Lesson video

Hi, welcome to today's lesson.

My name's Mr. Peters and in this lesson today, we're gonna continue to extend our understanding of multiplying fractions by whole numbers.

And this time we're gonna have a particular focus on where the product is greater than one whole.

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

So, by the end of this session today, you should be able to say that I can multiply a proper fraction by a whole number where the product is greater than one whole.

Throughout this session today, we've got five key words we're gonna be referring to throughout.

I'll say them first and then you can repeat them after me.

Are you ready? The first one is represent, your term.

The second one is unitise, your term.

The third one is mixed number, your term.

The fourth one is numerator, your term.

And the last one is denominator, your term.

So to represent something means to show something in a different way.

Unitising means treating groups that contain the same number of things as units or ones.

A whole number and a fraction can be combined into one known as a mixed number.

A denominator is the bottom number in a fraction.

It tells us how many equal parts a whole has been divided into.

And the numerator is the top number in a fraction.

It tells us how many parts we have.

Today's lesson is gonna be broken down into two cycles.

The first cycle will be representing calculations, and the second cycle will be unitising to support calculation.

If you're ready, let's get started with the first cycle.

Throughout our lesson today, both Alex and Laura will be joining us offering their thoughts and any questions that they have as we go throughout the lesson.

So let's start here then.

Alex is saying that his mom asks him every day after school how much water he's managed to drink at school.

Here you can see his water bottle and we split his water bottle into the bottle that he has for the morning and the bottle that he has for the afternoon.

Laura makes a good point.

She's only caring for you, Alex, and she wants to make sure that you drink enough so you don't get yourself a headache and feel poorly.

So Alex says that in the morning he drunk 3/4s of his water bottle, and then in the afternoon he emptied his water bottle and refilled it and drunk another 3/4s of his water bottle.

So I wonder how much he drunk altogether then.

Oh, Laura says, "You could say that you drunk two lots of 3/4s of a bottle." We could record that as an equation, two multiplied by 3/4s, or as we know, we could record it as 3/4s, two times.

That's right, Alex.

So you could say that you drunk 6/4s of a bottle of water.

But hang on, saying that doesn't sound quite right, does it? Is it possible to drink 6/4s of a bottle of water when a bottle only consists of 4/4s? It doesn't feel quite right, does it? Alex says it might be better if we converted it to a mixed number.

We can say that 6/4s is equivalent to one whole and two additional 1/4s, isn't it? Have a look at our water bottles here.

In both of the water bottles.

He had 1/4 left remaining, didn't he? And that's now represented on the right hand side.

So we can say that two multiplied by 3/4s is not only equal to 6/4s, it is also equal to one and 2/4s.

Let's say that together at the bottom, shall we? Two lots of 3/4s is equal to 6/4s or one and 2/4s.

Let's have a look at that on the number line.

Here you can see we've got a number line and it starts at zero and goes along to three.

So the whole is between zero and one, or one and two, or two and three.

Let's look between zero and one.

One whole has been divided into four equal parts.

So each one of those intervals would represent 1/4.

And if we did the same for between one and two and two and three, then each of the intervals on our number line actually represent 1/4, don't they? We have two lots of 3/4s, so let's represent that here then.

We can have one jump of 3/4s and we can have another jump of 3/4s, two lots of 3/4s.

Hopefully you can see that takes us to 6/4s, which is gonna be the same as one and 2/4s or one and a 1/2 bottles of water.

Again, let's use our sentences at the bottom to help us articulate that.

Two lots of 3/4s is equal to 6/4s or one and 2/4s.

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

Laura's talking about the pizza that her and her brother had for tea last night.

Here you can see on the screen Laura's pizza and her brother's pizza.

Laura's saying that they both only managed to eat five out of the eight slices that the pizza was divided into.

Let's have a look at that in a bit more detail.

There we go.

Hopefully you can see that Laura's pizza has now been divided into eight equal parts, and the blue is representing the five parts that Laura managed to eat.

Let's do the same for her brother.

Her brother also had a pizza and that was divided into eight equal parts and five of those parts were eaten by Laura's brother as well.

So I wonder how much we could say was eaten altogether.

Take a moment to have a think.

That's right Alex, we could say that two lots of 5/8s were eaten altogether, weren't they? And as we know we could record that as two multiplied by 5/8s or 5/8s two times.

So let's look at the representation more carefully then.

How many slices of pizza were eaten altogether then? That's right.

There were 10 slices of pizza altogether.

So we can say that 10/8s of the pizza were eaten altogether.

Let's use our stem sentence at the bottom to help us with that.

Two lots of 5/8s is equal to 10/8s, isn't it? Brilliant, well done if you've got that.

Again, typically we wouldn't say to somebody that we've eaten 10/8s of a pizza, we might describe that in a slightly different way.

So I think we might use mixed numbers to help us describe this.

We know that 10/8s would be equal to one whole and two additional 1/8s.

Have a look at our pizzas, watch what happens.

There we go.

We can see that three sections from the right hand pizza have moved to the left hand pizza to create one whole and they've swapped over.

So those three pizzas that were left from Laura's pizza have been moved over to Laura's brother's pizza.

