Lesson video

Hiya! My name's Miss Lambell.

Really pleased that you've decided to join me today to do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Ordering Fractions by Way of a Common Denominator." And that's within our unit, comparing fractions and decimals, including positives and negatives.

By the end of this lesson, you'll be able to compare and order fractions by converting fractions with a common denominator.

This is really useful for this lesson and as we move forward when we'll be looking at further things like adding and subtraction of fractions.

Today's lesson, I just thought we'd recap three words that we are going to be using.

They are numerator, denominator, and a phrase, lowest common multiple.

Numerator is the expression of a fraction that is written above the fraction line.

It's the dividend.

So if we look at my example here, we can see that the 2 is above the fraction line, and so therefore that's our numerator.

And the denominator is the expression in a fraction that is written below the fraction line.

It's the divisor.

So in this case, it's 3.

The denominator is 3.

The lowest common multiple is the lowest number that is a multiple of two or more numbers.

Remember sometimes we abbreviate this to LCM.

We are going to split today's learning into three separate cycles, the first of which we will be reviewing, so this should be things that are familiar to you.

We are going to make sure that we're really confident with those before we move on.

We'll then be looking at comparing fractions, and then finally we'll move on to making sure that we can write fractions in an order.

Let's get started on that first learning cycle.

You're already familiar with simplifying fractions using the highest common factor, sometimes abbreviated, remember, to HCF.

We can also create equivalent fractions using factors.

The most useful purpose of this is to create fractions that share a common denominator.

That's useful for comparing fractions, but like I said, also, it's going to be really, really important as we move on to looking at arithmetic operations with fractions.

Aisha and Jacob are trying to decide if 3/8 is equal to 21/48.

What do you think? How could they check using factor pairs? Let's see how they can do that.

This is what we're saying at the moment.

3/8 is equal to 3/8 multiplied by something.

Is that equivalent to 21 over 48? 3 multiplied by 7 is 21 and 8 multiplied by 6 is 48.

So Aisha says she doesn't think they are equivalent.

Do you agree with Aisha? Jacob says he agrees.

To be equivalent, you have to have multiplied by 1.

That's right, isn't it? In order not to change the value of something, the only thing we can multiply it by is 1.

And we know that there are an infinite number of ways of writing 1 where the numerator and the denominator of the fraction are the same.

So Aisha says, "Yes, 7/6 is not equivalent to 1." So those two things are not equal.

Well done, Aisha, for spotting that 7/6 is not equal to 1.

It is greater than or less than 1? Greater than.

Brilliant.

Well done.

If we want to make 3/8 equivalent to a fraction with a denominator of 48, what would the numerator be? So just have a think at the moment.

What do you think it would be? So this time I'm telling you I want the new denominator to be 48.

In order to make it equivalent to 3/8, what would the numerator be? We'll take a look at this then.

3/8, we want the denominate, sorry, we want the denominator to be 48.

This time we're going to multiply 8 by 6 to make 48.

Remember, that has to be equivalent to 1, doesn't it? So it doesn't change the value of the original fraction.

What does the numerator need to be then so that that is going to be equivalent to 1, and that's 6.

3 multiplied by 6 is 18.

This means that 3/8 is equal to 18/48.

We're now going to have a go at sorting the following fractions into examples and non-examples of fractions that are equivalent to 2/5.

This means that if it is equivalent, it goes in the example, and if it's not equivalent, it goes in the non-examples.

Let's go through each of them in turn and I'm going to give you a moment to think about each one before I put it into the box.

12/30.

That is an example.

We can see, to get from 2 to 12, we multiply by 6, and to get from 5 to 30, we multiply by 6.

So we've multiplied by 1, therefore is an example of an equivalent fraction to 2/5.

Now let's look at 60/160.

It's a non-example.

2 multiplied by 30 is 60 and 5 multiplied by 32 is 160.

We've not multiplied by 1, therefore it can't be equivalent.

50/125.

That is an example.

