Lesson video

Hi, welcome to today's lesson.

My name's Mr. Peters.

And in this lesson today, we are gonna be extending our thinking about improper fractions and how we can multiply these with whole numbers.

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

So by the end of this lesson today, you should be able to say that I can represent the multiplication of improper fractions as either multiplication or repeated addition.

We've got three key words we're gonna be referring to throughout in this lesson.

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

The first one is repeated addition.

Your turn.

The second one is improper fraction.

Your turn.

And finally, the last word is represent.

Your turn.

So repeated addition is the process of adding the same equal group to itself multiple times.

An improper fraction is a fraction where the numerator is greater than the denominator.

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

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

The first cycle is to represent calculations, and then the second cycle is to find the product.

Let's get started with the first cycle.

In today's lesson, we're gonna be joined by Lucas and Aisha.

Again, they'll be sharing their thinking and any questions they've got as we go along.

So let's start here today.

Have a look at this image here.

How could you describe this image? Take a moment to have a think.

Aisha is describing it as two circles, and both of those circles have been divided into quarters.

Have a look at the circles now and have a look at the number of quarters that there are.

How many of them have been shaded? That's right, there are two circles and they've both been divided into quarters and five of them have been shaded.

And that's exactly what Lucas is pointing out.

What's happened to our image now? How would you describe this? Take a moment again to have a think.

That's right, initially we had 5/4, didn't we? We had two wholes and they were divided into quarters and five of them were shaded.

So we could describe that as 5/4.

And now we've got another 5/4 and we've got another 5/4, haven't we? We can use counters to represent each set of 5/4.

Hopefully you can see a blue counter, which represents each lot of 5/4.

So we could write this as a repeated addition if we wanted to know the total amount, couldn't we? So we could say that there are 5/4 plus another 5/4 plus another 5/4.

You might already be starting to think about how we could represent this in a different way now.

Again, take a moment to have a think.

That's right, Aisha repeated addition can be represented as multiplication, can't it? So we know that 5/4 plus 5/4 plus 5/4 is the equivalent of saying three lots of 5/4.

We could write that as three multiplied by 5/4.

Or you're right, Lucas, we could write it as 5/4, three times, couldn't we? And again, you can see how we've written that as 5/4 multiplied by three.

So let's think about what the numbers represent in these expressions.

What does the three represent in both of the expressions? That's right, Aisha.

The three represents the number of groups, doesn't it? And what about the 5/4 then? Yep, the 5/4 this time represents the value of each of the groups, doesn't it? There are 5/4 in each group.

And Aisha's making a good point here.

She's saying that when we are writing a multiplication equation, the numbers that are being multiplied together can be placed in any order, and we know that because of what we call the commutative law, where we can change the order of any of the numbers that are being multiplied together in a multiplication equation and the product would remain the same.

Let's have a look now about how we could represent this on a number line as well.

We are looking to represent three lots of 5/4.

So we can say we've got one lot of 5/4 and now we've got two lots of 5/4, and now we've got three lots of 5/4.

Hmm, take a moment to think.

What do the numbers represent, again, the expressions this time, on the number line? That's right, the three represents the number of groups, doesn't it? Or in this case, the number of jumps on the number line.

And the 5/4 represents the value of each one of those jumps.

Or in this case, the distance of each one of the jumps.

Let's have a look at another example now.

Take a look at our image here.

How would you describe this image this time? That's right, Aisha.

There are two bars, aren't there? And both of those bars have been divided into three equal parts, or should we say, each bar has been divided into thirds.

And we can see that there are four of those parts shaded in, so there are 4/3 thirds shaded in.

Have a look now at our image.

What's changed? That's right, we've got another set of two bars.

Now we've got another lot of two bars, which has been divided into thirds again.

And again, four of them are shaded.

So we've got one lot of 4/3 and we've got another lot of 4/3.

And we can use counters to show each one of those lots of 4/3.

Each of the counters are representing one lot of 4/3.

And as we know, we can now represent this as repeated addition.

We can say 4/3 plus 4/3.

Hmm, we know that repeated addition can also be represented as multiplication, can't it? Take a quick moment to think.

How would you write it as a multiplication expression? That's right, we could represent it as two multiplied by 4/3, couldn't we? Or we could represent it as 4/3 two times.

Take a moment again.

Have a think about what the numbers represent in these expressions.

Well, the two represents the number of groups, doesn't it? So we can see we've got two lots, haven't we? And the 4/3 represents the value of each one of those groups, isn't it? So we can say we've got two lots of 4/3, or we've got 4/3 two times.

And as we mentioned earlier, we know that we can rearrange the order of the numbers that are multiplying with each other in a multiplication equation because of the commutative law.

Let's have a look about how this now applies to our number line.

Have a look at the number line yourself, first of all.

Can you imagine where the jumps on the number line might be placed? That's right, we've got one lot of 4/3 and we can represent that here with this one jump.

And now we've got another lot of 4/3.

So we can say that we've got two lots of 4/3.

