Contextualization

The polygonal shape is a common representation in our real world, present in various perspectives of everyday life, from the structure of a tile to even in our architecture. Understanding the characteristics and properties of this group of geometric figures has important implications in the understanding of mathematics and interaction with our environment.

Among polygons, regular ones have special characteristics that make them particularly interesting. They are equilateral polygons (with all sides equal) and equiangular (with all angles equal). These characteristics give them a symmetry that is a great area of interest in mathematics and also in various other areas of knowledge.

Introduction

The study of Geometry is one of the fundamental pillars of mathematics, and regular polygons are key elements in this discipline. Regular polygons are flat geometric figures whose sides and internal angles are all equal. They are everywhere in our daily lives, in nature, and also in art.

The second paragraph of the introduction goes here. In this project, we will learn how to build regular polygons using three tools: our hands (with ruler and compass), our minds (to understand and describe the mathematical processes involved), and dynamic geometry software (to visualize and explore the constructions in an interactive environment).

In the third paragraph of the introduction, we will talk about the construction of regular polygons, that is, how to draw a regular polygon using a ruler and compass, or dynamic geometry software. In addition, it will be taught how to describe this construction process using an algorithm, which is a set of step-by-step instructions to perform a task.

Below are some useful references for understanding and deepening the subject:

1. Regular Polygons: what they are, properties, and applications - Mundo Educação - This is a reliable article that explains what regular polygons are, their types, and properties.
2. Construction of Regular Polygons - Educamais - This is a step-by-step tutorial on how to draw a regular polygon using a compass and ruler.
3. Construction of Regular Polygons - GeoGebra - On this platform, students can experiment with the construction of regular polygons.
4. Regular Polygons - Khan Academy - Here is a set of self-guided learning lessons on regular polygons.

Practical Activity

Activity Title: Building and Exploring Regular Polygons

Project Objective

• Develop skills for building regular polygons using a ruler, compass, and dynamic geometry software.
• Understand the process of building regular polygons and express it as an algorithm.
• Encourage collaboration and teamwork.
• Foster written communication skills through the elaboration of a detailed report.

Detailed Project Description

Students will be divided into groups of 3 to 5 and will be tasked with building regular polygons (for example, equilateral triangle, square, pentagon, hexagon) using a ruler and compass. After constructing the polygons on paper, students must replicate the constructions in dynamic geometry software. Finally, each group must elaborate a detailed report of the activity.

Required Materials

• Paper
• Pencil
• Ruler
• Compass
• Computer with access to dynamic geometry software (we suggest GeoGebra, which is a free and online tool)

Detailed Step-by-Step for Activity Execution

1. Divide into groups of 3 to 5 students.
2. Using the compass and ruler, each group must build regular polygons on paper. You can start with the equilateral triangle, then move on to the square, pentagon, and hexagon. Pay attention to keeping all sides and angles equal.
3. After drawing the polygons on paper, use GeoGebra (or any other dynamic geometry software you prefer) to reproduce the constructions made on paper.
4. Once you have successfully built the regular polygons on both paper and software, the next step is to describe the construction process in the form of an algorithm, that is, a set of step-by-step instructions to create a regular polygon.
5. Finally, each group must elaborate a detailed report on the project, which should contain the following elements:
   • Introduction: Contextualization of the theme, its relevance and application in the real world, as well as the project's objective.
   • Development: Explanation of the theory of regular polygons, detailed description of the activity, explanation of the methodology used, and presentation and discussion of the results obtained.
   • Conclusion: Recap of the main points of the work, explanation of the learnings obtained, and conclusions about the project.
   • Bibliography: Indication of the sources used to develop the project.

Project Deliverables

At the end of the project, groups must deliver:

1. A set of regular polygons built by hand and a replica of the same drawings using dynamic geometry software.
2. The algorithm for constructing each regular polygon.
3. A detailed report following the structure of Introduction, Development, Conclusion, and Bibliography, with all instructions mentioned above.

Each stage of the project must be clearly explained and justified in the report, showing even the attempts that were unsuccessful and what was learned from these errors. The report should be written in a clear and objective manner, demonstrating the group's understanding of the topic and the ability to work together and communicate their results.