Pyroxene

Achievement objectives

GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.

GM4-6: Relate three-dimensional models to two-dimensional representations, and vice versa.

Required materials

See Materials that come with this resource to download:

• Pyroxene activity (.pdf)

Activity

Pyroxenes are minerals found in cooled volcanic lava. They have an interesting geometric structure.

Identify and list the irregular polygons that make up the pyroxene crystal shown in the diagram.

Use this list to construct a net for a pyroxene crystal.

The following prompts illustrate how this activity can be structured around the phases of the mathematics investigation cycle.

Make sense

Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.

• Do I understand the situation and the words? (Students may need support to interpret the 2-dimensional picture as a 3-dimensional solid.)
• Where else in my life or in the world can I see this happen? (Students may be aware of the solid shapes of crystals, like salt.)
• Can I draw or sketch the situation? (Can students sketch the individual polygons that make up the solid?)
• What will a solution look like? (A net (flat pattern) that folds up to form the solid.) 

Plan approach

Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.

• What are the maths skills I need to work this out? (Students should recognise the importance of lengths and angles and the importance of using tools like rulers and protractors.)
• What strategies can I use to get started? (Drawing the individual shapes first before attempting to construct the net is a useful beginning.)
• Can I notice a pattern in how the solid is made? (Around each vertex (corner), there are usually three shapes. What shapes surround each vertex?)
• What side lengths will need to be equal? What will the angles be? How do you know?
• How can I use symmetry or partition the solid to simplify the problem?

Take action

Allow students time to work through their strategy and find a solution to the problem.

• Does my net work? If not, how can it be altered so it does work?
• Have I recorded my ideas in a way that shows how I worked things out?
• How does my net look different from, and the same as, the nets of others? Why could this be?
• Is there another possible answer or way to solve it? Are other nets possible that still work?

Convince yourself and others

Allow students time to check their answers, and then either have them pair share with other groups or ask for volunteers to share their solution with the class.

• What is a solution, or a set of solutions? 
• How would I convince someone else that I am correct?
• Could I have solved the problem in a more efficient way? What did I learn from this task?
• What connections can I see to other situations, and why would this be? (How would I use what I learned to create other nets?)
• Are there some properties of nets that are always true? (The importance of an angular gap around each vertex is important. The net must have the correct number of shapes and the correct arrangement of shapes around each vertex.)
• Have I tested my generalisation out on other cases, e.g., nets for Platonic solids.