We can now see that one whole and two slices were eaten altogether, leaving us with 6/8s of a pizza left to be eaten.

So let's use our stem sentence again underneath to help us here.

We can say that two lots of 5/8s is equal to 10/8s or one whole and 2/8s.

There is one whole and 2/8s of a pizza leftover.

And that's right Laura, you've got some leftover for your lunch or tea today, haven't you? As we know, multiplying a fraction by a whole number can also be represented on a number line.

Let's show that here then.

Again, our number line starts at zero and goes all the way to two this time.

The one, one whole is between zero and one or one and two, and that whole has been divided into eight equal parts, which would mean all eight parts in between the wholes would be worth 1/8.

So we're gonna represent that here like so.

Now that each interval represents 1/8, we can show that we have two lots of 5/8s.

You ready? Here's one lot of 5/8s and here is now two lots of 5/8s.

We know that as equal to 10/8s, or as we've already calculated, we know that as equal to one whole and two additional 1/8s.

Let's use our stem sentence again at the bottom to help us articulate this.

Two lots of 5/8s are equal to 10/8s or one whole and 2/8s.

Well done if you've got that.

Right time for you to check your understanding now.

Can you convert 5/3s into a mixed number? Take a moment to have a think.

That's right, it's a, isn't it? 5/3s is the equivalent of one whole and two additional 1/3s.

We know that one whole consists of 3/3s and we need another two additional 1/3s to make 5/3s.

Okay, and another quick check for understanding.

Can you write an equation to represent this image and give your answer as a mixed number? Take a moment to have a think again.

Okay, well done.

We can say that there are two lots of 4/5s here.

So two multiplied by 4/5s.

We know that would be equal to 8/5s or as a mixed number would be one whole and 3/5s.

Well done if you've got that one.

Okay, time for you to practise now.

What I'd like to do is write an equation for each of these representations, and then ask yourself what did you notice about the representations and the equations that you wrote? And then for task two, what I'd like to do is write a story to represent each one of these equations.

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

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

So for the first one, we can say that it's two lots of 6/7s.

Two multiply by 6/7s and that is equal to one whole and 5/7s.

The second one is two lots of 4/5s.

So that would be equal to one whole and 3/5s.

For the third one, we could say it is two multiplied by 5/6, and that would be equal to one whole and 4/6.

And finally for the last one, we've got three lots of 3/4s.

So we could represent that as 3/4s multiplied by three and that would be equal to two wholes and 1/4.

Well done if you've got all of those, did you notice anything in particular? That's right, each one of the wholes we had, had a full number of parts except for one part in each example, weren't they? So actually all we needed to do was add the number of wholes that we had together and subtract one part from each one of those wholes to find out the total amount that we had.

So as Laura's saying, she could have just subtracted the number of missing parts from the number of wholes that there were.

Well done Laura for thinking of another strategy to solve these other than just using the multiplication that we've been working on so far.

Okay, and then to task two then.

I wonder if we were able to write a story to represent these ones here.

The first one shows that one and 4/6 is equal to two multiplied by 5/6.

Here's Laura's example.

She says that she cycled 5/6 of a mile on Monday and cycled another 5/6 of a mile on Tuesday.

And altogether she cycled one whole and an additional 4/6 of a mile over those two days.

Nice example, Laura.

And here's Alex's example for the second question.

On Monday morning I had three lessons that all last 45 minutes.

Altogether I had two and 1/4 hours of lessons in that morning.

Another great example of thinking about how we could use time to represent these fractions.

Well done, Alex.

Okay, that's the end of cycle one.

Moving on to cycle two now then.

Unitising to support calculation.

So here the example says, each lunch time, pupils get 2/3s of an hour to play on the playground.

This happens every day during the school week.

How much time do they get to play on the playground over the course of the whole week? Alex is thinking, "How could we represent this?" Laura says, "We could draw circles and divide them all into three equal parts and shade in two of them each time." Laura saying she struggles to draw circles and divide them all up equally as you can see here.

It's not an easy thing to do without a compass.

Ah, so Alex has come up with the idea.

We could use a unitised encounter to represent each lunchtime, couldn't we? Let's have a look at that.

We know that each lunchtime the children get 2/3s of the time on the playground.

So instead of using a circle to represent it, split into three parts, we could use a counter and here each counter has a value of 2/3s.

We have unitised the amount of time that the children get to spend on the playground, and now our unit is 2/3s.

So each counter represents 2/3s of an hour.

There we go.

We could represent this as five lots of 2/3s.

Or we could represent it as 2/3s five times.

We could count up in 1/3s to help us identify this.

Are you ready? Maybe you could count with me.

2/3s, 4/3s, 6/3s, 8/3s, and 10/3s.

That's five lots of 2/3s.

So we can say that five lots of 2/3s is equal to 10/3s.

But again, 10/3s of an hour doesn't sound right, does it? That's not how we would generally talk to people about how much time we spend doing things, would we? So it might be useful again to think about how we can convert this into a mixed number to talk about the total number of hours that we spend on the playground.

10/3s is equivalent to three whole hours and an additional 1/3 of an hour.

