Lesson plan of Regular Polygons: Angles and Diagonals

Objectives (5-7 minutes)

1. Understand the definition of regular polygons: The teacher will ensure that students understand the difference between regular and irregular polygons, with a focus on the regularity of angles and sides. This can be done through a quick review of basic polygon concepts and exploring what it means to be regular.

2. Know the formula for the sum of the interior angles of a regular polygon: The teacher will explain and demonstrate how this formula is derived. Students should be able to understand the logic behind the formula and, most importantly, be able to use it to solve problems.

3. Apply the formula to calculate the measure of one interior angle of a regular polygon: Students will be able to apply the formula for the sum of the interior angles to calculate the measure of one interior angle in a regular polygon. The teacher should provide examples and guidance as they practice.

4. Understand the definition and formula for finding the measure of the sum of the diagonals of a regular polygon: The teacher will explain and demonstrate the formula for finding the sum of the measure of the diagonals of a regular polygon. Students should be able to understand the logic behind the formula and apply it to solve problems.

Additional Objectives:

• Develop critical thinking and problem-solving skills: Through the practice of regular polygon calculations, students will be encouraged to think critically and develop mathematical problem-solving skills.

• Promote active student participation: The teacher will facilitate the lesson in a way that promotes active student participation by encouraging them to ask questions, discuss ideas, and work together to solve problems.

Introduction (10-15 minutes)

1. Review of prior knowledge: The teacher should start the lesson by recalling the concepts of polygons, interior and exterior angles, and their sums. It is important for students to be familiar with these concepts before moving on to the topic of regular polygons. The teacher can reinforce these concepts through interactive activities, such as constructing polygons using geoboards or folding paper.

2. Problem situation: Next, the teacher can introduce two problem situations that will serve as a starting point for the development of the theory. The first one could be: "If I have a polygon with 5 sides, all sides are the same size and all interior angles are equal. How can I calculate the measure of each interior angle?" The second problem situation could be: "If I have a polygon with 5 sides, all interior angles are equal and the measure of each interior angle is 108°, how can I calculate the sum of the measures of the interior angles?"

3. Contextualization: The teacher should then contextualize the importance of regular polygons, explaining that they are widely used in geometry, physics, and other areas of science. One example of a practical application of regular polygons is their use in the construction of structures, such as bridges and skyscrapers, where the regularity of the polygons is fundamental to the stability of the structure.

4. Introduction to the topic: The teacher can introduce the topic of regular polygons with some interesting facts and applications. For example, they could mention that regular polygons are very common in nature, being found in crystal formations, honeycombs, and many other biological structures. They could also mention that regular polygons are widely used in art and design, due to their symmetry and harmony. The teacher could, for example, show images of Islamic art, which makes extensive use of geometric patterns based on regular polygons.

Development (20-25 minutes)

1. Activity "Building Regular Polygons": (10-12 minutes)

   • The teacher should organize the students into groups of 3 to 4 people.
   • Each group will receive a set of sticks with the same size and a task: build regular polygons with 3, 4, 5, and 6 sides.
   • The students must carry out the activity, discussing among themselves and taking the necessary measurements to ensure that all sides and angles are equal.
   • During the activity, the teacher should circulate around the room, observing and guiding the groups as necessary.
   • At the end of the activity, each group should present their polygon and explain how they concluded that it is regular.
   • This hands-on activity will help the students visualize and better understand the regularity of polygons, preparing them for the next activity.

2. Activity "Discovering the Formulas": (5-7 minutes)

   • Still in their groups, students will be tasked with trying to discover the formulas for the sum of the interior angles and for the sum of the measures of the diagonals of a regular polygon.
   • The teacher should provide hints and guidance as necessary, but the goal is for the students to arrive at the formulas on their own, through a process of investigation and discovery.
   • After a set amount of time, each group should present their "discovery" to the class, explaining how they arrived at the formula and why they believe it is correct.
   • This activity will promote collaboration among students, as well as critical thinking and problem-solving.

