Lesson video

Hello there.

My name is Mr. Tilston.

If I've met you before, it's nice to see you again, and if I haven't met you before, it's lovely to meet you.

Today's lesson is all about mixed numbers and I'm confident that you've learned about mixed numbers before and that you are ready to start applying those skills to an even higher level.

So let's do that.

If you are ready, let's begin.

The outcome of today's lesson is I can accurately label a range of number lines and explain the meaning of each part.

And specifically we're going to be dealing with number lines that involve mixed numbers.

Our keywords today, my turn, mixed number, your turn.

And my turn, interval.

Your turn.

I'm confident that you know what a mixed number is.

Maybe you're not quite so sure about interval.

Let's have a look.

Let's have a reminder on both.

A mixed number is a whole number and a fraction combining to one mixed number.

So for example, here we got one and a half, that's a mixed number.

And you might note that although I said and, you don't write and in the mix number.

You write it just like that.

And an interval is a space between two values or points.

So you might have had experience of intervals on graphs or number lines before.

Our lesson today is split into two cycles.

The first will be counting in fractions using number lines and the second, labelling number lines.

So let's begin by thinking about counting in fractions using number lines.

And in today's lesson, you're going to meet Jacob and Sophia.

Have you met them before? They're here today to give us a helping hand with the maths and very helpful, they are too.

So let's look at this number line.

What do you notice? Take your time.

Have a good look.

Do what good mathematicians do.

They notice things.

What can you see, anything at all? Well, there are five parts between zero and one, and that means that each part has a value of one-fifth.

So five parts, that means they're fifths.

So that will be one-fifth.

This will be two one-fifths, or two-fifths.

Here we've got three one-fifths or three-fifths.

Here we've got four one-fifths or four-fifths.

And finally, we've got five-fifths.

What do you notice about five-fifths? It's equivalent to one in this case.

Five-fifths is equivalent to one whole.

We say one and we know that one is composed of five-fifths.

So let's count up in fifths.

Zero.

One-fifth.

Two-fifths.

Three-fifths.

Four-fifths.

Now I could say five-fifths.

I'm not going to say that.

We're just going to say one.

The whole number part, we know that it means five-fifths.

So let's extend that number line beyond one.

Let's go beyond one.

What do you notice? So again, take your time.

Have a good look at that number line.

What can you see? What happens before one? What happens after one? What's different? What's the same? There are five parts between zero and one.

This means we are still counting in units of one-fifth.

So all the way across, we're counting in fifths.

So let's counting units of one-fifth.

So zero, one-fifth.

Can you join in with me? Two-fifths.

Three-fifths.

Four-fifths.

And then five-fifths is equivalent to one whole, but we're not saying five-fifths in this case.

We are saying one or one whole.

One whole is composed of five-fifths.

And then let's carry on.

So we've got one and one-fifth, one and two-fifths, one and three-fifths, one and four-fifths, two.

So one and five-fifths is equivalent to two wholes.

We wouldn't say one and five-fifths.

We would say two or two wholes.

Two wholes are composed of one and five-fifths.

Let's carry on.

Two and one-fifth, two and two-fifths, two and three-fifths, two and four-fifths, and then two and five-fifths.

We wouldn't say that, but it's equivalent to three wholes.

Three wholes are composed of two and five-fifths.

Three and one-fifth, three and two-fifths, three and three-fifths, three and four-fifths.

And finally, four.

Three and five-fifths is equivalent to four wholes.

Remember, we do not say three and five-fifths.

We say four.

Four wholes are composed of three and five-fifths.

We could carry on with that number line.

We could keep going forever, but let's stop there.

What do you notice about the numbers marked underneath the line? Well, Jacob says, "Numbers smaller than one only have a fractional part." Can you see that? So look at the numbers before one, yes.

One-fifth, two-fifth, three-fifth, four-fifth.

Just a fractional part.

Some numbers do not have a fractional part.

Did you notice that? On this number line, we've got one, two, three, and four for example.

Non-whole numbers greater than one are written as mixed numbers.

So the first one that we can see after one is one and one-fifth.

That's a mixed number.

