Lesson video

Hi, how are you today? Welcome to today's lesson.

My name's Mr. Peters, and in this lesson today, we're gonna be thinking about how we can multiply a whole number by a proper fraction.

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

So by the end of this lesson today, you should be able to say that I can multiply a whole number by a proper fraction.

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

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

The first one is relationship.

Your turn.

The second one is unit fraction.

Your turn.

And the third one is non-unit fraction.

Your turn.

We can describe a relationship as a connection between one or two more things.

A unit fraction is a fraction where the numerator is one, and a non-unit fraction is a fraction where the numerator is greater than one.

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

The first cycle is finding a unit fraction of an amount, and the second cycle is finding a non-unit fraction of an amount.

Let's get started with the first cycle.

In this lesson today, you're gonna meet both Jun and Izzy.

As always, they'll be sharing their thinking and any questions they have along the way to help us with our learning.

So we're gonna start our lesson here today.

Have a look at these two rods.

How could we describe the relationship between these two rods? You might like to use the blank equations underneath to help you with that thinking.

Take a moment to have a think.

Look carefully.

What did you notice? Yes, you may have come up with the idea that the light green rod is actually 1/3 of the whole of the blue rod.

We can represent that like this.

We know the blue rod has a value of 30 and the light green rod has a value of 10.

So we can say that 1/3 of the blue rod, which is 30, is equal to 10, which is the value of the light green rod.

I wonder how else we could record that.

Yes, that's right.

We could also say it is 1/3 multiplied by 30 is equal to 10.

And we know that because we can exchange the word of for the multiplication symbol, can't we? And these represent the same thing.

Let's think about the blue bar.

What if we were to say that the blue bar has shrunk in size? Take a moment to have a think.

How much will it have shrunk by? Yeah, that's right.

We can say that the blue bar has shrunk to 1/3 times the size.

We can multiply the blue bar by 1/3 and that will be equal to the green bar.

So hopefully, you can see here that when we multiply a whole number by unit fraction, then the whole number can also be said to shrink in size.

It becomes one-something times the size.

We can say that the whole number has shrunk or has become smaller.

So looking at this example more closely then, we can say that the 30 has shrunk in size.

So we can say that 10 is 1/3 times the size of 30.

Do you think you could say that with me? Are you ready? 10 is 1/3 times the size of 30? Let's have a look at another example here.

Take a moment to have another think.

Using the blank equations below, how would you describe the relationship between these two bars? Look carefully again.

That's right.

We could say that the white bar is 1/5 of the yellow bar, can't we? We can see that five of the white bars will be equal to the yellow bar and we have one of them.

So we can say that 1/5 of the yellow bar would be equal to the white bar.

However, the yellow bar has a value of 20 and the white bar has a value of four, doesn't it? So we can say that 1/5 of 20 is equal to four.

How else could we write this? That's right.

We know we could switch the language of of for the multiplication symbol.

So we'd have 1/5 multiplied by 20 is equal to four.

And let's have a think about how we can describe this multiplicatively then.

How much has the yellow bar shrunk by? That's right.

We can say that the yellow bar has shrunk 1/5 times the size, hasn't it? We know this because when we multiply a whole number by a unit fraction, that whole number becomes smaller, doesn't it? So we can say that 20 has shrunk in size, and therefore, four is 1/5 times the size of 20.

Have a look at these two rods now.

How would you describe the relationship between these two? Have a look.

Does that help you at all? That's right.

We know that the white rod is equal to 1/4 of the pink rod.

So we can describe this as 1/4 of the pink rod, or 12, in this case, is equal to three, which is the white rod.

We know that because you would need four lots of the white rod to be equal to the pink rod, and we only have one of them.

So it represents 1/4 of the whole.

We know we can write this as a multiplication equation.

We could write it as 1/4 multiplied by 12 is equal to three.

And describing this multiplicatively then, how could we do that? How much has our whole shrunk by? That's right.

Our whole has shrunk by 1/4 times the size.

We know that when we multiply a whole number by a unit fraction, it makes the whole number smaller.

So we can say this time that 12, which was the whole number, has shrunk in size.

