Lesson video

Hello there.

My name is Mr. Tilston.

<v ->I'm a teacher.

My favourite subject is maths.

</v> So it's a real pleasure and a real delight let me tell you to be here with you today to teach you this lesson, which is all about fractions.

You may have had some recent experience of converting between improper fractions and mixed numbers, and today we're going to do the opposite.

We are going to convert from mixed numbers to improper fractions.

So if you're ready to begin, I'm ready, let's go.

The outcome of this lesson is I can explain how a mixed number is converted into an improper fraction.

And our keywords today are, my turn mixed number, your turn.

And my turn, improper fraction, your turn.

I'm confident that you know what those words mean, but just in case, let's have a little check because they're very important to today's lesson.

A mixed number is a whole number and a fraction combined, so it's got a whole number part and a fractional part.

So for example, one and a half.

Can you think of a different example of a 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 eights.

Can you think of a different example of an improper fraction? Mixed numbers and improper fractions are very closely linked.

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

And we're going to explore that further today.

Our lesson is split into two cycles or two parts.

The first will be converting mixed numbers into improper fractions and the second using a generalisation.

So let's begin by converting mixed numbers into improper fractions.

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 some rods.

Maybe you've got something similar in your classroom.

What do you notice? What can you see? What fraction can you see? Can you see a mixed number? Can you see a whole number part and an extra fractional part? Well, these rods represent three and one sixth, and this is how we write that.

Three and one sixth is a mixed number, and I'm convinced that you've had lots of opportunities by this point to write mixed numbers and to read mixed numbers.

So I bet you are getting pretty confident at that.

How though would you convert it into an improper fraction? Hmm? What could you do? Have you got any strategies? The unit that we've established here, our denominator is sixths, so we will be thinking about groups of six.

Is that helpful? What would you do now? There are six sixths in one whole.

Aisha knows that she to divide her rods into six equal parts, this will support her to convert three and one sixth, that mix number into an improper fraction.

Aisha divides each of her rods into six equal parts.

Like so.

There are three groups of six one-sixths which is 18 one-sixths, and you could count them all or you could count that they're six, 12, 18, and then one extra sixth that we already had.

So that is 19 one-sixths.

So those three wholes have been converted into 18 one-sixths.

We say 19 sixths, and we write it like this.

That's 19 sixths.

Aisha summarises her work in a table.

So the mixed number that she's got is three and one sixth.

The prompt question is this, how many groups of six sixths can we make from three wholes? We can make three groups of six sixths, which is 18 sixths, and there's one more sixth.

So there's three one wholes became 18 one-sixths and the extra one.

So that gave us an improper fraction of 19 sixths.

Three and one sixths is equivalent to 19 sixths.

Jun also has some rods.

What can you see? What's he got? What mixed number has he got? What improper fraction has he got? These rods represent three and two one-sixths.

We write that just like this.

That's three and two-sixths.

Three and two sixths is a mixed number.

How would you convert that into an improper fraction? Think about what we did before.

What could you do with those three one wholes? Our unit is six, so we will be thinking about groups of six.

There are six sixths in one whole.

So each of those one wholes could also be expressed as six sixths.

So how many sixth does that? Jun knows that he needs to divide his rods into six equal parts.

This will support him to convert three and two one-sixths into an improper fraction.

Jun divides each of the rods into six equal parts, just like so.

Can you see how many sixths those three wholes became? There are three groups of six one-sixths which is 18 one-sixths and then two more sixths.

So that's 20 one-sixths.

We say 20 sixths and that's how we write that.

That's 20 sixths.

Jun adds his work to Aisha's table.

So three and two sixths.

Before our question was how many groups of six sixths can we make from three wholes? And we've got that same question again because we've got three wholes again, and we've got the same answer again.

We can make three groups of six sixths, which is 18 sixths.

This time though there are two more sixths because that's a fractional part two sixths.

And that's 20 sixths when we add those together.

So three and two sixths, the mixed number is equivalent to 20 sixths, the improper fraction.

Let's have a little check.

By using or drawing rods if you haven't got any of the in front of you, express the mixed number three and three sixths as an improper fraction and complete the table.

So lots of this is already done for you.

So we've got three and three sixths our mixed number.

And the prompt question is this, how many groups of six sixths can we make from three wholes? We can make hmm groups of six sixths which is hmm sixths, and then there are hmm more sixths.

And then add them together.

That will give you the improper fraction.

Pause the video and have a go.

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

Well, we can make three groups of six sixths, which is 18 sixths, and then there are three more sixths this time.

So altogether that's 20 one sixths.

It is time for some practise.

I think you're ready for this.

By using or drawing rods to help express these mixed numbers as improper fractions.

Some of them are connected and Jun says, "Remember to use the prompt question, how many groups of hmm can we make from hmm wholes and then add on the extra fractional part." And number two, by using or drawing rods to help express these mixed numbers as improper fractions.

Explain any patterns that you spot.

Good luck with that, pause the video, and I will see you soon for some feedback.

Welcome back.

How did you get on? Are you starting to feel confident about converting mixed numbers into improper fractions? Well, let's compare answers.

Number one, here's three and four sixths.

You might have drawn something like this or you might have created something like this.

That's what three and four sixths looks like.

And then when we divide those wholes up into sixths, like so we can see that there are three groups of six sixths, that's our three wholes, which is 18 sixths, and then four more sixths.

So that's 22 one-sixths.

We say 22 sixths.

And we write it just like that.

So Well done if you got that.

So let's have a look at some others.

So three and five sixths is 23 sixths, three and one seventh is 22 sevenths, three and two eighths is 26 eighths, three and three ninths, 30 ninths, three and four tenths, 34 tenths, four and one seventh, 29 sevenths, five and one seventh, 36 sevenths, six and one sevenths, 43 sevenths and seven and six sevenths is 55 sevenths.

