Lesson video

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Hello, my name is Mrs. Buchmire, and today I'm going to be teaching you about surface area conjectures.

Now don't worry if there's anything there you don't understand.

All should be clear by the end of the lesson.

So for now you have to make sure you have a pen and paper, and your mind is engaged and you're ready to learn.

Remember, pause the video whenever you like if you need more time, and rewind and rewatch bits that you don't quite understand.

Watching again can sometimes help.

Okay, so for your try this, here we have three cubes, and I want you to find the surface area of these cubes.

And then tell me what comes next in this pattern? Can you also tell me what is the surface area of the 10th shape in this pattern? Now, if you're like, "Oh, there's so many things here I can't do.

I have no idea," pause, and just wait a moment, and I will give you support.

If you're confident, then you should pause the video.

Okay, so firstly, what is surface area? So surface area is the area of the surfaces.

So if we take each of these we made out of the individual cubes being unit cubes, you can even say they're centimetre unit cubes, then to find the surface there, you can count the number of faces that are visible here.

So for the first one, for example, I would see that at the front there's one.

It's just one unit cube.

And then there're six faces, so it's going to be six units squared or I'm going to say they're centimetre one.

So centimetre squared.

Okay? Pause the video and have a go.

Okay, so I'm going to go through first the service area and this task, and then I'm going to talk to you about conjectures.

So first find the surface there.

So if you saw in support, I said this one was six.

And I did that because the front was one and there're six faces.

So one, I times it by six.

So it's six.

And I'm going to say they're centimetres unit cubes.

So centimetres squared.

Okay, and then this next one.

So there's one, two, three, four units at the front.

So the front area is four centimetres squared and there're six of them.

So four times six.

It is 24 centimetres squared.

And what about this one? here is nine.

So nine centimetres squared.

Times it by six 'cause there're six faces.

And because it is cubed, so all the faces are equal.

So the area of all the faces are equal.

So it's 54 centimetres squared.

Hmm, so what will be next? Did you have to draw it or do you just know? Right, I'm going to draw it.

Here's my lovely cube.

Oh, it's so great.

Look at this.

Oh, it was all going so well I thought.

Here we go, here we go.

I'm getting it back.

There you go.

Four by four, ach.

You're making me do the sides.

Okay, I'm having a go.

Here you go.

There we go.

On the top as well.

Right, your picture's probably way better than this.

Okay, so it's four by four by four.

So the front is 16.

There are six faces.

So 16 times six.

What did you get? Six times six is 36 plus 60.

So 96 centimetres squared.

Okay, so do you notice a pattern? How are you going to work out the 10th shape? Aah, all of them have times in my six, but it's not 10 times six, is it? 'Cause that would be 60, and that'd be smaller than our fourth one.

Aah, so first to get these numbers I'll circle them.

These are all squared numbers.

So this one is one squared, two squared, three squared because the faces are squared.

It makes sense, you know? So it should be 10 squared.

10 squared times six.

So 10 times 10 is 100 times six.

So it's 600 centimetres square for the 10th pattern.

Well done if you got that.

So what I was doing there is kind of trying to spot patterns.

That is what matters about math.

There are so many patterns there.

Mathematicians love finding patterns, and when you find a pattern, we might make a conjecture.

So I might say, "Oh, so for all cubes to find the surface area, you would do the edges squared times six." So if it was 100, I'd do 100 squared times six.

If it was 575, I'd do 575 squared times six.

And that would be a conjecture because I believe it's true, as in it's kind of an educated guess, but it's not maybe yet verified by me, but I could then prove it.

Okay, so conjectures, we can have a go at proving, but we haven't yet proven it because what we think, hmm, this is probably what's going on.

So in your independent task, I want you to do something quite similar.

So now I've gone through that try this task.

Hopefully, you can better understand find the surface of these cuboids and think about what comes next in the pattern.

So what's the fifth one? But then also try to find the surface area of the 10th shape.

And as a challenge, can you find the surface area of the nth shape? So for any, let's say for any random height or, sorry, length, what would it be? So by length, I mean here.

So for any length, if this was N what would the surface area be? This is quite tough, okay? So do pause the video and have a go.

Okay, so find the surface there first.

That's our first bit before we get onto the next part.

So the first one.

So the front was four and the back then is also four.

So four times two.

The left-hand side.

So here I can see two, and it's two on the other side as well, so four.

The front, sorry.

The top is two and the bottom is two.

So another four.

So all together they're 16.

For the next one, did you get 24? And then 32? And final one, I got 40.

Okay, what was the next one then? What do you think? Could you draw it? So maybe you could.