It is shrunk by 1/4 times the size.

So therefore, we can say that three is 1/4 times the size of 12.

So let's look at our three examples altogether now.

This should really highlight to us what happens to the whole number when we multiply it by the unit fraction.

Can you see the whole numbers represented by the top bar in each of our examples? And each time when we've multiplied it by a unit fraction, it has become one something times the size, hasn't it? And therefore, it has shrunk.

In the first example, we multiplied it by 1/3.

In the second example, we multiplied it by 1/5.

And in the last example, we multiplied it by 1/4, and our whole number shrunk each time.

We refer to this as scaling.

Okay, time for you to check for understanding now.

Can you fill in the missing numbers of the equation based on the bar model? Take a moment to have a think.

That's right.

When I was looking at this, I estimated that it would take about six of the white bars to be equal to the green bar.

So we can say that the green bar has been multiplied by 1/6, and now it's equivalent to the white bar.

So we could write this as 1/6 of 24 is equal to four or 1/6 multiplied by 24 is equal to four.

Okay, and here's some other equations that match the bar model.

Can you have a go at ticking them off and telling me which ones would represent the bar model? That's right, it's a and d, isn't it? We can estimate that the white bar is 1/5 of the whole here.

We would need five of those white bars to be equal to the orange bar.

So we can say that 1/5 of the whole, which is the orange bar, which, in this case, is 30, would be equal to the white bar, in this case, is six.

What did you notice about d? That's right.

It doesn't matter which order the numbers that are multiplying with each other are in, does it? So therefore, we know if 1/5 multiplied by 30 is equal to six, then we can also say that 30 multiplied by 1/5 is also equal to six.

Okay, onto our first task for today then.

what I'd like you to do is write an equation to represent each of the bar models.

And then, what I'd like to do for the second one is to draw the whole number for each of the bar models on the left-hand side that represent the equation on the right-hand side.

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

Okay, let's see how you got on then.

So for the first one, we can see that the whole has been multiplied by 1/5 to create the red bar.

So therefore, 1/5 multiplied by 50 is equal to 10.

For the second one, we can see that the whole is 30, and it's been multiplied by 1/3.

Therefore, we can say that 1/3 times 30 is equal to 10.

And for the last one, this time, what did you notice? That's right.

You may have started to notice that the whole number has shrunk to the same size each time, isn't it? However, the amount that it is shrunk by has changed due to the size of the whole number that we started with.

So this time, the whole number was 20, and now it has shrunk to the size of 10.

So we can say that the 20 has been multiplied by 1/2.

So we can say that 20 multiplied by 1/2 is equal to 10, or 10 is 1/2 times the size of 20.

Okay, and for the second task then, were you able to draw the whole for each of the bar models? For the first one, we can see that the white part would represent the whole number once it has been shrunk, and we know it's been shrunk by 1/4 times the size.

So the whole would need to be four times the size larger than the white bar, wouldn't it? So the whole would look like this.

We know that the original whole number would be 24, and therefore, the size of the whole number now, once it has been shrunk, is six, and it has been multiplied by 1/4.

For b, looking at the fraction, that's gonna help us to identify how many white parts would be equal to the whole.

We can see that this white part represents 1/3 of the whole, therefore, the whole would be three lots of the white bar.

And we can represent that like this.

The whole, we know, in the equation is 24 'cause that's what's been multiplied by the unit fraction.

And the unit fraction in this case was 1/3.

So we can say it's been multiplied, so it is 1/3 times the size, and that leads us with the whole now being eight, where it is shrunk in size.

And the last example then.

Well, the white part represents 1/8, therefore, that is one of eight equal parts.

So the whole would need to be equivalent to eight of those parts.

We know that the whole is 24 because that's what the unit fraction has been multiplied by, and we know that we've multiplied the whole by the unit fraction of 1/8.

So it's become 1/8 times the size.

And in this case, we can say that three is 1/8 times the size of 24.

Okay, that's the end of our first cycle.

Let's move on to cycle two now then, finding a non-unit fraction of an amount.

Have a look at these two bar models.

How would you describe the relationship between these two bar models here? That's right.

