Lesson video

Hello, everyone, my name is Ms. Coo, and I'm really happy you've chosen to learn with me today.

In today's lesson, it might be easy in some places and tricky in others, but I am here to help.

You will come across some new keywords and maybe some keywords you've already come across before.

I hope you enjoy the lesson, so let's make a start.

In today's lesson from the unit Comparing and Ordering Fractions and Decimals with Positive and Negative Numbers, we'll be checking and securing understanding of fractions, and by the end of the lesson, you'll be able to represent fractions as numbers.

So let's have a look at some keywords, starting with the word numerator.

Well, a numerator is the expression in a fraction that is written above the fraction line and it's the dividend.

We'll also be looking at the word denominator.

And a denominator is the expression in a fraction that is written below the fraction line and it's known as the divisor.

For example, if we had 3/4, the numerator is the three and the denominator is the four.

Today's lesson will be broken into two parts.

Firstly, we'll be using fractions to describe diagrams and then we'll look at fractions as numbers on the number line.

So let's make a start looking at using fractions to describe diagrams. What fraction of the bar do you think is shaded? And really emphasising it's the fraction of the entire bar.

What do you think? Well, it's quite difficult to know what fraction of the whole bar is shaded, because some people might say it's 1/3, some people might say it's 1/4.

So the best thing to do is to split the whole into equal parts.

So what I'm going to do, I'm gonna split it into 10 equal parts, as you can see here.

Looking at the diagram, you'll be able to identify what fraction of our whole has been shaded.

The denominator represents the total number of equal parts the whole has been divided into, and the numerator counts the number of equal parts that the fraction represents.

So this diagram shows 3/10 of our bar has been shaded.

I want you to look at this diagram now.

What fraction of the whole rectangle is shaded? Well, hopefully you can spot its 3/10.

We've broken our rectangle into 10 equal parts and we've identified three out of the 10 are shaded.

Now let's have a look at the same rectangle and what I want you to do is have a think about what fraction of the whole rectangle is shaded.

You might notice I haven't split them into equal parts.

I want you to ask yourself, "Is this also 3/10?" Have a little think and press pause if you need more time.

Well, hopefully you've spotted it is 3/10.

Yes, there are 10 parts, and all of those 10 parts have an equal area.

So we can say 3/10 is shaded, And in this particular question, it's about the equal area as opposed to subdividing that rectangle into equal parts.

Now we need to have a look at this rectangle.

We still have our whole rectangle and I want you to have a look to see is the fraction shaded? Is it also 3/10? Have a little look and press pause if you need more time.

Well done.

Well, hopefully you can spot.

It's difficult to tell.

Yes, there are 10 parts in the rectangle, but they don't have an equal area, so we cannot say that this is 3/10 shaded.

Next, let's have a look at this square.

It's been drawn on a dotty grid and the dots are equally spaced.

Can we say what fraction of the square is shaded? Press pause if you need more time.

Well done.

Well, hopefully you can spot, we can count the area shaded by imagining the grid made by the dots.

So if you have a look at these little grids and split them into little squares, we can count, because we have equal parts.

We can see that there are 12 whole squares and four and a half squares shaded.

So 14 small squares are in total.

This means out of the 36 squares in the larger square, we have 14 out of the 36 which is shaded.

Simplifying this a little bit more, this is equivalent to 7/18.

Well done if you got that one right.

Now what I want you to do is have a look at this rectangle.

And what fraction of the whole rectangle is shaded? See if you can give it a go and press pause if you need more time.

Well done.

Well, hopefully you can spot it's 6/12 or 1/2.

You may have used the grid created by the dots to count the six smaller squares shaded out of the 12.

Or you may have noticed that the shaded pieces can be arranged to fit into half of our rectangle.

I'm gonna simply move our pieces around so you can see it's still a half of our rectangle shaded.

Now for a check question, which of these correctly represents 3/4 of this rectangle being shaded? See if you can give it a go and press pause if you need more time.

Great work, so let's see how you got on.

Well, hopefully you can spot it's A, C, and D.

You may have found this out by counting all the shaded areas.

By counting them all, you would've got 18 out of our 24 shaded, and this is the same as 3/4.

Alternatively, you may have moved the shaded areas around if you had dotted paper.

So a huge well done if you got this one right.

So now I'm going to show you a question.

Which is larger, 1/5 or 3/5? Have a little think.

Okay, so now what I'm going to do, I'm going to show you a diagram, and this diagram shows 1/5 of this bar and this diagram shows 3/5 of this bar.

Now do you think that changes your answer to the original question where I said, which is larger, 1/5 or 3/5? It does change your answer, because what's really important is that you pay attention to what is defined as the whole, as this is important when using fractions to describe diagrams, number lines, or even quantities.

So if we're looking at 1/5 or 3/5 of the same bar, it's clear that 3/5 is bigger and looking at different wholes in a diagram means we can't compare easily just by looking at the fractions.