The fraction one-fifth appears after each whole number.

Let's have a check for understanding, shall we? Let's see how you are getting on.

Look at the number line, which mixed number is missing? So it's the number in between two and two and two-fifths.

Pause the video and have a go.

Did you spot it? The mixed number two and one-fifth is missing.

And that is how we write two and one-fifth.

Well done if you got that.

The fraction one-fifth appears after each whole number on this number line.

Did you notice? So we've got one that's whole number then one and one-fifth.

Two, that's whole number, two and one-fifth.

Three, that's whole number, three and one-fifth.

Let's look at this number line.

What do you notice? So take some time.

So you notice you might want to have a little count.

This is what Jacob's noticed.

Let's see if you agree.

He says there are seven marks between zero and one.

The number line is increasing in units of one-seventh.

What do you think? Well, now he's not right there.

He is incorrect.

But can you explain why? Have a little think.

What could you say to Jacob if he was here with you now? What would you say? Can you see what mistake he's made? Let's have a look more closely.

We need to count the equal parts in between zero and one and not the marks.

So it's the parts that we are counting, not the marks.

I'll show you what I mean.

Here we've got the marks, that's a mark and there are seven of them, and that's what Jacob's counted, but that's not what you're supposed to count.

We count the parts.

These are the parts, and there are eight of these.

So one, two, three, four, five, six, seven, eight.

So Jacob says we're counting in sevenths.

What do you think? It's not sevenths, is it? What is it? It's eighths.

So we need to count the equal parts in between zero and one, not the marks.

There are eight equal parts between zero and one.

That means that the number line is increasing in units of one-eighth.

Yes, it is.

So let's counting units of one-eighth.

Do this with me please.

One-eighth, two-eighths, three-eighths, four-eighths, five-eighths, six-eighths, seven-eighths, one.

Remember we could say eight-eighths, but we don't.

We'll say one the whole number.

So eight-eighths is equivalent to one.

We say one, but we know that one is composed, in this case, of eight-eighths.

Now let's going.

One and one-eighth.

One and two-eighths.

One and three-eighths, one and four-eighths.

One and five-eighths.

One and six-eighths.

One and seven-eighths.

What am I going to say next? Careful on this one.

It's not one and eighth-eighths, is it? What do we say? Two.

One and eight-eighths is equivalent to two wholes.

Two wholes is composed of one and eight-eighths.

We would not say one and eight-eighths.

So we would say two.

Let's keep going.

Two and one-eighth.

Two and two-eighths.

Two and three-eighths.

Two and four-eighths.

Two and five-eighths.

Two and six-eighths.

Two and seven-eighths.

And then the tricky part, what's next? Three.

So if you said two and eight-eighth, that's not correct, but three is.

Two and eight-eighths is equivalent to three wholes.

Three wholes is composed of two and eight-eighths.

Let's have a little check.

Look at the number line, which mixed number is missing? So it's in between two and two and two-eighths.

What is the missing mixed number? Pause the video and have a go.

Did you get it? Did you spot it? The mixed number two and one-eighth is missing.

The fraction one-eighth appears after each whole number on this number line.

So just like before, after one is one and one-eighth, after two is two and one-eighths.

So we can't see it.

But what would be after three? Three and one-eighth.

It's time for some practise.

Complete these number lines.

So have a look very carefully at what you can see each time.

Look at the denominator, look what it's going up in.

Look at the numerator, look at what's missing.

Number two, circle the greater number in each of these pairs and give reasons for your choice.

Jacob says, "When we count on, we say smaller numbers first.

Right, pause the video, and off you go.

Welcome back.

How did you get on with that? Number one, complete these number lines.

So in the first example, we were looking at values, all of them under one.

So we're just looking for a fraction, and that fraction is three-fifths.

Three-fifths is in between two-fifths and four-fifths.

And for the next one it goes one third, two thirds, one.

And then remember, after one we're going to have one for the numerator in the mixed number.

So that's one and one third, one and two thirds, two, two and one third.

What's next? Two and two thirds, three.

And then the greater number in each of these pairs and give reasons for your choice.

