Lesson video

Hi there.

Welcome to today's lesson.

My name's Mr. Peters, and in this lesson, we're gonna start to extend our thinking about multiplying unit fractions by whole numbers and applying this to a range of different problems. If you've got your stuff ready and you're ready to get started, let's get going.

So, by the end of this lesson today, you should be able to say that I can use knowledge of multiplying a unit fraction by a whole number to help me solve a range of problems. In this lesson today, we've got two key words we're gonna be using throughout.

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

The first one is equation.

Your turn.

And the second word is strategy.

Your turn.

Thinking about what these mean then.

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

And a strategy is a plan or a method to achieve a specific goal.

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

The first one is finding a fraction of a larger amount, and the second one is solving problems in context.

Let's get started with the first cycle.

In this lesson today, we've got both Jacob and Andeep who'll be joining us.

They'll be keen to share their thinking and any questions or solutions that they've come up with for themselves.

So, let's start our lesson here then.

Our problem is 1/3 multiplied by 249.

Hmm, how could we start this off? Jacob's gonna start us off by representing this as a bar model.

So, let's start by drawing the two equal bars to start us off with, and then let's place the whole in the top bar, 249.

We now need to divide the whole into a set number of equal parts, don't we? We know that we're gonna divide it into three equal parts, because that's the size of the denominator in the unit fraction that we're multiplying by.

Not only can we write it as 1/3 multiplied by 249, we can also write it as 249 divided by 3, can't we? Hmm, Jacob's looking at that and thinking, well, it's quite a large whole number.

I might need to use a short division strategy to help me solve this.

Let's have a look at how Jacob might go about doing that.

He's represented his equation as a short division and as always, with a short division, we start with the hundreds, don't we? So, we can see that we've got two hundreds at the moment, and Jacob's asking, can I make any groups of three hundreds? Well, we know we can't do that.

We don't have enough hundreds at the moment to make any groups of 300.

Therefore, we're gonna place a zero in the hundreds column at the top, and then we're going to need to regroup them, those two hundreds that we've got into tens.

So, now, we've got 24 tens, don't we? In the tens column.

So, how many groups of three tens can we now make with 24 tens? Well, Jacob says he knows that three tens multiplied by 8 would be equal to 24 tens, because he knows 3 times 8 is equal to 24.

So, three tens multiplied by eight would be equal to 24 tens.

Therefore, we can say we can make eight groups, can't we? And then, finally, looking at the ones.

We've got nine ones here, haven't we? And how many groups of three ones can we make from nine ones? Well, we know that three multiplied by three is equal to nine.

So, we can also describe that as three ones multiplied by three is equal to nine ones.

So, altogether, we could make three groups of three ones, couldn't we? There we go.

We found the total, it's 83.

So, we can say 1/3 of 249 is equal to 83, can't we? Ah, Andeep's got an idea though.

He's saying he doesn't think we needed to use short division for that strategy there, Jacob.

We could have done it mentally using a partitioning strategy.

Let's have a think about what that might have looked like.

Well, we know the whole is 249 and we can partition our whole into 24 tens and 9 ones, can't we? Now that we know that 249 is composed of 24 tens and 9 ones, we can divide both of these parts by three, can't we? We know that 24 tens divided by three would be equal to eight tens.

And we know that nine ones divided by three is equal to three ones.

So, we can now say that altogether, we've got eight tens and three ones.

And if we recombine that altogether, that's the same as saying 83, isn't it? I think you're right, Andeep.

I think that was a more efficient method to do that mentally, wasn't it? And write it down as a short division.

Great thinking, you.

Well done.

Okay, here's another example now then.

Let's have a look here.

We've got 356 multiplied by 1/4.

Should we ever think about how we could represent this again? You might like to start drawing this out for yourself as we go along.

Well, first of all, we draw the two equal bars, don't we? And then, we write the whole in the top bar, which is 356 this time.

Now, we look at the denominator of the unit fraction that we're multiplying by which is four.

So, we're gonna divide that into four equal parts, aren't we? And now that we know that 356 multiplied by 1/4 is the equivalent of saying 356 divided by 4, we can represent this bar model in two different ways.

So, to find out what 356 multiplied by 1/4 is, we can use our division equation to help us here.

Jacobs looked at the numbers again and he thought, oh, do you know what? I might need to use a short division strategy again here.

Yeah, nice thinking, Andeep.

He's encouraging Jacobs to have a think, can he do it without using a short division strategy? Let's see what we can do.

Jacob knows that 356 is 44 away from 400.

So, we can represent that in our bar model like this.

Jacob knows that 400 is 100 lots of 4, and he also knows that 44 is 11 lots of 4.