We know that 3/3s make up one hour, 6/3s would make up two hours, 9/3s would make up three hours, but we've got 10/3s, so that's an additional 1/3 of an hour.

So 10/3s is equal to three whole hours and an additional 1/3 of an hour.

So each week on the playground the children get three hours and an third of an hour to play with their friends.

That sounds like they've got enough time to have loads of fun on the playground together.

And let's have a look at one more example here.

Laura's saying that she uses 3/8s of a pint of milk on her cereal every morning.

How much milk does she need to buy for the week ahead? Well, there are seven days in the week, so we're going to need seven counters, aren't we? And on each one of those days, she drinks 3/8s of a pint of milk.

So each counter represents 3/8s of a pint of milk, and we have seven of these counters.

So we can record this as seven multiplied by 3/8s or 3/8s seven times.

Let's have a look at this on a number line once more.

Again, if you'd like to join me, let's count up in lots of 3/8s.

Are you ready? 3/8s, 6/8s, 9/8s, 12/8s, 15/8s, 18/8s, and 21/8s.

So we could say that Laura drinks 21/8s of a pint of milk.

Again, as we know, it doesn't quite sound the right way.

We would tell somebody how much milk we need to get for the week, does it? So we'd be better off converting it to a mixed number again to find out the full amount of pints we'd need each time.

If you look at the number line carefully, you can see that we go past two, don't we? And we've got an additional 5/8s.

So altogether we can say that Laura would need two pints of milk and an additional 5/8s of a pint of milk.

It might be best to buy three pints of milk, Laura and then you've got a little bit left over if you need a bit more one day.

Well done if you manage to realise that for yourselves as well.

Okay, so let's have a look at our two calculations we've just worked through.

Take a moment to ask yourself, what do you notice? Well, that's right.

Once again, the denominator stays the same throughout, doesn't it? The denominator in our proper faction, which multiplies with our whole number, is equal to the denominator in the product.

And this case, our product is an improper fraction, isn't it here? So that applies for both of our examples.

And in order to work out the numerator of the product, which is an improper fraction, we can multiply the whole number by the numerator of the non-unit fraction.

We've got five multiplied by 2/3s.

Well, we know that five multiplied by two is equal to 10, so five multiplied by 2/3s is equal to 10/3s.

And again, you can see how we've represented that here.

Let's look at the example underneath.

We've got seven multiplied by 3/8s.

We know that seven multiplied by three is 21, so seven multiplied by 3/8s is equal to 21/8s.

Yap, and that's exactly right, Alex.

As we've been doing all the way throughout this lesson, we've been converting the products that we've created when they're an improper fraction into a mixed number, as it often helps us makes more sense of the context that we've been looking at.

So 10/3s is equivalent to three wholes and 1/3 and 21/8s is equivalent to two wholes and 5/8s.

Right, time for you to check your understanding again now then.

Can you tick the expressions that represent this image? Take a moment to have a think.

That's right.

It could have been a, c or d.

Our image shows four lots of five/9s.

We could record that as 5/9s multiplied by four.

We could record that as four multiplied by 5/9s, or we could record it as something is equal to four multiplied by 5/9s.

Well done if you've got those three.

Another quick check.

Can you write the product for this equation here? You need to write it as an improper fraction first and then write it as a mix number too.

Take a moment to have a think.

That's right, as an improper fraction, it would be 20/9s, wouldn't it? The denominator stays the same and we multiply the whole number by the numerator.

Four multiplied by five is equal to 20 and 20/9s is equivalent to two wholes and two additional 1/9s.

Well done if you've got both of those.

Okay, time for us to practise now.

For this first task here, what I'd like to do is write an equation for each representation shown here and then find the product for these as well.

And then for this task here what I'd like to do is using the numbers not to nine only once.

How many different ways can you solve this problem here? Now there is a number five in here that's been used once already and it is okay for you to repeat that number five once more.

Good luck with those two tasks, I'll see you again shortly.

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

The first one is two lots of 4/7s, so we can record that as two multiplied by 4/7s.

As an improper fraction that'll be equal to 8/7s or as a mix number, that would be equal to one whole and 1/7th.

The second one is four lots of 5/6.

Again, as an improper fraction, that would be equal to 26 And as a mix number, that would be equal to three Wholes and 2/6s.

And then find the last one.

We've got six lots of 4/5s, as an improper fraction that would be equal to 24/5s and as a mix number that would be equal to four wholes and 4/5s.

Well done if you've got all of those.

And then here's an example of how you may have tackled this problem here.

We've decided to represent it as four multiplied by 2/5s, and that would be equal to one whole and 3/5s.

That was just one solution.

And as we noticed, the denominators had to stay the same throughout, didn't they? I wonder if you managed to find any other ways.

If you did, share them with someone nearby to you and see how many ways they might come up with as well.

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

Well done for taking part.

Let's summarise what we've learned.

We know that when multiplying a proper fraction by a whole number, the denominators stay the same throughout.

You can multiply the numerator of the proper fraction by the whole number to find the numerator of the product.

And it's often helpful to convert an improper fraction into a mixed number to help understand the context of the product more clearly.

Well done for today.

Thanks for joining me.

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