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 Simplifying Fractions, and that comes within our unit, Comparing and Ordering Fractions and Decimals, including positives and negatives.

By the end of this lesson, you'll be able to simplify fractions by dividing both the numerator and denominator by common factors, but most importantly, understanding why this works.

Some key words that we'll be looking at in today's lesson, all of which you should be familiar with, are listed here.

They're factor, highest common factor and expressing a number as a product of its primes.

If you feel that you need to recap on what each of those means, you could pause the video now and read through them and unpause the video when you are ready to move on with our learning.

We are going to split today's lesson into two separate learning cycles.

In the first one, we are going to be looking at simplifying fractions using factor pairs.

Factor pairs is something that you are really familiar with and we're going to apply that in a slightly different situation.

When we've done that, we're then going to move on to simplifying using prime factors.

So again, this is something that you are familiar with, but you may not have seen it in the context of simplifying fractions before.

So by the end of this lesson, we need to be able to simplify things like 25 over 30, simplify that, but also understand why and how it works.

Let's get started on the first learning cycle.

So remember we are looking at simplifying fractions and we're going to be using factor pairs.

Today we're going to be working alongside Izzy and Alex.

Now, Alex says, "When you multiply two positive integers together, your answer will always be bigger." Just like you to pause a moment and have a think about whether you think that Alex is right.

Let's see what Izzy's got to say in return.

Izzy says she doesn't think it's true.

Is that what you decided too? Let's see why Izzy thinks that that is not true.

Oh, actually, Alex has had a little think about it a bit more.

He says, "Oh Izzy, you're right.

I forgot about multiplying by one." What happens when we multiply by one? Yes, anything we multiply by one, the answer remains the same, and that's going to be really, really important for understanding how we are going to simplify our fractions today.

So remember, anything multiplied by one doesn't change its value at all.

Can you think of any other ways of writing the number one? Now that may seem a really silly question.

Surely everybody writes the number one in the same way.

Okay, it might look a little bit different 'cause some people put a little tail on it, but essentially it's the same thing.

He's decided that one can be written in lots of different ways.

Izzy says, "It can.

Any fraction with the same numerator and denominator is one." Any fraction with the same numerator and denominator is one.

So Alex says, "Yes, yeah," he agrees.

Two over two.

I've got a bar split into two parts and I shaded two of them, I've shaded the whole, I've shaded one, five fifths, 11 elevenths, 24 24ths.

You get the idea.

They're an infinite number of others, we could carry on forever and we would never list them all.

We can use this fact though and factor pairs to help us to be able to simplify a fraction.

Now, you may have seen simplifying fractions before and you might be really familiar with simplifying them, but remember, I really want you to understand why it works, why we do what we do and not just do it because somebody showed us how to.

Alex and Izzy are using factor pairs to simplify 12 over 18.

So the fraction they want to simplify is 12 over 18.

Let's take a look at what they've done.

Here's Alex's method.

So Alex has written the 12 is six multiplied by two and the 18 is six multiplied by three, and then he simplified that to two thirds.

We'll now have a look and see what Izzy's done.

So Izzy has written the 12 as two multiplied by six and 18 as two multiplied by nine.

She's then decided that that's equivalent to six ninths and then she's written six ninths as three multiplied by two, over three multiplied by three, which is two thirds.

They've both got the same answer.

Two thirds.

Whose method do you prefer? Now, personally I preferred Alex's method.

That's my own personal preference.

Why do you think I might have preferred Alex's method? The reason I preferred Alex's method, because it's more efficient.

If we look, Alex has done it in one step, whereas Izzy has done it in two.

What is special then about Alex's method? What makes it more efficient? So here's Alex's method.

Like I said, he's managed it in one step and Izzy took two.

Why was that? Have a think, why was that more efficient? Izzy says, "I can see why, it's because you chose your first factor pair to include the highest common factor." So if we look, Alex chose six and six as his common factor, and six is the highest common factor of 12 and 18.

But Alex says, tries to make Izzy feel better.

"It doesn't matter though, 'cause you've got the answer right anyway." So it doesn't matter if we do it in one step or two or three, but sometimes it could be useful to look for the most efficient method.

And that's what Izzy's decided.

She's decided that she's going to try and use Alex's method from now on because it's more efficient.

She could go back to her previous method if she needed to at any point, remember.

Now this is the important bit, it's the why.

So why does six multiplied by two over six multiplied by three equal two thirds? Hear this term a lot and Alex is saying here, "Why can we just get rid of the sixes?" I really don't like just get rid of, okay, but I hear it a lot and Alex has said it here and it does certainly look like I've just got rid of those sixes, doesn't it? But why? Why are those sixes? Why do they appear to have just disappeared? Izzy says, "Because we know that six over six is another way of writing one." "Yes," Alex says, "And anything multiplied by one stays the same." So we could write six, oh, sorry, we could write six multiplied by two, over six multiplied by three.

