I recommend having the students make a paper model of the dodecahedron to compare with the icosahedron. The faces of the icosahedron correspond to the vertices of the dodecahedron, and the faces of the dodecahedron correspond to the vertices of the icosahedron. Students often notice that the dodecahedron looks like a soccer ball. A soccer ball is, in fact, a truncated icosahedron. The cubeoctahedron is an Archimedean solid.
Construction Take six toothpicks and six gumdrops.
Stick them together to make a hexagon. Place the hexagon on the desk. Take 6 toothpicks and 3 gumdrops. Stick a toothpick into each of the gumdrops in the hexagon. Join pairs of toothpicks at the top with the three gumdrops to form three triangles sticking up.