Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Rowlandson and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of plans and elevations.

This lesson is called Representing 3D Shapes in 2D, and by the end of today's lesson we'll be able to use isometric paper to draw a solid made from cuboids.

This lesson will introduce two new keywords.

One of those keywords is isometric.

Isometric means distances between points stay the same, and in this lesson we're going to use isometric paper, which usually has equally spaced dots or points.

It looks a bit like the example you can see on the screen.

Also, we'll introduce the word solid.

A solid is a shape that has three dimensions.

For example, it might have a width, a height, and a depth, as an example of two solids on a screen, a cuboids and a cylinder.

We'll see plenty of examples of these during today's lesson.

This lesson contains two learning cycles, and we're going to start with drawing cubes and cuboids with isometric paper.

Here we have Aisha, Jacob, Sam, and Laura.

They are all looking at a pencil sharpener from different points of view.

Let's take a look at what they can see.

Aisha says, "I'm looking at it from above." And this is what Aisha can see from above.

Jacob says, "I'm looking at it from the side." And this is what Jacob can see from the side.

And Sam says, "I'm looking at it from the front." And this is what Sam can see.

Laura says, "I'm looking at it from a position that allows me to see three faces at once." And this is what Laura can see.

So each person can see different parts of this pencil sharpener, depending on where they are looking at if from.

The pencil sharpener is a solid.

A solid is a shape that has three dimensions.

Now here's a photograph of the pencil sharpener, but this photograph is not itself three dimensional, it's two dimensional.

Quite often we find ourselves looking at pictures of 3D shapes, for example, on screens or in pages of books or on posters.

But those pictures themselves are not three-dimensional.

They are 2D representations of three-dimensional shapes, but because the way they're drawn, we know that they are three dimensional, and sometimes the context can help us see that it's three dimensional as well.

A solid may be represented in two dimensions in different ways, depending on which point of view is being presented.

We saw that with those examples of what each person saw in the previous slide.

This point of view allows you to see the sharpener in three dimensions.

You can see its width, its height, and its depth.

However, even though we know each face of this pencil sharpener is a rectangle, it doesn't quite look like that in the photograph.

Laura says, "If you trace over the photo, each face looks like a parallelogram or a trapezium." For example, if we trace over the front face, that shape I've just drawn, that is not a rectangle.

It doesn't contain any right angles, nor does this one or this one.

But we know that the sides of this pencil sharpener are actually rectangles.

So why don't they look like rectangles when you trace over them? "Well," Laura says, "that's because of the perspective.

All the faces are actually rectangles." They just don't look like that when we take the picture of them from the perspective we have here.

Isometric paper can be a useful way to create 2D representations of a solid.

Isometric paper contains a grid made out of equilateral triangles, or it may contain dots arranged into equilateral triangles.

And here are some examples of what isometric paper may look like.

You can use either of these.

Laura measures the pencil sharpener and then represents it as a cuboid on isometric paper.

Let's do that together.

We can first start with the front face.

That front face has a height of one centimetre and a width of one centimetre, so let's draw one line segment going up one centimetre, then one going across one centimetre.

Then we can join the rest of it up to complete that front face.

It looks like a rhombus on our isometric paper, but it represents the rectangle or the square, actually, from this particular perspective.

And then we can draw the face at the side, that is one centimetre by two centimetres.

We've drawn one of the sides already.

Let's draw a two centimetre side, and then one centimeter's going up, and then two centimetres coming back.

And then she says, "Finally I'll draw the top face." That is one centimetre by two centimetres.

We've drawn two of the sides already, let's draw the other two.

And here we have our completed true representation of a cuboid.

The drawing on the isometric paper does differ to the photograph a little bit.

In the photograph, the edges that are near the front appear bigger than the ones at the back.

That's what happens with perspective.

Things that are further away look smaller than things that are closer.

For example, if you imagine yourself looking at a field filled with cows, all those cows may be the same size, but the ones which are closer to you will appear bigger than the ones which are further away.

On the isometric paper, equal lengths are consistent, and that's why that's not necessarily the case on our 2D drawing of the cuboid isometric paper.

No matter where you look at those one centimetre edges, they'll be the same length.

That's not necessarily the case in the photograph.

Jacob says, "To me, the drawing just looks like three parallelograms." Doesn't look like a cuboid to Jacob.

So Laura says, "I could shade the faces to make it look more like a solid with light shining on it, like so." The light is shining on it from above and it's catching the front face a little bit, and then the face that's on the side is the shadiest.

Laura says, "This isn't really necessary though.

It can be helpful to help you see it, but it's not necessary." Let's now work through an example of drawing a cube, and then you can add one similar to try yourself.

