Objectives (5 - 7 minutes)

1. Understand the concept of a triangle and its elements: Students should be able to identify the elements that make up a triangle (three sides, three internal angles, and three vertices). They should also be able to measure the sides and angles of a triangle using appropriate tools, such as a ruler and protractor.

2. Identify the condition for the existence of a triangle: Students should learn the rule that the sum of any two sides of a triangle must always be greater than the third side. They should understand that this rule is crucial for determining if a set of segments can form a triangle.

3. Apply the condition for the existence of a triangle in practical problems: In addition to understanding the rule, students should be able to apply it in practical situations. They should be able to determine if a set of segments can form a triangle by analyzing the measurements of the sides.

   Secondary objectives:

   • Develop critical and analytical thinking skills: By solving problems involving the condition for the existence of a triangle, students will be encouraged to think critically and develop analytical skills.

   • Improve problem-solving skills: Through practical problems, students will have the opportunity to enhance their problem-solving skills by applying theory in a practical way.

   • Promote teamwork: Group activities will promote collaboration among students, encouraging them to work together to solve problems.

Introduction (10 - 15 minutes)

1. Review of necessary content: The teacher should start the lesson by quickly reviewing previous concepts that are essential for understanding the topic of the lesson, such as the definition of polygons and the sum of the internal angles of a polygon. This will serve as a solid foundation for introducing the new content.

2. Problem situation 1: The teacher can present the following situation: "Imagine you have three segments of different lengths and the challenge is to form a triangle with them. What do you think is necessary for this to be possible?" This question should prompt students to think about what they already know and try to formulate a hypothesis.

3. Contextualization of the subject's importance: The teacher should explain that the condition for the existence of a triangle is a fundamental rule in geometry and is used in a variety of contexts, from civil construction to science and engineering. They can provide practical examples, such as the construction of bridges and buildings, where verifying the condition for the existence of a triangle is crucial.

4. Curiosities about the subject: To spark students' interest, the teacher can share some curiosities about the subject. For example, they can mention that the condition for the existence of a triangle was discovered by ancient Greek mathematicians and is one of the first concepts that geometry students learn. Another interesting curiosity is that the condition for the existence of a triangle can be generalized to other shapes, such as quadrilaterals and regular polygons.

5. Problem situation 2: The teacher can present another situation: "Suppose you have three segments with the following measurements: 3 cm, 4 cm, and 7 cm. Is it possible to form a triangle with these measurements? Why?" This question should serve as a hook for introducing the concept of the condition for the existence of a triangle.

Development (20 - 25 minutes)

1. Theory explanation (10 - 12 minutes):

   1.1. Triangle Definition: The teacher should start by explaining that a triangle is a polygon with three sides, three internal angles, and three vertices. They can draw a triangle on the board and point to each of its elements.

   1.2. Theorem of the Sum of Internal Angles: Next, the teacher should review the Theorem of the Sum of Internal Angles, which states that the sum of the internal angles of a triangle is always equal to 180°. They can demonstrate this by dividing a triangle into two smaller triangles and showing that the sum of the internal angles of each one is equal to 180°.

   1.3. Condition for the Existence of a Triangle: The teacher should then present the condition for the existence of a triangle, which states that the sum of any two sides of a triangle must always be greater than the third side. They can illustrate this by drawing three segments on the board and showing that if the sum of two segments is less than or equal to the third, it is not possible to form a triangle.

2. Practical activities (10 - 12 minutes):

   2.1. Triangle Construction Activity: The teacher should divide the class into groups and distribute rulers and segments of different lengths to each group. Students should try to construct triangles with the provided segments, applying the condition for the existence of a triangle. The teacher should move around the room, helping the groups and clarifying doubts.

   2.2. Problem-Solving Activity: The teacher should propose some problem-solving problems involving the condition for the existence of a triangle. For example, they can ask: "Suppose you have three segments with the following measurements: 3 cm, 4 cm, and 7 cm. Is it possible to form a triangle with these measurements? Why?" Students should discuss in their groups and present their answers to the class.

   2.3. Discussion Activity: Finally, the teacher should promote a classroom discussion about the solutions to the problems. They should highlight the importance of the condition for the existence of a triangle in solving the problems and reinforce the idea that the condition is a fundamental rule in geometry.

