Lesson video

Hello there.

My name is Mr. Tilstone.

How are you? Are you having a good day? I'm having a great day, and it's about to get even better.

Do you know why? Because I'm going to teach you a lesson all about fractions.

And fractions is perhaps my favourite part about maths.

So if you are ready, I'm ready.

Let's begin.

The outcome of today's lesson is this.

I can express a quantity as a mixed number and an improper fraction.

And we've got some keywords.

My turn.

Mixed number.

Your turn.

My turn.

Improper fraction.

Your turn.

Are you familiar with those words? I think you probably are, but let's have a reminder anyway just in case.

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

For example, 1 1/2.

Can you think of a different mixed number? An improper fraction is a fraction where the numerator, that's the top number, is greater than or equal to the denominator, that's the bottom number.

So for example, 5/3 and 9/8 are improper fractions.

Could you maybe think of another improper fraction? Our lesson today is split into two cycles.

The first will be express mixed numbers as improper fractions and the second problem solving.

So for now, let's focus on expressing mixed numbers as improper fractions.

And in today's lesson, you're going to meet Sofia and Jacob, have you met them before? They're here today to give us a helping hand with the maths.

Sofia and Jacob recap some of their prior learning to help them make connections to this new learning.

Always a good idea.

When a whole is divided into four equal parts, the unit we work with is quarters.

And you knew that, didn't you? I'm sure you did.

There are four quarters in one whole.

Here there are two wholes.

What else can you see apart from two wholes? This is two groups of four quarters, which is eight quarters.

To determine the number of quarters in two wholes, we use our four times table.

That's a denominator that we've established.

Two fours are eight.

Jacob and Sofia continue to recap some of their prior learning to help them make connections.

When a whole is divided into five equal parts, the unit we work with is fifths.

You knew that, didn't you? I think you did.

There are five-fifths in one whole.

Here there are two wholes.

How else could we describe that? Hmm.

This is two groups of five-fifths, which is 10-fifths.

Jacob says, "To determine the number of fifths in two wholes, we use our five times table." So because it's fifths, it's a five times table.

Two fives are 10.

That's why it's 10-fifths.

I wonder what unit we will think about if wholes are divided into six equal parts? Hmm, six equal parts.

We will think about sixths and our six times tables.

And what about wonders Jacob, if wholes are divided into seven equal parts? I think you know the answer to this.

We will think about sevenths, so Sofia and our seven times tables, of course.

Let's do a little check.

If a whole has been divided into 10 equal parts, which unit do we need to think about? Do we need to think about quarters or fifths or eighths or tenths? Hmm.

Pause a video.

Hopefully, that was a nice quick, easy one for you.

It's tenths.

Let's look at this mixed number.

So it's 6 and 1 something.

So what do we know and what don't we know? We know the whole number part.

That's six.

We know the numerator.

That's one.

We don't know the denominator.

We don't know the number of equal parts that it's been split into.

How would we express this as an improper fraction? Well, Jacob says, "We can't determine the improper fraction from this because we need to know the unit that we are working with.

The denominator tells us the unit that we are working with." Let's look at this mixed number.

What do we know and what don't we know? Well, we do know the whole number.

That's three.

And we do know this time the denominator.

That's nine.

What we don't know is the numerator.

How would we express this as an improper fraction? Jacob says, "We can't determine the improper fraction from this because we need to know how many parts we already have." And we don't quite know that.

We almost know how many parts but not quite.

And little bit of information missing.

"The numerator tells us how many parts we already have." We can use Jacob's learning to form a generalisation.

We can calculate the number of parts in an improper fraction by multiplying the whole number by whatever the denominator is.

So in this case, 6 times something.

We dunno there's something yet.

And in this case 3 times 9.

And then we add the numerator.

And we do know that in this case that's 6 times something plus 1 for the first example.

And for the second, 3 times 9 plus something.

So we've got a little bit of missing information.

Let's look at this mixed number.

What's this? This is 3 1/7.

How could we write this as an improper fraction? Based on what we've just learned and discovered.

We could represent this.

Is a representation of 3 1/7? So seventh is our denominator.

That's a unit that we're working with.

Here's our three lots of 7 sevenths and there's our one extra one seventh.

This is three groups of 7 sevenths.

And one more seventh.

This is 22 sevenths.

7, 14, 21, 22.

3 1/7 is equivalent to 22/7.

So now, we've got a mix number 3 1/7 and an equivalent improper fraction, 22/7.

Another strategy is to use a stem sentence.

Let's start by determining the unit that we are working with.

What is the unit that we're working with? Hmm.

Think about the denominator.

The denominator of the fractional part is a seven.

So we are working with sevenths and the seven times table.

The denominator of the improper fraction will be a seven.

So there are mm groups of mm sevenths, which is mm sevenths, and one more seventh, so that is mm sevenths.

Let's see if we can fill that in.

There are three groups of 7 sevenths, that's our whole number, which is 21 sevenths.

