Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson on comparing fractions using equivalents and decimals.

Fractions are great, I love working with fractions, and so I hope during this lesson, you're going to see that fractions are really friendly and things that can really help us with our mathematical thinking.

So, let's get started.

So in this lesson, we're going to be using the relationship between the numerator and the denominator to simplify fractions.

We're also going to look at those fraction families again.

So, let's make a start.

We've got lots of keywords in our lesson today.

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

So I'll take my turn to say them and then you can say them.

So my turn, common factor.

Your turn.

My turn, simplest form.

Your turn.

My turn, simplify.

Your turn.

My turn, scale up and down.

Your turn.

Some of those words might be new to you, but we're going to learn more about them as we go through the lesson, but let's just look at what they mean, because that might help us as we're talking through the first parts of our lesson.

So we probably all know what factors are, those numbers that we multiply together.

And when you are comparing the factors of two numbers, a common factor is one shared by both numbers.

So we're gonna be looking out for common factors today, factors that are shared by both the numerator and the denominator.

When a fraction is in its simplest form, the numerator and the denominator only share a common factor of one.

So that's definitely something to look out for in the lesson today.

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

So we've talked about using scaling to create equivalent fractions.

This is creating the simplest equivalent fraction that we can for a set of fractions or for a particular fraction.

And to scale up or down is to multiply or divide by a given number or factor.

So let's have a look at what's in our lesson today.

There are two parts.

In the first part, we're going to identify and explain equivalence between fractions.

And in the second part, we're going to simplify fractions.

So let's make a start on Part 1.

And we've got Jun and Sophia helping us with our learning today.

So how would you fill in the gaps in these fraction families? You may have come across fraction families before, they've got a times table link in the numerators and the denominators, and they're all equivalent fractions.

So how would you fill in the gaps in these fraction families? You might want to pause and have a think about it.

Sophia says, "I can see 1 times 9 equals 9," in that first fraction there, in that 1/9." So she can see that the numerator multiplied by nine is equal to the denominator.

So she says, "So in all the fractions, the denominator must be nine times the numerator." Ah, that's good thinking, Sophia.

Well done.

And what about the second family? Jun says, "I can see the relationships across the numerator and denominators of these fractions." So he can see a times two relationship between the numerator of 3/7 and 6/somethings.

So we know that there must be the same relationship between the denominators of those two fractions.

And he can also see 7 multiplied by 4 is equal to 28.

So the numerator of that fraction must be 4 times the numerator of the 3/7.

So they found some relationships to work with.

So Sophia says, "Okay, so 1 times 2 is equal to 2, so the next denominator must be 18, 2 times 9 is equal to 18." So 1/9 is equivalent to 2/18.

And the other way she spotted it, that 1 times 9 was equal to 9, well, 2 times 9 is equal to 18 as well, isn't it? So that works too.

Jun says the next numerator must be a three as there's a pattern.

But also, 9 times 3 is equal to 27, so 1 times 3 is equal to 3.

Can you see the pattern that Jun has spotted? So can you use the fact that the denominator must be nine times the numerator to complete the other fractions in this family of equivalent fractions? Pause the video, have a go, and we'll come back and share our answers.

How did you get on? You could just have spotted some patterns that the numerator were going 1, 2, 3, so the next one must be 4.

But remember, we are looking to check that those fractions are equivalent, so it's always worth using another way of checking.

We knew that 1 times 9 was equal to 9, so that relationship had to be true for all the fractions.

So is 4 times 9 equal to 36? Yes, it is, so we have found another equivalent fraction.

You may have seen in the denominators, that we've got that pattern of the nine times table, 9, 18, 27, 36, so the next denominator would be 45.

But again, let's use that checking, 1 times 9 is equal to 9, 5 times 9 is equal to 45, so yes, it is equivalent.

Carrying on that pattern, the next denominator must be 54.

But our numerator is 6, 6 times 9, we can check, is 54.

And then using the pattern in the numerators, we can see that that final missing numerator must be seven.

But if we look at that relationship within the fraction, 7 times 9 is equal to 63, so we know those fractions are equivalent.

And Jun says, "The numerators increase by one each time and the denominators by nine." But Sophia says, "The fractions are all equivalent to 1/9, as the denominators are 9 times the numerators." So it's good to see a pattern and be able to follow it, but it's also good to check and make sure that we really understand that those fractions are equivalent, they represent the same proportion of our whole.

So what is the relationship between the numerator and denominator in 3/7? 3 times something is equal to 7, but it's easier to use the relationships across the numerator and denominators.

