Lesson video

Hello there, my name is Mr. Tilston, I'm a teacher.

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

Today, we're going to be converting between improper fractions and mixed numbers, and you might have had some very recent experience of doing that with quarters.

Today we're going to be looking at that skill with fifths.

So if you are ready, I'm ready, let's begin.

The outcome of today's lesson is this.

I can convert an amount of fifths from an improper fraction to a mixed number.

And we've got some keywords.

My turn, mixed number, your turn.

My turn, improper fraction, your turn.

What did those words mean? I'm sure you've encountered them before, but it's worth a recap.

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

For example, one and a half.

Could you think of a different mixed number? An improper fraction is a fraction with a 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.

Could you think of a different example of an improper fraction? And can you think of any examples from real life where you get mixed numbers and improper fractions? Our lesson today is split into two parts or two cycles.

The first will be improper fractions and mixed numbers, and the second converting improper fractions into mixed numbers, all the time focusing on fifths.

Let's start by thinking about improper fractions and mixed numbers.

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

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

Aisha has five, one fifth of a metre long, pieces of string.

Can you see that? They're each one fifth of a metre and she's got five of them.

How would you express that as a fraction? How could you say that? Hmm.

Well, the unit that we are working with is fifths.

So we've established our unit.

We've established our denominator, it's five.

And we have five of those parts, all of them.

This means that the numerator is also five.

The whole has been divided.

That's the fraction bar into five equal parts.

That's the denominator, and we have five of those parts, that's a numerator.

And that's what this fraction looks like.

This is an improper fraction.

Aisha says, I notice something.

Five fifths is equivalent to one whole.

Because five equal parts make one whole.

Did you notice that too? Well done if you did.

So we could say five fifths of a metre, or we could say one metre.

Jun has six of those pieces.

Six one fifth of a metre long pieces of string.

What's the same, and what's different? How could you express that as a fraction? The unit we are working with is still fifth, so it hasn't changed.

So the denominator is still five, that hasn't changed.

This time though, what has changed is we've got six of those parts.

We had five before, we've got six now.

So the numerator has changed and it's six.

How could we write that as a fraction? The whole has been divided, that's your fraction bar, into five equal parts, that's your denominator.

And we have six of those parts, that's your numerator.

And it looks like this.

This is six fifths of a metre, it's an improper fraction.

Jun says, I noticed something.

Hmm, I wonder what Jun's noticed.

Have you noticed anything? That's what good mathematicians do, they notice things.

Six fifths is an improper fraction, yes, it is.

So it could also be expressed as a mixed number.

Hmm.

Looking at that image, can you also see a mixed number that's got a whole number part and a fractional part? Let's use a part-part-whole model to support us to convert from the improper fraction to a mixed number.

Mixed numbers are composed of a whole part and a fractional part.

So have a look at this part-part-whole model.

We can partition that improper fraction, the six fifths into a part that's equivalent to the whole number, and then the remaining fractional part.

So the part that's equivalent to the whole number is five fifths, and then we had an extra bit than of fifth.

So six fifths could be thought of as five fifths plus one fifth.

Five fifths is equivalent to one whole.

So instead of saying five fifths, what could we change that to? We could say that's one, that's one whole.

Now can you start to see a mixed number forming? We also have one more fifth.

This is one and one fifth.

Six fifths is equal to one and one fifth.

So that improper fraction, six fifths is equivalent to that mixed number one and one fifth.

Six fifths are equivalent to one whole and one more fifth.

Six and fifth of a metre is equal to one and one fifth of a metre.

And we can see both in that image.

This makes it clearer that we've got a whole number part that's the one and the extra fifth.

Now this time we have seven of those one fifth of a metre pieces of string.

What has changed? What has stayed the same? How would you express that as a fraction? The unit that we are working with is still fifths, so the denominator is still five.

And we're going to think about fifths through this whole lesson.

We have seven of those parts.

This means that the numerator is seven.

How could we write that as a fraction? The whole that's your fraction bar has been divided into five equal parts, that's your denominator.

And we have seven of those parts, that's your numerator.

And this is how that looks.

