Lesson video

Hiya, my name's Ms. Lambau.

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 key words that we are 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 un-pause the video.

We're gonna move on now.

Today's lesson we are 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 'cause 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.

We're 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 representation of 11/19, and here is a reputation 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 gonna have a go now.

22 over 23 is greater than 22 over 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 over 23, or B, the parts are larger in 22 over 25.

Pause the video.

When you've got an idea, come back, and remember, no guessing.

You should've come 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 over 44, and we've got 101 over 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 over 44 can actually be simplified.

Now you've done simplifying previously, and you're really good at it, so remember, what we're doing here is we're looking for at the highest common factor of 11 and 44, which is 11.

We write that as 11 over 11 'cause, remember, that's 1, and if I multiply something by 1, 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 4 multiplied by 11 is 44.

But remember, 11 over 11 is equivalent to 1, 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 of 1/4.

So four multiplied by something is 400.

That's 100.

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

So I'm multiplying by 1 and, therefore, not changing the value of 1/4, and this is 100.

Now we can compare them.

I've got 101 over 400, and I've got 100 over 400.

So now we can see which is greater.

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

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

We recognise that 11 over 44 can be simplified, and then we used a common denominator approach.

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

Which is greater? Again, I'm gonna 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 over 19 and 11 over 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 gonna 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.

5 multiplied by 11 is 55.

I'm now comparing 11 over 55 with 11 over 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, 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 numerator rather than the denominators? So that's what we've been concentrating on, 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 a 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 3 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, 8 is a factor of 64, so that would be easier to compare those denominators.

3 is not a factor of 25, so that's gonna 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, 9 is a factor of 27, and 15's not a factor of 7.

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 numerator 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 statement 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 gonna answer the other questions.

I'm not going to read all of those.

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 1A, sorry, 13 over 21 is less than 13 over 15, B, 8/15 is greater than 8/19, C, 7/11 is less than 7/9, D, 11 over 315 is less than 11 over 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 over 35, 10 over 31, 10 over 18, 10 over 15, B, 234 over 518, 234 over 512, 234 over 511, and 234 over 500.

Maybe stop and think, look what's happening to the denominators as those number get larger, but only remember because the numerator's the same.

And then finally, onto 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 over 23, 9 over 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 example, 17 over 60 and 7 over 24, and E, this time there is only one answer because it was asking for the fraction that was exactly halfway between the two, and that is 11 over 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 ranks.

Sam scored 7 out of 8 in his round, well done, Sam, and Andeep scored 5 out of 6 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.

Sorry, "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 from fraction bars to help us have a visual look at what's going on.

Sam scored 7/8 of his questions.

So there's his bar split into eight equal parts because remember, he 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 'cause 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 gonna pause a moment and let you have a think about that.

Which is greater, 7, sorry, 17/20 or 19 over 22? Which is greater? Now in here, remember, we said we're gonna 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 a maths competition.

The first person did a round of 20 questions.

They 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 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 1, 19 over 22 is closer to 1 as its parts are smaller.

Now it's quite a lot to get your head around 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.

Whether you're sticking with me or coming back if you went back to just recap on some things, remember, that's absolutely fine, and we can here, we just need to finish off by putting in our symbol, which is 17 over 20 is less than 19 over 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 over 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 enumerators, so looking at enumerators 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 over 44, we simplified to 1/4 first, and then we made an equivalent fraction, and then we've also looked at comparing proper fractions to 1.

So just because something is the same number of parts away from 1 doesn't mean that it is the same size.

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

So I'm gonna give you two fractions, I'm gonna put them all up on the screen but one at a time, and then I'm gonna 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 a 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 over 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 3 is a factor of 15 and there in the denominators.

The next one we will compare to 1.

8 over 11 is three parts away from 1.

16 over 19 is also three parts away from 1.

So we can use the compare to 1 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 1.

9/11 is two parts away.

13/15 is two parts away from 1.

So we're going to be comparing to 1.

13/15 is two smaller parts away.

If I've split my unit to 15 parts compared to 11 parts, the 15 parts are going to be smaller, so, therefore, it is closer to 1.

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 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 gonna 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 over 22 is larger than 22 over 25.

Good luck with this one.

Nice sentences please, capital letters and full stops.

When you're ready, come back, and we'll 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's less than one where the numerator is smaller than the denominator.

You're gonna answer these questions using those numbers.

Good luck with this.

May be 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 'em all right? Amazing.

Let's check question number two.

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

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

So although they are both three parts away from 1, the parts are larger in 19 over 22, so it's further from 1, or you may have the other side of that.

You may have said that the parts are smaller in 22 over 25, so that is closer to 1.

And then the final question, A, 14 over 15 was closest to 1, 10 over 15 was closest to zero for part B, C, you may have written 11 or 12, D, 13, 14, or 15, 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.

The 11/17 is greater than 11/19, and we also looked at being able to compare fractions by considering their distance from 1, 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.