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 explaining the relationship between numerators and denominators across equivalent fractions.

You may well have been looking at equivalent fractions recently.

So this is just another way to add into our strategies for finding out whether fractions are equivalent or not for creating equivalent fractions and for comparing fractions.

We've got keywords in this lesson.

We've got numerator, denominator, equivalent fraction, and scale or scaling.

So I'll say them and then you can have a turn.

My turn, numerator.

Your turn.

My turn, denominator.

Your turn.

My turn, equivalent fraction.

Your turn.

My turn, scale or scaling.

Your turn.

Excellent.

I'm sure you know quite a few of those words, but let's just remind ourselves of what they mean 'cause they are going to be really useful to us in our lesson today.

So the numerator is the top number in a fraction, and it shows how many parts we have.

The denominator is the bottom number in the fraction, and it shows how many equal parts the whole has been divided into.

Equivalent fractions are fractions which have the same value, even though they may look different.

And scaling is when a quantity is made hmm times the size.

So perhaps it's been made three times the size.

So scaling is to do with multiplication.

Look out for those words as we go through today's lesson.

So there are two parts to today's lesson.

We're going to be explaining equivalence using scaling, and then comparing fractions and identifying equivalence.

So let's make a start.

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

So what do you notice about these fractions? Jun says, "They are equivalent fractions." 1/5 is equal to 3/15.

Sofia says, "How do you know?" And Jun says, "The denominator is five times the numerator in both fractions." One times five is equal to five, and three times five is equal to 15.

And this is one way of knowing that we have equivalent fractions.

The fractions have the same value.

"The numerator is the same proportion of the denominators in both fractions," says Sofia.

The numerator is 1/5 of the size of the denominator, or the denominator is five times the size of the numerator, so the proportion is the same.

Can you see another relationship between the fractions? Sofia says, "I'm going to look at the two numerators and the two denominators," and she's going to add two because one plus two is equal to three and five plus 10 is equal to 15.

Does that look right? Jun says, "I don't think this is about addition.

Could you try multiplication?" That would get very complicated if we were adding different values to the numerator and the denominator.

I'm not sure we can see that things are staying the same in that way.

So perhaps Sofia can think about multiplication.

Ah, she's found one times three is equal to three for the numerators, and five times three is equal to 15.

Sofia says, "That makes sense.

Both numbers have been scaled up by the same factor of three." Both of them have been multiplied by three and that's what scaling up by the same factor means.

Jun says, "They both still represent the same fraction of one, which is the whole." Can you see another relationship between the numerators and the denominators? Well, Sofia says, "Both numbers have been scaled by the same factor of three," but we can also say that they've been scaled down by the same factor of three.

Three divided by three is equal to one, and 15 divided by three is equal to five.

So we can see that scaling relationship going in a multiplying way and in a dividing way as well.

In equivalent fractions, the numerators and denominators have been scaled by the same factor, and that's something that's really worth remembering and using as we go through the lesson to test whether fractions are equivalent or not.

One is the same proportion of five as three is of 15.

Can you see equivalence in the bar models? We've got two equal holes here, one divided into five equal parts and one divided into 15 equal parts.

And the same parts are shaded, 1/5 and 3/15.

And Jun says, "There are three times as many parts in the whole." So when we have fifteenths rather than fifths, there are three times as many parts in the whole.

And Sofia says, "So you need three times as many parts to have the same proportion of the whole." We need three times as many of the smaller parts to have the same proportion as one of the bigger parts.

The numerator and the denominator have both been scaled by a factor of three.

One times three is equal to three and five times three is equal to 15.

Can you see the equivalence on the number line? So we've got a zero to one number line.

On the top, we've divided the number line into five equal parts, and on the bottom, we've divided it into 15 equal parts.

One out of five is the same proportion of the whole as three out of 15.

And Sofia says, "I can see that the two fractions sit at the same position on the number line." Another way of showing that 1/5 and 3/15 are equal.

They are equivalent fractions.

Time to check your understanding.

What relationship is shown by these arrows and how does it prove that the fractions are equivalent? So pause the video, have a think, and then we'll discuss our answers.

How did you get on? Ah, we can put a multiplied by four on both of those arrows.

One times four is equal to four, three times four is equal to 12.

So the denominator is four times the value of the numerator in both fractions.

One out of four is the same proportion of the whole as three out of 12.

And you might be able to picture that using bar models or a number line.

And all of this means that the fractions are equivalent.

Time to check your understanding again.

