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 use our understanding of equivalent fractions to solve problems. You may have been doing lots of work on equivalent fractions recently, so this is a chance to put all of that work together and see if we can use our understanding to solve some different sorts of problems involving equivalent fractions.

So let's make a start.

We've got some keywords in this lesson and you're probably not surprised that these are the keywords, seeing as this is a lesson about equivalent fractions.

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

My turn, equivalent fraction.

Your turn.

My turn, simplify.

Your turn.

My turn, scale up, down.

Your turn.

I hope they're words that are familiar to you and you've been using them a lot in your learning recently, but let's just remind ourselves what they mean.

We will be using them a lot in today's lesson.

So equivalent fractions are fractions which have the same value even though they may look different, they describe the same part or proportion of the whole.

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.

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

So our lesson is in two parts today.

In the first part, we're going to think about simplifying fractions, and in the second part, we're going to think about using equivalents to solve problems. So let's make a start on part one.

And we've got Aisha and Laura helping us with our learning today.

So let's start by thinking about using our knowledge of equivalent fractions to fill in the missing numerators and denominators.

And you might recognise this as a sort of fraction family perhaps.

Can you see some things in common with the numerators that we do have and with the denominators that we do have? So you might want to take a little time here just to have a think yourself and think about how you might go about filling in those missing numerators and denominators before we have a look at it together.

Okay, so we've filled in the missing values for you, and I really want you to think about explaining the thinking you did to work out those missing values.

And you may well have had a go at doing them yourself, but we're really gonna focus in on the explanations that we can give and you may have got something different for those last two fractions.

These are just suggested answers.

Obviously, we had no numerator or denominator, so there were lots of different possibilities.

So let's have a focus on the thinking that we're doing to work out the missing values.

So let's take the answers away again and have a think through.

So Aisha says, "I spotted the relationship between these numerators." So she saw a numerator of 15 and a numerator of three, and she realised that she could multiply the three by five to equal 15.

So if three times five equals 15, what does that tell us about the missing denominator? Laura says, "We can think of the inverse." Ah, that's interesting, Laura.

Yeah.

So she could say that if the numerator has been scaled up by a factor of five, we could scale that known denominator down by a factor of five to create the equivalent fraction.

So she says, "25 divided by five is equal to five, so the missing denominator is equal to five." So 15/25 must be equal to 3/5.

So one numerator has been scaled up by a factor of five, so the other denominator needs to be scaled down by a factor of five.

We could also check it by using the multiplication route.

If we know that the numerator three has been multiplied by five to equal 15, then we know that the denominator five in 3/5 must be multiplied by five to equal 25.

So we can see that that works as well.

So we've got our 3/5 now.

I wonder which relationship you spotted next.

Let's see what Laura and Aisha came up with.

Ah, so Laura's used that denominator that we've just put in, and she spotted a times two relationship.

So she's seen that from the denominator five to the denominator 10, she spotted the relationship between the denominators was a multiply by two, and so she could then apply that to the numerators.

Three times two is equal to six, so 3/5 must be equivalent to 6/10.

The denominator has been scaled up by a factor of two.

So the numerator also has to be scaled up by a factor of two for the proportion to remain the same.

Ah, Aisha again has used the numerator from the 6/10 and she's spotted a relationship between 6/10 and 24 somethings.

She says, "I spotted the relationship between the numerator and applied it to the denominators." So she's seen a multiply by four, so she's going to multiply the denominator by four, scale it by the same factor.

The numerator has been scaled up by a factor of four, so the denominator also has to be scaled up by a factor of four.

So 6/10 must be equivalent to 24/40.

And Laura says, "Each fraction represents the same proportion of the whole." And you might be able to picture that on a number line divided into tenths on one side and 40ths on the other, or maybe looking at two bars the same length, one divided into 10 equal parts and one divided into 40 equal parts.

And seeing that 6/10 would be equal to 24/40.

Time to check your understanding.

Can you explain the thinking you did to work out the remaining missing values? Pause the video, have a go, and then we'll come back to share some feedback.

How did you get on? So Aisha says, "I spotted the relationship between the numerators and applied it to the denominators." So there's a times four there.

The numerator has been scaled by a factor four, so we do the same to the denominator.

Five times four is equal to 20, so 3/5 must be equivalent to 12/20.

Ah, and then Laura says, "Well, this numerator must be 12 as well because we've got the same denominator of 20, so that numerator must be 12." So we had two fractions that were the same in the middle of this family of fractions.

Did you spot that too? Ah, then Laura spotted a division relationship, so scaling down by a factor.

