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Hello, and welcome to lesson 16.

I'm sure you got everything that you need ready for this lesson.

So you'll need some pen and paper again, your work from the previous lesson, and if you been using the multiplication square tab, you make some links between the timestables, and the highest common factor, and thinking about our language and multiples and factors.

Then perhaps you could've had that to handle as well.

So get everything you need and I'll see you in a moment.

The last lesson, Mrs Hussein, left you with some calculations to solve.

As she asked you to give the answer in its simplest form.

I'd like us just to think about the generalisation you used to help you do those up.

Do you remember this one? Let's say it together, when adding fractions with the same denominator, just add the numerators.

Could that really help you with your independent task? Let's try the first one.

So, five, one twelfths plus five, one twelfths, is equivalent to how many twelfths? What did you do? Did you say that was the same as ten twelfths? Really, really well done.

Why did he say it was the same as ten one twelfths? That's right, it's because our unit is one twelfth.

So, we're counting in one twelfths.

So if we have five, one twelfth and we add another five, one twelfth, we've got ten twelfths altogether.

Now, there's an equal sign there.

So, did you go further and get that simplified? What do I have to do to simplify it? So, I need to find the highest common factor don't I? Can we see that quite quickly? I'm looking at it.

I'm seeing that they're both even numbers.

So, I know that I've got a common factor of two straight away.

Is that the highest common factor though? No, they're not.

I can't find a higher one.

I looked at three, but of course, ten isn't a multiple of three.

Brilliant, so it must be two.

So what do I need to do? I can multiply or divide.

So we're going to divide the numerator and denominator by the same highest common factor, which is two.

Brilliant Did you do that? And therefore, ten divided by two is five, and twelve divided by two is six.

So, your answer should be five sixth.

Ten twelfths and five sixth are equivalent fractions.

So, they would take up the same proportion of the whole, and that has been expressed in its simplest form.

Excellent.

Did you do the same with the next ones? So, we've got seven sixteenths and five sixteenths.

So our unit is one sixteenth.

So seven, one sixteenths plus five, one sixteenths is the same as twelve, one sixteenths.

Fantastic.

Now again how are we going to find the highest common factor with these? I can see that they're both even, so I know that twelve and sixteen are in the two times tables.

Are there any others? Is that the highest common factor? No, that's right.

Four is a common factor of those, because twelve is divisible by four and sixteen is also divisible by four.

So both the numerator and denominator are in the four times tables, and both have a factor of four.

So, I can divide the numerator and the denominator by the same common factor, and highest common factor will get me the simplest form.

So, twelve divided by four is three.

Well done.

And sixteen divided by four is four.

Fantastic.

And so that's expressed in its simplest form.

Well done.

How did he get home with the next one? Let's have a see.

So our unit is one fifteenth.

So four, one fifteenths, plus two one fifteenths, is the same as six, one fifteenths.

Excellent.

And can we simplify that? Do you think there's a common factor between those two? Can you see that? So one is a common factor.

Is two, a common factor this time? No, it isn't is it? Because fifteen is isn't even.

Three is common? Yes, it is! Three is a common factor, because they are both divisible by three, so six and fifteen are in the three timestables.

Is that the highest common factor? Yes! So we can divide the numerator and the denominator by three.

And so our simplest form becomes two fifths.

Really well done.

And two fifths and six fifteenths are equivalent fractions, because they are the same proportion of the whole.

And that proportional relationship between the numerator and denominator is maintained as well, isn't it? Then you had to challenge, and you were asked to have a look at a subtraction.

How did you get on with that? Did you find that you could use the same generalisation, because your unit was the same, so you could subtract the numerators? So eight ninths subtract two, one ninths, is the same as six ninths.

Well done.

Now, is there a highest common factor between the numerator and denominator there? Yes, that was quite easy to see isn't it? They're both divisible by three.

So three is a factor of the numerator, and three is a factor of the denominator.

So I can divide the numerator and the denominator by the same common factor, which is the highest common factor.

And, I get six divided by three is two, and nine divided by three is three.

So two thirds is the simplest form.

Excellent.

Well done if you've got all those correct! So we're going to continue thinking about the simplest form today and calculating,.

