Lesson video

Hiya! My name's Ms. 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 in different ways." This is 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 using a range of techniques.

Some of these will be familiar to you.

Some may be new this lesson.

Here are some keywords that we're going to be using throughout today's lesson, and they are denominator, numerator, and proper fraction.

If you need a reminder of what any of these are, I suggest you quickly pause the video and have a read through, and then when you're ready, you can come back and unpause the video.

We're going to move on now.

Today's lesson we're going to split into two separate learning cycles, in the first of which we are going to be comparing fractions using a common numerator.

So we're very, very familiar with using a common denominator but less familiar with using a common numerator.

And in the second learning cycle, we're going to use another way of comparing fractions is by looking how they compare to one.

Let's get started on that first learning cycle then, which is comparing fractions using a common numerator.

I might have to stop myself saying common denominator through this lesson because I'm more used to saying that.

Going to be thinking a lot about which is greater? Which is lesser? Mainly greater.

We've got 11/19 and 11/17.

Which is greater? Is it always necessary to find a common denominator? What do you think? Or maybe at the moment, you would be saying yes.

Or using visual representation here to help us decide which of these is greater so we don't have to go down the route of finding a common denominator.

Here is a representation of 11/19 and here is a representation of 11/17.

Notice both of my bars are the same length.

This means that I can compare them one like for like.

In the first bar, it is split into 19 sections and 11 have been shaded.

And the second bar has been split into 17 sections and again, 11 have been shaded.

But it's clear now, from these two diagrams, which is the larger.

Really important that we remember that both of the rectangles or both of our fraction bars must be the same length in order to make that comparison.

And we can see that the bottom one is clearly larger.

That means 11/19 is smaller than 11/17.

Or we can say it the other way round, 11/17 is larger than 11/19.

So we didn't need to change each fraction.

So it had a common denominator.

We were able to compare by looking at the numerator.

You're going to have a go now.

22/23 is greater than 22/25.

Do you think that is true or false? Make your decision now.

And you said true.

Of course you said true because true is the correct answer.

But as always, I'm not happy with just a true or false.

That's a 50-50.

You need to convince me that you understand why that's right.

Now I'd like you to pause the video when I've read through the two justifications, and make your decision.

So our two justifications are here.

A, the parts are larger in 22/23.

Or B, the parts are larger in 22/25.

Pause the video.

When you've got an idea, come back.

And remember, no guessing.

You should have gone with A, parts are larger, and this is because the unit has been split into fewer parts and so therefore, each part is larger.

Well done.

So sticking with our theme then, which is greater? This time, we've got 11/44 and we've got 101/400.

Which is greater? I'm just going to pause a moment to let you have a think about that.

Which of those do you think is greater? Right, so what could we do here? Hmm, I'm having a think.

Here actually, I've recognised that 11/44 can actually be simplified.

I've done simplifying previously and you're really good at it.

So remember, what we're doing here is we're looking for the highest common factor of 11 and 44, which is 11.

We write that as 11/11 because remember that's one.

And if I multiply something by one, I'm not changing its value.

Here, I would have to multiply by 1/4.

Let's just check that.

One multiplied by 11 is 11 and four multiplied by 11 is 44.

But remember, 11/11 is equivalent to one.

So this is just 1/4.

Now we can create an equivalent fraction with a denominator of 400 and we've done that previously too.

So we know what we want our new denominator to be.

We want it to be 400.

We know we're starting with the fraction a quarter.

So four multiplied by something is 400.

That's a hundred.

Remember, the numerator and denominator have to be the same.

So I'm multiplying by one and therefore not changing the value of 1/4.

And this is a hundred.

Now we can compare them.

I've got 101/400 and I've got 100/400.

So now we can see which is greater.

Which is greater? That's right, 101/400.

Here, we didn't first jump into finding a common denominator.

Recognised that 11/44 can be simplified and then we used a common-denominator approach.

This time, my two fractions are 1/5 and 11/37,238.

Which is greater? Again, I'm going to pause a moment.

Let you have a think.

Have you got any ideas of strategies we could use here? Do you think it would be sensible to try and find the LCM, the lowest common multiple of five and 37,238? I'm not sure that I'd want to sit there listing all of those out.

So let's have a look at an alternative.

In this situation, it's going to be much more efficient to convert 1/5 into an equivalent fraction with a numerator of 11.

So if we think back to that very first example we looked at, I think it was 11/19 and 11/17, because they had the same numerator, we could then just consider the size of the parts.

So it's going to be much easier to do that here.

So I'm going to take my 1/5.

This time, I know that I want my numerator to be 11.

One multiplied by 11 is 11.

Remember, we have to be multiplying by one.

Otherwise, we're going to change the value of that 1/5.

Well, we've got those equal symbols there so that can't happen.

So I need to multiply by 11 on my denominator also.

Five multiplied by 11 is 55.

I've now comparing 11/55 with 11/37,238.

Now it should be obvious to see which is greater.

And it's 1/5.

If I've got a unit split into 55 parts and I've got the same unit split into 37,238 parts, those parts are going to be tiny compared to the one that's split into 55.

So therefore, that 1/5 must be larger.

What I'd like you to do now is to decide, in the four following examples, which of them would it be easier to compare the numerators rather than the denominators? So that's what we've been concentrating on.

We've got this idea of actually we can make comparisons using the numerator, whereas before, we've been very, very focused on using the denominator.

Which of those would it be easier to compare the numerators? Pause the video and come back when you've got your answer for me.

Great.

Let's have a look then.

Hopefully, you decided on the second one.

And the reason for that is that three is a factor of 12.

If we look at the top one, 1,200 is a factor of 8,400.

So it'd be easier to compare the denominators.

In the third one, eight is a factor of 64.

So that would be easier to compare those denominators.

Three is not a factor of 25.

So that's going to be quite difficult to look at.

Not impossible but just much more challenging.

And remember, we're all about making our life as easy as possible.

And then the final one, again, nine is a factor of 27 and 15 is not a factor of seven.

So it's really useful to look at and compare before you even think about doing any conversions, whether you are going to be looking at comparing the numerators or the denominators.

Now you're ready to have a go at the first task.

You need to decide which of those symbols needs to go in between the fractions to make the statements true.

And then you're going to use the word smaller and larger to complete the sentence in part E.

Pause the video.

Good luck with this.

You'll be absolutely fine.

Look forward to seeing you when you come back.

Well done.

Question number two.

Write the following in order from smallest to largest.

And remember, there's no need to find a common denominator.

Okay, you're using the common-numerator approach.

Pause the video.

Good luck.

Come back when you're ready.

And finally, question number three in Task A.

You're going to place the words larger and smaller into the sentence and then you're going to answer the other questions.

I'm not going to read all of those out.

I'm going to let you pause the video, read them through at your own pace, and then when you've got your answers, come back.

Good luck.

You can pause that video now.

Great work.

Should we check those answers? Okay, let's go.

So one A.

13/21 is less than 13/15.

B, 8/15 is greater than 8/19.

C, 7/11 is less than 7/9.

D, 11/315 is less than 11/311.

The larger the denominator, the smaller the parts, and that's a really important fact to remember.

Moving on to question two.

This is the correct order.

So A, 10/35, 10/31, 10/18, 10/15.

B, 234/518, 234/512, 234/511, and 234/500.

Maybe stop and think.

Look what's happening to the denominators as those numbers get larger, but only, remember, because the numerator is the same.

And then finally, on to question three.

3/11 is larger than 3/12 but smaller than 3/10.

Jacob says, "3/11 is exactly halfway between 3/10 and 3/12." Is he correct? No, he's not correct.

It is closer to 3/12.

Find another fraction between 3/10 and 3/12.

And these are just some examples here.

There are others.

I've written 6/23 and 9/35.

You could always get your calculator out and check to make sure that your answer is between those two.

Change them into decimals and then check.

D, again, just examples, 17/60 and 7/24.

And E, this time, there is only one answer because it's asking for the fraction that was exactly halfway between the two.

And that is 11/40.

Those questions there were so much more challenging, so well done if you got those right.

If you didn't, don't worry because we're going to move on now and you'll have an opportunity to come back to some of those things at another time.

We're now going to look at this idea of comparing to one.

So looking at our fraction and using one as our comparison.

Sam and Andeep took part in a maths competition and they both entered different rounds.

Sam scored seven out of eight in his round.

Well done, Sam.

And Andeep scored five out of six in his round.

Well done, Andeep.

Sam says, "That means we both did as well as each other because we both only got one wrong." I wonder what your thoughts are on that? Andeep says, "I think you did better, as you had more questions." What are your thoughts on that? So now you're ready to decide do you agree with Sam or Andeep? Sam has decided that it might be useful to write his score as a fraction and he would've scored 7/8.

And Andeep, his as a fraction would've been 5/6.

So if he was right, so Sam was saying on the previous slide that because they both got one wrong, they both did as well as each other.

