# Lesson video

In progress...

Hello, my name is Mrs Buckmire.

And today's lesson title is counting cubes 2.

Now, make sure you have a pen and paper.

Remember, pause the video whenever you like.

I'll be asking you to pause sometimes to practise, but also pause if you need some more time as well and do rewind if you need to listen to something again, and that's absolutely fine.

Okay, let's begin.

So, this task is quite a substantial task if I'm honest, okay? It's going to take some time, but it will be really worth it, okay? So Yasmin has built this cube out of smaller green and blue cubes.

The questions are, how many blue cubes could there be? What if the cube is the same on all sides and the inside is hollow, then how many blue cubes are there? And what if the cube is the same on all sides and the inside is full of blue cubes, then how many are there? Okay? So, I do want you to spend some time, pause the video.

I'd even recommend actually put it next to the bottom and go into the worksheet, have a look at this more work to do.

Take time on this.

I've also labelled on the left hand side green and blue and that's the pattern all sides if you can need it.

Okay.

So pause the video and have it go now.

Okay.

So how did we do? Okay, how was that? So let's go through the first one.

So how many blue cubes could there be? Well, I can see there's at least one, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

So the minimum is 12.

What would the maximum be? Hmm, well, the maximum would be, how many cubes that are all together and then you can subtract all the green ones we see.

Okay.

So how many green ones do we see? So at the front I see one, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

So the back then, well actually, there's 12 at the front, let's write that down actually.

It's a 12 here and at the back I see, also I see sorry.

13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25.

So all together, we can even put up here, there are 25 green ones on display.

That's all we can see, there's at least 25.

Now, so we want to find how many cubes they're all together, the little ones, and we can subtract 25, and then we know the most blue ones there could be.

Hmm.

So how many cubes are there altogether? Well, we know it's a bigger cube.

So this layer, is the same as this layer, the same as this layer, the same as this layer.

So the top layer, there are four, eight, 12, 16.

Say 16, so two layers they're 32, three layers they're 48, four layers they're 64.

So 64 all together.

So the most blue there could be is 64 take away that 25 which equals 39.

So 39 is the maximum number of blues.

Well done, if you've got that.

We could write as inequality.

So if n is the number of blue cubes, then n is less than or equal to 39 and greater than or equal to 12.

And they'll have to be integer, so you could even add some more constraints to it.

Well done if you have anything like that.

Okay.

What if the cube is the same on all sides and the inside is hollow? Hmm.

Okay.

So it is the same on each side, then that means each side, there are one, two, three, four, and a cube.

Hmm.

How many faces does a cube have? Wait, what's a face? Faces are the flat surface of 3D shape.

So this one big cube has six faces, and each face has four blue cubes.

So four times six, equals 24.

So there'd be 24 blue cubes 'cause the inside is hollow so it doesn't matter what's happening on the inside.

Well done if you've got that.

Okay.

The final question for the try this, was what if the cube is the same on all sides and the inside is full of blue cubes? Right, there are two ways to do this.

Now the previous answer was 24, so that was how many on the outside.

So now we're thinking okay, to that one, and the inside, cause it's full of blue cubes.

So how many are on the inside? Now if you're really good at visualising, you might believe to kind of cut off the top, open it up and see inside.

There is a smaller cube that is two by two by two, so eight.

So that means, actually the answer is going to 24 plus eight to give us 32.

Well done if you got 32.

There's one more method i think maybe some of you guys might have done.

And that's seeing, how many cubes that are all together, and then taking away kind of the outside green ones.

Okay.

So let's try that.

Well, actually we already know all together, there were 64, altogether.

So how many green ones were on the outside all there? So let's just look, kind of this column.

So there's four there and then five, six, seven, eight, nine, 10, 11, 12, so 12 in this front column.

That means there's also the 12 in the back column.

So that makes 24 green, 24 green ones.

Now, if we take the next column, there's one, two, and there's two on that side so that's another four.

And the next column one, two, three, four.

So another four, so all together, four plus four is eight plus 24, what is it? It is the 32.

And then we could do 64 take away 32 equals 32.

So that's another method, did you do a different way? Oh my god! There's so many different ways, I could spend all day just exploring different methods and things to do, okay? But different strategies even, but well done if you've got that answer, hopefully if you didn't, you understand it a bit better.

Let's get on with our learning.

Okay, so below is a solid cuboid.

Now, this person says, I think four of my fists takes up the same space as the cuboid.

This person thinks, I think the volume of this shape is 20 cm³ because 20 of my unit cubes would take up the same space as the cuboid.

