Lesson video

Hi, my name's Mr. Peters.

Thank you for joining me for this lesson today.

In this lesson, we're gonna be thinking about using our understanding of volume now and applying this to solve a range of problems in different contexts.

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

Okay, so by the end of this lesson today, you should be able to say that I can use my knowledge and understanding of volume to solve problems in a range of different contexts.

Throughout the session today, we're going to be using these two key words.

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

The first one is deduce.

Your turn.

The second one is dimension.

Your turn.

Let's have a think about what these mean in a bit more detail.

So if we use existing facts and logic, things that we already know to help us find new information, we can say that we have deduced the new fact, the new piece of information.

And a dimension is a measurement of an edge in one direction.

So, for example, we could class the length, width, or a height of a 3D shape to be a dimension of the shape.

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

The first cycle, we'll think about deducing missing dimensions, and the second cycle, we'll think about finding dimensions for a known volume.

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

So have a look at this shape here.

If we were to try and find the volume of this shape, we would need to probably think about splitting the shape into two separate cuboids and then finding the length, the width, and the height of each of those cuboids to find the volume of each of those and then to recombine them.

That would be one strategy which we could use to solve this.

However, to do that, we need to make sure that we do already have the length, the width, and the height for both of those cuboids.

Looking at the information we've already got on our compound shape, I'm not sure that we're gonna have those dimensions ready for us to tackle this problem.

So a good thing to do to start us off with is to write down all of the information that we can that we already know about this shape.

Let's see what we can figure out.

So let's start here then.

Can you see the four-centimeter dimension at the top of the shape? Well, if we look carefully, we can see that that dimension there will be the same all the way down throughout the shape.

We know that squares and rectangles have opposite sides that are the same length.

So we could say that opposite to that four-centimeter dimension at the top, we've got a dimension that runs across the middle of the shape, which is also four centimetres, and we've got four centimetres at the bottom of the shape, which represents some of the 10-centimeter dimension at the bottom.

So using our knowledge that squares and rectangles we know have opposite lengths that are the same, then we can deduce that all of these dimensions here are four centimetres long.

Let's see what else we can find.

Ah, there's another four centimetres, isn't there? We can see the rectangle on the top of the shape, can't we? If the dimension nearest to us is four centimetres, then the dimension furthest away from us is also four centimetres in length.

So that's a really good start, isn't it? Have a look for yourself.

Can you deduce anything else? Let's see what we came up with.

Well, we can see at the bottom of the shape, we've got a width of eight centimetres.

That dimension of the width is eight centimetres long, and we can see that actually, that eight centimetres would run all the way through throughout the shape as we move across to the left-hand side.

So that width is exactly the same width as the top of the compound shape.

So we could also say that those two there would be eight centimetres long.

And again, it's also a rectangle which has opposite sides, so they would be the same in length.

And that would also be the same for the rectangle down on the bottom right.

If we know that eight centimetres at the bottom of that rectangle is the dimension and also the dimension at the top of that rectangle would also be eight centimetres, do you think you could see eight centimetres anywhere else? That's right.

This dimension here would also be eight centimetres.

Again, we know that because there's a rectangle formed on the left-hand side here, and we already know that the dimension at the top of that rectangle is eight centimetres, and so, therefore, the opposite length at the bottom of that rectangle would also be eight centimetres.

We're nearly there.

We've nearly got all of the dimensions to this shape.

There's just a few more left.

Can you spot which ones we can try and figure out? That's right.

It might be useful for us to figure out this length here.

We already know the full length of the compound shape, and we know that part of this length is also four centimetres.

So we can now figure out the remaining length of this section here.

