Lesson video

Hi, 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 when a calculation represents either scaling down or repeated addition.

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

So by end of this lesson today, you should be able to say that I can explain when a calculation represents scaling down or when a calculation represents repeated addition.

Throughout this lesson today, we've got four keywords 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 scaling.

Your turn.

The second one is repeated addition.

Your turn.

The third one is numerator.

Your turn.

And the last one is denominator.

Your turn.

Let's have a think about what these mean then.

Scaling is when something is transformed through either shrinking or enlarging by a scale factor.

Repeated addition is when the same object or value is added to itself more than once.

The numerator is the number on the top of a fraction.

It tells us how many parts we have, and the denominator is the number on the bottom part of the fraction.

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

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

The first cycle, think about multiplying non-unit fractions and whole numbers.

And the second cycle, think about choosing appropriate strategies.

Let's get started with the first cycle.

In this lesson today, we've got Aisha, Andeep, and Laura.

They'll be sharing their thinking and any questions that they've got throughout the lesson as we go along.

Okay, so let's start our lesson here today then.

How would you represent this? You might like to grab a piece of paper and a pencil and have a go at drawing something to represent this equation.

Okay, let's have a think about this in a bit more detail then.

Do you think the product of this equation would result in a smaller or a larger number than the whole number? That's right, it would give us a smaller number than the whole number, wouldn't it? We know from previous learning that multiplying the whole number by a non-unit fraction or a unit fraction will make the whole number smaller.

Let's have a look at how Aisha represented it here.

We can see that she's drawn a bar model with the whole representing six, and she's devised the whole into three equal parts and she's shaded in two of those parts.

This represents 2/3 of six or 2/3 multiplied by six.

Here's how Andeep has represented it.

Andeep's drawn his bar model again.

He's drawn six as the whole, and then he's drawn 2/3 of six as a bar underneath and he's represented it as multiplication.

He said that the six has been made 2/3 times the size.

Therefore, the whole number has shrunk to the size of the purple bar, and we've represented that with our arrow and our multiplication symbol.

Let's have a look at how Laura's represented it.

She's represented it as a sequence of bars and she's got six parts in her whole here, and each part has a value of 2/3, so she actually has six lots of 2/3.

She could also represent this as 2/3 six times, and Laura then went on to represent it as a number line as well.

Have a look.

Do you see what Laura's done? Each of her jumps is a jump of 2/3 and she's done that jump six times.

So we can count up in lots of 2/3.

Maybe you could count up with me, 2/3, 4/3, 6/3, 8/3, 10/3, and finally, 12/3.

So we can again say this is six lots of 2/3 or 2/3 six times.

So look at the four representations that we have to represent that one equation.

What do you notice about both sets of expressions either side of the line? Well, let's have a look at the equations on the left-hand side to start off with.

Andeep is saying that both of these representations show 2/3 of six In the top one, the whole's been divided into three equal parts, and two of them have been shaded.

And in the bottom one, our six has been multiplied.

So it is 2/3 times the size and that gives us the size of the purple bar, which is 2/3 of the whole.

Let's have a look at the other two representations then.

Laura's saying that both of these representations show six lots of 2/3 and she's right.

Each of the parts in both of the models have a value of 2/3, and there's six lots of those 2/3 in both of the models.

So I think that's a really interesting point here, Andeep.

Actually, when we look at both of these, we know that they both give us the same product as well.

Let's have a look at how we can calculate these using both of these models.

So we can start here by saying that six has been divided into three equal parts.

Each one of those parts has a value of two, and we have two lots of those parts.

So we can say two multiplied by two is equal to four.

So 2/3 of six is equal to four.

We've represented it with this model.

Let's have a look at it with one of Laura's models now.

Well, we're looking now at six lots of 2/3.

Here we go.

We've got 2/3 plus another 2/3 plus another 2/3, and we've got that six times.

We know already that that gives us 12/3, and we know that 12/3 is an improper fraction.

We can now convert this into a mixed number by dividing that by three.

So 12/3 is equal to four as well.

So both of the products here are both equal to four, aren't they? That same equation can represent 2/3 of six or six lots of 2/3.

