Lesson video

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

In this lesson we're gonna be thinking about multiplying proper fractions by whole numbers, and we're gonna take a particular focus on examples where the product is less than one whole.

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 multiply a proper fraction by a whole number where the product is within one whole.

We've got four key words today that we're gonna be referring to throughout.

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

The first one is represent.

Your turn.

The second one is numerator.

Your turn.

Third one is denominator.

Your turn.

And the last one is unitize.

Your turn.

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

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

A denominator is the bottom number of a fraction.

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

A numerator is the top number of a fraction.

It shows us how many parts we have, and unitizing means treating groups that contain or represent the same number of things as units or ones.

Throughout this lesson today we've got two learning cycles.

The first learning cycle, we'll think about finding the product, and the second learning cycle, we'll think about unitizing to find the product.

When you're ready, let's get going.

We've got three students with us today, Izzy, Sam, and Aisha, and it'll be really interesting to see what they have to say and contribute to their learning throughout the lesson.

So let's start here then.

As you can see on the screen, we've got this shape here, and this shape represents one whole.

Watch carefully what happens to the shape.

Hmm, how many equal parts has it been divided into? Let's use our sentence stem below to help us.

Ready? The whole has been divided into seven equal parts.

Each part is 1/7 of the whole.

Hmm, so if each part is 1/7, how could we describe this here then? Well, as we know that each part is 1/7 and we can see that each colour has two parts that are shaded, we can say that each one of the colours represent 2/7 of the whole, don't they? And how many colours are there? That's right, there are three colours, aren't there? Which have 2/7.

So we can say that there are three lots of 2/7, can't we? Hmm.

So how many sevenths are shaded altogether then? That's right.

We know that three X two is equal to six.

So three X 2/7 is equal to 6/7, isn't it? That means there are 6/7 of the whole that are shaded.

Let's say that together underneath.

Are you ready? Three lots of 2/7 are equal to 6/7.

Let's look at that on a number line now.

As you can see, we've got zero at one end of our number line and one at the other end.

So our number line represents one whole and the whole has been divided into seven equal parts.

Each one of those parts or intervals represents 1/7.

And as we know, we have three lots of 2/7.

So let's see how we can represent that on here then.

Ready? Here's one lot of 2/7.

Here's two lots of 2/7 and finally three lots of 2/7.

We know that three lots of 2/7 is equal to 6/7 because three lots of two is equal to six.

So three lots of 2/7 is equal to 6/7.

Again, let's say that together underneath.

Ready? Three lots of 2/7 are equal to 6/7.

Have a look at these images here then together.

How do they relate to each other? Well, that's right, they're both wholes, aren't they? And they're both divided to seven equal parts.

Let's see how we can link our area shape on the left-hand side to our number line.

Izzy's saying that each jump on the number line represents 2/7, or each colour shaded in in our whole.

So what happens if we change the colours of our jumps here? The first jump could represent the two yellow segments.

There we go, that's 2/7.

The second jump could represent the two orange segments.

Again, that represents another 2/7.

And then finally, the two blue segments could be represented by another jump here of 2/7.

And we know altogether that is equal to 6/7.

So at this point here, we've shown three lots of 2/7 is equal to 6/7 in two different ways.

We've shown it with our area model on the left-hand side and we've shown it with our number line.

We can also represent it as an equation.

We know that three lots of 2/7 is equal to 6/7.

So we can write this like this: 3 X 2/7, or three lots of 2/7, = 6/7.

And using our knowledge that rotating the factors in a multiplication equation will mean the product will remain the same, we could write it as 2/7 X 3 = 6/7 as well.

Let's look at a different example here.

Now you can see a bar, can't you? This time the bar represents one whole.

Look carefully; how many equal parts has the whole been divided into? Let's say our stem sentence to help us with this.

The whole has been divided into 12 equal parts.

Each part is 1/12 of the whole.

Have another look.

How have they been shaded in this time? How could we describe this? Well, as we know, if each part represents 1/12, then we can say that each colour has five parts that are shaded, can't we? There are two separate colours that are shaded in with five equal parts.

So altogether we can say that there are 10 parts that have been shaded in.

Let's say it together.

Are you ready? Two lots of 5/12 are equal to 10/12.

Let's show that on the number line again.

Take a moment to have a think.