I can even add onto this one and imagine it.

So what's going on when I add it on.

So I'm adding on four cubes.

So I'm getting now the four original back ones are being covered, but then replaced with the four, the new back ones.

So that makes sense.

But I'm adding additional two here.

Two on the back, on the kind of right-hand side, left-hand side there.

Two at the top and two at the bottom.

So I'm adding on additional eight more visible surfaces.

That's what it looks like when I'm doing it here.

Let's see here.

The numbers 16 plus eight does get to 24.

24 plus eight does get to 32.

32 plus eight does get to 40.

So our next one, it does seem is eight more, and it's 48.

Hmm, did you think of the nth term? For the challenge, okay.

Let's first do the 10th shape.

So the 10th shape is going to be, mm.

It's hard to do that at the nth term I think.

You could keep going, but let's see.

If it was the nth term.

So for nth term, I said, "Considering the length, what would it be?" So the first one was one length, and we got to 16.

The second one was two.

The third one was three.

The fourth one was four.

The fifth one would be five.

So for the 10th one, we know the length is going to be 10.

Now we can see actually this forms a sequence, and it's a linear sequence 'cause it goes up by eight each time.

So think about the nth term or just in general, what sequence of numbers goes up by eight each time? Yeah, the eight times table.

So if I think about eight times tables, so let's do it in blue.

So eight times tables would be eight, 16, 24, 32, 40, but it's not the eight times table.

The first one's 16.

So what do we have to, how do we have to shift the times tables? Good, we have to add eight.

So each time I have to add eight, and then I get to my actual answer.

So eight plus eight is 16.

16 plus eight is 24.

24 plus eight is 32.

So actually the nth term was 8n plus eight.

Really, really well done if you got that for our challenge.

So to find the 10th term, maybe you did it like that, maybe you just worked it out.

The 10th shape would be to the surface area of the 10th shape would be 88.

And depending what units, if you use centimetres, it would be centimetres squared.

And for the challenge, it'd be eight 10 plus eight will give you the surface area of any cuboid in this pattern.

So with this particular pattern.

Okay, so for the explore, these students have made different conjectures.

And I want you to tell me if you agree or disagree, and if you disagree, can you kind of explain why? So Anthony says, "The volume of this shape is 27 centimetres cubed." And he's talking about this cube below.

Cala says, "The surface area of this shape is 54 centimetres squared." Talking about this cube.

Zaki says, "So the volume of each of these shapes on the right-hand side, the three of them.

You've got each one is 27 centimetres cubed divided by three to find out the volume." Xavier says, "The surface of each shape is 54 centimetres squared divided by three for each of those ones." Do you agree or disagree? Pause the video.

Look at the worksheet if you prefer and have a go.

Okay, so let's go through it.

The volume of the shape is 27 centimetres cubed.

Now I like to just work it out.

If I work it out, I know is it true or not.

So it's a cube, and it's got three by three by three.

So I can find the volume.

The front face is nine, and then there's three layers.

So it's going to be 27 centimetres cubed, nine times three.

So yes, tick.

It is true.

What about Cala? The surface area of the shape is 54 centimetres squared.

Well, I remember from the try this, that actually we did the front face, which was nine, and he times it by the number of faces, which was number face of cube, six.

So nine times six is 54 centimetres cubed.

So, true.

Okay, Zaki.

The volume of each of these shapes is 27 cubed divided by three.

Well, it does look like he's divided them into three equal piles, yeah? So yeah, I think that would be right.

27 divided by three is nine.

Yeah, true.

There's nine in each one.

That is correct.

Xavi said, "The surface area of each shape is 54 centimetres cubed, squared, sorry, divided by three." Hmm, that seems strange because surface area is how many, like the faces that are visible.

And here it seems like when we cut it, they'll be more visible, hmm.

54 divided by three.

What is that equal to? Okay, 18.

So does each one have 18 centimetres squared as the surface area? Let's see.

So this one's got one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, and then 13, 14, 15, 16, 17, 18, 19 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

Ah, he's not correct.

I think each one should have 30 centimetres squared.

Did you get the same? Well done.

So I disagree with Xavier.

Really, really well done today, everyone.

So I know that actually sometimes it's a bit hard to kind of think when things aren't right in front of you because I think about the 10th firm, think about the nth term is a bit more tricky.

But as mathematician, we just love creating conjects and creating patterns.

So I really, really hope you enjoyed that as well.

Do you have a go at the exit quiz.

It's an ideal time to test what you understand, and it will give you a chance to make a conjecture as well, and just test some of your knowledge from today's lesson.

Have a good day.