Show a cube (and perhaps also a tetrahedron) and ask what is so special about it.
Listen to different opinions and discover the "important" property that these shapes look the same way from all directions (technically, they are vertex-transitive, edge-transitive and face-transitive but you don't need to scare the people)
Then ask what other shapes are there like this?
Discover that all faces would need to be the same poylgon.
What could this polygon be? It would need to be a regular polygon.
Now take the prepared vectro-infused regular triangles, squares, pentagons, hexagons and octogons (kit is here at the school)
And just try to connect them.
After a while of experimentation realize that the only thing that specifies which solid we get other than the face polygon is the number of polygons adjacent to each vertex. And then see that there are only 5 options, which are the platonic solids.