So our whole number is worth 21 sevenths.

And then one more seventh.

That's our fractional part.

So that is 22 sevenths.

That stem sentence was very helpful.

So 3 1/7, the mixed number is equivalent to 22/7, the improper fraction.

They're worth the same.

Let's have a little check.

So we're going to use a stem sentence to express the mixed number as an improper fraction.

So I'll do one more example and we'll leave that on.

And then you can have a go at your own example.

So our mixed number is 3 1/6, and we are finding our improper fraction that's equivalent to that.

Let's use that stem.

The denominator of the fractional part is mm.

What's the denominator? The unit we are working with is mm.

Use the denominator for that.

There are mm groups of mm, which is mm and mm more mm, so that is mm.

That's a lot of mms. Let's fill those mms in.

The denominator of the fractional part is six.

So the unit that we are working with is sixths.

So there are three groups of 6 sixths, that's our whole number part, which is 18 sixths.

So our whole number, the three is worth 18 sixths.

And then we've got one more sixth, that's a fractional part.

So that's 19 sixths.

3 and 1 sixths, the mixed number is equivalent to 19 sixths, the improper fraction.

Now, let's see if you can do the same thing with the mixed number 3 1/8.

Here's that stem sentence, pause a video, and off you go.

Let's see.

The denominator of the fractional part is eight.

The units we are working with is eighths.

So it involves our 8 times table.

There are three groups of 8 eighths.

That's our whole number part, which is 24 eighths.

So that three is equivalent to 24 eighths.

And then our fractional part, one more eighth, so that's 25 eighths.

Did you get 25 eighths? If you did, very well done.

You are on track.

A third strategy is to use the generalisation.

So here we've got 3 1/7 and mixed number.

First, we multiply the whole number by the denominator and then we add the numerator.

Let's try that.

So 3 multiplied by 7 sevenths plus 1 seventh is equal to what? That's 21 sevenths plus 1 seventh is equal to 22 sevenths.

And we can write that as an improper fraction.

Let's try that method then.

I'll do another example.

And then you have a go at one of your own.

So use that generalisation to express the mix number as an improper fraction.

Our mixed number is 2 3/9.

First, we multiply the whole number by the denominator.

So the whole number 2, the denominator's 9.

And then we add the numerator.

So that's 2 lots of 9 ninths plus 3 ninths equals something.

So 18 ninths.

So our 2 is worth 18 ninths in this case, plus that extra 3 ninths is equal to 21 ninths.

That's our improper fraction.

Okay, see if you can do that.

Here's 2 2/7.

That's our mixed number.

And here is your generalisation.

Pause the video.

Off you go.

Did you like that way of doing it? Did you find that helpful? Let's have a look.

So 2 multiplied by 7 sevenths plus 2 sevenths equals something.

So 2 multiply by 7 sevenths gives us 14 sevenths.

So the 2 is 14 sevenths plus 2 sevenths equals 16 sevenths.

And this is how we write 16/7.

Did you get that? Very well done if you did.

It's time for some practise.

Express a following mixed numbers as improper fractions.

Use your preferred strategy.

Which way did you like best? Personally, I like the stem sentence best, but you might like the generalisation best.

You could then prove you are correct using a different strategy.

You might get to like the other one better.

Number two, what could the denominator be in this improper fraction? Write an equation using the shapes to show how you would calculate the numerator.

There are so many possibilities here, so don't stop at one.

Do as many as you possibly can.

Pause the video.

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

Welcome back.

Let's have a look.

So number one, express the following mixed numbers as improper fractions.

Whatever method you chose 4 1/8 is equivalent to 33/8.

6 4/9 is equivalent to 58/9.

3 11/12 is equivalent to 47/12.

And 8 2/3 is equivalent to 26/3.

You might have chosen to represent these with a drawing and then preview are correct using the generalisation.

And number two, what could the denominator be in this improper fraction? Write an equation using the shapes to show how you would calculate the numerator.

Well, the denominator would have to be the same as that in the fractional part of the mixed number.

So this is the green square.

That's the denominator.

The denominator could have a value of nine.

To calculate the numerator, we can use a stem sentence.

There are mm groups of nine which is mm multiplied by 9 ninths, and mm more ninth.

So that is mm times 9 ninth plus mm ninths.

To calculate the numerator, we could also have used a generalisation.

First, we multiply the whole number by the denominator and then we add the numerator.

This will be mm times 9 ninths plus mm ninths.

It's time for the next cycle, which is problem solving.

Jacob has two oranges.

Sofia has half an orange.

Can you visualise that? Can you picture that? Two oranges and half an orange? I can picture that.

It looks like this.

How can we write how many oranges we have altogether? Well, we have 2 1/2 oranges in total and that's what that's showing.

We've got 2 1/2 a mixed number.

Two's our whole number and 1/2 is our fractional part.

Jacob cuts both of his oranges in half.

How many orange halves will there be? Hmm.

