Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

I really hope you're looking forward to it and you're ready to try your best.

So today's lesson is about being able to relate a solid to its net, being able to draw the net from a solid and being able to draw the solid from its net.

So let's make a start.

So on the screen there is the key word of a net and the definition of one, you will have met these before.

So it may be that you wish to pause the video, read through it and be familiar before we make a start because we will be using that a lot during the lesson.

So our lesson about checking and securing understanding of nets of solids is gonna be in two parts.

So the first one is to know the solid that matches the net and the second learning cycle is to be able to draw the net for a given solid.

So let's make a start at being able to match up solids and their nets.

So here is a 3D object.

Can you remember it's formal name? So this is a cuboid.

So the net of a 3D object is a 2D representation of the faces of that said object, so in this case, this cuboid.

And here would be a net for this cuboid.

So have a look, can you see why this is the net for the cuboid for this particular 3D solid? It is unfolded, so a net you can think of as the faces unfolding into a flat 2D representation and some of the edges will have been cut in order for it to unfold.

So Alex is trying to work out which net belongs to this solid.

So the solid in the top right, maybe you can name that one.

What type of solid is it? So there are three nets available and Alex is trying to figure out which one matches with that solid.

So firstly he says, "The solid does not have any triangular faces, so it cannot be net B." So if you look at the solid, none of those faces are triangular, whereas the net here has got five triangular faces.

So B is not correct.

Then he says, "The solid does have two pentagonal faces, so it cannot be net C." If you look at net C, there are no pentagonal faces, so that means it is net A.

Net A has the five rectangular faces and the two pentagonal faces of the solid.

So if you check the solid to the net, it matches up with the correct number of each type of face.

So using the shapes of the faces and the number of them is a great way to identify the correct net for the solid.

So if we were looking just to match up one net to its solid, using the shapes of the faces and also using the number of them is a great way to be able to match them up.

So here is a check for you.

How do you know that this is not the net for this solid? So the solid's on the left and the net is on the right.

How do you know that these do not match? Pause the video and when you're ready to check, press play.

So the solid does not have any triangular faces.

And if you look at the net, there are two triangular faces.

So here's another check for you.

So this is a solid that's created by three different regular shapes.

Remember, regular means that all of the edges and the angles are the same within that shape.

So what are the three regular shapes? Pause the video, study the solid and think about the faces that are on that solid and then write down the names of those shapes.

Press play when you're ready to check.

So there are pentagons, squares and equilateral triangles.

So the square is the regular quadrilateral, we've got regular pentagons and the regular triangle is the equilateral triangle.

And actually this is the net for that solid, a very complicated net, it's got many faces, but you can identify the three different regular polygons on there.

So we've got the pentagon, the square, and the equilateral triangle.

So this solid unfolded can be represented as a net and here is the net.

So here's another check for you.

A solid is created from regular hexagons and regular pentagons only.

Which of these nets could be for that solid? So pause the video and then when you're ready to check your answer, press play.

It was B, so A only had regular pentagons, it didn't have pentagons and hexagons, and C had a square in there as well.

Or not just one square, many squares.

And the text told us that it only had regular pentagons and regular hexagons.

B is our regular hexagons and regular pentagons.

So we're onto the first task of the lesson and you need to match the nets to the correct solid.

So the solids are at the top and then the four nets are at the bottom, so you need to match them up.

So remember, use the faces and the amounts of each faces to help you match those.

You might want to try and name them as well.

Press pause and then when you're ready for question two, press play.

Question two.

I'd like you to select all of the nets that would fold to create this solid.

Something we haven't discussed is that there isn't necessarily only one net that would fold to create that solid.

So here we've got multiple nets and some of them would fold to create that solid and some of them would not.

So I need you to go through and think about whether they would fold up to create the solid there, it's a hexagonal prism.

So press pause whilst you go through each one, making a decision whether it is a net for that hexagonal prism or whether it is not.

And then press play when you're ready to go through the answers.

So here is the matches.

I've moved the nets to line up with the solids.

So the pentagonal prism clearly needed pentagonal faces or there needed to be pentagonal shapes on the net.

So that was the first one.

A pentagon has five edges and therefore there are five rectangles that wrap around this prism, the collar, if you like.

The next one is a square-based pyramid and we can see the square base and four triangular faces that meet there.