But you can see in the bar model here, with our 7 and 7, that the 7 is made up of 2 lots of 3 and then 1/3 of 3.

So 3 times 2 and 1/3 is equal to 7, and that will be true for all the fractions equivalent to 3/7.

But as Jun says, it's much easier in this situation to see the relationships between the numerators and the denominators across the pairs of equivalent fractions.

So as you can see, 3 times 2 is equal to 6, so we'd be able to multiply the denominators by two, and if 7 times 4 is equal to 28, then to find the equivalent number of 1/28 that are equal to 3/7, we'd have to multiply our numerator of three by four as well.

Those relationships are easier to see because we are multiplying by a whole number rather than multiplying by a fraction when we look at the relationship between the numerator and the denominator in each fraction.

So how would you fill in the gaps in this fraction family? So this family of fractions are all related to 3/7.

How would you go about filling in the gaps? You might want to have a bit of a think before we carry on.

Well, Sophia spotted that 3 times 2 is equal to 6, so she says, "The next denominator must be 14.

3 times 2 is equal to 6 and 7 times 2 is equal to 14." So we've scaled up the numerator by a factor of two, we must scale up the denominator by a factor of two, twice as many pieces in the whole, so we need twice as many of those smaller pieces to keep the proportion of the whole the same.

What else can you spot? Well, Jun says, "The first missing numerator must be 12.

7 times 4 is equal to 28, so 3 times 4 is equal to 12." So he spotted that there was a multiply by four relationship between the denominator of 7 and the denominator of 28, and if we've scaled up the denominator by a factor of four, we must do the same for the numerator.

So 12/28 is equivalent to 3/7.

Time to check your understanding now.

Use the relationship across the numerators and denominators to complete the other fractions that are equivalent to 3/7.

Pause the video, have a go, and then we'll come back and discuss the answers.

How did you get on? Well, Sophia says the next denominator must be 21, 3 times is equal to 9, 7 times 3 is equal to 21.

We need to scale both the numerator and the denominator by the same factor, and the factor was three.

Jun says, "The next missing numerator must be 15." He's seen that 7 times 5 is equal to 35, 3 times 5 is equal to 15, so 15/35 is equivalent to 3/7.

And he says, "The final missing numerator must be 18.

7 times 6 is equal to 42, and 3 times 6 is equal to 18." So what about that final denominator? Ah, Sophia says it must be 49, 3 times 7 is equal to 21, 7 times 7 is equal to 49.

So 21/49 is equivalent 3/7, it represents the same proportion of the whole.

Time for you to do some practise.

Sophia and Jun are discussing how to show these fractions on a number line.

Who do you agree with and why? Jun says, "We will need a really long number line.

12/108 has a really big denominator." And Sophia says, "We only need a 0 to 1 number line to show the position of these fractions." Who do you agree with and why? So for question 2, use a number line, a bar model, and the language of scaling to show that these pairs of fractions are equivalent.

Think about what we've done in the lesson.

We've looked at some bar models and we've looked at positions on a number line, and we've also thought about the idea of the numerator and the denominator being multiplied by the same factor, being scaled up or scaled down.

So use all those ideas to show that these pairs of fractions are equivalent and have exactly the same value.

Pause the video, have a go, and we'll come back with some feedback.

How did you get on? So here, they were deciding what the number line would look like in order to show that the fractions were equivalent.

Jun thought it would need to be a really big number line because of the 108 in the denominator, and Sophia said, no, it only needs to be 0 to 1.

And I hope you discovered that Sophia is correct, the fractions are both less than one, so it can be positioned on a 0 to 1 number line.

If we were going to show all 108 divisions, then yes, we would have a lot of lines on that number line.

But if we look along there and think about ninths, we can see that our number line on the bottom is labelled from 0 to 1, but in ninths.

But we can see that every 1/9 is worth 12/108, because our numerator has been scaled up by a factor of 12 and our denominator has as well.

1 times 12 is equal to 12 and 9 times 12 is equal to 108.

Sophia says these fractions are equivalent and sit at the same position on the number line.

And Jun says, "12 out of 108 is the same proportion of the whole as 1 out of 0." And if you imagine each of those divisions of 1/0 divided into 12 equal parts, we can see that 12 out of the 108 is the same as 1 out of 9.

So for part 2, you are proving all those different ways that the fractions were equivalent.

So we can see here, that 3/7 is equal to 27/63, because 3 times 9 is equal to 27 and 7 times 9 is equal to 63.

Both the numerator and denominator have been scaled up by a factor of nine.

On a bar model, we've got our bottom bar showing 3 parts out of 7, and our top bar showing 27 parts out of 63.

