Contextualization

Mathematics is an essential science in our society and, among its concepts, the interior eccentric angles stand out. They are a particular type of angle which appears in multiple everyday situations, even if they may go unnoticed at first sight. Therefore, it is crucial to have a sound understanding of this concept, not only for academic performance but also for its practical and theoretical application.

Broadly speaking, an interior eccentric angle is an angle where its vertex does not coincide with the center of the circumference and its interior to it. This concept is quite significant in various areas of knowledge, such as Engineering, Architecture, Design, and even in everyday activities. For example, in Architecture, the interior eccentric angle can be used to create daring shapes, like a circular dome. Similarly, in Engineering, these angles are used in several situations, such as to calculate the trajectory of a satellite around the Earth.

Introduction

To progress in the comprehension of the interior eccentric angles, some fundamental concepts will be explored and they will help with the development of the project. Firstly, it will be essential to understand what an angle is, as well as its diverse classifications and properties. The idea of an angle is one of the main notions of Geometry. Basically, an angle is formed by two half-lines with the same origin. The initial half-line and the final half-line are the sides of the angle and the common origin is the vertex of the angle.

Later on, the concepts of circumference and circle will be approached. They are essential figures to build an interior eccentric angle. We will also work on the concept of the radian, a unit of angle measure widely used in Mathematics and its applications, especially in Trigonometry. Finally, based on these concepts, we will describe what eccentric angles are and their properties.

As a resource to deepen your knowledge on these concepts, we suggest the reading of specific chapters of the book "Fundamentals of Elementary Mathematics: Plane Geometry" by Gelson Iezzi and Carlos Murakami (Editora Atual) and the consultation of the website Só Matemática (https://www.somatematica.com.br/ensmedio/circulo2.php ), where it is possible to find detailed explanations and exercises on the subject.

Practical Activity - "Rediscovering the Cosmos: a journey through the interiors of the eccentric angles"

Project Objective

This project aims to provide the students with an in depth comprehension of the interior eccentric angles through the exploration of their use in Astronomy and other disciplines. Students will work in teams of 3 to 5 members and the duration of the project is approximately 15 to 20 hours per student.

Project detailed Description

In this activity, the students will create a simplified model of the solar system, which will serve as a tool to explore and better understand the concept of interior eccentric angles. At the end, the students must be able to identify where these angles are formed in their model, calculate the measurement of these angles, and discuss its significance.

The project connects to the Physics discipline by exploring the planetary orbits, providing a scientific vision about the formation of the angles. Additionally, it develops the teamwork skills, creative thinking, time management, and problem solving.

Necessary Materials

• Styrofoam balls of different sizes (representing the planets)
• Wooden stick (representing the sun)
• Nylon lines (to connect the planets to the sun)
• Protractor (to measure angles)
• Ruler (to measure distances)
• Pencil and paper (for notes)

Step-by-step detailed instructions to carry out the activity

1. Each team must research about the planetary orbits and write down their findings (estimated time: 4 hours).
2. Next, each team will build their solar system model. Each planet must be attached to a nylon thread that, in turn, must be anchored to the "sun". The distance from each planet to the "sun" in its model must be proportional to the real life distance (estimated time: 4 hours).
3. Now, the teams must explore where the interior eccentric angles form in their model. Next they must use the protractor to measure these angles (estimated time: 2 hours).
4. Then, the teams must discuss the significance of these angles and how they relate to the planetary orbits. For example, they can simulate the trajectory of a planet and observe how the angle changes over time (estimated time: 4 hours).
5. Finally, each team must write a detailed report (estimated time: 5 hours).

Report Submission and Elaboration

The report must be divided into four main sections:

1. Introduction: Contextualization of the project theme, relevance of the interior eccentric angles and angles in general and how they apply to the real world. The project's objective and its relevance for learning must be explained, as well.

2. Development: In this section, the students must explain the concept of interior eccentric angles in detail, describe how the solar system model was put together, how they measured the angles and the results obtained. The relationship between the measured angles and the trajectory of the planets must be explained.

3. Conclusion: The students must revisit the main points of the report, describe their learnings, highlight the importance of the interior eccentric angles and the conclusions reached by the project.

4. Bibliography: In this section, the students must list all resources and materials they used to carry out the project (books, websites, videos, etc.).

The final delivery must include the solar system models and the written report, detailing all of the creation process, the knowledge acquired and the discoveries made during the project.