Hm.

What is the same and what is different? Pause the video now and have a think.

Okay, so what did you think about? What's the same? Okay, and what's different? So both of them are talking about taking up space.

So they're the person on the left is saying, I think four of my fist takes up the same space, person on the right is talking about actually unit cubes.

So both, doesn't mean that's the same.

What else is the same? If you approximate that person's fist to five unit cubes, then actually, five times four is 20.

So they're talking actually about the same amount.

So, 20 cm³ volume, okay.

What's different? Yeah, so the first face a very informal strategy.

And sometimes we do is in formal strategy to find out volume.

And that's what they've done while the second person is thinking, actually we know that the cube is one centimetres cubed, and actually how many exactly fits inside and it was 20.

So volume is a measure of three-dimensional space, okay.

So, it's how much space something's taking up.

Compose it and write that down.

So we know we can find volume by counting the centimetres cubed, and each of these shapes, when you see this cube, it's a one centimetre cube.

So one centimetre, one centimetre, one centimetre, okay.

And I want you to order the first shapes from lowest to highest volume.

The second is to find the volume of the cuboid.

The third is about reasoning and the fourth question.

I want to know how many different cuboids can you make with a volume of 12 cm³? And you can just do little sketches of them, okay? So do have a go.

I recommend looking at the worksheet for this, because I think it'd be a bit clearer, but, make sure you just have it done and write down what answers you think? Okay, so let's have a go.

Let's see how you did.

So this first one, what was the volume? So we can count the number of centimetres cubed, unit cubes.

So one, two, three, four, five, six, seven, eight, that's 8 cm³.

The next one's one, two, three, four, five, so 5 cm³.

Then one, two, three, four, five, six, seven, eight, again, centimetres cubed.

And one, two, three, four, five, 6 cm³.

For lowest, this one is the lowest, this is number two and these are joint third, lowest they are the highest.

What is the volume of this cuboid? Okay, so I'm going to count these one, two, three, four, five, six, seven, eight.

And how many lots of eight do we have? Well, that's the first column.

One, two, lots of eight, three, lots of eights.

Four, lots of eight.

So I get it to be 32 cm³.

Zaki thinks his shape has a greater volume than the one in question two, because it is taller.

Do you agree? But did you agree? No, so the reason he's incorrect is actually if you just work it out and see, so we could see that if we count.

So here we have one, two, three, four, five, six, seven, eight, nine, 10, 11 in this front, and all the others have 10 'cause that one's not there.

So plus 10, plus 10.

So we get 31 centimetres cubed, that is smaller.

So even though it's taller, it does not necessarily mean it has a greater volume.

And actually he is incorrect.

How many different cuboids can you make with a volume of 12 cm³? Okay, let's have a look one.

So, you could actually have, now these are not going to scale in any way, but you could have this being six centimetres, this been two centimetres and there's been one centimetre.

That would work, because then we would have all together.

We would have the 12 cm³.

Can you think of any other ways? Yeah, so we could have 12 centimetres, one centimetre and one centimetre.

So if these were just 12 cubes going across? Yeah, maybe you'd bought actually some different one, where actually you have two centimetres, two centimetres and three centimetres.

Good job.

Okay.

So what I want you to do, is find the volume of these cubes and tell me what comes next in the pattern.

So you find the pattern and tell me what comes next.

Then it'll be a bit of challenge, how many cubes are there and the 10th shape in this pattern? Okay, I believe you can have a good go with this.

Okay, so do pause and just, you know, just have fun with it.

Write some things down and have a go.

Okay, so what was the volume of each cube? So, this first one was 1 cm³.

This next one had eight, so 8 cm³.

This next one had nine in the first one, then another nine, 18, then another nine so, 27 cm³.

So what would the next one have? So here at the front, it would be four and four to make 16.

And it would go 4D.

So 16 times four is 64.

Oops! Okay, so this was the first one.

And in this one, each side length was one.

This was the second shape and in it, each side length was two.

This was the third one, the third cube and in it, each one was three, and this was the fourth.

So do you notice a pattern? So this is one times one times one, two times two times two, three times three times three, four times four times four.

There's a very special name for these numbers, these answers you've got, what's the name of these? These numbers one, eight, 27, 64? Yes, they are cube numbers and they form cubes, surprise, surprise.

So what is the 10th shape in this pattern? X in every 10 times, 10 times 10.

So, 100 times 10 is 1000.

So there'd be 1000 cm³.

Really well done, if you got that.

Okay, what would you say was the main thing you need to remember today? Okay good, write that down.

Excellent.