We are just, we are going to look at one aspect of that.

You were really confident in your practise activity there.

The calculating with fractions, and expressing them in the simplest form.

So we're going to continue with that.

But, I also want us to think back to some terminology that you know.

I've mixed numbers and improper fractions.

I'm also maybe going to investigate whether they can be expressed in their simplest form too.

Firstly though, have a look at this bar.

Can you tell me how many equal parts the whole has been divided into? That's right, it's been divided into ten equal parts! Fantastic! And what would one of those equal parts be? It will be one tenth.

What about this second bar? How many tenths are represented by the yellow portion? Yes.

There're three tenths So we can say come and let's say this together.

The whole is divided into ten equal parts.

One part represents one tenth of the whole.

So three parts represents three tenths of the whole.

Really well done.

If I show you a bar now, Can you tell me how many tenths are represented by the yellow portion? How many tenths the represented there? Six tenths well done! How many more tenths is that more than three tenths? How many more tenths is that? So if we were counted in tenths, how many more tenth would we need? Three more tenths.

Can we express that in a calculation like we were in the last session? You think you could write that down now? Pause now, and tell me how many tenths I have altogether using an addition calculation.

How did you get on? Did you say that three tenths, plus three tenths is the same as six one tenths.

Excellent, well done! I'm going to show you another bar now.

I'd like you to tell me how many tenths there are altogether.

But could you write it as a calculation? So pause me now.

I'm going to put the bar up, and right now is the calculation.

How did you get on? How many tenths did you say there were altogether? Nine one tenths? What calculation did you write? So I've for three, one tenths, plus another three one tenths.

It's the same.

Oh, and another three one tenths is the same as nine tenths.

So we've got three tenths added to three tenths, added to another three tenths is nine tenths.

Have I got a whole yet? No, it's less than a whole, isn't it? How many more tenths would I need to make one whole, I would need one more tenth, wouldn't I? Okay, I've got a little bit of a challenge for you here.

If you think that you could add another three tenths to this calculation, but also represent it pictorially.

So, I want you to pause me, draw bars that represent pictorially, how many tenths you have altogether, and write me the calculation.

Pause me now.

How did you get on? That was quite a challenge actually.

So we said didn't we, there were three tenths, plus another three tenths, plus another three tenths, was the same as nine tenths.

What did you have for your pictorial representation? Did you have something like this? Huh, yes, excellent! So we had to have two bars.

Didn't we? Why have we got to have two wholes now? Why have we got to have two wholes? Because, I needed three more tenths.

WelL done.

Did you write this calculation? Excellent, let's have a look.

Three tenths, also another three tenths, plus another three, one tenths, and now I need three more one tenths don't I? One, two, three more one tenths.

So, that made twelve tenths altogether.

Fantastic! So did you find, you needed you to draw two wholes, and split both of those wholes into ten equal parts to make twelve tenths? What could you tell me about twelve tenths? Can you remember how we describe that? The terminology we can use to express it? Yes, I can hear you screaming at me.

It's an improper fraction.

Let's say this together.

An improper fraction is when the numerator has a greater value than the denominator.

Why does the numerator have a greater value, than the denominator? It's because I've got more equal parts.

Haven't I.

So I only needed ten equal parts for my whole, because my whole is divided into ten equal parts.

We said that over here didn't we? The whole is divided into ten equal parts.

What if I add on three more tenths? Then actually, now I have twelve tenths.

So, I have two extra tenths then one whole.

Is one other way I can write that? So if I look at those wholes, I have one hole and two tenths.

Really well done! And so, this is described as a mixed number.

Do you remember both those terms? Let's say it together mixed numbers are quantities made up of both whole numbers and fractional parts.

So what was the same, and what's different? Can you tell me what's same, what's different? Pause me now, have a little think about the different images we can see there.

What did you think? What's the same? We got the same number of equal parts altogether.

So we've got twelve one tenths all together.

Fantastic! What's different? `The way we've written it.

So the way we've expressed it.

So here we've expressed it as an improper fraction and we can see we've got twelve, one tenths, and here we've expressed those twelve, one tenths, as wholes and parts.