That's seven fifths of a metre.

That's an improper fraction.

What else could we say? Seven fifths is an improper fraction, you are correct here.

So it could also be expressed as a mixed number.

You are correct here.

Can you see what mixed number it's going to be? How many wholes we've got and how many extra parts? Well, again, let's partition it using this really useful part-part-whole model.

Seven fifths, how can we partition that so that we've got a whole number part and an extra part? Well, that could be thought of as five fifths, that's a whole number, plus two fifths.

Five fifths is equivalent to one whole.

So let's say one instead of five fifths.

And we have two more fifths.

So this is one and two fifths, that's our mixed number.

Seven fifths, the improper fraction, is equal to one and two fifths, that's the whole mixed number.

Seven fifths are equivalent to one whole and two more fifths.

So let's have a look at that.

Can you see seven fifths of a metre? And can you see one and two fifths of a metre? They are equivalent.

That's exactly the same amount of string.

Let's have a little check, see if you are getting this.

I think you are.

Use the part-part-whole model to write the mixed number as an improper fraction.

So most of the work's been done for you, you just gotta finish it off.

Eight fifths has been partitioned into five fifths and three and a fifths.

So therefore we could say eight fifths equals what? What improper fraction? Pause the video.

Well that five fifths is one whole.

So we can say that's one and three fifths.

Well done if you said that you are on track.

Let's look at this sequence of improper fractions in their equivalent mixed numbers.

So we've got five fifths is equal to one, six fifths is equal to one and one fifth.

Seven fifths is equal to one and two fifths.

Eight fifths is equal to one and three fifths.

Nine fifths is equal to one and four fifths.

10 fifths is equal to two.

11 fifths is equal to two and one fifth.

12 fifths is equal to two and two fifths.

13 fifths is equal to two and three fifths.

14 fifths is equal to two and four fifths.

And 15 fifths is equal to three.

Okay, what kind of things did you notice? There was so much to notice there, what did you spot? What's the same, and what's different? Well, this is one of the things that could be noticed.

This is what Jun saw.

He says, I noticed that the fractional part of the mixed numbers and the denominator of the improper fractions are the same.

So we'd established our unit, we are working with fifths.

So the denominator is five, and you can see that denominator in two places.

You can see it in the improper fraction and you can see it in the mixed number, in the fractional part.

Let's have another check.

True or false, 16 fifths expressed as a mixed number is three and one quarter.

True or false? Pause the video.

Did you hear it when I said it? 16 fifths is expressed as a mixed number as three and one quarter.

Well, did you spot that something wasn't right there? The denominators weren't the same, it's false.

The denominator of the fractional part of the mixed number and that of the denominator of the improper fraction should be the same.

In this example, one is expressed as fifths and the other quarters, so they're not the same.

Something else you might have noticed.

What do you notice here? Aisha spotted something.

Five fifths equals one or one whole.

10 fifths equals two or two wholes.

And 15 fifths equals three or three wholes.

I notice she says that when the numerator is a multiple of the denominator, the quantity is equivalent to a whole number.

So our denominator is five, and five is a multiple of five.

In the second one, the numerator is 10, and 10 is a multiple of five because five, 10.

And in the third example, the numerator is 15, and 15 is a multiple of five.

So five, 10, 15.

So when the numerator is a multiple of the denominator, we've got a whole number and nothing extra.

Five fifths is one full group of five fifths.

10 fifths is two groups of five fifths.

And 15 fifths is three four groups of five fifths with nothing else left over.

So let's have a check.

Which of these improper fractions are equivalent to a whole number? Think about what I've just said about the numerator being a multiple of the denominator.

Okay, pause the video.

Did you spot any that are equivalent to a whole number? Well, 12 isn't a multiple of five, is it? Because five, 10, 15, it's in between those, 15 is a multiple of five because five, 10, 15.

21 is not a multiple of five, 'cause five, 10, 15, 20, 25, it falls between those numbers.

But 40 is a multiple because five, 10, 15, 20, 25, 30, 35, 40.

So therefore 15 fifths is a whole number.

In fact, it's equal to three.