Now what I want to do is just show you this lovely imagery using cuisenaire rods, and this represents fractions in a really nice way.

This is a beautiful way to show the relationship between each rod.

For example, if you're looking at the pink rod P, it's half of the tan rod T, and you can see that in the diagram or by counting the squares.

Now, I want you to have a look at the yellow rod and the orange rod, and you might be able to spot the yellow rod is half of the orange rod and you can see that by the diagram or counting those squares.

Counting the squares is more efficient and more accurate.

But I really want to draw your attention to the fact that the pink and yellow rods both represent 1/2, but in relation to different wholes.

P represents a half of the tan rod and yellow represents a half of the orange rod.

Yes, they still represent 1/2, but it's in relation to different wholes.

Now I want you to have a look at the dark green rod.

I'm gonna say the dark green rod represents a length of one, so we're calling that our whole.

And what I want you to do is identify what fraction does the light green rod represent, which is G? Have a little think.

Well, hopefully you can spot the light green rod fits twice along the dark green, so that means the light green represents 1/2.

Well done if you got that one right.

Now I'm going to ask, what other fractions can you represent using these cuisenaire rods? See if you can find some more and press pause for more time.

Well done.

I'm just gonna show you some examples here.

For example, if you were looking at the yellow rod to represent one, this means the white rod would represent 1/5 and the red rod would represent 2/5, all using the yellow rod to represent one whole.

You may have used, for example, the dark blue rod as the whole.

That means G, which is the light green rod, represents 1/3 of this, and D, which is the darker green rod, represents 2/3, using the dark blue rod as a whole.

So it's always important what you're referencing as your whole.

So now we've looked at fractions which are less than one.

Let's have a look at a fraction which is greater than one.

So starting with the rod P, I'm also going to add on the rod G, but we're going to refer to rod P as our whole length of one.

I want you to have a little think.

What do you think the length of rod P and G is together if we're referring to rod P as one? Have a little think.

This is a great question, and remember, P has a length of one.

So I would look at it and compare it to the white rod, you might notice four white rods fit into our pink rod, so that means W, which is our white rod, represents 1/4.

Now using that W, we know G is 3/4 given the fact that W represents 1/4.

So that means P and our 3/4 gives us the mixed number of 1 3/4.

Or you might like to say it as 7/4 written as an improper fraction.

This was a good question, so well done if you got that one right.

Now let's check for understanding.

I want you to use the dark green rod to represent a length of one.

What I want you to do is identify what is the length of the light green rod, what is the length of the red rod, and what is the length of the yellow rod? Press pause if you need more time.

So let's see how you did.

Well, hopefully you spotted the light green rod represents 1/2 of the dark green rod and the red rod represents 1/3 of the dark green rod.

And we know the yellow rod represents 5/6 of that dark green rod.

Well done if you got that one right.

Now let's move on to your task.

For question one, I want you to identify what fraction of each diagram is shaded? See if you can give it a go and press pause if you need more time.

Great work.

So let's move on to the second part of question one.

See if you can give these a go and press pause if you need more time.

Really well done.

So let's move on to question two.

Question two shows us those cuisenaire rods again, and what I want you to use is the tan rod to represent the length of one.

So if you know the tan rod represents a length of one, what is the length of the red rod, what's the length of the pink rod, and what is the length of the yellow rod? See if you can give it a go and press pause for more time.

Great work, so let's move on to question three.

Now, question three uses the orange rod and the red rod to represent our length of one.

So knowing this, what do you think the length of the dark green rod would be, what do you think the length of the pink rod would be, and what do you think the length of the dark blue rod would be? See if you can give it a go and press pause for more time.

Well done, everybody.

So let's move on to these answers.

Question one, A identifies 5/9 has been shaded.

You may have split that diagram into nine equal squares to identify 5/9 has been shaded.

For B, 1/8 has been shaded.

Same again, you may have split the diagram into quarters and then realise you need to split it again, so splitting it into eighths would've helped you out there.

For C, we worked out 3/18 or 1/6 has been shaded.

This was a great question, because you had to divide it into six sections as you could see here.

And from those six sections, we have equal parts, so it's clear that 1/6 has been shaded.

Really well done if you got that one right.

For the second part of question one, let's see how you got on.

For D, eight over 16 or 1/2.

For E, three over 12 or 1/4.

And for F, six over 16 or 3/8.

So question two refers to the tan rod as having a length of one.

So what length represents the red rod? Well, this would be 1/4.

What length would represent the pink rod? Well, this would be 1/2.

What length would represent the yellow rod, which is 5/8.

Very well done if you got that one right.

For question three, the orange and the red rod all make a length of one.

So what fraction or length represents the dark green rod? Well, it's 1/2.

What fraction or length do you think represents the pink rod? It's 1/3.

And what fraction or length do you think is represented by the dark blue rod? Well, it's 3/4.

A huge well done if you got that one right.

Great work.