Well, six-eighth is greater.

While counting on in eighths, we say three-eighths before six-eighths.

So six-eighths must be greater.

And for this one, three and eight-ninths is greater.

When counting on in ninths, we say three nights before eight-ninths.

So eight nights must be greater.

And here two and one third is smaller than two.

So two and one third is greater than two.

It is further along the number line.

You might have mentioned something like that.

You're doing very, very well and I think you are ready for the next cycle, which is labelling number lines.

Sophia has been asked to label this number line.

Have a look at it.

What would you do if you were labelling this number line? Where would you write the fractions? She says, "First we need to determine what unit we are working with." Now, can you remember from before what not to do and what to do? What should you count? What should you not count? Remember the mistake that Jacob made earlier on.

And Sophia's got a good idea here.

She says, "Let's use a stem sentence to help.

The stem sentence is this.

The line is divided into equal parts.

This allows us to count in.

Okay, let's see if we can fill that in.

How many equal parts is that line divided into? You're not counting the marks, counting the equal parts.

So there's one, two, three, and four.

So it's divided into four equal parts.

And that means we are counting in quarters.

The number line can now be labelled working systematically.

So we're not just going to add numbers to it randomly.

We're going to be systematic.

So let's start here.

That's one quarter, two quarters, and you've guessed it, three quarters.

Now what would you do for the next number? What could you write? What's correct? We could say four quarters, but four quarters is equivalent to one whole.

So that's what we're going to do and that will help us when we're writing the mixed numbers.

One whole is composed of four quarters.

Sophia says, "Did you notice that the denominator is the same as the number of equal parts from zero to one Sophia's, right? There are four equal parts and the denominator is four.

Let's have a quick check for understanding, shall we? Look at this number line, determine the unit we are working with by using the stem sentence and then label it working systematically.

So the stem sentence is this.

The line is divided into equal parts.

This allows us to count in.

Pause the video and have a go.

How did you get on? Let's have a look.

Well, it's divided into seven equal parts.

So that means we're counting in sevenths, that's our fraction.

So one-seventh, two-sevenths, three-sevenths, four-sevenths, five-sevenths, six-sevenths, and then one, which we know is the same as seventh-sevenths.

The denominator is a seven, which is the same as a number of equal parts between zero and one.

Let's look at a different number line.

Have a look.

What do you notice this time? How's it changed? Sophia says, "Let's use that stem sentence to determine the unit that we are working with." Great idea, Sophia.

It worked before.

And the stem sentence is this.

This slide is divided into equal parts.

This allows us to count in, Well, this is what Sophia says.

Have a listen and see if you agree.

She says the line is divided into 12 equal parts.

This is allows us to count in twelfths.

Is that right? Is that what you would've said? No, she's incorrect.

Can you explain why? Where's she gone wrong? I can see the 12 equal parts.

We need to adapt our stem sentence because it's going beyond one.

This is an interval.

It's a space between zero and one.

So we call that an interval.

That's another interval and another interval.

So there are three equal intervals that are equal to one.

There they are.

Each marked interval is divided into the same number of equal parts.

So the marked intervals are one, two, and three.

So we need to adapt our stem sentence.

Each interval between the whole numbers is divided into equal parts.

This allows us to count in.

let's see if we can make a little change.

An interval on this line is a space between a consecutive whole number.

So it does not matter which two intervals we choose, but we are looking for how many equal parts there are between those two intervals.

"Sophia says an interval on this line is the space." between a consecutive whole number.

So it's a space between one and two or two and three or zero and one.

They're consecutive whole numbers.

Each interval between the whole numbers is divided into four equal parts.

This allows us to count in quarters.

Yes.

So shall we label this number line? Count along with me.

We've got one quarter, two quarters.

three quarters, and then we could say four quarters, but we're going to stick to our whole number.

That's one.

One in one quarter, one in two quarters, one in three quarters, two, not one in four quarters, two.

And then what's after two, two and one quarter.

So one is always the numerator after the whole number when we use mixed numbers.

Two and two quarters, two and three quarters and three.

We don't say two and four quarters.

