Lesson video

Hi, my name's Mr. Peters and in this lesson today, we're gonna be thinking about extending our understanding of finding the volume of shapes to compound shapes.

When you're ready to get started, let's get going! Okay, so by the end of this lesson today, you should be able to say that I can explain how to calculate the volume of compound shapes.

Throughout this lesson, we've got two keywords we're gonna be thinking about.

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

If you're ready, the first one is compound shape.

Your turn.

The second one is strategy.

Your turn.

Let's think about what these mean then.

A compound shape is a shape created with two or more basic shapes.

And a strategy is a planned approach used to solve a problem.

Watch out for these words throughout our lesson as we go.

Our lesson today is gonna be broken down into two cycles.

The first cycle is gonna be thinking about a partition and add strategy, and the second cycle will be thinking about other strategies used for finding the volume.

Let's get started.

Throughout our lesson today, Aisha, Jun, and Lucas are gonna be with us and they're gonna help us along the way, sharing their thinking and their questions as we go.

So, let's start here then.

Aisha is asking, I wonder how we could work out the volume of this shape? Lucas thinks, well, surely we can just multiply the length by the width, by the height, and that will give us the total volume of this shape.

Here are the dimensions of the shape.

Lucas is saying that the length is nine centimetres, the width is seven centimetres, and the height is eight centimetres.

So, we can multiply nine centimetres by seven centimetres by eight centimetres, and that should hopefully give us the volume of this shape.

Aisha is questioning this.

She doesn't think this is gonna work because this isn't a cuboid, is it? It's a different shape.

So we're gonna need to think slightly differently about how we can tackle this.

She says it looks like two cuboids that have been joined together.

And where two basic shapes are joined together, we call this a compound shape.

So, let's have a look a bit more closely then.

Lucas is saying, yes, I understand now.

I can see why you were saying you think it looks like two cuboids put together, 'cause we could partition it here, couldn't we? So there we go.

We've now got two separate cuboids, haven't we? So, now that we know that this compound shape could be thought of as two separate cuboids, let's put it back together and think about the measurements that we know already.

Aisha's saying we can calculate the cuboid on the right hand side by multiplying the length by the width by the height.

She's managed to find these measurements already.

The length would be five centimetres, the width would be seven centimetres, and the height would be four centimetres.

So, the volume would be equal to five multiplied by four, multiplied by seven.

In order to do this, we can do five multiplied by four first.

That would be equal to 20, and then we need to multiply that by seven.

So the volume would be equal to 140 cubic centimetres.

Now, let's have a think about the cuboid on the left hand side, shall we? Aisha's saying, so far we only have the height.

We only have the height at eight centimetres.

We don't actually know the length or the width just yet.

I wonder how we can work these ones out then.

Ah, well Aisha's saying that the length could be found here at the top of the shape.

If that length is moved down to the bottom, you can see that they represent the same length.

So actually, the length would be four centimetres, wouldn't it? So, now that we know that the height is eight centimetres and the length is four centimetres, all we need to do now is find the width.

Can you have a look at the shape? Where do you think we can find the distance of the width from? Well, if we separate the shapes again, hopefully you'll see that the distance of this width here is actually going to be the same as the distance of the width here on the cuboid from the right hand side.

And we can make that really obvious by overlaying the shapes with each other.

Let's have a look at that.

There we go.

We can now see that the cuboid on the right hand side overlaid to the cuboid on the left hand side now shows that those widths were exactly the same length.

So the width would be seven centimetres.

Therefore, in order to work out the total volume of this cuboid, we would need to do eight centimetres multiplied by four centimetres multiplied by seven centimetres.

So, we could do the eight multiplied by the four first.

That would give us 32.

And then we can multiply that by seven, altogether, the volume would be 224 cubic centimetres.

Now we can start thinking about adding these volumes together, can't we? We know that the volume of the cuboid on the right hand side was 140 cubic centimetres, and we know that the volume of the cuboid on the left hand side was 224 cubic centimetres.

So when we add these two together, the total volume of this compound shape would be 364 cubic centimetres.

Lucas has given a name for this strategy.

He says we can call it the partition and add strategy.

We partitioned the compound shape at first into two separate shapes and then we added the volumes back together at the end, didn't we? Okay, time for you to check your understanding now.

Can you choose the compound shape that would be best to use a partition and add strategy to solve it? Take a moment to have a think.

That's right, it's A and B, wasn't it? We could partition the cuboid on the left into a smaller cuboid on the top and a larger cuboid underneath it.

And for B, we could partition a cuboid into a larger cuboid on the left and a smaller cuboid on the right hand side.

Well done if you managed to get that.

Okay, and another quick check for understanding then.

