Lesson video

Hi, thank you for joining me today.

My name is Mr. Peters, and this is the first lesson in our sequence, which begins to think about multiplying fractions by whole numbers.

In this lesson today, we're gonna be thinking about how we can develop our understanding and explain the relationship between the repeated addition of unit fractions and the multiplication of unit fractions.

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

So by the end of the assessment today, you should be able to say that I can explain the relationship between the repeated addition of unit fractions and the multiplication of unit fractions.

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

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

The first one is repeated edition, your turn.

The second one is unit fraction, your turn.

And the last one is represent, your turn.

So thinking about what these mean in a bit more detail, repeated addition is the process of adding the same equal group to itself multiple times.

A unit fraction is a fraction where the numerator is 1.

And to represent something is to show something in a different way.

Throughout our lesson today, we've got two cycles.

In our first cycle, we're gonna be thinking about linking repeated edition with multiplication, and in our second cycle we're going to be multiplying unit fractions.

Let's get going.

Throughout this lesson today, we'll be joined by Laura and Jacob.

They've got lots of thoughts and questions to share as we go through, so it'll be interesting to see what they have to contribute.

So our lesson starts with Laura and Jacob discussing what their parents were talking about before they came to school.

Laura was saying that her dad was always asking her to tidy up her shoes.

Hmm.

I wonder if you or anybody else gets asked the same at home.

I know I do sometimes.

Well, Jacob's asking, well, how many shoes do you have? Laura's saying, not many.

Have a look here.

So we can see we've got three pairs of two shoes here, haven't we? We can record this as 2 plus 2 plus 2.

Laura says she's got two shoes and another two shoes and another two shoes.

And we can record that as an expression of 2 plus 2 plus 2.

You may have said this in a slightly different way.

You may have said that we have three pairs of shoes.

We know that a pair represents two items, so we could record this as 3 multiplied by 2.

That's three lots of two shoes or a pair of shoes.

Or we could record it as two three times or a pair of shoes three times.

So, so far we found three ways to represent Laura's shoes as an equation, haven't we? We have a repeated addition expression, 2 plus 2 plus 2.

We have a multiplication expression of 2 multiplied by 3.

And we also have another multiplication expression of 3 multiplied by 2.

And all of these are equal to one another.

And it's important to remember that it doesn't matter which order the factors are placed in the multiplication equations, does it? We could record it as 2 multiplied by 3 or we could record it as 3 multiplied by 2.

What's important to know is that they represent the same thing.

Let's have a think about what the numbers represent.

Take a moment for yourself to have a think.

Well, we could say that the 2 represents the number of shoes in a pair, doesn't it? And the 3 represents the number of pairs of shoes.

You may have spotted how I've placed a unitized counter underneath the pair of shoes.

This counter represents the two shoes each time.

And as you can see, we've got three lots of two, or two three times.

And as we know, it doesn't matter the order in which we place those factors.

The two will always represent the number of shoes in the pair, and the three will always represent the number of pairs of shoes that there are.

Jacob was saying that he didn't get a chance to speak to his dad this morning 'cause he was already up and out going for a run, which he's doing for charity.

Jacob's saying that he raises more money for every 2 kilometres that he runs.

So that leads us to asking how many kilometres has he been running? Well, he usually runs 4 lots of 2 kilometres each day.

Let's have a look.

We could represent this as repeated addition, couldn't we? We could say that this is 2 kilometres plus 2 kilometres plus 2 kilometres plus another 2 kilometres.

Or we could write this as a multiplication expression, couldn't we? We could write this as 4 multiplied by 2 or 4 lots of 2 or that's right, we could record it as 2 multiplied by 4, two four times.

Again, we know that all of these represent the same thing, and it doesn't matter the order in which the factors are placed.

We can write it as 2 multiplied by 4 or 4 multiplied by 2 for the multiplication expressions.

Take a moment again to have a think.

What does the 2 represent and what does the 4 represent? That's right.

The 2 represents every set of 2 kilometres that he runs, and the 4 represents the number of sets of 2 kilometres that he ran altogether.

So we can see he ran 4 sets of 2 kilometres or 4 lots of 2 kilometres.

Again, we've used our counters to represent this.

So we can say that there are 4 lots of 2 kilometres, or Jacob's dad runs 2 kilometres 4 times.

Okay, time for you to check your understanding now.

Can you tick the expression that represents this image? Take a moment to have a think.

That's right.

It's A, B, and D, isn't it? We know that this image can represent 4 multiplied by 2 or 2 multiplied by 4.

It doesn't matter the order of the factors in a multiplication expression.

And we know also that multiplication expressions can be represented as repeated addition as it is down the bottom.

And we can see that's 2 plus 2 plus 2 plus 2.

Well done if you've got that.

Here's another one.

Now, which images match the expression? Take a moment to have a think.

