Objectives (5 - 7 minutes)

1. Understand the concept of internal and external angles of a triangle, focusing on the sum of the internal and external angles of a triangle.

   • Students should be able to identify and distinguish between the internal and external angles of a triangle.
   • They should be able to use the property that the sum of the internal angles of a triangle is always 180 degrees to solve problems.

2. Develop skills to calculate the sum of the internal angles of a triangle.

   • Students should be able to apply the property of the sum of the internal angles to find the measure of an unknown angle in a triangle.

3. Apply the concept of the sum of the internal angles to solve problems and practical situations.

   • Students should be able to apply the concept of the sum of the internal angles to solve geometry problems involving triangles.
   • They should be encouraged to think critically and apply the property effectively to arrive at correct solutions.

Secondary Objectives:

• Stimulate logical thinking and problem-solving.
• Promote collaboration and group discussion to enhance understanding of the content.
• Develop communication and argumentation skills when explaining the problem-solving processes.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher should briefly review the concept of angles, the sum of the angles of a quadrilateral (360 degrees), and the classification of triangles according to sides and angles.
   • Quick questions can be asked to verify students' understanding of these concepts and clarify any remaining doubts.

2. Presentation of problem situations:

   • The teacher can present two problem situations involving the sum of the internal angles of a triangle. For example:
     • 'If the measure of two internal angles of a triangle is 40 degrees and 80 degrees, what is the measure of the third angle?'
     • 'If the sum of the internal angles of a triangle is 180 degrees, what are the possible measures for the three angles of a triangle?'
   • These problem situations should be presented in a way that instigates students' curiosity and motivates them to learn the concept.

3. Contextualization:

   • The teacher should explain the importance of the concept of the sum of the internal angles of a triangle, highlighting that this property is fundamental for solving various geometry problems.
   • Some practical applications of this concept can be mentioned, such as determining angles in constructions, cartography, and navigation.

4. Introduction of the topic:

   • To capture students' attention, the teacher can introduce the topic with interesting facts or applications. For example:
     • 'Did you know that the concept of the sum of the internal angles of a triangle is so fundamental in mathematics that it is used to prove other properties and theorems?'
     • 'Have you ever wondered why the sum of the internal angles of a triangle is always 180 degrees, no matter the size or shape of the triangle? Let's find out today!'
   • The teacher can also introduce the topic with a historical curiosity, such as the fact that the discovery of non-Euclidean geometry in the 19th century led to a reevaluation of the concept of the sum of the internal angles of a triangle.

Development (20 - 25 minutes)

1. Activity 'Walking with Triangles' (8 - 10 minutes):

   • Instructions: The teacher should divide the class into groups of three. Each group will receive three pieces of rope of different lengths. They must form triangles with the lengths of the ropes and then measure the internal angles with a protractor. Finally, they should sum the measures of the internal angles and record the results.
   • Objective: This activity aims to allow students to experience and visualize the sum of the internal angles of a triangle. By performing the activity, they should realize that, no matter the size or shape of the triangle, the sum of the internal angles is always 180 degrees.
   • Tips: The teacher should circulate around the room, guiding the groups as needed. It is important for students to use a protractor correctly and record the measures of the internal angles accurately.

2. Activity 'Triangle Treasure Hunt' (8 - 10 minutes):

   • Instructions: The teacher should prepare cards with triangles of different shapes and sizes in advance. Each card should have a problem involving the sum of the internal angles of the triangle written on it. The cards should be hidden around the classroom. The groups of students must search for the cards and solve the problems together.
   • Objective: This activity aims to promote problem-solving in a playful and engaging way. By solving the problems, students will have the opportunity to apply the concept of the sum of the internal angles of a triangle in a practical and contextualized manner.
   • Tips: The teacher should ensure that the problems on the cards are of appropriate difficulty and varied. Additionally, it is important for students to discuss their problem-solving strategies and explain their reasoning to the rest of the group.