So 15 divided by three is equal to five.

So she says, "I can scale up the numerator 'cause I know there's that divide three times three relationship so I can multiply by three," she says.

Three times three is equal to nine.

And Aisha says, "I could use many relationships to calculate that the final missing denominator is equal to 50." Lots of different ways she could have looked at that.

She could have used the 3/5 and seen them multiply by 10, couldn't she? She could have used the 15/25 and seen a multiply by two.

She could have used 6/10 and seen a multiply by five.

Lots of different ways to work out that that final fraction in our family is 30/50.

I hope you were able to use all that you understand about equivalent fractions to explain how you worked out those missing values.

It's really important to know why things work and not just how to do them.

So what about the last two fractions? We've got no numerator or denominator there.

So how could we think about those? They need to be equivalent to all the other fractions though in our family.

So Laura says, "I know that the numerator is three times a number and the denominator is five times the same number." Ah, that's interesting.

Let's just have a think about that.

So we've got 3/5 in there.

So let's have a look at 30/50.

Well, that's three times 10 for the numerator.

And then she said the denominator is five times the same number, so it was three times 10, so the denominator would be five times 10, which it is, it's 50.

Does that work for 12/20? So the numerator is three times the number.

Well, 12 is three times four, and the denominator is five times a number.

Five times four is 20.

So that works.

Good thinking, Laura.

So she thinks she can use this to fill in the missing fractions.

Aisha says, "I will pick six, the numerator and the denominator will be three times six for the numerator, which is 18 and five times six for the denominator, which is 30." Three times six is 18, five times six is equal to 30.

So she's created the fraction 18/30.

Okay, so what about the last one then? Laura says, "I will pick 16." Oh, gone for a bigger number here.

So she says, "The numerator and denominator will be three times 16, which is 48 and five times 16, which is 80." So three times 16 is 48 for the numerator, and then five times that same number.

Five times 16 is 80 for the denominator.

So she's created another fraction, 48 80ths.

And actually we could check that one, couldn't we? We've got 24 40ths.

I quite like that.

We've scaled down by a factor of two.

48 divided by two is 24.

80 divided by two is 40.

So yes, that works there.

Can we find one to check with 18/30? Well, let's see.

We've got a 6/10, haven't we? 30 divided by three is equal to 10.

18 divided by three is equal to six.

So we can see that, in different ways, we can think about the relationships between the numerators and the denominators and between the numerator and numerator and denominator and denominator of equivalent fractions.

Aisha says there's an infinite number of ways to complete these fractions.

I think she's right.

We could multiply three and five by any whole number.

And we know that those whole numbers keep going.

We can just add one more, add one more, add one more.

Okay.

So let's think about simplifying fractions.

Which fraction in the list cannot be simplified? That's right, it's three fifths.

So what do we mean by cannot be simplified? Laura says, "We can't express this fraction with smaller numbers." Three and five are the smallest value digits that we can use to express this part of a whole or this number.

And Aisha says, "The highest factor of both three and five is one." They're both prime numbers, aren't they? The only common factor they have is one.

If we listed all the factors of three and all the factors of five, the only one they would have in common, the only one they share is one.

So using that thinking, which of these fractions are in their simplest form? Pause the video, have a think, and we'll come back together for some feedback.

How did you get on? Did you spot that these fractions are not in their simplest form? In these fractions, the numerator and denominator share a factor greater than one.

In 8/12, it's four, in 4/8, it's two, and in 9/12, it's three.

What about the other fractions then? These fractions are in their simplest form, the highest factor of both the numerator and the denominator is one.

And there's something we can spot about these fractions.

Can you see that all of them have a prime number? At least one prime number.

So 7/12, seven is a prime number.

3/5, three and five are both prime numbers.

5/6, five is a prime number.

And 3/8, three is a prime number.

And we know that prime numbers only have factors of one and themselves.

So if our denominator is not a multiple of our prime number, then that fraction will be in its simplest form.

Something to watch out for and to spot when you're looking for fractions in their simplest form.

Time for you to do some practise.

So can you simplify these fractions? And in part two, you're going to use your knowledge of equivalent fractions to fill in the missing numerators and denominators in this fraction family.

When you've done that, which fraction in the list cannot be simplified? And for four, what is the relationship between the numerator and the denominator in each fraction? So we're doing lots of looking across numerators and across denominators.

But for four, I want you to concentrate on the relationship between the numerator and the denominator in each fraction, what can you see? Pause the video, have a go, and we'll come back for some feedback.

How did you get on? So in their simplest form, these fractions look like this.

