Grade Eight Waldorf Geometry Block

Exploring the Platonic solids, their Duals, and Deriving Formulas for the Volume of Common Geometric Solids

We are working on our Grade Eight Geometry Block, the highlight of which is the platonic solids. We’ve done some review work with constructions, angle proofs, and quadrilaterals, as well as the pentagon and golden ratio. As we move along the review of this material from years’ past, we are adding in new material. I thought I’d share some of our resources and how I am teaching the block.

In this post, you’ll find:

Analysis and Transformations of the Solids

As part of the main work of our block, we are analyzing, forming, and transforming platonic solids. There are five platonic solids – the tetrahedron, the octahedron, the cube, the dodecahedron, and the icosahedron. I formed these five solids by drawing the faces — the equilateral triangle, the square, and the pentagon – and using net patterns in my sources to draw nets on cardstock paper with tabs for gluing. I scored the paper to make the folds crisp, and then folded and glued.

We took these solids and did a ‘notice and wonder’ where we looked at the number vertices, faces, and edges of each. We also looked at the shape of the faces. We looked for the patterns in these numbers and shapes and saw several. These of vertices and faces correspond with the duals of the solids. For instance, an octahedron which has eight faces and six vertices is the dual of a cube which has the reverse number of faces and edges: six faces and eight vertices. The same is true of the icosahedron and the dodecahedron. The tetrahedron is its own dual.

We later explored this relationship with the clay. Beginning with the form of a cube, we pressed the corners into our clay forming a truncated cube. We then continued to press until the corners of the triangles touched forming a cuboctohedron. Then we continued to press into the clay until we had a truncated octahedron, and then finally until we had a fully formed octahedron. In theory, this can also be done with a dodecahedron and an icosahedron, but that is beyond my skill level. We did quickly do a transformation of a tetrahedron which dual is itself. The transformation of the cube was drawing and labeled in our Main Lesson Books. This exploration of duals continues as we explore our shapes in other ways.

Folding the Circle

An aspect that is new to us is work from Bradford Hansen-Smith from Whole Movement with Folding the Circle. This is somewhat new material for me as I’ve not used this much with my son. I’m really loving how it is integrating with our block. Much of the triangle and angle work we have been able to see in our folding project for the formations of the solids. We are not only constructing the platonic solids, we are also able to explore their relationships. For instance, when arranged the tetrahedron in a tetrahedron form, the negative space between them was an octahedron. Conversely, when we arranged the octahedron in a tetrahedron form, we were able to observe that the negative space formed was a tetrahedron.

We are also further able to explore the duals in a similar faction. We stellated an octahedron with several of our tetrahedron to form a stellated octahedron. We then connected each of the vertices from the stellation with colored yarn. What we found is that it sits within a cube, the dual for an octahedron. When we look closer, we can see that the vertices of the octahedron underneath are the center of each of the faces of the cube. This can also easily be done with the icosahedron and its dual, the dodecahedron.

Folding the Circle Video

Deriving Formulas for Volume

Another aspect of this area of geometry is to derive formulas for the volume of common solids – prisms, pyramids, cones, and spheres. We started with talking about the volume of a cube or a rectangular prism and how we are essentially finding the area of the face and then multiplying it by the height. Then we moved on to see that we can do this with other prisms with different faces such as the triangular prism or the cylinder.

This opened the pathway for us using this prism area and the relationship of other solids to derive formulas for those solids. We first looked at a pyramid with a square base. What is the relationship between a pyramid with the same height as its base and the cube with the same base? From Making Math Meaningful with Jamie York, I used the net to draw and from three tilted pyramids. We know from our ‘sheer and stretch’ picture proofs that if we move a pyramids apex to one corner, leaving us with a tilting pyramid, it has the same area as the original pyramid. Because three pyramids fit within the cube, we know that the area of the pyramid is a third of the cube. We can then derive a formula, by taking a third of the rectangular prism that the pyramid would fit within.

Note: ‘Sheer and stretch’ is a method to demonstrate area model proofs such as Pythagoras’s theorem or the area of a trapezoid. See this demonstrated in the video below.

From this point, we then looked at the relationship between a cone, a sphere, and a cylinder. This was a little more difficult because they cannot be easily fit into the cylinder. We first found the volume of the cylinder by calculating it from the other work on prisms we had done. Then we found the volume of the sphere by water displacement. This was great review of our earlier chemistry block. We found that a sphere is 2/3 volume of a cylinder with the base the same diameter as the sphere.

With the water displaced, we filled a cone that had the same diameter of the cylinder and the sphere. We were able to fill this twice which means that the cone was half the volume of the sphere with the same diameter. Knowing that the cone was ½ of the sphere and the sphere was ²/₃ of the cone, we were able to reason that the cone would be ¹/₃of the cylinder.

Video of Volume Work

Resources for the Block

I used a number of resources for our geometry block. The main resources that I used was from Jamie York in his Making Math Meaningful for the Middle Grades. This provided me with most of the material that I wanted to convey in the block. Folding the Circle books 1 and 2 by Bradford Hansen-Smith were also instrumental. We are doing many of the folds and exploration in this block. They are a wonderful fit as they are experiential in nature and conveys much of the information we are exploring through other means. There are others that are sprinkled in. I go over them in the video below with some details of our block. There are also a number of books on geometry in The Online Waldorf Library, some of which are free downloads.