Okay, time for you to check your understanding now.

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

That's right.

It could be A, B and D, couldn't it? We've got four lots of 3/5, so that could be B.

We could represent that as four multiplied by 3/5, or we could say it's 3/5 four times.

Let's have a look, another check for understanding now then.

Can you tick the images that represent the expression 2/5 of 10? Take a moment to have a think.

Yeah, that's right.

It's both A and C, isn't it? The purple bar in A represents 2/5 of the whole and the whole is 10.

And in C, we can see that the whole has been divided into five equal parts, and we have two of those parts shaded.

B actually represents 10 lots of 2/5, doesn't it? Which we're saying can be represented differently to 2/5 of 10.

Okay, and onto your first task for today now then.

What I'd like you to do is draw an image to represent both of the examples below.

And then once you've done that, what I'd like you to do is group the images that show either 4/5 of 10 or 10 lots or 4/5.

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

Okay, welcome back.

Let's see how you got on then.

On the left-hand side, you can see I've drawn a representation of eight lots of 3/4.

I've done this on a number line, and you can see that I've got 3/4 and another 3/4 and another 3/4 and another 3/4, and then another four lots of 3/4 as well, which gives us eight lots of 3/4.

You may have also done it as a bar model as well.

And then on the right-hand side here, you can see that I've drawn my whole of eight and we've multiplied this.

So it's become 3/4 times the size and therefore, we've got a bar underneath that, which represents the new whole of 3/4 of eight.

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

Okay, and grouping these images here.

Yep, as you can see on the left-hand side here, we've got two representations, which show 4/5 of 10, and on the right-hand side, we've got two representations, which show 10 lots of 4/5.

Well done if you managed to get those as well.

Okay, onto cycle two now then.

Choosing appropriate strategies.

So let's have a look at this equation here.

How could we calculate this equation? Take a moment to have a think.

Well, Laura's saying that she would read it as 3/4 of 12.

So she's saying that she would divide the whole into four equal parts.

There we go.

And each one of those parts would have a value of three.

So, so far, we can say we've done 12 divided by four, and now that we know the value of the unit fraction or each one of the parts, we can now identify how many parts we need.

So now that we know the size of the unit fraction, we can actually look at the numerator and see that we need three of these parts.

So we can multiply this by three now, and there we go.

We can write that in our equation.

And that gives us a total of nine, doesn't it? So we can say 12 divided by four multiplied by three is equal to nine.

We divided it into four equal parts and then we multiplied it by three 'cause we needed three of those parts and that gave us a total of nine all together.

So 3/4 of 12 is equal to nine.

Yeah, Laura, and you're also right there as well.

We could say that the whole has shrunk in size, hasn't it? We can say that the whole has shrunk, so it is 3/4 times the size.

So we can say that 12 has been shrunk.

So it is 3/4 times the size, or we can say that nine is 3/4 times the size of 12.

We call this a scaling strategy.

Let's have a look about how Andeep would go about tackling it.

Andeep says he read this as 12 lots of 3/4.

Here we go.

We can see 12 lots of 3/4 here.

We've got 12 wholes and they've all been divided into quarters, and three of the parts in each whole have been shaded.

We could represent this as unitized counters.

Each one of these wholes represents 3/4.

So each one of the counters represents 3/4.

So to work out how many quarters that is then, we can say 12 multiplied by 3/4 is equal to 36/4.

And then we can divide this by four to convert from the improper fraction into the mixed number.

36 divided by four is equal to nine.

So that gives us nine wholes all together.

So the previous strategy was a scaling strategy, and this strategy here we can call a repeated addition strategy.

Let's have a look at them next to each other.

On the left, we've got our scaling strategy, and on the right, we've got our repeated addition strategy.

They both gave us the same product, didn't they? And is there anything else that you notice in particular? Ah, well, Andeep's noticed on the left-hand side, we had the 12 to start off with and then we divided it by four and then multiplied it by three.

And that gave us the product of nine.

And on the right-hand side, we had 12, but we multiplied it by the three first and then we divided it by the four, and that was also equal to nine.

