Lesson video

Hi, my name's Mr. Peters.

In this lesson today, we're gonna be thinking about different ways that we can calculate the volume of cuboids or cubes.

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

So by the end of this lesson today, you should be able to say that you can calculate the volume of a cuboid or a cube.

Throughout this lesson today, we've got a couple of keywords we're gonna be referring to.

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

The first one is cubic centimetre.

Your turn.

The second one is cubic metre.

Your turn.

The third one is cube number.

Your turn.

A cubic centimetre is the volume made up by a cube with one centimetre on each edge.

A cubic metre is the volume made up by a cube with one metre on each edge.

And finally, a cube number is the product of multiplying three repeated numbers.

This lesson today is broken down into three cycles.

The first cycle, we'll think about defining cuboids.

In the second cycle, we'll be thinking about calculating cuboid volumes.

And in the third cycle, we're gonna start thinking about cube numbers.

Let's get started with the first cycle.

Throughout our lesson today, Jun and Sam are gonna join us and share their thinking throughout.

Okay, so let's start our thinking here then.

All of these shapes below that you can see are cuboids.

What are the features that they all have? Take a moment to have a think.

Well done.

So you may have noticed that they're all different sizes.

You may have noticed that they're different colours as well, but these are all still cuboids.

So what is it that is the same about these cuboids? Well, you may have noticed that cuboids all have six rectangular faces.

A face is a part of the shape that you should be able to rub your palm of your hand over in order to feel it.

If you have shapes available with you, you might like to try that now.

All cuboids also have 12 edges.

The edges join the corners of the cuboids all together.

And again, you should be able to run your finger along each of those edges.

And finally, all cuboids also have what's called eight vertices.

You can see the arrows pointing to one vertex.

When there's one of them, we call it a vertex.

When there's more than one of them, we call them vertices.

You should be able to point to each of these on each of the shapes.

Jun asks a really good question here.

He says, "Isn't this a cube instead of a cuboid?" It's a good point, Jun.

Well, actually, a cube is a special type of cuboid.

Instead of having rectangular faces like a cuboid does, the cube has square faces, and a square is a special type of rectangle.

So we can say that cubes have six square faces, they have 12 edges, and they also have eight vertices as well.

Okay, time for you to check your understanding now.

True or false? A cube is a special type of cuboid.

Take a moment to have a think.

That's right.

It's true, isn't it? And which one of these justifications helps you to answer that why? That's right.

It's A, isn't it? Cuboids and cubes have the same number of faces, the same number of vertices and the same number of edges, don't they? The only difference is that a cube has square faces.

Okay, another check.

All cuboids have A, the same size edges, B, six edges, C, eight vertices, D, six faces.

Take a moment to have a think.

That's right.

It's C and D, isn't it? All cuboids have eight vertices and they also have six faces.

Okay, onto your first task then.

Here are a range of different cuboids.

What I'd like to do is draw in any missing edges that you can't see, as well as drawing a dot on each of the vertices to mark where these are.

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

Welcome back.

Let's see how we got on.

So to draw in the missing edges, you may have done this.

We can now see on this shape here all of the 12 edges required.

And then I've drawn a dot on each of the vertices as well.

So that is what the first one would look like.

Here's the second one, here's the third one, here's the fourth one, and here's the fifth one.

It's quite tricky to draw some of those edges on, isn't it? Particularly to make the shape look like they're 3D.

Well done if you managed to get those.

Okay, onto cycle two now then.

We're gonna start thinking about calculating cuboid volumes.

Okay, so as you can see on the screen, we've got the cuboid.

And the question is how could we calculate the volume of this cuboid.

Jun thinks that we could count each one of the cubes.

Each one of the cubes represents one cubic centimetre.

So we could go through and count each individual cube.

Sam thinks differently.

Sam's saying that might be quite tricky.

Not only 'cause we can't see all of the cubes, but actually he thinks there might even be a quicker way.

Sam says we should think about finding the volume of one of the layers first.

Let's have a look at one of the layers of our cuboid.

There are two rows of six cubes, and we know that two multiplied by six is equal to 12.

So there are 12 one cubic centimetres in this layer.

We can say that this layer has a volume of 12 cubic centimetres.

Now we can start thinking about adding on the layers, can't we? So we know that one layer is 12 cubic centimetres.

In this image, we can see that there are four layers all together.

So we have four layers of 12 cubic centimetres.

That must mean all together, we can find out the volume of this cuboid by multiplying four by 12.

Four by 12 we know is 48.

So the volume of this cuboid is 48 cubic centimetres.

Here's another shape.

Jun's now learning.

He thinks actually, it's gonna take too long to count them individually again.

So maybe we're better off using Sam's method again.

So let's find the volume of one layer first then.

Maybe you could have a go at saying the same sentence with me this time.

