Lesson video

Hello, my name's Mrs. Hopper, and I'm really looking forward to working with you in this lesson from our unit on addition and subtraction of fractions.

Now, fractions are one of those things that some people absolutely love, and I hope you are in that camp.

I love fractions.

I think they're really clever ways of expressing numbers and parts of wholes.

So, if you're ready to make a start, let's do some work on adding and subtracting fractions.

In this lesson, we're going to be looking at how we can explain how to write a fraction in its simplest form.

I wonder whether you've come across that before.

Let's have a look and see what's going on in our lesson today then.

We've got two key words in our lesson.

We've got equivalent and common factor.

So I'll take my turn to say them, and then it'll be yours.

My turn, equivalent.

Your turn.

My turn, common factor.

Your turn.

Excellent.

I hope you've heard about equivalent fractions before, and you've possibly done some work on factors and maybe even common factors.

But let's look at what those words mean 'cause they're going to be really useful for us today in our lesson.

If two numbers or expressions have the same value, they are said to be equivalent.

And when we compare lists of factors of two or more numbers, any factors that are the same are called common factors.

So look out for those in our lesson today.

There are two parts to our lesson.

In the first part, we're going to be thinking about equivalent fractions in their simplest form, and in the second part, we're going to be finding the highest common factor.

Let's make a start on part one.

And we've got Alex and Izzy helping us in our lesson today.

Oh, here's a fraction wall.

What do you notice? Hmm, you might want to have a think about this before Izzy and Alex share what they've noticed.

Izzy's noticed something.

She's noticed that four sixteenths are equivalent to three twelfths.

Can you see in the sort of bottom left of our fraction wall, there are highlighted four sixteenths and three twelfths, but they're the same fraction of the bar.

They're the same proportion of the bar.

And she says both of these are equivalent to two eighths and one quarter.

Can you see that shaded section of the wall? There's a vertical line, isn't there? Showing that all of those fractions represent the same proportion of the length of the wall, one quarter, two eighths, three twelfths, and four sixteenths.

And there they are recorded as one fraction rather than a group of unit fractions.

So four one-sixteenths is four sixteenths, three one-twelfths is three twelfths, and two one-eighths is two eighths.

Alex says, "So they all have the same value.

They are all a form of one quarter." They're all another way of writing one quarter.

So we can say that one quarter is in the simplest form.

Hmm, can you think about what we mean by simplest form then, from what Alex has said? We can also say that these fractions are equivalent.

And Izzy says, "Because they all have the same value and are equivalent, we can record them like this." One quarter is equal to two eighths is equal to three twelfths and is equal to four sixteenths.

They are equivalent, they are all the same.

They have exactly the same value.

So we can use that equal sign in between them.

Look at these ones.

These fractions are also equivalent.

This time we've shown fractions of circles.

So what can you see? Well, Izzy says, "Here's another example.

All of these fractions have the same value.

They are all a form of one-third." Can you see that the shaded area of each circle is the same? It's just described using a different fraction.

One third, two sixths, three ninths, and four twelfths.

But they are all representing the same proportion of the whole, the same part of the whole.

And Alex says, "So we can say that one third is the simplest form of all these fractions." Oh Alex, you're gonna have to tell us more about this simplest form.

So how can we find the simplest form of a set of equivalent fractions? We've established that all of those fractions in the fraction wall were equivalent to a quarter, and all of the fractions in the circles were equivalent to one third.

Izzy says, "I think we have to make the fraction as small as possible." Do you agree with Izzy? Is that what's happening here? What do you think? Alex says, "Not quite.

The size of the fraction doesn't change so we don't make it smaller." The size of the fraction isn't changing.

All those fractions are equivalent.

They're representing the same proportion of the whole.

They have the same value.

They've just been written down using different numerators and denominators.

Ah, Alex says, "We just scale down the numbers used for the numerator and the denominator." So can you see? You might well have created equivalent fractions before by scaling up and down numerator and denominators by the same factor.

So how can we find the simplest form of a set of equivalent fractions? Izzy says, "It looks like the simplest form has to have a numerator of one." Do you agree? It is the case for the two sets of fractions we've got here, but is it always true? "Ah," says Alex, "Lots of fractions in their simplest form do have a numerator of one.

But there are examples that do not have a numerator that is one.

So we can't just look for a numerator of one to find a fraction in its simplest form.

Although if the numerator is one, then the fraction will be in its simplest form.

So here's another set.

The simplest form of a fraction does not have to have a numerator that is one.

Here, the simplest form is two thirds.

So two thirds is equivalent to four sixths, is equivalent to six ninths, and is equivalent to eight twelfths.

And we've shown that by shading the same proportion of each circle.

So Alex says, "For example, the simplest form of all the fractions above is two thirds," so it doesn't have one as a numerator.

So there must be another way of knowing that we've got a fraction in its simplest form.

Izzy says, "Oh, I see.

So we just need to scale the numerator and denominator to be as small as they can whilst still being a whole number." Ah, that's an interesting way of thinking about it.

