixed number.

Can you think of a different mixed number? Our lesson today is split into two parts or two cycles.

The first will be making connections to prior learning and the second comparing mixed numbers that have the same whole.

So let's begin by rewinding a little bit and looking at some previous learning that's going to help with today's lesson.

In this lesson, you're going to meet Aisha and Andeep.

Have you met them before? They're here today to give us a helping hand with the maths.

Andeep and Aisha are trying to compare some numbers, and here are the numbers they're comparing.

What kind of numbers are these? These are mixed numbers.

They've got a whole number part and a fractional part.

What do you notice about them? What's the same? What's different? Have a good look, have a good think.

Do what a good mathematician does and notice things.

Here's what Andeep has noticed.

Both of these are mixed numbers with the same whole number part.

Did you spot that? So not only are they both mixed numbers, they've got the same whole number part, which is one.

How would you compare these numbers, these mixed numbers? How would you say which is greater, which is smaller? What would you do? Well, Aisha says, let's start by looking at a simpler case.

Yeah, let's rewind a little bit.

Let's look at these proper fractions.

They're quite similar to the other ones, but they're not mixed numbers, just proper fractions.

So this is four sixths and four fifths.

So they've got the same fractional part as those mixed numbers.

Andeep in Asia revisit their learning on proper fractions.

Aisha says, I know one sixth is smaller than one fifth, so I know that four lots of one sixth is smaller than four lots of one fifth.

Would you agree? Andeep says, so four sixths is the smallest number.

They're right.

Let's prove that though.

We can represent each fraction as a model to prove that.

So on the top bar we've got four sixths, and on the equalised bottom bar we've got four fifths.

And you can see that four sixths takes up less space than four fifths.

It takes up less of the hole, so is the smaller fraction.

The hole is the same size each time, but there are more equal parts, which means the parts must be smaller.

We can use an inequality symbol here.

We can say that four sixths is smaller than four fifths, or four sixths is less than four fifths.

We can also represent this on a number line to prove it.

So just like before, notice that the number lines are equally sized.

The top and the bottom one are the same size, but the top one is split into six equal parts.

And each part is smaller.

And the bottom one is split into five equal parts.

And each part is slightly bigger.

So there's four sixths, you can see that's closer to zero than four-fifths is.

Or you could look at it the other way around.

Four-fifths is closer to one than four sixths is.

And that's what Aisha says.

Four sixths is the smallest because it's near it to zero.

And that's what Andeep says.

Four-fifths is the greatest because it's closer to one.

We can use a generalisation to help us compare these proper fractions.

Are you ready? Here it is.

I'll say it.

We'll say it then you'll say it.

When we compare fractions with the same numerator, the smaller the denominator, the greater the fraction.

This is really important, and this is at the heart of today's lesson.

So let's say it again.

We're going to say it together this time.

Can you read it with me? I'll do it nice and slowly.

Are you ready? Let's go.

When we compare fractions with the same numerator, the smaller the denominator, the greater the fraction.

Very good.

Now just you say it in three, two, one, go.

So this is a very important piece of learning today, and we'll come back to this time and again.

Six is greater than five.

So sixths are smaller than fifths because the whole has been split into more equal parts.

Therefore, each of those parts must be smaller.

Right.

Let's see if you've understood the learning so far.

You're going to compare these proper fractions.

I want you to write an inequality symbol on your whiteboard and a word.

So four ninths is, hmm, than four sevenths.

Before you do that, notice that the numerator is the same.

The denominator is different.

Think about what you've just learned.

You might want to do a little sketch to help you out.

That's up to you.

Pause the video.

I wonder if you sketched two bar models there, just like the ones we saw before.

If you did, you would see that the bar split into nine equal parts has smaller equal parts than the bar split into seven equal parts.

So therefore, four ninths is smaller than or less than four sevenths.

And that is the inequality symbol that we would use.

So well done, if you've got that, you're on track.

If the numerator are the same, we compare the denominators.

And in this case, the numerators are the same.

Throughout this whole lesson, the numerators are going to be the same as each other.

The greater the denominator, the smaller the fraction.

Both these fractions have the same numerator, so we compare the denominators.

Nine is greater than seven.

So four ninths must be smaller.

Let's see if you can put that into practise.

Number one, shade the fractions and then compare them.

Number two, compare these numbers using those inequality symbols.

And remember what we said about the size of fractions, size of the denominator.

So the bigger the denominator, the smaller each equal part is.

Number three, starting with the smallest, order these sets of numbers.

And notice once again, please, that the numerator in each set are all the same.

So in A, they're all three.

In B, they're all six.

And in C, they're all 10.

It's the denominators that are different.

Remember that rule about what happens to the size of the fraction when the denominator increases, right.

I think you are ready, and I think you're going to do really well on this.

