What is the best mathematical definition of a conjecture?

What is the surface area of this 1cm unit cube?

Two 1cm unit cubes have been combined to form the solid shape below. What is the surface area?

The pattern is continued so that we have 3 unit cubes stuck together as a stick. What is the surface area?

The pattern is continued so that we have 13 unit cubes stuck together as a stick. Which calculation successfully works out the surface area?

Surface Area Conjectures

﻿Hello. My name is Mrs. Buckmire. And today I'll be teaching you how to find the surface area of cuboids. And first, make sure you have a pen and paper. Remember you can pause the video whenever you like. So if you need a bit more time with something, do just pause it. Also remember you can rewind it, so if you don't quite understand something I said or want to hear it again, then please do pause it, and rewind it even and then listen to it again sometimes can be super helpful. Okay, let's begin. So your try this task, now, if you've been with me and following along with this unit the whole way, you might seen this one before. And it says, how many ways can these groups of rectangles be arranged to make the net of a cuboid? That's what we did last time. But now we went to the ones that's further, and tell me, What would the total area of the nets be? So draw a net using these rectangles and then work out its area. Remember how to work out the area of a rectangle? Good! It's length times width. Well done. There will be more support on the next page. But pause it now, and have a look at the worksheet. If you feel confident. Okay, if you haven't try this task before, you need a bit reminder, I am here to do that. So let's see, let's say that this side was six centimetres. And let's actually use this one here. So this also six centimetres. Now I know it is not drawn to scale. But it does not matter. It should really be a square but that's fine. So this side is six centimetres then well this one also has to be six centimetres. And where's that. So we could use this, so we already have a six centimetres and a eight centimetre one. And if this one is going to be opposite, this one, so this one must be the same dimensions. So use that in reasoning, to then work out all the other lengths, and then work out the total area. Okay, I will go through it. But have a little go yourself first. Okay, So I'm going to start with the support one. And we're trying to find the area. So this one was six centimetres. So the area here is 36 centimetres squared to that face. It's the same as this face. So this face is going to be eight times six. What's that? Good. 48 centimetres squared, which is going to be the same as this one that's going to be opposite. And this one, now this is also eight and this is also six. So it's actually also 48 centimetres squared. And 48 centimetres squared here. So if we add them all up, we have 48 plus 48, plus 48, plus 48, plus 36, plus 36, and then what do we therefore get? Good. We get 264 centimetres squared. So that's the area of that net. So here's another one, that I'm going to quickly show you. So we can find the area of these two that are opposite of each other. Of these two, and area of this one. And the total, is, 236 centimetres squared for that net. Now you might have had a different answer. And like you might had made a different net, and that's fine. Check your answers really carefully. Okay so, we've just found the area of nets. And that is actually the same as the surface area of a cuboid. For cuboid nets. Okay. So here, I just wanted to show you how they link up. Now you might have heard me say, oh it's the same as the opposite, same as the opposite. And it's because, if let's say this is the front, it's going to be the same as the back. And our 3D shape when it's drawn like this, we can only see the front, but our net, we can't see the front and back. Here, we have the right hand side shown. But we can see it's the same as the left hand side. And same with blue. So if I was going to give on my 3D diagram. If I was going to write in some lengths. This side would be eight centimetres. This height would be five centimetres. And here would be six centimetres. So the area of the front and back, is going to be 40 centimetres squared, each. So here I would actually have two lots of 40 to get 80 centimetres squared. Let me make that a bit clearer. So that's an eight for 80 centimetres squared. So let's left hand side, right hand side, so six times five is 30. So it's going to be 30 centimetres squared for one. Box leave, well we do have two. The left hand side and the right hand side so it's two times 30, which equals 60 centimetres squared. And then the top and bottom. So it's eight times six, which was 48. So I have two lots of 48. Which equals 96. So the total of 80, plus 60, plus 96, gets us to that 236 centimetres squared. So we don't have the net, we can still work it out from the 3D shape. Remember surface area, which is the area of all the face of the area, of all the faces that are visible. Okay, so oh, Antoni has made a mistake when trying to work out the surface area of this cuboid. Spot the mistake and correct it. So look really carefully at his work out and see, where did he go wrong. Pause the video and have a go, now. Okay, did you pause it? Yeah? Did you spot the mistake? What did Antoni do? Ahh yeah, he only found the face that's shown, is the mistake that I think that he might