Can you see that we've scaled the numerators and denominators down to be as small as they can be, but still being a whole number.

So thinking about that, it's time to check your understanding.

Can you tick the fraction that is the simplest form of five tenths? So is it A, B, C, or D? And you've got a bar there with five tenths highlighted to show you.

So pause the video, have a go, and when you're ready for some feedback, press play.

What did you reckon? Did you reckon that it was a half? You can sort of see it looking at the bar.

The simplest way to describe that shaded area is one half of the whole bar, isn't it? And also with Izzy saying that we've scaled down the numerator and the denominator so that they are represented by the smallest numbers, and we can see that one half represents that shaded part of the bar.

Time for another check.

Two fifths is the simplest form.

What fractions are equivalent to it? So we've got two fifths, and we've got a bar showing two fifths.

So what other fractions are equivalent to two fifths? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Izzy says, "If I scale the numerator and denominator by the same whole number, I can create equivalent fractions like these." So she's created four tenths, six fifteenths, and eight twentieths.

And if you think about those scale factors, two fifths is equivalent to four tenths because the numerator and denominator have both been scaled up by a factor of two.

Two times two is four, five times two is 10.

We may have twice the number of pieces in the whole, but we've got twice as many of those pieces.

So the proportion remains the same.

Time for you to do some practise now.

Look at the list of equivalent fractions and write each list in its simplest form and explain how you know.

So there's a gap with each of those lists for you to write the equivalent fraction in its simplest form.

And can you explain how you know that it's in its simplest form? Think about what Izzy said during the lesson.

Pause the video, have a go, and when you're ready for the answers and some feedback, press play.

How did you get on? Izzy says, "If you find the highest common factor of the numerators and denominators in any of the equivalent fractions, you can find the simplest form of the fractions." Ah, so we are looking for the highest common factor of the numerators and denominators.

That's interesting, isn't it? Let's see what this looks like with these missing fractions.

So for our first row, one third was in its simplest form.

We can scale any of those other fractions down by the same factor, the numerator and denominator, to be equivalent to one third.

Two sixths? Well, if we halve the numerator and the denominator, we get one third.

Three ninths, we can divide the numerator and denominator by three.

And for four twelfths, we can divide by four.

And if we scale the numerators and denominators down by the same factor, we will get the simplest form of the fraction.

So in that case, it was one third.

What about the next set? Well, it was one half, wasn't it? We can also see there that relationship between the numerator and the denominator.

The numerator is half the value of the denominator.

And the simplest way we can express that with whole numbers is one half, one is half of two.

What about the next set? Well, that was our one fifth.

And we can also see that when we have a unit fraction, we can test that relationship between the numerator and the denominator.

So for one third, the denominator was three times the numerator, and for one fifth, the denominator will be five times the numerator.

So that works if our simplest form is a unit fraction.

What about the others though? Ah, for this one it isn't, is it? But we can find a common factor of two between four and six.

And so four divided by two is two, and six divided by two is three.

And now we haven't got another common factor, have we? Two and three are the simplest way that we can record that proportion of a whole.

Two thirds is the simplest form.

Two fifths is the simplest form for the next set.

Ah, and one fifth again.

Ah, interesting.

We've got another set with one fifth.

Did you spot there again that the denominator was five times the numerator in all of those fractions? And the simplest way we can represent that as a fraction is with one fifth.

Well done if you found all of those and if you were thinking about common factors.

We're going to look more at common factors in the second part of our lesson.

And here we are, finding the highest common factor.

Let's have a look.

So what is the simplest form of this fraction? We've got the fraction four twelfths.

So you might want to have a think before Izzy And Alex share their thinking.

Izzy says, "If we divide the numerator and denominator by the same value, we can find simpler forms. So we can scale down the numerators and denominators.

But she says, "To do this, we need to find common factors of both four and 12." The number we divide by has to be a factor of both the numerator and the denominator if we're going to end up with a whole number answer.

Alex says, "We can use our factor bugs to help." Have you met factor bugs before? They help us to list the factors of numbers.

So here are the factors of four.

There's our factor bug.

So we've got one and four and two are factors.

One times four is equal to four, and two times two is equal to four.

So those are the factors of four.

What about 12? And here are the factors of 12.

We've got one, two, three, four, six, and 12.

And if we think about factor pairs, one times 12, two times six, and three times four.

Alex says, "One, two, and four are common factors of both numbers." So, one, two, and four are factors of four and factors of 12.

So what about this simplest form? How are we going to use that knowledge to find the simplest form? Izzy says, "So which factors should we divide both the numerator and the denominator by?" Which do you think? Alex says, "Well, if I divide them both by one, the numerator and the denominator stay the same.

That doesn't look right." Well, we've not changed the way we've expressed the fraction, have we, by just dividing by one? Because anything divided by one is equal to itself.

If we divide them both by two, what happens? Well, we can, that's two sixths, but then we can divide them both by two again to simplify the fraction further.

Ah.

So if we simplified by a factor of two, four twelfths is equal to two sixths, and we can do that again, take the factor of two again, and it's equal to one third.