Then we've got the cylinder.

So a cylinder has got circular faces, that's the top and the bottom of the cylinder in the way that this is orientated.

And then the curved surface, when unfolded, lies flat into a rectangle.

And then lastly, we've got a triangular prism.

So we need two triangular faces and three rectangles if this is a right prism in this case.

Question two, you needed to select all the nets that would fold to create this hexagonal prism.

A would fold, so you've got the six rectangles and then a hexagon at either end, which would complete the hexagonal prism, close it up.

Then B would also fold up.

So here it's thinking about one of the hexagons being like the base of the shape and unfolding all of those rectangular faces down, sort of making that sort of star shape.

And then you still need the other end of the hexagonal prism, the other cross-sectional face.

C does not fold.

So it has got the correct amounts of each face.

So we've got six rectangles and two hexagons.

But the reason this would not fold is because if you look at the three that are joined together, that third one would be overlapping with a rectangle that's already attached to the hexagon marked C.

So there would be an overlap of two faces and therefore there would be a missing space.

D is also not a net for this hexagonal prism and that's because it doesn't have enough faces.

So it needs two hexagonal faces and six rectangular faces.

There are only five rectangular ones.

And then E is not a net and that's because we've got the six rectangles all lined up next to each other, which again would wrap around like a collar of the prism.

But we've got both hexagons on the same side and therefore they would overlap.

We need to have one hexagon on one end of a rectangle and we need the other hexagon on the other end of a rectangle in order to close up the prism.

And then lastly, F is a net.

So this one would unfold and fold back up to be our hexagonal prism.

How did you get on? It is quite challenging sometimes to think about how they fold.

So we're up to the second learning cycle and this time we're looking at drawing the nets for each solid.

So the solid we're gonna focus on to begin with is the cube.

So think about the properties of a cube.

You're very familiar with that as a solid.

So what do you know about it? And that's gonna help us as we go through.

So we've got Andeep, Jun and Sofia, and they're each going to draw the net for this cube.

They're all drawing a net for this particular cube so their dimensions will all be the same.

So maybe you wanna think about what would your net for this cube, if you unfolded this into its 2D representation, what would it look like for you? Well, Andeep has drawn it like this.

Jun has done it like this and Sofia has done it like this.

So they've come up with three different nets for this cube and remember, that there aren't only one way of unfolding the shape.

So they've found a different one each.

It's quite hard on some of them maybe to visualise how that would fold back to be the cube.

Have they got them correct? So I've done Andeep and Jun in animation.

So we've got Andeep and we've got Jun's net there, you can see it, and then we're gonna animate it to fold up to show you that it does create the cube.

So Andeep, so if we start with Andeep's.

Keeping one as like the base of the cube, you can see it folds up and closes the box.

So imagine that all of those faces have now created that cube.

And Jun's similarly, it does fold to create the cube.

So if you're not sure, you could try and draw yourself an accurate net of a cube, making sure you're thinking about the properties and cut it out and see if it folds up.

So what's the same in each of the nets? So their three nets are for the same cube and what makes them similar? So pause the video and think about that for a moment.

They're similar to each other because they've all got six square faces.

And when you thought about the properties of a cube earlier, you probably thought about the fact that all of the faces are square.

So they've all got six square faces.

How are the nets different from each other? Again, you may wish to pause the video so you can think about that or discuss it with somebody nearby.

Well, the configuration of the faces, so they look different because of the way that the six square faces are configured, the way that they're attached to each other.

But having six square faces is not the only condition for it to be a net of a cube.

So this one here has six square faces, but the way it is configured would not fold to make a cube.

So it is not a net.

Can you see why that's not a net of a cube? Again, you may wanna pause and discuss that or think about it.

Why would that not fold to create a cube? So two of the faces will be in the same position.

So we've got four square faces in a line making a long rectangle.

And again, you can think about them folding around the outside or the collar, making a tube-like structure of the cube.

But then we've got these two square faces that are both on the right-hand side and so they would both be trying to occupy the same space and so they're in the same position, and so the other side would be open.

The structure would look like a cube, you would get a structure that looks like a cube.

However it would be an open box, there would be a missing face.

And it's worth just mentioning here that if you were to take apart a cereal box, a tissue box, any kind of cardboard box, you might find that there are faces in the same position.