Well, that would mean then that 7/8 and 5/6 are equal.

Andeep says, "Yes, but we know they're not equal.

We know those two fractions aren't equal." We can write each one as a common.

"We can write each fraction with a common denominator to see which is larger." So Sam's remembered from previous learning that if you write both fractions with a common denominator, then you're able to compare them.

Maybe he was with us when we did the lesson on basketball and penalty shooting.

Andeep says, "We could, but your fraction is larger, as it's closer to one." Now that's an interesting thought, Andeep.

Your fraction is larger, as it is closer to one.

Hmm.

Let's take a look at some fraction bars because sometimes, it is useful to use some fraction bars to help us have a visual look at what's going on.

Sam scored 7/8 of his questions.

So there is his bar split into eight equal parts because remember, he's got 7/8 and he got seven right.

So I've represented that by shading in the seven that he got right.

Now Andeep scored 5/6 of his questions.

Really important that the two bars are equal in length.

They both represent one unit so therefore, they have to be the same length.

Andeep's bar is split into six parts.

And I've shaded in five because he got five questions right.

Can we now see who did better? Here, we can see the difference in both of their fraction bars and we can see that both fractions are less than one but actually, Andeep's fraction is further away from one.

So we can see that to get from the end of Andeep's green bar to the end is longer than getting from Sam's bar to the end.

This means that actually, Sam did better in the competition.

So initially, Sam thought that they did the same because they both got one wrong.

Then Andeep decided he thought that Sam was better because he had more questions.

And here, we can see a visual representation agreeing that actually, Sam did do only very slightly better.

Now I want you to think about is it possible to decide which proper fraction is the greater fraction by comparing it to one without a fraction bar? I'm going to pause a moment and let you have a think about that.

Which is greater? 17/20.

One 19/22.

Which is greater? Now in here, remember we said we're going to try and do this without a fraction bar because I don't really want to draw out a fraction bar, split it into 20 parts, and then a second fraction bar split into 22.

If the numbers have been smaller, I'd be happier to do that.

But here, I want us to see if we can come up with a much more efficient method for solving that problem.

Remember, you could always go back to the bars if you needed to.

These are both three parts away from one.

So we think of it with regards to the maths competition.

The first person did a round of 20 questions and got 17 right.

So that was three wrong.

And the second person did a round of 22 questions and got 19 right.

That's also three wrong.

So I think we can see from this that they are both three parts away from one.

But which fraction will have the larger parts? We can now see how this links very closely back to what we were doing in the first learning cycle about the size of parts.

And if you remember, I said it was a really important fact to remember.

17/20 will have larger parts, as the unit is split into fewer parts.

So if we have our fraction bar and it's split into 20 parts, those parts are going to be larger than if it was split into 22 parts.

So this means although they're both three parts away from one, 19/22 is closer to one, as its parts are smaller.

Now it's quite a lot to get your head round there, this idea of parts and smaller.

So if you need to, it's absolutely fine to pause the video and just read through again what's written on the screen.

There's no problem with doing that at all.

There's also no problem with rewinding the video and going back to that first learning cycle where we were talking about the size of parts and which fractions had the larger parts and the smaller parts.

Well done for sticking with me or coming back.

If you went back to just recap on some things, remember, that's absolutely fine.

And here, we just need to finish off by putting in our symbol, which is 17/20 is less than 19/22.

So here, they're the same number of parts away but we need to consider the size of the parts.

Now you can have a go.

I want you to list the following in order from which has the smallest parts to the largest parts.

Pause the video, write down your answer, and then come back when you're ready.

Great work.

This should be the order.

So 1/312, 1/85, 1/28, 1/9, 1/3.

The more parts we split our unit into, the smaller those parts are going to be.

Notice that the denominator is decreasing as the parts get larger.

Which of the following is in correct order? So this is another one for you to have a go at and I've ordered them from smallest to largest.

Well, I say that.

I've only ordered one from smallest to largest and your job is to find out which one.

Pause the video.

Come back when you've got your answer.

Great work.

The correct answer was C.

Well done if you got that right.

We've actually now, in the last sequence of lessons, covered four different methods for comparing fractions.

So what I'd like us to be able to do now is to recap what those are but also think about which is most appropriate in different situations.

So we've used the common-denominator approach.

We've used comparing numerators.

So looking at the numerators and then the denominators and thinking about the size of parts.

We've looked at simplifying and comparing.

So if I think back to the one we did in this lesson, I think it was 11/44, we simplified to a quarter first, and then we made an equivalent fraction.