Alex says, "However, by the time I've done that, it would've been easier just to divide by four, which is the highest common factor." It was one, two, and four were the common factors, and four is the highest.

So four divided by four is equal to one, and 12 divided by four is equal to three.

So we can scale down the numerator and denominator by the same factor, and we create the equivalent fraction of one third.

And that is in its simplest form, it has that numerator of one.

There are no other common factors higher than one.

Izzy says, "So to find the simplest form of a fraction, divide the numerator and the denominator by the highest common factor." That's right, Izzy.

And that's a really important thing to remember and to think about why that works.

Time to check your understanding.

Alex is trying to find the simplest form of five fifteenths.

What is the highest common factor he should use between five and 15? Is it A, B, C, or D? Pause the video, have a think, and when you're ready for some feedback, press play.

What did you think? It's five, isn't it? Five has factors of one and five.

So as long as 15 has a factor of five, that's got to be the highest common factor.

And we know that the factors of 15 are one, three, and five.

So five is the highest common factor.

There we go.

Time to check again.

Here's a true or false.

The highest common factor is always the number represented by the numerator.

Is that true or false? And why? Pause the video, have a think, and when you're ready for some feedback, press play.

What did you think? It's false, isn't it? Well, if you divide the numerator by itself, it would leave the numerator as one, but not all simplest forms of fractions have that numerator that is one, do they? And it's not always going to be the case that the numerator will be a factor of the denominator.

And that's the key, isn't it? We've got to divide the numbers by something that is a common factor of the numerator and the denominator.

Well done if you've got that reasoning right.

It's really important to think about.

And time for you to do some practise.

Can you find the highest common factor and give each fraction in its simplest form, and we've given you some arrows there to think about what you're going to be scaling the numerator and denominator down by.

And for question two, tick the statements that are true.

So we've said that one fraction is the simplest form of another.

Is that statement true or not? And can you correct the incorrect statements? Pause the video and have a go at the two questions.

And when you're ready for the answers and some feedback, press play.

How did you get on? So did you spot that the common factor between nine and 18 and nine eighteenths is nine? So we can divide both the numerator and the denominator by nine, scaling them down to create the equivalent fraction of a half.

What about nine twenty-sevenths? Ooh, did you notice something here? Yes, it was nine again, wasn't it? So we could scale down the numerator and the denominator by nine, and we get the equivalent fraction of one third.

So whilst we've said that it's not always the numerator that gives us the highest common factor, in this case, it looks to be, doesn't it? Nine thirty-sixths, same again, highest common factor is nine.

And in nine forty-fifths, again, the highest common factor is nine.

So this time we were creating unit fractions each time that were equivalent to nine somethings.

What about the bottom row then? Well, four tenths is equivalent to two fifths.

So this time our highest common factor was not the value of the numerator, was it? The highest common factor, in fact, was two.

So the factors of four are one, two, and four.

The factors of 10 are one, two, five, and 10.

So two is the highest common factor.

What about the rest of them? Well, in fact, for all of them, the highest common factor was two in each case.

So the highest common factor shared by the numerator and the denominator in all of those fractions in the bottom row was two.

So we created the equivalent fractions of two fifths, four fifths, six fifths, and eight fifths, all in their simplest form.

The numerators and denominators in those fractions do not share a common factor other than one.

So in question two, you were ticking the statements that were true.

So the first one is true.

One fifth is the simplest form of three fifteenths.

The highest common factor is three there.

What about the next one? The next one wasn't true, was it? One quarter is not the simplest form of five twenty-fifths.

Five and 25 share a common factor of five that would give us an equivalent fraction of one fifth.

What about the next one? It's true.

One half is the simplest form of 500 one-thousandths.

500 is half of a thousand, so the fraction must be equivalent to a half.

So what about the next one? Well, three ninths is equivalent to six eighteenths, but it's not in its simplest form.

We'll come back to that one when we look at correcting the incorrect statements.

And what about the last one? Yes, four sevenths is the simplest form of 16 twenty-eighths.

The highest common factor is four.

16 divided by four is four, and 28 divided by four is equal to seven.

Four and seven do not share a common factor higher than one.

So for the second one, I think we mentioned this as we went through, you may have corrected one quarter to one fifth.

One fifth is the simplest form of five twenty-fifths.

And yes, these ones were equivalent.

Three ninths is equivalent to six eighteenths, but it's not the simplest form.

The simplest form is one third.

There's a common factor of six between six and 18.

Well done if you reasoned your way through those and used your knowledge of common factors to think about fractions in their simplest form.

And we've come to the end of our lesson.

We've been explaining how to write a fraction in its simplest form.

What have we learned about? We've learned that all equivalent fractions can be expressed in their simplest form.

At some point, there's a way of writing a fraction where the numerator and the denominator only share a common factor of one.

And to find the simplest form of a fraction, you can divide the numerator and denominator of the fraction by the highest common factor that they share.

Thank you for your hard work and your mathematical thinking in this lesson, and I hope I get to work with you again soon.

Bye-bye.