Objectives (5 - 7 minutes)

1. Understanding the triangle existence condition: The teacher must ensure that students fully understand the triangle existence condition. This includes understanding that the sum of the measures of two sides of a triangle must always be greater than the measure of the third side. Additionally, students should be able to identify situations where this condition is not satisfied.

2. Problem-solving skills: Students should be able to apply the triangle existence condition to solve practical problems. This includes the ability to identify if a set of segment lengths can form a triangle and, if so, what type of triangle would be formed.

3. Development of Critical Thinking: Through the exploration of the concept of the triangle existence condition, students should be encouraged to develop their critical thinking skills. They should be able to justify their answers and explain the reasoning behind their problem-solving solutions.

Secondary Objectives:

• Active Engagement: The teacher must ensure that students are actively engaged in the learning process. This can be achieved through the use of practical activities, group discussions, and immediate feedback.

• Real-world Application: Students should be encouraged to make connections between the triangle existence condition and real-world situations. This can help increase the relevance of the topic and students' motivation to learn.

Introduction (10 - 15 minutes)

1. Review of related content: The teacher should start the lesson by briefly reviewing the concepts of line segments and triangles. This can be done through questions to the students, such as 'What is a triangle?' and 'How do you determine the measure of a line segment?'. This review will help prepare students for the new content that will be presented and ensure that everyone is on the same page.

2. Problematic situations: The teacher can then present two problematic situations to the students involving the triangle existence condition. One situation could be: 'If a triangle has sides of lengths 3 cm, 4 cm, and 8 cm, can it exist? Why?'. The second situation could be a real-world scenario, such as 'If a builder has three wooden boards with lengths of 5 meters, 7 meters, and 12 meters, can he build a triangle? Why or why not?'. These situations will help spark students' curiosity and demonstrate the relevance of the content to the real world.

3. Contextualization of the topic: Next, the teacher should contextualize the importance of the triangle existence condition. Examples of its application in various areas, such as architecture, engineering, and design, can be mentioned. For example, in architecture, the triangle existence condition is essential for the stability of triangular structures, such as those used in bridges. In engineering, the triangle existence condition is used to determine if a set of segments can be used to form a polygon, which is crucial in the design of many devices and structures.

4. Introduction to the topic: To capture students' attention, the teacher can share some curiosities about the triangle existence condition. For example, it can be mentioned that in a triangle, the sum of the measures of the internal angles is always equal to 180 degrees, regardless of the size of the sides. Or that in an equilateral triangle, all sides and angles have the same measure. This can help spark students' interest in the topic and prepare them for the detailed presentation of the content.

Development (20 - 25 minutes)

1. Activity 1 - Building Triangles (10 - 12 minutes): The teacher should divide the class into groups of three or four students. Each group will receive cardboard paper, a ruler, and a compass. The group's task will be to build triangles using the available materials, following the triangle existence condition. The teacher should give clear instructions on how to use the compass and ruler and how to apply the triangle existence condition. After constructing the triangles, the groups should measure the sides and angles of the created triangles and record their observations.

   Activity step-by-step:

   • The teacher divides the class into groups and distributes the necessary materials.
   • The teacher explains again the triangle existence condition and how to use the compass and ruler.
   • The groups build the triangles and measure the sides and angles.
   • The groups record their observations and conclusions.
   • The teacher circulates around the room, providing help and feedback as needed.

2. Activity 2 - Real-world Application (5 - 7 minutes): After the construction activity, the teacher should ask the groups to apply the triangle existence condition in a real-world scenario. For example, they may be asked to determine if a set of segments of different lengths can form a triangle and, if so, what type of triangle would be formed. The groups should discuss the question together and present their conclusions to the class.

   Activity step-by-step:

   • The teacher presents a real-world scenario that requires the application of the triangle existence condition.
   • The groups discuss the question and present their conclusions to the class.
   • The teacher facilitates a class discussion about the different solutions proposed by the groups.

3. Activity 3 - Practice Problems (5 - 6 minutes): To consolidate learning, the teacher should provide the groups with a series of practice problems involving the triangle existence condition. The problems should vary in difficulty and complexity, allowing students to apply the concept in different ways. The groups should work together to solve the problems and present their solutions to the class.