2 multiplied by 25 is 50 and 5 multiplied by 25 is 125.

What about 20/45? This is a non example.

2 multiplied by 10 is 20, 5 multiplied by 9.

So we've not multiplied by 1, therefore it's a non-example.

10/25.

That is an example because we've multiplied by 5/5, which is 1.

22/55.

That's another example because we've multiplied by 11/11, which is 1.

8/15.

Non-example because we've multiplied by 4/3, which is not equivalent to 1.

One final one.

And that's a non-example because we've multiplied by 13/12, which is not equivalent to 1.

Your turn now.

True or false? 4/7 is equal to 12/21.

And the correct answer is true.

What I'd like you to do now is to decide which is the correct justification for that.

Remember the justification is important.

You could have just guessed through there.

So you're gonna pause the video and decide which of those two justifications, a or b, is the correct one.

Pause the video and come back when you've got your answer.

The correct answer was b.

If we look at the first one, you have multiplied by 3, so it must get 3 times bigger.

We haven't actually multiplied by 3, we've multiplied by 3/3, which is equivalent to 1, which is why the second justification, justification b, is the correct one.

You're now ready to do some independent work.

In this task, what I'd like you to do please is to identify the missing numerator or denominator.

So this time the equals are all there because they are equal fractions.

Your job is to work out what's missing, what's the missing numerator and denominator.

Pause the video, come back when you're ready.

Good luck.

And now we can move on to question 2.

Which of the following are true? So this time I've given you two fractions.

Now I've put an equal symbol between them but actually some of those are not true, so they shouldn't have an equal symbol between them.

So your job is to show me which of those are true, which are equal and which are not equal.

And you could put the not equal to symbol in between the ones that aren't, if you'd like to.

So pause the video, give these a real good go, come back when you're ready.

Great work.

Let's check our answers then.

So the missing numerator and denominators.

So in 1a, the missing numerator was 90.

In b, the missing denominator, 25.

In c, the missing numerator was 28.

In d, the missing denominator was 36.

In e, the denominator was 64, f, it was 60, g was 156, and h was 3.

Did I trick you with that final one, which was an improper fraction? Hopefully not.

And then moving on to question 2, which of the following are true? So the first one, the multiplier on the top is 15 and the bottom is 16.

That's not equal to 1, so therefore that is false.

The next one, I've multiplied by 4/4, which is equivalent to 1, so that's true.

The next one we've multiplied by 3/3.

That's equivalent to 1, so that's true.

Next one, 7/6.

That's not equivalent to 1, so false.

e is 40.

I've actually, here, I've put in a divide one.

Same thing applies, so divide by 48, divide by 48, we've still divided by 1.

What happens when we divide by something by 1? Its value doesn't change, so that one was true.

And then the final one, I divided by 16/15, so that is false.

So I just put in some, we hadn't done any examples of when we were dividing, but I'm sure they didn't trouble you at all.

So the last one was false.

We're now ready to move on to our next learning cycle.

Now we're confident with writing equivalent fractions with a common denominator, we'll be able to compare those fractions.

Which is greater? Let's look at a diagram.

Here's my diagram for 11/17 shaded, and here's my diagram for 13/17 shaded.

Now, it's really obvious to see which is the larger, so I should have my symbol pointing to the left.

11/17 is smaller than 13/17.

Now, you probably didn't need the visual to work that one out.

You might have thought of it as a test score.

If somebody got 11 out of 17 and somebody got 13 out of 17 and the person who got 13 out of 17 got more right.

Here we have Aisha and Jacob and they love to practise their penalty shooting skills.

You may, too.

They want to know who had the greatest success rate.

Unfortunately, though, they didn't shoot the same number of penalties.

Oh, that's gonna cause us a bit of a problem maybe.

Aisha says, "I shot 8 penalties and scored 5," and Jacob says, "I shot 12 and scored 7." Aisha now says, "We have the same success rate." Do you agree with Aisha? We'll use our knowledge of equivalent fractions to see if this is true.