Use isometric paper to draw a cube with edges of length five units.

Let's start by drawing one of the faces.

We draw an edge which is five units along, and then an edge which is five units up, and then across, and then down.

We've drawn one of our faces.

Let's now draw another face.

You can draw an edge, which is five units, then going up, then another, then going down to join it up.

And then finally we have one more face that we can see from this perspective.

The face at the top.

Let's join that one up with two more edges that are five units along.

Here's one for you to try.

Use isometric paper to draw a cube with edges of length two units.

Pause the video while you do this, and press play when you're ready to see what the answer looks like.

Here's what your answer should look like.

Here we have Lucas.

Lucas is using isometric paper to draw a cuboid with lengths one unit by two units by three units.

He says, "There are different ways that I could draw this solid, depending on its orientation.

For example, I could draw it so that the one by two rectangle is at the front." And it would look a bit like this one here.

"However," he says, "this isn't the only way I could draw the one by two rectangle at the front.

I could also draw it like this.

I could rotate the solid through 90 degrees." We still have a one by two rectangle at the front of it, and the depth going backwards is three units for both those examples.

Lucas says, "Or I could draw the one by three rectangle at the front.

That way two would be the depth.

Again, there are two different ways to do this." I could have the one by three rectangle in the top orientation, or the one by three rectangle in the bottom orientation.

Each one is a 90 degree rotation of the other.

Lucas says, "Finally, there are two ways that I could draw the three by two rectangle at the front, the way at the top, and the way at the bottom." Once again, one is a 90 degree rotation of the other, and both of them have a depth of one unit.

All six of these cuboids are congruent.

They're all one unit by two units by three units.

Let's now do one of these together, and you can try one yourself afterwards.

Use isometric paper to draw a cuboid with lengths one unit by two units by five units.

Now there're different orientations we could have this cuboids, but here's an example.

We could have the one by two unit rectangle at the front, and draw that one first.

It'll look a bit like this.

We could then draw the two units by five units rectangle along the side, and it would look a bit like this.

And finally, draw the one unit by five units rectangle on top, and it would look a bit like this.

Here's one for you to try.

Use isometric paper to draw a cuboid with lengths one unit by one unit by four units.

Pause the video while you do this, and press play when you're ready to see what an answer might look like.

Here's what an answer could look like.

In this one, we have the one unit by one unit square at the front, and the depth is four units, but you could have it in a different orientation.

Okay, it's over to you now for task A.

This task contains two questions, and here is question one.

You need to use isometric paper to draw cubes with the following lengths for it edges: a one-unit cube, a two-unit cube, a three-unit cube, and a four-unit cube.

Pause the video while you do this, and press play when you are ready for question two.

And here is question two.

Use isometric paper to draw six congruent cuboids with the following dimensions, two units by three units by five units, and draw each cuboid in a different orientation.

Pause the video while you do this, and press play when you're ready to see what the answers look like.

Question one.

When we draw our cubes, it should look something a bit like this.

Pause the video while you check this against your own, and then press play when you're ready for more answers.

Then question two.

When we draw our six cuboids in different orientations, it should look something a bit like this.

Yours may be in a slightly different order, but hopefully you have the same six cuboids somewhere on your page.

Pause the video while you check this against your own, and then press play for the next part of this lesson.

Fantastic work so far.

Now let's move on to the next part of this lesson, which is drawing connected cubes and cuboids.

Here we have Sofia.

Sofia has drawn a cube on isometric paper.

She wants to draw another cube next to her first one.

Now with this cube here, we can see three of its faces, but we know it has six faces, so, when we want to join another cube to it, there are six different places where that cube can go, and that will affect how it looks.

Sofia says, "If my second cube connects here, then one of its faces will be covered by the first cube, so I'll only need to draw two of its faces." It'll look a bit like this.

She says, "If my new cube goes below the first one, then its top face would be covered, so I would only need to draw two faces." And it would look a bit like this.

Sofia then says, "I would also only need to draw two more faces if my new cube went here." It would look a bit like this.

But there are three more places where this additional cube can go.

Let's take a think about what happened in these situations.

Sofia says, "If my new cube went above the first one, then you would be able to see three of its faces, but it would cover the top face of my first cube." It would look a bit like this from a drawing.

Now if the two cubes were made out of glass, then you would be able to see all the edges for both of the cubes.

But if they're not made out of glass, if they're not transparent, or if they are opaque, then that wouldn't be the case.

Sofia says, "I could erase the edges that I would no longer be able to see in this situation." So for example, we could rub out those edges there, and now we can see one cube above the other.

Sofia says, "If my new cube went here, then you'll be able to see all three of its faces, but it would cover a face of my first cube." For example, if would put it here, and then we'd need to rub out, or erase, to the edges, like so.