   Activity step-by-step:

   • The teacher distributes the practice problems to the groups.
   • The groups work together to solve the problems.
   • Each group presents their solutions to the class, explaining their reasoning. The teacher provides feedback and clarifies any doubts that may arise.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should gather all students and conduct a group discussion. Each group will have up to 2 minutes to share the solutions or conclusions they reached during the activities. This will allow students to see different approaches to the same question and help reinforce the concept of the triangle existence condition. The teacher should encourage students to ask questions and give feedback to each other.

   Activity step-by-step:

   • The teacher gathers all students and each group shares their solutions or conclusions.
   • The teacher facilitates the discussion, asking questions to students and encouraging them to give feedback to each other.

2. Connection to Theory (2 - 3 minutes): After the group discussion, the teacher should connect the practical activities with the theory. The teacher should highlight how the triangle existence condition was applied in the activities and how the solutions to the practice problems relate to the theory. This will help students see the relevance of what they have learned and understand how they can apply these concepts in different situations.

   Activity step-by-step:

   • The teacher highlights the main applications of the theory in the practical activities.
   • The teacher explains how the solutions to the practice problems relate to the theory.
   • The teacher answers any questions students may have and clarifies any misunderstandings.

3. Individual Reflection (2 - 3 minutes): To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. The teacher can ask questions like 'What was the most important concept you learned today?' and 'What questions do you still have?'. Students should write their answers on a piece of paper and hand them to the teacher. This activity will allow students to consolidate their learning and help the teacher identify any areas that may need reinforcement in future lessons.

   Activity step-by-step:

   • The teacher proposes that students reflect individually on what they have learned.
   • Students write their answers on a piece of paper.
   • Students hand in their answers to the teacher.

4. Teacher's Feedback (1 minute): The teacher should provide overall feedback on the lesson, highlighting strengths and areas that need reinforcement. The teacher should also answer any questions that have not been addressed and clarify any misunderstandings.

   Activity step-by-step:

   • The teacher provides general feedback on the lesson.
   • The teacher answers any unanswered questions.
   • The teacher clarifies any misunderstandings.

Conclusion (5 - 7 minutes)

1. Recap of Key Contents (2 - 3 minutes): The teacher should start the Conclusion by recapping the main points of the lesson. This includes explaining the triangle existence condition (the sum of the measures of two sides of a triangle must always be greater than the measure of the third side) and the practical application of this concept in the activities carried out. The teacher should ensure that students fully understand these ideas before moving on.

   Activity step-by-step:

   • The teacher reviews the triangle existence condition and its practical application.
   • The teacher asks students questions to verify understanding.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should explain how the lesson connected the theory, practice, and applications of the triangle existence condition. The teacher should highlight how the practical activities and group discussions allowed students to apply the theory in a meaningful and relevant way. Additionally, the teacher should reiterate how the triangle existence condition is relevant to various areas, such as architecture, engineering, and design.

   Activity step-by-step:

   • The teacher explains the connection between theory, practice, and applications.
   • The teacher highlights again the relevance of the triangle existence condition in various areas.

3. Suggestion of Additional Materials (1 minute): The teacher should suggest additional reading or viewing materials for students who wish to deepen their understanding of the triangle existence condition concept. This may include math books, educational videos online, or math practice websites. The teacher should encourage students to explore these materials at their own pace and to bring any questions they may have to the next lesson.

   Activity step-by-step:

   • The teacher suggests additional reading or viewing materials.
   • The teacher encourages students to explore these materials and bring questions to the next lesson.

4. Relevance of the Topic to Everyday Life (1 minute): To conclude, the teacher should emphasize the importance of the triangle existence condition concept in everyday life. The teacher can mention examples of how understanding this condition can be useful in everyday situations, such as assembling furniture, measuring distances on a map, or designing a garden. This will help solidify the relevance of the topic in students' minds and motivate them to continue exploring the subject.

   Activity step-by-step:

   • The teacher emphasizes the importance of the triangle existence condition concept in everyday life.
   • The teacher gives examples of how understanding this condition can be useful in everyday situations.