We need to find the lowest common multiple, or LCM, of 8 and 12.

This is going to allow us to create two fractions that have a common denominator so that we can compare them.

So effectively, say, if they took the same number of goes each, what would we expect them to have scored? Now, Aisha's written her score as a fraction.

"I scored 5/8 of my shot," so 5 out of 8 shots.

And Jacob scored 7/12 of his shots.

Here are my multiples of 8 and my multiples of 12, and remember I'm looking for the LCM, the lowest common multiple.

In this case, that's 24.

Now in the first learning cycle, we got really confident with rewriting a fraction with a given denominator.

That's just what we're going to do here.

We want to make the new denominator 24.

What do I multiply 8 by? 3 to get to 24.

Remember, I need to multiply by 1, so that has to be a 3 there.

Otherwise, I'm gonna change the value of the 5/8.

15 multiply by 3, it's 15.

Now let's do the same with Jacob's.

Right, this time, I'm gonna ask you to pause the video and give this one a go before we go through it together.

I'm sure you can do it.

Come back when you're ready.

Great.

So we wanted the denominator to be 24.

We started with 7/12.

12 multiplied by what is 24? That's 2.

Therefore, the numerator has to be 2, so we're multiplying by 1 and not changing the value.

7 multiplied by 2 is 14.

Can we now see who had the better success rate? Aisha, therefore, had a better success rate.

How do we know that Aisha had a better success rate? Well, we know this because 15/24 is larger than 14/24.

If I've got a rectangle cut into 24 parts, equal parts remember, and I shade 15 of them, and then in another identical rectangle cut into 24 equal parts, I shade in 14 of them.

The area that I would've shaded in the first one would be greater.

You could think of it also as a test score.

15 out of 24 is better than 14 out of 24.

That was a quick turnaround in that learning cycle, and that's because you did so well in that first one.

You were so good at finding fractions with, sorry, finding equivalent fractions with common denominators that you were able now to very quickly move that learning on to comparing two fractions.

So that's what you're going to do here.

You're going to decide whether the one on the left or the one on the right is the greater, and you're gonna use the greater than and less than symbols.

But remember, you are not going to just guess.

You need to show me, I want to see written down on your page your common denominator, your conversion of the two, so that you can tell me which is the larger.

Good luck with that.

Pause the video and I look forward to seeing you when you come back.

Question 2.

This time, we've got a little bit more thinking maybe involved here.

Some of you might find this a little bit more challenging, but you'll be fine, honestly.

You're gonna place the words greater and less in the sentence.

And then it says, Jacob says, "3/11 is exactly halfway between 4/11 and 2/11." Is he correct? And as always, what I'd like there is a justification, an explanation as to whether you think Jacob is correct or incorrect.

Pause the video and come back when you're ready.

Here are our answers.

So a, we should have less than, b, greater than, c, greater than, d, less than, e, less than, f, less than, g, greater than, h, greater than.

Well done.

I'm sure you've got all of those right.

And then, like I said, a little bit more challenging here maybe.

You should have 3/11 is less than 4/11 but greater than 2/11.

And Jacob was correct, as there is 1/11 between 3/11 and 4/11, as well as between 3/11 and 2/11.

So he had found the halfway point.

Well done, Jacob.

Moving on now.

So now we can compare two fractions, we will be able to now write fractions in order from smallest to largest or largest to smallest.

Let's have a look at an example.

We're back with our sport.

So Aisha, Lucas, Laura, and Jacob, this time they've been practising their basketball shooting skills.

We want to write them in order of their performance.

Aisha scored 7/12 of her shots, Lucas scored 5/8, Sophia, 9/16 of her shots, and Jacob, 13 out of 24 of his shots.

Here are those four fractions.

We're going to need to find the LCM, lowest common multiple, of 12, 8, 16, and 24.

Now what I don't really want to do is to start to list out all of the multiples of 12 and 8 and 16, et cetera.

So I'm going to share with you a slightly more efficient way of doing it, and that's this.

