Lesson video

Hi, my name is Mr. Peters and welcome to today's lesson.

In this lesson we're gonna be thinking about how we can extend our understanding of the repeated edition of fractions and how this relates to multiplication of fractions.

If 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 explain the relationship between the repeated addition of fractions and the multiplication of fractions Throughout the session today, we've got four 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.

Are you ready? The first one is unit fraction.

Your turn.

The second one is unitise.

Your turn.

The third one is non-unit fraction.

Your turn.

And finally the last one is represent.

Your turn.

So let's have a little think about what these mean in a bit more detail.

A unit fraction we can say is a fraction which has a numerator that is one.

Unitising means treating groups that have the same number of objects or things in them as units or ones.

A one-unit fraction is a fraction that has a numerator that is greater than one.

And finally, to represent something, it means to show something in a different way.

Throughout this lesson, we've got two cycles.

The first cycle, we'll think about grouping unit fractions, and the second cycle we think about unitising to multiply.

Let's get started.

Throughout this lesson today, you're gonna meet both Jun and Sophia, and as always, they'll be sharing some of their thinking and any questions that they've got throughout the lesson as we go.

So our lesson starts here, have a look.

We've got an image of a cake, haven't we? And this cake represents our whole.

We can divide this cake into equal parts.

Let's have a look.

There we go.

It is quite a big cake, isn't it? And that means we can divide it into a number of equal parts here.

Let's use our stem sentence below to help us say how many equal parts we have and what each of those parts represents.

Are you ready? The whole has been divided into nine equal parts.

Each part is one ninth of the whole.

Well done if you managed to say that too.

So we can record multiple unit fractions as either repeated edition or multiplication.

Let's have a look how we'd record it as repeated edition.

Here we go.

Look at this.

It says one ninth plus one ninth plus one ninth plus one ninth plus one ninth.

And it carries on going until we've got nine lots of one ninth.

That's a lot of one ninth, isn't it? And it would take a long time to write that down so it's more efficient to write this as a multiplication.

Let's have a think about what that would look like.

We know that we have nine lots of one ninth, so we can record that as nine multiplied by one ninth.

Or we could say that we have one ninth nine times, so we could record it as one ninth multiplied by nine.

Take a moment to think.

What do the numbers represent in these expressions? That's right.

We know that the nine represents a number of slices that we have altogether that made the whole of the cake.

And the one ninth represents the size of each one of those slices.

You may have noticed that we've written the factors of those expressions either way around.

In the first expression, we've got nine multiplied by one ninth, and in the second expression we've got one ninth multiplied by nine.

We know that these represent exactly the same thing.

What's important to know is what the numbers themselves represent.

And as we said, the nine represents the number of slices of cake, and the one ninth represents the size of each one of those slices of cake.

Have a look at our image now, what'd you notice this time? That's right.

Our one ninths have been placed, haven't they? Onto three trays.

How many one ninths are on each tray? Yep, that's right.

We've got three one ninths on each tray, haven't we? So we could record that like this, couldn't we? Using our multiplication expressions, we could say that we have three lots of one ninth, so we could write that as three multiplied by one ninth.

And we have that three times.

On the left hand tray, we've got three slices of cake.

So that's three multiplier by one ninth, because each slice of cake has a size of one ninth, we have the same repeated and the middle tray, and we have the same repeated on the right hand side tray.

So we could say that each tray has three ninths on it.

Three ninths is a non-unit fraction.

A non-unit fraction is a fraction which has a numerator greater than one.

And we can see that here in our repeated edition expression.

We've now got three ninths plus three ninths, plus three ninths.

As we know, we can record repeated addition as multiplication as well.

Take a moment to have a think.

How might we write this as an expression? That's right.

We write this as three multiplied by three ninths.

We have three lots of three ninths.

Or we could write it as three ninths three times.

So we could write it as three ninths multiplied by three.