There are three rows of three cubes.

We know that three multiplied by three is equal to nine.

So there are nine lots of one cubic centimetre in this layer.

So we can say all together that this layer has a volume of nine cubic centimetres.

Now we can start thinking about adding the layers, can't we? We know that one layer has a volume of nine cubic centimetres, and all together, there are six layers.

So there are six layers of nine cubic centimetres.

We know that six multiplied by nine is equal to 54.

So the volume of the cuboid is 54 cubic centimetres.

Jun's saying that he's noticed something.

He's saying that we multiplied two numbers, first of all together to find the volume of the first layer, and then we multiplied the product from that by the number of layers that there were.

So we're effectively multiplying three numbers together, aren't we? Sam's saying that when we multiplied those two numbers, first of all, we were multiplying the length by the width.

We can see here on this example that the length is four centimetres and the width is two centimetres.

So now we would multiply the products of these two, which would be eight centimetres by the height, the number of layers that there are.

The height is three centimetres.

So in order to work out the volume of the cuboid, we would need to multiply the length by the width by the height.

This will give us the total volume of the whole cuboid.

And as Sam's pointing out, when we do write this down, make sure we put this is equivalent to V 'cause V represents volume, doesn't it? Jun's now asking, "Do we always have to do it in that order? Do we always have to multiply the length by the width and then by the height to get the total volume?" Hmm, let's have a think.

What'd you notice? Take a moment to have a think.

That's right, we've rotated the cuboid, haven't we? It was still exactly the same cuboid, we've just rotated it round.

So it sat slightly differently now.

That means the length has now changed to three centimetres.

The width has stayed the same at two centimetres.

But the height now is four centimetres.

Is it still the same cuboid though? It is, isn't it? So actually, are we still gonna get the same volume? We will, won't we? And we know that because four multiplied by two multiplied by three is the same as saying three multiplied by two multiplied by four.

And we can say that is the same 'cause we know that multiplication is commutative, and that means we can rearrange the order of the factors and the product will remain the same.

So in order to find the volume of a cuboid, we could do the length multiplied by the width multiplied by the height.

We could do the width multiplied by the height multiplied by the length, or we could do the height multiplied by the length multiplied by the width.

It doesn't matter which order we multiply these in, we will still find the volume.

You might think of some other ways to order the factors, and that's absolutely fine as well.

Okay, time to check your understanding now.

To find the volume of a cuboid, you can multiply A, B, C, or D.

Take a moment to have a think and see if you can explain why.

That's right.

It's all of them, isn't it? It doesn't matter which order we place these factors in, does it? We know that the order of the factors can be changed around and the product will remain the same.

Therefore, if we multiply the length by the width by the height in any order, we will still get the same volume of the cuboid.

Okay, here's a quick go for you as well.

Can you find the volume of this cuboid for me? Well done.

Sam's saying that he multiplied the length by the height by the width this time.

So the length would be five centimetres, the height would be eight centimetres and the width would be three centimetres.

All of that would be equal to the volume.

So five multiplied by eight would be equal to 40.

And then 40 multiplied by three would be equal to 120.

So the volume of this cuboid would be 120 cubic centimetres.

Well done if you got that.

Okay, onto your next task then.

I'd like you to have a go at finding the volume for each of the following shapes.

And you can write the answer in underneath each one of the shapes.

And then for task two, I'd like you to think about how many different cuboids can you make with a volume of 12 cubic centimetres? And for task three, I've got a question for you.

Always, sometimes, or never? The taller the cuboid, the greater its volume.

Maybe you could have a go at justifying that and creating an explanation showing why.

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

Okay, welcome back.

Let's go through the answers to these then.

So the volume of the first cuboid was 48 cubic centimetres.

The volume of the second cuboid was 48 cubic centimetres, and the volume of the third one was also 48 cubic centimetres.

Did you notice that too? All of those different cuboids had exactly the same volume.

What was happening each time? That's right.

They were just being rotated around, weren't they? Cuboid D has a volume of 180 cubic centimetres.

Cuboid E has a volume of 120 cubic centimetres.

And finally, cuboid F has a volume of 480 cubic centimetres.

Well done if you got all those.

Okay, and for question two then, how many different cuboids could we have made using 12 cubic centimetres? Here's one example for you.

Here's another example and here's another example.

Hopefully you came up with those three for yourself.

And finally, for question three, well, the answer to that is sometimes.

You can see in the middle here that B has the largest volume of the three shapes.

However, A and C are both taller than B.

So actually, we can see here that it doesn't have to be taller in order for it to have a greater volume.

However, both A and C actually are the same height, although C has a lesser volume than A, so that therefore goes to prove that the height of the cuboid does not necessarily dictate that the volume of a cuboid would be greater.

Okay, onto our last cycle now.

Cube numbers.

