Lesson video

Welcome back.

It's Mr. Kendall again.

This is the final lesson in our series about finding equivalent fractions and simplifying.

I'm going to look back at the question sets, at the end of your previous lesson, and then we'll review the main points from the whole series.

Do you have your work with you? Have you got a pencil and paper? Of course you have, because you knew I was going to ask you.

Here are the calculations that Mrs. Furlong left you with in the last lesson.

She certainly kept you busy, and me.

Did you have a go at finding the answers as mixed numbers? Mrs. Furlong, also wanted you to put them in order of difficulty.

And you might've been able to decide, which were easiest and which were hardest without doing the calculations.

And the ones that you thought the hardest might be different to mine or someone else's, and that's okay.

Let's have a look at the first one.

Did you remember the generalisation in the yellow box? That multiplying a fraction by a whole number, the numerator of the fraction is multiplied by the whole number, and the denominator remains the same.

Did you partition? So that you multiplied the whole number five by two, and then the fraction 5/6 by two.

So we've got five multiply by two, nice and easy.

That's just 10, 5/6 multiplied by two.

Multiply the numerator five by two, and the denominator remains the same.

So that's 10/6.

Hmm 10, and 10/6.

Can you see a problem? That's it.

We can't have an improper fraction within a mixed number.

Mixed numbers are made up of a whole number and a fraction, a proper function.

Okay, so 6/6 is one whole.

So 10/6 is one whole and 4/6.

Now we can simplify that 4/6 to 2/3.

So now we have one and 2/3 plus 10.

11 and 2/3.

Is that what you got? Well done if you did.

Let's look at question b.

Again, we can partition.

Did you add the whole numbers four plus two is equal to six? And then add the fractions, 8/9 plus 5/9 is equal to 13/9? We'll need to sort out that improper fraction again, won't we? Cause we can't just leave it, the six and 13/9.

9/9 is one whole.

So 13/9 is one whole and 4/9.

So altogether we have seven and 4/9.

Now we need to remember to do a final check to see if we can simplify.

Well, four and nine have no common factors, other than one.

So we can't simplify it any further than that.

Did you get that one too? What about question c? Well, this looks quite straightforward.

Can you remember the generalisation? You found this one with Mrs. Hussein.

When adding fractions with the same denominator, just add the numerators.

You remember that the generalisation works for subtraction too? When subtracting fractions with the same denominator, just subtract the numerators.

So we'll just subtract the numerators 15 minus nine equals six.

That gives us 15/4 minus 9/4 equals 6/4.

Hmm, 4/4 is one whole, so 6/4 is one whole, and 2/4.

Hang on, what could we do now? Aah yeah, of course.

2/4 is equivalent to 1/2.

So our final answer is one and 1/2.

Have you got that one? Well done.

It's time for question d.

This one has three fractions to add.

Remember to just add the numerators again.

Did you get the improper fraction 31/6? I think we can simplify.

No, that's right, the 31 and six have no common factors.

So now we just need to convert to a mixed number.

Can you tell me what you did? Pause the video while you tell me.

Was it like this? 6/6 make one whole, so how many groups of 6/6 are there in 31/6? Well, using the six times table, I know that five whole groups of 6/6 make 30/6.

So 31/6 is equal to five and 1/6.

That's great if you got that one.

Finally for these questions, we have six and 3/8, minus two and 7/8.

What did you think? Well already I'm thinking this looks tricky because the 7/8 in the subtrahend is larger than the 3/8 in the minuend.

There's going to need to be some bridging.

Now there are quite a few different methods you could use here.

Mrs. Furlong used a number line.

So let's see that method again.

We could start at six and 3/8, and then if we subtract the whole number parts, so we're going to subtract the two, that takes us to four and 3/8.

Now we can use the four to bridge.

So we're going to subtract 3/8 and then we need to subtract another 4/8.

Which takes us to three and 4/8.

Well now three and 4/8, what do you think there? I'm going to put the three in, on this point on the number line.

Is there a simpler way of writing that? Well look, 4/8, three and 4/8 is halfway between three and four.

That's it, you've got there already.

Three and 4/8 is equivalent to three and 1/2.