And that's because, practically, they need to glue them.

They may have added tabs so they can put glue to form and hold the structure.

But when we are drawing our nets, we are not going to add any of those elements, an overlapping space so that there is glue between them or a tab.

So we only want to draw the faces connected at the right places to ensure that the box or whichever solid you're making is created with everything closed.

So this is a check for you.

Why is this not a net of a cube? So pause the video and then when you're ready to check whether you came up with the same answer, press play.

This isn't a net of a cube because there are not enough faces.

A cube has six square faces, this has only got five.

So this would fold, but it wouldn't close up the cube, there would be an open face.

So if you wanted an open cube-like structure, then this would be a net for that.

But it isn't a net for a cube.

So a cuboid is a 3D shape that has six rectangular faces.

A cube is a special case of that where all the faces are square rather than rectangular and therefore the net of the cuboid must have the same properties.

So think about what those properties are.

We've got the solid and we've got a net of the same cuboid.

Well, those properties are that there needs to be six faces in total and that there are three pairs of congruent faces.

So congruent meaning exactly the same size and shape.

So let's have a look at this.

If we think about calling some of the faces the back and the front, the one that is at the back of the solid diagram and the one at the front of the solid diagram, they are a pair of congruent faces.

Those ones are identical in terms of their size.

And I've highlighted them there on the solid diagram so we know which pair we're talking about.

And in terms of the layout of the net, there is a face in between the back and the front.

So if you went around it vertically, you'd go front, top, back, bottom.

If you went around it sort of horizontally, you would say front, side, back, side.

So there is one face in between the back and the front.

They're not adjacent to each other on the net.

Similarly, the bottom and the top faces, they have got a gap in between them.

Again, if you went round vertically and starting at the bottom, you'd say bottom, back, top, front, so there would be this one in between.

And lastly, the last pair of congruent faces is the right and left side.

So again, if I just highlight that on the solid for you, there is a gap between them, so they're not adjacent to each other in the net.

And it's important that we think about this when we are unfolding our solid, that which faces are next to each other and which faces are one away from each other, how are they gonna connect back together? So a quick check, which of these are possible nets of a cuboid? So pause the video, go through them one by one is how I'd encourage you to do it.

Think about if it's got all of the properties of a cuboid, so six faces in total and three pairs of congruent faces.

Press play when you're ready to check.

B and E were the only possible nets of a cuboid up there.

So why were A, C and D not possible nets? Well, A just didn't have enough faces, there are only five faces there, so it's missing a face, therefore it can't be a cuboid.

C has got too many faces.

There are seven drawn there, we only want six.

D has got six faces.

So it's not that property that this one fails at.

It's the three pairs of congruent faces which are rectangles for a cuboid.

Because if you have a look, there is only one three by three, whereas there are three two by three rectangles.

So it doesn't have the property of having three pairs of congruent rectangles.

So two edges of two rectangular faces that touch need to be the same length in order for it to fold up to be a cuboid.

So if you look at the net for this cuboid, if you look at the front and the right rectangles, the right has got an edge of five and the front has got an edge of five, and when they join together then they are an edge of five length.

If we focus on this edge that's now marked with the black line, it's at the bottom of the right face and it's the side of the bottom face and both of those are four units long.

But if this edge was a different length for the bottom and right faces like this, so if somebody accidentally drew this as the net, what would it look like when folded up? So just think about what would happen if your net was drawn like this where the edges that are adjoined, the ones that are touching, are not the same length.

There would be a hole or a gap in your 3D shape.

So it's really important when we think about cutting along these edges and unfolding into our net that the edges that will meet upon folding again have the same length in order for there to be no holes or gaps when you create the solid.

So another check.

Which of these are possible nets of a cuboid? So again, pause the video, think about all of the things we've said up to this point in terms of the properties of a cuboid and how that needs to reflect on the net, and how also if the edges that meet are different lengths, then it won't be the nets because the solid would have a gap or a hole.

Press play when you're ready to check.

So A and D would create a cuboid when folded up.

If we have a look at why B and C do not, so B has got six faces in total.

There are three pairs of congruent faces, however two of the pairs are actually the same size and that's because we've got these square ends, if you like, the square cross section of the cuboid.

The reason that's not a net though, is because of the way that that rectangle has been drawn vertically rather than horizontally.