3. Feedback and clarification of doubts (3 - 5 minutes):

   3.1. Feedback on activities: The teacher should ask students to share their experiences and learnings during the activities. They should praise the students' efforts and provide constructive feedback.

   3.2. Clarification of doubts: The teacher should open a space for students to ask questions and clarify their doubts. They should respond to students' questions clearly and concisely, ensuring that everyone has understood the concept of the condition for the existence of a triangle.

   3.3. Reinforcement of the subject's importance: The teacher should reinforce the importance of the subject, explaining that the condition for the existence of a triangle is a powerful tool that students can use to solve a variety of problems in geometry and other disciplines.

Return (8 - 10 minutes)

1. Connection with everyday life (3 - 4 minutes):

   1.1. Practical applications: The teacher should ask students to think about everyday situations where the condition for the existence of a triangle can be applied. For example, in the construction of a bridge or a building, it is necessary to verify if the steel segments that will be used to form the triangles of the structures satisfy the condition for the existence of a triangle. Another example would be in the decoration of a room, where checking the condition for the existence of a triangle can be useful to ensure that furniture and objects are well distributed.

   1.2. Relevance of mathematics in practice: The teacher should highlight how mathematics, specifically geometry and the condition for the existence of a triangle, is a practical science that can be applied in various everyday situations. This can help students understand the relevance of the subject and the motivation to learn.

2. Review and reflection (2 - 3 minutes):

   2.1. Reflective questions: The teacher should ask some questions for students to reflect on what they learned in the lesson. For example, they can ask: "What was the most important concept you learned today?" or "What questions have not been answered yet?".

   2.2. Classroom discussion: Students should be encouraged to share their answers with the class. This not only helps consolidate what was learned but also promotes active participation and the exchange of ideas.

3. Feedback on the lesson (2 - 3 minutes):

   3.1. Students' feedback: The teacher should ask students to provide feedback on the lesson. They can ask: "What did you like most about today's lesson?" or "What could be improved next time?".

   3.2. Explanation of the next steps: The teacher should briefly explain what will be covered in the next lesson and how the topic connects with what was learned today. This helps prepare students for the next topic and maintain interest and motivation for learning.

This Return is a crucial part of the lesson plan, as it allows the teacher to assess the effectiveness of teaching, provides students with an opportunity to consolidate what they have learned, and helps establish a bridge between theory and practice.

Conclusion (5 - 7 minutes)

1. Summary of key points (2 - 3 minutes):

   1.1. Triangle Definition: The teacher should remind students that a triangle is a polygon with three sides, three internal angles, and three vertices.

   1.2. Theorem of the Sum of Internal Angles: The teacher should reinforce that the sum of the internal angles of a triangle is always equal to 180°.

   1.3. Condition for the Existence of a Triangle: The teacher should reiterate that for a set of segments to form a triangle, it is necessary that the sum of any two sides is always greater than the third side.

2. Connection between theory and practice (1 - 2 minutes):

   2.1. Review of activities: The teacher should remind students of the practical activities carried out during the lesson, where they had the opportunity to apply the theory learned to build triangles and solve problems.

   2.2. Importance of the condition for the existence of a triangle: The teacher should emphasize that the condition for the existence of a triangle is not just an abstract concept, but a practical tool that can be used to solve everyday problems and in various areas, such as civil construction, decoration, and science and engineering.

3. Extra materials (1 minute):

   3.1. Recommendation of additional resources: The teacher should suggest some additional resources for students who wish to deepen their knowledge on the subject. This may include math books, educational websites, explanatory videos, and interactive geometry apps.

4. Application in everyday life (1 - 2 minutes):

   4.1. Relevance of the subject: The teacher should conclude the lesson by reinforcing the importance of the subject and reminding students that mathematics, specifically geometry and the condition for the existence of a triangle, can be applied in various everyday situations.

   4.2. Practical examples: The teacher can give concrete examples of how the condition for the existence of a triangle can be useful in everyday life, such as checking the safety of a bridge or organizing furniture in a room.

At the end of the lesson, students should have a solid understanding of the concept of the condition for the existence of a triangle and be able to apply it to practical problems. They should also have an appreciation of the importance of the subject and how it applies to the real world.