Let's have a look here.

We've got a shape on our screen now.

How would we work out the volume of this shape? Jun doesn't think we've got enough information here.

He can only see the length of the shape.

And in order to work out the volume of the shape, we're also going to need the height and the width, aren't we? Sam disagrees.

Sam thinks we do have enough information, don't we? In order to find the volume of this shape.

Sam says, "What'd you notice about the faces?" That's right.

The faces are square, aren't they? And therefore, what do we know about the length of each of the edges of a square face? That's right.

They'd be all the same length, wouldn't they? So actually, we do have enough information.

Each edge actually is two centimetres long.

So we can add these on, can't we? The width would be two centimetres and the height would also be two centimetres.

So thinking about this cube then, how would we work out the volume of it? Well, just like before, we'd multiply the length by the width by the height, wouldn't we? We know that the length is two centimetres, the width is two centimetres, and the height is two centimetres.

So two multiplied by two multiplied by two is equal to eight cubic centimetres.

Okay, and what did you notice this time? Look carefully at each of these cubes.

What did you notice? That's right, the length, the width and the height of each cube moving to the right was increasing by one centimetre, wasn't it? And in order to work out the volume of each of those, we had to multiply the length by the width by the height each time.

And the length, the width and the height were all the same, weren't they? So Jun's pointing out the volume of a cube is the product of three numbers that are the same multiplied with each other.

Take a moment to think about the difference between a cube which has two centimetres in length, width, and height, and a cube that has four centimetres in length, width and height.

The total amount of cubic centimetres required for the smaller cube was eight.

And the total amount of cubic centimetres needed for the larger cube with four centimetres length, width, and height was 64 cubic centimetres.

Jun says he can't believe how many more cubes were needed to make that larger cube on the right-hand side, considering the length, width and height have only changed from two centimetres to four centimetres.

And we call the product of three numbers that are the same multiplied with each other cube numbers.

And here you can see these numbers here are cube numbers.

Two multiplied by two multiplied by two is equal to eight.

So eight is a cube number.

Three multiplied by three multiplied by three is 27, and 27 is a cube number.

Four multiplied by four multiplied by four is 64.

So 64 is a cube number.

And another example here, 10 multiplied by 10 multiplied by 10 is 1,000.

And we know that 1,000 is also a cube number.

So when we think about measuring volume, cube numbers can be represented using either cubic centimetres or cubic metres.

Let's have a look.

Here we can see a cube which has three centimetres in length, in width, and in height as well.

So we multiply three centimetres by three centimetres by three centimetres, which would give us a product of 27 cubic centimetres.

So the volume of this cube would be 27 cubic centimetres.

Now look closely.

We've now changed the length, the width and the height of this cube, haven't we? Bearing in mind this is not to scale, we've now changed the length, the width and the height into metres rather than centimetres.

So our cube now has a length of three metres, a width of three metres and a height of three metres.

Therefore, we record that as three metres multiplied by three metres multiplied by three metres, which is equal to 27 cubic metres.

Okay, final check for understanding then.

A cube number is the product of A, B, or C? Take a moment to think.

That's right.

It's B, isn't it? A cube number is the product of three numbers that are the same.

Okay, and another quick check for understanding then.

Can you fill in the missing numbers? There we go.

Jun's saying that a cube number is the product of multiplying three numbers together that are the same.

And we know that one of these numbers is three metres, therefore the other two numbers must also be three metres.

And we know that three multiplied by three multiplied by three is equal to 27.

So the answer to this one, the volume here would be 27 cubic metres.

Okay, onto our last task for today then.

Can you fill in the missing numbers in our grid here? And then at the bottom, I'd like you to create two of your own examples to fill in as well.

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

Okay, so now I've filled in the answers for you just to go through and have a quick look for yourselves.

Well done if you managed to get all of those.

Let's just have a quick look at the examples we made for ourselves at the bottom.

So my examples were 12 cubed, so that's 12 multiplied by 12 multiplied by 12, and that was equal to 1,728, or I put 100 cubed as well.

So that would be equal to 100 multiplied by 100 multiplied by 100, and that would be equal to one million.

Well done if you managed to come up with some of your own.

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

Hopefully you're feeling a lot more confident when it comes to finding the volume of a cuboid or a cube.

To summarise our learning, we can say that to find the volume of a cuboid or a cube, you can multiply the length and the width and the height together.

We also know that it doesn't matter the order in which you multiply those three things together.

You could multiply the length by the width, by the height, or you could multiply the height by the width by the length.

It does not matter.

We will still get the same volume.

We know that the volume of a cube is the product of three numbers that are the same multiplied with each other.

And as a result, that product that we create from multiplying those three numbers that are the same together, we call this a cube number.

Thank you for learning with me today.

Hopefully again, you're feeling a lot more confident with your maths.

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