So the edges that are going to meet are of different lengths, it wouldn't work.

And then if we look at C, C has got six faces once again, but it doesn't have three pairs of congruent faces.

It's got five square faces and one rectangular face.

We've got edges that are already connected, so when we fold, there's gonna be some gaps as well.

So another way that we can represent solids is actually using plans and elevations.

And this is a solid drawn on a one centimetre grid.

So each square is one by one.

So plans and elevations is another 2D representation of a solid.

So we can have a net that is a 2D representation of a solid.

We can have plans and elevations that are a 2D representation of a solid, a 3D shape.

So we can draw the net for this solid here.

Can you see what solid it is? Well, it's a cuboid.

So we've just been working with cuboids, this is just looking at it from a different way.

So our net would be drawn and again we'd have our opposite faces, the left and the right, one away from each other, the top and the bottom, one away from each other, the back and the front, one away from each other.

We can use the elevations and the plan to get our dimensions of our cuboid.

So we can see the plan, so that's from looking straight from above, so our top view, it's four by two.

So we've got our length and our width, and our front and our side elevation have both got a height of three.

So on our net we've got those three pairs of congruent faces.

When it's folded, the ones that are touching are the same length already in the net.

And also unfolded, the ones that will touch are the same length.

Here is the plans and elevation for another solid, drawn again on a centimetre grid.

So we've got a plan view, so remember that's from above, front elevation and a side elevation.

Izzy's going to draw a net for this solid.

She decides first to sketch the solid from the plans and elevations first.

So she's gonna go from a 2D representation, she's gonna sketch a 3D image figure and then she's gonna go from that to her net.

So can you tell from the plans and elevations which solid it is? So pause video and think about it.

Think about what it means to look at it from the front elevation and the side elevation.

And then when you're ready to check, press play.

So it is an isosceles trapezoidal prism.

So it's a prism, it's trapezoidal, its cross sections are both trapeziums and an isosceles trapezium, we can tell that from the front elevation that we've got this symmetry to it.

So an accurate net is to be drawn.

So it was on a centimetre grid so we could read off some of those measurements.

The slant ones you may need to calculate using Pythagoras's theorem.

So we've got edge of the isosceles trapezium is root 10 centimetres.

I'm leaving it as root 10 to be exact.

Then we've got four centimetre and a six centimetre parallel sides of the trapezium and five centimetres is the length of our prism.

So Izzy's gonna start by thinking about her net.

So unfolding it by thinking from the bottom face, the one that it's sort of sat on.

So she's gonna start with that one.

And that is a rectangle with the dimensions of five, which is the length of the prism and six centimetres, which we know is one of the edges of the isosceles trapeziums. Next she's gonna imagine dropping down those trapezium faces.

So they are our cross-sectional faces of this prism and she's gonna sort of imagine them folding down.

So she's added those and we've got that edge of six centimetres, which is an edge of the trapezium as well as the rectangular face.

That's where they join up.

And then we know that the isosceles trapezium has got these edges of root 10 and the top edge is four centimetres.

We could mark all of those dimensions on each of the trapeziums, but then it becomes a bit over complicated.

So you only need to mark as much as you need.

So she's unfolded it so far from dropping the sort of cross-sectional faces down.

And next she's gonna unwrap the tube section.

So she's looking from this on the side of the isosceles and then the top.

And then lastly, the other side.

We know because it's an isosceles trapezoidal prism that two of those faces that are rectangular are the same.

They are congruent because of the isosceles nature of the trapezium.

Once again, you could mark more dimensions onto this net, but this is sufficient.

It tells you enough information.

So here's a check for you.

Which of these solids are shown by this plan and elevations? So you've got the plan, front elevation and side elevation.

Which of the solids, if you were gonna sketch the solid before drawing its net, which of those would be the correct solid? Press pause and then when you're ready to check, press play, It would be C.

The first one is a cuboid.

And we can see from the front elevation that there is a triangular face.

B is a pyramid of some sort.

So we have got a triangular face, but we are looking to have rectangular faces as well.

And so C is our triangular prism that gives us both triangular faces and rectangular faces.

So complete the net for this solid.

I've started, there are three of the faces drawn there.

So pause the video and complete the net for this solid.

Press play when you're ready to check.

So this could be the completed net.