Can you picture that? That's represented as a bar model.

So on the top part of the bar model, we've got the 2 1/2 oranges, and then on the bottom 2 1/2 oranges again.

But this time they're all halves.

So what can we see? On the top, we've got 2 1/2 and that's equivalent to 5 halves.

There will be five orange halves or five halves of orange.

What other strategy could we have used to express this as an improper fraction? We could use that stem sentence.

I really like that stem sentence.

There are two groups of two halves, which is four halves, and one more half.

So that's five halves.

And that's how we write five halves.

Or we could use a generalisation that was good too.

First, we multiply the whole number by the denominator and then we add the numerator.

So in this case, 2 multiplied by 2 halves plus 1 half is equivalent to, 4 halves plus 1 half, which equals 5 halves.

So 2 1/2 is equal to 5 halves.

It's more efficient to use the generalisation, but it's important that we understand the structure behind the generalisation.

Let's have a little check.

True or false? Jacob has four whole pizzas and a third of a pizza.

If the whole pizzas are also cut into thirds, there will be 13 thirds of pizza.

Is that true or is that false? You might like to read that again and take some time to understand it before deciding.

Okay, pause the video.

What did you think? Was that true or false? That's true.

And why? You might have used the stem sentence.

There are four groups of three thirds, which is 12 thirds and then one more third, so that is in fact 13 thirds.

That's correct.

You might have used a generalisation instead.

First, we multiply the whole number by the denominator and then we add the numerator.

And as we said before, that generalisation is quicker and more efficient.

Here we go, 4 multiply by 3 thirds plus 1 third equals 13 thirds.

And that is how we write 13/3.

It's time for some final practise tasks.

Number one, solve these problems by identifying the mixed number and expressing it as an improper fraction.

At a party, there are five whole pizzas and 2/6 of another pizza.

Could you write that as a mixed number? If the whole pizzas are each cut into sixth, how many sixth would there be in total? Hmm.

And b, Sofia's mum takes 6 2/4 oranges to hand out at half-time in Sofia's netball match.

She cuts each of the six whole oranges into quarters.

How many quarters does she have in total? So again, think about that mixed number.

Jacob has a piece of string that is four metres long.

He also has a piece of string that is 4/5 of a metre long.

He cuts all the string into pieces that are 1/5 of a metre in length.

How many pieces does he have? So read that carefully.

Take your time and think about that before you solve it.

I always say that if you can work with a partner, do it.

Because then you can bounce ideas off each other, you can compare strategies.

One of you might like the generalisation more.

One of you might like the stem sentence more.

And between you, you might get a really good system going.

Okay, well, pause the video and off you go.

Welcome back.

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

So a, at a party, there are five whole pizzas and 2/6 of another pizza.

If the whole pizzas are each cut into sixths, how many sixths will there be in total? You might have started by identifying the mixed number whole.

So that's 5 2/6.

That's a good start.

And then use the generalisation.

First, we multiply the whole number.

That's 5.

By the denominator.

That's 6.

So that's 5 times 6.

And then we add the numerator.

5 times 6 sixths plus 2 sixths is equal to, 32 sixths.

32 sixths was the answer.

Well done if you got that.

And then Sofia's mum takes 6 2/4 oranges to handout at half-time in Sofia's netball match.

She cuts each of the six whole oranges into quarters.

How many quarters does she have in total? Well, you might have started by identifying that mixed number whole at 6 2/4.

And then we can use that nice efficient generalisation.

We multiply the whole number by the denominator and then we add the numerator.

So this is 6 multiplied by 4 quarters plus 2 quarters.

That's 24 quarters plus 2 quarters, and that is equivalent to 26 quarters.

And that's how we write 26 quarters.

There are 26 orange quarters.

And see Jacob has a piece of string that's four metres long.

He also has a piece of string that's 4/5 of a metre long.

When we combine those, we've got a mixed number.

He cuts all the string into pieces that are 1/5 of a metre in length.

How many pieces does he have? So let's look at that mixed number, that's 4 4/5.

And then lets use that generalisation.

Multiply the whole number by the denominator and then add the numerator.

So that's 4 multiplied by 5 fifths plus the extra 4 fifths.

That's 20 fifths plus 4 fifths, which is equal to 24 fifths.

And that's how we write 24 fifths.

Well done if you said they were 24 pieces of string.

Excellent work.

We've come to the end of the lesson.

Today, we've been expressing a quantity as a mixed number and an improper fraction.

I think you've made so much progress today, and I hope you feel the same way.

A mixed number can be written as an improper fraction of equal value.

And that useful generalisation, first, we multiply the whole number by the denominator then we add the numerator can help us express mixed numbers as improper fractions.

And you've done lots and lots of examples of that, and you got better and better as you went along.

It's been a great pleasure spending this math lesson with you.

I've enjoyed it so much and I'd love to do it again soon.

In the meantime, have a great day and be successful at whatever you are doing.

Be the best version of you.

Take care and goodbye!.