What do you think it would look like for yourself? Well, we can see our whole here, starts at zero, ends at one, and the whole has been divided again into 12 equal parts.

Each one of the intervals represents 1/12.

And we know that one lot of 5/12 can be represented here by this jump.

And then two lots of 5/12 can be represented by another jump here.

5/12 and another 5/12 will be equal to 10/12.

So two lots of 5/12 are equal to 10/12.

Can you have a go at saying that? So let's have a look at these two together again, and how they've relate to one another.

We've now represented it as an area model again and on a number line.

And we also know that we can represent it as an equation, can't we? How do you think we could record this as an equation? That's right.

We have two lots of 5/12, don't we? So we can record that as 2 X 5/12, and we know that is equal to 10/12.

Or we could record it as 5/12 X 2.

There we go.

5/12 X 2, and again, we know that's equal to 10/12 because in multiplication we can reorder the factors and the product would remain the same.

So let's have a look at both our examples so far.

What do you notice? Well, you may have noticed what Izzy did.

Izzy noticed that the denominators stayed the same throughout.

The denominator of the proper fraction that we X the whole number is the same denominator that we have in our product, isn't it? And that's the same for both examples.

In the first example, we were doing 3 X 2/7, and our answer became 6/7.

So the denominator stayed the same.

And in the second example, we were doing 2 X 5/12, and that was equal to 10/12.

And again, the denominators were the same there, weren't they? Was there anything else that you noticed? That's right, Aisha has spotted that to form the numerator of the product, we can multiply the whole number by the numerator, can't we? Let's have a look.

Here we can see if we multiply the whole number by an numerator of two, we said earlier 3 X 2 = 6.

So 3 X 2/7 = 6/7.

So you can see that the six is represented on the top with our numerator here.

Again, look at the other example; 2 X 5.

2 X 5 = 10, so 2 X 5/12 = 10/12.

And again, we can represent that here with our numerator.

So how could we generalise what we've just learned in a sentence? Well, we could say that when multiplying a fraction by a whole number, we multiply the numerator by the whole number and the denominator stays the same.

Shall we say that together? Ready? When multiplying a fraction by a whole number, we multiply the numerator by the whole number and the denominator stays the same.

Izzy and Aisha think they've spotted something else.

Izzy says, "It sounds like something similar to what we did when we were looking at the addition of fractions." Let's have a look.

Aisha agrees.

She says, "Yes, when adding two fractions together with the same denominator, that we just add the numerators together and the denominators stay the same," don't they? So let's link these multiplication equations to these addition equations.

Here we can say that 3 X 2/7 = 6/7.

And we know that is the same as saying 2/7 + 2/7 + 2/7, and that is also equal to 6/7.

Look carefully, when we're adding these fractions together and the denominators are all the same, then the denominator and the product remains the same.

And when we add these fractions together, we need to add the numerators together to form the numerator for the product.

Let's have a look at the other example.

2 X 5/12 = 10/12.

We can record that as 5/12 + 5/12 = 10/12.

Once again, the denominators from the fractions are the same.

And therefore, when we add these two fractions together, the denominator of the product remains the same.

And when we're adding these fractions together, the numerators need adding together.

Well done if you managed to spot that for yourself as well and make that connection.

So Sam's come along now.

Sam's saying, "I think that when you're multiplying a fraction by a whole number, you can multiply both the numerator and the denominator by the whole number." His example is below.

He says, "2 X 3/8 = 6/18." Hmm, what do you think about that? Aisha disagrees with Sam.

She's saying that it's only the numerator that would change and therefore the numerator would be multiplied by the whole number to form the new numerator.

Let's have a look why that's the case then.

Aisha's saying 2 X 3/8 would be equal to 6/8.

Here we can see our whole.

Our whole is divided into eight equal parts.

And at the moment we have got three parts that are shaded.

That's one lot of 3/8.

And if you were to multiply both the numerator and the denominator by the two, which you were saying, Sam, we would get 6/16, which would look like this.

Hmm, what'd you notice about these two? That's right.

They're exactly the same, aren't they? 3/8 = 6/16, isn't it? In fact, all we've done is show that each block is now worth two blocks, but the block sizes have had to shrink so that the wholes are the same size altogether.

So you can see here that for every 1/8, that's equivalent to 2/16, and therefore 3/8 would be equivalent to 6/16s.

