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 how dividing by a whole number can be related to multiplying a whole number by a unit fraction.

Again, this is a really useful connection to make, which will help you along the way with thinking more efficiently about your calculations as well as when working with problems in your everyday lives.

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

So by the end of this lesson today, you should be able to say, I can explain how dividing by a whole number relates to multiplying by a unit fraction.

Throughout our lesson today, we've got three key words we're gonna be referring to.

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

The first one is expression, your turn.

The second word is equation, your turn.

And the last word is denominator, your turn.

Thinking about what these mean then.

An expression contains one or more value where each value is separated by an operator.

An operator might be, for example, an addition, subtraction, multiplication, or division sign.

An equation is used to show one number, calculation or expression is equal to another.

And the denominator is the bottom number in a fraction.

This lesson today is broken down into two cycles.

The first cycle is linking multiplication and division, and the second cycle is solving numerical problems. When you're ready to get started with the first cycle, let's get going.

In this lesson today, we'll be joined by both Aisha and Sofia.

As always, they'll be sharing their thinking, and any questions that they've got to help further our thinking along the way.

So let's start our lesson here today then.

We're gonna start thinking about how we can represent this equation here.

Our equation shows one third multiplied by 15 is equal to something.

Aisha is suggesting that we start by representing it as a bar model.

Let's see how we would do that.

First, we'd need to draw two equal bars.

Now we need to write the whole into the bar.

We're gonna use the top bar to represent the whole.

Once we've done that, we now need to divide the second bar into the number of equal parts that we need.

There we go.

We've now got three equal parts.

How did you know we needed three equal parts, Aisha, I wonder? That's right.

We used the denominator from the fraction that we're multiplying by, don't we? We can see that the fraction is telling us to divide the whole into three equal parts and that's represented by the denominator of the fraction, and therefore we now divide the hole into three equal parts.

Ah, yeah, that's a nice idea, Aisha, isn't it? Now that we know how many parts the whole needs to be divided into, we can write this as a division equation to help us, can't we? So we could say that 15 divided by three is equal to something.

This will help us identify the value of each of the bars.

There we go.

We can use our times table knowledge and division facts to help us here.

We know that three multiplied by five is equal to 15, therefore 15 divided by three is equal to five.

Great.

That means each one of the parts has a value of five.

Now that we know that, we can fill in our equation at the top, can't we? We can say that one third multiplied by 15 is equal to five because we can look at our bar model and look at the parts.

And if we look at one third, which is one of the three parts that we've divided the whole into, it has a value of five.

Therefore one third of 15 is equal to five, or one third multiplied by 15 is equal to five.

So look carefully at these two equations now then.

What do you notice? Ah, nice thinking, Sofia.

You've said that you've noticed that the result in both of the equations is a smaller number, isn't it? Than what we started with.

In the first equation, we started with the whole of 15, didn't we? And we multiplied that by one fifth.

And in the second equation we started with the whole of 15 and we divided it by three.

And it's given us a smaller number each time, hasn't it? Not only that, these two examples have given us the same number, haven't they? For the product.

So we can say in this example here that the whole has been made one third times the size, hasn't it? In the top example, we multiplied the whole by one third and that gave us five.

And in the second example, we divided the whole by three, therefore leaving us with three parts and one of those parts is one third of the whole.

Therefore we can say that part is one third times the size of the whole and the product is the same number in this example, isn't it? So we can now say that both of the expressions are equal to each other.

There we go.

One third multiplied by 15 is equal to 15 divided by three.

Let's have a look at a different example now.

This time we've got 1/4 multiplied by 20.

Let's represent this as a bar model again.

How are we gonna start that off? Maybe you could have a go at starting to draw this for yourself as well whilst we do.

Well, first we're going to need to draw two equal bars, aren't we? And then we're going to need to write the whole into one of the bars.

We're gonna write that in the top bar in this example.

Once we've done that, we can now look at our fraction and decide how many equal parts the fraction is asking us to divide the whole into.

We can see that the denominator is four, so the whole has been divided into four equal parts.

So we're now going to do that here.

And of course, now that we know that, we can start thinking about how we can write that as a division equation again, can't we? So we can say that 20 divided by four is equal to each one of the parts and the size of the parts is five.

So each part has a value of five.