Let's have a think again now then what do each of these numbers in our expressions represent this time? That's right.

The three represents the number of trays this time, doesn't it? And the three ninths represents the value of the group this time on each tray or the combined value of the slices on each individual tray.

And as we already know, we know that multiplication is commutative.

So we can reorder the factors in these expressions.

And they mean the same thing, don't they? Hmm? Jun's clearly gotten hungry.

He says he's gonna eat one of the slices now.

Watch carefully.

Now there are only eight ninths left aren't there? At the moment, our groups are no longer equal.

I wonder if we could rearrange this into equal groups.

What do you think that might look like? That's right.

We can now rearrange it into two groups.

And we've got four slices of cake on each of those trays, haven't we? Hmm.

Again, take a moment.

How would you write this as an expression now? Well, that's right.

Each tray has four lots of one ninth on it, doesn't it? So we can say this is four multiplied by one ninth.

And because each tray has four lots of one ninth, we now know that that represents four ninths and one-unit fraction.

So we can record this as four ninths plus four ninths.

There are four ninths on the first tray plus another four ninths on the second tray.

And as we've been looking at so far today, we know we can record repeated addition as multiplication as well.

What do you think that might look like? That's right.

We've got two lots of four ninths, haven't we? So we can record it as two multiplied by four ninths, or we could record it as four ninths multiplied by two.

Let's go back to what these numbers represent again, the two in each of the expressions represents the number of trays, isn't it? And again, the four ninths represents the combined value of the slices of cake on each tray.

We know the combined value is four ninths or four lots of one ninth.

And we know because we're using multiplication, we can rearrange the order of the factors and the expressions will mean exactly the same thing.

Okay, here's a slightly different context now.

Jun's dad is training for a triathlon.

During one of his training sessions he divides his time up like this, for two sixth of the time of his training he goes for a run.

For another two sixth of the time of his training, he goes for a swim.

And for the final two six of his training, he gets on his bike.

So we can see that represented here on our number line like this.

The first two sixth represents the amount of time he runs for.

The second two sixths represents the amount of time he swims for.

And the third two sixth represents the number of time that he cycles for.

Sophia's saying that the amount of time each exercise takes throughout the training session lasts for two one sixths.

And we know that we can record two one sixth as a non-unit fraction of two sixth.

So we can say that we've got two sixth plus two sixth plus two six, haven't we? This expression would represent total amount of time spent on all of those exercises.

Looking at our addition expression, we can see that it's repeated addition.

It's the same value being added to itself several times.

So we can record this as multiplication.

We know that this is the same as saying three multiplied by two sixth or two sixth multiplied by three.

That's right, Jun.

I think he is training really hard.

It takes a lot of effort and mental strength to do a triathlon.

So well done to him for giving it a go.

Let's remind ourselves and her think about what these numbers represent in these expressions.

Well, the three represents the different exercises as Jun's dad does.

He goes running, swimming and cycling.

So that's three different exercises and that's represented by the three.

And the two six represents the value of each group or the amount of time spent on each of the exercises throughout his training session.

As we've already mentioned before, we know that multiplication is commutative so we can write these expressions either way.

So we can write the factors in these expressions either way around.

Okay, time for you to check your understanding now.

Which expressions match this image correctly? Take a moment to have a think.

That's right.

It's A and D, isn't it? We can see that we have three lots of two sixth in our bar model.

One group is the purple section, one group is the green section, and another group is the blue section.

Each one of those groups represents two sixth 'cause the whole is divided into six equal parts and each of them have two that are shaded.

So we can say there are three lots of two sixths or two sixth, three times.

Okay, and then have a quick check.

What value of the Pentagon is orange? That's right, it's A and B, isn't it? We can see that the whole has been divided to 10 equal parts and two of those parts are shaded orange.

We can say that as two tenths, or we could record that as one 10th multiplied by two, one 10th two times.

We could also say it's D, couldn't we? Why would it be D as well? That's right.