Hmm, this is interesting, isn't it? So, we can now work out how many lots of four make 356 by finding the difference between 100 lots of 4 and 11 lots of 4.

So, if we were to subtract 11 lots of 4 from 100 lots of 4, that would leave us with 89 lots of 4.

There we go.

So, we can say that the answer is 89, can't we? So, now that each of the parts have been divided into lots of 4, we can say that it's 89 lots of 4, which would be the product.

So, 89 is the product here.

What a great strategy, Jacob.

You were thinking really carefully about that there, weren't you? And great question for other people as well.

Can you think of your own strategy for how to tackle this? Okay, time for you to check your understanding now.

Can you tick the short division layout, which best represents this bar model? Take a moment to have a think.

That's right.

It's B, isn't it? We can see that the whole is 498 and the whole has been devided into six equal parts.

Therefore, we would represent it as B, wouldn't we? The dividend is 498 and the divisor in this case is 6.

And for the second one here, can you now calculate 525 multiplied by 1/5? Take a moment to have a think.

Brilliant.

You're right.

The answer was 105.

I wonder how you worked it out.

Let's have a look how Andeep did it.

Andeep used a partitioning strategy.

He said that he partitioned 525 into 50 tens and 25 ones.

Therefore, 50 tens divided by 5 is equal to 10 tens.

And 25 ones divided by 5 is equal to 5 ones.

And when you recombine all of that together, 10 tens plus 5 ones, that's equal to 105, isn't it? Well done if you managed to get that as well.

Okay, time for you to check your understanding now through some practise.

What I'd like to do is solve this problem in a number of different ways, and then decide which one is the most appropriate strategy to use.

For task two, I'd like to have a go at calculating using one of those efficient strategies to solve each of the following problems. Good luck with those two tasks, and I'll see you back here shortly.

Great, well done.

Let's have a look through them then.

I wonder what different strategies you came up with to solve this.

On the left-hand side, we've got a partitioning strategy where we've got 816 divided into 80 tens and 16 ones.

80 tens divided by 4 is equal to 20 tens, and 16 ones divided by 4 is equal to 4 ones.

20 tens and 4 ones altogether is the same as 24.

20 tens and 4 ones altogether is the same as saying 204.

We also use short division strategy.

And as you can see, we've worked through that and found the same quotient, 204.

Hmm, which one do you think was the most appropriate for this calculation? Yeah, I agree.

I think it was the partitioning strategy on the left-hand side.

Well done if you managed to come up with that for yourself as well.

Okay.

And then, to calculate these ones here, a bit of practise for you.

650 multiplied by 1/5, that's equal to 130.

897 multiplied by 1/3, that's equal to 299.

1/8 multiplied by 432, that's equal to 54.

And the last one, 1,075 multiplied by 1/25 is equal to 43.

I wonder what strategies you decided to use.

That's the end of cycle one now.

Onto cycle two, solving problems in different contexts.

Here's our first problem for today.

It says there are 36 sweets in a bag.

1/3 of the sweets are eaten.

Exactly how many sweets were eaten altogether then? Hmm.

Well, we know we can represent this as a multiplication equation, can't we? Where we multiply the unit fraction of 1/3 by the whole, which is 36 in this case.

We're gonna draw a bar model to help us represent this.

There we go.

And we know that 1/3 multiplied by 36 is the same as saying 36 divided by 3.

36 divided by 3 we know is equal to 12.

So, each part has a value of 12.

Therefore, we know that 1/3 of 36 would be one of those parts and one of those parts has a value of 12.

So, 12 sweets were eaten altogether.

Nice thinking, Andeep.

Here's a second problem to have a look at.

A relay race takes place over 1,500 metres.

Each athlete runs 1/4 of the distance before passing the baton to another member of the team.

How far does each athlete run? Take a moment to have a think.

How might you represent this as a multiplication equation? That's right, we can represent it as 1/4 multiplied by 1,500 metres, can't we? We know that represented as a bar model.

That means we would need to divide into four equal parts.

So, this is the same as saying 1,500 divided by 4.

And as a result of that, we can now calculate 1,500 divided by 4, and that is equal to 375.

I wonder what strategy you used to solve that.

So, we can say in total that each athlete runs 375 metres.

That's quite a long way to sprint, isn't it, in the race? I wonder who won.

And one more example here.

A kettle has a maximum capacity of 1.

7 litres.

The kettle is 1/2 full.

How much water is in the kettle? How do you think we'd write this as a multiplication equation? So, we could represent that as 1/2 multiplied by 1.

7 litres.