Well we know that six over six is one, so one multiplied by two thirds and we know that multiply anything by one leaves the value the same.

So that's two thirds.

So we don't just get rid of those two sixes, we recognise that they're equivalent to one, then we don't need to include them in our calculation.

The highest common factor of two integers is the greatest number, which is a factor of both integers.

Now like I said, when we looked at those keywords at the beginning, you are familiar with this, but we'll go through this together just as a quick reminder of how we find the highest common factor.

We need to simplify 18 over 24 and we really want to try and use Alex's method because it was more efficient, remember? So we need to identify the highest common factor of both 18 and 24.

Factors of 18.

Here are my factors of 18.

So remember they come in pairs, we do them systematically and we make sure that we've got them all.

And 24, same way.

Notice I've done them in pairs, I've used a system, then I know I've got them all.

Which factors could I use to simplify 18 over 24? Just take a moment to think about that.

Use one or could I? Hmm, maybe that's one to think about.

I could use two, that appears in both.

I could use three, that appears in both.

I could use six because that appears in both.

But which one of those factors is going to give me the most efficient method for simplifying the fraction? Yep, you're right.

It's six isn't it? Because six is the HCF, remember HCF just stands for highest common factor.

We will use six to simplify this fraction.

Let's have a look at how we do that then.

So we've got 18 over 24.

This is the fraction we are simplifying.

We've identified that the highest common factor is six.

So what we're going to do is we're going to write 18 using its factor pair of six and then three and 24 using the factor pair of six and four.

Remember six over six is equivalent to one.

So this really means one multiplied by three quarters, which we know is just three quarters because anything multiplied by one, its value doesn't change.

This means that 18 over 24 in its simplest form is three quarters.

Now you may also hear people talk about cancelling down, they mean the same thing, but we'd like to use the term if we can.

Simplest form.

Let's have a look at another one.

We've got 16 and 48.

Here are my factors of 16, systematically and in pairs remember.

And here are my factors of 48, again systematically and in pairs.

16 is the highest common factor.

It's the highest one that appears in both my lists.

So that is the number that I'm going to use.

That's my common factor.

So we can see 16 multiplied by one.

And I can use my list of factor pairs here.

Notice that's the where I've highlighted in the 16 in the factors of 16 and then 16 multiplied by three.

Notice I've written them the other way round.

They appeared in their factor lists.

Does that matter? And remember it doesn't.

We can do our multiplication either way round.

This 16 over 16.

Remember anything that has the same numerator as denominator is equivalent to one.

And so this means one multiplied by one third, which is just one third.

16 over 48 in its simplest form is one third.

Now I'd like you to have a go at this one.

Which of the following factors of 40 and 56 is the best choice to efficiently simplify 40 over 56.

Pause the video, have a go at this and then come back when you are ready.

Great work, which was the best then? And the best one was eight.

I'm sure you listed out all of your factor pairs systematically and identified eight was the highest that appeared in both.

Eight was the highest common factor of 40 and 56.

Four is a common factor, but it's not the highest common factor.

So you will get the right answer, but it's not the most efficient method.

So it's a bit like Izzy did earlier on.

She got the right answer, she just had to do it in more steps.

Remember this is not a problem, but if we can, let's try and use that most efficient method.

Eight is the highest common factor of 40 and 56, and 14 is a factor of 56 but it's not a factor of 40.

So remember, we must choose a factor that is common to both the numerator and the denominator.

So A would've given us the right answer just in more steps.

A, B, sorry would've given us the most efficient.

And C, unfortunately wouldn't work because it needs to be a factor of both the numerator and the denominator.

Gonna do a question together now and then you are gonna have a go at a question on your own independently on the right hand side, let's have a look at this one first.

I've identified the eight is the highest common factor of 40 and 56.

So I'm going to use my factor pairs to rewrite the numerator and the denominator.

40, I'm gonna rewrite as eight multiplied by five, and 56, I'm going to rewrite as eight multiplied by seven.

Remember eight over eight is one.

So that's one multiplied by five sevens.

Now I want you to think, would it matter if I left that step out? And it wouldn't.

It's absolutely fine so let's miss that step out.

So the answer is five sevens.

I'd like you now please have a go at this one.

Try if you can to use the highest common factor.

It's okay to make those lists of factors and then look for the highest one in both.

When you are done, you can come back.

So pause the video, good luck.

Like I said, try and use the highest common factor 'cause we wanna be as efficient as possible, but if you can't find the highest common factor, don't worry, because you will get there, won't you.

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

Good luck.

Great work.

Here we are then.

So hopefully you identified that 12 was the highest common factor of both.

So we've rewritten 24 as 12 multiplied by two, 60 as 12 multiplied by five.