Could you draw two dotted lines to show two different ways that we could partition this compound shape into two separate cuboids? Take a moment to have a think again.

Okay, so here's one example.

If we draw a dotted line here, we could partition the cuboid on the left with the cuboid on the right hand side, or we could partition the line here, couldn't we? So we could slice off a cuboid from the top and separate that from the bottom cuboid, couldn't we? Well done if you manage to see those.

Okay, onto task one then for today.

What I'd like you to do is partition these compound shapes into smaller cuboids that would help you to calculate it more easily.

And for question two, what I'd like you to do is calculate the volume of these compound shapes.

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

Okay, welcome back.

Let's see how you got on.

Cuboid A could have been partitioned like this.

Cuboid B could have been partitioned like this or like this.

And cuboid C could have been partitioned into three separate cuboids, couldn't it? And you can see how we've done that here.

Or we could have partitioned it into three separate cuboids just like this as well.

Well done if you managed to get those.

And then, working out the volumes of these different compound shapes.

Well, the volume of cuboid A was 108 cubic centimetres.

The volume of cuboid B was 126 cubic centimetres.

And the volume of cuboid C was 14 cubic centimetres.

Let's take a closer look at how we worked out cuboid C.

You can see how we've partitioned it into three separate cuboids.

You've got the two tall standing cuboids at either end, and then the cuboid lying down in the middle.

Now, there's a length that we need to work out here and that is the length of the cuboid lying down in the middle.

Now, we can see that the total length of the whole shape at the bottom would be six centimetres.

And at the top, we can see that the length of the two standing towers were both one centimetre in length.

Therefore, if we subtract both of those two lengths of one centimetre from this total amount of six centimetres, we know that this lying down length in the middle here is four centimetres in length.

So, to work out one of the towers then, we know it has a width of one centimetre, a length of one centimetre, and a height of five centimetres.

So the total volume here would be five cubic centimetres.

And we obviously need two of those, don't we? So that would be 10 cubic centimetres altogether, plus the lying down one.

We know it has a length of four centimetres now and it has a width and a height of both one centimetre.

So again, that would be equivalent to a volume of four cubic centimetres.

So the total volume of the whole of the shape would be 14 cubic centimetres.

Well done if you managed to combine all of the skills from that first cycle to help you solve that one.

Okay, onto cycle two then.

Thinking about other strategies to find the volume of compound shapes.

So Aisha is asking, I wonder if we can think of another way in order to tackle this.

Lucas is saying, well, it's funny you should say that because instead of it looking like two separate cuboids being joined together, Lucas felt that it looked like one big cuboid with a big chunk missing out of it.

Here's an example of what he meant.

Aisha's saying, gosh, I didn't think of it like that.

Well, let's have a go using Lucas's strategy in order to tackle this.

So, we can now see our compound shape with the additional section added onto it.

We can work out the volume of the complete cuboid now by multiplying the length by the width by the height.

So, nine multiplied by seven multiplied by eight.

And that would give us the volume of the complete cuboid.

So, if we were to do nine multiplied by eight, multiplied by seven, I would do eight multiplied by seven here.

That would give me 56.

So now I've got nine multiplied by 56, and that would be equal to 504 cubic centimetres.

So now we know the volume of the complete cuboid.

Now what we need to do is subtract the volume of the smaller cuboid that we added on to complete the shape in the first place.

So, we can see here that the length of this cuboid would be five centimetres.

The width remains the same at seven centimetres and the height we can see is placed here at the back, and that's four centimetres.

So to find the volume of this shape then, we can multiply five by seven by four, can't we? Now, if this was me, because we know that multiplication is commutative, I would multiply the five and the four first.

So five multiply by four is going to give me 20.

And then I'm gonna multiply this by seven.

So altogether, this is gonna give me a volume of 140 cubic centimetres.

We know that the total volume of the large cuboid altogether would be 504 cubic centimetres.

And we know that the section that was added on originally has a volume of 140 cubic centimetres.

And to find out the volume of the compound shape, we would need to subtract that smaller amount, won't we? So, 504 cubic centimetres minus 140 cubic centimetres would be equal to 364 cubic centimetres.

And there we go.

So, the volume of this shape is 364 cubic centimetres, which is exactly what we had earlier on.

Lucas has given his strategy a title as well.

He's called it the complete and subtract strategy.

We first of all need to complete the shape by putting on an extra section, don't we? And then, we need to subtract that section from the total volume at the end, don't we? In order to find the volume of the original compound shape.

Jun says, I love that strategy, but he's also found his own little different way of tackling it and he's wanting to share that with us as well.

Should we have a look at what he did? Jun is calling this the slice and slide strategy.

Hmm, I wonder what this means.

Let's have a look.

Well, did you notice what happened? That's right.

