Objectives (5 - 10 minutes)

1. Understanding the concept of inscribed polygons: Students should be able to understand what inscribed polygons are, recognizing their characteristics and properties. Moreover, they should be able to differentiate inscribed polygons from other types of polygons.

2. Recognizing and characterizing inscribed polygons in practical situations: Students should be able to identify inscribed polygons in drawings and figures, as well as in everyday situations. They should be able to describe the properties of inscribed polygons that allow for their identification.

3. Solving problems involving inscribed polygons: Students should be able to apply the concept of inscribed polygons to solve mathematical problems. They should be able to determine measures of angles and sides in inscribed polygons, as well as to calculate areas and perimeters of these figures.

Secondary objectives:

• Stimulating critical thinking and problem-solving: In addition to mastering the content, students should be encouraged to think critically and to apply acquired knowledge to solve complex problems.

• Developing the ability to communicate mathematics: Students should be encouraged to articulate their mathematical ideas in a clear and logical way, both orally and in writing.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should begin the class by recalling some basic geometry concepts that are essential for understanding the current topic. This may include a brief review on polygons, circles, angles, and their properties. The teacher can use practical examples or problem situations to reinforce these concepts. (3 - 5 minutes)

2. Problem situation 1: "The clock and the bicycle": The teacher can present the students with an image of a clock and a bicycle, and ask: "What geometrical figure is inscribed in the clock? And in the bicycle wheel? How can you prove it?". This problem situation aims to arouse students' curiosity and to get them to start thinking about the relationship between polygons and circles. (3 - 5 minutes)

3. Contextualization of the theme: The teacher should emphasize the importance of inscribed polygons, showing examples of how they are used in practice. For example, the teacher can mention that inscribed polygons are used in the construction of bicycle wheels, car tires, satellite dishes, among others. Moreover, the teacher can mention that the idea of inscribed polygons is applied in several areas, such as architecture, engineering, and even in computer games and 3D animations. (2 - 3 minutes)

4. Problem situation 2: "The mystery of the Great Circle": The teacher can present the students with a problem that involves the concept of inscribed polygons. For example, the teacher can ask: "If I tell you that a great circle divides a sphere into two equal parts, and that a regular polygon is inscribed in this great circle, how can you determine the number of sides of this polygon?". This problem situation aims to challenge students to apply what they have learned about inscribed polygons to solve a complex problem. (3 - 5 minutes)

Development (20 - 25 minutes)

1. Activity 1: Constructing Inscribed Polygons (10 - 12 minutes)

   • Materials needed: Sheet of paper, compass, ruler, pencil, and eraser.

   1. Instructions: The teacher should divide the class into groups of 3 or 4 students and provide each group with the necessary materials. Next, the teacher should explain that each group must construct a regular polygon inscribed in a circle, using the materials provided.

   2. Step-by-step of the activity:

      • Step 1: Each group should draw a circle of any size on the sheet of paper, using the compass.

      • Step 2: Next, they should choose any point on the circle's circumference and mark it.

      • Step 3: Still using the compass, they should mark other points on the circumference of the circle, in such a way that the distance between each pair of points is the same.

      • Step 4: They connect the points on the circumference of the circle, forming a regular polygon.

   3. Discussion: After each group finishes constructing their polygon, the teacher should promote a discussion in the classroom, where each group presents their polygon and the other students should check if it is actually inscribed in the circle. This activity aims to allow students to visualize and manipulate inscribed polygons, reinforcing the concept of inscription.

2. Activity 2: Applying the Central Angle Formula (10 - 12 minutes)

   • Materials needed: Sheet of paper, compass, ruler, pencil, and eraser.

   1. Instructions: The teacher should continue with the groups formed in the previous activity and provide each group with a new inscribed polygon, but without the measure of the angles. The challenge is to determine the measure of each interior angle of the polygon.

   2. Step-by-step of the activity:

      • Step 1: Each group should measure the central angle of the polygon with the compass.

      • Step 2: Next, they should divide the measure of the central angle by the number of sides of the polygon to find the measure of each interior angle.

      • Step 3: Finally, they should check if the measure of the obtained interior angle is consistent with the definition of a regular polygon.

   3. Discussion: After each group finishes the activity, the teacher should promote a discussion in the classroom, where each group presents the measure of the interior angle of their polygon and the other students should check if the measure is correct. This activity aims to allow students to apply the central angle formula to find the measure of the interior angles of an inscribed regular polygon.

3. Activity 3: The Magic Polygon Challenge (5 - 7 minutes)

   • Materials needed: Sheet of paper, compass, ruler, pencil, and eraser.

   1. Instructions: The teacher should present the students with an inscribed polygon with an unknown number of sides. The challenge is to determine the number of sides of the polygon, using only the compass and the ruler.

   2. Step-by-step of the activity:

      • Step 1: Each group should measure the distance between two non-adjacent vertices of the polygon with the ruler.

      • Step 2: Next, they should measure the radius of the circle inscribed in the polygon with the compass.

      • Step 3: Finally, they should divide the measure of the circle's circumference by the perimeter of the polygon to find the number of sides of the polygon.

   3. Discussion: After each group finishes the activity, the teacher should promote a discussion in the classroom, where each group presents the number of sides of the polygon they found, and the other students should check if the number is correct. This activity aims to challenge students to apply what they have learned about inscribed polygons to solve a complex problem.

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes)

   • After the activities are concluded, the teacher should gather all students in a large circle for a group discussion.
   • Each group should briefly share their solutions or conclusions on the activities carried out, describing the process they used to reach these results.
   • The teacher should ask clear and directed questions to each group, encouraging them to explain their reasoning and to justify their answers.
   • This discussion aims to promote the exchange of ideas and collaborative learning, allowing students to see different approaches to the same problem and to learn from each other.

2. Connection with the Theory (3 - 5 minutes)

   • After the group discussion, the teacher should make the connection between the practical activities carried out and the theory on inscribed polygons.
   • The teacher can make a brief summary of the main ideas discussed during the class, highlighting how they relate to the practical activities.
   • The teacher should emphasize that the goal of the practical activities was to allow students to visualize and manipulate inscribed polygons, reinforcing the concept of inscription.
   • Moreover, the teacher should point out that the practical activities also allowed students to apply the central angle formula and the concept of perimeter to solve problems involving inscribed polygons.

3. Final Reflection (2 - 3 minutes)

   • To conclude the class, the teacher should propose that students reflect for a minute on the following questions:
     1. What was the most important concept learned today?
     2. What questions have not yet been answered?
   • After the minute of reflection, the teacher can ask some students to share their answers with the class.
   • This activity aims to make students think about what they have learned and what they still need to learn, helping them to consolidate the acquired knowledge and to identify possible gaps in their understanding.

4. Feedback (2 - 3 minutes)

   • Finally, the teacher should provide feedback to students on their performance during the class.
   • The teacher should praise the students' efforts, highlighting the strengths and improvements observed.
   • In addition, the teacher should identify any common mistakes or areas of difficulty that arose during the class, and provide additional guidance or resources to help students overcome these difficulties.

Conclusion (5 - 10 minutes)

1. Summary of the Content (2 - 3 minutes):

   • The teacher should make a brief summary of the main points covered during the class, reinforcing the concept of inscribed polygons, their properties, and how they can be identified and manipulated.
   • Students should be reminded that inscribed polygons are geometrical figures that have all their vertices on a circle.
   • The teacher should highlight how inscribed polygons are used in several areas of knowledge, such as in the construction of bicycle wheels, car tires, satellite dishes, among others.

2. Connection of the Theory with Practice (1 - 2 minutes):

   • The teacher should reinforce how the practical activities carried out during the class helped to solidify students' theoretical understanding of inscribed polygons.
   • The teacher can highlight examples of how the central angle formula and the idea of perimeter were applied to solve problems involving inscribed polygons.
   • It should be emphasized that practice is essential for the development of mathematical skills, and that students should continue practicing what they have learned at home, through exercises and problems.

3. Complementary Materials (1 - 2 minutes):

   • The teacher should suggest some additional study materials for students who wish to deepen their knowledge on inscribed polygons. This may include mathematics books, educational websites, explanatory videos, among others.
   • The teacher can, for example, suggest that students research the "Thales' Theorem" and how it relates to the concept of inscribed polygons.
   • Moreover, the teacher can recommend that students practice constructing inscribed polygons and determining their measures using dynamic geometry software, such as Geogebra.

4. Relevance of the Subject (1 - 2 minutes):

   • Finally, the teacher should highlight the importance of knowledge about inscribed polygons for students' everyday lives.
   • It can be mentioned, for example, that the ability to identify and draw inscribed polygons can be useful in several situations, from solving practical problems to understanding more advanced concepts in geometry and other areas of mathematics.
   • The teacher should encourage students to perceive mathematics as a powerful and relevant tool, capable of providing a better understanding of the world around them and of helping them to develop important skills, such as logical thinking and problem-solving.