Lesson video

Hi there.

Welcome to today's lesson.

My name's Mr. Peters, and in this lesson today we're gonna be thinking about how we can explain finding a unit fraction of a quantity relates to multiplying by a unit fraction.

If you're ready to get started, let's get going.

So by the end of this lesson today, you should be able to explain how finding a fraction of a quantity relates to multiplying by a unit fraction.

In this lesson today, we've got two keywords we're gonna be referring to throughout.

I'll have a go at saying it and then you can repeat after me.

The first one is relationship and the second one is unit fraction.

A relationship is a connection between two or more things, and a unit fraction is a fraction where the numerator is one.

In this lesson today, we've got two cycles.

The first cycle, we'll be thinking about describing relationships, and then the second cycle we'll be thinking about reasoning with expressions.

Let's get started with the first cycle.

In our lesson today, we'll be joined by both Izzy and Jacob.

They'll have lots of questions to ask as well as some great thinking to share, to help us along the way with our learning.

So our lesson starts here then.

We've got two rods.

We've got the orange rod at the top, and we've got the yellow rod underneath that one.

Take a moment to have a think.

How can you describe the relationship between both of these rods? Hmm, well, you may have started by looking at that and thinking how many yellow rods are potentially equal to the orange rod.

And if that's the case, then what's the size of the yellow rod in comparison to the orange rod? Well, we could say that 1/2 of the orange rod is equal to the yellow rod.

Can you see that? The yellow rod comprises of half of the orange rod, doesn't it? So we can say that 1/2 of the orange rod is equal to the yellow rod.

If that's the case, we could also say that two lots of the yellow rod would be equal to the orange rod, wouldn't they? And we can record it like that.

Two yellow rods are equal to one orange rod.

Okay, have a look at the rods this time.

How do you think you might describe the relationship between these two rods now then? Take a moment to have a think.

Well, once again, you might be thinking about how many of the red rods are equal to the green rod? And therefore what's the size of the red rod in comparison to the green rod? Well, we could say that one third of the green rod is equal to the red rod, and therefore we can say that three lots of the red rod are equal to the green rod.

Here we go, here's another example.

Have a look here.

How would you describe the relationship between these two rods? That's right, we can say here that the white rod is actually one fifth of the yellow rod.

So we could describe it as one fifth of the yellow rod is equal to the white rod.

Hmm, so if the white rod is one fifth of the whole yellow rod, then how many white rods are gonna be needed to make the yellow rod? That's right, five lots of the white rod is equal to the yellow rod, isn't it? Okay, time for you to check your understanding now.

How can you describe the relationship between these two rods here? The top rod is the dark blue rod and the rod underneath is the light green rod.

Take a moment to have a think.

That's right, we can describe the light green rod as one third of the dark blue rod.

So one third of the dark blue rod is equal to the light green rod.

Or how else could we describe this? That's right, we can say that three lots of the light green rod is equal to the dark blue rod.

Okay, so let's have a look how we can nudge this on a little bit further now.

What would happen if we started giving each of the colour of the rods a set value? Hmm, we've got the orange rod now, which has a value of 10, and the yellow rod, which has a value of five this time.

How would we describe the relationship between these two now? Well that's right.

Instead of saying that 1/2 of the orange rod is equal to the yellow rod, we can now say that 1/2 of 10 is equal to five, can't we? And we can apply that to the stem sentence as well underneath that, we could say that two lots of five are equal to 10.

Okay, have a look at these two rods now then this time.

This time the green rod has a value of 30 and the red rod has a value of 10.

We knew from last time that we could describe it as one third of the green rod is equal to the red rod.

Therefore, how could we describe it using the numbers this time? That's right.

We can say that one third of 30 is equal to 10, isn't it? And again, how else could we describe this below? That's right, we would need three lots of 10 for it to be equal to 30, wouldn't we? Well done if you've got that one, and the last one as well then here, what's the relationship between the white rod and the yellow rod? Well that's right, we know that the white rod is one fifth of the yellow rod, don't we? So we can say that one fifth of 20 this time, which is the value of the yellow rod, is equal to four, which is the value of the white rod.

We can say that five lots of four are equal to 20.

We know that five lots of the white rod would be equal to the yellow rod.

So five lots of four would be equal to 20.

Well done if you've got that.

Okay, another check for understanding now.

Have a look at the values here.

How would we describe the relationship between these? That's right.

We could say that 1/5 of 50 is equal to 10.