Sofia says, "Here's another place where I'd be able to see all three faces of my new cube, but it would cover a face of my first cube, like this." Let's do an example of drawing something like this together now, and then use one for you to try yourself.

A solid is made from three cubes, and we can see a picture of the solid here.

Draw this solid on isometric paper.

So let's start by drawing the front cube, where we can see all three of its faces from this perspective, like so.

And then, we can draw the cube that goes behind it, like this.

And then we can draw the cube that goes on top, or that last one, like this.

But we'll need to rub out, or erase, two of those edges that can no longer be seen, like so.

And there's our answer.

Here's one for you to try.

The solid here is made from four cubes.

Three cubes can be seen, one cube is completely covered by the others.

Draw the solid on isometric paper.

Pause the video while you do this, and press play when you're ready to see what the answer should look like.

Here's what the answer should look like.

With all three cubes, you can see all three faces from this particular perspective.

You may want to shade in parts of your cubes to make it clear to see which faces are at the top, and which faces are towards the front, and which ones are towards the side, but we don't have to do that.

Here's another example.

This solid is made from two congruent cuboids, and we want to draw it on isometric paper.

Let's start by drawing the cuboid that's at the front.

Now they are congruent, so it means the cuboids at the front will have the same measurements as the one on the back, which means it'll be one unit by two units by three units.

Let's draw that first, and it would look a bit like this.

And then we could always draw the other cuboid doing it one face at a time.

For example, we could draw this face, which is one unit by three units going up, and then we can draw the top face, which is one unit by two units going to the side, and then we can draw in the final face together, like this.

Here's one for you to try.

Here's a solid that is made from two congruent cuboids.

Please could you draw it on isometric paper? Pause the video while you do this, and press play when you're ready to see what the answer looks like.

Here's what your answer should look like.

Pause the video while check this against you own, and press play we're ready to continue.

Okay, it's over to you now now for Task B.

This task contains three questions, and here is question one.

We have Alex who has designed a 3D model, and there's a picture of the model at the bottom.

Use the measurements given in Alex's description to draw the model accurately on isometric paper.

He says, "The design is based on stacking cubes in a pyramid-like structure." And you can see the units for one of those cubes is one unit.

All the cubes are the same.

Pause the video while you do this, and press play when you're ready for question two.

And here is question two.

We have Izzy this time, who has designed a 3D model.

You need to use the measurements given, and Izzy's description, to draw a model accurately on isometric paper.

She says, "The design uses four congruent cuboids." Look carefully at how these cuboids overlap of each other.

Pause video while you do this, and press play when you're ready for question three.

And here is question three.

This time we have Jun who has designed a 3D model.

He says, "My model is a cube inside a hollow cube inside a hollow cube, with some of the faces removed." Please use Jun's description of the model, and the measurements that are given to you, to create an accurate drawing of this on isometric paper.

And once again, you can shade in parts if it helps you see things more clearly, but you don't have to do that.

Pause the video while you do this, and press play when you ready to see what the answer should look like.

Let's take a look at some answers.

For Alex's model, it should look something a bit like this.

You can just have it all unshaded, but the shading helps you see which face is at the top and which face is at the side, and that helps you see how the faces connect together to make each individual cube.

But you don't have to do that, so long as yours are drawn in the same sort of way.

And let's take a look at question two.

Your answer should look something a bit like this.

Look very carefully at where the edges are for each cuboid to ensure that they are overlapping in the correct ways.

If you draw an edge in the wrong place, you might end up with one cuboid being longer than another.

Let's now take a look at Jun's model.

It looks something a bit like this.

If you haven't shaded yours in, it might just look like a lot of hexagons, one inside the other, with some other line segments in between.

But it should represent a cube inside a hollow cube with some faces removed, which is also inside a hollow cube with some faces removed as well.

Fantastic work today.

Now let's summarise what we've learned in this lesson.

The world is filled with 3D objects.

We're surrounded by them, but quite often we find ourselves looking at representations of 3D objects in 2D, or we may be creating images ourselves that are 2D representations of 3D objects.

And that's what we're focusing on during today's lesson.

Isometric paper aids the drawing of a solid constructed from cubes or cuboids.

While the drawing is made up of parallelograms, each parallelogram actually represents a rectangular face.

If you look at each face individually, carefully, yes, it does look like a parallelogram, but if you look at the object as a whole, usually the perspective, and our own experiences of cuboids in the 3D shapes in the past, well help us recognise that each one is a rectangle.

The drawing therefore will look 3D.

And the final picture has three distinctive viewpoints.

Well done today, and I hope you have a great day.