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.
A polygon is a two-dimensional shape with straight sides. A polyhedron is a fully enclosed three-dimensional object with faces that are polygons. A Platonic solid is a special type of polyhedron, made of identical, regular (all angles equal and all sides equal), polygonal faces with the same number of faces meeting at each vertex. The Platonic solids were named after the Greek mathematician Plato (though actually proved by Euclid).
There are 5 Platonic solids, the tetrahedron (4 triangles, 3 meeting at each vertex), the cube (6 squares, 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).
Terms commonly used to describe the attributes of polyhedra include:
In the 1750’s Leonhard Euler discovered a famous relationship between these three properties. The number of vertices, plus the number of faces take away two equals the number of edges.
E = V + F - 2
For a more detailed and comprehensive unit that covers the Platonic solids and extends the ideas further, see Polyhedra (3D shapes).
The learning activities in this unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. Ways to support students include:
This unit is focussed on the construction of specific geometric shapes and as such is not set in a real world context. There are ways that it could be adapted to appeal to the interests and experiences of your students. For example, students could be given the opportunity to decorate a model of their favourite Platonic solid in a style of their choice for a class display. This could make reference to cultural motifs, patterns, colours, and images that are relevant to your students, or those featured in an artist study. Groups of students could also be asked to transform a selection of platonic solids into an object (e.g. a robot, a futuristic city, a new school - consider how links could be made to your current context of learning and your students’ interests).
Te reo Māori vocabulary terms such as āhua (shape), tapa (edge), mata (face of a solid figure), ahu-3 (3 dimensions) and the names for individual shapes could be introduced in this unit and used throughout other mathematical learning. Students might be engaged in looking at how the te reo Māori kupu for shapes describe the form of a shape (i.e tapatoru - triangle. Tapa means edge and toru means three, therefore tapatoru means ‘three sides’ and describes the form of a triangle).
Construct a chart to record results for the cube. This chart will be used further as the week's investigations continue.
Polyhedron | Vertices | Edges | Faces |
Cube | 8 | 12 | 6 |
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 | Prefix |
4
| tetra-
|
Dear family and whānau,
This week we have been exploring polyhedrons. Ask your child to explain how these shapes have faces, edges and vertices. For homework your child has been asked to either:
a) find photographs of different polyhedrons from magazines, junk mail or websites and create a poster page for their maths book
or
b) use materials found at home to construct an icosehedron.
Printed from https://nzmaths.co.nz/resource/shapes-sticks at 12:44am on the 30th March 2024