So 1/5 of the orange rod would be equal to the red rod.

And then underneath that we could say that we would need five lots of the red rod, which would be equal to the orange rod.

So five lots of 10 would be equal to 50.

Okay, time for you to start practising now.

What I'd like to do is complete the stem sentence for each one of the images on the left hand side.

You might like to think about drawing the number of parts underneath the whole each time to help you identify how many parts the hole is comprised of.

For task two then, I've given you a rod.

Each rod represents a part.

I haven't given you the whole each time.

I have however, given you the relationship between the parts and the whole.

So what you need to do is to draw the whole to go on top of each one of the rods on the left hand side.

You're going to need to estimate that to decide how big the whole rod would need to be.

It doesn't need to be exact, but it needs to be roughly accurate.

Good luck with those two tasks, and I'll see you back here shortly.

Okay, welcome back.

Let's have a look then.

So for the first one, we can say that one quarter of 12 is equal to three, or four lots of three is equal to 12.

For the second one, we can say that 1/6th of 30 is equal to five.

So six lots of five is equal to 30.

And for the last one we can say that 1/8th of 32 is equal to four, therefore eight lots of four are equal to 32.

For task two then, have a look at the size of the rods that we ended up drawing here.

We know the light green rod was one third of the whole, so we know to draw the size of the whole, we would need three light green rods in a row and the length of that would be equal to the size of the whole.

For the second one, we know that the yellow rod is half the size of the orange rod.

Therefore we would need two yellow rods to find the size of the whole of the orange rod.

And then for the last one, we need to find the whole.

We know the part is 1/7th of the whole, so we need seven sevenths, don't we? To make one whole.

So if we put seven of the white rods next to each other in a row, that would be equal to the black rod, which represents the hole.

Well done if you've got all of those.

Okay, onto task two now then, reasoning with expressions.

So have a look here at the relationship between the blue rod and the light green rod.

We know that 1/3rd of the blue rod is equal to the light green rod.

So we can describe that as 1/3rd of 15 is equal to five.

Or we can say that three lots of the light green rod are equal to the blue rod.

So three lots of five is equal to 15.

That's right, Izzy, we can say that three lots of five is the same as saying three multiplied by five, isn't it? So actually we can exchange the language of lots of for the multiplication symbol, can't we? So instead of saying three lots of five is equal to 15, we can say that three multiplied by five is equal to 15, can't we? Knowing this then, it means we can extend this to our understanding of when we talk about the language or finding a fraction of an amount.

So here we can say, if we know that three lots of five is equal to 15, therefore three multiplied by five is equal to 15, we can say 1/3rd of 15 is equal to 1/3rd multiplied by 15.

There we go.

So we can see, we can use the multiplication symbol to replace the language of "of" in one of our equations.

So that's right, Izzy, when we've multiplied by this unit fraction, it feels like the blue bar has shrunk in size.

And that's the relationship we can describe here, can't we? We can say that the green bar is 1/3rd times the size of the blue bar.

It's been multiplied by 1/3rd, it has been made one third the size.

Let's have a look at another example from earlier on again then.

We described the relationship here between the orange and the yellow rod, didn't we? The orange rod had a value of 10 and the yellow rod had a value of five.

So we could say that 1/2 of 10 is equal to five or two lots of five is equal to 10.

We know we can now replace the "of" with the multiplication symbol.

So instead of saying two lots of five is equal to 10, we can say that two multiplied by five is equal to 10.

And once again, we can do the same using our unit fraction in our equation.

So instead of saying 1/2 of 10, we can say 1/2 multiplied by 10.

There we go.

So we can say 1/2 multiplied by 10 is equal to five.

That's right, we could also describe it as this time the orange bar having shrunk in size.

It feels like the orange bar has shrunk to half the size, hasn't it? So we can describe it as the yellow bar being 1/2 times the size of the orange bar.

So take a moment to have a look now then carefully.

These are the three examples we've looked at so far and we've changed the language of "of" with the multiplication symbol.

What'd you notice happen to the whole number each time in our examples? That's right Izzy, the whole number gets smaller each time, doesn't it? So in the first example, the whole was 15, wasn't it? And then when we multiplied it by 1/3rd it became five, didn't it? For the second one we can say the whole number was 10, and then once we multiplied that by half, that became five, so it shrunk in size again.