And let's move on to the second part of our lesson, which is fractions as numbers on the number line.

So what I'm gonna do is I'm going to show you a number line and what I want you to do is mark the position of the number three on this number line.

Well, for me, I'm going to put it around about here and now what I want to ask you to do is to mark the position of the number one.

Well, for me, I'm going to mark it around about here.

But the next question I'm going to ask is where is the position of the number two on your number line? Now, it's really important to recognise three could have been placed anywhere on our number line.

One could have been placed anywhere on our number line as long as it was to the left of the number three, but two must go halfway between the one and the three.

So what I'm really emphasising is the fact that once you have two numbers on a number line, any other numbers should be positioned correctly relative to these numbers.

For example, if we were given a number line and we had the number zero and seven on this number line, where do you think the number five would be? Well, if you were to break that number line into seven equal sections, we'll be able to identify where the number five sits, because we've divided our line equally into seven parts as our number goes between zero and seven.

This means each section will be worth one, so our five can be placed here.

And this is important to recognise once you have two numbers on a number line to divide that number line according to those two numbers.

So the next question I'm going to ask is which arrow do you think shows the position of the number three on this number line? See if you can give it a go and press pause if you need.

So hopefully you can spot B is where the number three lies.

Given that the number line starts from zero to five, splitting into five equal sections identifies that each section is one, so therefore B must be three.

Now we need to have a look at these number lines and I want you to put the position of the number 1/4 on these number lines.

So this is a different question to what we've looked at before where we're looking at the whole and identifying a quarter of that.

Now we're looking at the number, which is 1/4.

And I want you to identify where is the number 1/4 on these number lines? See if you can give it a go.

Press pause if you need more time.

Well, to identify the number 1/4 on the first number line, as our number line goes from zero to one, I'm going to break it into four equal sections.

1/4 would have to be here.

Now for the second number line, it goes from zero to 10.

So breaking it into our 10 equal sections, I can see the number one is the first section.

So then I'm going to break that into 1/4, identifying 1/4 is here.

Next, I've got from zero to 3/4.

So same again, I'm going to identify this into three equal sections, because each section would represent 1/4, thus giving me the 3/4.

This is a really nice question as we're focusing on the number 1/4.

So now let's have a look at a check question.

Here I want you to mark the position of 1/4 on the number line.

See if you can give it a go and press pause if you need more time.

Well done.

Let's see how you got on.

Well, firstly, I know it goes between zero and two, so I'm going to do two equal sections, thus identifying the number one.

Now because I have the number one, I can break it into a further four sections for each whole.

That means I know my quarter is here.

So it's all about subdividing the number line according to the question.

Now let's have a look at another check.

Here's a number line and I want you to mark the position of 17/10 on our number line.

And for B, I want you to mark the position of 8/5 on our number line.

See if you can give it a go and press pause if you need.

Great work.

So let's see how you got on on the first one.

Well, if we're looking at 17/10, that means for each whole, let's break it into 10 equal sections.

So you'll see 10 equal sections between zero and one and 10 equal sections between one and two.

Identifying 17/10 means our answer is here, For B, we had to mark the position of 8/5 on our number line.

This would mean I need to break each whole into fifths.

So you can see from zero to one, I broke it into five sections and from one to two, I broke it into five sections.

Now I can count 8/5, which is here.

Huge well done if you got that one right.

Let's have a look and another check question.

Jacob has marked this fraction on the number line and he says, "I've counted 11 lines, so it must be in the 11ths.

This is the eighth line along, so it's 8/11." Do you agree with Jacob? See if you can give it a go and press pause if you need.

Hopefully you can spot it's the lines that divide the number into 10 equal parts, so he's not correct, because the arrow is pointing to 7/10.

Great work, everybody, so let's move on to the task.

Label the fraction the arrow is pointing to on each number line.

See if you can give it a go and press pause if you need.

Well done, so let's move on to the next question.

Here for parts C and D, you need to label the fraction the arrow is pointing to on each number line.

Great work.

Now let's move to question two.

Question two wants you to fill in the missing numbers on these number lines.

See if you can give it a go and press pause for more time.

Well done.

So let's go through our answers.

Here you should have identified 12 over 20 or six over 10, or even three over five.

Well done if you got this one right.

For B, it was 3/8.

For 1c, you should have got 1 2.

9, but 11 over nine is exactly the same as an improper fraction or you would've got 1 2/5 or seven over five.

Great work if you got that one right.

For question two, here are our missing answers.

A huge well done if you've got any of these right.

So in summary, a numerator is the expression in a fraction that is written above the fraction line.

It's the dividend, and the denominator is the expression in a fraction that is written below the fraction line and it's known as the divisor.

Fractions can be used to describe diagrams, number lines, quantities when we define what part of it represents the whole.

And the denominator must represent the number of equal parts when the whole is divided into.

Now fractions are numbers that can be placed on a number line by dividing lines equally.

A huge well done.

It was great learning with you.