In this unit students make and investigate a variety of three dimensional shapes. By examining a wide range of shapes and looking at the relationship between the numbers of faces, edges and vertices they see whether they can “discover” Euler’s famous formula.
- Construct models of polyhedra using everyday materials.
- Use the terms faces, edges and vertices to describe models of polyhedra.
A polyhedron is a three-dimensional solid object which consists of a collection of polygons, usually joined at their edges. Terms commonly used to describe the attributes of polygons include:
Vertex: a point of intersection of two or more lines – a corner
Edge: A line that connects 2 vertices
Face: One surface of a solid figure
In the 1750’s Leonhard Euler discovered a famous relationship between these three attributes. The number of vertices, plus the number of faces take away two equals the number of edges.
E = V + F - 2
The platonic solids are one group of polyhedra, they are polyhedra all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex.
In other words, a platonic solid is a three dimensional shape where each face is an identical flat shape with all sides and angles the same, and the same number of these faces meet at each corner.
There are 5 platonic solids, the cube (6 squares, 3 meeting at each vertex), the tetrahedron (4 triangles, 3 meeting at each vertex), the octahedron (8 triangles, 4 meeting at each vertex), the dodecahedron (12 pentagons, 3 meeting at each vertex), and the icosahedron (20 triangles, 5 meeting at each vertex).
This unit involves a lot of exploration with three dimensional shapes and would be ideal as a lead in to the unit Building with Triangles, a level four unit which goes on to look at a group of polyhedra, the platonic solids, in more detail.
- Cube models (dice or similar)
- Play-dough, blu-tack (or similar)
- Variety of polyhedra models: a polyhedron dice set is ideal if it is available
- Straws, Toothpicks or Pipecleaners
- Paper and pens for recording
Invite the students to look at one of the dice or other cube models that you have available. Show them the materials they have to work with and ask
Can you make a model of this shape using these straws (or toothpicks) and play dough?
Students work in pairs to explore the materials and make models of cubes.
Once most groups have made a cube model successfully hold a group discussion, focusing on attributes of the shape.
What is one part of this shape we could count?
What are the other parts of this shape we can count?
Accept the terminology the students use and introduce the mathematical terms vertex, edge and face as needed.
Construct a chart to record results for the cube. This chart will be used further as the week's investigations continue.
Cube 8 12 6
Ask the students to make some more shapes with the materials available, making it clear that they can cut the straws to make polyhedra with shorter sides. Some of the simpler examples are shown below.
Record results for some of the other shapes students create on the chart and discuss.
Does anyone have one with the same numbers that looks different?
Can we see a pattern in these numbers?
Who has a polyhedron with the same number of edges? faces? vertices?
Who has a different polyhedron?
What did you discover as you made your polyhedron?
Over the next few days have students use the materials to create a variety of polyhedra. Students can use the geoblocks or polyhedra dice as models for the shapes they build or create their own unique examples. If you are using a dice set remove the icosahedron as this shape will be the focus for the last session of work.
As they work record some of the shapes created on the chart and encourage them to look at the numbers of faces edges and vertices each shape has.
Can anyone find a pattern with these numbers?
Explain that over 250 years ago a famous mathematician named Euler discovered there was a relationship between these numbers and challenge them to see if they can find it.
As a variety of shapes are made ask students to name their shapes and introduce the mathematical terminology. Each shape has a prefix according to the number of faces it has, followed by “hedron.”
Number of faces
Students can use these names for their shapes or create their own. Students could also use the internet to research the names of other larger polyhedra. There is a systematic naming system and many good sites outline this. One good example is:
Throughout each session find one model to use as a focus for discussions at the end of the session. To conclude the session, hide the model from the students and tell them the number of edges and vertices. Challenge them to predict the number of faces.
Who can work out how many faces this mystery shape will have?
How could you work it out?
Can you see a pattern in the numbers of other shapes that could help you?
Have students build models to help them as they try to predict the number of faces. Show them the model at the end to confirm the correct number of faces.
Who predicted the correct number of faces?
How did you work out your answer?
Encourage students to explain their thinking.
To conclude the week's investigations show students a model of an icosahedron. If you don’t have a set of polyhedron dice containing this model you could build one using the copymaster.
As you show them the model explain that they are going to predict the number of faces this shape has and then build a model to check whether they are right. Tell them the shape has 30 edges and 12 vertices. Discuss.
Look at the other numbers on our chart. See if you can find a relationship between these numbers that will help you make your prediction.
Students write down their prediction of the number of faces before they construct their model to check results. Hide the model as they work.
If students have trouble building an icosahedron, remind them that the faces are made of triangles and there are five triangles meeting at each vertex.
Once students have finished working, have a class discussion to compare results:
Did you manage to build an icosahedron?
How many faces did you predict it would have?
Was your prediction correct?
How did you make your prediction?
What pattern can you see in the numbers of edges, faces and vertices ?
If students have not come up with Euler’s Formula to describe the relationship between these numbers, tell them what it is and have them check whether it works for some of the shapes on the chart.
To conclude the session have students choose and photograph their favourite model from the ones they have constructed this week, and write down what they have learnt about it. These could form a class book of polyhedra.