Polyhedra – Mathematical process It is important that the children are encouraged to use their own process and strategies to investigate the situation. It may be helpful for children to have access to the models of the 3D shapes that are to be investigated. The net of each shape is available as a printout for the children to construct. Problem1 Process/strategy
Be systematic ·
o Start with the simpler shapes and count up the faces, edges and vertices of each shape. It is important that children keep track of the faces/edges/vertices that they have already counted. If they are not using the interactive program then it is recommended that they be allow to mark the shapes in some way so that they can keep count accurately. Hexagonal prism
· Faces ­ 8
2 3 7 4 8
5 1 6 ·
Edges – 18
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Vertices – 12 ·
Tabulate results Shape Number of faces Number of vertices Number of edges 6 8 12 6 8 12 Triangular prism 5 6 9 Hexagonal prism 8 12 18 Square based pyramid 4 4 6 Hexagonal based pyramid 7 7 12 4 4 6 8 6 12 12 20 30 20 12 30
Cube Cuboid Tetrahedron Octahedron Dodecahedron Icosahedron ·
Look for a pattern in results If the results have been tabulated in this way then the relationship between the faces, vertices and edges should be fairly easy for the children to identify. (The less able worksheet is worded in a way that leads them towards finding the relationship) Cube 6 + 8 ­ 2 = 12 8 12 Cuboid 6 Triangular prism 5 + 6 ­ 2 = 9 Hexagonal prism 8 12 18 The pupils should notice that the number of edges can be found by adding together the faces and vertices and then subtracting 2.
Solutions
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To find the number of edges add the number of faces and the number of vertices and then subtract 2.
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NB Children will most probably use specific values to explain their logic, not a generalised one.
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To find the generalised statement; Let the faces = f Let the vertices = v Let the edges = e So if the number of edges is equal to the number of faces added to the number of vertices subtract 2 we can write this as (f + v) – 2 = e
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To extend the learning of more able children you may ask them to find either the number of faces or vertices if they have the data for the other two.
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Number of faces This could be tackled by looking at the table of results Shape Number of edges Number of vertices Number of faces Cube 12 ­ 8 + 2 = 6 Triangular prism 9 6 5 Hexagonal prism 18 12 8 It could also be tackled by rearranging the formula (f + v) – 2 = e f – 2 = e ­ v (taking v away from both sides) f = e + v + 2 (adding 2 to both sides) Our general term is f = (e ­ v) + 2 They should notice that the number of faces can be found by taking the number of vertices away from the number of edges and adding 2.
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Number of vertices This could be tackled by looking at the table of results Shape Number of edges Number of faces Number of vertices 6 8 Cube 12 ­ + 2 = Triangular prism 9 5 6 Hexagonal prism 18 8 12 They should notice that the number of vertices can be found by taking the number of faces away from the number of edges and adding 2. It could also be tackled by rearranging the formula (f + v) – 2 = e v – 2 = e ­ f (taking f away from both sides) v = e + f + 2 (adding 2 to both sides) Our general term is v = (e ­ f) + 2 Problem 2 Process/Strategy
· Be systematic NB It should be noted that these results are based on the use of the 3D shapes that can be constructed from the nets within the pack. The results will vary if you use different 3D shapes (e.g. our pentagonal prism is made up of pentagons and squares. You may have a pentagonal prism that is made up of pentagons and rectangles.) Each of the 3D shapes needs to be studied and the name of the different 2D shapes that make up its faces need to be recorded. The children could use the interactive program to study the shapes or the nets of the shapes can be printed out and constructed. 5 4 3 2 1 1
Pentagonal based pyramid is made up of 5 triangles and 1 pentagon. ·
Tabulate results Shape Triangular face Square Rectangular face face Pentagonal Hexagonal face face × × × P × × P
P P
P × × × P × × P × P × × × P P
P P × × × P × × × P × P Cube × Cuboid × Square based pyramid Pentagonal based pyramid Hexagonal based pyramid Triangular prism Pentagonal prism Hexagonal prism × × Look for shapes that share a face Triangular face Square face Rectangular face Pentagonal face Hexagonal face Square based pyramid Pentagonal based pyramid Hexagonal based pyramid Triangular prism Cube Cuboid Cuboid Hexagonal prism Pentagonal based pyramid Pentagonal prism Hexagonal based pyramid Hexagonal prism
Square based pyramid Pentagonal prism Triangular prism Making branching trees for each shape may help Cube Cuboid Square based pyramid Pentagonal prism Triangular prism Hexagonal Pentagonal Hexagonal based based prism pyramid pyramid
Follow the lines of the branching tree to complete the polyhedral chain Cube Cuboid Pentagonal based pyramid Square based pyramid Pentagonal prism Hexagonal prism Triangular prism Hexagonal based pyramid There are many solutions to the polyhedral chain. The children should be encouraged to adopt their own strategies and explain their reasoning. Some possible solutions are as follows: Square based pyramid – Pentagonal based pyramid ­ Pentagonal prism ­ Cube – Cuboid – Hexagonal prism – Hexagonal based pyramid –Triangular prism – Square based pyramid Hexagonal based pyramid – Square based pyramid – Triangular prism – Pentagonal based pyramid ­ Pentagonal prism – Cube – Cuboid –Hexagonal prism ­ Hexagonal based pyramid Pentagonal prism – Cuboid – Hexagonal prism – Hexagonal based pyramid – Pentagonal based pyramid – Triangular prism –Cube –Square based pyramid – Pentagonal prism Triangular prism – Hexagonal based pyramid – Hexagonal prism – Cuboid – Cube ­ Pentagonal prism –Pentagonal based pyramid – Square based pyramid – Triangular prism