So the dividing by four references the denominator and the multiplying by three references the numerator.

And it doesn't matter which order you decide to do that in.

You could divide by the denominator, then multiply by the numerator, or you can multiply by the numerator, then divide by the denominator.

You'll still get the same product each time.

That's a really interesting way of looking about this.

Good spot, Andeep.

So which of these strategies did you prefer then? Was there one you preferred over the other? I think it's more dependent on the calculation that is asked of you.

Let's look at a few more examples where we might prefer to use a scaling strategy or a repeat addition strategy.

Laura is saying for this calculation, she preferred the scaling strategy because the denominator of the fraction was a factor of the whole number.

We can see that the whole number was 12 and it could be easily divided into four equal parts.

Therefore, if it's easily divisible by the denominator, then actually, using a scaling strategy is often more appropriate.

Let's look at a few more examples now where either using a scaling strategy or a repeated addition strategy is more important based on the calculation that we have to calculate.

Hmm, which strategy should be used? Should we read it as five lots of 3/4, which would be a repeated addition strategy, or should we read it as 3/4 of five, which would be a scaling strategy? Hmm.

Well, Aisha's saying that she thinks we should do five lots of 3/4 because she doesn't know what 3/4 of five is equal to.

So let's use our repeated addition strategy then to help us with this.

Here we've got a representation of that.

We've got five lots of 3/4, and we can represent that as our unitized counters again.

To work out how many quarters that is all together, we need to do five multiplied by three.

So that's five lots of 3/4, which is equal to 15/4.

And therefore, to convert that into a mixed number, we can divide by the denominator, can't we? Which is four in this case.

So 15 divided by four is equal to 3 and 3/4.

So we found the answer to that calculation relatively quickly, haven't we? Using a repeated addition strategy.

You may have also noticed that the denominator here isn't a factor of the whole number that we started with.

Four is not a factor of five.

Therefore, examples like that often lend itself better to use a repeated addition strategy.

Let's have a look at a different example this time.

We've got 444 multiplied by 3/4 this time.

Hmm, well, should we see this as 444 lots of 3/4 or should we see this as 3/4 of 444.

What do you think? Well, Laura's saying, "I think we should use a scaling strategy here: 3/4 of 444." And I wonder why.

Well, she says the denominator, which is four is a factor of the whole number of 444, so that would be useful for us to use a scaling strategy, but also trying to represent 444 lots of 3/4 would take a long time, wouldn't it? So actually, this strategy would be a better off we think.

So for our scaling strategy then, we know we can draw our whole of 444 and we can divide it into the number of equal parts that we need, which is four equal parts 'cause that's what the denominator tells us.

So we've got 444 divided by four.

Each one of those parts has a value of 111.

Therefore, the unit fraction is 111.

And how many of those parts do we need? That's right, we need three of those parts.

So we can now multiply by the numerator or three of those parts.

There we go.

So now we've got 444 divided by four, and then multiply that by three.

And that gives us a total of 333, doesn't it? There we go.

So we can say that 3/4 of 444 is equal to 333.

We can also say at this point that 444 has been multiplied, so it is 3/4 times the size.

We can say that 333 is 3/4 times the size of 444.

Brilliant, and yeah, we've used our scaling strategy here to help us with that, haven't we? Okay, time for you to check your understanding now.

Can you choose the most appropriate strategy? You can ask yourself, "Should I do three lots of 2/6, or should I do 2/6 of three?" Take a moment to have a think.

Well, Aisha's reasoning this by saying that the denominator of the fraction is not a factor of the whole number.

So it's probably better to use a repeated addition strategy.

Let's have a look.

We've got three lots of 2/6 here, haven't we? We can do three multiplied by two to give us the number of six that we need, which is 6/6, and now that we've got 6/6, we know that 6/6 is also equal to a whole, or we could divide it by the denominator, which is six in this case.

So that would also be equal to one whole, wouldn't it? Here's another example.

Choose the most appropriate strategy for this calculation and then solve it.

Take a moment to have a think.