Because, we have, one whole and two, one tenths as well.

Excellent.

We're going to continue looking at that now, and thinking about the simplest form too.

Let's have a look at counting in three tenths on a number line.

Take a look at this number line.

What can you tell me about the divisions? What do the divisions represent? So if we look at these yellow lines here, what are the divisions between those yellow markers.

Yes, that's right there're one tenth.

Fantastic.

So, each of these divisions represents one tenth because the whole has been divided into ten equal paths.

Really well done.

Do you think you could show me with your finger, where three tenths is on this number line? Go on, put your finger at where you think three tenths is.

Did you say it was there? Excellent.

So the whole has been divided into ten equal parts, and three of those parts is represented by the numerator.

And we can see that can't we? Three of those parts I shaded orange.

Where would six tenths be? So, if I added on three more tenths now, where would six tenths be? Can you show me where that is with your finger? So if I add three tenths I would find that six six tenths is there.

Is that where you got it? Fantastic.

So six tenths, the whole has been divided into ten equal parts and six of those parts are represented by the numerator.

Fantastic.

But we can see, can't we, is that less than, or greater than a half? It's greater than a half, isn't it? And we can see that more and half of the whole is shaded.

And what would half of that whole be? It would be five tenths wouldn't it? It would be five tenths here.

Excellent.

And, five tenths is an equivalent fraction to a half.

Okay, let's continue skip counting in three tenths.

So three temps plus three tenths is six tenths.

Can you put your finger, where you think the next fraction is going to come? The fraction, if I add on three tenths will be there, nine tenths.

Fantastic.

And that is the whole is divided into 10 equal parts.

And nine of those parts are shaded and represented by the numerator.

Have I got a whole yet? No.

How many more parts do I need to make a whole? I need one tenth don't I? We can see that would that blue part, that's not shaded there.

So, nine tenths, plus another tenth, would make ten tenths, which is a whole.

We're not going to add one tenth though are we, we're going to add another three tenths.

Do you think you can put your finger where that will be, and tell me how many tenths we have altogether? So, nine tenths plus another three more tenths, is twelve tenths.

Fantastic! And so what do we call that? How did we express that? We said it was an improper fraction, because the numerator is greater than the denominator.

So the whole has been divided into ten equal parts, which is represented by the denominator, but we have twelve of those parts.

So, we need two more parts.

So, the whole has been divided into ten equal parts.

Ten of those parts makes one whole, two more tenths, gives us one whole and two tenths.

And if I were to write it as one and two tenths, how would I be expressing it now? I'd be expressing it as a mixed number.

So, we can see can't we? As an improper fraction, the numerator greater than the denominator, because I have more equal parts than one whole.

And if I write it as a mixed number, I can say, I have one whole and two tenths.

Super! Now that you can tell me what an improper fraction is, and show me what that looks like, we said, we'd have to see if we could express that in its simplest form as well.

Do you think we can? Let's have a see and revise how to find the simplest form.

So, how many tenths do we have altogether here? The sum of three tenths and three tenths is, six tenths.

Well done! And how do we find the simplest form? We divide by the highest common factor, which is two.

Excellent.

So our equivalent fraction is three fifths, because the numerator and denominator have been divided by the same common factor.

And the highest common factor is two, because six and ten, both the numerator and denominator, are in the two times tables, and two is a factor of both six and ten.

Now, do you think we can apply that, to this improper fraction? Can you find a common factor? Pause me now.

Have a see if you can find a common factor, and have a go at dividing the numerator and denominator by the same common factor, and see what you get.

See you in a second.

How did you get on? Did they have a common factor? Yes.

What was the common factor? It's two again, isn't it? So, if I divide the numerator denominator by two, what do I get? Six fifths, really well done.

Are these six fifths equivalent to twelve tenths? Yes, it is.

Well, shall I have a look, to see if we can prove that? So, here we can say that that's the same as one and a fifth, because how many fifths did we need for a whole? We needed five fifths, and if we look at one and a fifth, wouldn't that be the same proportion of the whole as twelve tenths? Because, each of these fifths here will be two tenths, wouldn't they? So, I'd have two, four, six, eight, ten, twelve tenths.

Fantastic.

