Objectives (5 - 10 minutes)

1. Understanding the Theorem of Thales: Students should be able to understand and explain the Theorem of Thales, including the concept of proportion and how it is applied in the theorem.

2. Application of the Theorem of Thales: Students should be able to apply the Theorem of Thales to solve problems of proportionality in plane figures, such as triangle similarity.

3. Relationship between Theorem of Thales and Triangle Similarity: Students should be able to relate the Theorem of Thales to the concept of triangle similarity, understanding how one applies to the other and how both are used to solve problems.

Secondary Objectives:

• Development of Logical Thinking: Through problem-solving involving the Theorem of Thales, students should develop their logical thinking and mathematical reasoning skills.

• Encouraging Active Participation: The lesson plan includes group activities and discussions aimed at encouraging active student participation, promoting a collaborative learning environment.

• Strengthening Mathematical Communication Skills: Students should be encouraged to express their ideas, questions, and solutions clearly and coherently, thus strengthening their mathematical communication skills.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by briefly reviewing the concepts of triangle similarity and proportion, which are fundamental for understanding the Theorem of Thales. This can be done through a quick oral or written review, using simple examples to illustrate the concepts. (3 - 5 minutes)

2. Problem Situation 1: The teacher can propose the following situation: "Imagine you have two similar triangles, and one of them has a side that is twice the length of the corresponding side in the other triangle. How could you use this information to calculate the proportion of the other sides?" This situation will prepare students for the Introduction to the Theorem of Thales. (2 - 3 minutes)

3. Contextualization of the Subject's Importance: The teacher should emphasize the importance of the Theorem of Thales, explaining that it is widely used in various areas such as architecture, engineering, maps, and even in digital games to create the illusion of depth. Additionally, the teacher may mention that the theorem was discovered by the Greek mathematician Thales of Miletus, one of the seven sages of Ancient Greece, which may spark students' interest in the history of mathematics. (2 - 3 minutes)

4. Curiosity 1: The teacher can tell students that, according to legend, Thales of Miletus used the Theorem of Thales to measure the height of one of the pyramids in Egypt without having to climb it. He would have waited until his own shadow was the same height as the pyramid, and measured the length of the pyramid's shadow and his own shadow. Using the Theorem of Thales, he would have been able to calculate the height of the pyramid. This curiosity can help spark students' interest in the subject. (2 - 3 minutes)

5. Curiosity 2: The teacher can share with students that the Theorem of Thales applies not only to triangles but to any set of parallel segments intersected by two transversal lines. This more general application of the theorem can be illustrated with practical examples, such as the construction of perspectives in drawings and paintings. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity 1 - "The Mystery of the French Garden" (10 - 15 minutes)

   • Problem Situation: The teacher should present students with a drawing of a French garden, with its symmetrical and precise characteristics. Then, the teacher should propose the following problem situation: "Suppose you are a gardener and need to replant the rose bushes in this garden. However, you only have a piece of string and need to know the distance between the rose beds. How could you use the Theorem of Thales to solve this problem?"

   • Development: Students, organized in groups of 4 to 5 people, should discuss and propose a strategy to solve the problem. They should draw a diagram of the garden, identifying reference points and the lines that will be used. Then, they should apply the Theorem of Thales to calculate the distance between the rose beds. The teacher should circulate among the groups, guiding and clarifying doubts.

   • Presentation of Solutions: At the end of the activity, each group should present their solution to the class, explaining step by step how they arrived at the answer. The teacher should promote a classroom discussion, highlighting the main points and correcting possible errors.

2. Activity 2 - "Building a Proportional Map" (10 - 15 minutes)

   • Problem Situation: The teacher should challenge students to build a proportional map of a neighborhood in the city, starting from a non-proportional sketch. The teacher should provide students with a sketch of the map, with some known measurements and the indication that the map needs to be proportional. The question to be answered is: "How can you use the Theorem of Thales to build a proportional map of the neighborhood?"

   • Development: Again, students should be organized into groups and guided to discuss and propose a strategy to solve the problem. They should identify the parallel segments and transversal lines on the map, and apply the Theorem of Thales to calculate the unknown measurements. The teacher should circulate among the groups, guiding and clarifying doubts.

