Objectives (5 - 7 minutes)

1. Understand and apply the concept of triangle similarity, using its properties to solve mathematical problems.
2. Develop skills to identify similar triangles in different geometric contexts, including everyday situations and practical applications.
3. Practice using proportions in problems involving triangle similarity, reinforcing the application of this important topic in Mathematics.

Secondary Objectives:

• Stimulate critical thinking and problem-solving through the application of triangle similarity.
• Foster the understanding that triangle similarity is a fundamental property of geometry and has applications in various fields of science and engineering.
• Encourage active student participation through practical and interactive activities.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the lesson by reviewing fundamental geometry concepts necessary for understanding the lesson topic. Concepts of angles, sides, and triangle properties will be reviewed. The teacher may also briefly review proportions, which will be used in solving problems involving triangle similarity.

2. Problem situation: The teacher proposes two situations involving triangle similarity but does not reveal the solution. The first situation could be: "If a 10-meter tall pole casts a 5-meter shadow on a day when the sun's elevation angle is 30 degrees, what would be the height of a building casting a 20-meter shadow on the same day and at the same time, considering the sun's elevation angle is the same?" The second situation could be: "If two triangles have proportional sides, are they similar? Why?"

3. Contextualization: The teacher explains that triangle similarity is a fundamental concept in geometry and has practical applications in various fields such as architecture, engineering, and physics. In architecture, for example, triangle similarity is used to design scale models of buildings. In engineering, it is used to calculate inaccessible heights, as in the example of the problem situation.

4. Introduction to the topic: To spark students' interest, the teacher may present some curiosities about triangle similarity. For example, they may mention that the Greek mathematician Thales of Miletus, who lived in the 6th century BC, was one of the first to study triangle similarity and use this concept to measure inaccessible heights, such as the height of the Egyptian pyramids. Another curiosity is that the concept of triangle similarity is used in various works of art, such as the famous painting "The Last Supper" by Leonardo da Vinci, which has a vanishing point that forms a triangle with the eyes of Jesus.

Development (20 - 25 minutes)

1. Theory of triangle similarity (10 - 12 minutes):

   • The teacher begins by explaining that triangle similarity is a property that states that if two triangles have all corresponding angles equal, then the corresponding sides are proportional and vice versa.
   • Next, the teacher explains the fundamental reason for triangle similarity, which is the ratio between the corresponding sides. For example, if the ratio between the corresponding sides of two triangles is 1:2, then the ratio between the areas of these triangles will be 1:4, and the ratio between the volumes will be 1:8.
   • The teacher should also emphasize that triangle similarity is a transitive property, meaning if triangle A is similar to triangle B and triangle B is similar to triangle C, then triangle A is similar to triangle C.

2. Methods to prove triangle similarity (5 - 7 minutes):

   • The teacher presents the three main methods to prove triangle similarity: AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side).
   • The teacher describes each of these methods and provides examples of how to use them to prove triangle similarity. For example, in the AA method, if two triangles have two corresponding angles equal, then they are similar. In the SAS method, if two triangles have a corresponding side proportional to a corresponding angle, then they are similar. In the SSS method, if the three corresponding sides of two triangles are proportional, then they are similar.

3. Problem solving involving triangle similarity (5 - 6 minutes):

   • The teacher presents some problems involving triangle similarity and guides students in solving these problems, step by step.
   • The teacher emphasizes the importance of identifying similar triangles and using proportions to solve the problems.
   • The teacher should also remind students to check if the answers make sense in the context of the problem. For example, if the problem involves the height of a building, the answer should be a positive and reasonable number, not a negative or very large number.

4. Practical activity (5 - 6 minutes):

   • The teacher proposes a practical activity in which students must identify similar triangles in various geometric figures.
   • Students should work in groups and use a ruler and compass to measure the sides and angles of the figures.
   • The teacher circulates around the room, assisting the groups and clarifying doubts.
   • At the end of the activity, each group presents their findings to the class, and the teacher provides feedback on the work done.

Return (8 - 10 minutes)

1. Group discussion (3 - 4 minutes):

   • The teacher promotes a group discussion on the solutions found by students for problems involving triangle similarity.
   • Each group has up to 2 minutes to share their resolution strategies and conclusions.
   • The teacher should encourage the participation of all students, asking questions and requesting additional explanations when necessary.
   • The goal of this discussion is to allow students to see different approaches to solving problems involving triangle similarity and to realize that there are several ways to arrive at the same answer.

2. Connection with theory (2 - 3 minutes):

   • After the discussion, the teacher revisits the theoretical concepts presented at the beginning of the lesson and shows how they apply to the discussed problems.
   • For example, the teacher may recall that the ratio between the corresponding sides of two similar triangles is the same for all sides and that the corresponding angles are equal.
   • The teacher may also highlight the importance of identifying similar triangles and using proportions to solve problems.
   • The goal of this stage is to help students understand the relevance and applicability of theoretical concepts in solving practical problems.

3. Individual reflection (1 - 2 minutes):

   • The teacher suggests that students reflect individually on what they learned in the lesson.
   • The teacher asks questions like: "What was the most important concept you learned today?" and "What questions have not been answered yet?".
   • Students have a minute to think about these questions.
   • The goal of this stage is for students to consolidate their learning and identify possible gaps in their understanding that can be addressed in future lessons.

4. Feedback and closure (1 minute):

   • At the end of the lesson, the teacher provides overall feedback on student participation and performance.
   • The teacher praises students' strengths and offers constructive suggestions for improvements.
   • The teacher also answers any final questions from students and concludes the lesson, reinforcing the key concepts and learning objectives.

Conclusion (5 - 7 minutes)

1. Summary of contents (1 - 2 minutes):

   • The teacher starts the Conclusion by summarizing the main points covered in the lesson. They reinforce the definition of triangle similarity, the importance of the ratio between corresponding sides, and the transitivity of similarity.
   • The teacher also recalls the three methods to prove triangle similarity: AA, SAS, and SSS.
   • Finally, they emphasize the practical application of these concepts in problem-solving and interpreting everyday situations.

2. Theory-practice connection (1 - 2 minutes):

   • The teacher highlights how the lesson connected theory, practice, and content applications. They recall the practical examples used, such as calculating inaccessible heights and identifying similar triangles in geometric figures.
   • The teacher emphasizes how understanding the theory allowed students to apply knowledge effectively in solving practical problems.

3. Additional materials (1 - 2 minutes):

   • The teacher suggests additional materials for students who wish to deepen their knowledge of triangle similarity. They may recommend geometry books, educational videos online, math exercise websites, and interactive learning apps.
   • The teacher may also suggest that students practice identifying similar triangles in their everyday environment by observing, for example, the shapes of buildings, trees, poles, etc.

4. Importance of the subject (1 minute):

   • Finally, the teacher emphasizes the importance of the subject presented for daily life and other disciplines. They reinforce that triangle similarity is a fundamental tool in various fields such as architecture, engineering, and physics.
   • The teacher may also mention that developing logical thinking and problem-solving skills, stimulated by the study of mathematics, are essential competencies for life and success in other areas of study and work.