We can do that by finding the difference, can't we? We know that four plus something is equal to 10 centimetres, or 10 centimetres minus four centimetres would be equal to six centimetres, wouldn't it? So we can see that 10 centimetres was the whole here, and a part of that was four centimetres, so we are looking for this missing part, and therefore this missing part could be calculated by saying four centimetres plus something centimetres is equal to 10 centimetres, or 10 centimetres minus four centimetres would be equal to six centimetres, wouldn't it? So we can deduce that this length here, using the known knowledge that we have of the 10-centimeter dimension and the four-centimeter dimension, we know that this dimension here would represent six centimetres in length.

Now we can deduce a few more, can't we? If we know this length, what other lengths do we know? Well, we also know this length using our knowledge and understanding of squares and rectangles.

We also know this length, and we also know this length.

Gosh, we found out so much information about this shape so far, haven't we? It looks like we've got most of the information, haven't we? There's a lot of measurements now in this compound shape, isn't there? There's a couple more sets left to find, though.

We know that the height of this section of the compound shape is three centimetres tall.

So, what else do we know? Well, we also know that this is three centimetres tall, again using our understanding of squares and rectangles having opposite edges that are the same in length.

And if we know that the whole height of the compound shape is eight centimetres and we know a part of this height here is three centimetres, then we can start to work out the remaining heights, can't we? And again, we can do that by finding the difference, can't we? We know that the whole height is eight centimetres.

We know a part of the height is three centimetres, therefore, the remaining height would be equal to five centimetres.

And again, we could have worked that out by saying three centimetres plus something centimetres is equal to eight centimetres, or we could have said eight centimetres minus three centimetres is equal to five centimetres.

We've now actually found all of the dimensions for this shape, haven't we? Now we just need to calculate the volume for each section.

Andeep is going to use the partition-and-add strategy.

We're gonna partition this shape into two cuboids here, and now we can apply the measurements for each dimension that we've worked out.

We know the length of this cuboid is four centimetres.

We know the width of this cuboid is eight centimetres, and we know the height of this cuboid is eight centimetres.

We know the length of the cuboid on the right is six centimetres, we know the width is eight centimetres, and again, we know the height is three centimetres.

Now, using our knowledge of how to calculate the volume of a cuboid, we need to multiply each of these dimensions together, don't we? So we're gonna multiply the length, multiply by the width, multiply by the height.

I think is actually easier here to use the commutative law to multiply the two eights together, so we're going multiply the eights together to make 64, and then we're gonna multiply 64 by four, which leaves us with a volume of 256 cubic centimetres.

Let's work out the volume of the other cuboid.

Again, the length, the width, and the height.

We've got six multiplied by eight multiplied by three.

I think it's easier to multiply the six and the eight together first.

That's gonna give us 48, and then we can multiply that by the three.

48 multiplied by three is equal to 144 cubic centimetres.

So we've now got both of the volumes of the shapes, haven't we? As we know, it's called a partition and add strategy, so we need to recombine these shapes together to find the complete volume of the compound shape.

So 256 cubic centimetres plus 144 cubic centimetres gives us a total volume of 400 cubic centimetres.

We can say the volume of the compound shape is 400 cubic centimetres.

Okay, there we go.

Let's check our understanding now.

What is the size of dimension A? Take a moment to have a think.

That's right.

It's four centimetres, isn't it? And how did you deduce that? Yep, that's right.

We know that the face at the end of this cuboid is a rectangle.

It has measurements of four centimetres and three centimetres in its dimensions.

And we know that in a rectangle, the opposite dimensions are the same in length, so that would also be equal to four centimetres.

Well done if you got that.

Okay, here's another one now.

Look carefully.

Can you find the size of dimension B? Brilliant.

Well done.

And again, how did you deduce that? Yep, exactly that.

That face is another rectangle, isn't it? And again, one of the dimensions is four centimetres, so the opposite dimension must also be four centimetres.

Well done if you got that.

Okay, onto our task for today then.

What I'd like you to do is deduce the missing dimensions for each of these shapes.

And then, once you've done that for question two, what I'd like you to do is calculate the volume of both of these compound shapes.

Good luck with that.

And when you're done, come back, and we'll go through the answers.

