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**KS3 Mathematics S3 3-D shapes**

The aim of this unit is to teach pupils to: Use 2-D representations, including plans and elevations, to visualise 3-D shapes and deduce some of their properties Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp S3 3-D shapes

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**S3 3-D shapes Contents A S3.1 Solid shapes A**

S3.2 2-D representations of 3-D shapes A S3.3 Nets A S3.4 Plans and elevations A S3.5 Cross-sections

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**3-D shapes 3-D stands for three-dimensional.**

3-D shapes have length, width and height. For example, a cube has equal length, width and height. How many faces does a cube have? 6 How many edges does a cube have? Face Explain that two faces meet at an edge and two or more edges meet at a vertex. 12 How many vertices does a cube have? 8 Edge Vertex

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**Three-dimensional shapes**

Some examples of three-dimensional shapes include: A cube A square-based pyramid A cylinder Explain that just as a square can be described as a special type of rectangle, a cube can be described as a special type of cuboid. For each solid shape ask pupils to tell you how many faces, edges and vertices it has. Ask pupils to tell you the shape of each face (ask if this is possible for a sphere). For the cylinder, ask pupils to imagine that the curved face is ‘unrolled’ and laid flat. Explain that if all the faces of a solid shape are polygons, the shape is called a polyhedron (plural polyhedra). Ask pupils which of the solid shapes shown on the board are polyhedra. If all the faces are regular polygons, the solid is a regular polyhedron. Ask pupils which solid shapes shown on the board are regular polyhedra. There are only five regular polyhedra, of which the cube, with six square faces, and the tetrahedron, with four equilateral triangular faces, are two. The other three are the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (20 triangular faces). Tell pupils that a prism is a shape with a constant cross-section. The triangular prism is one example, ask pupils to identify the other two (the cube and the cylinder). Nets of each of these solids (except the sphere) can be found in S2.3 Nets. A triangular prism A sphere A tetrahedron

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**Describing 3-D shapes made from cubes**

For this activity you will need a set of interconnecting cubes. Ask a volunteer to stand with their back to the board while you construct a 3-D shape using the virtual cubes on the board. Members of the class must describe the shape shown on the board to the volunteer who must create the shape using real multilink cubes.

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**Equivalent shape match**

For each shape, challenge pupils to find the same shape shown in a different orientation.

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**S3.2 2-D representations of 3-D shapes**

Contents S3 3-D shapes A S3.1 Solid shapes A S3.2 2-D representations of 3-D shapes A S3.3 Nets A S3.4 Plans and elevations A S3.5 Cross-sections

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**2-D representations of 3-D shapes**

When we draw a 3-D shape on a 2-D surface such as a page in a book or on a board or screen, it is called a 2-D representation of a 3-D shape. Imagine a shape made from four interlocking cubes joined in an L-shape. On a square grid we can draw the shape as follows: We start by drawing the L-shape. From each vertex we draw a 45 º sloping line (point out that a line that slopes one square along for one square up slopes at an angle of 45° to the horizontal). We then complete the drawing by joining the end-points of the sloping lines. Notice that there are three sets of parallel lines: horizontal lines, vertical lines, and 45º sloping lines. We can use shading to differentiate between the faces that are facing forwards, the faces that are facing to the side and the faces that are facing upwards. The disadvantage of using a square grid to draw shapes made from cubes is that it is not possible to make the edges the same length (the 45º sloping edge is shorter). This view is sometimes called an oblique view.

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**Drawing 3-D shapes on an isometric grid**

The dots in an isometric grid form equilateral triangles when joined together. When drawing an 2-D representation of a 3-D shape make sure that the grid is turned the right way round. The dots should form clear vertical lines.

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**Drawing 3-D shapes on an isometric grid**

We can use an isometric grid to draw the four cubes joined in an L-shape as follows: Again the diagram has three sets of parallel lines: one set is vertical, and two sets are 30º from the horizontal in opposite directions. The advantage of drawing shapes made from cubes on isometric paper is that all the edges are the same length.