Our whole remains the same, but we've divided our whole into seven equal parts and then into 63 equal parts, 9 times as many parts, so we need nine times as many of those smaller parts to keep the proportion the same.

There are nine times as many parts in the whole.

And we can see that 3 jumps out of the 7 is the same as 27 jumps out of 63.

So here, we've got a number line marked from 0 to 7 and also from 0 to 63, not quite the same as the number lines we've had before, but you can see that if we count to 63 in 7 equal jumps, that that represents 27 out of 63, which represents the same as 3 out of 7.

You might have drawn a number line with the sevenths up to one and the sixty-thirds up to one whole.

And for part B, we were looking at the equivalence of 7/42 and 1/6.

And here we can see that the numerator and denominator have both been scaled up by a factor of seven if we look from the unit fraction to the non-unit fraction, or you may have said they were scaled down by a factor of seven.

7 divided by 7 is equal to 1, 42 divided by 7 is equal to 6.

And again, on the bar model, one part out of six is equal to 7 parts out of 42.

There are seven times as many parts in the whole, each of those sixths has been divided into seven equal parts, so we have seven times as many parts in the whole, so we need seven times as many of those small parts to keep the proportion the same.

And again, we've used this slightly different number line.

We've got a number line going from 0 to 6 and 0 to 42.

But if we divide that number line into six equal parts, we can see that a jump of 1 on the number line, to 6, is the same as a jump of 7, on the number line, to 42.

1 jump out of 6 is the same as 7 jumps out of 42 along the number line, so the proportion is the same and those fractions are equivalent.

Okay, so into the second part of our lesson, and we're going to think in more detail about simplifying fractions.

I wonder if having thought about what we've done in part 1, you've got an idea of what this is all about.

Let's have a look.

Okay, so what fractions can you create that are equivalent to 3/12? So what are you going to do to create the fractions? And what do you notice about the numbers that you use? You might want to pause here and have a little think about this.

We are gonna talk about it together though.

So here's what Sophia and Jun think.

Ah, so Sophia says, "I've created 6/24.

I've scaled up the numerator and denominator by the same factor." Can you see what that factor is? And she says, "I can keep scaling up the numerators and denominators by the same factors." So she used a factor of two, 3 times 2 is equal to 6, 12 times 2 is equal to 24, and then a factor of three, 3 times 3 is equal to 9, and 12 times 3 is equal to 36.

Jun says, "I think I can scale the numerator and denominator down by the same factor." Ah, so what has he spotted? That's right, he's spotted that you can divide both the numerator and the denominator by three.

3 divided by 3 is 1, 12 divided by 3 is 4, and he's created a fraction, 1/4.

He says, "I can't scale this fraction down as the only common factor is one." Ah, that's interesting.

So when we think about dividing, we are looking for that factor that is shared by both the numerator and the denominator.

So in 9/36, we've actually got a shared factor of nine, haven't we? 36 and 9 are both in the 9 times table, and that's a really good way to think about those factors that are common, which times table are both the numerator and the denominator part of? So these fractions are all equivalent, but 1/4 shows the fraction in its simplest form.

There are no more common factors other than one.

So let's think about the common factors within the fractions.

3 is a factor of 3 and 12, so 3 is a common factor in 3/12.

3 is a common factor of 6 and 24 as well.

Three 2's are 6, and three 8's are 24.

2 and 6 are also common factors of 6 and 24.

So we could scale the numerator and denominator down by a factor of two, or we could scale them down by a factor of six.

6 divided by 6 is equal to 1, and 24 divided by 6 is equal to 4, and that gets us our equivalent fraction of 1/4.

In 1/4, the only common factor is 1.

1 divided by 1 is 1, and 4 divided by 1 is 4, so we've only got a common factor of one, and that doesn't change the numbers that we're using within the fraction.

So when the only common factor between the numerator and the denominator is one, the fraction is in its simplest form, it's written with the smallest value digits we can use to represent that proportion of the whole.

So time to check your understanding, is this fraction in its simplest form? Explain how you know.

Pause the video, have a go, and we'll come back with some feedback.

What did you think? No, the fraction is not in its simplest form, is it? We can see lots of common factors there, I expect, between the numerator of 12 and the denominator of 36.

So we could see that 12 and 36 are even numbers, so two must be a common factor.

All even numbers share a common factor of 2.

So at the very least, we can scale the numerator and denominator down by a factor of 2.

12 and 36 are also both in the 4 times table.

3 times 4 is 12, 9 times 4 is 36, so four must be a common factor as well.

3 and 6 are also common factors of 12 and 36.

