Objectives (5 minutes)

1. Understand the concept of the sum of the interior angles of a triangle: Students should be able to understand that the sum of the interior angles of a triangle is always 180 degrees. They should be able to apply this concept to different types of triangles, such as equilateral, isosceles, and scalene.

2. Develop skills to calculate the sum of the interior angles of a triangle: Students should be able to calculate the sum of the interior angles of a triangle, even if they do not know the exact values of each angle. They should understand that if two angles of a triangle are known, the third one can be calculated by subtracting the sum of the first two from 180 degrees.

3. Apply the concept of the sum of the interior angles of a triangle in problematic situations: Students should be able to use the concept of the sum of the interior angles of a triangle to solve geometry problems involving triangles. They should be able to identify the information given in the problem, apply the correct formula, and arrive at a solution.

Secondary Objectives:

• Foster critical thinking: When solving problems involving the sum of the interior angles of a triangle, students should be encouraged to think critically, make assumptions, and test different strategies.
• Promote active learning: Students should be encouraged to actively participate in the class by asking questions, discussing concepts, and solving problems in groups.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by reviewing the basic concepts of angles and triangles, emphasizing the different types of triangles (equilateral, isosceles, and scalene) and their properties. This includes reviewing the interior and exterior angles of a triangle, as well as the sum of the angles of a triangle (which totals 180 degrees). Students may be asked to share what they remember about these concepts.

2. Problem situation: To introduce the topic and spark students' interest, the teacher can propose two problem situations. The first one may involve constructing a triangle on paper and measuring its interior angles, followed by the question: 'If we add the interior angles of any triangle, what is the result?'. The second problem situation may involve a mathematical challenge, such as: 'Suppose you know the values of two interior angles of a triangle. How can you find the value of the third angle?'.

3. Contextualization: The teacher should highlight the importance of the topic, explaining that the sum of the interior angles of a triangle is a fundamental property of geometry. It is used in many practical applications, such as engineering, architecture, graphic design, and physics. The teacher can cite specific examples to illustrate the importance of the topic.

4. Introduction to the topic: To capture students' attention, the teacher can share some curiosities or stories related to the topic. For example, they can mention that the discovery that the sum of the interior angles of a triangle is always 180 degrees was a milestone in the history of mathematics. Additionally, the teacher can present a puzzle or a mathematical challenge involving the sum of the interior angles of a triangle, so that students can try to solve it during the lesson.

Development (20 - 25 minutes)

1. Presentation of theory (10 - 12 minutes): The teacher should start the formal explanation of the concept of the sum of the interior angles of a triangle. They can use the following structure:

   • Definition of the concept: The teacher should explain that the sum of the interior angles of a triangle is always 180 degrees. This is true regardless of the size or shape of the triangle.
   • Demonstration of the concept: The teacher can draw different triangles on the board, mark the interior angles, and demonstrate that their sum will always result in 180 degrees. They can use a ruler and a protractor to measure the angles and prove the statement.
   • Examples of application: The teacher should provide examples of how this concept can be applied. For example, they can show how to find the value of an unknown angle in a triangle if the other two angles are known. Or, they can show how to use the sum of the interior angles to verify if a given set of angles forms a valid triangle.

2. Discussion and clarification of doubts (5 - 7 minutes): The teacher should encourage students to ask questions and share their doubts. They should clarify all doubts clearly and precisely, ensuring that all students have understood the concept.

3. Practical application of the concept (5 - 6 minutes): The teacher should propose some practical exercises for students to apply the concept of the sum of the interior angles of a triangle. For example, they can provide a set of angles and challenge students to identify if they form a valid triangle. Or, they can ask students to find the value of an unknown angle in a triangle if the other two angles are known.

4. Group activity (optional, 5 - 6 minutes): If time allows, the teacher can propose a group activity to reinforce the concept. For example, they can divide the class into groups and give each group a set of angles. The challenge will be for the students, in groups, to identify if the set of angles forms a valid triangle and then find the value of the unknown angle, if possible. The teacher should circulate around the room, assisting groups that have difficulties and ensuring that all students are involved and understand the concept.

Return (10 - 15 minutes)

1. Group discussion (5 - 7 minutes): The teacher should promote a group discussion to allow students to share their solutions and conclusions with the class. This can be done in various ways, depending on the number of students and the available time. The teacher can ask students to share their answers to the proposed exercises, or they can ask each group to present their solutions for the group activity. During the discussion, the teacher should ensure that all answers are respected and that errors are seen as learning opportunities.

2. Connection with theory (3 - 5 minutes): The teacher should guide the discussion to connect students' solutions with the presented theory. For example, they can ask students how they used the sum of the interior angles to solve the problems. Or, they can ask students to explain, in their own words, what they understood about the sum of the interior angles of a triangle. The goal is to ensure that students feel confident in applying the concept, both in similar problems and in real-world situations.

3. Final reflection (2 - 3 minutes): To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. They can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'. Students should have a minute to think about their answers. Then, the teacher can ask some students to share their reflections with the class. This not only allows the teacher to assess the effectiveness of the lesson but also gives students the opportunity to think critically about their own learning process.

4. Homework assignment (optional): If time allows, the teacher can propose a homework assignment to consolidate learning. For example, they can ask students to solve more exercises on the sum of the interior angles of a triangle. Or, they can ask students to research and write a brief report on how the sum of the interior angles of a triangle is used in a specific area, such as architecture or engineering. The teacher should provide clear guidance on the assignment and be available to clarify doubts, if necessary.

Conclusion (5 - 7 minutes)

1. Summary of key points (2 - 3 minutes): The teacher should summarize the main points covered during the lesson. They can reiterate the definition of the sum of the interior angles of a triangle, emphasizing that it is always equal to 180 degrees. The teacher should emphasize that this concept is valid for all types of triangles, regardless of size or shape. Additionally, they should recall how to calculate the sum of the interior angles of a triangle when two angles are known.

2. Connection between theory, practice, and applications (1 - 2 minutes): The teacher should explain how the lesson connected the theory of the sum of the interior angles of a triangle with practice, through exercises and group activities. They can also reinforce the practical applications of this concept in various areas, such as engineering, architecture, and graphic design. The teacher should ensure that students understand the relevance of what was learned and how they can apply this knowledge in everyday situations.

3. Extra materials (1 - 2 minutes): The teacher should suggest additional materials for students who wish to deepen their knowledge of the sum of the interior angles of a triangle. These materials may include math books, educational websites, explanatory videos, and geometry apps. The teacher can also suggest extra exercises for students to practice at home.

4. Importance of the subject and next steps (1 minute): To conclude, the teacher should emphasize the importance of the presented topic and how it connects with other concepts in mathematics and other disciplines. They can mention that understanding the sum of the interior angles of a triangle is fundamental for more advanced topics in geometry and trigonometry. The teacher should encourage students to continue studying and prepare for the next lesson, where new concepts will be presented.