What relationship is shown by these arrows and how does it prove that the fractions are equivalent? Remember, some of our keywords in this lesson might help you with this.

So pause the video, have a think, and then we'll get back together to discuss our answers.

How did you get on? Did you notice that there was a times three relationship? One times three is equal to three and four times three is equal to 12.

The numerator and the denominator have both been scaled by a factor of three.

One times three is equal to three, four times three is equal to 12.

One out of four is the same proportion of the whole as three is out of 12.

This means that the fractions are equivalent.

Can you use the relationships to work out the missing numerator and denominator in these pairs of equivalent fractions? Think about what the arrows represent.

How can you work out those missing values? So you might have spotted in this first pair that the denominator is 12 times the numerator, and it must be in both fractions if the fractions are equivalent.

So one times 12 is equal to 12 and something times 12 is equal to 48.

And we can also see that 12 multiplied by four is equal to 48.

So one times four must be equal to our missing numerator.

The numerator and denominator have both been scaled up by a factor of four.

And all that leads to the fact that our missing number must be four, our missing numerator.

1/12 is equal to 4/48.

So what about the denominator in the second pair? Gosh, we've got some big numbers there, big denominators.

So eight times something is equal to 240.

Well, eight times three is equal to 24.

So eight times 30 must be equal to 240.

So one times 30 must equal our missing denominator.

The denominator is 30 times the numerator in both fractions.

And if we look at that relationship between the numerator and numerator and the denominator and denominator, again, we can see that one multiplied by eight is equal to eight.

So something multiplied by eight must equal 240.

The numerator and denominator have both been scaled by a factor of eight.

And all that leads to the fact that our missing denominator must be 30.

8/240 are equal to 1/30.

Over to you to check your understanding.

How can you show and explain that the fractions are equivalent? Think about the arrows and what they're telling you about the relationships between the numerator and denominator in a fraction and the numerators and denominators across the two fractions.

And can you use the stem sentence? Hmm out of hmm is the same proportion of the whole as hmm out of hmm.

And it might help you to visualise a bar model or a number line.

Pause the video, have a go, and we'll get back together to share some feedback.

How did you get on? Did you spot those relationships or did you imagine some jumps on a number line or equal bars? So in the first pair of fractions, we can see that if we look within the fractions, one times 12 is equal to 12 and four times 12 is equal to 48.

And if we look across the fractions, looking at the numerators, one times four is equal to four, and we're looking for that times four relationship again, and 12 times four is equal to 48.

So we can say that one out of 12 is the same proportion of the whole as four out of 48, and those fractions are equivalent.

And then what about the second pair? So we can see that eight times 30 is equal to 240 and one times 30 is equal to 30.

So the relationship within the fractions is the same.

And then the relationship across the fractions is the same.

One times eight is equal to eight for the numerators, and 30 times eight is equal to 240 for the denominators.

So we can say that eight out of 240 is the same proportion of the whole as one is out of 30.

So therefore, our fractions are equivalent.

Time for you to do some practise.

What are the missing numerators and denominators in these pairs of equivalent fractions? Use the stem sentences to explain how you know.

And for the second part, we've given you some descriptions about the proportions.

So which equivalent fractions could be represented by these sentences? And then which pairs of equivalent fractions could be represented by this sentence at the bottom? The numerator and denominator have both been scaled by a factor of nine.

Which pairs of equivalent fractions could you write that would be explained by that sentence? Pause the video, have a go at your tasks, and we'll get together for some feedback.

How did you get on? So A, B, C, and D just had one possible answer.

But in E, F, G, and H, we could have had different answers, more than one solution, and G and H could have had different factors for scaling as well.

But let's look at the individual pairs of fractions.

So in A, 1/30 is equivalent to 3/90.

The numerator and denominator have both been scaled by a factor of three.

One times three is equal to three, 30 times three is equal to 90.

What did you spot in B? Did you spot that eight times eight is equal to 64? So therefore, our missing numerator must be eight.

The numerator and denominator have both been scaled by a factor of eight.

So in C, we can see that 15 and 30 relationship, there's a doubling there, isn't there? So if 15 times two is equal to 30, then one times two is equal to two.

So 2/30 are equivalent to 1/15.

The numerator and denominator have both been scaled by a factor of two.

Scaled up if we look from right to left and scaled down if we look from left to right.

What about D? Again, we could look at division or multiplication.

Let's look at the multiplication route going from the second fraction with the missing denominator.