8/12 is equivalent to 2/3.

The numerator and denominator have a common factor of four.

So we have scaled the numerator and denominator down by a factor of four, and now we have 2/3.

And 2/3 cannot be simplified.

The only common factor shared by two and three is one.

What about 4/8? 4/8 is equal to one half.

Once we got a unit fraction, we know that fraction is in its simplest form, and you might have spotted that the numerator was half the value of the denominator, so therefore, the fraction had to be equal to a half.

And what about C? Well, in C, the numerator and the denominator have a common factor of three.

Nine divided by three is equal to three.

12 divided by three is equal to four.

So 9/12 is equal to three quarters, and three is a prime number.

Four is not in the three times table.

So we know that that fraction, so we know that that fraction can be, so we know that that fraction is in its simplest form.

So how did you get on with these? So you might have started by seeing that times will divide by three relationship between those two numerators.

So therefore, there's a times or divide by three relationship between the denominators.

So 9/63 must be equal to 3/21.

Or you might have seen a different relationship.

You might have seen that times divide by nine relationship between the seven and the 63.

And then you could apply that to the numerator.

So you'd have one seventh as equivalent to 9/63.

Now you have enough information to use scaling up and down of the numerators and denominators to complete the remaining fractions.

So there we have our missing numbers filled in.

And then, of course, we have the fractions on the end, which had no numerator or denominator.

So these are just some examples of what you might have found for the first and last fractions.

Question three asked you to find the fraction that cannot be simplified.

Well, 1/7 cannot be simplified as it is a unit fraction.

So there can be no common factor greater than one.

And then for question four, you were asked to look at the relationship between the numerator and the denominator in each fraction.

And you'll have found that in all the fractions, the denominator is seven times the numerator as they are all equivalent to one seventh.

So when we've got a unit fraction as our simplest form, it's really easy to see that relationship between the numerator and the denominator.

So if we multiplied any of those numerator by seven, we would get the value of the denominator.

So I'm gonna pick my favourite times table fact, which is eight times seven or seven times eight.

So eight times seven is 56.

So we can see that the numerator of eight has been multiplied by seven to get the denominator of 56.

So that fraction must be equivalent to one seventh.

Okay, and on into the second part of our lesson.

We're going to be using equivalents to solve problems. So we've got some images here.

Can you complete the equivalent fractions represented by the shaded areas in these images? So you might wanna have a little look before we look at the answers together.

Okay, so what can you see? Well, four out of six parts are shaded.

So that fraction must be 4/6.

What about the middle shape? Not as obvious to see, but we've got a three by three square there.

We've got three rows of three or three columns of three.

So we've got nine in total, which we can see, and six of them are shaded.

So six parts out of nine are shaded, so the fraction is 6/9.

And for the final one, eight out of 12 parts are shaded.

So we've got a fraction of 8/12.

What is the fraction represented in its simplest form? Hmm.

Are those fractions in their simplest form? I don't think they are, are they? Well, we can see that because the one with the smallest value digits is 4/6 and that's two even numbers.

So we know that's not in its simplest form.

Aisha says, "The highest common factor of four and six in 4/6 is two.

So we can simplify the fraction to 2/3." So we can divide both the numerator and the denominator by two.

So they've both been scaled down by the same factor to create the fraction 2/3.

How can you show that these fractions are equivalent to 2/3? Well, Laura says, "I can shade the shapes to show that they all represent 2/3 of the whole." So let's have a look.

So we can clearly see that two out of three equal parts there are shaded.

If we think of each equal part as the columns, we can see that two columns out of the three columns are shaded.

4/6 is equivalent to 2/3.

What about the middle one? Oh, and again we've got that idea of the two columns of three and then the one column not shaded.

So we can see that 6/9 is also equivalent to 2/3.

What about the final one? I've got four there, so I'm not seeing my columns here, but I can see that two out of the three rows are shaded.

So 8/12 must be equivalent to 2/3.

So by just re-imagining how the shading was, we can clearly see that 2/3 of each of those shapes are shaded, and so therefore, the fraction that is shaded, if we count all the individual parts, is equal to 2/3.

I wonder if you can do this the other way round.

Can you sketch and shade the rectangle to show another fraction that is equivalent to 2/3? So have a look.

We've given you 2/3 there.

Can you adapt that image or draw a new image to show another fraction equivalent to 2/3? Pause the video, have a go, and we'll come back for some feedback.

How did you get on? Laura says, "I can add extra lines across the rectangle to show fractions equivalent to 2/3." She's put lots of lines in there, hasn't she? She's added four extra lines.