Or actually, as you can see here, we've made it into 1,700 millilitres, haven't we? We've converted it from litres into millilitres, which is exactly what Jacob was pointing out there.

It might be useful to convert it to help us work with a whole number more easily.

So, we know the whole is 1,700 millilitres and that whole needs to be divided into two equal parts, isn't it? Because the denominator of the fraction is two.

So, 1,700 divided by 2 is equal to 850.

So, we can say that there is 850 millilitres of water in the kettle altogether.

Well done if you managed to get that for yourself as well.

Okay, it's time for you to check your understanding now.

Can you write an equation that would represent this problem.

Andeep swims nine lengths of the pool, which in total would be 225 metres altogether.

What is the length of the pool? Take a moment to have a think.

That's right.

We could represent it as 1/9 multiplied by 225 metres.

We are trying to divide the total length that Andeep swam into nine equal parts and each one of those parts would then test the distance of one length that he swam.

Well done if you managed to write an equation for that as well.

Okay, onto our last task for today then.

What I'd like to do is solve the following worded problems. Once you've done that, I'd like to have a go at drawing a bar model to represent each of these problems, and then solve them as well.

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

Okay, welcome back.

Let's have a look at the first problem then.

Jacob's dad is using a power hose to clean the garden patio.

His patio has 40 paving slabs.

He has cleaned 1/5 of them so far.

How many has he cleaned? Okay.

Well, we can represent this as 1/5 multiplied by 40 or 1/5 of 40.

We know that's the same as saying 40 divided by 5, and that would be equal to eight.

For the second problem, the school raised 1,356 pounds at the school fair.

1/3 of the money that they raised will be put towards new playground equipment.

How much money can they spend on playground equipment? Well, we can represent this as 1/3 multiplied by 1,356.

Again, we know that's the same as saying 1,356 divided by 3, and that gives us a total of 452 pounds.

That's quite a lot of money to spend on outdoor playing equipment.

They should have a great time, shouldn't they? And then, C, Andeep's brother is a quarter of a century old.

His little sister is 1/5 of his brother's age.

How old is Andeep's sister? Well, how do we represent this then? We know that Andeep's sister is gonna be 1/5 of the age of Andeep's brother.

And if Andeep's brother is 1/4 of a century old, we know that a century is 100.

So, 1/4 of a century would be 100 divided into 4 equal parts, which gives us 25.

So, to find out how old Andeep's sister is, we need to find out what 1/5 of 25 is.

We know that 1/5 of 25 is the same as saying 25 divided by 5, which is equal to 5.

So, Andeep has got a little sister who's five years old.

Well done if you managed to get those three.

Okay, and for these ones here, they get to draw a bar model to represent it, and then solve each equation as well, didn't we? So, the first one says there are 18 chocolate chips on 1/3 of the cupcake.

How many chocolate chips are there on the whole cupcake altogether? Well, we can draw a bar model here.

We don't know the whole, do we? But we know what 1/3 of the whole is.

So, we can divide the whole into three equal parts, and then write in what 1/3 is.

We know that is 18.

So, as in the multiplication equation, we can write it as 1/3 multiplied by something or 1/3 of something is equal to 18.

We can rearrange this to a division equation, so it's something divided by 3 is equal to 18.

And we can now multiply the divisor and the quotient together to find the original dividend.

So, we can multiply 3 and 18 together, which would give us a total of 54.

So, there were 54 chocolate chips on the cupcakes altogether.

The second problem says there are 56 children in a year group.

14 children are planning to go to the same secondary school.

What fraction of the year group is this? Hmm.

So, let's draw a bar model again here then.

We know the whole, we know it's 56 children, and we know that one part of that is gonna be 14, don't we? Those 14 children represent the ones who want to go to the same secondary school.

But we don't know the size of this part, do we? In comparison to the whole.

Let's write it as a multiplication equation.

We can write it as one something multiplied by 56 is equal to 14.

We know we can write this as division equation, 56 divided by something is equal to 14.

So, 56 divided by something is equal to 14 is the same as saying 14 multiplied by something is equal to 56.

Hmm.

What multiplies a 14 to make 56? That's right, it's four.

So, we can say that 1/4 of the amount of children want to go to the same secondary school.

So, 14 is 1/4 times the size of 56 or it is 1/4 of 56.

Right.

That's the end of our lesson for today.

Let's summarise what we've learned.

So, we can say that you can apply your understanding of multiplying a unit fraction by a whole number using larger numbers.

It's important to consider the most appropriate strategy when multiplying a unit fraction by a whole number.

And you can also apply your understanding of multiplying a whole number by a unit fraction in a range of different contexts.

Thanks for joining me today.

Hopefully you enjoyed yourself.

I know I did.

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