Now that we know that the answer is three and 1/2, I think you could try some of your own subtraction strategies.

See if you can find another method for doing this question, maybe even a more efficient one.

Okay, so here are our answers.

Which calculation did you choose as the easiest? They all needed some thinking about, this is what Mrs. Furlong and I thought, but don't worry if you disagree.

We thought that c was the easiest, as it was quite easy to subtract the numerators and the simplifying was okay.

Then we chose d.

Adding the numerators was quite easy and them changing to a mixed number was a little bit harder than the previous one.

We thought b was next, as once we'd add the fractions, we needed to convert to a mixed number.

But there was no simplifying required.

We chose a, next as the multiplication was quite easy, but the simplifying from 10 and 10/6 to 10 and 5/3 and then 11 and 2/3, that's a good bit of working out didn't it? Finally, we chose e, as the hardest one.

What did you think? Subtracting needed bridging, and then we have to simplify it as well.

If you chose a different order, that's fine.

It's up to you.

Now we're going to review the lessons in our series.

In the first three lessons, Mrs. Lambert and Mrs. Holmes showed you examples of fractions that had different numerators and denominators, but shared the same value.

Can you remember what they were called? Chest out to me, come on louder, you really do know it.

Okay, let's put that word into this stem sentence.

Sometimes two fractions have the same value.

We call these equivalent fractions! That's right, equivalent fractions.

We're going to find out how many equivalent fractions we can make using these number lines.

Have a look at the number lines you've got here and tell me what the whole has been divided up into for each one.

I'm going to pause the video so you can do that.

Let's check how each line has been divided up.

The top line has three equal parts between zero and one.

One, two, three.

We can say the stem sentence together.

The whole has been divided into three equal parts, each equal part is 1/3 of the whole.

What about the middle line? The middle line has one, two, three, four, five, six equal parts.

So can you say that stem sentence for yourself about the middle line.

Come on, I want you to say it.

The whole has been divided into six equal parts.

Each equal part is 1/6 of the whole.

How many equal parts did you count in that bottom line? Did you count twice to make sure? Hopefully you counted 18 equal parts.

Let's say that stem sentence together.

The whole has been divided into 18 equal parts.

Each equal part is 1/18 of the whole.

Well done.

Our activity now is to see how many equivalent fractions we can find using these number lines.

So for example, 1/3 sits at the same position on the number line as 2/6.

What about the bottom line? How many equal parts did we say they were? That's right, they were 18.

So 1/3 sits at the same position on the number line as, one, two, three, four, five, 6/18.

So 1/3 is equivalent to 2/6 and they're both also equivalent to 6/18.

How many more equivalent fractions can you find using these number lines? Any positions on the number lines.

Pause the video and see how many you can find for yourself.

Welcome back.

Did you find some more equivalent fractions? I hope so.

You might have them in a different order to me because from 1/3, I went to 2/3.

So 2/3 on the number line sits at the same position as 4/6, and that's the same position on the bottom number line as 12/18.

Did you get that one? Did you find some more equivalent fractions? I hope so.

I'm moving my blue line now to here.

Now I don't have an equivalent fraction in thirds on my top line, but here's 1/6, and 1/6 is in the same position as 3/18.

So 1/6 and 3/18 are equivalent fractions.

From 1/6 we can go past 2/6 because you saw that that was 1/3 earlier on.

And now we're at 3/6.

Now 3/6 doesn't have an equivalent fraction in thirds, but it does have an equivalent fraction on our bottom line of eighteenths.

What was that one? 9/18, so that's right.

3/6 and 9/18 are equivalent fractions.

Is there another fraction that we can see that those two are both equivalent to.

Come on chest down to me, louder! Yeah, that's right.

It's 1/2, isn't it.

1/2 is equivalent to 3/6 and 9/18.

It's half way between zero and one.

We can see that quite clearly using our number line.

Here is an octagon.

And my question for you is, what fraction of the octagon is shaded blue? This is a bit like a question that Mrs. Holmes gave you at the end of lesson two.

Have a pause for a think, and come back when you're ready.

What have you decided? Well, we could divide the whole into eight equal parts.