The two represents a number of groups or jumps on the number line, and the 4/3 represents the size of each group, doesn't it? Or the distance of each of the jumps on the number line.

Okay, time for you to check your understanding now.

Which expression represents this image? Take a moment to have a think.

That's right, it's four multiplied by 5/2, isn't it? If you have a look at each row, you can see we've got a rectangle that has to been divided into two parts, and one of those rectangles has been shaded.

Then we've got five of those half-rectangles that have been shaded.

So we've got 5/2 in each row and we've got 5/2 four times because there are four rows altogether of 5/2.

And another quick check.

Tick the image that represents this expression.

Is it A, B, or C? That's right.

It's A, isn't it? You can see that we've got two circles representing two wholes, and each one of those circles is divided into quarters and seven of those quarters are shaded.

And we've got that set of 7/4 three times along the line.

Well done if you managed to get that one too.

Okay, and what I'd like to do now for your first task today is to complete this image here.

I'd like you to represent each one of the expressions on the left-hand side, either as an area model, so you might like to do that with the circles or with a square or with a rectangle depending on what you think is best.

And then I'd like you to represent it as counters like we did in the previous examples.

And then I'd also like you to represent it on a number line too.

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

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

The first one is two multiplied by 6/5 or two lots of 6/5.

So we could represent that here.

I've used circles for this.

So we've got two wholes for each one, haven't we? And we can see that those wholes have been divided into five equal parts, six of them have been shaded, and we've got that two times.

We could represent that as counters here of two lots of 6/5.

And we could represent it on a number line, which looks a bit like this.

Let's have a look at the second example.

6/5, this time it's 6/5 multiplied by three.

Well there we go.

You can see that the image stayed the same with regards to the number of wholes and the number of fifths that were shaded for those wholes.

But this time we've got three sets of them, haven't we? In the first example, we had two sets, whereas now we've got three sets of them.

So how's that gonna change our counters? That's right, we would need one extra counter this time, won't we? One extra counter of 6/5.

And how would that change our number line? That's right, we need one extra jump on our number line as well.

An additional 6/5 jumped along on our number line.

And then the last one, 6/4 multiplied by three this time.

What did you notice changed here? Well, that's right.

The circles that we used had to be changed in the number of parts that they're divided into, weren't they? Previously, we were dividing into five equal parts, whereas this time we need to divide them into four equal parts.

So each one of our wholes was divided into four equal parts and we needed to shade in six of them.

So we could see that we've got two wholes represented by two circles, and they're both divided into quarters, and six of those quarters have been shaded.

And we needed those 6/4, three times, didn't we? So let's have a look at that as counters now then.

That's right, we still needed three counters and each one of those counters represents 6/4 this time rather than 6/5.

And on a number line then.

That's right, our wholes needed to be divided into four equal parts rather than five equal parts this time.

So you can see between zero and one or one and two that each one of those wholes has been divided into four parts.

Whereas if you look at the number line above it, you can see that the distance between zero and one or one and two, for example, has been divided into five equal parts.

Well done if you managed to get all of those.

Okay, moving on to our second cycle today now, finding the product.

So let's revisit our first example here.

We can see that we've got three lots of 5/4 and we could represent this on our number line again, and that will help us to count up in lots of 5/4.

So let me demonstrate this to you and then you can repeat it after me.

Here we've got 5/4, now we've got 10/4.

Two lots of 5/4 is equal to 10/4.

And then we add on another 5/4, that would mean we've got 15/4 altogether, that's right.

So we could say that three multiplied by 5/4 is equal to 15/4 altogether.

Hmm, Aisha has noticed it might have been quicker and easier rather than drawing a number line and representing it like that, for us to think about our times table knowledge to help us.

She's saying that she knows three lots of five is equal to 15.

So three lots of 5/4 is equal to 15/4.

I wonder if you could say that with me.

Are you ready? We know that three lots of five is equal to 15, so three lots of 5/4 is equal to 15/4.

Well done, let's have a look at another example here this time.

Let's represent it on the number line again and have a quick think what's the value of each of the jumps that we're gonna be taking on our number line.

That's right, it's 4/3, isn't it? So I wonder if we can count up together with a number of 4/3 increasing each time.

Are you ready? So we're going to make two jumps, aren't we? Here is 4/3, and now we add on another 4/3, which would give us 8/3 altogether.

Hmm, you may have realised, again, similar to Aisha has, that would've been easier to potentially use our times tables and a lot quicker as well.

We know that two lots of four is equal to eight, so two lots of 4/3 would be equal to 8/3, wouldn't it? And there we go, as we showed on our number line, two multiplied by 4/3, or 4/3 two times, is equal to 8/3.

Here's one more example.

Have a look at the images this time.

How would you describe those images? There you go, you can see that we've got two sets of two cylinders, haven't we? And in each cylinder, the whole has been divided into eight equal parts and we can see that the combined number of eighths that have been shaded are 12/8 in each set.

So we've got two lots of 12/8 here, or 12/8 two times.