So we've now got five rows of three.

So we've got 15 parts in the whole altogether and 10 of them are shaded.

So this shows that 10/15 is equivalent to 2/3.

So now can we use all the cards to create three equivalent fractions? You might want to pause and have a think about this before we look at it together.

Ah, well, we could try fractions equivalent to a half.

I can see we've got a one and a two there.

So let's put those in to make one half.

So can we now make another fraction equivalent to a half? Oh, 4/8 is equivalent to a half, isn't it? Because the numerator must be half the value of the denominator.

Or we could look at the fact that both the numerator and denominator have been scaled up by a factor of four.

One times four is equal to four, two times four is equal to eight.

So we've got four and 16 left.

Ah, but 4/16 is not equivalent to a half.

If we think about four out of 16, four isn't half of 16 and four times one is four, but eight times two is 16.

So we haven't scaled the numerators and denominators by the same factor.

So that can't be the right fraction to start with.

I wonder what else we could try.

Well, let's start with another unit fraction.

Let's see about fractions equivalent to a quarter, shall we? So let's put a quarter in.

Now what can we see? Ah, well we could have 2/8.

That's equivalent to a quarter, isn't it? So we could look and see that both the numerator and denominator have been scaled by a factor of two.

So we can either think scaling up one times two is two and four times two is eight, or scaling down, two divided by two is one and eight divided by two is equal to four.

Now we're left with 4/16.

Is that going to work? Yes, that does, doesn't it? If we start with the one quarter, we can see that the numerator and denominator have been scaled by a factor of four.

One times four is four, four times four is 16.

Or we could start with the 2/8 and see that that factor is now two, two times two is four, eight times two is 16.

These fractions are all equivalent to one quarter as the denominators are four times the numerator.

And that's the other way to check, isn't it? In all of those fractions, the numerator multiplied by four is equal to the denominator.

Time to check your understanding.

Can you use all the cards to create three equivalent fractions? Think about what we've just talked about with the last example.

See if you can apply that thinking to this set of cards.

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

How did you get on? Where did you start? You might have started with 1/3 thinking, "Well, let's make a unit fraction." You might have started with a quarter or a six or a half.

We could have made lots of different unit fractions, but why is it good to start with a third? Let's just have a look at the numbers we've got.

We've got three, six, and 12.

They're all multiples of three and then one, two, and four.

Well, they're the smaller numbers, but we've got that family idea there going on with the three, six, and 12.

So I wonder if they will end up being our denominators.

So 1/3 is equivalent to 2/6, which is equivalent to 4/12.

And we could have looked at scaling up and down the numerators and denominators, but we could also have looked at that relationship within the fraction 1/3.

These fractions are all equivalent to 1/3.

The numerators are 1/3 the value of the denominators.

So three divided by three is one.

Six divided by three is two.

12 divided by three is equal to four.

Or we could have looked at the multiplications.

I hope you were successful with those.

So what do you know about equivalents that will help to order these fractions from the smallest to the largest? Again, you might want to just pause and have a little think about that before we think about it together.

Let's see what Laura and Aisha think about.

Aisha says, "I've noticed that 7/6 is the only fraction greater than one.

So it must be the largest." "Oh yes", says Laura.

"The numerator is greater than the denominator." And we know that that means that we've got more than a whole.

6/6 would be equal to one as we would have all the parts of the whole.

So let's put that one at the largest end.

So what do you know about equivalents that will help you to order the last two fractions? Aisha says, "We could compare them to a half." That's a good idea.

Laura says, "Two quarters and 4/8 would equal one half.

So 3/4 and 5/8 are both a bit bigger than a half." And Aisha says, "5/8 is one eighth more than a half, and three quarters is one quarter more than a half." I wonder if that's gonna help us.

Oh, Laura says, "One quarter is greater than one eighth, so three quarters will be greater than 5/8." Ah, brilliant thinking.

Well done, you two.

So how can understanding equivalents help us to complete these equations? You may have spotted that they're all equal to zero.

And Aisha says, "Any number subtracted from itself is equal to zero." Ah, that's something, isn't it? So there's definitely something about equivalence here.

Laura says, "We have to find an equivalent fraction to complete the equations." So let's look at the first one, 11/44.

What's that equal to? It's equal to one something.

So what can we see here? We've got 11 as a numerator and one as a numerator.

Well, one times 11 is equal to 11.

So something times 11 must be equal to 44.

And we know our 11 times table, so we know that must be a quarter.

So 11/44 is equal to one quarter.