It's an octagon, eight equal parts.

Each equal part is 1/8 of the whole, and we've got six parts shaded.

So, we could say, that 6/8 of the octagon is shaded.

Is there another way we could express that? What if, we moved this blue triangle here, to this position here.

Does that help us to see anything? Oh, hang on, I've got here.

This isn't an eighth anymore, is it? That's not an eighth anymore, what's that? It's 2/8, it's like these 2/8.

But also, that looks like a different fraction as well.

How many equal parts are there now? There are one, two, three, four equal parts.

So each equal part is 1/4 of the whole.

How many of those are shaded? Three equal parts.

So we could say 3/4 of the octagon is shaded blue, and we've seen that 6/8 and 3/4 are equivalent fractions.

Is that what you came up with? Well done if you did.

Are you ready for a challenge? You've done quite a few of these now.

I'm going to introduce you to my friends, Mr. Luke from Shanghai.

This is us when I was in Shanghai a few years ago.

Our daughters have become friends too, and they send messages and photos to one another.

This is Mr, Luke's cat in her Christmas outfit.

Just likes dressing her up.

As well as photos of his cats, sometimes he sends me maths puzzles called Dong nao jin.

Use your head puzzles, and that's what you're going to do now.

We know that 1/4 and 3/12 are equivalent fractions.

So can you read these equations and figure out what goes in the missing number box.

Pause the video, and come back when you have a solution.

Did you solve it? Did you find out what goes into this missing box? Well, if 1/4 and 3/12 are equivalent, then 1/4 and 3/12 is the same as 1/4 plus another quarter.

So 1/4 plus 1/4 will be 2/4.

And here then if 1/4 minus 3/12.

what's that going to be? Well, that's just the same as 1/4 minus 1/4, and that comes to zero.

Could you create some similar calculations using this pair of equivalent fractions? I think you can.

You can pause the video now, or you can come back later.

Perhaps you could test out your calculations on somebody in your home.

With Ms. Heaton, you noticed that some equivalent fractions can be found when there is a vertical multiplicative relationship between the numerator and the denominator.

So the denominator is a multiple of the numerator.

For example, here's our number line again.

1/6 is at the same position on the number line, as 3/18.

So 1/6 and 3/18 are equivalent fraction.

For both fractions, the denominator is six times the numerator.

One times six is six, three times six is 18.

We can use a vertical multiplicative relationship to find an equivalent fraction to a unit fraction, really easily.

The denominator is always eight times the numerator and the numerator is always 1/8 of the denominator.

I can scale my numerator as much as I want to.

I could have a numerator of five, five multiply by eight is 40, 40 is 1/8 of five.

I could have a numerator of 100.

100 multiplied by eight is 800, 1/8 of 800 is 100.

I could have a numerator of 1 million.

1 million multiplied by eight is 8 million.

1000000/8000000.

All these fractions are equivalent to 1/8.

For which of these non unit fractions can we easily use the vertical multiplicative relationship to find an equivalent fraction? Pause the video for some thinking time.

What have you found out? What did you decide? Did you decide that 2/8 would work easily? And 4/8 would work easily? What were your reasons? I think there are at least a couple of reasons, aren't there? We look at the denominators.

Eight is a multiple of two, and eight is a multiple of four.

So those ones work easily.

2/8 is equivalent to 1/4.

So for 1/4 and 2/8 the denominator is always four times the numerator.

What fraction is equivalent to 4/8.

Chest it out to me, go on loudly.

That's right.

4/8 is equivalent to 1/2.

All fractions equivalent to 1/2, the denominator is two times the numerator.

That's right.

So why can't we use the vertical multiplicative relationship easily to find an equivalent fraction for 3/8.

That's right.

Because eight is not a multiple of three.

So it won't work easily.

We need a different method.

with Ms. Heaton, you also looked at equivalent fractions that had a horizontal multiplicative relationship.

In this example, the numerator and denominator have both been scaled up by the same factor.

Can you spot what that is? That's right.

They've both been scaled up by a factor of six.

So we can say 4/5 and 24/30 are equivalent because both the numerator and denominator have been scaled by a factor of six.

