Lesson plan of Polygons: Sum of Angles

Objectives (5-7 minutes)

1. Develop students' understanding of the concept of polygons and interior angles, and how they relate to the total sum of angles in a polygon.
2. Enable students to apply the general formula for the sum of interior angles of a polygon and the formula to calculate the measure of each interior angle of a regular polygon.
3. Encourage students to solve practical problems involving the sum of interior angles of polygons and the measure of interior angles in regular polygons.

• Secondary Objectives:
• 1. Stimulate students' critical thinking and problem-solving skills by applying the concept of sum of interior angles of polygons to practical situations.
• 2. Promote collaboration among students by encouraging them to work in groups to solve problems and discuss their solutions.
• 3. Foster students' interest in mathematics by demonstrating the relevance and applicability of the concepts learned.

Introduction (10-15 minutes)

1. Review of Prior Concepts: The teacher begins the lesson by reviewing the concepts of angles and polygons, emphasizing the characteristics of each. It is important that students are clear on what polygons are and how to identify their interior and exterior angles. (3-5 minutes)

2. Problem Situations:

   • The teacher presents two problem situations:
     • The first situation can be a challenge of assembling a polygon-shaped puzzle, where students should observe and discuss the sum of the interior angles of the completed polygon.
     • The second situation can be a practical application, such as constructing a polygon-shaped building and needing to calculate the measure of the interior angles to ensure the stability of the structure.
   • The aim of these situations is to spark students' curiosity and show the applicability of the content that will be covered in the lesson. (5-7 minutes)

3. Contextualization:

   • The teacher contextualizes the importance of the topic by explaining that the sum of the interior angles of a polygon is a fundamental property in geometry.
   • They can cite practical examples, such as using this concept in architecture (to design and build stable structures) and in cartography (to measure and represent areas of land and countries). (2-3 minutes)

4. Introduction to the Topic:

   • To introduce the topic and capture students' attention, the teacher can share some curiosities about polygons.
     • For example, the curiosity that the sum of the interior angles of a polygon with n sides is always equal to (n-2) * 180 degrees, or that the sum of the interior angles of a quadrilateral is always 360 degrees.
   • Another curiosity could be the relationship between the number of sides of a polygon and the measure of each of its interior angles in a regular polygon.
     • For example, in a regular hexagon, the measure of each interior angle is 120 degrees, while in a regular octagon, the measure of each interior angle is 135 degrees. (3-5 minutes)

Development (20-25 minutes)

1. Activity with Tangram (10-12 minutes)

   • The teacher distributes tangrams (sets of geometric pieces that, when put together, form a square) to each group of students.
   • The proposed challenge is for students to assemble a square using all the pieces of the tangram, leaving no empty space.
   • After assembling it, the teacher guides students to observe the square that was formed, emphasizing that they have just created a new polygon.
   • Next, the teacher asks students how many interior angles this new polygon has and how they could calculate the sum of these angles.
   • After the group discussion, each team presents their solution to the class, allowing all students to share their strategies and solutions.

2. Polygon Construction Activity (10-12 minutes)

   • The teacher provides each group of students with a sheet of paper, a protractor, and a ruler.
   • The challenge is for the students to draw a polygon on the sheet of paper, with the number of sides of their choice, but that is regular.
   • After constructing the polygon, the teacher guides the students to measure the amplitude of each interior angle of the polygon and to calculate the sum of the interior angles, using the formula (n-2) * 180, where n is the number of sides of the polygon.
   • The students then verify if the sum of the interior angles they calculated is equal to the sum of the interior angles of the polygon they drew.
   • This activity allows students to visualize the relationship between the number of sides of a polygon and the measure of each of its interior angles, reinforcing the concept of regular polygons and the application of the formula for the sum of interior angles.

3. Problem-Solving Activity (5-7 minutes)

   • The teacher proposes problems involving the sum of the interior angles of polygons and the measure of the interior angles of regular polygons for the students to solve.
   • The problems may include determining the number of sides of a polygon from the measure of one of its interior angles, determining the measure of each interior angle of a regular polygon from the sum of the interior angles, among others.
   • Students solve the problems in groups, discussing their strategies and solutions. The teacher circulates around the room, assisting groups that encounter difficulties and encouraging everyone's participation.

Return (8-10 minutes)

1. Group Discussion (3-4 minutes)

   • The teacher gathers all the students and promotes a group discussion about the solutions found by each team during the practical activities.
   • At this stage, the teacher should encourage students to share their experiences, difficulties, and strategies. This allows them to learn from each other and realize different ways to approach the same problem.
   • The teacher can ask specific questions to stimulate students' reflection, such as: "How did you use the sum of the interior angles formula in the tangram activity?" or "How did constructing the regular polygon help you understand the relationship between the number of sides and the measure of the interior angles?".

2. Connection with Theory (2-3 minutes)

   • After the discussion, the teacher makes the connection between the practical activities carried out and the theory presented in the Introduction of the lesson.
   • The teacher emphasizes how the formulas for the sum of the interior angles of a polygon and for the measure of the interior angles of a regular polygon were applied in the activities.
   • They also reinforce the importance of the concept of polygons and interior angles in the context of geometry and in various everyday applications.

3. Individual Reflection (2-3 minutes)

   • The teacher proposes that students reflect individually on what was learned in the lesson.
   • To do this, they formulate questions such as: "What was the most important concept learned today?" and "What questions have not yet been answered?".
   • The students have one minute to think about the questions and are then invited to share their answers with the class.
   • This reflection step is crucial for consolidating learning and identifying possible gaps in students' understanding, which can be addressed in the next lessons.

Feedback and Closure (1 minute)

• The teacher thanks everyone for their participation and provides brief feedback on the lesson, highlighting the positive points and areas that can be improved.
• They emphasize the importance of continuous study and practice for consolidating mathematical knowledge.
• The teacher also announces the topic of the next lesson and, if any, the necessary preparations, such as reading a chapter from the textbook or solving specific problems.

Conclusion (5-7 minutes)

1. Summary of Content (2-3 minutes)

   • The teacher begins the Conclusion by summarizing the main topics covered during the lesson.
   • They reinforce the concept of polygons, interior and exterior angles, and the general formula for the sum of the interior angles of a polygon.
   • In addition, they recall the formula to calculate the measure of each interior angle of a regular polygon, and how to apply it in practical situations.
   • The teacher also highlights the skills developed during the practical activities, such as problem-solving ability, critical thinking, and group collaboration.

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

   • Next, the teacher emphasizes the importance of the practical activities performed for understanding the theoretical concepts.
   • They highlight how the construction of polygons and the resolution of practical problems allowed students to visualize and apply the concepts learned in a concrete and meaningful way.
   • The teacher also reinforces the applications of the content learned, such as in architecture, cartography, and in various everyday situations that involve the understanding and use of angles and geometric shapes.

3. Supplementary Materials (1 minute)

   • The teacher suggests complementary study materials for students to deepen their knowledge on the subject.
   • This may include reading textbook chapters, solving more problems, conducting virtual experiments, or watching explanatory videos on the internet.
   • The teacher can, for example, recommend the use of interactive geometry applications, which allow students to explore the properties of polygons and angles in a fun and intuitive way.

4. Importance of the Subject (1 minute)

   • Finally, the teacher reinforces the importance of the subject for students' lives, highlighting its applicability in various everyday and professional situations.
   • They also encourage students to continue studying and making an effort, remembering that mathematics, despite being challenging, is a powerful tool for understanding the world and for developing skills such as problem-solving and logical thinking.