Contextualization

Polyhedra are three-dimensional solids formed by faces, edges, and vertices that have properties and characteristics that fascinate mathematicians and students around the world. For a mathematics teacher, polyhedra can be a very valuable didactic resource, as they allow students to visualize and manipulate abstract mathematical concepts. Understanding their structure requires the application of various thinking, logic, and calculation skills.

Polyhedra are classified into two main categories: regular and irregular. The former are formed by identical faces and have equal angles; notable examples are the cube and the dodecahedron. Irregular polyhedra, on the other hand, do not have equal faces or angles. To understand and calculate the properties of these solids, the Euler's formula (V+F=E+2) is often used, where V is the number of vertices, F is the number of faces, and E is the number of edges.

Importance of Polyhedra

The importance of polyhedra goes beyond the realm of mathematics textbooks. They are present in various fields of study and in many aspects of everyday life. In the field of architecture and engineering, polyhedra are used in the design of constructions and structures for providing solid and stable frameworks. In nature, regular polyhedra like the dodecahedron and the icosahedron can be found in crystalline structures, viruses, and some pollens. Therefore, understanding polyhedra is essential not only for our mathematical understanding but also for our perception and appreciation of the world around us.

In art, polyhedra have been a source of inspiration and raw material for numerous artists. From Picasso and Braque's cubism to modern sculptures, polyhedra have been used to explore shapes and patterns. Therefore, understanding polyhedra can also enrich our understanding and appreciation of art.

Practical Activity

Activity Title: Building and Exploring Polyhedra

Project Objective:

The objective of this activity is to allow students to learn about polyhedra in a practical way, building their own polyhedra models and using Euler's formula to calculate their properties, while practicing socio-emotional skills such as teamwork, creativity, and responsibility.

Detailed Project Description:

Groups of 3 to 5 students will be tasked with building at least 3 different models of polyhedra (a tetrahedron, a cube, and a dodecahedron), using recyclable materials.

After building their models, they must explore their properties by calculating the number of faces (F), vertices (V), and edges (E) of each one, and verifying if Euler's formula (V+F=E+2) applies to all models.

Required Materials:

• Popsicle sticks or straws
• Modeling clay or styrofoam balls
• Tape or glue
• Cardboard paper
• Colored pencils, pens, and markers
• Ruler
• Scissors

Detailed Step-by-Step for Activity Execution:

1. Each group must select the materials they will use to build their polyhedra models.
2. Next, they should research the structures of the polyhedra they will build, identifying the number of faces, vertices, and edges of each one.
3. After conducting the research, the groups should start building their models, being creative in choosing colors and designs.
4. After construction, each group must measure and record the number of faces, vertices, and edges of each of the models they built.
5. Then, they should verify if Euler's formula applies to the models they built, recording the results.
6. Finally, each group must prepare a presentation of their results for the class, explaining how they built each model, their properties, and if Euler's formula applies to all of them.

In addition to the practical deliverables, each group must produce a report documenting their findings and reflections on the project.

Project Deliverables:

Practical Part:

• Each group must deliver the three polyhedra models they built.

Written Document:

Each report should consist of the following topics:

1. Introduction: the group must contextualize the theme, its relevance, and real-world applications. It is important for students to explain the purpose of the project and why they chose to build the models they built.

2. Development: the group must detail the steps they took to build their models, describe the properties of each model, and discuss if Euler's formula applied to each of them. They should also discuss any difficulties they encountered and how they resolved them.

3. Conclusion: the group must summarize their main points, explain what they learned from the project, and how this learning will apply to other areas of their lives and studies.

4. Bibliography: the group must indicate all sources of information they used to carry out the project, such as books, videos, websites, etc.

The project must be delivered within a week, and the total time each student should dedicate to the project is two to four hours.