Well done.

You think you could do this one on your own? So, if you can find the sum of seven tenths and eight tenths, put this number line here to help you, so you can find out where seven tenths is.

Add eight tenths, more, find out what the sum is, and then express it in its simplest form by dividing the numerator and denominator, by the highest common factor.

And see if you can go a little bit further, and tell me what the mixed number would be.

So pause me now, have a go with those, and use the number line to help you.

How did you get on? So, what did you find was the sum of seven tenths and eight tenths? So if my unit is tenths, and I've got seven one tenths, and I add another eight one tenths, then I have fifteen tenths altogether.

Fantastic! Did you get that too? Really well done.

So, I wanted you to express this in its simplest form.

What's the highest common factor, of the numerator and denominator? It's five, well done! Because fifteen is a multiple of five, and ten is also a multiple of five.

So, I can divide the numerator denominator by the same highest common factor, which is five.

And what did you get as your equivalent fraction? Three halves.

Fantastic.

And it might help to have a think about that pizza now.

If I look at a pizza and I cut it into halves, how many halves is three halves? How many pizzas would I have? I would have one pizza and another half of a pizza.

So sometimes thinking about the proportion of the whole, helps us to express it as a mixed number, doesn't it? Because I've got one whole and one more half.

Did you all get that? Excellent, really well done! Can you find the difference between twenty seven sixths and five sixths? So quite a challenge for your here.

I'm going to put a number line on to help you again.

I want you to notice one thing, where does my number line start? So it starts at one whole it doesn't start at zero.

So how many sixths do I already have? I already have six sixths.

Really well done.

So I want you to keep that in mind and in finding the difference between these two fractions, you're going to have to show me where twenty seven sixth is on the number line, and subtract five of those one sixth.

So pause me now, and have a go with that.

And then express your answer in the simplest form.

How did you get on? Okay where did you put twenty six sixths Did you put it here, twenty seven sixths, did you put it there? Yes.

Why did you put it there? We had to count up in six sixths, didn't we? So let's do that.

Six sixths, twelve sixths, eighteen sixths, twenty four sixths, and then twenty five, twenty six, twenty seven sixths.

Fantastic! So we know we got that in the right place, and then I'm going to subtract five sixths.

Where did you put that? Yes, if I subtract five sixths, I end up there don't I, with twenty two sixths, which is between twenty four sixths, and eighteen sixths, which we counted before.

Okay.

How did you simplify it? Did you find the highest common factor with the numerator and the denominator? And what is that? It's two again, isn't it? Yes, so twenty two and six, are both in the two times tables.

Two is a factor of the numerator.

So if I divide it twenty two by two, I get eleven Well done.

And six divided by two is three.

So I had eleven thirds.

Where would that be on the number line? Where would it be on the number line? Where did you put it? Yes, it's in the same position as twenty two sixths, isn't it? Because they are equivalent fractions and they take up the same proportion of the whole.

Do you think we could prove that? So could I write eleven thirds as a mixed number? How many thirds do I need to make a whole.

Three thirds, excellent.

So, nine thirds would be three wholes, and I have two more thirds.

Excellent.

Now, how can I prove that that's free in two thirds on the number line? So, I've got three here.

How can I split between three and four into thirds? Let's have a go so three and.

One for third, we can there can't we? Two thirds and that would be my 3rd third.

So, eleven thirds is in exactly the same place on the number line as twenty two sixth.

So, you have proven that those are equivalent too.

As well as expressing it in its simplest form, which was eleven thirds, which is still an improper fraction, but then you went on to express that as a mixed number, which is a whole number and parts.

Whew, that was an awful lot of work that we did there, so let's move on to our independent practise.

We've worked really hard in that session, and I think we've made an awful lot of connections.

We've been really thinking about our understanding of improper fractions and mixed numbers and calculating and expressing the simplest form too.

So I'd like you to apply what we've done, to these three calculations.

You've got two additions and a subtraction, and don't forget you can draw number lines and express these pictorially, to help you with your understanding.

So, I hope you've enjoyed today.

You can look back at the generalisations to help you throughout the lesson and previous sessions.

And we'll see you again.

Bye! Bye!.