3. Activity 'Drawing Triangles' (4 - 5 minutes):

   • Instructions: The teacher should provide each group of students with a sheet of paper and a compass. The students must use the compass to draw three circles of different sizes on the paper. Then, they should connect the intersection points of the circles to form three triangles. Finally, they should measure the internal angles of each triangle and verify if the sum of the internal angles is 180 degrees.
   • Objective: This activity aims to reinforce the idea that the sum of the internal angles of a triangle is always 180 degrees, regardless of the size or shape of the triangle.
   • Tips: The teacher should provide clear guidance on how to use the compass correctly. Additionally, it is important for students to understand that even if the triangles they draw are not perfect, the sum of the internal angles should still be 180 degrees.

4. Group Discussion (5 - 7 minutes):

   • After the conclusion of the activities, the teacher should facilitate a group discussion where each group shares their findings and reflections. This will allow students to consolidate their understanding of the concept of the sum of the internal angles of a triangle and see how it can be applied in different situations.
   • The teacher should act as a moderator, asking questions to stimulate discussion and clarify any misunderstandings. Additionally, they should reinforce the main ideas and concepts discussed during the activity.

Return (8 - 10 minutes)

1. Sharing Solutions (3 - 4 minutes):

   • The teacher should ask each group to share their solutions or conclusions from the activities. Each group will have up to 3 minutes to present. It is important for students to be encouraged to explain not only the answer but also the problem-solving process.
   • During the presentations, the teacher should encourage other students to ask questions and offer constructive comments. This will help promote interaction and discussion among students.

2. Connection to Theory (2 - 3 minutes):

   • After all presentations, the teacher should give a quick review of the theoretical concepts covered in the lesson, highlighting how they were applied in the activities. The teacher should emphasize the importance of the sum of the internal angles of a triangle and reinforce that this property is valid for all triangles, regardless of their size or shape.
   • It may be helpful to use the solutions or conclusions of the groups to illustrate these points. For example, if a group found the sum of the internal angles of a triangle that was not exactly 180 degrees, the teacher can use this as an opportunity to discuss common errors and reinforce the correct concept.

3. Individual Reflection (2 - 3 minutes):

   • The teacher should then propose that students reflect for a minute on the following questions:
     1. What was the most important concept learned today?
     2. What questions have not been answered yet?
   • After the minute of reflection, students should be invited to share their answers with the class. The teacher should listen carefully to students' responses and note any questions or concepts that seem to be causing difficulties.

4. Closure and Homework Assignments (1 minute):

   • To conclude the lesson, the teacher should summarize the main points discussed and reinforce the importance of the sum of the internal angles of a triangle.
   • The teacher should then assign homework that reinforces the learned concept. For example, students may be asked to solve a series of problems involving the sum of the internal angles of a triangle.
   • The teacher should remind students that if they have any doubts or difficulties while doing the homework, they should seek help in the next class.

Conclusion (5 - 7 minutes)

1. Recap of Content (2 - 3 minutes):

   • The teacher should start the Conclusion by recalling the main concepts presented during the lesson. This includes the definition of internal and external angles of a triangle, the property that the sum of the internal angles of a triangle is always 180 degrees, and the application of this concept to solve problems.
   • A brief review of the activities carried out and the solutions proposed by students can be helpful, reinforcing the correct application of the concept of the sum of the internal angles.
   • The teacher should ensure that all students understand these fundamental concepts before moving on.

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

   • The teacher should then explain how the lesson connected theory, practice, and applications.
   • It should be emphasized that the theoretical understanding of the concept of the sum of the internal angles was reinforced through practical activities, where students had the opportunity to experience and visualize the property in action.
   • Additionally, the teacher should highlight how the property of the sum of the internal angles is used in practice, for example, in solving geometry problems and in real-world applications such as construction and cartography.

3. Extra Materials (1 minute):

   • The teacher should suggest extra materials for students who wish to deepen their understanding of the topic. This may include math books, online learning websites, explanatory videos, and educational geometry games.
   • It is important that these extra materials are accessible and suitable for the students' level of understanding.

4. Importance of the Topic (1 - 2 minutes):

   • To conclude the lesson, the teacher should emphasize the importance of the topic covered. It should be explained that the property of the sum of the internal angles of a triangle is fundamental for understanding and solving many other concepts and problems in geometry.
   • The teacher may also briefly mention some practical applications of the topic, such as defining navigation routes, building construction, and solving design problems.
   • Finally, the teacher should encourage students to remain curious and continue exploring the wonderful world of mathematics.