Lesson video

Hello there.

My name is Mr. Tilston.

How are you? I hope you're having a good day.

I certainly am because today I get to teach you about probably my favourite part of maths, which is fractions.

Specifically, today's lesson is about converting between improper fractions and mixed numbers, and I think you've probably already got quite a lot of experience doing that.

So let's see if we can take you even further on that journey today.

If you're ready, I'm ready, let's begin.

The outcome of today's lesson is this.

I can solve problems involving converting between mixed numbers and improper fractions.

And we've got some keywords, my turn, mixed number, your turn.

And my turn, improper fraction, your turn.

Can you remember what they mean? Shall we have a reminder? A mixed number is a whole number and a fraction combined.

So for example, one and a half.

Can you think of a different mixed number? And 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, five thirds and nine eighths are both improper fractions.

I wonder if you could give me a different example of an improper fraction.

Remember, there's a very strong link between mixed numbers and improper fractions.

Any mixed number can be expressed as an improper fraction, and vice versa.

Today's lesson is split into two parts.

The first will be comparing mixed numbers and improper fractions, and the second, problem solving.

So let's begin by thinking about comparing mixed numbers and improper fractions.

And in this lesson you're going to meet Aisha and Jun.

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

Aisha represents two and three quarters using rods.

You might have some rods in your classroom, so you might be able to do this as well.

That's two and three quarters.

Jun represents 10 quarters using rods.

Can you see 10 quarters there? And the question is this, who has represented the greater number? Which is worth more, two and three quarters, or 10 quarters? Well, it's quite hard to see just looking at them, isn't it? So we're going to have to delve a little deeper.

Aisha says, "Mine is the greater number because it has two ones or wholes".

So she thinks that's what makes it the bigger number.

And Jun says, "Mine is a greater number because it has ten one quarters".

So Jun thinks it's the number of quarters in this case that's making it bigger.

Hmm.

What do you think? Who do you agree with? To be able to compare a mixed number with an improper fraction efficiently, we can write the numbers in the same way.

There's two ways we can do this.

Let's start with one of them.

This means that if we write both numbers as mixed numbers for example, or both as improper fractions, a different way, they will be easier to compare.

Absolutely.

So let's convert two and three quarters into an improper fraction.

We've got two and three quarters.

What could that be as an improper fraction? What could we do to those two wholes? The unit we are working with is quarters, and we've established that because that's the fraction that we can see, quarters, one quarters.

To find the amount in quarters, first we multiply the whole number by the denominator, and then we add the numerator.

Hopefully that rings a bell.

Hopefully you've been doing that quite recently.

So in this case, it would be those two wholes multiplied by the four quarters, and then adding the extra three quarters, two multiplied by four equals eight, so that's eight quarters.

And then add the extra three quarters on.

Eight plus three equals 11.

So there are 11 quarters there.

So we can say two and three quarters, or we can say 11 quarters.

Now this should make it easier to compare, shouldn't it? Because Jun expressed his value as an improper fraction, and now we've got an improper fraction to compare it with.

So two and three quarters is equivalent to 11 quarters.

Now to determine our greater number, we need to write the numbers in the same way.

So let's do that.

Now we've got 11 quarters or 10 quarters, and now it's really easy, isn't it, to see which one's greater? When the denominator is the same, the fraction with the largest numerator is the greatest, and that's what we can see there, the denominator is the same, it's quarters, but one of them has got 11 for the numerator, so it's slightly bigger, slightly greater.

And that's how we can show that, using the greater than symbol, 11 quarters is greater than 10 quarters.

So two and three quarters must be greater than 10 quarters.

Yes.

Let's do a check.

Let's see if you've followed.

So by converting the mixed number to an improper fraction, compare these numbers.

I'll do one more, and then you have a go.

So we've got four and two thirds, and 15 thirds.

Quite hard to tell just looking at those fractions, which is greater.

So let's break them down.

Let's make them both improper fractions.

Well, the unit we're working with is thirds, we can see that.

First we multiply the whole number, that's four, by the denominator, that's three and then add the numerator, that's two.

Four multiplied by three, add two, equals 14.

So therefore we've got 14 thirds.

So let's not say four and two thirds, let's say 14 thirds.

Ah, now we can compare them look.

Now we can say which is a greater, and we can use this inequality symbol.