And now that can help us to understand what 1/4 multiplied by 20 is equal to.

We know that 1/4 of 20 is equal to five, and as we know because both of these equations both equal to five, they give us the same value, we can say that they are equal to each other.

So there we go.

We can say that 1/4 multiplied by 20 is equal to 20 divided by four.

That makes our life a lot easier, doesn't it? When we're trying to multiply a whole number by unit fraction, doesn't it? Okay, time for you to check your understanding now.

Can you fill in the missing number? That's right.

The missing number was five, wasn't it? And where did you get that from? Yeah, that's right.

We took that from the denominator of the unit fraction, didn't we? The denominator of the unit fraction tells us how many parts we need to divide the whole into and therefore if the whole was 20, we need to divide that whole into five equal parts.

Well done if you managed to get that.

And the second check then, can you tick the expressions that represent the bar model? Take a moment to have a think.

That's right.

It's both A and B, isn't it? The whole has been divided into four equal parts.

So we can say 24 divided by four would be a correct expression for this.

So we give that one a tick.

And we also know that the whole has been divided into four equal parts, therefore each part is 1/4 and one of those parts, 1/4, has a value of six.

So we could say that six is 1/4 of 24.

Although we're only writing it as an expression and we're not including the product here, we can therefore say that 24 multiplied by 1/4 or 1/4 multiplied by 24 is representative of this bar model.

Okay, time for you to have a go at practising now.

What I'd like you to do is match the expressions both on the left hand side and the right hand side to the bar models in the middle.

Each bar model should have two expressions that matches to it.

And then once you've done that, I'd like you to have a go at filling in the missing numbers for each example here.

And then for the last one, what I'd like to do is circle the expressions that are equivalent to 1/6 multiplied by 864.

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

Okay, welcome back.

I'm gonna plot on which expressions match to each bar model.

Hopefully you can tick them off and see how you got on with those.

We can see that the top bar model is 28 and that 28 has been divided into four equal parts.

Therefore that can represent 1/4 multiplied by 28 or 28 multiplied by 1/4, which is the bottom one on the left hand side.

And it can also represent a division equation of 28 divided by four.

For the second bar model, we've got the whole as 28, and this time it's been divided into seven equal parts, hasn't it? That's the same as saying 28 multiplied by 1/7, isn't it? Or we could also describe it as 28 divided by seven.

And then the last one is 24 and that's been divided into six equal parts this time, hasn't it? So once again, we can represent that as a multiplication equation as 24 multiplied by 1/6, or we can represent it as a division equation, saying 24 divided by six.

Well done if you got that.

For task two then, we've got to fill in the missing numbers.

The first one is 1/3 multiplied by 21.

That would be equal to 21 divided by three, wouldn't it? The second one, we know we've got 72 divided by eight, so that would be equivalent to saying 1/8 multiplied by 72.

In the one underneath that now then, oh, we haven't got a full expression here to help us, but we know that in one of the expressions we're multiplying by 72.

And then the other expression we're gonna be dividing by eight.

The dividing by eight tells us the size of the fraction, doesn't it, that we need? So that would be 1/8 multiplied by 72.

And the 72 that we have gives us the whole for the division expression on the right hand side of the equals sign.

So that would be 72 divided by eight.

Next one is 30 multiplied by 1/6.

That's equal to 30 divided by six, isn't it? The one after that is something multiplied by 1/7 and that would be equal to 42 divided by something.

Well, we know that the whole is 42, so we can say that 42 multiplied by 1/7 would be equal to 42 divided by seven.

That's correct.

The denominator of the unit fraction that we're multiplying by shows that the whole needs to be divided into seven equal parts.

And then the last one, hmm, what's acting as the whole number in this example? That's right, the circle would be the whole number.

Therefore, we need to do the circle multiplied by the unit fraction of one over a triangle.

And the triangle is acting as the denominator in this case.

We know that the denominator tells us how many equal parts the hole needs to be divided into.

So we can say that this expression is equal to the circle being divided by the triangle.

Well done if you managed to get that as well.

And then the last task then here, can you circle the expressions that are equivalent to 1/6 multiplied by 864? Well, hopefully you circled these two here.

We know that the position of the numbers within the expression can be rotated around.

