Patterns in the faces, edges and vertices of polyhedra — 11–13
Count, spot a pattern, write it with a letter, test it and explain why it works. Mathematics, 11–13 years.
A rule for a whole class of solids
A conjecture is a rule you suspect is always true. Take prisms whose base has n sides. Vertices: n in each base, so 2 × n. Edges: n in one base, n in the other and n side edges, so 3 × n. Faces: n side faces and 2 bases, so n + 2. You can say it in words or with the letter n. Then you test it on more prisms, and explain why it works.
Counting once, answering for every prism
Counting edges one by one works for a small solid, but not for a prism with a 10-sided base. A rule gives the answer at once. And testing is not enough: a rule that works on five prisms could still fail on the sixth. Explaining why (every vertex on one base has a partner on the other, so the double) shows it works for all. Another class has another rule: a pyramid with an n-sided base has n + 1 vertices and 2 × n edges.
From a table to a rule
Base with n = 3, 4, 5, 6 sides. Vertices: 6, 8, 10, 12. Edges: 9, 12, 15, 18. Faces: 5, 6, 7, 8. Pattern: V = 2 × n, E = 3 × n, F = n + 2. Test with a 10-sided base: V = 20, E = 30, F = 12. Check the edges by explaining: 10 + 10 + 10 = 30. A pentagonal pyramid, by its own rule, has 5 + 5 = 10 edges and 5 + 1 = 6 vertices.
One example is not a proof
A triangular prism has 6 vertices and 9 edges. Someone sees this and guesses “edges = vertices + 3”. It fits, and it feels right. But a cube has 8 vertices and 12 edges: 8 + 3 = 11, not 12. The guess fails. A conjecture built on a single case may be just a coincidence. Always test it on other solids, and look for the reason that makes it work.
Building models with straws
If you build a model of a pentagonal prism with straws and modelling clay, the rule tells you what to buy before you start: 2 × 5 = 10 balls of clay for the vertices and 3 × 5 = 15 straws for the edges. It has 5 + 2 = 7 faces. The same idea helps you plan any kit or model: find the rule once, use it for any size of base.
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