In fact, I know you are.

So good luck and I'll see you soon for some feedback.

Welcome back.

How did you get on with that first set of practise questions? Would you like some answers? Yes, of course, let's do that.

So number one, shade the fractions and then compare them.

So this is what three fifths looks like, and that's what three quarters looks like.

And you can say three fifths is smaller than or less than three quarters.

And for B, that's what the fractions will look like when shaded in.

That's what the bars look like.

That's four sixths, four eighths.

You can quite clearly see that four sixths takes up more room.

So therefore four sixths is greater than four eighths.

So remember the numerators were the same, the denominators were different.

When the denominators increased, the size of the parts decreased.

And then here you didn't have bars, but hopefully you knew the rule by this point.

So compare these numbers using those symbols, those inequality symbols.

A, we've got four eighths is greater than four tenths.

B, three eighths is less than three sevenths.

C, 99, two hundredths is less than 99 one hundredths.

D, two thirds is greater than two quarters.

E, six 20ths is less than six fifteenths.

And finally eight one thousandths is less than eight 999ths only by a very, very small amount.

But it is.

Number three.

Starting with the smallest, order these sets of numbers.

So that would go three eighths because eight is the largest denominator there.

The numerator are all the same.

So that would mean that the parts will be the smallest.

So three of those will be smaller than three fifths or less than, and they will be smaller than three quarters.

And then for B, these are all six something.

So six 20ths is less than six twelfths is less than six sevenths.

And for C 10 25ths is less than 10 fifteenths, which is less than 10 elevenths.

So you might have noticed that the denominators were decreasing in size.

You're doing very, very well.

And I think you are ready for the next cycle, which is comparing mixed numbers that have the same whole.

So far we've been comparing proper fractions that have the same numerator, but different denominators.

Let's see if we can extend that and apply that to mixed numbers.

Let's revisit our mixed numbers.

Here they were.

We had one and four sixths and one and four fifths.

Because we know that four sixths is smaller than four-fifths, one and four sixths must be smaller than one and four fifths.

Would you agree? What do you think? How could we prove that? What could you do? Could you draw a model? For example, what's about a part, part whole model? They're always really good for looking at the composition of fractions.

So let's look at the composition of these mixed numbers.

One and four sixths.

That's the whole that's split into two parts.

We've got a whole number part, that's one and a fractional part, that's four sixths.

One and four fifths has also got two parts, a whole number part, which is one again, and a fractional part, which is four fifths.

So the whole number parts are the same.

So it's not really helpful to compare those.

They're the same.

We need to compare the fractional parts.

The fractional parts of both mixed numbers have the same numerator, so we need to compare the denominators.

Sixths are smaller than fifths.

We've already proved that today.

So one and four sixths must be smaller than one and four fifths.

We can also represent this using a different model.

Let's prove it in a different way.

Look at these circles.

What do you notice about them? Well, the first one's split into six equal parts.

So they're sixths.

And the second one's split into five equal parts.

So they're fifths.

And we've got all six of those equal parts, all five of those.

So that's representing the one part.

That's the whole number part.

And then we've got a fraction part.

So here we've got four sixths, and here we've got one and four fifths, making the mixed numbers one and four sixths and one and four fifths.

So as we said before, it's not very helpful to compare the whole number parts because they're the same.

It's more helpful to compare the fraction parts 'cause they're different.

Less of the second whole is shaded for one and four sixths, so it must be smaller.

And hopefully you can see that.

You can use an inequality symbol to express that.

You can say one and four sixths is smaller than or less than one and four fifths.

What have you noticed about how we compare mixed numbers that have the same whole number parts? Is there a rule do you think that we can start to formulate here? If we have the same whole number parts as we did in this case, then it's the size of the fractional parts that determines which number is greater.

Andeep says, first, we compare the whole number part, if this is the same and it was in this case.

Then we compare the fractional part.

And in this case it was different.

Look at these numbers, which is the greatest? Before you do that, have a look.

See what you can notice about the numbers, what's the same, what's different, and then which is the greatest? You might want to express that with words or inequality symbols or both.

Pause the video and have a go.

Which was the greatest? Let's have a look.

It was this one, but why? Did you come up with a good reason? Could you explain that? The whole number parts are the same.

So they were both three.

So we need to compare the fractional parts.

They're different, but they've got something in common, the fractional parts, haven't they? That's the numerator.

The numerators are the same.

They're both 12.

So we need to look at the denominators.

They are different.

The smaller the denominator, the greater the fraction.

We've proved that time and again, haven't we? This means that 12 20ths are greater than 12 30ths.

12 20ths would take up more space in a model, proving that it's greater.

But in this case, we didn't need a model, did we? We've got a rule.

So it was nice and quick and efficient.

I think you are ready for some final practise.