D is actually equal to both A and B, isn't it? And we know that we can rotate the order of the factors, can't we? So this time, instead of saying one 10th, two times, we've got two lots of one 10th.

Okay, time for you to have a go at some practise now for me.

What I'd like to do is write three expressions for each of the following images.

And then once you've done that, what I'd like you to do here is draw an image to represent each of these expressions.

When you do that, have a little think is there anything that you noticed when you had a go at drawing those images.

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

Okay, let's have a look then.

First image here represents that each hole has been divided into four equal parts and two of them are shaded in each of the holes.

So we can say this is two quarters plus two quarters plus two quarters.

We could also say that this is three lots of two quarters or three multiplied by two quarters or two quarters multiplied by three, two quarters, three times.

For question B, there are two wholes here.

And in each of the wholes there are seven watermelons out of a possible eight watermelons.

So we could record that as seven eighths plus seven eighths.

We know we can record this repeated edition as two lots of seven eighths or seven eighths two times.

And then finally for the last one, we've got a vertical number line here.

Our whole this time has also been divided into eight equal parts.

However, each group each jump on the number line represents two eighths.

So we've got two eighths plus two eighths plus two eighths, plus two eighths.

And that would be equal to four lots of two eighths or two eighths four times.

Well done.

If you've got all of those.

Let's have a look at the images we could have drawn potentially for these ones here, we're gonna use hands to represent this here.

So here we can see we've got three lots of two fifths.

Each hand has five fingers, and we're showing two of them to you here on each hand.

So that's two outta five fingers on each hand.

And we've got that three times.

So that would represent three multiplied by two fifths.

This time we need three hands.

And on each hand we need to show three fingers, three outta the five fingers.

So that's three multiplied by three fifths.

And for the last one, we need four hands, don't we? And again, we need three fingers showing on each hand.

So you've got four multiplied by three fifths.

'cause that's three outta the five fingers showing.

Did you notice anything when you did this? Jun said that for the second example when moving on from the first example, he had to increase the value of each group, whereas moving from B to C, Sophia noticed that she had to change the number of groups and the size of the groups stayed the same this time.

Well done if you manage to notice that as well.

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

Moving on to our second cycle now, Unitising to multiply.

Let's revisit our example with the cake slices.

Here you can see we've got, again, three trays.

And on each of the trays we've got three slices.

And each one of those slices represents one ninth of the cake.

So we could record that as three lots of one ninth on each of our trays.

We know that multiplying a unit fraction by a whole number can result in a non-unit fraction.

So we can record this as three ninths.

Each tray has a value of three ninths, and we've created a unitised counter to show this.

The counter shows the value of the group on each of the trays.

Each tray has a value of three ninths.

We call this unitising.

Previously the value we were working with were one ninths.

We were looking at having three lots of one ninth, or the whole had nine lots of one ninth.

Now the value of the unit has changed.

We're not working in one ninths anymore.

We're now gonna be working in three ninths.

So we can say that we have three lots of three ninths.

The unitised value is a non unit fraction of three ninths.

So we can now say that we've got three lots of three ninths or three ninths three times.

Here's a different example.

Have a look here.

What do you notice about our eggs? That's right.

There are six eggs in the whole box.

So each egg represents one sixth of the box, isn't it? And you may have noticed that they've been split into different colours.

We've got two pale brown, two white ones, and two darker brown eggs, haven't we? Because each one represents one sixth, we know that we can group the eggs by their colours and say that we have two lots of one sixth of each colour.

We know that two lots of one sixth is same as saying two sixth.

So let's unitise this expression now, and we can now represent each set of two one sixths as a counter, which represents two sixth.

Each group of the same colour of eggs has a value of two sixth.

And we have three lots of those two sixth, don't we? So whilst our unit is now two sixth, we can record this as multiplication expressions.

We have three lots of two sixth, or we have two sixth three times.

One more example.