Therefore 1/6 multiplied by 864 is the same as saying 864 multiplied by 1/6.

And we can also write this as a division equation, can't we? We can say this is 864 divided by six, can't we? If you want to extend yourself even further, you might be able to try and explain why the other three are incorrect examples.

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

Let's move on to cycle two, solving numerical problems. So let's start here then.

This is our first problem here that we're going to look to solve.

1/4 multiplied by 12 is equal to something.

How would we go about this then? That's right, Aisha.

It's a good idea to help draw a bar model to help us to start us off with, isn't it? So let's do that here.

Let's draw our two equal bars and let's write the whole in at the top.

We know that we're multiplying by 1/4, therefore the whole needs to be divided into four equal parts because the denominator is four.

And as a result of that, we can create a division equation to help us solve this.

We know that 12 divided into four equal parts would give us a value of three, wouldn't it? So each part has a value of three, and therefore that helps us to solve 1/4 multiplied by 12, because that is also equal to three.

Have a look this time.

What'd you notice here? That's right, Sofia.

The order of the numbers in the equation have been rotated round, hasn't it? In the previous example, it was 1/4 multiplied by 12, whereas now it's 12 multiplied by 1/4.

Does this matter? No, it doesn't, does it? We know that because multiplication is commutative.

So we can rotate the order of the numbers around the multiplication symbol to help us calculate this problem a little bit easier if that helps.

So now that we know that the values can be placed in any order, let's have a closer look at how both of these might be represented.

So this equation here says 12 multiplied by 1/4.

It sounds a little bit like 12 lots of 1/4.

Here we go, let's count up in quarters.

1/4, two quarters, three quarters, four quarters.

That's one whole.

And then another four quarters, five quarters, six quarters, seven quarters, eight quarters.

There we go.

And then keep going.

Nine quarters, 10 quarters, 11 quarters, 12 quarters.

Ah, that's three wholes altogether, isn't it? So we can say that 12 lots of 1/4 or 12 quarters is equal to three wholes, can't we? However, if we read it the other way round, it might feel that we can say it as 1/4 of 12.

There we go.

Look at our 12 counters here.

Our 12 counters have now been divided into four equal parts and we want one of those parts, don't we? So one of those parts is 1/4 of the whole, and that 1/4 has a value of three, doesn't it? So we can say that 12 lots of 1/4 is equal to three, and 1/4 of 12 is also equal to three.

So we can see that the order of the numbers in the expression or the equation doesn't matter there does it? It still gives us the same product.

We can also still apply our division equation to both of these, can't we? 12 divided by four.

On the left hand side, the whole is 12 quarters, and those 12 quarters have been divided into groups of four.

So we can see that that's three groups of four quarters altogether.

And on the right hand side, the 12 represents the 12 counters altogether, and this time they've been divided into four groups.

And in each one of those groups, that gives us a value of three in each group.

So once again, this division equation is equal to both of the expressions that we've looked at so far with using the multiplication symbol.

Here's one more example for us to have a look at then.

How would you solve this? Take a moment to have a think.

1/300 multiplied by 1,200.

How would you do this then, Aisha? Well, Aisha's suggesting that she draw a bar model again.

Sofia shut her down pretty quickly there.

She said no way should we draw a bar model! With 300 parts needed, that would take ages to draw, wouldn't it? So there we go, Sofia.

Yeah, you're right.

That's a good idea, isn't it? Actually we know we could use our understanding that multiplying by a unit fraction is the same as dividing the whole number by the denominator to help us solve this, couldn't we? So we can say 1/300 multiplied by 1,200 is equal to 1,200, which is the whole divided by the denominator, which is 300.

Is that easier to calculate now? We've got 1,200 divided by 300.

Well, we know that 12 divided by three is equal to four.

So 1,200 divided into groups of 300 would mean that we'd have four groups as well.

So we can now say 1/300 multiplied by 1,200 is equal to four.

That was quick and easy, wasn't it? Rather than drawing out a bar model of 300 parts.

Okay, time for you to check your understanding now.

Can you have a go at solving this equation here, please? That's right.

The product is 12, isn't it? And we know that because we can use our division fact to help us here.

36 divided by three is equal to 12.

Okay, and time for you to practise now then.

What I'd like you to do is have a go at solving each of the following equations here.