Okay, so should we do 120 lots of 2/6, or should we do 2/6 of 120? Well, Aisha this time is saying that the denominator is a factor of the whole number.

Six is a factor of 120, so we could use a scaling strategy here.

We can have our whole of 120 and we can divide it into six equal parts.

Each one of those parts has a value of 20.

So, so far, we've done 120 divided by six, and then we can find out how many parts we need by looking at the numerator.

We need two of those parts.

So two of those parts would be equal to two lots of 20, and that is equal to 40.

So we can now do 120 divided by six, then multiply that by two, and that gives us 40 all together.

Well done if you managed to get that too.

And of course, we can represent that as the whole number having been scaled, can't we? 120 has been made 2/6 times the size, or we can say that 40 is 2/6 times the size of 120.

Onto our final tasks for today then.

What I'd like to do here is group the calculations that would be best solved as using either a scaling strategy or a repeated addition strategy.

There might be some which suit both strategies, which you could place in the middle of our Venn diagram.

And then for task two, what I'd like you to do is solve the calculations using the appropriate strategy as well.

And then for task three, what I'd like you to do is come up with your own calculations, one calculation that would be best suited for a repeated addition strategy and one calculation that would be better suited for a scaling strategy.

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

Okay, let's run through the answers to these then.

So for the scaling strategies, we're looking for ones which have the denominator, which would be a factor of the whole number.

So we can see 140 multiplied by 3/7.

Well, seven is a factor of 140 'cause we know seven is a factor of 14.

So that would work.

We've got 2/9 multiplied by 27.

Well, we know that nine is a factor of 27, so that would also work.

And we've got 40 multiplied by 5/8.

Eight is a factor of 40, so that one would also work.

On the right-hand side, we're looking for ones with repeated addition where we've got eight multiplied by 2/12 here.

Now, the denominator of 12 is not a factor of eight, so it might be better to use a repeated addition strategy here of eight lots of 2/12 rather than trying to find 2/12 of eight.

And then finally, we've got some that was fit in the middle, which might be suitable for both strategies.

We've got 2/3 multiplied by six.

Well, the denominator of 2/3 is three, and that is a factor of six, isn't it? But also, we could represent this as six lots of 2/3.

Where we don't have so many groups, it might also be useful to represent it as a repeated addition.

So we only have six groups here, don't we? So that might be easier.

Whereas on the left-hand side, we had 140 groups, 27 groups and 40 groups.

They might take a lot longer, mightn't they? And then the bottom one, we've also got 10 multiplied by 4/10.

Well, again, we know 10 is a factor of 10, isn't it? So that could be used as a scaling strategy, but it could also be used as a repeated addition strategy.

10 groups of 4/10, 10 groups isn't a lot of groups.

So we could represent that as 10 groups of 4/10.

Well done if you got those.

Okay, onto calculating each one of the calculations then.

Well, for the first one, we knew it was a scaling strategy, didn't we? So that gave us a total of 60.

For the second one, we used a repeated addition strategy.

That gave us one and 4/12.

For the third one, we used a scaling strategy and that gave us a product of six.

For the fourth one, we did 40 multiplied by 5/8.

Well, again, we used a scaling strategy for this one, which was equal to 25.

We then did 2/3 multiplied by six.

We could have used either strategy for this one, and that gave us a product of four.

And then finally, we could have used either strategy for 10 multiplied by 4/10.

That gave us a product of four as well.

Well done if you've got all of those.

Okay, here's an example that we came up with, which could use either a repeated addition or a scaling strategy.

We've got 10 multiplied by 4/5.

We could read this as 10 lots of 4/5, which isn't a great amount of groups, which is why we could do it as a repeated addition strategy, but we could also use it as a scaling strategy because five is the denominator here and that is a factor of the whole number of 10.

So we could use either strategy for this.

I wonder what ones you came up with.

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

Let's have a think about how we could summarise what we've learned.

So you can represent multiplication of a whole number by a non-unit fraction as either scaling or repeated addition.

And it is often more appropriate to solve problems where the denominator is a factor of the whole number using a scaling strategy.

Thanks for joining me today.

I hope you enjoyed yourself.

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