And we can say the whole, in the last example, that the whole was 20, wasn't it? And then once we multiplied that by 1/5th it shrunk in size and became four, didn't it? So we can generalise to say that when you multiply a whole number by a unit fraction, obviously, which is less than one, it makes the whole number smaller in size.

Have a go at saying that generalisation for yourself.

Okay, time to check our understanding now.

Which expression is the largest here? Use your inequalities to describe the relationship between the two.

That's right, we can say that 1/3rd multiplied by 20 is less than 20, isn't it? The whole in both the examples was 20.

However, we know that when we multiply a whole by a unit fraction, the whole number would become smaller, wouldn't it? So we know that multiplying 20 by 1/3rd would be a smaller expression than simply 20 itself on the right hand side.

Okay, and our second check now then.

Can you match the expressions on the left hand side with the expressions on the right hand side? Take a moment to have a think.

There we go.

The top one on the left hand side is linked to this bottom one on the right hand side.

The second one down is linked to the third one down on the right hand side.

The third one down on the left hand side is equal to the top one on the right hand side, and the last one on the left hand side is equal to the second one on the right hand side.

How did you decide that these were the same as each other? That's right, we just replaced the language of "of" with the multiplication symbol each time, didn't we? And the numbers stayed the same.

Okay, once again, time for you to practise to finish off our lesson for today now then.

What I'd like to do is complete the stem sentences for each image, and then once you've done that, I'd like to use your inequalities to compare the expressions on either side of the circles.

And then finally, I've got one more for you to think about.

If the triangle and circle represented numbers, then which number would be smaller? Would it be the triangle or the circle? Have a go at trying to explain how you knew that.

Good luck with those tasks and I'll see you back here shortly.

Okay, let's go through this then.

For the first one then we can say that one quarter of 12 is equal to three.

So one quarter multiplied by 12 is equal to three.

We can also say that three is one quarter times the size of 12.

For the second one, we can say that 1/6th of 30 is equal to five, so therefore 1/6th multiplied by 30 is also equal to five.

And then we can also say that five is 1/6th times the size of 30, can't we? It has shrunken size and it is 1/6th times the size of the whole.

And for the last one we can say that 1/8th of 32 is equal to four.

1/8th multiplied by 32 is equal to four.

And of course we can also describe that as four being 1/8th times the size of the hole, and the hole in this case is 32.

Well done if you got those.

Okay, now we can go on to think about reasoning the size of the expressions using our inequalities.

The first one says 1/3rd multiplied by 12 and 12 on the right hand side We know that one third multiplied by 12 is less than 12 because we're multiplying the 12 on the left hand side by a unit fraction.

For the second one we've got 24, and then we've got 1/8th multiplied by 24.

Again, this time we can say that 24 is greater than 1/8th multiplied by 24, that's because 1/8th multiplied by 24, we are multiplying by a unit fraction, don't we? And we know that always makes the number smaller.

The third one, we've got 50 on the left hand side and then 50 multiplied by 1/2.

Even though we're multiplying by the unit fraction as a second part of our multiplication expression here, it doesn't matter, the order there, does it? We know that 50 multiplied by 1/2 is the same as multiplying by a unit fraction.

Therefore that would be less than the 50 on the left hand side.

And then the last one, 482 multiplied by 1/100th.

Well, once again, it doesn't matter the size of the whole number that we're using to start off with, does it? As long as that's the same, if we are multiplying the whole number by unit fraction, it's gonna become smaller in size.

Therefore 482 multiplied by 1/100th, which of course is a unit fraction, is going to be less than 482.

Well done if you got all of those.

And then the last one then, what did you come up with? Did you decide that this triangle or the circle would be the larger number? That's right, Izzy, the circle has to be the smaller number, doesn't it? Because the triangle would be representing the whole, and then when we multiply the whole by a unit fraction, we know it makes the product smaller, doesn't it? Therefore the circle is acting as the product and the circle would therefore be smaller than the triangle.

Well done if you're able to reason that and come up with an explanation for that for yourself.

Okay, that's the end of our learning for today.

Hopefully you've enjoyed that lesson and once again, feeling a lot more confident about how we can interchange the language of "of" with the multiplication symbol.

To summarise our learning, we can say that when using the word "of," we can replace this with a multiplication symbol.

Our generalisation for today is that when you multiply a whole number by unit fraction, less than one obviously, the whole number becomes smaller in size, doesn't it? And we can describe how much smaller it becomes saying that the whole number becomes one something times the size.

That's the end of our lesson.

Thanks for joining me, take care, and I'll see you again soon.