So they're the same thing.

So multiplying 3/8 by two cannot be 6/16 'cause they would be the same thing.

So Aisha's saying that if you were to multiply a positive fraction by a whole number then it would increase in size.

That's right, Sam.

Let's have a look at an example of what it should look like.

Here we go.

We can now see that our bar model shows one lot of 3/8 with the yellow, and another lot of 3/8 with the orange.

So we can see altogether that there are two lots of 3/8 and that would be equal to 6/8.

Well done if you're able to explain that to somebody else near you as well.

Okay, let's check our understanding now.

Have a quick look here.

2/9 X 3.

Can you tick the correct answers for this? Off you go.

That's right, it's B isn't it? And how do you know that? Well, we know that the denominator stays the same, doesn't it? And we can multiply the numerator of the proper fraction by the whole number to find the numerator of the product.

And therefore 2 X 3 would be equal to six.

So 2/9 X 3 would be equal to 6/9.

Here's another quick check.

Which expressions are equal to 8/10? Take a moment to have a think.

That's right, it's A, C, and D, isn't it? We know that 2 X 4 = 8.

So 2/10 X 4 would be equal to 8/10.

We know that 1 X 8 = 8, so 1 X 8/10 = 8/10.

And finally we know that 4 X 2 = 8, so 4/10 X 2 = 8/10 as well.

Well done if you've got both of those.

Okay, onto our tasks for today, then, to start us off with.

What I'd like you to do is shade in the shapes to show the calculation and write in the product as well for each of them.

And for task two, what I'd like you to do is draw in your own representation for the equation that's shown.

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

Okay, let's have a look then.

2/5 X 2.

We know that's equal to 4/5.

And here's a representation of that shaded in.

3 X 2/9.

Notice here how the order of the factors are in a different order here.

It doesn't matter though how we go about tackling this, does it? So 3 X 2/9 = 6/9.

And again, here's an image using the circle to represent that.

And finally, something is equal to 3 X 4/12.

Notice here how the product is at the beginning of our equation.

Again, it doesn't matter how we go about calculating this.

3 X 4/12 = 12/12.

So in that case, the whole shape would need to be shaded in, wouldn't it? With each colour representing each lot of 4/12.

And for task two, I wonder how you got on with drawing.

Here's an example of what you could have come up with.

Here I've drawn an oval to represent the whole and within the oval I've got some different coloured dots.

There are eight dots altogether in the whole.

And as you can see there are three lots of two dots that are in different colours.

Two of them are yellow, two of them are orange, and two of them are blue.

So we could say here that the shaded amount is represented here by 2/8 X 3 and that would be equal to 6/8.

Right, onto cycle two now; unitizing to find the product.

So we can also represent multiplying a proper fraction by a whole number by using unitizing counters.

Let's have a look here.

Each counter here represents 3/10.

These are a lot easier to draw than drawing out different shapes showing 3/10, aren't they? It would take quite a while to draw a shape and divide each of those into 10 equal parts and shade in three of them.

So we can use a counter to make our life easier, and to show that unitized amount in the counter.

Our counters show three lots of 3/10.

So we can record that as 3 X 3/10.

And we know we can also represent it as 3/10 X 3.

Now to find what this is equal to, we can multiply the numerator of the proper fraction by the whole number.

And we can do that for both examples here.

So 3 X 3 would be equal to 9.

So the numerator would be nine.

And we know that the denominator stays the same throughout.

So the denominator would be 10.

That means 3 X 3/10 or 3/10 X 3 = 9/10.

Let's have a look at that on our number line as well then.

Here we've got our whole divided into 10 equal parts.

And we can see that here.

Each interval represents one 10th.

We've got three counters here, each with 3/10 in them.

So we can do one jump to show 3/10.

That's one lot of 3/10.

Here's another lot of 3/10; two lots of 3/10.

And here's another lot of 3/10; three lots of 3/10.

We know that three lots of 3/10 = 9/10.

Linking that to an equation, then, we can write it as 3/10 X 3 = 9/10.

Here's a slightly different way of looking at it.

The product is 8/12.

What could the fraction and the whole number be? Take a moment for yourself to have a quick think.

Izzy's pointing out that she knows that the denominator must stay the same throughout the equation, mustn't it? So we can place a 12 in here to represent the denominator of the proper fraction.