So 14 thirds is less than 15 thirds.

Okay, let's see if you can do the same thing with five and one thirds and 17 thirds.

Can you compare those, by converting the five and one third into an improper fraction? Pause the video and have a go.

How did you get on? Did you manage to convert five and one third into an improper fraction? Let's see.

Well, the unit we're working with is thirds.

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

So there's five wholes multiplied by three, that gives us 15 thirds, add the extra one third, equals 16 thirds.

And 16 thirds is written like this.

Now we can compare them, can't we? We can say 16 thirds is less than 17 thirds.

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

To compare the numbers that he and Aisha had represented, Jun converted the mixed number to its improper fraction equivalent.

Let's see what happens if we'd chosen to convert the improper fraction to its mixed number equivalent, so going the other way.

The unit that we are working with is quarters.

So let's see if we can turn this 10 quarters into a mixed number.

Find out how many groups of the denominator can be made from the numerator by dividing.

So how many whole groups of quarters can you see? I can see two.

And then what's the extra fractional part? Two quarters.

So 10 divided by four, equals two remainder two.

This is two full groups of four quarters and two more quarters.

But we're not going to say remainder, we're going to turn it into a fraction.

That's two and two quarters.

10 Quarters is equivalent to two and two quarters.

So when we express them both as mixed numbers, again they become easy to compare.

The whole number part is the same.

So we compare the fractional parts.

Two and three quarters is greater than two and two quarters.

And we can use that inequality symbol to express that.

Two and three quarters must be greater than 10 quarters.

That only became easy when we made them the same.

When they were both mixed numbers, it became easy to compare.

Let's have a check.

So let's see if you can convert the improper fraction to a mixed number.

I'll do one more.

So 13 fifths, and two and four fifths.

Very hard to say just by looking which is greater, so let's delve deeper, and let's convert.

The unit we're working with is fifths.

You can see that in both denominators.

Find out how many groups of the denominator can be made from the numerator by dividing.

So this is 13 divided by five.

How many times does five go into 13? Two remainder three.

That's two full groups of five fifths and three more fifths, and let's express that as a mixed number.

That's two and three fifths.

Now we can compare, two and three fifths is less than two and four fifths.

Right, your turn, this time we've got 17 fifths and three and one fifth.

Can you convert that 17 fifths into a mixed number? Pause the video.

How did you get on? Did you manage to turn 17 fifths, the improper fraction, into a mixed number? Well, let's see.

The unit that we are working with is fifths.

We've established that, that denominator is present in both fractions.

Find out how many groups of the denominator can be made from the numerator by dividing.

So this time we're doing 17 divided by five.

How many times does five go into 17? If we're counting in fives, how many can we reach before we make 17? So 17 divided by five is equal to three remainder two, because 5, 10, 15, that's three, but then with two leftover.

Two what leftover? Two fifths.

So this is three full groups of five fifths, and two more fifths.

So then it's three and two fifths.

And now we can compare them.

Three and two fifths is greater than three and one fifth.

Very well done if you said that, you're on track.

It did not matter if we converted the improper fraction to the mixed number, or vice versa, it results in the same comparison.

Did you have a favourite way to do it? Which way did you prefer? So two and three quarters is greater than 10 quarters, two and three quarters is greater than two and two quarters, or 11 quarters is greater than 10 quarters.

They're all exactly the same.

"It was easier to compare the numbers after they were written in the same way".

I fully agree with you, Aisha there, it definitely was.

So it's time for some practise.

Number one, complete these equations using the inequality and equality symbols.

So is it less than, greater than, or equal to? And it doesn't matter which way you choose to convert them.

You might turn them both into improper fractions, you might turn them both into mixed numbers, that's entirely up to you, whichever way you prefer.

Aisha says, "Remember to write the numbers in the same way to help comparison".

And then number two, starting with the smallest, write these numbers in order of size.

So smallest first.

You're going to have to do some converting, because I can see some improper fractions, and I can see some mixed numbers, hmm.

And Aisha says, "Remember to write the numbers in the same way to help comparison".

So once again, which way do you prefer? Do you like converting mixed numbers to improper fractions, or do you converting improper fractions to mixed numbers? Your choice.

Right-E-O, pause the video and away you go.

Welcome back.

How did you get on with that? Let's have a look, let's give you some answers.

