Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson from our unit on addition and subtraction of fractions.

Now, fractions are one of those things that some people absolutely love, and I hope you are in that camp.

I love fractions.

I think they're really clever ways of expressing numbers and parts of holes.

So if you're ready to make a start, let's do some work on adding and subtracting fractions.

In this lesson, we're going to be using knowledge of fractions in their simplest form when solving addition and subtraction problems. You might have had some experience of fractions in their simplest form, thinking about common factors.

And now we're going to have a look at that and see how it can help us when we are adding and subtracting fractions.

We've got some keywords.

We've got common factor, simplify and simplest form.

So I'll take my turn to say them and then it'll be your turn.

Are you ready? My turn, common factor.

Your turn.

My turn.

Simplify your turn, my turn.

Simplest form, your turn.

Excellent.

I'm sure there words you're familiar with.

Let's just double check their meanings because they're going to be really useful to us in our lesson.

So when we compare lists of factors of two or more numbers, common factors are those shared by all the numbers.

To simplify a fraction is to identify the highest common factor shared by the numerator and denominator in that fraction and to scale down both by that factor.

So we're keeping the fraction the same value, but we're just representing it with smaller value digits.

And a fraction is in its simplest form when the numerator and denominator cannot be any smaller while still being whole numbers.

And we can find that out that they are in their simplest form when we realise that the numerator and denominator only share a factor of one.

They can't be scaled down by a number greater than one.

And obviously dividing by one leaves the number unchanged.

Okay, so there are two parts to our lesson today.

In the first we're going to be simplifying solutions.

And in the second part of our lesson we're going to be thinking about improper solutions.

So improper fractions.

So let's make a start on part one and we've got Lucas and Jun helping us in our lesson today.

So how would you calculate this? You might want to have a little look at it before Lucas and Jun share their thoughts.

Well, Lucas says, I've noticed that both of the denominators are the same.

So we're dealing with the same type of fraction here.

Jun says, and we know that when the denominators are the same we can add the numerators.

So let's just think about that.

We've got 3/24ths and 5/24ths.

So we're adding like denominators.

So we can just think about the numerator because we know that three of something plus five of something must be equal to eight of something.

And in this case our somethings are 24ths, so our sum will be eight 24ths and that's what June agrees.

Lucas says it's quite hard to picture how much eight 24ths of something is.

He's right, isn't it? We don't often have things in 24 equal parts, do we? Lucas says, I think we can simplify the fraction further.

I think he's right.

Can you see a common factor between eight and 24? Jun says the highest common factor of eight and 24 is eight.

So we can divide the numerator and the denominator by eight to find its simplest form.

We're going to scale them both down by a factor of eight and we create the equivalent fraction one third.

And we know that that's in its simplest form because the numerator is one, it's a unit fraction.

So one third is the simplest form of eight 24ths.

And Jun says that's much easier to visualise, isn't it? You can visualise a third of something.

So we know that when a fraction is converted into its simplest form, it's often easier to visualise.

Lucas says, let's look at another example.

We've got 96/384ths.

That fraction is really difficult to visualise, says Lucas, he's right.

I can't really visualise 384 of something, let alone 96 of them.

Oh my goodness Jun, thank goodness you've done some thinking for us.

Jen says the highest common factor of 96 and 384 is 96.

So we're going to be able to create a unit fraction.

Can you estimate what our unit fraction's going to be? 96 is quite close to 100.

384 is quite close to 400.

So what do you think it's gonna be? We divide the numerator and denominator by 96, scaled them both down by the same factor.

We get the equivalent fraction of a quarter.

The simplest form is one quarter, which is a lot easier to visualise.

Absolutely.

I can think of a quarter of something much more easily now.

I know I can think of 96, 384ths as a quarter as well.

Let's look at the subtraction example.

So we did some addition.

Let's think about subtracting.

So we've got nine eighteenths, subtract three eighteenths.

What do we notice? Well we can just think about the numerator 'cause we know we are counting in eighteenths.

So we've got nine eighteenths and we're subtracting three of those eighteenths.

