Lesson video

It's Mr. Whitehead ready for your maths lesson.

You're going to need a couple of things pen or pencil and some paper and your practise task from last lesson.

Press pause while you get your things together and come back when you're ready.

Let's take a look at the task.

Can you hold it up for me so I can see how you got on with those three problems from last lesson.

Looking good.

Let's take a look at each part.

So four inequality symbols to fill in.

I've left the generalisation from last time on the page.

Give it a read with me.

One, two, three.

When comparing unit fractions, the greater the denominator, the smaller the fraction.

Did you keep that in your mind while you were solving these problems? Let's take a look at what the missing symbols are? Greater than.

Less than.

Less than.

Less than.

Is that what you got? Wait, which one? The last one is not right.

1/7 is less than 1/12.

When comparing unit fractions the greater the denominator, the smaller the fraction.

Ah, so it should be, 1/7 is greater than 1/12.

Good spots.

Second part.

Arrange the following numbers in order.

So again the generalisation is there.

That's what I was thinking about as I was ordering the fractions and this is what they got.

1/3, 1/7, 1/9, 1/10 and 1/15.

Is that what you got.

You didn't? When comparing unit fractions, the greater the denominator, the smaller the fraction.

So, I've got my larger fractions on the left and they get smaller.

1/3 is the largest and they start to get smaller.

Why is that wrong? Ah, look at the question Mr. Whitehead.

Arrange the following numbers in order from smallest to largest.

So is this what you've got? Is that better? Fantastic.

Thanks for your help.

Third part.

So we have two inequalities, two statements to complete with missing denominators.

What did you get for 1/6 is less than something? Say it again.

Ah, there's lots of options.

How many options are there though? How many different denominators could we fill in? So 1/6 is less than.

So it couldn't be a denominator of seven.

It couldn't be 1/7 because 1/7 is smaller than 1/6.

So the denominator has to be five, four, three or two.

Any of those is your denominator would be correct.

How about the second one? One something is greater than 1/8.

So again what could the denominators be? Tell me what you got for this one.

Say it again.

Again I've had lots of different options but how many options could there be? How many do you think they could be? What could the denominator not be? It couldn't be a denominator that's greater than an eight, a larger number than eight because then 1/9 for example is less than 1/8.

So the number has to be smaller than eight.

So it could be 1/7, 1/6, 1/5, 1/4 1/3 or again 1/2.

Any of those would be correct and well done if you were able to spot how many possibilities there were.

Right, this lesson up until now we have been looking at unit fractions and comparing unit fractions.

But what is a unit fraction? How would you define it? Tell me.

Good.

A fraction with a numerator of one.

So, in this lesson we're going to be working with non-unit fractions.

If that's what a unit fraction is, what would you change in the definition to give me a definition for a non-unit fraction? What would you change? Super.

A non-unit fraction is a fraction with a numerator greater than one.

So today in this lesson we're going to be comparing non-unit fractions.

We'll be using our symbols greater than and less than.

Here's our first problem.

Now let me tell you the problem has been solved.

Your task is going to be to prove it is correct.

What does it mean to prove something.

You're going to give me some reasons.

You're going to be giving me some evidence, some proof that what's in the problem is right.

Let's take a look at it.

Can you read it with me? Izzy and Olivia each had an identical juice bottle, identical, the same.

Izzy's juice is 2/3 full.

Olivia's juice is 2/4 full.

Izzy has more juice than Olivia.

Suggest ways to prove that 2/3 is greater than 2/4.

Now notice the fractions in previous lessons they were unit fractions.

In this lesson they're non-unit fractions.

Why? Because they have numerators greater than one.

What else do you notice about those fractions? Yes, they are non-unit fractions but what else do you notice? Good.

The denominators are different but the numerators are the same.

They're both two.

Okay, In a moment press pause to support your thinking as you give me some proof that 2/3 is greater than 2/4.

In previous lessons you have used diagrams, number lines and verbal reasoning.

At least one of these please, at least one of these use to give me your proof that 2/3 is greater than 2/4.

Challenge yourself by using more than one piece of evidence, by using more than one methods.

Press pause and come back when you're ready.

How did you get on.

Anyone use method one, a diagram to prove that 2/3 is greater than 2/4.

If you use the diagram, hold it up for me.

Let me have a look.

Looking good.

You're showing me 2/3 is greater than 2/4.

Here's mine.

I started off my diagram making it look like a bottle because of course it's about juice, isn't it? We're comparing amounts of juice.

So one bottle divided into three equal parts and the other into four equal parts.

There's Izzy's juice, two of those equal parts and Olivia's juice, two of the four equal parts.

And we can see Izzy's juice is greater than, there's more of it than Olivia's.

Now I noticed some of you used circles.

For one of the circles we've divided it into three equal parts and the other circle into four and we're going to represent 2/3 and 2/4 to help us compare and to gain see which of those fractions is Largest Of course, 2/3.

How about method two? Who used the number line? Hold up for me.

Let me take a look at the number lines.

Super.

Two number lines the same length because we're comparing two fractions that represent an amount of liquids, an amount of juice in the same sized bottle.