Let's have a check.

look at this number line.

Determine the unit that we are working with by using the stem sentence and then label it working systematically.

So the stem sentence is this.

Each interval between the whole numbers is divided into equal parts.

This allows us to count in.

So be very careful.

Try not to make that mistake that Sophia made initially.

Pause the video and have a go.

<v ->Let's see.

</v> I wonder If you counted the equal parts between zero and one or the equal parts between one and two.

Both are fine.

So each interval between the whole numbers is divided into six equal parts.

And this allows us to count in sixths.

So that's the answer.

Well done if you got that one.

One-sixth, two-sixths, three sixths, four-sixths five-sixths, six-sixths, one, and one and one-sixth, one and two-sixths, one and three-sixths, and one in four-sixths, and one in five-sixths, two.

The denominator is a six, which is the same as a number of equal parts in each interval.

Time for some final practise.

Label these number lines.

Use the stem sentence to help.

Each interval between the whole numbers is divided into equal parts.

This allows us to count in.

Now you might notice that for these, they all go from zero to one.

So we're not looking for any mixed numbers.

And number two, label these number lines.

Use a stem sentence to help each interval between the whole numbers is divided into equal parts.

This allows us to count in.

You might notice something slightly different about one of those number lines.

Can you spot it? Number three, Jacob has labelled this number line incorrectly.

Explain his mistake.

You might need to have a little bit of time looking at that and seeing if you can spot Jacob's mistake.

And you might like to use a stem sentence to help you explain the mistake.

Number four, how much water is in this container? Write your answer as a mixed number.

So it's just like a number line all over again, isn't it? Except it's on a container.

So good luck with that.

If you can work with somebody else, I always recommend that.

Then you can share ideas with each other.

Pause the video and off you go.

Welcome back.

Let's have a look at some answers.

So number one, label these number lines.

Use a stem sentence to help.

So each interval between the whole numbers is divided into five equal parts.

This allows us to count in fifths for the first one.

So one-fifth, two-fifths, three-fifths, four-fifths.

And then for the next one, it's divided into eight equal parts.

And this allows us to count in eighths.

Ready? One-eighth, two-eighth, three-eighth, four-eighth, five-eighth, six-eighth, seven-eighth, one.

And for the final one, this time it's divided into nine equal parts.

This allows us to count in ninths.

Ready? One-ninth, two-ninths, three-ninths, four-ninths, five-ninths, six-ninths, seven-ninths, eight-ninths.

And number two, label these number lines using that stem sentence to help.

So let's start with A.

Each interval between the whole numbers is divided into five equal parts.

And this allows us to count in fifths.

For B, each interval between the whole numbers is divided into eight equal parts and that allows us to count in eighths.

For C, each interval between the whole numbers is divided into three equal parts and this allows us to count in thirds.

And then Jacob made a little mistake here, didn't he? Did you spot it? Could you explain to him where he went wrong and how he could improve it? But you might have used that stem centre to help you explain the mistake.

Each interval between the whole numbers is divided into five equal parts.

This allows us to count in fifths.

So Jacob has counted incorrectly in quarters.

Jacob has not understood that one is composed of four quarters.

So well done if you said anything like that.

And then how much water is in this container? Write your answer as a mixed number.

Each interval between the whole numbers is divided into five equal parts.

This allows us to count in fifths.

There is one and four-fifths of a litre of water in the container.

And that is how we write that, one and four-fifths.

Well done if you got that.

We've come to the end of the lesson.

It's gone so quickly.

Today's lesson has been accurately labelling a range of number lines.

Mixed numbers are numbers that can be located on a number line.

The stem sentence, each interval between the whole numbers is divided into equal parts.

This allows us to count in, supports us to determine the unit that we are working with.

And remember, don't make that mistake that Jacob made earlier on where he counted the marks.

You need to count the parts between the intervals and that's it.

I think you need to give yourself a little pat on the back.

It's well deserved.

You've been amazing.

I hope I get the chance to spend another math lesson with you in the near future.

But until then, enjoy the rest of your day.

Whatever you've got in store, be successful at whatever that is.

Take care and goodbye.