This is similar to what we looked at before, isn't it? We can see that this is 1/3 of 30 is equal to 10, or we could describe that as 1/3 multiplied by 30.

And that would also be equal to 10, wouldn't it? We can say that the whole number of 30 has been made 1/3 times the size and that leaves us with 10.

And therefore, we can say that 10 is 1/3 times the size of 30, or that 30 has shrunk 1/3 times the size.

Have a look at this example now carefully in comparison to the next example.

What do you notice this time? Take a moment to have a think.

That's right.

You may have noticed that we don't have one green part on the bottom anymore, do we? We actually have two of those parts? Hmm, I wonder how we could represent this as an equation then.

Well, we know from the last example, that each one of those parts represents 1/3 of the whole.

And how many of those parts do we have now? That's right, we have 2/3 of the whole, don't we? We have two of those parts.

So we have 2/3 of the whole.

We could write that like this then.

2/3 multiplied by 30, which is the whole, what would that be equal to then? That's right.

It would be equal to two lots of 10, wouldn't it? Hmm, how else could we write this as a multiplication equation then? That's right.

We could write it as 2/3 multiplied by 30 would be equal to, again, two lots of 10.

What did you notice now? That's right.

We exchanged those two bars, didn't we? For one longer bar now, one longer bar of 20.

So we can exchange those two lots of 10 for 20 in our equations now.

So our equations now read 2/3 of 30 is equal to 20 or 2/3 multiplied by 30 is equal to 20.

So let's think about this relationship multiplicatively again now.

How could we describe how much the whole number has shrunk by this time? Take a moment to have a think.

That's right.

In the last example, we multiplied the whole number to make it 1/3 times the size.

So we multiplied it by 1/3.

However, we know that this 20 represents 2/3 of the whole, doesn't it? So we can say that the whole number has shrunk.

So it is 2/3 times the size.

We can say that 20 is 2/3 times the size of 30.

Or we can say that 30 has shrunk 2/3 times the size.

Let's have a look at another example here then.

How would we describe this relationship again? That's right.

We know from earlier that the red bar represents 1/5 of the whole.

So we can say that 1/5 of 50 is equal to 10, or we could write this as a multiplication equation.

1/5 multiplied by 50 is equal to 10 as well.

We know that we can say that 10 is 1/5 times the size of 50, or we can say that 50 has shrunk 1/5 times the size.

Take another very careful look between this example and the next example.

What do you notice? That's right, in the first example, we only had one red part, didn't we? Whereas now we've got two of these parts.

How could we describe this again then? That's right.

We know that each one red part represents 1/5 of the whole.

And how many fifths do we have? Yep, we have 2/5, don't we? So we could say that 2/5 of 50 is equal to two multiplied by 10.

Each one of the fifths had a value of 10, and we have two of them.

So we can say that 2/5 of 50, which is the whole, is equal to two lots of 10.

We know we could also record this as a multiplication equation, 2/5 multiplied by 50.

And that would be equal to, again, two lots of 10, wouldn't it? Hmm, what's another way of representing two lots of 10 then? Yep.

We know that two lots of 10 is equal to 20, don't we? So you can see how the bar has now changed from those two lots of 10 to represent one bar of 20.

And there we go.

We've adjusted our equations as well.

So how would we describe this multiplicatively again then? Take a moment to have a think.

How much has the whole been multiplied by to allow it to shrink? That's right.

In the first example, the whole was multiplied by 1/5, wasn't it? So that it became 1/5 times the size.

However, how much is it gonna be multiplied by now? That's right.

It's gonna be multiplied by 2/5, isn't it? It's going to become 2/5 times the size.

These equations also represent this.

So we can say that 20 is 2/5 times the size of 50, or we can say that 50 has shrunk.

So it is 2/5 times the size.

Let's get ready to look at the next example.

What do you notice this time? Yeah, this is slightly different this time, isn't it? In the first example, we had one red bar.

In the second example, we had two red bars, whereas now we've got three red bars this time.

Hmm.

How would we record this this time then? Take a moment to have a think again.

That's right.

We know that each one red bar represents 1/5 of the whole.