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**2-D representations of 3-D objects**

There are several different ways of drawing the same shape. Are these all of the possibilities? You may wish to have a model of this shape made of interconnecting cubes in class. You can invite pupils to think logically about all of the different possible orientations there are and use the model to demonstrate these. Challenge pupils to draw these on isometric paper. Can you draw the shape in a different way that is not shown here? How many different ways are there?

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**Drawing 3-D shapes on an isometric grid**

Use this activity to practice and to demonstrate isometric drawings of 3-dimensional shapes made from cubes. Use the pen tool, set to draw straight lines, to draw the required shape on the grid. As a more challenging exercise ask pupils to draw the given shape in different orientations of with extra cubes added in given positions.

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**Making shapes with four cubes**

How many different solids can you make with four interlocking cubes? Challenge pupils to find all of the solids that can be made from four cubes. The solution is shown on the next slide. Pupils may use real cubes if they need to, but should record their results as drawings on isometric paper. Point out that when we say ‘different’ shapes, we do not include rotations and reflections of the same shape. Suggest to pupils that if they work systematically, they can be more certain of finding all the shapes. For example, there are only two different shapes that can be made from three cubes. Pupils could start with one of their two shapes and make shapes from four cubes by moving a single cube to different positions. They should draw each one ignoring reflections and rotations of the same shape. These shapes are called tetracubes. Make as many shapes as you can from four cubes and draw each of them on isometric paper.

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**Making shapes with four cubes**

You should have seven shapes altogether, as follows: Pupils can compare their answers with these pictures. They should be able to match shapes that they have drawn in a different orientation.

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**Making shapes from five cubes**

Investigate the number of different solids you can make with five interlocking cubes. Extend the activity to five cubes. Pupils could start by finding all of the shapes that are made up of a single layer. There are 12 of these, called pentominos. Pupils can then move on to using more layers to make pentacubes. There are 11 shapes made from more than one layer, not including rotations and reflections. If reflections are allowed, there are 29 possible shapes altogether (not including reflections there are 23). Make as many as you can and draw each of them on isometric paper.

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**Opposite faces Here are three views of the same cube.**

Each face is painted a different colour. This problem shows different isometric views of the same cube. Establish that the yellow and blue faces are opposite each other, as are the green and the pink faces, and the orange and the purple faces. Ask pupils to justify their reasoning. What colours are opposite each other?

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**S3 3-D shapes Contents A S3.1 Solid shapes A**

S3.2 2-D representations of 3-D shapes A S3.3 Nets A S3.4 Plans and elevations A S3.5 Cross-sections

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**Can you tell which 3-D shape it would make?**

Nets Here is an example of a net: This means that if you cut this shape out and folded it along the dotted lines, you could stick the edges together to make a 3-D shape. This net is of a square-based pyramid. Ask pupils to describe or sketch other possible nets for the same shape. Challenge pupils to construct the net of a square-based pyramid of base length 3 cm, and sloping edge of length 4 cm, using a ruler and a pair of compasses. Links: S6 Construction and loci – constructing nets. S8 Perimeter, area and volume – surface area. Can you tell which 3-D shape it would make?

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Nets This animation shows how the net can be folded up to make to make a pyramid.

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**What 3-D shape would this net make?**

Nets What 3-D shape would this net make? Ask pupils to describe or sketch other possible nets for the same shape. Challenge pupils to construct the net of a cuboid of length 5 cm, width 3 cm and height 2 cm, using a ruler and a protractor, a set square or a pair of compasses. Links: S6 Construction and loci – constructing nets. S8 Perimeter, area and volume – surface area of a cuboid. A cuboid

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**What 3-D shape would this net make?**

Nets What 3-D shape would this net make? Ask pupils to describe or sketch other possible nets for the same shape. Stress that the slanting edge of the triangular face must be the same length as the edge of the rectangle that will join onto it. Challenge pupils to construct the net of a triangular prism with a length of 5 cm and a triangular cross section whose base is 2 cm and whose slanting edge is 3 cm. They should use a ruler and a pair of compasses. Link: S6 Construction and loci – constructing nets. A triangular prism

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**What 3-D shape would this net make?**