You're doing very, very well.

So number one, complete these equations using the symbols less than or greater than.

So have a look.

Look at all of these examples.

See what you notice.

Is there something that each set has got in common? Hmm.

And what's different about that bottom set in C? Number two, Andeep has spilt his drink on his homework.

Oh dear, Andeep, accidents do happen.

A, what could the missing digit be? And there's more than one possibility for that.

So see if you can find more than one.

And B, what could the missing digit not be? And see if you can explain why as well.

Why could it not be that? Number three, starting with the smallest, put these numbers in order and then explain your reasoning.

Why have you put them in the order that you've chosen? Before you start, look carefully at the numbers.

What do you notice? Good luck with that.

I'm sure you're going to be absolutely amazing.

If you can work with a partner, I always recommend that.

Then you can bounce ideas off each other.

Pause the video and I'll see you soon.

Welcome back.

How did you get on with that final set of practise questions? Let's give you some answers so you can see.

So number one, complete these equations using those inequality symbols.

And you might have noticed in A, the numerators were all five.

So five eights is less than five sixths.

So therefore one and five eights is less than one and five sixths.

Now they've got the same whole number part.

So we don't even need to look at the whole number part.

We can just focus on the fractional part.

And then 10 and five eights is less than 10 and five sixths.

You could have had any number there, couldn't you? 47 and five eighths and 47 and five sixths, for example, doesn't matter.

We've already proved that it's a fractional part that we're comparing.

And then for B, starting with that proper fraction, again, four sevenths is greater than four ninths.

So therefore three and four sevenths is greater than three and four ninths.

And then we could pick any number, couldn't we here, for the whole number? Because it's the same.

99 and four sevenths is greater than 99 and four ninths.

So once again, we're proving that when the whole number is the same, it's the fractional part that we need to pay attention to when comparing.

And then we had three fractions here.

So five elevenths is greater than five fifteenths, which is greater than five nineteenths.

They're the proper fractions.

And then once again, we could have chosen absolutely any whole number as long as it's the same.

Five and five elevenths is greater than five and five fifteenths, which is greater than five and five nineteens.

The whole number part was the same.

It's the fractional part that we're comparing.

And number two and Andeep has spilt his drink on his homework.

We've all been there, haven't we? Hopefully it was water.

What could the missing digit be? Well, we're looking for a digit that is less than 11, but greater than five that would fit there.

So it could be 10, nine, eight, seven or six.

They're all the possibilities.

Well then if you've got any of those, but the special well done if you've got all of those.

And what could the missing digit not be? Well, the missing digit could not be less than or equal to five or greater than or equal to 11.

So well done, if you've got that, and well done, if you explained that nice and clearly.

And number three, starting with the smallest, put these numbers in order and explain your reasoning.

Now, I don't know about you, but one of those jumped out at me and that was the three.

So that was a whole number without a fractional part with it.

So that had to come first.

And then did you notice that we've got three fractions that have got three for the numerator? So we've got three and three eighths, three and three fifths, and three and three ninths.

So we need to start thinking about those.

Well, the next one then had to be three and three ninths because nine is the largest of those denominators.

So therefore the parts are the smallest.

So three and three ninths would take up a smaller part, and then it will be three and three eighths, and then it will be three and three fifths.

And then did you notice that the next one had the same denominator but different numerator? Well, three and four fifths, hopefully you would agree, is slightly greater than three and three fifths.

It's got one more in the numerator part.

And then finally five, which is greater than all of the others because it's a greater whole number.

So your explanation might have been slightly different.

You might have said something like this.

Three had to be the smallest because it has no fractional part and will be first on a number line.

All the mixed numbers except three and four fifths have the same whole number part and the same numerator in their fractional part.

So the denominators were compared.

Remember this rule, the greater the denominator, the smaller the fraction.

That's been a helpful rule today, hasn't it? Three and four fifths could then be compared with the other mixed number that had the same denominator.

The greater the numerator, the greater the fraction.

So that was a slightly different rule.

We've come to the end of the lesson.

You've been fantastic.

Today, we've been comparing mixed numbers where the denominators of fractional parts are different.

When comparing mixed numbers, if they have the same whole number parts and you've seen that time and time again today, then it's the size of the fractional parts that determines which number is greater.

When we compare fractions that have the same numerator, the greater the denominator, the smaller the fraction.

And again, you prove that time and again with bar models and circles and all that kind of thing.

And the knowledge of comparing proper fractions can help when comparing mixed numbers.

And you've applied those skills today very nicely.

Well, I think you've been incredible today.

So I think you should give yourself a pat on the back.

I've thoroughly enjoyed spending this maths lesson with you, and I hope to get the chance to spend another maths lesson with you in the near future.

But until then, have a great day, whatever you've got in store.

Take care and goodbye.