That's right.

It's A and B, isn't it? Can you explain why? Well, A has 3 dice, doesn't it? So the 3 represents the 3 dice, and on each dice there are 4 dots being shown.

So that's 3 lots of 4 or four three times.

For B, there are three counters aren't there? So the 3 represents a number of counters and each counter has a value of 4.

So we have 3 lots of 4 or again, four three times.

Why wasn't it C? Yeah, that's right.

We don't have equal groups do we? We have 2 groups of 3 here and we have 1 group of 4.

So we don't have equal groups, so we can't record that by only using a multiplication symbol.

Okay, your first task for today, then what I'd like to do is write three expressions for each of these images.

I'd expect you'd probably write a repeated addition expression as well as two multiplication expressions.

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

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

Okay, welcome back.

Let's see how we got on then, shall we? So for the first one, we can see that we have 4 pairs of shoes, don't we? That's 4 lots of 2 or 2 lots of 4.

So we can record that as 2 plus 2 plus 2 plus 2.

We could record that as 2 multiplied by 4 or we could record that as 4 multiplied by 2.

For the second example, we've got 3 bundles of 10 straws, so we could record that as 10 plus 10 plus 10.

We could also record that as 3 multiplied by 10, or we could record that as 10 multiplied by 3.

And then finally, we've got 4 trays here, haven't we? And on each tray we've got 5 cupcakes, so we could record that again as 5 plus 5 plus 5 plus 5.

That's 5 cupcakes on each of the trays.

And we've got that 4 times.

We could also record it as 4 multiplied by 5, 4 trays with 5 cupcakes on each one.

Or we could record it as 5 multiplied by 4.

That's 5 cupcakes on each of the 4 trays.

Well done if you've got those.

Okay, and here's some examples that you might have come up with for question two.

For 3 multiplied by 5, you can see we've got 5 tricycles here.

Each tricycle has 3 wheels.

So the 3 represents the number of wheels that each tricycle has, and there are 5 tricycles altogether.

So as we know, we can record that as 3 multiplied by 5 or 5 multiplied by 3.

For B, We can see we've got two vases, haven't we? And in each vase we've got 3 flowers.

So the 3 represents a number of flowers in each vase, and the 2 represents a number of vases that we have.

We can record that as 3 multiplied by 2.

And then finally the last one, we've got 2 logs and we've got 2 frogs on both of those logs.

So we could record that as 2 multiplied by 2.

Hmm, which 2 represents each part of the image? Well, it doesn't exactly matter as long as you know which one you are saying represents what.

So let's say that the first 2 represents the number of logs and the second 2 is going to represent the number of frogs on each of those logs.

Well done if you come up with something similar to that for yourself.

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

Let's move on to cycle two now, multiplying unit fractions.

So we're gonna start here with this really delicious looking cake.

I like cake.

Do you like cake? What's your favourite? I think mine is probably a vanilla sponge with a strawberry and buttercream layer in the middle.

Might have to go make 'em one later on.

Anyway, back to this cake.

We can divide this cake into equal parts.

So have a look here.

We can now see that this cake has been divided into 9 equal parts, and each one of these parts is a unit fraction of the whole.

Let's use our stem sentence to describe this.

Have a go at saying it with me.

The whole has been divided into 9 equal parts.

Each part is 1/9 of the whole.

Well done if you've got those.

We're going to also represent each one of these parts as a unitized counter.

Each one of these counters represents 1/9 of the whole or 1 equal slice of the cake.

Laura's clearly hungry.

She's saying, what if we both had a slice of cake each, Jacob? That would mean we would eat 1/9 and another 1/9 of the cake.

We could record that as a repeated addition, couldn't we? We could say it's 1/9 plus 1/9.

Hmm.

We could also say that it's 2 lots of 1/9.

Here we go.

We can record it like this, can't we? Hmm.

Is there another way we could record it as well? That's right.

We could also record it as 1/9 two times, couldn't we? So we can write that as 1/9 multiplied by 2 as well.

As we know, all of these will be equal to one another, don't we? We know that we can represent multiplication expressions as repeated addition, and the multiplication expressions that we write, it doesn't matter the order of the factors in which they're placed in, and that's because we know multiplication is commutative.

So let's have a think about what these numbers and our expressions represent.

Well, we know that the 1/9 represents the size of each slice of cake, and we know that the 2 represents the number of slices of cake.

Let's look at a slightly different example here.

This time, our whole is the bar at the top and the bar has been divided into several equal parts.

Let's use our stem sentence.

Can you say it with me? The whole has been divided into 5 equal parts and each part is 1/5 of the whole.

Well done.

If you've got that, have a look now at our whole, Laura is saying that three of the parts have been shaded.

We can use the counters to represent each one of those parts that has been shaded as well.

