# What’s So Special About These Solid Polyhedrons? ## Third Graders Discover the Attributes of Platonic Solids as They Searched for “Rules” That Define These Special Polyhedrons.

### “They have the same number of edges that come together at each vertex”

These were the two rules that third graders came up with regarding these special solids after carefully testing many other conjectures.  In the process of their reasoning and proof explorations, students refined their ideas about geometric concepts such as congruence, edges, faces, and vertices.

## Looking For Mathematical Relationships in the Data After each student completed a table listing the number of faces, vertices, and edges, they looked for mathematical patterns. Many noticed that there were repeat numbers that came up. They also noticed a mathematical relationship within the number of each platonic solid’s faces, vertices, and edges. Several students noticed that when they added the number of faces to the number of vertices of each solid, the sum was always two more than the number of edges. Students tested this out and realized that this was true for all of these solids.

# V – E + F = 2

The mathematical relationship that students discovered was Euler’s formula. Euler’s formula is the relationship among the number of vertices (V), edges (E), and faces (F) of a polyhedron. The formula can be written several different ways. Through this exploration, the concept of an unknown variable was introduced, as well as a formula. Third graders were very excited to learn that this was a rule that could be applied to these different polyhedrons!