Contextualization

Theoretical Introduction

In mathematics, inscribed angles are a fundamental concept in geometry. They exist in a circumference when we have an angle whose vertex is on the circumference and its sides are two chords of this same circumference. This concept of inscribed angles is key to establishing relationships between arcs, central angles, and inscribed angles in the circumference.

As such, the inscribed angle helps us explore the properties of circles and other shapes that contain them. Inscribed angles are used in various mathematical theorems, such as the Inscribed Angle Theorem.

Mathematicians have developed formulas and methods to calculate inscribed angles in a circle. Solving problems involving inscribed angles requires an understanding of the properties of angles and circles, including the concept of circle arcs.

Contextualization

Inscribed angles are used in many real-life situations. In civil engineering, for example, inscribed angles are used in the construction of bridges and buildings. In architecture, inscribed angles are used in project design. Even in fields like physics and astronomy, the concept of inscribed angles is applied.

Furthermore, the ability to understand and apply inscribed angles is an important skill not only in mathematics but in many areas of life. This may include solving everyday problems, such as calculating distances or determining viewing angles. Understanding inscribed angles can also be the basis for further study of complex mathematical topics.

Practical Activity

Activity Title: The Challenge of Inscribed Angles

Project Objectives

The main objective of this project is to apply the acquired knowledge about inscribed angles in a practical and engaging scenario. Students will build models of circles with inscribed angles and corresponding arcs, and estimate the values of these angles. Subsequently, these estimates will be verified through precise calculations.

Detailed Project Description

For this project, students divided into groups of 3 to 5 people, will create physical and digital models of circles with inscribed angles. Each group will collaboratively work to draw, measure, and calculate the inscribed angles in their models.

Required Materials

1. Graph paper
2. Compass
3. Ruler
4. Pencil and eraser
5. Protractor
6. Dynamic geometry software (GeoGebra or similar)

Activity Steps

1. Each group should draw on graph paper several circles of varying sizes, each with 3 inscribed angles. The inscribed angles should be of different sizes and positions in the circle.

2. Using the protractor, students will estimate the values of the inscribed angles in each circle.

3. Students will recreate their physical models in the dynamic geometry software, inserting the circles and inscribed angles as drawn on paper.

4. Using the software tools, students will calculate the exact values of the inscribed angles.

5. The estimated and calculated values of the inscribed angles should be recorded in a table for comparison.

6. Finally, each group should prepare a presentation explaining the process used, and discussing their observations and conclusions.

Project Deliverables

The project delivery should consist of:

1. The physical model of circles with inscribed angles;

2. The table with the estimated and calculated values of the inscribed angles;

3. The digital model of circles and inscribed angles created in the dynamic geometry software;

4. A detailed project presentation, which should include:

   • Introduction: The contextualization of the problem, explaining the relevance of studying inscribed angles;

   • Development: The explanation of the process used to create the models and calculate the inscribed angles. It should include the errors made and the corrections made to achieve the final results;

   • Conclusion: The conclusions drawn regarding the work, what they learned, and how this can be applied in real situations;

   • Bibliography: The list of sources consulted during the work, such as books, web pages, videos, etc.