See you shortly.

Okay, let's work through these then.

For compound shape A, we know that part of the height is three centimetres, so this height would also be three centimetres.

We know that the width is seven centimetres, so this width, this width, this width, and this width would also be seven centimetres.

We know the height of the other section of the compound shape is five centimetres, so we can say that this is also five centimetres.

And we also know that a part of the width of the cuboid at the bottom is eight centimetres, so this would also be eight centimetres.

And at the face on the top of our compound shape, we know that the length is two centimetres, so this would also represent two centimetres.

And to deduce the total height of the shape, well, we know that the heights combined would be three centimetres and five centimetres, so altogether, that would be eight centimetres.

Well done if you managed to get all of those.

For the second shape, I'm gonna click through one at a time, and you can tick them off to see if you've got them.

Well done if you managed to get all of those.

Okay, and we're off to task two then.

Now that we've deduced all of the dimensions, we can start thinking about calculating it, can't we? We're gonna partition this compound shape into these two parts here.

So let's start working on the compound shape to the left-hand side.

We know the length is two centimetres, we know that the width was seven centimetres, and we knew that the height was eight centimetres.

So that's the compound shape to the left-hand side.

And we also know that on the right-hand side that the length is eight centimetres this time.

The width is seven centimetres, and the height is three centimetres.

So let's calculate these then.

We can see in both sets of three numbers that are being multiplied together that there is a seven and an eight.

So let's multiply these together first.

That gives us a 56, isn't it? So now we've got on the left-hand side of our equation, two multiplied by 56, and then we're gonna add to that 56 multiplied by three, or three lots of 56.

Two lots of 56 is 112, and three lots of 56 is 168, and if you combine that together, that gives us a total of 280 cubic centimetres.

Well done if you got that one.

Okay, and then for cuboid B, We're actually gonna use a different strategy here.

We're gonna use a complete-and-subtract strategy.

So if we were to complete the shape here, we know the length would be eight centimetres, the width would be eight centimetres, and the height would be four centimetres.

And then, we would need to minus the volume of the section that we added on.

The volume of that section can be represented by two multiplied by three multiplied by four.

That's because it's two centimetres in length, three centimetres in width, and four centimetres in height.

So calculating each set of three numbers that are being multiplied, I would do the eight times the eight first of all, that was giving me 64.

And on the other side, I would do the three times the four, which would give me 12.

So now I've got 64 multiplied by four, and then we're going to minus two multiplied by 12, or two lots of 12.

64 multiplied by four would give us 256.

And then minus two lots of 12, which is 24, that gives us a total volume of 232 cubic centimetres.

Well done if you managed to get that as well.

Okay, onto cycle two now, finding dimensions with a known volume.

So have a look at our shape here.

What dimensions do we already know? Jacob's right.

We already know the length, don't we? And we already know the width, but we don't know the height, do we? And actually, we know the volume as well.

We know the volume has been given to us, hasn't it? So how can we work out what the height is? Andeep's suggesting that we record it as an equation to help us.

And you can see that I've written this equation underneath.

We've written three centimetres, which represents the length, multiplied by five centimetres, which represents the width, multiplied by something, which is representing the height, and all of that is equal to the volume in this case, which is 30 centimetres.

So we could say that three multiplied by five multiplied by something is the same as saying 15 multiplied by something is equal to 30 cubic centimetres.

Hmm, can we work this out now, do you think? Andeep's saying, "That's right." Two fifteens make 30, doesn't it? So that must mean that the height of this shape would be two centimetres.

Andeep used a times table fact to help him with that.

Whereas Jacob's saying we could have also used division.

We could have also said that the 30 divided by 15 would leave us with two, therefore, the height would be two centimetres.

Okay, here's another example.

Have a look at the information.

What do we know already? Well, this time we know the height, don't we? The height is five centimetres, and we also know the width.

The width in this case is four centimetres.

And we also know the volume.

The volume is 40 centimetres.