And we have three of those bars, don't we? So it's gonna represent 3/5 of the whole.

We can say that 3/5 of 50 is equal to three lots of 10, isn't it? We can write this as a multiplication equation as well.

3/5 multiplied by 50 is equal to three lots of 10.

And once again, we could exchange those three red bars for one long bar, couldn't we? So we can adjust our equations as well.

Our equations now read that 3/5 of 50 is equal to 30 or 3/5 multiplied by 50 is also equal to 30.

Let's take a moment to have a think about the relationship now between these two bars.

What happens to the orange bar in order for it to shrink to be the size of the green bar? That's right.

In the first example, it was multiplied by 1/5 times the size.

In the second example, it was multiplied by 2/5 times the size.

And in this example, it's been multiplied, so it is 3/5 times the size.

We can say that 30 is 3/5 times the size of 50, or we can say that 50 has been shrunk so that it is 3/5 times the size.

Let's look at all of these carefully once again now then.

What do you notice this time? Well, you may have noticed that in the first example, we multiplied it by a unit fraction.

And then in the second and third examples, we multiplied it by a non-unit fraction.

And what happened to the whole each time.

That's right.

The whole shrunk as well, didn't it? So we can say that when you multiply a whole number by a non-unit fraction, then the whole number will also shrink in size.

It would also become smaller.

Okay, time for you to check your understanding now.

Can you fill in the missing parts in our blank equation here to represent this bar model? Take a moment to have a think.

Okay, so by looking at that red bar then, I don't think any multiple of that red bar will be equal to the yellow bar.

So, actually, I think that red bar is the combination of a number of parts.

Actually, I think if I split that red bar in half, then it would give me two parts, wouldn't it? And if I had more of those parts, that would be equal to the yellow bar, I think.

So each one of the parts would represent 1/5.

And this time, we could say that the red bar is gonna represent 2/5.

So we could say that 2/5 multiplied by yellow is equal to the red bar.

Or we could say that the yellow bar has been multiplied so it is 2/5 times the size.

Here's another example.

Look carefully at the numbers this time.

Well, we know that no amount of whole tens would be equal to 25, would it? Multiples of 10 are 10, 20, 30, 40, et cetera.

So something's happened to the 10 here, isn't it? It must be more than one part of the whole.

What could our 10 be divided down into? Well, we know that 25 can be divided into five equal parts, and each one of those parts would be five, wouldn't it? So 10 would represent two of those parts if that was two lots of five.

So altogether, the whole could be made up of five lots of five and that would be equal to 25.

But we only have two lots of those five.

So we could say that 2/5 multiplied by 25 is equal to 10, isn't it? We can say that 25 has been multiplied, so it is 2/5 times the size, or 10 is 2/5 times the size of 25.

And onto our last task for today then.

Could you have a go at filling in the equations for each of the bar models on the left-hand side.

You should be able to use the first one to help you with the remaining bars as you go through.

And for the second task, what I'd also like you to do is fill in the blank equations on the right-hand side for each of the bar models on the left-hand side.

And for the third task then, I'd like you to do another and another and another.

What I'd like to do is use the whole number as 100, and I'd like you to find three different ways that you could shrink 100.

And you can write that in using the blank equations underneath each of the bar models.

Izzy set you a challenge.

She's asking, "Can you come up with a way that nobody else would come up with?" Good luck with that, and I'll see you back here shortly.

Okay, welcome back.

Let's go through the answers then.

So the first one, the white bar would be equal to 1/4 of the whole, wouldn't it? So we can say that 1/4 multiplied by 16 is equal to four.

16 could be shrunk so it is 1/4 times the size, and that part would be equal to four or four is 1/4 times the size of 16.

What did you notice about the second example? This time, the whole has been shrunk so it becomes eight.

We know if 1/4 is equal to four, then 2/4 would be equal to eight.

So we could say that 2/4 of 16 or 2/4 multiplied by 16 is equal to eight.

For the third example, we know that 1/4 of 16 is equal to four, 2/4 of 16 is equal to eight, but 3/4 of 16 would be equal to 12.