So well done if you got those.

And number two, did you spot any patterns? Four and one eighth is 33 eighths.

So there's four wholes become 32 eighths.

And then that one extra eighth makes 33 eighths.

Four and two eighths is 34 eighths.

What's happening here? Four and three eighths is 35 eighths.

Hmm, four and four eighths is 36 eighths.

I think we can probably guess what's coming next.

I'll work it out.

Four and five eighths is 37 eights.

What was going on there? What was the pattern? What did you notice? You might have spotted that the numerator of the fractional part of the mixed numbers increased by one each time.

So they had the same whole number part, they had the same denominator, but the numerator of that fractional part was increasing by one.

And this resulted in the amount of eights increasing by one each time too in the improper fraction.

Well, you're doing ever so well, I think you're ready for the next cycle, which is using a generalisation.

Let's see if we can do that without the drawings or the rods.

Look at this mixed number, four and three something.

What's missing? Well, the denominator's missing.

We know the whole number part.

We know the numerator but not the denominator.

How would we express that as an improper fraction? Could we do that? We can't determine the improper fraction from this because we need to know the unit that we are working with.

That's a denominator.

We haven't established that.

Look at this mixed number, four and something sevens.

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

That's four.

We know the denominator.

That's seven.

But we don't know the numerator.

We dunno how many of those sevens we've got.

So how could we express that as an improper fraction? Can we do that? No, we can't.

We can't determine the improper fraction from this because we need to know how many parts we already have.

And we'll know that when we know that numerator.

We can use Aisha's learning to form a generalisation.

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

So in this case it's four by whatever the denominator is.

And in this case, in the first one, we don't know the denominator.

In the second one we do, it's sevenths, and then adding the numerator.

So four multiplied by something plus three is equal to something, and then four multiplied by seven plus something is equal to something.

Let's use the generalisation to express this mixed number as an improper fraction.

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

Okay, well the whole number is five and the denominator is seven.

Can you multiply five by seven? Five multiplied by seven sevenths plus the extra three sevens equals 38 sevenths.

Five lots of seven or 35 plus three, 38.

So that's got 38 sevenths altogether.

That was quick, wasn't it? We didn't need to draw the rods.

38 sevenths.

Okay, let's use that generalisation.

Let's check, see if you've got that.

Use it to express the mixed number as an improper fraction.

So we've got nine and one third.

I'll do another one for you and then see if you can do a different one.

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

That makes 27 plus one, 28.

So we've got 28 thirds.

27 thirds plus one third equals 28 thirds, and that's how we write it.

Okay, let's see if you can do that.

This time we've got the mixed number six and two thirds.

So remember, multiply the whole number by the denominator and then add the numerator.

Pause the video.

Did you get that nice and quickly? It's helpful if you know your times tables off by heart, isn't it? If you know them automatically, it's even quicker.

So this time we've got six multiplied by three thirds, which is 18 thirds, plus that one third equals 20 thirds.

And that's how we write that.

If you've got 20 thirds very well done, you're on track.

Time for some practise.

Let's see if you can use that generalisation to help you match these mixed numbers to their equivalent improper fraction.

You may notice the denominator's eight each time.

And number two, Jun expresses two and five eighths as an improper fraction, and he shows his workings like this.

Let's see if he's right.

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

Is that okay so far? So he says two, multiplied by five eighths plus five eighths is equal to? 10 eighths plus five eighths, which is equal to 15 eighths.

Two and five eighths equals 15 eighths.

Give him some feedback.

Is that right or is that not? Well, actually, no.

Jun has made a mistake.

See if you can spot the mistake and correct it.

Okay, pause the video and away you go.

Welcome back.

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

Number one, we're using that generalisation, so we multiplying the whole number by the denominator and adding the numerator.

So three multiplied by eight is 24 plus six is 30, so that's 30 eighths, and that's there.

Three multiplied by eight is 24 plus one is 25, so that's 25 eights.

Five multiplied by eight is 40 plus the extra four, so that's 44 eights, three multiplied by eight is 24 plus the seven is 31, so that's 31 eighths.

And five multiplied by eight is 40 plus the extra two eights is 42 eights.

And then this is how Jun expressed two and five eights.

He said two multiplied by five eights plus five eights equals something.

So he thought that was 10 eighths plus five eights, which is 15 eighths.

Not quite is it? What mistake did he make? You might have spotted that Jun multiplied the whole number by the numerator.

It's supposed to be the denominator, wasn't it? He should have multiplied the whole number by the denominator.

And then you might have corrected his calculation like this.

Two multiplied by eight eighths, is 16 eights plus five eights equals 21 eights.

And that's how we write that.

So very well done if you've got that.

We've come to the end of the lesson.

You've been fabulous today.

The lesson has been explaining how a mixed number is converted into an improper fraction.

So now you've done it both ways.

You've converted improper fractions to mixed numbers and mixed numbers to improper fractions.

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

They are equivalent.

The generalisation.

First, we multiply the whole number by the denominator, and then we add the numerator can help to convert mixed numbers into improper fractions.

So you did lots of work with rods and drawing rods and that kind of thing, but as soon as you've got that, the quickest way to do it is to use that generalisation.

That is an efficient way to convert a mixed number into an improper fraction.

Well done on your achievements and your accomplishments today, you truly are a maths superstar.

Why don't you give yourself a pat on the back and say, well done me.

I hope I get the chance to spend another math lesson with you in the future because this has been fun.

In the meantime, have a great day.

Whatever you've got in store, try your best, be the best version of yourself, and you'll be amazing.

Until the next time, take care and goodbye.