So how could we record this as an equation? Well, we could say the length is unknown, so we can use a question mark for that.

And we're gonna multiply that by the width, which is four centimetres, and then multiply that by the height, which is five centimetres, and that is all equal to 40 cubic centimetres.

Hmm, how would we tackle this next then? Something multiplied by five multiplied by four is the same as saying something multiplied by 20 is equal to 40 cubic centimetres.

So, hmm, do you know any facts that could help you to solve that? That's right.

If we use division, we could say that 40 divided by 20 is gonna be equal to two.

So we now know that the length of this shape is going to be two centimetres.

Or, as Jacob said, we could have used our understanding that two lots of 20 is equal to 40, or two lots of two is equal to four, so two lots of 20 is equal to 40.

And as a result, that would mean the length again was also two centimetres.

Well done if you managed to spot that.

Okay, here's a slightly different one now.

Have a look for yourself.

Have a look at our cereal packet.

What do we know, and what don't we know? Well, in this example, we know the volume, don't we? We also know the length, and we also know the height.

But this time, it's the width that we don't know, isn't it? Let's record this as an equation again then.

So that would be 20 multiplied by something multiplied by 30 would be equal to the volume.

Hmm, how would we tackle this one then? Well, we could multiply the 20 and the 30 together here, couldn't we? So 20 multiplied by 30 would be equal to 600, wouldn't it? 'Cause 20 multiplied by three is 60, so 20 multiplied by 30 is 600.

So we could say this is the same as saying, 600 multiplied by something is equal to the volume, in this case, 3000 cubic centimetres.

So again, we've got two strategies.

We can either use a fact if we know any facts, or we could use some division.

Andeep is saying that 3000 divided by 600 is equal to five centimetres.

So he is saying that the width of this cereal packet is five centimetres.

Jacob's opted for a fact this time.

He says he knows that five multiplied by six is 30.

So five multiplied by 600 is equal to 3000, isn't it? So there we go.

Two different strategies to work out the width of this cereal packet.

Well done if you managed to get that as well.

Okay, time for you to check your understanding now.

Which shape has an unknown length? A, B, or C? That's right.

It's C, isn't it? A has an unknown width, B has an unknown height, and C has an unknown length.

And, okay, for this one, can you tick the calculations that would help you to solve the missing dimension? Take a moment to have a think.

That's right.

It's actually in fact all of them, isn't it? If we were to write the calculation as an equation, we know the volume, we know the height, which is nine centimetres, and we know the length, which is six centimetres.

It's the width that we don't know.

And we know because multiplication is commutative, it doesn't matter which order we place the length, the width, or the height when we multiply 'em together.

So nine multiplied by six multiplied by something is equal to the volume of 162 cubic centimetres.

That would work.

Something multiplied by six multiplied by nine would also work just as well.

We know that six times nine is 54.

So 54 multiplied by the missing dimension would also give us the volume.

And then finally, we could use division, couldn't we, to help us solve that? We could divide the complete volume by the known length so far, which would leave us with the missing dimension.

So the volume of 162 cubic centimetres divided by 54 would be equal to the missing dimension.

Well done if you spotted that as well.

Okay, onto our final task for today then.

Can you find the missing dimensions for each box? And then, once you've done that, have a go at solving these worded problems. And another worded problem for question three here, we'll have to think a little bit more deeply.

Good luck with those.

And when you're ready, come back, and we'll go through them.

Okay, welcome back.

Let's go through these first ones then.

So for the first one, we have a missing width, don't we? So we can write an equation which represents five multiplied by four multiplied by something equal to the volume.

This would mean that we've got 20 multiplied by something is equal to 120, which therefore would be six.

20 multiplied by six is 120.

For the second one, we've got a missing height, haven't we? So we've got seven multiplied by one multiplied by something is equal to 35, therefore, seven multiplied by one is seven.

So seven times something is equal to 35.