As you're going through, you might like to ask yourself what it is that you notice about each of the equations as you work through them.

And then a similar example for task two here.

Work through the equations and ask yourself what is it you notice each time.

And then for task three, what I'd like you to do is have a go at filling in the missing boxes each time.

Good luck with those tasks.

I'll see you back here shortly.

Okay, welcome back.

Let's work through these then.

Can we solve the following equations? One half multiplied by 60? Well, we know that would be the same as saying 60 divided by two, which gives us 30.

The second one is 1/3 multiplied by 60.

That would be the same as saying 60 divided by three.

So that would be 20, wouldn't it? The next one is 1/4 multiplied by 60.

That's the same as saying 60 divided by four.

That gives us 15.

And the last one is 1/5 multiplied by 60.

That's the same as saying 60 divided by five, which is equal to 12 each time.

Well did you notice here then? That's right, Sofia As the number of parts we divided the whole into increased, the size of each part decreased in size, didn't it? Great spot.

Well done you.

For task two then, I've put the answers in here for you.

Let's work through them.

The first one is 1/3 multiplied by 12, and that would be equal to four because that's the same saying 12 divided by three.

The next one is 1/3 multiplied by 24.

24 divided by three is equal to eight.

The next one is 1/3 multiplied by 48.

So 48 divided by three would be equal to 16.

And the last one, 1/3 multiplied by 96, that's equal to 96 divided by three, which again would be equal to 32.

What did you notice this time? Ah, that's right.

Well, the fraction that we are multiplying by stayed the same each time.

However, the whole number that we've multiplied the fraction by doubled in size each time, didn't it? So we know that one of the numbers that are being multiplied is doubled in size, then the product also needs to be doubled in size.

Well done if you got all of that.

And the last one then.

Let's have a look here quite carefully at these.

The first one says 50 multiplied by 1/5 multiplied by something else is equal to 50.

Hmm, what did you put for that? Well, you're right, the answer was actually five.

We know that when we multiply 50 by 1/5, we make it 1/5 times the size.

We are dividing it by five, aren't we? So if we are dividing a number by five and we need to get back to the original number, we therefore need to undo what we've just done.

We've just divided by five, haven't we? So we're now going to need to multiply it back by five to get back to the original number.

So therefore 50 multiplied by 1/5, that's the same as saying 50 divided by five, and then multiply that by five again, that would lead us back to 50 where we started.

What about the second example? 32 multiplied by something multiplied by four is equal to 32.

Once again, we started with a number, a whole number of 32, and we're ending with the same whole number.

So we need to go through a process of doing an undoing, don't we? We've got 32 multiplied by something multiplied by four.

Well, if I multiplied 32 by four, that would make it four times the size, wouldn't it? However, we need to then bring it back down to 32.

So we'd need to divide it by four, wouldn't we? We know that dividing by four is the same as saying multiplying by 1/4.

So the missing numbers here would be a one for the numerator and a four for the denominator to show 1/4.

And then the last one as well.

Something multiplied by four, multiplied by five is equal to five.

What about this? Well, it might be worth looking for the whole number that's the same on both sides of the equal sign, and that happens to be five, doesn't it? We've got a five on the right hand side of the equals sign and a five on the left hand side of the equals sign.

We know that if we multiply that five by four, that's gonna make it four times the size.

Therefore we need to undo that to get it back to be the original size of five, don't we? So if we've multiplied it by four, we therefore need to divide it back by four, don't we? One way of doing that is to say multiply by 1/4, so therefore 1/4 is the missing fraction required for that equation.

Well done if you managed to get all of those.

Brilliant.

That's the end of our learning for today.

I myself am feeling a lot more confident about the connection between multiplying a whole number by a unit fraction and dividing the whole number by the denominator of the unit fraction.

Hopefully you feel just as confident too.

To summarise what we've learned today then, we can say that you can represent multiplying a unit fraction by a whole number using a bar model.

We know that multiplying a unit fraction by a whole number is the equivalent of dividing the whole number by the denominator of the unit fraction.

And when multiplying a unit fraction by a whole number, you can swap the position of the numbers that are being multiplied with each other to either side and the product will remain the same.

Thanks for joining me again today.

Take care and I look forward to seeing you again soon.