You may have missed that next step out, the one multiplied by two fifths, that's absolutely fine, but you should have got a final answer of two fifths and hopefully in one step.

You are now ready to have a go at some independent work.

Now this task is a little bit different, so I'm just going to talk you through how it works here.

So here is a section of the maze that you are going to be using and you need to find your way through the maze.

You need to find a path through the maze by simplifying fractions.

So we're going to start, no surprise, on the start square and we're going to take 12 over 30.

Now we've got two options here that is either going to simplify to two fifths or three fifths.

We're going to do just as we did on the previous slide, we are going to find the highest common factor.

We are going to simplify that fraction and it will show us that the answer is two fifths.

That means that the next square I move on to is the one with two fifths.

I now simplify the fraction in the bottom right hand corner of this space to see where I go next.

So we're gonna simplify 26 over 48.

The answer is either going to be 10 over 25 or three quarters to tell us whether we're going to be moving left or right or up or down.

In this case, right or down.

You are gonna make your way through the maze.

So I've started you off and then what I'd like you to do, just as a little bit of an extra challenge, is when you get to the end, I want you to write down what simplified fraction should go in the end box 'cause you'll notice the end box is empty.

I'm hoping that you're going to enjoy working your way through this maze, pause the video and then come back when you're ready.

Well done.

That was super quick.

We're now gonna move on.

And there is a second question in task A.

In this question, I would like you please to identify which one of the fractions does not simplify to the fraction given in the first column.

The one that I've put the purple box around.

So all of those five fractions simplify, four of them simplify to a quarter and one of them doesn't.

Remember, I don't want you guessing.

I want to see all of your simplifications written out really carefully using your factor pairs and hopefully the highest common factor.

And remember no calculators, what I'd like you to do now is pause the video, then you can come back when you're ready.

Good luck.

Amazing work, well done.

Let's check our answers now then.

So here is your pathway through the maze.

So I'm not gonna read out the answers.

What I'm gonna say is pause the video, mark it, and then come back when you're ready.

Just going to mention though that N square, we should have had six sevenths as the final answer.

Well done if you've got that.

So pause the video, mark it, and then come back when you're ready.

And now the answers to question two.

A, the odd one out was four twelfths.

B, eight twentieths.

C, 60 over 126.

D, 42 over 104, and E, 63 over 153.

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

Very, very similar to our first learning cycle, but we're going to be concentrating on prime factors as opposed to just factors.

And this is really useful when we start getting into fractions that have large numerators or denominators, but like I said, very similar to what we've just been doing.

So you'll be absolutely fine.

Here we've got Alex and Izzy and Alex says, "I've just realised that we can find the highest common factor of two integers using the product of prime factors." So he's remembered back to some previous learning where we used prime factors to find the highest common factor of some numbers and he's obviously thought to himself, hang on, we've just been using the highest common factor to simplify fractions.

So there's a distinct link between these two things.

Izzy says, "Oh yes, you're right.

This is really useful when we start to look at fractions which have large numerator or denominators." Going to start with one that actually hasn't got large numerators or denominators.

But sometimes it's nice to start off with a fairly basic problem so that we can understand the structure of what we are doing.

So we're going to start with 30 over 42.

We could list the factor pairs for 30.

We could list the factor pairs for 42.

30 written as a product of its prime factors is two multiplied by three multiplied by five and 42 as a product of its prime factors is two multiplied by three multiplied by seven.

Now just as we did previously, we recognised that if we had two over two, that was one.

Here we've got two multiplied by three, over two multiplied by three.

So that is equivalent to one because if we calculate it, it would be six over six.

Anything over the same numerator, over the same denominator, remember is equivalent to one.

That means that this simplifies to five sevenths.

Now this is when this method really comes into its own 'cause I don't about you but I certainly wouldn't want to sit and list systematically the factor pairs of these two large numbers.

We use the product of the prime factors 'cause effectively that is how we identify the highest common factor.

294 written as a product of its prime factors is two multiplied by three, multiplied by seven, multiplied by seven.

Remember there's that button on your calculator that will convert an integer into a product of its prime factors.

And if you've used the calculator to do this, your answer may say seven squared rather than seven multiplied by seven.

But remember they mean the same thing.

546 is two multiplied by three, multiplied by seven, multiplied by 13.

Now remember what we are trying to do here is we are trying to look for common factors in the numerator and the denominator so that we can make them equivalent to one.

And here we can see that both the numerator and the denominator have two multiplied by three, multiplied by seven, that is equivalent to one.

And so therefore we are just left with seven over 13.

That's certainly much, much quicker than it would take us to list all of those out.

Let's take a look at another one, 70 over 462.

Yeah, agreed, 70 wouldn't be too bad to write the factor pairs out, but 462 would be a bit more challenging.