have made. So from the two times five, that is only the front. If you're going to find front and back, it would be 10 times two. Or two times 10. Cause he's only found this front, but we actually know there's also a back. So two times 12, so that relates to this one here. The side, but actually there's a right hand side, and a left hand side. So left hand side, and right hand side, would be 12 times two. And here, this is the top but you also need the bottom, so it should really be top and bottom. Which is, 30 times two. So all together, what should you have got. Yes, should have got 94, 60... I'm long on time. What's 44 plus 60? There you go, 104 centimetres squared. And that makes a lot more sense, because actually he got 52 and really he only needed half of it. So we want to actually double it. And then we will get our correct answer. So it should be 104, so if that doesn't look like a four, do you want me to do that again? 104 centimetres squared. That's the surface area. Well done if you spotted the mistake, and got the right answer. Okay, time for you to practise. So I've got this independent task. I have full faith that you can do every single question. Okay. So pause the video. Have a look at the worksheet. And you can do off the video, but I think it's a bit small. So look at the worksheet and try you're best with it. Okay, let's go through it. So this first one is kind of similar to what you've done with the surface area, and with area before, with counting squares on a centimetres grid. So we want to find the area of each face. And we can see that this top one, is six, as this one is also six. So it's two times six equals 12. We have eight because we have this one, and this one, are both eight. And we have two times 12 because we have the last two faces are 12 each. So the total is 54 centimetres squared. Well done if you got that. So surface area of each shape here, so starting with cube, was three times three equals nine. So that's the front face. And what's special about a cube, yeah all the edges are equal. And all the faces are equal. So the faces have equal areas so, hmm how many face on a cube again? Ahh six. So it's six times nine, which equals 54, so it is 54 centimetres squared. When we total it, cause that's what it is. And now for the cuboid, so we can do the front and back, so the front it six centimetres squared. And the back is also six, so it's 12. We can do the left hand side and right hand side. Which is going to be 10 each, so two times 10. And the top and bottom. Three times five is 15. And we have to times it by two to get 30. So the total, is 62 centimetres squared. Okay, so which one has the greater surface area? Good. It's 'b'. Which one has a greater volume? Oh I didn't work that out yet. So volume of the cube, equals three cubed. Cause three times three times three. Which is 27 centimetres cubed. And for the cuboid, Which is, which is five times three times two. Or two times three times five. Equals six times five is 30, so 30 centimetres cubed. So it is also 'b' has the bigger volume as well. Well done if you got those correctly. Okay, so the final question of the independent task. The volume of this cuboid is centimetres cubed. Every side length is a prime number. The base has an area of 35 centimetres squared. What is the surface area? Okay, so the base has an area of 35 metres squared, and the volume is 70. Well then I know then the height, is going to be two metres. Hmm what two prime numbers might apply to get 35? Well done. Seven, and three metres. So one, if you got lengths, that's a very good start. If you didn't, you can feel free to pause it and have a quick go at finding the surface area. So I'm going to find the surface area of this face first, which is two times seven, which is 14. So the front and back, equals 14 times 2. And then we have the left hand side and right hand side. Left hand side, same as right hand side, they're both six. So left hand side, and right hand side, equals six times two, which is 12 metres squared. And finally the top is, seven times three, which is 21. So the top and bottom equals 21 times two, which equals the 42. So all together, eight, nine, 10, 11, 12. One, five, six, seven, eight. So 82 metres squared. Is the surface area. Well done if you got that. Okay, my favourite task. I want you to explore this. Now each student is thinking of a cuboid with integer dimensions. Now Binh, says my cuboid has a volume of 120 centimetres cubed, and two squared faces. Yasmin says, my cuboid has a surface area greater than 250 centimetres squared. And Xavier, my cuboid has at least one edge measuring three centimetres. Now I want you to tell me, give me even, the dimensions of a cuboid that satisfies exactly one of the statements. And does not satisfy the other two statements. So maybe it's true for Binh, but not for Yasmin and Xavier. True for Yasmin, not true for Binh and Xavier. Give the dimensions of a cuboid that satisfies exactly two of the three statements. So it's true for Binh and Yasmin, but not for Xavier. And finally, the dimensions of a cuboid that satisfies all three statements. Now maybe you've seen this type of task before in which case you might feel a bit more confident. And therefore you can pause the video, and feel free. If you're not sure either about using and thinking about volume and surface area. Or not sure about the task. Do hold on a moment. I will explain it and give a little more