One times 25 is equal to 25, so something times 25 is equal to 50.

And you might also have spotted that 25/50 relationship.

Our missing denominator is two.

The numerator and denominator have both been scaled by a factor of 25.

But we can also look at that relationship between the numerator and denominator in each fraction to see that if our first fraction has the numerator half the value of the denominator, that has to be true for the second fraction as well.

So indeed, here's an example of one answer.

Looking at the denominators, we can see that scale factor of six, 10 times six is equal to 60.

So as long as our numerators had a scale factor of six, then we would've been all right.

We put it one and six, so 1/10 is equal to 6/60, but we could have had 2/10 is equal to 12/16 and so on.

Similarly for F, we can see that one times three relationship between the numerators.

So as long as the denominator of our second fraction was three times the denominator of our first fraction, the fractions would be equivalent.

We've put in 1/3 is equal to 3/9, but we could have had 4 and 12 as the missing denominators or 5 and 15 as the missing denominators as long as they were related by a scale factor of three.

Then for G, we've put in 1/7 and 3/21.

So there, we went for a scale factor of three.

So we went for three divided by three is equal to one to give us a unit fraction.

And then we knew that seven had to be multiplied by three to give us our denominator of 21.

But we could have used other scale factors if we'd wanted to.

And for G, well, the world's your oyster there, isn't it? All we've got is a numerator of one.

So we could have had any unit fraction as our first fraction and then we could have had an equivalent non-unit fraction as our second one.

We picked 1/6 and 5/30.

So lots and lots of choices, in fact, possibly an infinite number of choices for G.

I hope you had fun playing around with those.

And in two, we asked you to find the equivalent fractions that could be represented by these sentences.

So one out of seven is the same proportion of the whole as three out of 21.

1/7 is equal to 3/21.

5 out of 30 is the same proportion of the whole as hmm out of 90.

Ah, we've gotta think about that, haven't we? So 30 times three is equal to 90, so five times three is equal to 15, so 5/30 is equal to 15/90.

Seven out of 35 is the same proportion of the whole as one out of hmm.

Well, we've got a one times seven link there, haven't we? So we could do a divide by seven.

35 divided by seven is equal to five.

So 7/35 is equivalent to 1/5.

And then for D, we had to say which pairs of equivalent fractions could be represented by this sentence? The numerator and denominator have both been scaled by a factor of nine.

So we could have had 1/8 is equal to 9/72.

There are lots of possible answers to D.

As long as the second fraction you write has the numerator and denominator of the first fraction scaled up by a factor of nine, then our fractions will be equivalent.

So let's move on to the second part of our lesson.

We're comparing fractions and identifying equivalence.

So Jun says, "I think these are all pairs of equivalent fractions." Is Jun correct? Sofia says, "I'm going to check the numerators and denominators." Good thinking, and we're thinking across those fractions as you can see with those arrows.

You can identify equivalence by making sure that the numerator and denominator have been scaled by the same factor.

Jun is still saying, "I think these are all pairs of equivalent fractions." So Sofia has identified in the first one that the numerator and the denominator have been scaled by the same factor.

One times 80 is equal to eight and she said 48 divided by eight is equal to six.

So we can look at it both ways.

As long as that scale factor is eight, it doesn't matter if we scaled up by multiplying or scaled down by dividing.

So yes, Jun's right, this first pair of fractions are equivalent.

What about the next pair? Well, one times six is equal to six, but 11 divided by six is not equal to six, is it? "Oh," says Jun, "So this pair is not equivalent." Sofia says, "I think you tried to use addition and subtraction here, Jun, and that doesn't work." Over to you now.

Are the last two pairs equivalent fractions? Can you use your knowledge and the thinking that Sofia and Jun have been using to work it out? Pause the video, have a go and we'll come back and discuss our answers.

How did you get on? So in this pair, the numerator and denominator have been scaled by the same factor.

Nine divided by nine is equal to one and eight times nine is equal to 72.

"So yes, this pair is equivalent," says Jun.

What about the final pair? Well, nine divided by nine is equal to one, but eight times nine is not equal to 81, is it? The numerator and denominator have not been scaled by the same factor.

"Oh," he says, "So this pair is not equivalent." "So Jun, you were right on two, but you weren't right on the other two, I'm afraid." So which symbol is needed between these pairs of fractions? Are they equivalent or is one greater than or less than the other? So can we use our knowledge of equivalence to help us? Jun says, "Let's find out if they are equivalent fractions." Have the numerators and denominators been scaled by the same factor? Well, one times five is equal to five, but eight times four is equal to 32, so they haven't, have they? These are not equivalent fractions.