   • Presentation of Solutions: At the end of the activity, each group should present their proportional map to the class, explaining how they used the Theorem of Thales to reach the solution. The teacher should promote a discussion, highlighting the difficulties encountered and the strategies used to overcome them.

3. Final Discussion and Conclusions: (5 - 10 minutes)

   • Discussion: The teacher should lead a final discussion about the activities carried out, highlighting the main points and clarifying any doubts. Students should be encouraged to ask questions and express their opinions, thus promoting reflection and consolidation of learning.

   • Conclusions: The teacher should conclude the lesson by emphasizing the importance of the Theorem of Thales and proportionality in various contexts, from the construction of gardens and maps to problem-solving in architecture, engineering, and other areas of knowledge. The teacher should also reinforce the learning objectives of the lesson and assess if they were achieved.

Return (10 - 15 minutes)

1. Connecting Theory to Practice: The teacher should ask students to reflect on the activities carried out during the lesson and how they connect with the presented theory. He can ask questions like: "How was the Theorem of Thales applied to solve the problems of the French garden and the proportional map?" and "In what ways can the understanding of the Theorem of Thales and the ability to solve proportionality problems be useful in everyday situations?" Students should be encouraged to express their opinions and share their experiences, thus promoting a deeper reflection on the learned content. (3 - 5 minutes)

2. Review of Key Concepts: The teacher should review the key concepts of the lesson, reinforcing the definition and use of the Theorem of Thales, and the relationship between the theorem and triangle similarity. He can do this interactively, asking students to complete sentences or answer questions about the concepts. For example, the teacher can say: "The Theorem of Thales states that in a set of parallel segments intersected by two transversal lines, the transversal lines divide the segments in a...?" and wait for students to complete the sentence with "proportional". (2 - 3 minutes)

3. Individual Reflection: The teacher should propose that students make an individual reflection on what they learned in the lesson. He can ask students to write on a piece of paper their answers to questions like: "What was the most important concept you learned today?" and "What questions have not been answered yet?" This individual reflection activity allows students to consolidate what they have learned and identify areas where they still have doubts. (3 - 5 minutes)

4. Sharing Reflections: The teacher should give the opportunity for some students to share their reflections with the class. This can be done voluntarily, and students should be encouraged to express their ideas clearly and coherently. The teacher should value all contributions, promoting an environment of respect and appreciation of different perspectives. (2 - 3 minutes)

5. Feedback and Closure: Finally, the teacher should thank the students for their participation, provide overall feedback on the lesson, and announce the topic of the next lesson. He can also suggest additional study materials, such as videos, online games, or extra exercises, so that students can deepen their knowledge of the Theorem of Thales. (1 - 2 minutes)

Conclusion (5 - 10 minutes)

1. Summary of Contents: The teacher should start the Conclusion by summarizing the main points covered in the lesson. This includes a recapitulation of the definition of the Theorem of Thales, the relationship between the theorem and triangle similarity, and the application of the theorem to solve problems of proportionality in plane figures. The teacher should ensure that students understand these fundamental concepts before moving on. (2 - 3 minutes)

2. Connection between Theory, Practice, and Applications: The teacher should then highlight how the lesson connected theory, practice, and applications. He can recall the practical activities carried out, such as the "Mystery of the French Garden" and the construction of the "Proportional Map", and explain how they illustrate the application of the Theorem of Thales in solving real problems. The teacher should emphasize that mathematics is not just a set of abstract rules, but a powerful tool for understanding and solving real-world problems. (1 - 2 minutes)

3. Additional Materials: The teacher should suggest additional study materials for students. This may include textbooks, educational websites, explanatory videos, online math games, and extra exercises. The teacher should encourage students to explore these materials on their own to deepen their understanding of the Theorem of Thales and practice applying the theorem in different contexts. (1 - 2 minutes)

4. Importance of the Theorem of Thales: The teacher should conclude the Conclusion by reinforcing the importance of the Theorem of Thales. He can explain that the theorem is a fundamental tool in many disciplines and professions, including architecture, engineering, cartography, graphic design, and physics. Additionally, the teacher can highlight that the Theorem of Thales is an example of how mathematics can be used to solve problems in a general and systematic way, thus developing students' logical thinking and problem-solving skills. (1 - 2 minutes)