So therefore, 11/44 subtract one quarter must be equal to zero because those fractions have the same value.

They look very different, don't they? But with all the work you've been doing on equivalent fractions, you can prove that they have the same value.

What about the next one? We've got 3/9 and we're subtracting 3/9 from something and it's got to be the same value, hasn't it? So which equivalent fraction can we create? Well, we can see a three times three is equal to nine, both in the relationship between the numerator and the denominator and between those two denominators.

So if we use that multiply divide by three, we can see that 3/9 is equal to 1/3.

So zero is equal to 1/3 subtract 3/9.

We've subtracted the same value from itself.

So in the last one, we've got to find a fraction equivalent to 3/15.

It can be anything we want it to be.

And Aisha says, "Any fraction equivalent to 3/15 or a fifth will complete this equation." If we look at the relationship within the fraction we've got, 3/15, three multiplied by five is equal to 15.

So we know that if the denominator is five times the numerator, then that fraction will be equivalent.

And the simplest way we can do that is 1/5.

So there we are, 1/5.

Subtract 3/15 is equal to zero.

So what about this? How can our understanding of equivalents help us to complete these equations? This time, we've got equations equal to one and we're adding.

And Laura says, "4/8 and 50/100 are equivalent to a half and a half plus a half is equal to one." So this time we are looking for fractions equivalent to a half.

So we can have 5/10, which is equal to a half, and we could just have a half, or we could have any fraction equivalent to a half for the second one.

But in each of those, the sum will be one because our add ends are both equal to a half.

Time to check your understanding.

Which of these equations is correct and explain how you know.

Pause the video and we'll come back for some feedback.

How did you get on? Did you spot that it was C that was correct? How did you explain your thinking? Here's how we explained it.

5/9 is greater than a half.

So if we add it to a half, the sum will be greater than one.

I hope you got that thinking as well.

Time for you to do some practise.

You are going to choose two fractions from the set and use them to complete one of the equations.

Use equivalents to explain your thinking.

And you can use the fractions more than once in the different equations.

And for question two, how many different ways can you complete each fraction? The fractions must be proper fractions, so less than one.

So pause the video, have a go at your tasks, and we'll come back for some feedback.

How did you get on? So for the first one, we were looking for two fractions that when you subtracted them, the answer was zero.

So the fractions needed to be equivalent.

So we chose 4/12 and 3/9.

So for the two where our expression had to be either greater than or less than zero, we were looking for fractions that were not equivalent.

For the answer to be greater than zero, the fraction we subtracted had to have a smaller value than the fraction we were starting with.

And for the answer to be less than zero, then obviously the first fraction had to have a smaller value.

We were subtracting more so we'd have a negative number as our answer.

So for the fractions to be equal to one, in this case, they both had to be equal to a half.

Obviously, if we had fractions equivalent to 1/4 and 3/4, we could have used those to be equivalent to one.

But in this case, our fractions needed to be equivalent to a half.

So for the fractions that had to be greater than or less than one, again, we could use that idea of comparing to a half.

So that if we had a fraction equal to a half, we could add a fraction greater than a half or two fractions greater than a half to be greater than one and two fractions less than a half to be less than one as the total.

And in part two, we were looking for proper fractions.

So fractions less than one to complete these two pairs of equivalent fractions.

So as five is a prime number, the fractions can only be 5/25 and 1/5.

There's nothing that we can divide five by to create a numerator that's a whole number smaller than five.

So it had to be five divided by five is equal to one.

So for the second pair of fractions, 12 is not a prime number.

In fact, the numerator of the second fraction could be one, two, three, four, or six.

So we could divide 12 by 12 to give us a numerator of one.

So then 12/144 would be equivalent to 1/12.

12 divided by six would give us a numerator of two.

So 12/72 would be equal to 2/12.

12/48 would be equal to 3/12, and 12/36 would be equal to 4/12.

And 12/24 would be equal to 6/12.

So because 12 has many factors, we had far more options with the second pair of equivalent fractions than we did with the first.

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

We've been using understanding of equivalent fractions to solve problems. So we've learned that fractions which have been simplified cannot be represented with lower numbers.

The highest factor of the numerator and the denominator is one.

You can use knowledge of when fractions are equivalent or not to solve problems. Scaling numerators and denominators up and down helps us to simplify fractions and to create equivalent fractions.

And knowing if fractions are close to or equivalent to one half can help us to order fractions.

Wow, there was a lot of really good thinking going on in that lesson.

Thank you very much for all your hard work.

I hope you enjoyed it.

I certainly did, and I hope I get to work with you again soon.

Bye-bye.