Representing that on the number line, we can show 12/8 and then adding another 12/8 would be equal to 24/8.

Yep, and again, spot on there, Lucas, we could have used our times table knowledge again.

We know that two lots of 12 is equal to 24, so two lots of 12/8 is equal to 24/8.

So here's the three examples we've looked at so far.

Take a moment to have a look.

What'd you notice about each of the examples? Is there anything that stays the same? Is there anything that's different? That's right, Aisha.

It's exactly the same as when we multiply a proper fraction by a whole number, isn't it? So we can see, we can multiply the whole number by the numerator of the improper fraction and that would give us the numerator of our product.

And that applies for every example here.

And would you notice about the denominator? That's right, the denominator stays the same throughout, doesn't it? There we go, so we can extend that generalisation then of when we are multiplying a whole number by proper fractions to when we multiply whole numbers by improper fractions.

Hmm, I wonder if you could use that to help you for the rest of our tasks today.

Okay, time for you to check your understanding.

Can you tick the correct product for the following equation? That's right, it's 14/5, isn't it? We've got two multiplied by 7/5, so two lots of 7/5.

We know that we can multiply the whole number by the numerator of the improper fraction to get the numerator of the product.

So two multiplied by seven would be equal to 14, and the denominator stays the same.

So that would be 14/5.

I know two multiplied by seven is equal to 14, so two multiplied by 7/5 is equal to 14/5.

And true or false here.

When you multiply a whole number by an improper fraction, you multiply the whole number by both the numerator and the denominator.

Take a moment to have a think.

That's right, it's false, isn't it? And have a look at these justifications here.

Which one of these helps you to reason why it was false? That's right, it's B, isn't it? You only need to multiply the numerator by the whole number and we can use our stem sentence to help us, can't we? So like the previous example, I know that two multiplied by seven is equal to 14.

So two multiplied by 7/5 is equal to 14/5.

Okay, and onto our last task for today then.

What I'd like to do here is to fill in the missing boxes.

And then for task two, what I'd like you to do is have a go at filling in the multiplication grids.

And have a quick think as you go through, what is it that you notice as you're filling them in? Good luck with those two tasks and I'll see you back here shortly.

Okay, let's go through these together then.

The first one is two multiplied by 9/5.

I know that two multiplied by nine is 18, so two multiplied by 9/5 is 18/5.

The second one is 8/7 multiplied by five.

Well, we know that numbers have been rotated here because of the commutative law and we can rotate them back if we wanted to.

So we could write it as five multiplied by 8/7.

I know that five multiplied by eight is equal to 40, so five multiplied by 8/7 is equal to 40/7.

The next one is four multiplied by 15/10.

Well four multiplied by 15 is equal to 60, so four multiplied by 15/10 is equal to 60/10.

And the bottom one is 100/4 multiplied by 72.

Hmm, well I know that 72 multiplied by 100 is 7,200, so 72 multiplied by 100/4 is 7,200/4.

That feels like a lot of quarters, isn't it? And for row B then, we've got two multiplied by 6/4, two multiplied by six is equal to 12, so two multiplied by 6/4 is equal to 12/4.

What did you notice about the second one? That's right, the whole number stayed the same, but the number of quarters increased by one, didn't it? So we've gone from 6/4 to 7/4.

So now we've got two lots of 7/4.

Hmm, we had two lots of 6/4.

So I think we're gonna need to add another 2/4.

So therefore we'd have 14/4 altogether.

Hmm, what'd you notice about the third one this time? That's right, the number of quarters stayed the same, didn't they? But the number of lots of those quarters increased.

We had one extra lot of 7/4.

So we previously had two lots of 7/4, but now we've got three lots of 7/4.

So we actually, we just need to add on another 7/4.

So we had 14/4, now we've got 21/4.

And then for the last one, what did you notice this time? Ah, we've got an increase in the number of quarters and we've got an increase in the number of lots of those quarters, haven't we? Let's use our generalisation to help us from earlier on.

We can say that I know that four multiplied by eight is equal to 32, so four multiplied by 8/4 is equal to 32/4.

Well done if you managed to get all of those.

Okay, and here are the complete grids.

I'll let you take a moment to have a look at it for yourself and tick off if you managed to get all of those.

Was there anything in particular that you noticed? Aisha saying it's just like your times tables.

However, instead of counting up in groups of ones, we're actually counting up in groups of quarters.

And those groups of quarters either have 2/4, 3/4, or 4/4.

Or for task B, it had 3/12, 4/12, or 5/12 in each of the groups.

Well done if you managed to spot that for yourself too.

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

So to summarise our thinking then, we can say that repeated addition of improper factions can be represented as multiplication.

You can represent the multiplication of an improper faction either using area models, unitized counters, or on the number line.

And our generalisation for today.

When you multiply a whole number by an improper fraction, you multiply the whole number by the numerator and the denominator stays the same.

There we go, that's the end of our learning for today.

Hopefully you're able to connect that learning to some previous understanding around multiplying proper fractions by whole numbers.

Take care and I'll see you again soon.