3. Activity "Solving Problems with Regular Polygons": (5-6 minutes)

   • The teacher should provide a series of problems involving regular polygons, including calculating interior angles and the sum of the measures of the diagonals.
   • The students, in their groups, should work together to solve the problems, using the formulas they discovered and discussing the solution strategies.
   • The teacher should circulate around the room, assisting the groups as necessary and providing feedback.
   • At the end of the activity, the teacher should review the solutions to the problems with the class, clarifying any doubts and reinforcing the concepts learned.

4. Activity "Applying the Concepts": (3-5 minutes)

   • To consolidate learning, the teacher should propose an activity to apply the concepts learned. For example, students could be challenged to calculate the sum of the measures of the angles of a regular polygon with a large number of sides or to calculate the sum of the measures of the diagonals of a regular polygon with an odd number of sides.
   • This activity will allow students to apply the concepts learned in a more challenging and contextualized way.

Debrief (8-10 minutes)

1. Group Discussion (3-4 minutes)

   • The teacher should gather all students and lead a group discussion about the solutions to the problems and the results of the activities carried out.
   • Each group should share their conclusions and solution strategies, allowing other students to learn from their approaches.
   • The teacher should facilitate the discussion, asking questions to stimulate critical thinking and deepen the understanding of the concepts.

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

   • After the group discussion, the teacher should make the connection between the hands-on activities carried out and the theory presented at the beginning of the lesson.
   • The teacher could, for example, highlight how the formula for the sum of the interior angles and the formula for finding the sum of the measures of the diagonals were applied to solve the problems.
   • The teacher should also reinforce the importance of the regularity of polygons and how it affects their properties.

3. Individual Reflection (2-3 minutes)

   • To finalize the lesson, the teacher should ask the students to reflect individually on what they learned.
   • The teacher could ask questions like: "What was the most important concept you learned today?" and "What questions still remain unanswered?"
   • The students should have a minute to reflect and then can share their answers with the class if they wish.
   • This final reflection will allow students to consolidate what they learned and identify any gaps in their understanding that still need to be addressed.

4. Teacher Feedback (1 minute)

   • The teacher can conclude the lesson by asking the students for quick feedback about the lesson.
   • The students could be asked what they found most interesting, what was most challenging, and what they would like to learn more about the topic.
   • This feedback will be valuable for the teacher to assess the effectiveness of the lesson and plan future lessons on the topic.

The Debrief is a vital step in the lesson plan as it allows the teacher to assess the students' understanding of the topic and identify any areas that may need reinforcement in future lessons. Furthermore, the Debrief provides students with an opportunity to consolidate what they have learned and to reflect on their own learning process.

Conclusion (5-7 minutes)

1. Summary of Content (2-3 minutes)

   • The teacher should begin the Conclusion by recapping the main points covered during the lesson. This includes the definition of regular polygons, the formula for the sum of the interior angles, the formula for finding the sum of the measures of the diagonals, and how these formulas can be applied to solve problems.
   • The teacher can reinforce these concepts through a short quiz or question-and-answer game, ensuring that students have understood and retained the information.

2. Connection Between Practice and Theory (1-2 minutes)

   • Next, the teacher should explain how the lesson connected theory, practice, and applications.
   • The teacher can highlight how the hands-on activities, such as building regular polygons and solving problems, helped to illustrate and deepen the theory.
   • Additionally, the teacher can reiterate the practical applications of regular polygons, reinforcing the importance of what was learned.

3. Supplementary Materials (1 minute)

   • The teacher should suggest additional study materials for students who wish to further their understanding of the topic. This could include math textbooks, online videos, educational websites, and extra exercises.
   • The teacher could, for example, recommend that students explore constructing regular polygons using dynamic geometry software, such as GeoGebra.

4. Relevance of the Topic (1 minute)

   • Finally, the teacher should emphasize the importance of the topic for everyday life and for other disciplines.
   • The teacher could revisit the practical applications of regular polygons, such as in architecture and biology.
   • Additionally, the teacher could explain how the study of regular polygons connects to other math topics, such as trigonometry and analytic geometry.

The Conclusion is a crucial step in the lesson plan as it allows the teacher to restate the key concepts, draw connections between theory and practice, and encourage students to continue studying the topic. Furthermore, the Conclusion helps to solidify students' learning by recapping the concepts and highlighting their relevance.