Objectives (5 - 10 minutes)

1. Understanding the concept of inscribed polygons: The teacher must ensure that students have a clear understanding of what an inscribed polygon is and how it is defined. This includes understanding that a polygon is inscribed in a circumference if all its vertices are on the circumference.

2. Identification of properties of inscribed polygons: Students should be able to identify the specific properties of inscribed polygons. This includes understanding that the sum of the interior angles of an inscribed polygon is equal to 360 degrees, regardless of the number of sides of the polygon.

3. Application of the inscribed angle theorem: Students should be able to apply the inscribed angle theorem to solve related problems. They should understand that the angle inscribed in an arc is half the central angle that subtends the same arc.

   Secondary objectives:

   • Encouragement of critical thinking and problem-solving: Through the application of the inscribed angle theorem, students should be encouraged to think critically and solve problems effectively.

   • Promotion of active participation and teamwork: Classroom activities should be designed to promote active participation and teamwork among students, encouraging discussion and collaboration.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the fundamental concepts necessary to understand the topic of the lesson, such as circle, circumference, polygons, and angles. This can be done through a brief recap or a quick quiz to assess students' prior knowledge.

2. Problem situations: To spark students' interest, the teacher can present two problem situations involving the concept of inscribed polygons. For example:

   • Imagine you are drawing an 8-sided polygon inside a circumference. How can you ensure that all the vertices of the polygon are on the circumference?

   • Suppose you have an inscribed polygon in a circumference and want to find the sum of the interior angles. How can you do this without measuring the angles?

3. Contextualization: The teacher should explain the importance of inscribed polygons in real life. For example, in architecture and design, inscribed polygons are often used to create symmetrical patterns and structures. Additionally, in science and engineering, they are used to model natural phenomena and design complex structures.

4. Capturing students' attention: To make the topic more interesting, the teacher can share some curiosities or practical applications of inscribed polygons. For example:

   • The Pythagorean Theorem, one of the fundamental principles of geometry, can be proven using an inscribed polygon in a circumference.

   • In Ancient Greece, inscribed polygons were considered sacred, and it is believed that the gods used them to create symmetrical shapes in nature.

Development (20 - 25 minutes)

1. Theory presentation (10 - 15 minutes):

   1. Definition of Inscribed Polygons: The teacher should explain that a polygon is inscribed in a circumference if all its vertices are on the circumference. The formal definition and visual examples can be used to illustrate the concept.

   2. Properties of Inscribed Polygons: The teacher should mention that the sum of the interior angles of an inscribed polygon is always equal to 360 degrees, regardless of the number of sides of the polygon. This property can be demonstrated mathematically and visually with the help of a drawing.

   3. Inscribed Angle Theorem: The teacher should present the inscribed angle theorem, which states that the angle inscribed in an arc is half the central angle that subtends the same arc. This theorem can be demonstrated with the help of a drawing and explained why it is true.

2. Practical activity (10 - 15 minutes):

   1. Construction of Inscribed Polygons: Students should be divided into groups and provided with compasses, rulers, and paper. Each group should construct an inscribed polygon in a circumference. They should start with a circle and then use the compass to mark the vertices of the polygon on the circumference. Students should count the number of sides of the polygon and measure the sum of the interior angles to verify if it is equal to 360 degrees.

   2. Measurement of Inscribed Angles: Next, students should measure the inscribed angle in each arc of the polygon and the central angle that subtends the same arc. They should record the measurements and verify if the inscribed angle theorem is valid for the polygon they built.

   3. Group Discussion: After the activity, groups should discuss their findings and present to the class. The teacher should guide the discussion, highlighting the main ideas and correcting any misconceptions.

3. Problem Solving (5 - 10 minutes):

   1. Inscribed Polygon Problems: The teacher should propose some problems involving the concept of inscribed polygons. The problems should vary in difficulty and application to challenge students and allow practice of different skills. For example, students may be asked to find the measure of an inscribed angle or the sum of the interior angles of an inscribed polygon.

   2. Discussion and Correction: Students should work in groups to solve the problems and then discuss their solutions with the class. The teacher should correct errors and provide constructive feedback to improve students' understanding.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher should promote a group discussion where each team shares their solutions or conclusions from the practical activities and proposed problems. During the discussion, the teacher can ask each team to explain the process they used to reach the solution, any difficulties they encountered, and how they overcame them.

   • The teacher should encourage all students to actively participate in the discussion, asking questions to clarify any doubts and requesting the opinion of different students on the solutions presented.

2. Connection with theory (3 - 5 minutes): After all teams share their solutions, the teacher should make the connection between the practical activities and problems with the theory presented at the beginning of the lesson. The teacher can highlight how the construction and measurement of inscribed polygons help visualize and understand the properties and the inscribed angle theorem.

   • The teacher can also reinforce the importance of understanding and correctly applying the theory to solve the proposed problems, and how the practical activities help reinforce the understanding of the theory.

3. Final reflection (2 - 3 minutes): To conclude the lesson, the teacher should propose that students reflect silently on what they have learned. The teacher can ask questions such as:

   1. What was the most important concept you learned today?
   2. What questions have not been answered yet?
   3. What strategies did you use to solve the problems?
   • After a minute of reflection, the teacher can ask some students to share their answers with the class. This can help identify any gaps in students' understanding and provide valuable feedback for planning future lessons.

4. Teacher Feedback (1 - 2 minutes): Finally, the teacher should give overall feedback on students' participation and performance during the lesson. The teacher can praise students' efforts, highlight areas where they made progress, and identify any areas that still need more practice or review.

   • The teacher should encourage students to continue studying the topic at home, reviewing the theory, practicing more problems, and exploring other sources of learning, such as videos, textbooks, and educational websites.

Conclusion (5 - 10 minutes)

1. Summary of contents (2 - 3 minutes): The teacher should summarize the main points covered in the lesson. This includes the definition of inscribed polygons, their properties, and the inscribed angle theorem. The teacher should reinforce that the sum of the interior angles of an inscribed polygon is always equal to 360 degrees and that the inscribed angle is half the central angle that subtends the same arc.

2. Connection between theory and practice (1 - 2 minutes): Next, the teacher should highlight how the theory presented in the lesson was applied in the practical activities and problems. The teacher can recall the construction of inscribed polygons and the measurement of inscribed angles, and how these activities helped visualize and understand the properties and the inscribed angle theorem.

3. Extra materials (1 - 2 minutes): The teacher should suggest extra materials for students who wish to deepen their knowledge on the topic. This may include textbooks, educational websites, videos, and interactive activities. The teacher can prepare a list of links and references to share with students, facilitating access to the materials.

4. Importance of the topic (1 - 2 minutes): Finally, the teacher should explain the importance of the topic for daily life and other disciplines. It can be mentioned that understanding inscribed polygons is fundamental in various areas, such as architecture, design, physics, and engineering. Additionally, the teacher can highlight that the ability to solve problems involving inscribed polygons is a practical example of how mathematics can be used to solve real-world problems.

   • The teacher can illustrate this importance with concrete examples, such as the application of inscribed polygons in creating symmetrical patterns in art and architecture, or in modeling natural phenomena in science and engineering.