So number one, 31 fifths and six.

So which is greater, or are they the same? Well, 30 fifths is the same as six.

Six wholes is 30 fifths.

So we can say 31 fifths is greater than six.

For the next one, 31 fifths is equal to six and one fifth because that's 31 fifths, they both are.

Six wholes multiplied by five is 30 fifths, add the extra one fifth, is 31 fifths.

And then 31 fifths we could convert to a mixed number.

And when we do that, it becomes six and one fifth.

So you can see six and two fifths is slightly greater.

Nine halves is less than five and a half, 10 and three ninths is greater than 92 ninths, four and two sevenths is greater than 29 sevenths, four and five sevenths is less than 34 sevenths, and five and three sevenths is equal to 38 sevenths.

But we could only do all of that when we converted.

You might have done it a different way.

You might have made them all improper fractions, for example.

Or you might have made them all mixed numbers.

You might have done a little bit of a mix.

Number two, starting with the smallest, write these numbers in order of size.

We've gotta convert them first, haven't we? You might have decided to convert the mixed number to an improper fraction.

Maybe you did it the other way, but let's look at it this way.

The unit we're working with is sixths.

Even with the three, we can express that as a number of sixths.

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

So the three and two sixths, three multiplied by six is 18, plus two is 20.

So that's 20 sixths.

Now we can hopefully begin to compare them, just one more though.

You might have then decided to convert the whole number to an improper fraction, I would, that would help.

Then they're all improper fractions.

So how many sixths are the same as three, three wholes? Six sixths in one whole multiplied by three, becomes 18 sixths.

So that's 18 sixths, 20 sixths, 21 sixths, and 26 sixths, that's the correct order.

You might then have ordered the fractions in their original form.

So that's three, three and two sixths, 21 sixths, and 26 sixths.

Well done if you did that.

You're doing really, really well.

We are ready for the next cycle, which is problem solving.

Aisha has three and a quarter oranges.

Jun has cut up his oranges so that he has 14 quarters of orange.

Can you visualise that? Can you picture that? 14 Quarters of orange, and can you picture three and a quarter oranges? Well, that's what three and a quarter oranges looks a bit like.

And this is what 14 quarters looks like.

So what might the question be? Well, it might be who has the greater amount of orange? And I don't think I could say just by looking at them like that, could you? I think I do need to convert them.

So Aisha has three and a quarter oranges.

We can write that as three and one quarter, that's our mixed number.

Jun's cut his oranges up so that he has 14 quarters of orange.

We can write that as an improper fraction just like this.

We need to determine which is greater.

Is it three and one quarter, or 14 quarters? I definitely can't say just by looking at them, so I do need to convert.

What shall we do though? Shall we convert that mixed number into an improper fraction so that they're both improper fractions? Or shall we convert the improper fraction into a mixed number, so that they're both mixed numbers? What do you notice? What's the same? And what's different? Well Jun says, "I noticed that the unit we're working with for both numbers is quarters".

Did you notice that? So we've determined our unit, we've determined our denominator.

"One of these numbers is a mixed number and the other an improper fraction.

To compare them, we need to convert one of them so that they are the same type.

So let's convert the improper fraction to a mixed number".

Yes, that's one way of doing it, Jun.

So 14 quarters, let's turn that into a mixed number.

We could use division here, couldn't we? 14 Divided by four.

14 Divided by four, equals three, remainder two.

Four, eight, 12, that's three whole groups with two leftover so three remainder two.

And then we expressed that as a mixed number.

So three remainder two, in this case is the same as three and two quarters.

Using the conversion we can determine which is greater, three and one quarter or 14 quarters? 14 Quarters expressed as an improper fraction is three and two quarters.

The whole numbers are the same, so we can compare the fractional parts, and there wasn't much in it, was there? But one was slightly greater.

Three and one quarter is smaller than three and two quarters or three and two quarters is greater than three and one quarter.

And we can express that like this.

Three and one quarter is less than three and two quarters.

Right, true or false? Two and one quarter is greater than 10 quarters.

Hmm.

Hard to say just by looking at those different fractions, so convert one so that they're both the same.

And then explain.

Pause the video, and away you go.

What do you think? Was that true or false? That was false.

Why? Two and one quarter is smaller than 10 quarters.