What about the next set? One times four is equal to four and seven times four is equal to 28.

These are equivalent fractions.

So we can put the equal symbol between those because they are equivalent.

What about the last pair? One times 10 is equal to 10, but nine times nine is equal to 81.

So these are not equivalent fractions.

Okay, how do we use this information to compare these fractions? We know they're not equivalent, but which one is greater than or less than the other one? Jun says, "Well, to make them equivalent, there would be 5/40." So if we were looking at that multiplied by five, one times five is five, eight times five is equal to 40, so 1/8 would be equivalent to 5/40.

"5/32 is greater than 5/40 as there are fewer parts in the whole." Ah, that's good thinking, Sofia.

So we can think about our knowledge of parts in the whole.

That means 1/8 must be less than 5/32.

We could think of it another way.

To make them equivalent, there would be 4/32, so we could say eight times four is equal to 32 and one times four is equal to four, so 1/8 would be equivalent to 4/32.

4/32 is less than 5/32 as we have fewer parts of the whole.

So either way we've looked at it, that also means that 1/8 must be less than 5/32.

So we're going to put that symbol between those fractions.

Time to check your understanding.

Can you use that thinking to decide which is greater in the last pair of fractions? Pause the video, have a go, and we'll come back and talk about your answer.

How did you get on? So Jun says, "To make them equivalent, if we look at that one times nine bit," we know that nine times nine is 81, so one times nine would be nine.

So 1/9 would be equivalent to 9/81 and 10/81 is greater than 9/81 as we have more parts of the whole.

So 10/81 would be greater than 1/9.

You may have worked it out the other way.

You may have said one times 10 is 10, and nine times 10 would be 90.

Time for you to have some practise.

So in part one, you're going to order these fractions from the smallest to the largest, maybe thinking about comparing them, thinking about equivalence.

And then for B, you're going to write another five fractions equivalent to these and order them from the smallest to the largest.

What do you notice? And for C, describe how the fractions in A and B are equivalent using scaling of the numerators and denominators.

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

How did you get on? So Jun says, "I knew that 1/4, 3/8, and 3/7 are all smaller than a half as the numerator is less than half the denominator." Oh, great thinking, Jun, so he compared the fractions to a half.

And Sofia says, "3/5 is greater than a half as the numerator is more than half the denominator." "For the fractions with three as the numerator, Jun says, "The smaller the denominator, the larger the fraction." So he knew that 3/7 was going to be larger than 3/8 and 1/4 is smaller than 3/8 as 1/4 is equivalent to 2/8.

So they've used really good thinking there about equivalence to a half and equivalence to a quarter.

And what a larger denominator means when you've got the same numerator.

Really good thinking, Sofia and Jun.

I wonder if you used the same thinking.

In B, you had to write a set of fractions that were equivalent to the ones we'd ordered.

So this is one set of equivalent fractions that you might have written.

And if you notice, the order has to be the same because the fractions are equivalent to each other.

And in C, you had to describe how the fractions in A and B were equivalent using scaling of the numerators and denominators.

So you could use this stem sentence.

The numerator and denominator have both been scaled by a factor of.

So in the first case, they'd both been scaled by a factor of three.

For the next fractions, they'd both been scaled by a factor of two.

For the 1/2 and 50/100, they'd been scaled by a factor of 50.

3/8 and 6/16 had been scaled by a factor of two.

And 3/5 and 6/10 had also been scaled by a factor of two.

I hope you've enjoyed exploring equivalence using that relationship between numerators and denominators.

So what have we learned about today? We've got some fractions there just to remind us.

We've been explaining the relationship between numerator and denominators across equivalent fractions.

We found that you can create equivalent fractions by scaling the numerator and denominator by the same factor.

And you can see that in our little example there.

One times four is equal to four for the numerators, and 12 times four is equal to 48 for the denominators.

By scaling by the same factor, the numerators stay the same proportion of the denominators.

And we showed that by looking at bar models and number lines.

You can also see that the relationship between the numerator and the denominator remains the same, especially where you've got unit fractions.

It's really easy to see that relationship as well.

And you can use knowledge of equivalent fractions to compare and order fractions.

Thank you for all your hard work and your good thinking today.

I hope you've enjoyed exploring equivalent fractions again and thinking about comparing and ordering them.

Hope I get to work with you again soon.

Bye-bye.