Shapes With Sticks
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.
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.
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).
Getting started
-
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 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?
Can you find another part 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 weeks investigations continue.
|
Polyhedron |
Vertices |
Edges |
Faces |
| 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 but the possibilities are endless.
-
To conclude the session 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?
Exploring
-
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 |
Prefix |
| 4 5 6 7 8 10 12
|
tetra- penta- hexa- hepta- octa- deca- dodeca-
|
-
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 polyhdra. There is a systematic naming system and many good sites outline this. Two particularly good examples are:
http://www.georgehart.com/virtual-polyhedra/naming.html
http://www.math.com/tables/geometry/polygons.htm#names
-
Throughout each session find one model to use as a focus for discussions at the end of the session. To conclude the sessions, 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.
Reflecting
-
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 their favourite model from the ones they have constructed this week and write down what they have learnt in their investigations. Alternatively you could use a digital camera to record the student’s models and publish their writing in a class book of polyhedra.
We have been exploring different shapes in class this week. Take some time with your child to look through the book they have made with a group of other students. Can you suggest any additions they could make to the book?
| Attachment | Size |
|---|---|
| icosahedron.pdf | 5.5 KB |
Similar Resources
Foil Fun
In this unit students will be having hands-on experiences with three-dimensional solid shapes. They will investigate the features of solid shapes and learn the names for them. They will be introduced to making nets for 3D shapes.
Visit ESOL Online for a version of this unit designed to support students for whom English is an additional language.
Keeping in Shape
This is a level 3 geometry activity from the Figure It Out theme series.
Space Tiling with Captain Planet
In this unit tessellations are used as an application of angle properties of polygons. Interior angle properties of polygons are used to justify the existence of the five platonic solids.
Building with triangles
In this unit, students will construct three-dimensional objects with triangles with emphasis on
- identifying properties of triangles
- constructing triangles (both equilateral and irregular), using either ruler and protractor or ruler and compass
- identifying three-dimensional objects which can be constructed using triangles
- designing the net and making three-dimensional objects
- naming three-dimensional objects
Arty Shapes
In this unit students take inspiration from several cubist artists then participate in a variety of art based activities to develop their knowledge of 2-dimensional shapes. They use their own language to describe their works and the shapes they have used.
