This learning sequence will assist students to develop the geometrical language and properties associated with shapes of prisms and pyramids and to construct geometrical models. By exploring monuments that are pyramid-shaped and found in the Asia region, students will develop an awareness of the achievements and contributions of the peoples of Asia.
Activity 1: Developing mathematical vocabulary
At the beginning of this activity find out what students already know about the vocabulary of prisms and pyramids. Geometric terms are provided for students to build students' knowledge about the terms and meanings fundamental to understanding the geometry of prisms and pyramids.
In preparation for working with prisms and pyramids it is a good idea to assemble a classroom collection. Foods such cereals and chocolate are often boxed in containers that are prism-shaped forms. Other products such as soaps, candles and sweets are sometimes boxed as pyramids. Students should be encouraged to contribute to the collection.
Expansion notes for Euler's formula
F + V = E + 2
This formula is known as Euler's formula, named after Leonhard Euler, a Swiss mathematician. It refers to the relationship between the number of faces [F], sides [S], and vertices [V] of a convex (one without dents) polyhedron.
For any convex polyhedron in the classroom collection check that the formula holds true.
Notice that for a square pyramid, F = 5, V = 5 and E = 8 and so F + V = E + 2.
Now imagine the square pyramid transforming into a pentagonal pyramid by the number of sides in the base increasing by 1. Consider what happens to F, V and E.
Clearly V will increase by 1 (as there will be a new vertex in the base). The new side in the base will result in there being an increase of 1 in the number of faces [F]. The new side in the base shape will result in an extra edge in the base and an extra edge in the triangular faces. So we get …
[F + 1] + [V + 1] = [E + 2] + 2
This is true for any chosen pyramid … If the number of sides in the polygon base is increased by 1, the number of vertices [V] will increase by 1 as will the number of faces [F] and the number of edges [E] will increase by 2.
Activity 2: Nets of prisms and pyramids
Nets of prisms and pyramids require you to become familiar with the 'language of shape' and prepare the materials for the accompanying activities..
Equipment for this activity includes geometrical models and light cardboard for model making, scissors, rulers and craft glue.
Activity 3: Pyramids ... Where in the world?
This activity develops the link between the Mathematics curriculum content and architectural pyramids found in the countries of Asia. The aim of the activity is to expand students' awareness that pyramids are also found in Asia. While research about Egypt's pyramids is extensive, there is very little known about the large pyramid-shaped structures in the Asian region, especially about their construction, history and cultural significance.
Students will use the internet to identify the location of some of these pyramid-shaped structures and investigate their relative size and cultural significance.
There is an opportunity to introduce students to the Tower of Hanoi (sometimes referred to as the Tower of Brahma) puzzle. The puzzle is simply constructed. It can be regarded as a stacked pyramid and its solution from simple two- or three-piece forms to a number of pieces (5, 6, 7 …) is challenging to complete and to describe. Links are provided to websites which contain interactive representations of the puzzle.
Activity 4: Geometrical origami
This activity takes students beyond the Japanese origami techniques of folding paper squares into different sculptural shapes such as animals and flowers.
Provide origami paper for students who will use origami techniques to make geometrical shapes and solids, some involving modular approaches requiring a number of sheets of paper. Links to websites are provided to display photographs of examples of such constructions and to access construction instructions.