Contextualization

Theoretical Introduction

Mathematics is a discipline built on fundamental concepts, and geometry is one of those foundations. Among the topics studied in geometry, one of great importance is the study of polygons, especially the study of geometric transformations that can be applied to them.

A polygon is a figure formed by line segments, which are called the sides of the polygon. Polygons can be classified as regular if all sides and angles are equal, or irregular if the sides and angles are not equal. In addition, polygons can also be classified according to the number of sides they have.

Geometric transformations, in turn, are changes that occur in the shape, size, or position of a polygon without losing its essential properties. The main geometric transformations are translations (displacements), rotations (turns), and dilations (increases or reductions in size).

Contextualization of the Theme

The study of polygon transformations is highly relevant as it allows the understanding of various everyday phenomena. For example, the reflection of an object in a mirror can be understood as a geometric transformation. In the fields of architecture and design, polygon transformations are used to create innovative shapes and structures.

Furthermore, polygon transformations are essential for various areas of technology, such as the development of games and virtual reality systems. In these cases, transformations are used to generate the three-dimensional graphics that form the basis of these technologies.

Activity

Activity Title: "Transforming Polygons: A Journey through Geometry and Art"

Project Objective

To understand the geometric transformations of polygons and their real-world applications, especially in the field of art, and to develop teamwork, time management, and critical thinking skills.

Detailed Project Description

In this project, groups of 3 to 5 students will immerse themselves in the world of geometry and art by transforming polygons and creating their own artworks based on polygons. To do this, they will use the concept of geometric transformations of polygons, involving translations, rotations, and dilations, and apply these concepts in the creation of their artworks.

Required Materials

• Pencils, colored pens, ruler, compass, graph paper or grid paper
• Calculator (optional)
• 2D graphics software (such as GeoGebra, which is free and easy to use)
• Mathematics textbooks and/or internet sites for reference
• Various art materials (such as paints, brushes, colored papers, etc.)

Step-by-Step Guide for the Activity

1. Study the concepts of polygons and their properties, as well as geometric transformations (translation, rotation, and dilation).
2. Choose a polygon (can be regular or irregular) and represent it on a Cartesian plane.
3. Perform transformations on this polygon (at least one translation, one rotation, and one dilation) and represent these transformations on the same Cartesian plane.
4. Calculate the area, sides, or perimeter of the original polygon and the transformed figures (using the property that the area and perimeter are multiplied by the square and the dilation factor, respectively).
5. Using the polygons and their transformations, create an original artwork on paper or on graphic software. The artwork should demonstrate the use of the learned geometric transformations.
6. Write a project report, which should include the following topics: Introduction (contextualizing the theme and its objective), Development (explaining the theory used, the activity performed, the methodology, and the results), Conclusions (summarizing main points and learnings), and Bibliography.

Project Deliverables and Connection with Activities

1. Graphic representations of the transformations performed and related calculations.
2. The artwork created using polygons and their transformations.
3. The project report, which should contextualize the work, present the theory used, describe the activity performed (with a detailed step-by-step of the polygon transformations and artwork creation), present and discuss the results (including difficulties encountered and how they were resolved), and list the sources consulted during the work.

This project aims not only to assess the knowledge acquired by students regarding the topics covered but also to assess their socio-emotional skills such as teamwork, time management, problem-solving, and critical thinking.