He sliced off a cuboid from the top, didn't he? And he moved that cuboid down towards the end, didn't he? Hmm, that seemed to work really well for you, isn't it? And what it's done is make the cuboid slightly longer, hasn't it? So, let's have a look at that in a bit more detail then.

Jun said that he noticed that the height and the width of the bottom cuboid was the same as the top cuboid and therefore, we could use his strategy of slice and slide.

If the height and the width weren't the same, then we wouldn't be able to because the shapes would not look the same and therefore, we'd not be able to use a simpler calculation in order to tackle that.

Jun's saying that he noticed that the height and the width of this cuboid on the bottom was the same as the cuboid on the top.

Let's have a look at that.

Well, here's the height of the cuboid on the bottom, and here's the height of the cuboid on the top.

They're both the same.

And here's the width of the cuboid on the bottom, and here is the width of the cuboid on the top.

And as these are the same, it means we can use the slice and slide strategy to solve the total volume of this shape.

So, let's do that now.

Let's use the slice and slide strategy.

There we go.

And now we have extended our shape in length, haven't we? He knows that the length of the larger cuboid on the bottom was nine centimetres.

So now we just need to add on the length of the cuboid that has been sliced and slid.

So in order to work that out, he knew he was given the length of the cuboid by looking at the top edge of the face on the cuboid.

And he says that he knows a square, which is what the shape of the face is, has all the same size edges, doesn't it? So, that must mean this length here must also be four centimetres.

Therefore, what we need to do is add on those extra four centimetres to the original nine centimetres that we had, which then makes the total length of 13 centimetres.

We now have the length, the width, and the height to work out the volume of the cuboid.

So, 13 centimetres multiplied by seven centimetres multiplied by four centimetres.

Again, because I know multiplication is commutative, I'm gonna multiply the 13 and the four first of all here.

That leaves me with 52 multiplied by seven.

So, the total volume of the whole shape again would be 364 cubic centimetres.

And as Jun's quite rightly pointing out, it gave us the same volume as the other two strategies as well, didn't it? That leads me to ask you this question then.

Which strategy did you prefer? Did you prefer the partition and add, the complete and subtract, or the slice and slide? You wouldn't necessarily be able to apply all of these to every single compound shape when working with it.

You may need to use one in particular for certain types of shapes.

So, it's really useful to have a really strong grasp for all three of these types of strategies.

Okay, time for you to check your understanding now.

Can you choose the shape that would be best to use a slice and slide strategy? Take a moment to have a think.

That's right, it's B, isn't it? B would be best to use a slice and slide strategy, because we could slice off the cube off the top and place it into the middle of the shape, couldn't we? Let's have a look at this one as well.

Can you choose the shapes that would be best to use a complete and subtract strategy on? Take a moment to have a think.

That's right, it's A and C, isn't it? You can see that on A and C, there's only a small section that would need to be added in order to complete the shape each time.

That's slightly different for B.

You could use a complete and subtract strategy for B, but you are adding quite a big section, so you're subtracting quite a large amount.

So it's a better strategy for when there's only a little part of a shape missing and you can add that little bit on quite easily.

Okay, and onto our final task for today then.

What I'd like you to do here is to calculate the volume of these shapes using either a complete and subtract strategy or a slice and slide strategy.

And then, once you've had a go at that, what I'd like you to do is think about which expression represents the correct volume of the shape below.

There might be more than one answer for this.

And finally, I'd like to have a go at creating your own shape, where any one of those strategies could be used to find the volume of the shape completely.

You might like to either draw it or create it with cubes yourself if you can.

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

Okay, welcome back.

Let's run through the answers then.

So, for compound shape A, the volume was 108 cubic centimetres.

For compound shape B, the volume was 150 cubic centimetres.

And for compound shape C, the volume was 29 cubic centimetres.

Let's have a little think here about which expressions would match the correct volume of this shape.

Okay, well done if you managed to get those.

And then finally, here's an example of a shape which could utilise all three of those strategies.

Hopefully, you came up with something similar for yourself.

Aisha is now saying to Lucas, if you let me know the size of the edges, I'll have a go at calculating it.

Well, good luck with that, Aisha.

Hopefully you can explore lots of different ways of solving the volume of this compound shape.

Okay, there we go.

That's the end of our learning for today.

Hopefully you're feeling loads more confident about the different strategies you could use to find the volume of different compound shapes.

To summarise our learning today, a compound shape is where two or more basic shapes are joined together to make a larger shape.

And the volume of a compound shape can be calculated using one of three different strategies.

You could use a partition and add strategy, you could use a complete and subtract strategy, or you could use a slice and slide strategy.

Well done for keeping up with that learning today.

And it would be really interesting to see if you can apply this to your everyday lives.

Take care and I'll see you again soon.