And we know that 7 times 5 is 35, so the missing dimension is five centimetres.

And for the last one, we've got a missing length, haven't we? So we've got something, the missing length multiplied by 12, which is the width, and the height, which is five.

We know that 12 multiplied by five is 60.

So something times by 60 is equal to 120.

And again, we know that two six of 12, so two sixties are 120.

Well done if you managed to get all of those three.

Okay, for the second task then.

Andeep has 24 cubic metres of sand to fill a sandpit.

He knows that the sandpit is two metres deep, but he doesn't know the length and the width.

He also knows that he's gonna have 24 cubic metres to fill in that sandpit.

So how could we work out the width and the length in this respect? Well, we can write it as an equation.

We know that two multiplied by something, multiplied by something else is equal to 24.

Hmm, what do you know? Why don't we try something out? Why don't we try three? Two multiplied by three would be equal to six, wouldn't it? So then we've got six multiplied by something which would be equal to 24, so that would be four, wouldn't it? So we could say that two multiplied by three multiplied by four is equal to 24 cubic metres.

You may have also decided to go down the route of dividing 24 cubic metres by the depth.

So 24 divided by two, which would leave you with 12.

And then you need two numbers that multiplied together to make 12.

So it could have been three or four, or it could have been six and two as well.

So we could have had two multiplied by three multiplied by four.

We could have had two multiplied by two multiplied by six.

We could have also had two multiplied by one multiplied by 12, because one times 12 is equal to 12.

And then multiply that by two would equal to 24 as well.

Which one of those do you think were the most appropriate for us to have for the size of the sandpit? That's right, probably the first or the second one.

The two by three by four or the two by two by six.

Having a 12-meter long sandpit is probably a bit long for most people's gardens.

And then, for question B, a swimming pool has 300 cubic metres of water in it.

What might the dimensions of the pool be? Well, we're looking for three numbers that multiply together to make 300 cubic metres.

A good way to start with this is look at the volume.

The volume is an even number, so we know that two could be one of the factors.

So if two was a factor, we could divide 300 by two, which would leave us with 150.

And then we need two numbers that would multiply together to make 150.

So we could have two multiplied by two lots of 75, or you could have two multiplied by three lots of 50, or we could have six multiplied by 25.

And as Andeep's pointing out, I think there'd be lots of different solutions that we could do for that.

I wonder how many different ones you found? Okay, and then the last question as well.

Jacob makes a small cardboard box in Design and Technology.

Andeep's box is the same in length, double the height of Jacob's, and has a width of six centimetres.

What's the difference in volume of these two boxes? Well, we can work out the volume of Jacob's box by multiplying four by seven by two.

Those, as I mentioned, is given to us are the length of the width and the height.

That gives us a total volume of 56 cubic centimetres.

And we can work out the volume of Andeep's box by thinking about what those lengths, and widths, and heights would be.

So we know that they have the same length.

So the seven is the same.

The width is six centimetres, so we can add that one in.

And we also know that the height is double that of Jacob's box.

So if Jacob's box height is four centimetres, that must mean Andeep's box height is eight centimetres.

So to work out the volume of Andeep's box, we can do eight multiplied by seven multiplied by six.

That gives us a volume of 336 cubic metres.

And then, to find the difference between the two, we can subtract the volume of Jacob's box from Andeep's box.

So 336 cubic centimetres minus 56 cubic centimetres is equal to 280 cubic centimetres.

Well done if you managed to get that.

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

Hopefully you're feeling a bit more confident in deducing missing dimensions as well as finding a dimension when you know the volume.

To summarise our learning, we can deduce the missing dimensions of a compound shape by using other known dimensions to find the difference.

If you know the volume of a shape and you also have a missing dimension, then we can use our understanding of multiplication and division to help us find that missing dimension.

And we also know that unknown lengths, widths, and heights can be applied to many different contexts in our everyday lives.

Thanks for learning with me today.

Hopefully you've enjoyed yourself.

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