And 40 fifths is a whole number is equivalent to eight.

These improper fractions will be equivalent to a whole number.

Let's do some practise, let's put those skills into action.

Number one, match these improper fractions to their whole number equivalents.

Number two, Aisha says that 13 fifths expresses a mixed number would be two and three eighths.

Aisha is incorrect.

Explain why she's incorrect.

See if you can do that nice and clearly using some mathematical vocabulary.

Okay, good luck with that.

Pause the video, and I'll see you soon for some feedback.

Welcome back.

How did you get on? Let's give you some answers.

So 20 fifths, five, 10, 15, 20, that's four lots of five.

There we go.

25 fifths, well, I know my times tables and I know that five times five equals 25.

So 25 fifths is equal to five.

Five wholes.

30 fifths well, five lots of six are equal to 30.

So therefore 30 fifths is equal to 6.

35 fifths, well, seven lots of five are equal to 35.

So 35 fifths is equal to seven wholes and then 40 fifths while eight lots of five are equal to 40.

So 40 fifths must be equivalent to eight or eight wholes.

And Aisha says that 13 fifths expressed as a mixed number would be two and three eighths.

That's not correct, why? Well, the denominator of the fractional part of the mixed number, and that's the denominator of the improper fraction should be the same.

But are they the same? She says that 13 fifths and two and three eighths are the same.

In this example, the denominator of the improper fraction is a five, that's fifths, but the denominator of the fractional part of the mixed number is an eight, eighths.

The denominator of the fractional part of the mixed number should be a five or fifths.

So that either both have to be fifths or both have to be eighths.

You are ready for the next part of the learning, I think, and that's converting improper fractions into mixed numbers.

Aisha now has 12 of one fifth of a metre long pieces of string.

Can you see that? 12 of them.

Let's represent the string using rods, okay, good idea.

How about this? We can see those fractions very clearly now.

How many fifths have we got? How do we write 12 fifths as an improper fraction? Well, the unit that we are working with is fifths.

So we've established our unit.

The denominator is therefore five.

We have 12 of those parts, this means that the numerator is 12.

The whole has been divided, that's a fraction bar into five equal parts, that's a denominator.

And we have 12 of those parts, that's a numerator.

So our fraction here is 12 fifths.

That's our improper fraction.

What else can you see? Can you see something else there? A mixed number perhaps? Let's convert to our fifth into a mixed number.

How would you do that? Aisha says, let's start by identifying the unit that we are working with.

So what is that unit? What have we established? What's the denominator? The denominator is a five.

So each whole has been divided into five equal parts, and we saw that with the string and the fraction bars.

And five parts make one whole.

So every time we see five parts, we've got a new one whole.

The numerator is 12, and this means that we have 12 of those equal parts.

Let's use a part-part-whole model to represent this.

So how could we partition that 12 fifths? We need to determine how many wholes we have.

So if you are counting in fives, how close can you get to 12? We know we are working with fifths and five fifths make one whole, but I think we can keep going, don't you? There's more than one whole.

It's possible to make two full groups of five fifths, which is 10 fifths.

So the multiple of five just before 12 is 10.

So we can partition that into 10 fifths, and the extra two fifths.

There are two more fifths.

What could we say about that 10 fifths then? 10 is a multiple of five and is equivalent to two wholes.

So we'll say two instead of 10 fifths, let's call it two.

So 12 fifths is equal to what? The two fifths form the fractional part of the mixed number.

So it's two and two fifths.

12 fifths is equal to two and two fifths.

Do you think you're getting the hang of this? 12 one fifths of a metre long pieces of string are equal to two whole metres and two fifths of a metre.

So 12 and a fifths of a metre.

Can you see that? 12 one fifths of a metre is equal to what? Well, we could change those to one metre.

We haven't changed the amount or the length, but we can see now that that's two and two fifths of a metre.

12 and a fifths of a metre is the improper fraction.

Two and two fifths of a metre is the mixed number and they are equivalent.

Let's have a little check, write this improper fraction as a mixed number, so 17 fifths.

That's how a whole, sketch that part-part-whole model.

