Contextualization

Mathematics, in all its vastness and diversity, is a discipline that allows numerous approaches, whether from a purely theoretical perspective or focusing on its real-world applications. In the field of Geometry, polygons constitute an interesting and important category of geometric figures. Polygons are closed flat shapes bounded by straight segments, called sides. Their classification is based on the number of sides, and the study of these figures involves the understanding of various concepts, such as perimeter, area, and, centrally in this project, the sum of the interior angles.

The sum of the interior angles of a polygon is a fundamental concept that allows us to understand and work with the structure of these shapes. For a triangle, the sum of its interior angles is 180 degrees. The measure of the sum of the interior angles of an n-sided polygon is given by the formula 180(n-2), a result that originates from the thesis that any polygon can be divided into (n-2) triangles. Therefore, this formula constitutes an essential tool for solving various types of problems in Geometry.

The study of polygons and, in particular, the understanding of the sum of their interior angles, is not limited to the realm of Mathematics. These concepts find practical applications in various disciplines and areas of knowledge. From product design and architecture to physics and chemistry, the ability to work with polygons is an integral part of many professional fields.

For example, in Architecture, the use of different polygonal shapes in project design is common. Architects use the sum of the interior angles of polygons to plan the layout of rooms and create structures that are both aesthetically pleasing and functionally efficient. In Physics, polygons are often used to model complex phenomena, such as the movement of particles, where predicting trajectories is based on the concepts of reflection and refraction of light, which in turn are based on the geometry of angles.

To assist in your study, we suggest the following sources:

1. "Fundamentos da Matemática Elementar - Vol. 3: Trigonometria" by Gelson Iezzi and Carlos Murakami. This book provides a detailed introduction to polygons and the calculation of the sum of their interior angles.

2. Khan Academy - On this site, you will find a series of explanatory videos and exercises on polygons and the sum of their interior angles.

3. Mundo Educação - This online article provides a comprehensive review of angles in polygons, including the sum of interior angles.

These sources will provide a solid foundation for understanding and working with these concepts, but we encourage you to explore and find other sources that may help in your understanding.

Practical Activity

Poly-Art Project

Project Objective

This project aims to combine knowledge of Mathematics, focusing on the Geometry of Polygons and their properties, especially the sum of the interior angles, with artistic and creative expression in the disciplines of Arts and Design.

Students, in groups of 3 to 5, must create a two-dimensional artwork that expressively and consciously uses the shape and property of polygons, preparing a report that contextualizes the artistic production, explains the choice of polygons used, and demonstrates the understanding and application of the concept of the sum of the interior angles in the process of creating and assembling the artwork.

Detailed Project Description

Part 1 - Research: Students must initially conduct a bibliographic research on polygons, the sum of the interior angles, and their practical applications in the real world, as well as search for examples of artworks that use polygons in their composition.

Part 2 - Planning: After understanding the theory about polygons, the groups must plan their artwork, drawing a sketch and deciding which materials they will use.

Part 3 - Creation: Students must proceed to the execution of the artwork.

Part 4 - Documentation: Finally, they must write a detailed report of the artwork creation process, explaining their choices and how they applied the studied concepts.

Required Materials

The necessary materials will depend on the artwork to be created. Some examples of materials that may be used include, but are not limited to: colored paper, glue, scissors, markers, colored pencils, ruler, compass, etc.

Detailed Step-by-Step

1. Research polygons and the sum of their interior angles. Use the suggested references in the introduction and search for additional sources.
2. Research examples of artworks that use polygons in their composition.
3. Meet with your group to discuss the research findings.
4. Plan the artwork that the group will create. Sketch your idea and decide which and how many polygons will be used, as well as their respective materials.
5. Check if the necessary materials are available or need to be acquired.
6. When all materials are available, start creating the artwork.
7. Document each step of the creation process. Take photos, make notes, and sketches.
8. After completing the artwork, write a detailed report, explaining the choices made, how the concept of the sum of the interior angles was applied in the artwork, and its relation to the research conducted.

The project should last approximately twelve hours per student, including research, planning, artwork creation, and report writing. This period can be divided into several work sessions over two weeks, for example.

Project Deliverables

Groups must deliver the two-dimensional artwork along with the report documenting the entire creation process. The report should be divided into four main sections:

1. Introduction - Contextualization of the theme, relevance of polygons and the sum of their interior angles in geometry and other areas, as well as the importance of the connection between art and mathematics.
2. Development - Detailed explanation of the research, planning, and creation process of the artwork. The methodology used, choices made, and how theoretical concepts were applied in practice should be explained.
3. Conclusions - Review of the main points, reflections on the project, lessons learned, and their relation to the theory studied.
4. Bibliography - Citation of all sources used to support the project realization.

This project aims not only to assess students' knowledge of polygons and the sum of their interior angles but also their research, planning, task execution, problem-solving, creativity, teamwork, and writing skills.