You could have, in a similar way to Izzy, thought about the base, then flapped down the triangular faces and then finished the tubing.

This is one way that you could have configured your net, but you may have done this.

So again, sort of thinking about those triangular faces falling down and then unwrapping the tube section in a different way, starting with a different face.

Or this could be a net for this triangular prism as well.

Or this and there are more.

So when we have our nets, depending on which edges we sort of imagine cutting along and how we unfold it, we might get different configurations.

What did they all have in common? They had two congruent triangular faces because they were all uniform cross-section.

And then we had the three different rectangular faces.

The reason they were different is because actually our right angle triangle is scaling.

So we're up to the last task of the lesson, Task B.

And so on question one, here are three non-congruent nets for a cube and there are more non-congruent nets for a cube.

What I mean by non-congruent is ones that we haven't already seen here, not just unfolding it into the same configuration.

Remember, to be congruent, it doesn't matter if you rotate it or reflect it, you need to think about that when you are drawing your net.

If your net rotated and you've drawn it again, then they are congruent to each other.

They're just a rotation.

So I need completely unique nets.

Sketch as many as you can.

Question two, a tetrahedron is a platonic solid.

It's made of four equilateral triangles.

Draw a net for a tetrahedron.

So pause the video whilst you're working through questions one and two.

So question one, remember there are more that you can find.

Sketch as many as you can find.

And question two, draw a net for a tetrahedron.

Press play when you're ready to move on in this task.

Okay, so question three, the plans and elevations are drawn on a centimetre grid.

Draw the net for the solid shown by the plan and elevations.

So pause the video, you may want to sketch the solid before trying to draw its net.

Remember to do that accurately thinking about the dimensions as it's on a centimetre grid.

Press play when you're ready for question four.

Question four, I'd like you to finish the net for this pentagonal prism.

So we've got one pentagonal face and two rectangles, I'd like you to complete the net.

Press pause whilst you're doing that, and then when you press play, we'll to question five.

Question five, which of the following rectangles can be used to create the net of a cuboid? So there isn't a net here, but if you were to be able to select the rectangles that could create a net for a cuboid.

Press pause and then when you think you're finished with question five, when you press play, we're gonna go through our answers.

So here is question one.

So I asked you to sketch as many as you can, and actually there were eight more that could be found.

So eight non-congruent.

So it doesn't matter if you rotate them or reflect them, you will not get the same image.

So pause the video and go through the ones that you found.

They needed to have six square faces to have any chance of being a net of a cube.

These are the eight further ones you could find.

Question two, a tetrahedron.

There were actually two nets for a tetrahedron.

So you can imagine the triangular face that's on the one it's laying on, the one it's sat on, being stuck and then just unfolding the other three.

Or, again, keeping one sort of in one place and then unwrapping it and we get an equilateral triangle is the shape of that net.

And a parallelogram is the shape of the other net, four equilateral triangles in that tetrahedron.

Question three, you needed to draw the net, and this is an example of the net that you could have drawn.

So again, because we could have unfolded it in a slightly different configuration, but you needed two isosceles triangles and three rectangles of which two were congruent because of the isosceles triangles.

Question four, here is an example of the net that you could have come up with to finish the net.

You needed to have five rectangles in total and two pentagons of which those pentagons needed to be congruent.

So you may have needed to use a piece of tracing paper to ensure that they were identical to finish the net.

So this is one configuration, you may have had it a slightly different way.

You could potentially trace over it, cut it out and see if it folds back to be the pentagonal prism.

And lastly, which of the following rectangles can be used to create the net of the cuboid? Were the three that are now sort of shaded or greyed out.

The reason these three would work is because we needed to have these pairs of congruent rectangles where we would have an overlap of the dimensions so that they match up when we fold it.

So if we look at the top left one is a two by three rectangle.

The one below it is a three by four rectangle.

So the three edge would be where they could connect without any gaps.

And then the other rectangle is two by four.

So it is got an overlap with the two and an overlap with the four on the other one.

So we can create the net where these are going to match up and fold to be a cuboid.

So to summarise today's lesson which was checking and securing understanding of nets of solids, a net is a 2D representation of a 3D solid and they're composed of the faces of that solid.

Folding the net along certain edges and joining others produces the solid, and some solids can have multiple nets.

Really well done today.

I look forward to working with you again in the future.