There are a lot of ways we could have used the factors of these numbers to create equivalent fractions, but what's going to get it to its simplest form? Well, 12 and 36 are both in the 12 times table, so 12 must be a common factor.

1 times 12 is 12, 3 times 12 is equal to 36.

So how can we make sure, now we've decided it isn't in its simplest form, that we have written the fraction in its simplest form? So what are the factors of 12? The factors of 12 are 1, 2, 3, 4, 6, and 12.

What are the factors of 36? Lots of factors here.

1, 2, 3, 4, 6, 9, 12, 18, and 36.

And we can work all those out when they're not in our times table knowledge by thinking about the sort of factor pair.

If we know that 1 and 2 are factors of 36, we know it's got to be one times something, and it's 1 times 36 and 2 times 18.

They're not in our times tables up to 12 times 12, but we need to think about them when we're thinking about the factor pairs.

So now we've got a list of all the factors of 12 and all the factors of 36.

So what is the highest factor that is common to both the numerator and the denominator? What's the largest number that they both share as a factor? Oh, that's 12, isn't it? So we need to scale down the numerator and denominator by the highest common factor, and that's 12.

So 12 divided by 12 and 36 divided by 12, and that gets us to 1/3.

The fractions are equivalent, and 1/3 shows the fraction in its simplest form.

The highest common factor now is one, we can't express that fraction with any smaller value digits.

So are 12, 36, and 1/3 equivalent fractions? Can we be sure? Well, we can.

Simplifying fractions is the same as creating equivalent fractions.

Scaling the numerator and denominator by the same factor is how we create equivalent fractions, and that's what we do when we simplify fractions as well.

We can see that 12 divided by 12 is equal to 1, and 36 divided by 12 is equal to 3.

So yes, the numerators and denominators have been scaled by the same factor.

Ah, we can also look at that relationship within the fractions.

1 times 3 is equal to 3, 12 times 3 is equal to 36.

So, yes, they're definitely equivalent.

In both fractions, the denominator is three times the value of the numerator.

So we can definitely say that those fractions are equivalent, and we could draw a bar model or a number line to prove it as well.

So, can you simplify this fraction? This is 9/15.

How could we go about simplifying it? Well, let's think about the factors.

The factors of 9 are 1, 3, and 9.

The factors of 15 are 1, 3, 5, and 15.

3 is the highest common factor of 9 and 15, so we can scale down the numerator and denominator by a factor of 3.

So let's do that.

9 divided by 3 is 3, 15 divided by 3 is 5, so we've got a fraction of 3/5.

So if this is in its simplest form, then the only common factor between the numerator and the denominator will be 1.

We haven't got to a unit fraction here, have we? 3/5 is 3 parts out of 5, but think about the factors of 3 and 5.

So the factors of 3 are 1 and 3, the factors of 5 are 1 and 5.

They're both prime numbers, aren't they? So the only factor that they share is 1.

So 3/5 is written in its simplest form, and 9/15 is equivalent to 3/5.

Time to check your understanding.

Simplify these fractions.

Look for the highest common factor of the numerator and the denominator, and see if you can write these fractions in their simplest form.

Pause the video, have a go, and we'll come back and look at the answers together.

How did you get on? What did you find? So in our first fraction, 4/12, the highest common factor of the numerator and the denominator is 4.

So we can scale the numerator and denominator down by the same factor of 4, and get the equivalent fraction of 1/3.

And 1/3 must be in its simplest form because the only factor they share is 1.

In fact, all unit fractions must be in their simplest form.

If we've got a numerator of 1, the only factor it can have is 1.

What about the highest common factor here? Well, it's four again, isn't it? 8 is in the 4 times table, 12 is in the 4 times table, but neither of them are in the times table higher, so four is the highest common factor.

So if we scale the numerator and denominator down by a factor of 4, we get the fraction 2/3.

Is 2/3 in its simplest form? Well, yes, because 2 and 3 are both prime numbers, and so therefore they only share a factor of 1.

What about 8/18? It looks as though this should simplify really well, doesn't it? But when we list those factors, the highest common factor of the numerator and the denominator is 2, so we can scale them both down by a factor of 2, and we get the fraction 4/9.

And the only factor that 4 and 9 share is 1.

What about this next one, 2,000/8,000? What did you spot here? Well, the highest common factor is 2,000, isn't it? If we actually sort of think, "Well, I've got 2 lots of 1,000, and 8 lots of 1,000, I can think about 2 and 8, can't I?" And the highest common factor is 2, or 2,000 in this case.

So we can divide them both by 2,000.

2,000 divided by 2,000 is equal to 1, and 8,000 divided by 2,000 is equal to 4, so we've actually got an equivalent fraction of 1/4, and we know that that is in its simplest form because it's a unit fraction.

Okay, so can this fraction be simplified? What do we think? Let's think about the factors.

So a fraction can be simplified when the numerator and denominator have a common factor other than 1.

All numbers have a common factor of 1, but we are looking for a common factor that isn't 1, so can we see that in 5/12? The 5 of the numerator and the 12 of the denominator? Well, the factors of 5 are 1 and 5, it's a prime number, isn't it? And the factors of 12 are 1, 3, 4, 6, and 12, so quite a lot of factors.

But what about common factors? Factors that are shared by 5 and 12? Nope, this fraction is already in its simplest form.

The only common factor is 1.

So we need to think quite carefully, we can't tell just by looking, does it have one in the numerator or are they both even numbers? We saw before with the other fraction, that one odd number and one even number, they can still share a common factor other than 1, that sometimes a fraction is written with quite large numbers, but it is actually in its simplest form because the numerator and the denominator only share a common factor of 1.

Time for you to do some practise.

Can you sort the fractions? Can they be simplified or not? And you've got two circles to sort them into.

And part B, simplify the fractions that can be simplified, so the ones that you've sorted into the can be simplified, simplify them, write them in their simplest form.

And then for C, explain how you know when a fraction is in its form.

And for part D, is a simplified form of the fraction equivalent to the fraction that you started with? And explain your answer.

So you're going to sort, you're going to simplify, you're going to explain how you know when a fraction is in its simplest form, and then you're going to think about the fractions that you've simplified and see them in their simplest form, and explain whether they make a pair of equivalent fractions or not.

And question 2, is Jun always, sometimes, or never true? He says, "To simplify a fraction, you just halve the numerator and the denominator." Explain your answer.

Pause the video, have a go at your questions, and we'll come back for some feedback.

How did you get on? So here are the fractions sorted into those that can be simplified and those that can't, so you might just want to pause here and check that you sorted your fractions correctly.

So then we asked you to simplify the fractions that are not in their simplest form.

So 3/9 was not in its simplest form, in its simplest form, it's equal to 1/3.

The highest common factor that the numerator and denominator shared was 3.

For 3/18, its simplest form is 1/6.

Again, the highest common factor shared by the numerator and denominator was 3.

What about 11/33? In its simplest form, it's 1/3.

The highest common factor shared by the numerator and the denominator was 11 in this case.

7/14 is equal to a 1/2.

So we could think about common factors, but we could also think, "Well, hang on.

7 is half of 14, or 14 is double 7." And we know that when the numerator is half the value of the denominator, the fraction must be equivalent to 1/2.

10/15 was equivalent to 2/3 in its simplest form, 12/30 is equal to 2/5 in its simplest form, and 12/33 is equal to 4/11 in its simplest form.

In C, we asked you to explain what it means when a fraction is in its simplest form.

So you may have written something like this: If a fraction is in its simplest form, then the only common factor between the numerator and the denominator will be 1.

And for part D, we asked you to explain how you knew if these fractions were equivalent or not when we'd written them in their simplest form.

And the answer, of course, is yes.

They have to be equivalent.

They're equivalent as the numerator and denominator have been scaled down by a common factor.

The fractions still represent the same proportion of the whole.

They'd sit at the same point on the number line.

And we could draw a bar model to show that all of those pairs of fractions are equivalent and have exactly the same value.

So is Jun's statement always, sometimes, or never true? He says, "To simplify a fraction, you just halve the numerator and the denominator." Well, Jun's statement is sometimes true.

When two is the highest common factor between the numerator and the denominator in a fraction, then halving them will create the fraction in its simplest form.

But if two isn't the highest common factor, then that won't work, so it is sometimes true.

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

So we've been using the relationship between the numerator and the denominator to simplify fractions.

And we've had a look at those fraction families again, haven't we? So what have we learned about today? Simplifying a fraction is the same as creating equivalent fractions, and it's really important to really explore that and understand that a simplified fraction is equivalent to the fraction that we've started with.

A fraction can be simplified when the numerator and denominator have a common factor other than one.

To write a fraction in its simplest form, the numerator and denominator are scaled down by their highest common factor.

And if the numerator and denominator in a fraction only have a common factor of one, then the fraction is in its simplest form.

I hope you've enjoyed more exploration of equivalent fractions, and learning about fractions in their simplest forms. Thank you for your hard work and all your good thinking today, and I hope I get to work with you again soon.

Bye-bye.