We write both numbers as a product of their prime factors.

Here they're not next to each other, but that doesn't matter.

We are going, we've got two multiplied by seven as a numerator and a two multiplied by seven as a denominator.

Remember those two things are going to be equivalent to one, which means we are left with five over three multiplied by 11, which is five over 33.

You'll notice the process is exactly the same, but this time instead of just having two factor pairs that are equivalent to one, in this case we've got two.

Let's take a look at another question.

300 over 2,250.

We know the drill now, I'm gonna write them out and this time I used my calculator.

So notice I have my, some of my prime factors in their exponent form.

What I like to do though is to take those out of exponent form, it just means that I find it easier to see what's going on.

So I've taken them out of the exponent form.

So two squared, I've written as two multiplied by two and five square is five multiplied by five and so on.

Because there's lots of numbers, I'm now gonna move all of the common ones to the front of my numerator and denominator.

You might find that you don't need to, but sometimes I get a little bit confused when there are quite a lot of numbers and I like to do this step.

So I've just rewritten it so that all of my common factors appear at the beginning of my products.

Now I can see I've got a two multiplied by three multiplied by five multiplied by five as a numerator and a denominator, that's equivalent to one remember.

So we're just left with two over three multiplied by five, so it's two fifteenths.

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

So maybe you might need to just rewind the video and have another look at this.

Don't worry if you do, you'll be fine.

Now let's have a go at one together and then the one on the right hand side in a moment you can do independently.

Using the prime factor button on my calculator, I have rewritten 140 over 1,260 as product of their prime.

So each individual numerator and denominator, I had to do separately.

I then take it out of exponent form and then I put all of my common factors in the numerator and denominator at the beginning.

I can now see that each numerator and denominator contains two multiplied by five multiplied by seven.

Hmm, that's interesting.

There's nothing left on the numerator.

What is my numerator gonna be then? Is it gonna be zero? Well, no 'cause if you think about it, two multiplied by two multiplied by five multiplied by seven multiplied by one is just the same thing.

So my numerator is going to be one and my denominator is going to be the product of three and three, which is nine.

So remember this is really important if you end up with the common factors as a numerator and denominator and that leaves you with nothing or what looks like nothing on the top actually there is a numerator of one.

Now you are ready to have a go at one by yourself.

I would suggest that you use the product of prime factors button on your calculator, just to save you a little bit of time.

Pause the video and then have a go.

When you are ready, come back.

We use the prime factor button on our calculator.

We should have ended up with two multiplied by three, multiplied by five squared as our numerator and two cubed multiplied by three multiplied by five multiplied by seven as our denominator.

Take it out of exponent form, rewrite it with the common factors at the beginning, and you should end up with five over 28.

If we look, we can see that two multiplied by three, multiplied by five is common to the numerator and the denominator, which left us with a factor of five as the numerator and two multiplied by two multiplied by seven as the denominator and two multiplied by two multiplied by seven is 28.

Well done.

Now you are ready to have a go at some of those independently.

Now, because I'm super kind, what I've done is I've given you all of the numerators and the denominators written as a product to their prime factors so you don't have to worry about that step.

That wasn't new learning for today.

So I just want you to concentrate on what our new learning for today was.

So I've given you those written at the top.

So remember our process, write the numerator and denominator as a product of primes, take it out of exponent form, collect together the common factors at the beginning.

Anything that's common to both remember is equivalent to one.

And then simplify what you are left with.

There's a lot there to do, but you'll be fine.

Remember, I don't want anybody using any calculators.

I want to see all of the steps of your working out.

Good luck, pause the video now.

I look forward to seeing you in a moment.

Great work.

Now let's check those answers.

A, it's 10 over 21.

B, one third.

C, one over 56.

D, seven eighths.

E, nine over 80.

F, four sevenths.

G, nine over 140, and H, 12 over five.

I hope me putting that little sneaky one at the end didn't confuse you and make you think you've got it wrong.

We can simplify fractions that are improper fractions into another improper fraction.

Well done on that.

Now we're ready to summarise the learning that we've done in today's lesson.

So it was about simplifying fractions remember, and I talked about at the beginning of the lesson that you may be familiar with simplifying fractions, but what I really wanted you today to concentrate on was the why.

Why does it appear that we can just get rid of, although remember I don't really like that term, but it does appear like we can get rid of things, and that's because we are looking at fractions that are equivalent to one.

So fractions can be simplified using factor pairs.

The most efficient way of doing that, remember, is to use the HCF.

So if the numbers are small, you might decide to list out those factor pairs, but each number and find the highest common factor.

If the numbers are larger, then I would definitely encourage you to use the product of prime factors.

And there's an example of each of those that we've done during this lesson today.

You've done fantastically well and I'm really glad you stuck with me throughout this lesson and I look forward to seeing you again.

Bye.