Contextualization

Mathematics, at its core, is a discipline of discoveries, connections, and relationships. One of the most intriguing and fascinating concepts that unites plane geometry and pure mathematics is that of inscribed angles. This concept, first introduced by the ancient Greeks, has applications in a variety of domains, from physics to engineering and astronomy.

In our study, we will focus on the inscribed angle in a circle. This angle, formed by points that lie on the circumference of the circle, has a consistent mathematical relationship with the arc it intercepts, which is surprisingly simple yet of great practical utility.

The importance of inscribed angles cannot be underestimated. They are one of the fundamental concepts in geometry and have applications in many areas of daily and professional life. The most obvious of these is navigation, where the use of angles is essential, whether in terrestrial or maritime orientation, in aviation, or in space exploration. They are also fundamental in civil construction, architecture, and many other fields.

Inscribed angles are also closely linked to other key mathematical concepts, such as trigonometric relationships and analytical geometry. That is why this project is not limited to just one concept, but to four key concepts.

The resources we suggest for further exploration of these concepts are:

1. Khan Academy: Inscribed Angles in Circles (https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-inscribed-angles/a/inscribed-and-central-angles)
2. Matemática Rio with Prof. Rafael Procopio: INSCRIBED ANGLE (https://www.youtube.com/watch?v=JUd0h7dMYnE)
3. Fundamentals of Elementary Mathematics - Volume 9: Plane Geometry - Gelson Iezzi, Osvaldo Dolce, Carlos Murakami, and Roberto Périgo

Practical Activity

Activity Title: "Exploring the Inscribed Universe: A Journey among Angles, Arcs, and Stars"

Project Objective

This project aims to provide students with a deeper understanding of the concept of inscribed angles, how they are formed, their relationship with the intercepted arc, and their practical applications. Additionally, the project seeks to develop teamwork skills, time management, critical thinking, and problem-solving.

Detailed Project Description

This project will be carried out in groups of 3 to 5 students and will have a total duration of 15 hours, distributed over weeks according to the class's convenience. It is an interdisciplinary project that involves, in addition to mathematics, the discipline of physics in understanding the application of angles in astronomy.

The activity will be divided into three main parts:

1. Theoretical study of inscribed angles and their relationships: In this phase, students will conduct a thorough review of the concept of inscribed angles, relationships between inscribed angles and central angles, as well as between inscribed angles and arcs.

2. Creation of a portable planetarium: Here, students must create a simple model of a celestial sphere using recyclable materials. They should mark the main constellations and trace the angles formed by the stars' locations in the constellations.

3. Project report: Finally, students must prepare a comprehensive report on the activity carried out, discussing the theory, methodology, results, and conclusions.

Required Materials

For the completion of this project, groups will need the following materials:

• Textbooks and internet access for theoretical study.
• Large Styrofoam ball (or similar) to serve as a celestial sphere.
• Toothpicks and colored markers to mark the stars and angles formed.
• Nylon thread to suspend the celestial sphere.
• Paper, pen, and text editing software for the report.

Detailed Step-by-Step for Activity Completion

1. Theoretical Study: (5 hours)

• Students should review the didactic material and watch the recommended videos to deeply understand the concepts of inscribed angles and the involved relationships.

2. Creation of the Portable Planetarium: (8 hours)

• Groups should research specific constellations and replicate them in the celestial sphere, marking the stars with toothpicks and the angles formed with colored markers.

• Each team must explain the relationship of the observed inscribed angles in their constellations, clearly demonstrating the practical application of the concept of inscribed angles.

3. Project Report: (2 hours)

• Finally, students must prepare a detailed report on the activity carried out. The report should be structured in the following order: Introduction, Development, Conclusions, and Bibliography used.

• In the Introduction, students should contextualize the theme, its relevance and application, and the objective of this project.

• In the Development, students should explain the theory behind inscribed angles, detail the activity, the methodology used, and finally present and discuss the results obtained.

• In the Conclusion, students should reaffirm the main points, explaining the learnings obtained and the conclusions drawn from the project.

• In the Bibliography, students should indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Project Deliverables

The final deliverable of the project will be the portable planetarium created by the students and the complete project report, which must be submitted in digital format. The report should be structured as a scientific document and needs to clearly connect the activities carried out with the studied theoretical concepts.

Students must ensure that the report's development section clearly explains how they carried out each step of the project, the reasoning behind their actions, and how the practical activities reinforced their understanding of inscribed angles concepts.