We're going to not list the multiples this time, like I said, and what we're going to do is we're gonna go through the multiples of 24, because that's the highest number, and consider if they're multiples of the other three.

So we always start with the highest denominator, in this case, that's 24.

So we're gonna start with 24.

24 is not a multiple of 16, so therefore we move on to the next multiple of 24, which is 48.

We're now gonna check 48.

Is 12 a factor of 48? Yes.

Is 8 a factor of 48? Yes.

And 16 is also a factor of 48.

48 is a multiple of 16, 8, and 12, and this is the LCM, the lowest common multiple, of all of the four denominators.

The LCM of 12, 8, 16, and 24 is 48, so this is going to be our common denominator.

We are going to convert our four fractions so that each have a common denominator of 48.

All right, you're super good at this now, so I'm just going to whiz through this quite quickly.

So we want to create a fraction with a denominator of 48, and then Lucas, we're gonna do exactly the same thing, and then Sophia.

Notice the order in which I'm doing these.

I'm just going to go through this last one just in case I lost any of you going through those fairly quickly.

So remember, 24 multiplied by what is 48? That's 2.

The numerator has to be the same so we're not changing the value, and that's 26.

Now, we'll be able to write those in order, in order from most successful to least successful.

So who was most successful? It was Lucas with 30 out of 48, Aisha, 28/48, Sophia, 27/48, and then finally, Jacob with 26/48.

But you can see actually all of their scores were very, very similar.

I'd like you now to have a go at doing the same thing here.

I'd like you to write each of those with a common denominator and then order them this time from smallest to largest.

So you're going to pause the video, have a go, and come back when you're ready.

Good luck with this.

You'll be fine.

Great work.

What did you decide on as your common denominator? I hope you went for 20.

I went for 20 because that's the lowest common multiple.

You may have gone for 40 or 60 or 80.

It doesn't matter.

I just want us to think about efficiency where we can.

3/4 is 15/20, 7/10 is 14/20, 4/5 is 16/20, and I skipped 17/20 'cause it's already in 20ths, and now I can write those in order.

Remember, we're going here from smallest to largest.

So in order, 7/10, 3/4, 4/5, and 17/20.

Again, they were all very similar once we converted them.

They didn't look very similar to start with, but they were very similar.

You're now ready to move on to Task C independently.

Write the following fractions with a common denominator and order them from smallest to largest.

So exactly the same as you just did in that Check for understanding.

Now, if you made any errors in that Check for understanding, you might decide to just quickly rewind the video and rewatch from the beginning of the third learning cycle.

But I'm sure you're ready to give these a real good crack.

So remember, you're looking for the LCM, ideally, 'cause that's the most efficient.

Then you're going to create your equivalent fractions, and then you're going to write them in order from smallest to largest.

You've got all of the skills that you need to be successful at this.

Good luck, and I'll see you when you come back.

You can pause the video now.

Let's have a look.

Here are our answers.

So the order for a should be 11/20, 3/5, 7/10, 3/4; for b, it should be 1/2, 9/16, 5/8, 3/4; c, 3/5, 13/20, 17/25, 7/10, and 3/4; and finally, d should be 2/3, 11/15, 9/12, 4/5, and 5/6.

How many of those did you get right? All of them? Amazing, well done! Now we can summarise our learning from today's lesson.

The main points of today's lesson are these, any fraction can be written as an equivalent fraction by multiplication of 1.

For example, 4/7, if we multiply that by 3/3 creates the equivalent fraction 12/21.

If I changed it to multiply by 4/4, which is still 1, I would get 16/28.

There are infinite number of equivalent fractions that we could write.

Fractions can be ordered by converting fractions with a common denominator.

By using a common denominator, we are easily able to compare those fractions.

The most efficient way of doing that, remember, is to find the LCM, the lowest common multiple of those denominators.

You've done fantastically well this lesson, and I'm glad that you managed to stick with me right through to the end.

Thank you for joining me, and I look forward to seeing you next time.

Bye.