So the space between zero and one, I'm dividing into three equal parts on one number line and four equal parts on the other number line.

I can represent 2/3, and I can represent 2/4.

And as I look at my number lines I can see 2/3 is closest to one and it's the furthest from zero.

What do you notice about 2/4? Is it closer to one or it's halfway between zero and one? However, 2/3 is closest to one, is furthest from zero.

It's the greater fraction.

Method three.

Now verbal reasoning, using your words to explain.

So maybe you wrote down a few things to help you with this.

And what I'd like to do now is show you a way of structuring the sentences you used to give your evidence.

Let me show you.

2/3 is two lots of 1/3.

Can you fill in the gaps on this sentence for the other fraction.

Say it.

2/4 is two lots of 1/4.

Now we already know comparing 1/3 and 1/4 that 1/3 is greater than 1/4.

We know that because when comparing unit fractions, the greater the denominator, the smaller the fraction.

And if we look at our containers and our juice in those bottles we can see 1/3 is greater than 1/4.

We know that already.

So, I know that two lots of 1/3 is greater than two lots of 1/4.

I can compare fractions that have the same numerator using what they know about unit fractions and I can explain that verbally like I've just done.

2/3 is greater than 2/4.

So our three methods, diagram, number line and verbal reasoning.

Is there one that you think was easiest for you to use? Smartest for you to use? Is there one that you found more challenging? I've got another problem for you.

Push yourself again this time when you pause to use at least two of the methods and try to make one of them verbal reasoning.

True or false? So this time you need to solve the problem and tell me if what's in the problem is true or false but again with some proof.

Give the problem a read with me.

Sam and Jay are both reading the same book.

Sam has read 4/6 of her copy.

Jay has read 4/5 of his copy.

Is it true or false that Sam has read less than Jay? That 4/6 is less than 4/5? Again, press pause and like I said push yourself to using at least two methods to prove whether this is true or false and try to make one of those methods method three.

Give me some verbal reasoning.

Press pause and come back when you're ready to take a look How did you get on? Who had to go at method one? hold up your diagrams. Show me the diagrams you've used to show me whether or not it's true or false that 4/6 is less than 4/5.

I see some circles like these.

I see some rectangles as well.

Okay, let's have a look.

So one of the circles is representing six divided into six equal parts and the other into five equal parts.

I am going to represent 4/6 and 4/5.

I can see 4/6 is less than 4/5.

So it's true.

Some of you used rectangles.

Again you divided your rectangles into six and five equal parts.

And we're representing four of those sixths, four of those fifths.

Now, when I looked at these rectangles next to each other, side by side it was a little bit difficult to see which one was greater.

So, we can also arrange them like this, vertically and we can clearly see that 4/6 is less than 4/5.

It's true.

If you had to go at method two hold up your number lines for me.

Those number lines equal in length.

Have you divided one of the number lines into six equal parts and the other into five equal parts? And have you marked one, two, three, four, six and one, two, three, four, fifths on your number lines.

So how does this help you to see whether or not it's true or false? 4/6 is closest to zero.

4/5 is furthest from zero.

Or we can say 4/6 is the furthest from one out of the two and 4/5 is closest to one out of the two fractions we're comparing.

The diagram and now the number line have proved that it is true, 4/6 is less than 4/5.

Now I did challenge you to all have a go at using verbal reasoning.

Here are the sentence stems to help you again.

Can you join me from the beginning in using verbal reasoning to prove whether 4/6 is less than 4/5.

We know it's true already but let's have a go with our verbal reasoning as well.

Okay, so 4/6 is four lots of 1/6, good.

Second fraction for this.

Ready? 4/5 is four lots of 1/5.

Super.

Now let's use what we know about unit fractions.

I know that something is less than something.

Ready? I know that 1/6 is less than 1/5.

How do we know that? On a number line 1/6 is less than 1/5 and with our generalisation, when we compare unit fractions read the rest of the sentence with me.

The greater the denominator, the smaller the fraction.

1/6 is less than 1/5.

So how does that help us with 4/6 and 4/5? So ready? Four lots of 1/6 is less than four Lots of 1/5.

Look on the number line, 4/6 is less than 4/5.

Fantastic.

So we have used three methods, a diagram, a number line and verbal reasoning to compare non-unit fractions.

Although those non-unit fractions of course have the same numerator.

For your practise activity to have a go at before the next lesson, I've given you three statements to compare and to fill in the missing inequality symbols.

The challenge that needs you to think when the denominator is a larger which method is the smartest to use to help you to compare the fractions and why? So we've used diagrams, number lines, verbal reasoning.

But for those for example in two and three, for those large denominators, which of the methods is the smartest to use.

I've left on the screen as well the sentence stems that we were using to explain verbally and the generalisation that we were using to compare unit fractions.

Use those to support your thinking as you solve your practise activity and challenge yourself as well if you're ready.

Well done for working so hard on this lesson where we've been comparing non-unit fractions.

Press pause, take a copy of the activity and have it ready for the start of the next lesson.

Bye.