So we can record this as 3/4 multiplied by 16 is equal to 12.

And lastly, hmm, the bars are equal, aren't they? They're the same.

And how many quarters of 16 would that be equal to? That's right.

That'd be 4/4 of 16, wouldn't it? So we can say that 4/4 of 16 or 4/4 multiplied by 16 is also equal to 16.

Have a look at these bar models then.

What did you notice here? Oh yes, the bars that the whole numbers were shrunk to the size of weren't at the beginning were they? They were placed at different places underneath the bar, weren't they? Do you think that makes a difference though? No, it doesn't, does it? Let's have a look at how we could represent each of these equations now then.

For the first one, the whole is 12, and one part of that whole is equal to two.

How many of those parts would be equal to the whole? That's right, it would be six, wouldn't it? So we can say that 1/6 of 12 or 1/6 multiplied by 12 is equal to two.

For the second one, this time, we know that 1/6 of the whole was equal to two.

So 2/6 of the whole would be equal to four, which is the red bar.

So we can say that 2/6 multiplied by 12 is equal to four.

For the third bar then, if 2/6 multiplied by 12 is equal to four, what do we notice about the third one? Ah, we've got six this time, haven't we? So that must be equal to 3/6 of the whole.

3/6 multiplied by 12 would be equal to six, wouldn't it? And then finally, the last one, we've got a 10 here, haven't we? Is that equal to 4/6? No, it's not, is it? If 3/6 was six, then 4/6 would be equal to eight, wouldn't it? So it can't be that.

Ah, I'm looking carefully though.

It looks like it's missing only one of the six.

And that would be 6/6 if it was the whole, so this must be 5/6.

This is 5/6 of the whole.

So we can say that 5/6 multiplied by 12, which is the whole, is equal to 10.

Well done if you've got all of those.

Okay, and onto this challenge here then.

Did you come up three different ways of representing how 100 could be shrunk like I have here? Let's look at the first one.

We can say that 100 has been shrunk so it becomes the size of 50.

Now that 50 is gonna represent more than one part, it's actually gonna represent two parts here.

So that means the whole would've been divided into quarters.

If that 50 is 2/4, then we can say this is 2/4 of 50 or 2/4 multiplied by 50, then that would be 2/4 of 100.

So we can say that is 2/4 multiplied by 100 is equal to 50.

For the next one.

Hmm, we've got an 80 underneath.

Hmm, how many parts could our whole have been divided up into then? That's right.

I know that 100 can be divided into multiples of 20, and that could be five equal parts.

100 divided into five equal parts would give us a value of 20 in each of those parts.

However, we don't have one part, do we? We have several of those parts.

How many twenties do we have? That's right, we'd have four lots of 20, wouldn't we? So if the wholes divide into five equal parts, each part would be 1/5.

That 1/5 represents 20.

But we don't just have 1/5, we have four lots of 20, don't we? So that's 4/5.

We can say that 4/5 multiplied by 100 is equal to 80.

And the last one then.

Hmm, we've got 100 here, hmm.

And the bar underneath is six.

Does six go into 100? It doesn't, does it? So that must mean, again, that the six represents more than one part.

Hmm.

What numbers make up six? Two, three.

Okay, let's think about three first then.

Is three a factor of 100? Nope.

Three isn't a factor of 100, is it? The closest number would be 99 again.

So maybe it's two.

Is two a factor of 100? Yes, it is.

Two is a factor of 100.

If we divided the whole into 50 equal parts, then each part would be a value of two.

So we can say that two is 1/50 of 100, isn't it? Well, we don't have two though, do we? We have six.

So how many lots of two is that? That's three lots of two.

So we can say that 3/50 of 100 is equal to six.

Well done if you manage to come up with three of your own like that.

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

To summarise what we've learned, we can say that the relationship between two numbers can be represented as multiplication, where the whole number is made something times the size by a non-unit fraction.

And when a whole number is multiplied by a non-unit fraction, the whole number becomes smaller.

Thanks for joining me today.

Hopefully, you are feeling a bit more confident about multiplying whole numbers using non-unit fractions.

Take care, and I'll see you again soon.