Can you find the parts and can you turn that into a mixed number? And Aisha says, remember to start by determining how many wholes there are.

So how close can you get to 70? Okay, pause the video.

Did you do it? Did you manage to turn that into a mixed number? Well, 15 fifths is as close as we can get.

So five, 10, 15, so that was three wholes.

And then there's an extra two fifths.

So we've got three wholes and two fifths, or three and two fifths, so 17 fifths.

The improper fraction is equal to three and two fifths, the mixed number.

Look at this improper fraction.

We've got 34 fifths.

Let's use stem sentences to support us to convert it to a mixed number.

Here's our stem sentence.

The denominator is, hmm, this means that each whole has been split into equal parts.

Equal parts make each whole.

Hmm.

So think about the denominator.

The numerator is mm, this means there are equal parts.

It is possible to make full groups of fifths.

This is fifths and there are more fifths.

Let's break that down a little bit.

Let's think about the denominator then.

The denominator is five.

This means that each whole has been split into five equal parts.

Five equal parts make each whole.

And we've got quite a few of those wholes.

The numerator is 34, this means there are 34 equal parts.

It is possible to make six full groups of five fifths, because six lots of five are equal to 30.

So this is 30 fifths and there are four more fifths.

30 fifths and four more fifths.

Now let's turn that 30 fifths into a whole number.

So how many fives are equal to 30? Six So that's six and four fifths.

34 fifths, the improper fraction is equal to six and four fifths, the mixed number.

Let's have a final check.

Use the stem sentences to convert the improper fraction to a mixed number.

46 fifths equals what? Use stem sentences and pause the video, off you go.

The denominator is five.

This means that each whole has been split into five equal parts.

Five equal parts make each whole.

The numerator is 46, this means there are 46 equal parts.

It is possible to make how many groups, how close can you get in fives to 46? Nine, nine full groups of five fifths, and that's 45 fifths.

And then there's one extra fifth left over.

So that's nine and one fifth.

That stem sentence was really helpful and we didn't have to do the part-part-whole model.

It's time for some final practise.

Number one, match these improper fractions to their mixed number equivalents.

And number two, convert these improper fractions to mixed numbers.

And in all of these cases, we are thinking about fifths.

Okay, pause the video.

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

Welcome back, how did you get on? Let's have a look.

So number one, match these improper fractions to their mixed number equivalents.

13 fifths, well, that's two wholes because two lots of five are 10.

And that's three fifths leftovers.

So that's two and three fifths.

56 fifths, well, you can get 11 wholes out of that because 11 lots of five are 55 with one fifth left over.

So that's 11 and one fifth.

18 fifths is equal to three and three fifths.

27 fifths is equal to five and two fifths.

61 fifths is equal to 12 and one fifth.

And 39 fifths is equal to seven and four fifths.

And then convert these improper fractions to mixed numbers.

12 fifths, well, two lots of five are equal to 10.

And that's as close as we can get to 12.

So we've got two wholes and two fifths left over.

So two and two fifths.

13 fifths is equal to two and three fifths.

14 fifths is equal to two and four fifths.

17 fifths is equal to three and two fifths.

27 fifths is equal to five and two fifths.

37 fifths is equal to seven and two fifths.

21 fifths is equal to four and one fifth.

36 fifths is equal to seven and one fifth.

And finally, 53 fifths is equal to 10 and three fifths.

We've come to the end of the lesson.

I think you've been incredible today, well done.

Today, we've been converting an amount of fifths from an improper fraction to a mixed number.

So by now you should have experience of converting quarters and fifths to mixed numbers.

There are other denominators and we will explore those.

Improper fractions can be converted to mixed numbers.

The denominator of the improper fraction will be the same as the denominator of the fractional part of the mixed number.

If the numerator is a multiple of the denominator, the improper fraction will be equivalent to the whole number.

And we've seen some examples of that today.

Well, you've been great today.

Give yourself a pat on the back is very well deserved and say, well done me.

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

But until then, have a fantastic day.

Be the best version of you and really succeed at whatever it is that you are doing.

Take care and goodbye.