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This problem builds on work done in finding Semi-Regular Tessellations and offers a great opportunity to create the Archimedean solids using students' knowledge of interior angles of polygons and angles around a point.
These printable worksheets may be useful: Which Solids Can We Make?
Which Solids Can We Make? - Extension
The problem has been structured in such a way that a class can spend some time answering each of the questions in bold, with time set aside to discuss approaches, thoughts and solutions.
The problem builds up to the search for Archimedean solids using what students learn about the angle deficit of Platonic solids. One way of introducing the concept of angle deficit could be to make a solid out of Polydron and then unfold it into its net and look at the gap at a vertex.
Vertex Form | Angle Sum | Angle Deficit | Number of Vertices | Total Angle Deficit | |
Cube | $4, 4, 4$ | $270$ | $90$ | $8$ | $720$ |
Tetrahedron | |||||
Octahedron | |||||
Icosahedron | |||||
Dodecahedron |
Starting with some two-dimensional work on Semi-Regular Tessellations may help to prepare students for the three-dimensional thinking required in this task.