So nine somethings subtract three somethings is equal to six somethings and these are eighteenths.

So Lucas says the difference between nine eighteenths and three eighteenths is six eighteenths.

So there's our answer, but we can simplify it.

And Jun says the highest common factor of six and 18 is six.

They're both in the six times table.

So we can divide the numerator and the denominator by six to find its simplest form, we can scale down the numerator and the denominator by the same factor.

So we keep the proportion the same.

Six divided by six is one and 18 divided by six is three.

So our fraction is equivalent to one third.

So nine 18ths, subtract three eighteenths is actually equal to one third and one third is the simplest form of six eighteenths.

Time to check your understanding.

Can you complete this edition and give the sum in its simplest form? Pause the video, have a go.

And when you're ready for some feedback, press play.

How did you get on? Did you spot that it was fifteenths that we were adding? So we've got three plus three and this time it's three fifteenths plus three fifteenths.

So it's equal to six fifteenths.

And did you then think about the common factor, the highest common factor of six and 15 is three.

So we can scale the numerator and the denominator down by a factor of three, dividing them both by three.

And we get the fraction in its simplest form, which is two fifths, not a unit fraction this time, but remember that two is a prime number, so it only has factors of one in itself and five is not a multiple of two.

So it does not have two as a factor.

Time to check again this time.

Can you complete the subtraction and give the difference in its simplest form? Pause the video, have a go.

And when you're ready for some feedback, press play.

How did you get on this time? Did you spot that it was 12th? And so we could just think about those numerator.

We've got five of something.

Subtract two of something and five twelfths subtract two twelfths is equal to three twelfths.

Did you spot some common factors? Aha, Lucas is there the highest common factor of three and 12 is three.

So we can scale the numerator and the denominator down by the same factor.

So we preserve the fraction and the proportion of the whole, three divided by three is one divided 12 divided by three is four.

So our equivalent fraction is a quarter.

Three twelfths in its simplest form is equal to one quarter time for you to do some practise.

In question one, you're going to calculate the following equations and give your answer in simplest form.

So think carefully about what you are adding and subtracting and then look for those highest common factors between the numerator and denominator to express the fraction in its simplest form.

And in question two, you've got some more to try again, calculate the answer to the following equations and give your answer in the simplest form.

Pause the video, have a go.

And when you're ready for the answers and some feedback, press play.

How did you get on? Let's look at question one.

So we had two ninths plus four ninths, which is equal to six ninths.

We've got a highest common factor of three and that simplifies to two thirds.

For the next one we had four fifteenths plus two fifteenths, which is six fifteenths.

We've got a highest common factor there of three, which gives us an answer of two fifths.

For the next one we have three sevenths, subtract one seventh, which is two sevenths.

Oh, have we got a common factor there? We haven't have we? Two sevenths is already in its simplest form.

Then we've got eight 24ths subtract five 24ths eight, subtract five is equal to three.

So we've got three 24ths.

Our highest common factor is three and that simplifies to one eighth.

Now we've got three add ends, but that's okay, we can add them all together.

They're all twelves.

Five twelfths plus five twelfths, plus two twelfths.

Five plus five plus two is equal to 12.

So that's 12 twelfths.

What do we know about fractions where the numerator and the denominator are the same? They're equal to one.

So 12 twelfths in its simplest form is one.

And what about the next one? We've got two thirteenths plus seven thirteenths and subtract four thirteenths while two plus seven is nine.

Nine subtract four is five.

We've got five thirteenths.

What do you know about five and 13? They're both prime numbers, aren't they? So the only factor they share will be one.

So that fraction is already in its simplest form, five thirteenths.

And question two, again, we were calculating and expressing the answer in its simplest form.

So we've got one 12th plus two twelfths, which is three twelfths, highest common factor is three.

So our answer is one quarter.

This time we've got two twelfths plus two twelfths.

Ah, can you see something going on here? We are only adding twelfths, aren't we? Two twelfths plus two twelfths is equal to four twelfths, four and 12 have a common factor of four.

So that's one third.

What about two twelfths plus three twelfths? Well two plus three is five, so that's five twelfths, five is a prime number, 12 is not a multiple of five.

So that fraction is already in its simplest form, five twelfths, three twelfths plus three twelfths.

Ah, so that six twelfths, do you know something about six twelfths? The numerator is half the value of the denominator.

So that must be equal to a half.

What do you think is gonna happen next? We've got three twelfths plus four twelfths.

Well, three plus four is seven, isn't it? Seven Twelfths, what do you know about seven? The prime number is 12, a multiple of seven.

It isn't.

So seven twelfths is already in its simplest form.

Let's look at the final one.

Four twelfths plus four 12ths.

So we're going to end up with eight twelfths.

So we know we're going to be able to simplify this because the numerator and the denominator are both even, eight twelfths.

What's the highest common factor? It's going to be four, isn't it? So we're going to get a simplified fraction of two thirds, not a unit fraction this time because eight is not a factor of 12.

So we couldn't get to a unit fraction.

I hope you were successful with all of those.

Let's have a look at part two then, we're going to be simplifying improper solutions, improper fractions where the numerator is larger than the denominator.

Let's have a look.

So how would you calculate this? You might would have a think before Lucas and Jun share their thinking.

Well, Lucas says both fractions still have the same denominators, so we can still add thinking about our numerator.

And Jun says just like earlier, we know that when the denominators are the same we can add the numerator because we know we're adding twelfths.

So nine plus 11 is equal to 20.

So we've got 20 twelfths.

Now that means our numerator is greater than our denominator.

So we call that an improper fraction.

It's greater than one.

Can we simplify it though? Yeah, we can.

Improper fractions can also be simplified.

Again, we are looking for those highest common factors.

So we can find the highest common factor of 12 and 20 and the highest common factor of 12 and 20 is four.

So we can scale down the numerator and the denominator by a factor of four 20 divided by four is five, 12 divided by four is three.

And we can leave it as an improper fraction.

It's five thirds.

So nine twelfths plus 11 twelfths is equal to five thirds.

So we can leave it as an improper fraction like this if we like.

Or says Lucas, we could convert the improper fraction to a mixed number to start with.

So we've got 20 twelfths.

Well 12th twelfths is one hole, so that's one hole and an extra eight twelfths.

So it's one and eight twelfths.

And now we can simplify the eight twelfths.

The highest common factor of eight and 12 is four.

So if we divide the numerator in the denominator by four, scale them both down by the same factor, we get a fraction of two thirds, but we mustn't forget that this is a mixed number.

So we've got our one as well and our one doesn't change when we simplify the fraction.

So nine twelfths plus 11 twelfths is also equal to one and two thirds.

So which strategy do you prefer? To begin with, we kept the fraction as an improper fraction and simplified it and we could then convert it to a mixed number.

And in the second strategy we converted the improper fraction into a mixed number straight away and then simplified the fraction.

I wonder which strategy you prefer.

I think if I had to make a choice, I'd go for the top one.

I think simplifying the fraction and then making the mixed number was an easier route for me, but you may prefer the other one.

There's no right or wrong answer to this is whichever suits your thinking best.

So we've done an addition here and we've got a sum of 14 12ths.

Can you give the sum in its simplest form and it doesn't matter which way you go about doing it.

Pause the video, have a go.

And when you're ready for some feedback, press play.

How did you get on? Which route did you go? Well, Lucas says the highest common factor of 14 and 12 is two.

So he's going to simplify the improper fraction.

So he's going to scale the numerator and the denominator both down by two.

Divide them both by two and he gets seven-sixths, it's greater than one, but what is it as a mixed number? And that's equal to one and one-sixth, 6/6 is one whole, and then we have one extra six.

Or you could have created a mixed number from 14 twelfths of one and two twelfths and then simplified the two twelfths to one sixth.

And you'd have the same answer of one and one sixth.

I wonder which way you did it? Another check, this time we've got a subtraction.

So can you work out the difference and then give it in its simplest form.

Pause the video, have a go.

And when you're ready for some feedback, press play.

How did you get on? Well, 1912, subtract five twelves is going to be equal to 14 twelfths.

So we could convert that into a mixed number one and two twelfths, and the highest common factor of two and 12 is two.

So we can then simplify the fractional parts of our mixed number by scaling both the numerator and denominator down by a factor of two, divided them both by two and we get a fraction of one and one six well done if you worked it out that way.

Well done also if you simplified 14-12 to seven-sixth and then created your mixed number, time for you to do some practise.

So in question one, you're going to calculate the following equations and give your answer in the simplest form.

And in question two, you're going to write three of your own equations where the simplest form of the solution is one and one third.

So you've got an addition of two numbers, a subtraction, and then you've got a final one where you're going to add two numbers and then subtract another one.

Can you give three where the answer in its simplest form would be one and one third? Pause the video, have a go.

And when you're ready for some answers and feedback, press play.

How did you get on? Let's look at question one.

So we started with nine sevenths plus five sevenths.

Well, nine plus five is 14.

That's 14 sevenths.

Ah, 14 sevenths, I think I can simplify that to two, can't I? Seven sevenths is one whole.

So 14 sevenths must be equal to two.

The next one we had eight fifteenths plus 12 fifteenths.

Well that's 20 fifteens now they look like multiples of five to me, 20 fifteenths.

So I think I can simplify that to four thirds.

And then as a mixed number, that's one and a third.

So four thirds and I can simplify that again to one and one third.

I've got three ninths plus two ninths plus 10 ninths.

Well, three and two is five plus 10 is 15.

So 15 ninths.

Okay, so I can think of that as a mixed number or I can simplify my fraction.

15 ninths is going to simplify to one and two thirds.

I might have simplified five thirds first and then created my mixed number.

Or I might have created my mixed number and simplified my fraction.

20 eighths.

Subtract six eighths is 14 eighths.

So I've got a common factor there of two, so seven quarters.

So I'm going to have one and three quarters.

23 quarters subtract five quarters.

So 23, subtract five is going to be 18.

18 quarters.

So a common factor of two.

So that's nine halves.

So that's going to be four and a half and 23 ninths.

Subtract five, ninths is going to be 18 ninths, oh, hang on, 18 ninths again.

That's two just like the first one was.

So we got one answer of two by subtracting and one answer of two by adding.

I hope you were successful with that.

And I hope you explored the two different ways of either converting to a mixed number and then simplifying or simplifying the improper fraction and then converting to a mixed number.

So lots of different possibilities for here.

Here with three that we came up with.

So we had 17 eighteenths plus seven eighteenths, so that gave us 24 eighteenths.

So we can simplify that to one and one third.

We also had 12 six, subtract four sixth, well that's two.

Subtract two thirds, I like that.

Two holes subtract two thirds is going to give us one and one third.

And then we had eight ninths plus seven ninths.

Subtract three ninths, well seven ninths subtract three ninths is four ninths, eight plus four is 12.

12 ninths, that's one and one third.

Fantastic.

I hope you had fun exploring those and coming up with some interesting solutions.

And we've come to the end of our lesson.

We've been using knowledge of fractions in their simplest form when solving addition and subtraction problems. And what have we found out? Well, we found out it can be helpful to simplify a fraction to its simplest form to help visualise the magnitude of the fraction, the size of the fraction, the scale of it.

Sometimes fractions that have what look like quite scarily big numbers actually simplify to something very simple.

Do you remember our one that's simplified to a quarter? Thanks to Jun's knowledge of the 96 times table.

When adding or subtracting fractions with the same denominator, the answers can often be converted into their simplest form.

And when adding and subtracting improper fractions with the same denominator, you can convert the improper fraction to a mixed number, which can make simplifying easier.

But equally, you can simplify the improper fraction and then convert it to a mixed number, whichever you find the simplest.

Thank you for all your hard work and your mathematical thinking in this lesson, and I hope I get to work with you again soon.

Bye-Bye.