So we can say as a repeated addition that we have 1/5 plus another 1/5 plus another 1/5.

Hmm.

How many lots of 1/5 is that? That's right.

That's 3 lots of 1/5, isn't it? So we could record it as 3 multiplied by 1/5, or as we know, we could record it as 1/5 multiplied by 3, 1/5 three times.

Take a moment again to have a think.

What does each one of those numbers represent in those expressions? Well, the 3 represents the number of parts that's been shaded, doesn't it? And the 1/5 represents the size of each one of those parts, doesn't it? Well done if you've got that.

Okay.

And here's one more example for us to look at.

The whole this time is the distance from Laura's home to the church.

Have a look at our number line, can you use our stem sentence to help us articulate how we could describe the distance between these two? That's right.

We could say that the whole has been divided into 7 equal parts and therefore each part is 1/7 of the whole.

So let's have a look now.

Let's see how far Laura travels along the way towards the church.

And there we go.

Laura's saying that she's walked 1/7 of the way 4 times.

We can record that as repeated addition, 1/7 plus 1/7 plus 1/7 plus 1/7.

Or we can record this as a multiplication expression, can't we? We know this is the same as saying 4 lots of 1/7, or it is equal to 1/7 4 times, as Laura said.

Right, time for you to check your understanding.

Now, tick the expressions that represent this image.

Take a moment to have a think.

That's right.

It's A and D, isn't it? The whole has been divided into 10 equal parts, and we can see that on the number line.

And we can see that there are 3 of those parts left of juice left available in the jug.

So we could record this as three lots of 1/10, 3 multiplied by 1/10.

Or we could record this as 1/10 three times, 1/10 multiplied by 3.

Okay, and another quick check.

Can you match the image to the expression? Let's see how you got on.

Well, we can say that 4 multiplied by 1/3 is the same as saying 4 lots of 1/3.

So that's the image at the bottom here.

This bar, isn't it? 4 multiplied by 1/4 represents the jug.

The number line on the jug has been divided into 4 equal parts and each one of those parts would represent 1/4.

And it's full, isn't it? So it's 4 1/4 full.

1/3 plus 1/3 plus 1/3 would be equal here to the watermelons in the box.

Each watermelon represents 1/3 of the whole box, doesn't it? And we have 3 lots of 1/3 there.

And finally 1/4 plus 1/4 plus 1/4 would be represented by the counters.

We've got 3 lots of 1/4 here, haven't we? And we know that 3 lots of 1/4 can be recorded as a repeated addition of 1/4 plus 1/4 plus 1/4.

Okay, and onto our final tasks for today, then what we'd like you to do again is write three different expressions for each one of these images here.

For task two, I'd like you to draw counters to represent each of these expressions.

And then for task three, we've got a true or false question.

If one of them is false, what I'd like to do is write the correct equation next to it.

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

Okay, welcome back.

Let's see how you got on then.

Well, for the first one here, we could record it as 1/12 plus 1/12 plus 1/12 plus 1/12.

Or we could write it as 4 multiplied by 1/12.

We could also write it as 1/12 multiplied by 4 or 4 multiplied by 1/12.

For the second one, we can see that the bus has travelled 3/5 of the way to school, isn't it? So we could record that as 1/5 plus 1/5 plus 1/5.

We could say it's 1/5 multiplied by 3 or it's 3 multiplied by 1/5.

Well done if you've got those.

Okay, for task two, let's see the counters that we could have used.

We need 2 lots of 1/12.

So there we go.

We've got 2 counters and both of those have a value of 1/12.

For B, we needed 5 counters with a value of 1/3.

And for C, we needed 3 counters with a value of 1/10.

And then finally, the true or false questions.

We know that the first one was true, the second one was also true.

However, the third one was false.

How did you go about writing the correct equation for the third one? That's right.

The reason why it was false is because it says 1/2 multiplied by 3 is equal to 1/3 plus 1/3.

Well, actually we've got 1/2 3 times, so our equation should say that 1/3 multiplied by 3 would be equal to 1/2 plus 1/2 plus 1/2.

Or you may have decided to correct it the other way around by saying that 1/3 plus 1/3 is equal to 2 lots of 1/3.

So you may have written your equation like that instead.

Well done if you've got either of those.

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

And hopefully you're feeling a lot more confident with thinking about how you can explain the relationship between the repeated addition of unit fractions and the multiplication of unit fractions.

To summarise our learning, then we can say that repeated addition can be represented as multiplication.

We can also represent repeated addition of a unit fraction as multiplication of a unit fraction by an integer.

And we know that we can reorder the factors in any multiplication equation and it'll still represent the same thing.

The meaning that we attach to each of the numbers in the multiplication equation would stay the same no matter what order they're placed in.

Thanks again for joining me.

I really enjoyed that lesson.

Hopefully you did too.

Take care and I'll see you again soon.