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're in that camp.

I love fractions, I think they're really clever ways of expressing numbers and parts of wholes.

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 thinking about reasoning around how to write a fraction in its simplest form.

You may have been looking at fractions in their simplest form already.

And today, we're going to do some reasoning about how we know how to write a fraction in its simplest form, and when to spot that one is in its simplest form.

We've got three keywords or phrases today.

We've got common factor, simplify, and equivalent.

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, equivalent, your turn.

Excellent, I'm sure they're words and phrases that you're familiar with, but let's just double check what they mean.

They're gonna be really useful to us in our lesson today.

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, and to scale down both by that factor.

And if two numbers or expressions have the same value, they are said to be equivalent, even if they don't quite look the same.

That's really important with equivalent fractions.

Okay, so with those keywords at our fingertips, ready to use, let's get into the lesson.

There are two parts to our lesson.

In the first part, we're going to be identifying fractions in their simplest form and in the second part, we're going to be using simplest form to compare fractions.

So let's make a start on part one.

And we've got Laura and Andeep in our lesson with us today.

So which of these fractions are in their simplest form? You might want to pause and have a think before Laura and Andeep share their thinking.

Laura says, "How can we find out if a fraction is in its simplest form?" What do you think? Andeep says, "If the highest common factor of the numerator and the denominator is one, then we know that the fraction is in its simplest form." We can't scale the numerator and denominator down any more than it is already if the common factor is one.

So knowing that, what do you think about these fractions? Let's look at them in more detail.

So what's the simplest form of this fraction? We've got 3/15.

Well, let's look at the factors, so we've got a factor bug here to help us.

So these are the factors of three, just one and three.

Three is a prime number.

It has two factors, one and itself.

So here are the factors of three, and here are the factors of 15.

So 15 has factors of one, three, five, and 15.

So what do you notice about three and 15 and their factors? Well, one and three are common factors of both numbers.

Well, one is always going to be a common factor, isn't it? But if one is the only common factor, then we know a fraction is in its simplest form, but we've got another common factor here.

So we've got three as a common factor.

So because three is the highest common factor of three and 15, we can further simplify the fraction.

We can scale the numerator and denominator down by the same amount, and the proportion of the whole stays the same, the fractions are equivalent.

So three divided by three is one, and 15 divided by three is five.

So we can create the equivalent fraction, 1/5.

So Laura says, "So 1/5 is the simplest form of 3/15." Well, it must be, mustn't it? We've got a numerator of one, and when we've got a numerator of one, we know a fraction must be in its simplest form because the only factor of one is one.

What's the simplest form of this fraction then, we've got 2/5? Well, let's look at the factors of two, here they are.

And the factors of five, here they are.

One is the only common factor of both numbers.

They're both prime numbers, aren't they? They've only got factors of one and themselves.

So one is the only common factor.

Because one is the highest common factor of two and five, the fraction stays the same.

Even if we divide the numerator and the denominator by one, we're going to create the same fraction, 2/5.

So 2/5 is the simplest form of 2/5.

One is the only common factor shared by two and five.

What about this one, 4/20? Well, we can see that both of those numbers are even, so there must be something in that, mustn't there? So here are the factors of four, one, two, and four.

And what about the factors of 20? We've got one, two, four, five, 10, and 20, lots of factors for 20.

What do you notice about common factors? Well, one, two, and four are common factors of both numbers.

So what's the highest common factor? Ah, four is the highest common factor of 20.

So we can further simplify the fraction.

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

Four divided by four is one, 20 divided by four is equal to five.

1/5 is the same value as 4/20.

One part out of five is the same as four parts out of 20, so they represent the same proportion of the whole.

So 1/5 is the simplest form of 4/20.

And when we look at 1/5, it's a unit fraction, so we can't simplify any further because the only factor of one is one.

Andeep says, "Also two is a common factor." And we can halve the numerator and the denominator, but we can do it twice, can't we? So we could halve them to make 2/10, and halve them again to make 1/5.

Laura says, "That's an interesting strategy which would work when the numerator and the denominator are even numbers." But she says, "I think it's more efficient to find the highest common factor though." I think it is, it's worth thinking about those numbers a little bit more.

We can always halve if they are both even, but we might be missing a trick.

So what about the simplest form of this fraction, we've got 25/36? So here are the factors of 25, one, 25, and five.

It's a square number, isn't it? And what about the factors of 36? Oh, loads of factors.

One, two, three, four, six, nine, 12, 18, and 36.

It's also a square number because six is sort of on its own there, isn't it? The factor pair for six is six.

What about common factors though? What do you spot? There's only one.

One is the only common factor of both numbers, of both the numerator and the denominator.

Because one is the highest common factor of 25 and 36, the fraction would stay the same, wouldn't it? Any number divided by one is equal to itself, so there's no point doing that.

And so Laura says, "We didn't even need to make factor bugs." There are no times tables with both 25 and 36 in, except for the one times table.

So if you know your times tables, you will know that there is no common factor for 25 and 36 other than one.

So one is the only common factor of both numbers.

And therefore, 25/36 is in its simplest form.

Time to check your understanding.

Can you give the simplest form of 8/24? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, Laura said, "Eight is the highest common factor of both numbers, so the simplest form is 1/3." We can divide the numerator and the denominator by eight, scale them both down by the same factor, and we create the equivalent fraction of 1/3.

So 1/3 is the simplest form of 8/24, they both represent the same proportion of the whole.

Time for another check, can you tick all the fractions that are in their simplest form? So, again, you're looking for those common factors of the numerator and the denominator.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, did you spot that 7/9 and 1/3 are both in their simplest form? 4/6 can't be in its simplest form because both are even numbers, so they must share a common factor of two.

And 3/30, well, we know that both of those are multiples of three, so they share a common factor of three.

But seven and nine only share a common factor of one.

And one and three, well, it's in its simplest form already, isn't it? Because the numerator is already one, it's a unit fraction.

And there we go, Andeep just reminding us, "One is the highest common factor of the numerators and denominators in these fractions," in 7/9 and 1/3.

Time for you to do some practise.

Can you group the fractions as to whether they are in their simplest form, or are not in their simplest form? So we've got two circles, one for fractions in their simplest form and one for fractions not in their simplest form.

And I wonder if you could add any others into those circles as well.

Pause the video, have a go, and when you're ready for some feedback, press play.

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

So 1/6 is in its simplest form, it's a unit fraction, isn't it? 11/20 is in its simplest form.

11 is a prime number, so unless the denominator is a multiple of 11, it will be a fraction in its simplest form, and it isn't, it's 11/20th.

18/30, well, they're both even numbers.

So at the very least, they share a common factor of two.

But in fact, they probably share a common factor of six, don't they? Six times three is 18, and six times five is 30.

23/30, well, they're not even and, in fact, 23 is another prime number, isn't it? So unless the denominator is a multiple of our prime number, then that will be a fraction in its simplest form.

7/21, both odd numbers, seven is a prime number, but this time the denominator is a multiple of seven, isn't it? Three sevens are 21, so we could simplify that fraction to 1/3.

9/17, well, this time the denominator is a prime number, isn't it? And so, our fraction must be in its simplest form because prime numbers only have factors of one and themselves.

And 5/15, well, we can see if we know our test of divisibility for five.

Both of those numbers are in the five times table with ones and the five, so they share a common factor of five.

So did you create your own fractions, I wonder, for those categories? I wonder what you put into those.

If you didn't have a go, maybe have a go now before we start the second part of our lesson.

And here is the second part of our lesson, and we're going to be using simplest form to compare fractions.

So which of these fractions is the greatest in value? Is it 3/18 or 5/20? What do you think? Laura says, "I think 5/20 is the largest, as it has a larger numerator and denominator." Hmm, is her reasoning right there, do you think? Andeep says, "I disagree, we can't tell easily as the numerators are different and so are the denominators." But they're not that far apart, are they? We've got 3/18 and 5/20.

Laura says, "Why don't we see if we can simplify each fraction to make them easier to compare?" Good thinking, Laura.

So they're going to simplify each fraction.

So we've got 3/18, so we're going to list the factors.

So our factor bug for three tells us that the factors are one and three.

Can you already see that there's going to be a common factor here? And here are the factors for 18.

We've got one, two, three, six, nine, and 18.

So can you see a highest common factor there? One and three are common factors of both.

It's three we're interested in, isn't it? 'Cause that's the one that's going to allow us to simplify the fraction.

So because three is the highest common factor of three and 18, we can further simplify the fraction.

We can scale the numerator and denominator down by a factor of three, divide them both by three, and we create the equivalent fraction, 1/6.

"So 1/6 is the simplest form of 3/18," says Laura.

What about 5/20? Well, five, again, is a prime number and its factors are one and five.

But 20 is a multiple of five, isn't it? So we're going to find a common factor.

Well, it must be five, mustn't it? So here are the factors of 20, one, two, four, five, 10, and 20.

So, yes, common factors of one and five.

And which is the one we're interested in? It's five, isn't it? So because five is the highest common factor of five and 20, we can further simplify the fraction.

We scale them both down by a factor of five, so we divide them both by five, and we get the fraction 1/4, which is equivalent to 5/20.

So as Laura says, "1/4 is the simplest form of 5/20." Ah, so we've created two unit fractions now.

So they're going to return to their original problem to compare.

She says, "I think this should now help us to compare these fractions." So 3/18 was equal to 1/6, and 5/20 is equal to 1/4.

So we've simplified the fractions, now what do you spot? Andeep says, "Yes, when the numerator are the same, the fraction with the largest denominator is the smallest." We've got one out of a larger number of parts in the whole, haven't we? So we can say that 1/6 is less than 1/4.

So therefore, 3/18 was less than 5/20.

Let's add another fraction into the mix here.

We know about 3/18 and 5/20, what about 30/35? Which of these fractions is greatest in value Laura says, "Why don't we do the same again, and convert to its simplest form?" We could do that, got any other thoughts? Andeep says, "I don't think we need to." Why does he think that, He says, "30/35 is very nearly one whole, and this is much larger than the other two fractions, which are equivalent to unit fractions." 1/3 and 1/5 and 30/35 is really close to one whole.

So this time we don't need to simplify, we can use our knowledge of fractions and their equivalents to a hole to help us.

So we knew 3/8 was less than 5/20, and we know that they're both less than 30/35.

And as Laura says, "It's always worth taking a really close look at the fractions." What do you know about fractions that could help you? You may not need to simplify, you may be able to reason it, just by knowing what the fraction represents.

Time to check your understanding.

Can you compare these two fractions? Which symbol should go in the circle, greater than, less than, or equals? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Did you simplify both fractions? Can you see that there's a common factor of three there in 3/27, so that would simplify to 1/9.

And there's a common factor of 4 in 4/32, so that would simplify to 1/8.

So we've got 1/9 and 1/8.

And as Andeep says, "When the numerators are the same, the fraction with the largest denominator is the smallest." So 1/9 is smaller than 1/8.

What about these three fractions? Can you simplify them to compare them? I think we might need to you simplifying here to help us to compare them.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? So did you manage to simplify them to 1/8, 1/7, and 1/9? So now we can see that 1/8 is less than 1/7, and 1/7 is greater than 1/9.

Simplifying these fractions made it easier to compare them.

So it's always a useful strategy to have in your fraction toolkit when you're comparing and ordering fractions, which is what you're going to do now in task B.

So in question one, you're going to compare each pair of fractions by converting them to their simplest form, remembering that the simplest form fraction is equivalent to the fraction you started with.

It's just expressed with the smallest value digits we can to express that same proportion of the whole.

And in question two, you're going to order these fractions in ascending order by simplifying them.

Ascending means from the smallest to the largest.

Pause the video, have a go at the two questions, and when you're ready for some answers and feedback, press play.

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

So you were comparing each pair of fractions by converting them to their simplest form, looking for that highest common factor shared by the numerator and the denominator of each fraction.

So the first pair of fractions can be simplified to 1/7 and 1/8, and we know that 1/7 is greater than 1/8 because the hole has been split into a smaller number of parts.

So each part will be slightly bigger than 1/8.

What about the next pair? Well, 22/55, they have a highest common factor of 11, so that's 2/5.

And six and 21 have a common factor of three, so that simplifies to 2/7.

Oh, now we haven't got a common denominator here, but we have got that common numerator again.

So just like the reasoning we had for 1/7 and 1/8, if we've got two of each fraction then, again, the fraction with the largest denominator represents the smallest proportion of the whole.

So 2/5 is greater than 2/7.

What about the final pair? Well, 14 and 18, well, they definitely have a common factor of two 'cause they're both even and, in fact, I think that's the best we're going to get, yeah, 7/9.

And 35/45, well, there's a common factor of five there.

Seven, oh, 7/9 and 7/9, so those fractions are equal.

So when we simplified them, we could see that they were equivalent fractions, even though they didn't look like it when they were 14/18 and 35/45.

And for question two, you're going to order the fractions in ascending order from smallest to largest by simplifying them.

And Andeep's saying, "When the numerator are the same, the fraction with the largest denominator is the smallest," just as we used in our previous slide.

So let's have a look at how these fractions simplify.

Three and 18 have a common factor of three, so we can simplify to 1/6.

5 and 20 have a highest common factor of five, so we can simplify to 1/4.

2/18, well, the highest common factor is two there, so 1/9.

4/8, well, we could look for that, but we could also see that the numerator is half the value of the denominator.

So that must be equivalent to 1/2.

4/12, well, we could look at 1/3 relationship there or divide both by four, which is the highest common factor, and we get 1/3.

And then 6/60 is equivalent to 1/10.

So now we've got unit fractions.

So remember what Andeep says, "The fraction with the largest denominator is the smallest," and we want to go from smallest to largest.

So our order will be 1/10, which is equivalent to 6/60, and then 1/9, which is equivalent to 2/18.

1/6 which is equivalent to 3/18, oh, we could have seen that, couldn't we? One more in our numerator, so it must be a slightly bigger fraction.

So that was 1/6, then we must have 1/4, which is 5/20.

Then 1/3, which is 4/12, and then 1/2, which was 4/8.

And it's always good practise to write the original fractions back out again when you're ordering.

Well done if you've got all of those correct, and well done for all your simplifying of fractions.

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

We've been reasoning about how to write a fraction in its simplest form.

To find the simplest form of a fraction, you can divide the numerator and denominator of the fraction by the highest common factor that they both share.

You could use factor bugs to help you to identify the highest common factor.

And you can simplify fractions to their simplest form to help you to compare them, and even to order them more easily.

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.