support. Okay, so to help you, put your information in a table. And tables are often super useful for clarifying mathematicians boards. So, as you can see the first row, Binh, it's true for Binh. Not true for Yasmin, not true for Xavier. So you need to think of a solution, where it's true in the volume of 120 centimetres cubed, does have two square faces, but it's surface area is smaller than 250, so it's false for Yasmin. And there are no edges that are three centimetres. Okay. And same here, so for the second row, Yasmin's is true. Third row, Xavier's is true. The next one, true and true. True and true, true and true. And finally all three are true. Okay. So if that was all concerning the task, then hopefully that would help a bit. About the math, so I think it's helpful to actually visualise, actually just draw a cuboid. So for all of these, when I was doing it, I had to draw a cuboid each time. And then I labelled length, height, and width. And that might help. So they're the three different dimensions that I expected for your to be thinking about. Now remember volume, how to find the volume of a cuboid? Good. We can times the length, and height, and width together. Surface area, we just worked out so, re-watch the video if you're not sure about that. Okay, and one edge, would any one of those to be three. Okay, so there a key information. So now I've actually just chosen some examples, that you can actually place in the grid. Now this bottom no sign means actually it's not possible and there might be one or two that aren't actually possible. So if you're thinking, oh, I can't do this, nothing's working. Maybe cause it's not possible. Now remember, I do want integer values, and these are all in centimetres. I just have them written in centimetres, so you can do similar, I think. It's more aligned thinking about the numbers and just have a go. Find the surface area and volume of different cuboids. Good luck Okay, so where do you place those dimensions? If you did those ones. This one, would go in here. So it is true for Yasmin. But the cuboid does not have two square faces. And it does not have a edge measured three centimetres. But the surface area is over 250 centimetres squared. And this one would go in here. So two times two is four, times 30 is 120. And now two edges are the same, so there is, now will be two square faces. And that we know three centimetres edges. So it fits in there. And the surface area being less 250, yeah I think, just about, it works out. Okay, so this one was impossible. So it's impossible to have a cuboid with a volume of 120 centimetres cubed, and two square faces, and have one edges measuring three centimetres. If you want all the edges to be integers. Cause if we have one being, for let's say this one is three, that means that well we need another one to be three as well to make this square. So this would also be three. And then three times three is nine. And there's no whole number, that times by nine to get 120. So even if, we said actually, we want the three to be the width and actually these two to be equal to each other. Well that means that 120 divided by three is 40. And 40 is not a squared number. So we can't square with 40 to get an integer whole. So this does not work. And cause this one does not work, it means that, this does not work as well. You could other answers as well than the ones I just showed you. And here's more answers, but then again, there are lots and lots of different ones. So don't worry if you didn't get those. These are just some examples of some answers. But you can check yours carefully, and have a think about whether you're sure it's correct. And maybe even check mines, and say, oh, am I definitely correct? Do check it. Okay, I absolutely love this kind of talk. Cause there's so many different answers you can come up with. And I really would love to be able to discuss those with you. But we can't through the screen. Okay, but if you're really proud of your work and you would liked to share it, then please ask your parent or carer to share your work in Instagram, Facebook, or twitter. Tag @OakNational #LearnwithOak. I would love to see it and this is one of my favourite tasks so I really think you can be creative with this so, I would love to see your work. Hope you have a lovely day. Do not forget to do the exit quiz. And I'll see you on our next lesson. Bye.

Well done for completing the lesson. Time to show off what you have learnt!

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Surface Area of Cuboids

In this lesson, we will learn how to calculate the surface area of cuboids.

Maths

Y8 Maths

Key Stage 3

Year 8

Surface area of cuboids

Volume and surface area of prisms

Geometry

THEME

UNIT

3-D Shapes

Prisms and cylinders

Nets of cubes

Nets of prisms

Counting cubes (Part 1)

Counting cubes (Part 2)

Cuboids

Volume of prisms and cylinders

Surface area introduction

Surface area conjectures

Surface area of cylinders

What is the volume of a cuboid with lengths 4cm, 2cm and 5 cm?

Work out the volume of this cube.

Is it true or false that to find the volume of a cuboid you always just multiply all the given numbers together?

Is it true or false that this shape has the same number of edges as vertices?

Yasmin says this is a prism because the top face and bottom face are both rectangles. Is this shape a prism?

Lesson overview:Surface area of cuboids

Core Content

Volume and surface area of prisms: