Lesson plan of Regular Polygons: Angles and Diagonals

Objectives (5 minutes)

1. Understand what regular polygons are, identifying their main characteristics, such as equal sides and angles.
2. Learn to calculate the sum of the interior angles of a regular polygon using the formula (n-2) * 180, where n is the number of sides of the polygon.
3. Understand how to calculate the measure of each interior angle of a regular polygon by dividing the sum of the interior angles by the number of sides.

Secondary objectives:

1. Develop the ability to solve mathematical problems involving regular polygons, applying the concepts learned.
2. Encourage active student participation, promoting discussion and questioning during class.
3. Stimulate students' critical thinking and problem-solving skills through mathematics.

Introduction (10-15 minutes)

1. Review of Previous Content (3-5 minutes): The teacher should begin the class by reviewing the concepts of polygons and their characteristics, such as sides and angles. It is important that students are already familiar with these terms before delving into the study of regular polygons. The teacher can ask students to briefly explain what polygons are and what their properties are.

2. Problem Situations (5-7 minutes): The teacher should then present two problem situations involving the calculation of angles and diagonals in regular polygons. For example, "If the sum of the interior angles of a regular polygon is 900 degrees, how many sides does the polygon have?" and "If the sum of the interior angles of a regular polygon is 1800 degrees, what is the measure of each interior angle?". These situations will motivate students to think about the subject and arouse their interest in the topic.

3. Contextualization (2-3 minutes): It is important that the teacher shows students the importance of regular polygons in our daily lives. For example, it can be mentioned that many shapes in nature, such as the cells of a honeycomb or the crystals of a snowflake, are regular polygons. In addition, many everyday objects, such as street signs and tiles, are also regular polygons.

4. Capture Students' Attention (2-5 minutes): To capture students' attention, the teacher can share some curiosities about regular polygons. For example, it can be mentioned that the number of diagonals that can be drawn from a vertex of a regular polygon is always equal to the number of sides minus 3. The teacher can also present the famous "Trisectrix Problem", which is an unsolved mathematical problem that involves dividing an angle into three equal parts using only a compass and a ruler, and which has a solution based on regular polygons.

Development (20-25 minutes)

1. Presentation of Theory (10-12 minutes): 1.1. The teacher should begin by explaining what regular polygons are, highlighting that they are plane figures formed by line segments (the sides), which are all congruent to each other, and that have all equal interior angles. 1.2. It is important to emphasize that in a regular polygon, the sum of the interior angles is always a fixed number, regardless of the size of the polygon. 1.3. The teacher should then introduce the formula to calculate the sum of the interior angles of a regular polygon: (n-2) * 180, where n is the number of sides of the polygon. 1.4. The teacher should then explain how to calculate the measure of each interior angle of a regular polygon by dividing the sum of the interior angles by the number of sides.

2. Practical Demonstration (5-7 minutes): 2.1. The teacher should then demonstrate the application of the theory in practice. To do this, he can use drawings of regular polygons and demonstrate how to calculate the sum of the interior angles and the measure of each interior angle. 2.2. The teacher should also show how to calculate the number of diagonals that can be drawn from a vertex of a regular polygon, which is always equal to the number of sides minus 3.

3. Problem Solving (5-6 minutes): 3.1. After the demonstration, the teacher should propose solving the problem situations presented in the Introduction. 3.2. Students should be encouraged to actively participate by asking questions, proposing solutions, and discussing the answers. 3.3. The teacher should guide the discussion, clarifying doubts, correcting errors, and reinforcing the concepts learned.

4. Practical Activity (3-5 minutes): 4.1. To conclude the Development of the class, the teacher should propose a practical activity. Students should be divided into groups, and each group should receive a set of regular polygons of different shapes and sizes. 4.2. The challenge for students is to calculate the sum of the interior angles and the measure of each interior angle of each polygon, and to verify whether the calculations correspond to the characteristics that they visually observe. 4.3. During the activity, the teacher should circulate around the room, helping groups that encounter difficulties and encouraging discussion among students.

Feedback (10-15 minutes)

1. Group Discussion (5-7 minutes): 1.1. The teacher should gather all students and promote a group discussion about the solutions found by each team in the practical activity. 1.2. Each group should be invited to share their findings and the challenges they faced during the activity. 1.3. The teacher should ask questions to stimulate students' critical thinking and reflection. For example, "Why is the sum of the interior angles of a regular polygon always a fixed number, regardless of the size of the polygon?" or "Why does the measure of each interior angle of a regular polygon decrease as the number of sides increases?". 1.4. The teacher should also reinforce the concepts learned, correcting possible misunderstandings, and highlighting the effective strategies used by the groups to solve the problems.

2. Connection with Theory (3-5 minutes): 2.1. After the discussion, the teacher should make the connection between the practical activity and the theory presented. 2.2. The teacher should emphasize how the calculations made by the students during the activity correspond to the mathematical formulas to calculate the sum of the interior angles and the measure of each interior angle of a regular polygon. 2.3. The teacher should also highlight the importance of practice for understanding theoretical concepts, reinforcing that mathematics is not only a theoretical discipline but also practical and applied.

3. Final Reflection (2-3 minutes): 3.1. To conclude the class, the teacher should propose that students reflect individually on what they have learned. 3.2. The teacher should ask questions to guide students' reflection, such as "What was the most important concept you learned today?" and "What questions have not yet been answered?" . 3.3. Students should have a minute to think about the answers. Then, the teacher can ask a few students to share their reflections with the class, fostering a collaborative and respectful learning environment.

4. Teacher's Feedback (1-2 minutes): 4.1. Finally, the teacher should provide general feedback on the class, praising the students' effort and participation, and highlighting the class's strengths and areas that need more practice or study. 4.2. The teacher should also reinforce the importance of the topic studied for everyday life and for learning mathematics, encouraging students to continue practicing and exploring the subject outside the classroom.

Conclusion (5-10 minutes)

1. Summary of Content (2-3 minutes): 1.1. The teacher should begin the Conclusion with a brief summary of the main points covered in the class. 1.2. It should reiterate the definition of regular polygons, their characteristic of having equal sides and interior angles, and the formula to calculate the sum of the interior angles and the measure of each interior angle. 1.3. The teacher should also recall the formula to calculate the number of diagonals of a regular polygon. 1.4. It is important that the teacher makes connections between all of these concepts, showing how they relate to each other and are applied in solving problems.

2. Connection between Theory, Practice, and Applications (2-3 minutes): 2.1. The teacher should then highlight how the class managed to connect the theory, practice, and applications of the subject. 2.2. It should reinforce how the theory presented was applied during the practical activity, and how the concepts learned are useful for solving mathematical problems and understanding the world around us. 2.3. The teacher can also mention again the importance of regular polygons in various areas, from biology and physics to architecture and design.

3. Suggestions for Extra Materials (1-2 minutes): 3.1. The teacher should suggest extra materials for students who wish to deepen their knowledge of the subject. 3.2. You may suggest math books, educational websites, videos, and math games that cover the topic of regular polygons. 3.3. The teacher can also propose some additional challenges or problems for students to solve at home, as a way to review and consolidate what was learned in class.

4. Importance of the Subject for Everyday Life (1-2 minutes): 4.1. Finally, the teacher should reinforce the importance of the subject studied for the students' daily lives. 4.2. Examples of regular polygons found in everyday life, such as street signs, tiles, and shapes in nature, can be mentioned again. 4.3. The teacher should emphasize that in addition to being useful and applicable, mathematics is also a fascinating and challenging discipline, which helps us understand and appreciate the beauty and complexity of the world around us.