And then we just need to find a whole number and a numerator that would multiply together, don't we? And that would be equal to the numerator of the product, wouldn't it? So this is where using our understanding of factors of eight would be really helpful, wouldn't it? We could represent our factors of eight on the factor bug.

We know that one and eight are factors of eight, aren't they? So we could say that the missing number would be one and eight or 1 X 8/12 would be equal to 8/12.

And we can see that with the unitized counter.

We could also use 1/12 as the unit and say that's eight lots of one/12.

So the whole number could be eight and one could be the missing numerator for the proper fraction.

We also know that two and four are factors of eight.

So we could say that it's two lots of 4/12.

Two would be the missing whole number and four would be the missing numerator.

Or we could change the unit again into 2/12s.

So we could say we've got four lots of 2/12s.

Four would be the missing whole number and two would be the missing numerator of the proper fraction.

Izzy's decided, "Let's go for 2 X 4/12," and there we go.

Okay, time for you to check your understanding again now.

Tick the expressions that the counters would represent.

Take a moment to have a think.

That's right, it's A and D, isn't it? We could represent this as 2/15 X 4 or 2/15 X 4.

Well done if you've got both of those.

Okay, and moving on now then to our final practise activities for today.

For task 1, what I'd like you to do is complete the table.

We've got three columns in our table.

The first one is the representation.

So if there's a missing box in there, you'll need to draw the counters to represent the unit.

In the middle column, we've got the expressions.

So I'd like you to write in the expression there.

And in the final column, I'd like you to write the product that expression would be equal to.

For task 2, what I'd like you to do is complete the calculations, and once you've done that, have a think.

What did you notice as you were going through those calculations? Good luck with those two tasks and I'll see you back here shortly.

Okay, let's go through these then.

Well, the first one was already done for us.

So let's have a look at the second one.

We've got two lots of 3/11 here, haven't we? So we could write that as 3/11 X 2 or 2 X 3/11.

And the product for that would be 6/11.

For the third one, we've got 3 X 3/15.

So to record that, we'd need to draw three counters, each one with a value of 3/15, and the product of that would be 9/15.

The whole number of 3 X the numerator of the proper fraction would give us nine.

So that's equal to the numerator of the product, and we know the denominator stays the same.

Hmm, the image on the left-hand side now has three counters, each representing 5/20.

So as an equation we could write that as 3 X 5/20 or 5/20 X 3, and the product of that would be 15/20.

And then finally the last one.

The product this time is 24/50.

Hmm, how could we record that then? Well, there's actually a number of ways you could have gone around it, and you may have a slightly different way to doing this and that's absolutely fine, because 24 has lots of different factors.

The one we're gonna go for is four lots of 6/50.

Four lots of 6/50 can be represented as 4 X 6/50.

We know the denominator stays the same.

And when you multiply the whole number by the numerator, 4 X 6 = 24, so the product is 24/50.

And as Izzy's saying, there are actually lots of different ways of solving that last one.

You could have used one and 24 as factors.

You could used two and 12 as factors of 24.

You could have also used three and eight as factors of 24.

So if you've got any of those, well done to you as well.

And for the last one, then.

Let's go through these ones, then.

The answer to the first one is 6/10, the second one is 6/100, the third one is 6/1000.

What did you notice about those three? That's right, the numerator stayed the same this time because we were multiplying the same whole number each time with the same numerator.

However, the denominator was getting 10 times the size each time, but that denominator stayed the same in the product of the answers as well, didn't it? So well done if you noticed that.

And then what about the last one? That's right, Aisha.

We know that when multiplying a proper fraction by a whole number, the denominator stays the same throughout the whole time.

So if the denominator of the proper fraction was a triangle, then the denominator of the product would also be a triangle as well.

Well done if you managed to get all of those.

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

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

We know that multiplying a fraction by a whole number can be represented either by using area models, number lines, or unitizing counters.

And a generalisation from today; when you multiply a fraction by a whole number, you multiply the numerator by the whole number and the denominator stays the same.

Thanks for learning with me today.

I really enjoyed that lesson and hopefully again, you're feeling a lot more confident and have got a nice generalisation to take away from you, which will help you tackle multiplying fractions by whole numbers.

Take care.

I'll see you again soon.