The fractions are different types, so they need converting so that they're the same.

One of them needs converting.

Two and one quarter expressed as an improper fraction, that's one way to do it, is nine quarters.

So this is smaller than 10 quarters.

Maybe you converted the other way, but either way it's false.

Now that the fractions are the same type, we can compare the amounts of oranges.

So we've got three and one quarters, we've got 14 quarters, or we've got three and two quarters, so 14 quarters is a equal to three and two quarters.

The whole numbers are the same.

So we compare the fractional parts.

Jun has the greater amount of orange, only by a little tiny bit, only by one quarter.

Okay, it's time for some practise, some final practise.

Solve these problems. Jun has five and two quarters of orange.

Aisha has 21 quarters of orange.

Who has the greater amount of orange? And I think it would be helpful if you converted one of those so that they were the same kind of fraction.

B, Aisha has 12 tenths of a metre length of string.

Jun has a piece of string that is one and three tenths of a metre long.

So different kinds of fractions there.

The first was an improper fraction, the second was a mixed number.

Who has the greater total length of string? Again, convert one so that they're both the same type of fraction, doesn't matter which, whatever you feel more comfortable doing.

And C, Aisha runs three and two fifths of a kilometre and Jun runs 14 fifths of a kilometre, who has run the furthest? They're different types of fractions, so make them the same.

I always recommend if you can, working with a partner and then you can share ideas and strategies.

So if you can do that, I would do that.

Pause the video, and away you go.

Welcome back.

How did you get on? Number one, Jun has five and two quarters of orange.

Aisha has 21 quarters of orange.

Who has a greater amount of orange? Well, you've got two options here, haven't you? You can make them both mixed numbers or you can make them both improper fractions.

Which did you do? Well we're working with quarters, to convert the mixed number to an improper fraction, first we multiply the whole number by the denominator and then add the numerator.

So that's five multiplied by four add two, which is 22, so that's 22 quarters.

And now they're both improper fractions and now we can them.

22 Quarters is greater than 21 quarters.

Jun has the greater amount of orange.

And then B, Aisha has 12 tenths of a metre length of string, and Jun has a piece of string that's one and three tenths of a metre long.

Who has the greater total length of string? When you write those down, you can see, that they're different kinds of fractions, 12 tenths is an improper fraction and one and three tenths is a mixed number.

So let's make them the same kind, doesn't matter which way you do this.

The unit we're working with is tenths.

To convert the improper fraction to a mixed number first find how many groups of the denominator can be made from the numerator by dividing.

So let's try that strategy this time.

So 12 divided by 10.

That's one remainder two.

10 Goes into 12 one time with two leftover, so that's one and two tenths.

Now we can compare them.

One and two tenths is less than one and three tenths.

So Jun has the greater total length of string.

And Aisha runs three and two fifths of a kilometre, Jun runs 14 fifths of a kilometre, who has run the furthest? Well, three and two fifths and 14 fifths are different kinds of fractions.

Let's make them both the same, doesn't matter which way you do that.

We're looking at fifths either way, that's our denominator.

To convert the mixed number to an improper fraction, first we multiply the whole number by the denominator, and then we add the numerator.

So that's three multiplied by five add two.

And that gives us 17, so we've got 17 fifths.

And now we can compare them.

17 Fifths is greater than 14 fifths.

Aisha has run the greater distance.

And I think that's my preferred way of doing it using that generalisation, turning the mixed number into an improper fraction.

We've come to the end of the lesson.

Wow, you've been amazing today.

You're doing so well with this.

Today we've been solving problems involving converting between mixed numbers and improper fractions.

Mixed numbers can be converted into improper fractions, and vice versa.

To solve problems involving comparing amounts given as different types of fractions, it is efficient to convert to the same type of fraction.

And it doesn't matter what you do, you can make them both improper fractions if that's your favourite way, or you can make them both mixed numbers if that's your favourite way.

When you do that, it becomes easy to compare.

You can use inequality symbols for that.

Well done on your accomplishments and your achievements today.

You are a superstar.

You definitely deserve a little pat on the shoulder.

Say, "Well done me".

Very good.

I hope we get the chance to spend another math lesson together in the near future because this has been great.

In the meantime, have a fabulous day.

And whatever you're doing for the rest of today, be the best version of you while you're doing it.

Take care, and goodbye.