Remember, fractions that are equivalent, have the same value.

They're the same proportion of the whole.

One of the most important aspects of fractions is being able to notice the horizontal multiplicative relationship between two fractions.

You really need to know those multiplication tables.

How can we use a horizontal multiplicative relationship to find the missing number to make the statement true? 3/8 is equivalent to something over 72.

Use your times table knowledge, to find the relationship between the denominators.

Can you then complete the stem sentence.

Pause and come back when you've got a solution.

What did you find? Did you find that the denominator has been scaled by a factor of nine? So the numerator also needs to be scaled by a factor of nine.

Multiplying the numerators, three multiplied by nine is 27.

Let's say the statement together.

3/8 and 27/72 are equivalent because both the numerator and denominator have been scaled by a factor of nine.

Can you say it for yourself? Well done, very good indeed if you got that one.

Next I want to know, is this statement always, sometimes or never true? When the numerator and denominator are multiplied or divided by the same number, the value of the fraction remains the same.

You read it.

What do you think? Pause and come back when you've made a decision.

Well, I hope you've decided that it's always true.

It's really important, isn't it? That equivalent fractions have the same value.

This is Mrs. Lambert's favourite statement.

And it's really the key to equivalent fractions.

Can you remember what we mean when we say that a fraction is in its simplest form? I did this with you in lessons 13 and 14.

Here was our generalisation.

A fraction is in its simplest form when the only common factor of the numerator and denominator, is one.

So which of these fractions are in their simplest form.

1/7, 6/10, 7/12, 12/16? Pause the video while you have a think and then come back.

What did you decide? Did you say that 1/7 is in its simplest form? That's right, because 1/7 is a unit fraction.

So it must already be in its simplest form.

Did you say 7/12 is in its simplest form? Seven and 12 don't have any common factors so 7/12 is already in its simplest form.

What do we need to do to write 6/10 in its simplest form? How will we write 12/16 in its simplest form? What do we do to convert to a simplest form.

Here we are, to write a fraction in its simplest form divide both the numerator and denominator by their highest common factor.

So, to write 6/10 in its simplest form we need to divide both six and 10, by their highest common factor.

Their highest common factor is two, that's right.

So 6/10 is equivalent to 3/5.

What's the highest common factor of 12 and 16? Is it two? It's four, isn't it? Four is the highest common factor.

So divide 12 by four and divide 16 by four.

Our equivalent fraction, our simplest form is 3/4, well done if you got those two.

Here's one final practise question.

Join the pairs of equivalent fractions.

One is done for you.

Well yes, it's the easy one that's been done for you, isn't it? 1/2 is equivalent to 2/4.

For the others, you're going to need to think about what you've learned about multiplicative relationships, vertical and horizontal.

Think about what you've learned about simplifying fractions.

You could go back in the video if you want to.

Pause now, and see if you can pair up the fractions.

Have you solved it? Did you find the horizontal relationship between 2/5 and 6/15? If we multiply the numerator two by three, we get six.

If we multiply the denominator five by three, we get 15.

So 2/5 and 6/15 are equivalent to each other.

Hmm, the others were trickier, weren't they? 8/12 and 10/12, well they can't be equivalent because they've both got the same denominator, but different numerators.

So what did you do? Did you simplify the fractions? That would be the best thing to do because the others there's no easy horizontal relationship between them to easily spot.

So to simplify then, did you simplify 8/12 to 2/3? And 10/12 to 5/6? So we're looking at which other these are the two fractions, which ones are those going to match up with? Which one is 2/3, Which one is 5/6? Which did you find? Did you get 14/21 is 2/3? So there's a matching pair? And 25/30 is 5/6.

So there's a matching pair.

Well done, very well done if you got that.

Well, here we go.

You have now completed this set of lessons about equivalent fractions and simplifying.

It would be great if you could write about what you've learnt so you could show your teacher.

Maybe, you could draw a mind map.

You could include the stem sentences and generalisations that you've been saying.

You could draw some images and write some examples.

All these things will help you to remember the work you've done in the future.

On behalf of all the teachers who've been teaching you in this section, we'd like to thank you for your hard work and for joining in so well, well done.