Platonic solids investigation

Teacher notes

Teachers should read through the following activity ideas and make their own risk assessment for them before proceeding with them in the classroom.

Platonic solids are the set of regular 3d shapes. Unlike regular polygons, of which there can be an infi- nite number, the group of fully regular solids is small.

In this investigation, pupils will find and construct this group of solids, finding out how we can be sure that there are only five.

www.rigb.org Supported by the Mercers’ Company Resources needed for the practical activity

• Dice made from templates (one cube and one cuboid) • Cardboard shapes (pre cut or printed from template) - 1. Equilateral triangles– 32 per pair of pupils 2. Squares – 6 per pair of pupils 3. Regular pentagons – 12 per pair of pupils 4. Regular hexagons – 3 per pair of pupils • Sticky tape

Introduction

• Explain that pupils will be investigating and building 3d solids. Does anyone know the names of any 3d shapes? Encourage pupils to discuss shape names that are familiar, and address any confu- sion regarding 2d and 3d shapes.

• Show pupils two dice made in cardboard from template. One is a cube and the other a cuboid. Which would pupils prefer to play with? Why? Pupils may realise that both would be fair if all players were using the same one. However, the cuboid would not give the different results with equal fre- quency. In fact some results would be very unlikely. Why is this? Encourage pupils to imagine that the dice is resting on one of its vertices – is there a face that on which it is more likely to land? The con- cept of ‘the view from the vertex’ will be very important throughout the investigation.

• Demonstrate constructing a cube by taping together cardboard squares, a vertex at a time. Make sure to demonstrate that at each vertex, there are three squares meeting. This will model the approach the pupils should take in order to find the shapes.

• Today, pupils will be looking for a particular type of solid, known as a ‘Platonic solid’. In fact, we will be trying to find them all and learning about how solids are named. Can pupils explain how 2d shapes are named, using the suffix _gon? e.g. 32-gon, 96-gon. What do pupils understand by the word ‘regular’? What is its special meaning when talking about shapes? Platonic solids are the group of regular 3d shapes, but what does regular mean? For a 3d shape to be regular the ‘view from the ver- tex’ must be the same from each vertex, so

1. Each face must be a regular polygon shape

2. All faces must be the same

3. The same number of faces must meet at each vertex

Investigation

• Pupils will use the cardboard shapes to investigate what solids can be built that fit the rules above. To start with, they should limit their investigations to triangles.

• It is good for the pupils to work in pairs, so that both can be checking that each vertex has the same number of faces meeting there.

• This should allow them to discover that it is possible to make a shape where three triangles meet at each vertex (tetrahedron, with four faces). Is it possible to make a 3d closed shape with fewer faces? How can they be sure?

• Following that, pupils should see what shape results when they have more faces meeting at each vertex. They should find that it’s possible to construct a shape with four faces meeting at a vertex (octahedron) or five (icosahedron), but if they attempt to put six at a vertex, they should notice that this creates a flat shape and not a convex vertex.

• Discuss with pupils how they can be sure that there aren’t any more that can be made from triangles. What might be a good shape to try next?

• Pupils’ investigations should lead them to realise that there is only one possible regular solid that can be made from squares, and that one can be made from pentagons (dodecahedron), but that no more can be made without relaxing our rules. Pupils should use the hexagons to demonstrate that there are none that can be made with them – two at a vertex is not enough, but three results in a flat surface (this provides a good starting point for looking at angles in shapes and how they relate – eg. Equilateral triangle having half the angle at the vertex that a hexagon has)

Find out more Construction of 3d shapes provides a wealth of activities that can inspire pupils and introduce or consolidate aspects of other areas of mathematics. The Ri mathematics team can provide resources for planning and delivering activities looking into semi-regular solids and tilings. For more informa- tion, please email [email protected]. 6 4

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