Nets What 3-D shape would this net make? Ask pupils to describe or sketch other possible nets for the same shape. Stress that each triangle is an equilateral triangle. Challenge pupils to construct the net of a tetrahedron with edge length 4 cm, using a ruler and a pair of compasses. Link: S6 Construction and loci – constructing nets. S8 Perimeter, area and volume – surface area. A tetrahedron

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**What 3-D shape would this net make?**

Nets What 3-D shape would this net make? Ask pupils to describe or sketch other possible nets for the same shape. Stress that the height of each rectangle must be the same lengths as edges of the pentagon. Challenge pupils to construct this net using a ruler and a protractor. The length of the completed prism should be 5 cm with the edges of the pentagonal faces of length 2 cm. Link: S6 Construction and loci – constructing nets. S8 Perimeter, area and volume – surface area. A pentagonal prism

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**Nets of cubes Here is a net of a cube.**

M N A B C D E F G H I J K L When the net is folded up which sides will touch? A and B C and N D and M E and L Pupils could make this net and fold it into a cube to verify which sides touch. F and I G and H J and K

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Nets of cubes Six different squares joined together are shown each time this activity is reset. Pupils must decide whether or not they represent the net of a cube.

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Nets of dice Use the fact that the opposite sides of a die add up to seven to drag the missing faces into place.

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**S3 3-D shapes Contents A S3.1 Solid shapes A**

S3.2 2-D representations of 3-D shapes A S2.3 Nets A S3.4 Plans and elevations A S3.5 Cross-sections

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Shape sorter A solid is made from cubes. By turning the shape it can be posted through each of these three holes: Can you describe what this shape will look like? Provide pupils with interlocking cubes and ask them to make a shape that could be rotated to fit through all three holes. Is there more than one possibility? Can you build this shape using interlocking cubes?

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Shape sorter A solid is made from cubes. By turning the shape it can posted through each of these three holes: Here is a picture of the shape that will fit: Here is one possibility; a few variations are possible (by moving the cube at bottom left).

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**Plans and elevations A solid can be drawn from various view points:**

2 cm 7 cm Plan view 2 cm 3 cm 2 cm Side elevation 3 cm 3 cm 7 cm Front elevation 7 cm Define the plan view as the view of a solid from directly above. In this example the plan view is a 2 cm by 7 cm rectangle. Define the front elevation as the view of the solid from directly in front. In this example the front elevation is a 3 cm by 7 cm rectangle. Define the side elevation as the view of the solid from the side. In this example the side elevation is a 3 cm by 2 cm rectangle.

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**Choose the shape Front elevation: Side elevation: Plan view: A: A: B:**

Ask pupils to choose the shape that corresponds to the three views given.

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**Choose the shape Front elevation: Side elevation: Plan view: A: B: C:**

Ask pupils to choose the shape that corresponds to the three views given.

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**Choose the shape Front elevation: Side elevation: Plan view: A: A: B:**

Ask pupils to choose the shape that corresponds to the three views given.

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**Choose the shape Front elevation: Side elevation: Plan view: A: B: B:**

Ask pupils to choose the shape that corresponds to the three views given.

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Plans Sometimes the plan of a solid made from cubes has numbers on each square to tell us the number of cubes on that base. For example, this plan: 2 1 represents this solid:

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**Drawing shapes from plans**

Ask a volunteer to draw the solid shown in the plan on the isometric grid. Use the pen tool, set to draw straight lines. Pupils may wish to build the shape first using interlocking cubes.

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**Shadows What solid shape could produce this shadow?**

The obvious answer is a cuboid, although any prism could be orientated to produce this shadow. We could also have a pyramid with a rectangular base.

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**Shadows What solid shape could produce this shadow?**

Possible answers include a tetrahedron, any pyramid (including a cone) or a triangular prism.

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**Shadows What solid shape could produce this shadow?**

Possible answers include a sphere, a cylinder or a cone.

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**Shadows What solid shape could produce this shadow?**

Possible answers include a cube, a cuboid with a square cross-section, a square-based pyramid or any prism whose length is equal to its height.

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© 2012 National Heart Foundation of Australia. Slide 2.

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