And then we've also looked at comparing proper fractions to one.

So just because something is the same number of parts away from one doesn't mean that it is the same size.

We're now going to have a think about which method is most appropriate in these different situations.

So I'm going to give you two fractions.

I'm going to put them all up on the screen but one at a time.

And then I'm going to give you a little moment to have a think about which do you think is the most appropriate method for working out which is the larger, which is the greater, or smaller? So first two.

And then the second two.

The third pair.

Fourth pair.

And finally, our fifth pair.

So I'm going to ask you to decide, of our four methods, which of those may be most appropriate in each situation? Pause the video.

And then when you've got an idea, you can come back.

Great work.

We'll go through these one by one.

So the first one would be a simplify and compare.

20/25 is actually 4/5 and then we can create that equivalent fraction.

The second one would be comparing the numerators.

They both have a numerator of 11.

So we just need to think about the size of the parts, given those denominators.

The next one would be a common-denominator method.

Notice that three is a factor of 15 and there in the denominators.

The next one, we will compare to one.

8/11 is three parts away from one.

16/19 is also three parts away from one.

So we can use the compare-to-one method.

And the final one, we would compare the denominator.

Which is greater? So here's a check for understanding for you.

9/11 or 13/15? Which is greater? Pause the video, have a think, think about which methods you're going to use to be most efficient, and then when you're ready, come back.

You can pause that video now.

Lovely.

Well done.

They're both two parts away from one.

9/11 is two parts away.

13/15 is two parts away from one.

So we're going to be comparing to one.

13/15 is two smaller parts away.

If I've split my unit into 15 parts, compared to 11 parts, the 15 parts are going to be smaller.

So therefore, it is closer to one.

So 9/11 is less than 13/15.

Now we're ready for you to come back and do some independent learning in Task B.

Question one.

You're going to choose the most appropriate method to decide which of the following statements are true.

So I don't want you just jumping straight into doing common denominator.

And looking at some of these, they would be pretty grim.

You're going to look carefully and think of the four methods, which of those is most appropriate? When you're ready, come back and we'll check your answers.

Good luck.

Super work.

Now I'd like you please to have a go at question number two.

So an explain question.

I love an explain question because it shows me whether you've really understood what you're doing.

Explain why Aisha is wrong.

19/22 is larger than 22/25.

Good luck with this one.

Nice sentences, please.

Capital letters and full stops.

When you're ready, come back and we will see what you've got and see if it compares with what I've written.

Good luck.

You can pause the video now.

A third question in Task B.

This question, you're going to use the numbers to complete the following proper fractions.

Proper fractions, remember.

What do we mean by a proper fraction? One that is less than one where the numerator is smaller than the denominator.

You're going to answer these questions using those numbers.

Good luck with this.

Maybe a little bit more challenging.

But remember, you have all of the skills because you've done so well throughout this lesson.

So you've got everything you need to be successful.

Good luck.

Pause the video.

I look forward to seeing you when you come back.

Well done.

Now we can check our answers.

So here, we were deciding whether these statements were true or false.

So A was true.

B, false.

C, false.

D was true.

E was true.

And F was false.

How did you get on with those? You got them all right? Amazing.

Let's check question number two.

Here, this is just an example of a correct answer, but you should have something said fairly similar.

So we were explaining why 19/22 is not larger than 22/25.

So although they are both three parts away from one, the parts are larger in 19/22 so it's further from one.

So if something or you may have the other side of that.

You may have said that the parts are smaller in 22/25 so that is closer to one.

And then the final question.

A, 14/15 was closest to one.

10/15 was closest to zero for part B.

C, you may have written 11 or 12.

D, 13 or 14.

E, 14 or 15.

And F, 10.

How did you get on with those? Well done.

Now let's summarise the learning that we've done today.

In fractions, the larger the denominator, the smaller the parts.

So remember, we're splitting a unit into more sections.

So therefore, each section or each part, I should say, will be smaller.

We also looked at the fact that we can compare fractions, not only by finding a common denominator, but also by comparing the numerators.

So there's the example that we used during this lesson, that 11/17 is greater than 11/19.

And then we also looked at being able to compare fractions by considering their distance from one.

So thinking again about the size of the parts.

That's been a really, really important theme of today's lesson is understanding what fractions have bigger or smaller parts.

And there again is another example.

Thank you so much for joining me today.

You've done fantastically well.

And I look forward to seeing you again to do some awesome maths learning.

Thank you very much.

Bye!.