What'd you notice this time? That's right.

We've got a different colour of eggs this time, haven't we? And each colour of egg has one additional egg in comparison to the last example.

I wonder what the unitised value of each of the groups would be.

Well, we know there are 12 eggs in the pack altogether, and each egg has a value of therefore one twelfth, and there are three of each colour aren't there? So we could say that it is three multiplied by one one twelfth, couldn't we for each colour? And three lots of one twelfth is the same as saying three twelfths.

So the unitised value for each of those colour of eggs is now three twelfths.

How many lots of three twelfths do we have? Yep, we've got four lots of three twelfths, haven't we? So we can record this as four multiplied by three twelfths, four lots of three twelfths, or we could write it as three twelfths four times.

Well done if you managed to get that as well.

Okay, time for you to check your understanding now.

Have a look at the examples and focus closely on the green spotted cupcakes.

Can you match each of the images to the correct expression? Take a moment to have a think.

Okay, well the first one matches the image at the bottom.

This represents two multiplied by three fifths.

There are two trays aren't there, and three outta the five cupcakes on each of the trays are green spotted ones.

So we can say that's two multiplied by three fifths.

For the second example, we have three trays, don't we? So that's what the three represents.

And two out of the five cupcakes on each of the trays are the green spotted ones.

So therefore that represents two fifths or two fifths multiplied by three.

And for the last one, we have five trays, don't we? And on each of those trays, there are only three cupcakes.

And two of those are the green spotted type, aren't they? So actually we've got five multiplied by two thirds this time.

Two thirds of the amount of cupcakes on each of the tray are green spotted ones.

Well done if you've got that.

And time for you to have your final practise for today then.

What I'd like to do is draw an image and write an expression to represent both of these counters and what they show you.

And then once you've done that, what I'd like you to do is have a look at this image here.

And what I'm gonna ask you to do is think about how many ways you can represent this image here as a whole number multiplied by a fraction.

Good luck with those and when you're ready, come back and we'll go through 'em together.

See you shortly.

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

Here's some examples you might have come up with.

On the left hand side, we had two lots of two sixth, didn't we? So I've drawn two wholes here represented by the circles.

And in each of those wholes I've got some smaller dots, haven't I? And two of those dots are shaded out of the six.

So they represent the two sixth.

And because we've got two groups, we can write this as two multiplied by two sixth or two six multiplied by two.

In the next example, I've made three different wholes here, and each whole is divided into four equal parts and three of those are shaded.

And that's what each of those counters would represent.

Three quarters of the hole have been shaded green.

So again, we can record this as three multiplied by three quarters or three quarters multiplied by three.

Okay, and for the second task then I wonder how you got on.

One way that Jun decided to divide this up was to say that we're gonna divide it into four groups and each group would have a value then of three twelves because there's 12 equal parts altogether in the whole.

Each one has a value of one twelfth, and in each group there are three of them.

So we've got four multiplied by three twelfths, or you may have decided to go for three multiplied by four twelfths, or did you notice we did there? That's right.

We changed the groupings.

Instead of working vertically, we went horizontally.

And now there are three groups, which is what the three represents, and in each one of those groups there are four twelfths.

I wonder if you came out of anything different for yourself, maybe share it with someone near to you to check that it's correct if you did.

Well done if you managed to find lots of different ways of doing that.

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

Let's have a quick think about how we can summarise what we've learned today.

We know that multiple unit fractions with the same value can be unitised to create a new unit.

That new unit can be called a non-unit fraction.

We know that multiple non-unit fractions can be represented using repeated edition or multiplication.

And finally, when representing multiple non-unit fractions with the same value as multiplication, we can write the factors either way around, can't we? If we can put them in either order and it'll still represent the same equation.

Thanks for joining me today.

Hopefully again, you're feeling a lot more confident as we continue to develop our understanding of multiplying whole numbers by fractions.

Take care and I'll see you again soon.