Objectives (5 - 10 minutes)

1. Understand the Theorem of Thales: The teacher must ensure that the students have a complete understanding of what the Theorem of Thales is. This includes the definition of the theorem, the explanation of its applications, and the importance of its understanding for mathematics and other disciplines.

2. Apply the Theorem of Thales to practical problems: Students must be able to apply the theorem to real-world situations or complex mathematical problems. This requires a deep understanding of the theorem and the ability to adapt it to different contexts.

3. Develop logical and analytical reasoning skills: The use of the Theorem of Thales requires students to develop logical and analytical reasoning skills. They must be able to identify the relevant information in a problem, apply the theorem correctly, and reach a logical solution.

Secondary Objectives:

• Stimulate students' active participation: The teacher must encourage students to actively participate in the class, asking questions, discussing ideas, and solving problems together.

• Promote critical thinking: The teacher must encourage students to think critically about the Theorem of Thales and how it applies to different situations. This may include discussing real-world examples and applying the theorem to complex problems.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher begins the class by recalling the concepts of reason and proportionality, which are fundamental to understanding and applying the Theorem of Thales. The teacher can do this through practical examples or quick questions to assess students' prior knowledge.

2. Initial problem situations: The teacher proposes two problem situations that naturally lead to the need for a theorem like that of Thales. The first could involve determining an inaccessible distance, such as the height of a tree, using the shadow of the tree and the shadow of an object of known height. The second could involve determining the size of an inaccessible object in an image, using the ratio between the actual sizes of the objects and their images.

3. Contextualization of the importance of the theorem: The teacher then contextualizes the importance of the Theorem of Thales, highlighting its applications in various areas of science and engineering. For example, the teacher could mention how the theorem is used to calculate inaccessible distances, determine the height of mountains, design structures, create computer animations, among other applications.

4. Topic introduction: The teacher then introduces the topic of the Theorem of Thales in a way that will arouse students' interest. For example, the teacher could tell the story of Thales of Miletus, the 6th century BC Greek mathematician who is credited with discovering the theorem. Another option is to present a curiosity, such as the fact that the Theorem of Thales is one of the first known examples of a similarity theorem of triangles, a fundamental concept in geometry.

5. Capture students' attention: To capture students' attention, the teacher can present a short video or an animation that clearly demonstrates the application of the Theorem of Thales in a real problem. The teacher can also propose a mathematical challenge involving the theorem, promising a small prize for the first student to solve it.

Development (20 - 25 minutes)

1. Presentation of the Theory (10 - 15 minutes):

   • Definition of the Theorem of Thales: The teacher should begin by explaining that the Theorem of Thales is a fundamental geometric principle that describes a property of line segments that are cut by parallel lines. The teacher can give the formal definition of the theorem and explain that it is often used to solve problems of similarity of triangles.

   • Application of the Theorem of Thales: The teacher should then explain how the theorem is applied in practice. The teacher can give examples of common problems that can be solved with the theorem, such as determining inaccessible distances, solving problems of proportionality and determining the sizes of objects in images.

   • Demonstration of the Theorem: Next, the teacher should demonstrate the proof of the theorem, explaining each step and ensuring that the students fully understand the demonstration. The teacher can do this using a whiteboard, a projector or an interactive math software.

   • Practical Examples: The teacher should then present practical examples of how to apply the theorem. For example, the teacher can show how to determine the height of a tree using the shadow of the tree and the shadow of an object of known height. The teacher can also show how to determine the size of an object in an image using the ratio between the actual sizes of the objects and their images.

2. Guided Practice (10 - 15 minutes):

   • Problem Solving Together: The teacher should then guide the students in solving problems that require the use of the Theorem of Thales. The teacher should start with simple problems and gradually increase the complexity of the problems as students gain more confidence and skill.

   • Feedback and Error Correction: During problem solving, the teacher should provide continuous feedback to the students, correcting errors and reinforcing important concepts. The teacher should encourage students to discuss their problem-solving strategies and to justify their answers.

   • Reinforcement of Learning: The teacher should finish the guided practice session by reinforcing the main learning points and clarifying any remaining doubts.

3. Practical Activities (5 - 10 minutes):

   • Individual Problem Solving: To consolidate learning, the teacher should propose that the students solve problems on their own. The teacher should provide a list of problems that vary in difficulty and encourage students to choose problems that match their skill level.

   • Discussion and Presentation of Solutions: After an allotted time to solve the problems, the teacher should promote a classroom discussion about the solutions. Students should be encouraged to explain their problem-solving strategies and to justify their answers. The teacher should provide feedback and correct errors as necessary.

   • Post-Activity Feedback: Finally, the teacher should provide feedback on the students' performance in the activity and clarify any remaining doubts.

Feedback (10 - 15 minutes)

1. Review of Key Concepts (5 - 7 minutes):

   • The teacher should begin the Feedback stage by reviewing the key concepts that were covered during the class. The teacher should recap the definition of the Theorem of Thales, the importance of proportion and similarity of triangles, and how the theorem can be applied to solve practical problems.

   • During the review, the teacher should emphasize the most important points and clarify any misunderstandings that may have arisen. The teacher can use different teaching methods to reinforce the concepts, such as drawings on the board, additional examples, or interactive demonstrations.

2. Connection between Theory, Practice and Applications (3 - 5 minutes):

   • The teacher should then help the students make the connection between the theory that was taught, the practice that was carried out, and the applications of the Theorem of Thales. The teacher can do this through a classroom discussion, asking the students how they would apply the theorem to real problems or to everyday situations.

   • For example, the teacher could ask the students how they could use the theorem to determine the height of a building, calculate the distance between two inaccessible points, or solve a problem of proportionality in their daily lives.

3. Reflection on Learning (2 - 3 minutes):

   • The teacher should then ask the students to reflect on what they learned during the class. The teacher can do this through reflective questions, such as: "What was the most important concept that you learned today?" and "What questions are still unanswered?"

   • The teacher should give the students enough time to reflect on the questions and then ask some volunteers to share their answers with the class. The teacher should praise the students' answers and encourage them to continue reflecting on what they have learned.

4. Feedback to the Teacher (1 - 2 minutes):

   • Finally, the teacher should ask the students to provide feedback on the class. The teacher can do this through a quick survey, asking the students what they liked about the class, what they found difficult, and what they would like to learn more about.

   • The teacher should take the feedback from the students into account to improve their future classes and to adjust their teaching plan if necessary.

Conclusion (5 - 10 minutes)

1. Summary of the Contents (2 - 3 minutes):

   • The teacher should begin the Conclusion by summarizing the main points covered during the class, recapping the definition of the Theorem of Thales, the importance of proportion and similarity of triangles, and how the theorem can be applied to solve practical problems.
   • The teacher can do this in an interactive way, by asking the students to share what they consider to be the most important points that they learned.

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

   • The teacher should then reinforce the connection between the theory, the practice, and the applications of the Theorem of Thales. For example, the teacher could highlight how solving practical problems during the class helped to illustrate the application of the theorem in everyday situations.
   • Additionally, the teacher could once again mention the theorem's real-world applications, such as determining inaccessible distances and solving problems of proportionality, to emphasize the relevance of what was learned.

3. Complementary Materials (1 - 2 minutes):

   • The teacher should suggest additional study materials for students who would like to further their understanding of the Theorem of Thales. This could include mathematics textbooks, educational websites, explanatory videos, and practice exercises.
   • For example, the teacher could recommend a YouTube video that explains the theorem in a different way, or a math website that offers additional practice problems with step-by-step solutions.

4. Relevance of the Subject (1 - 2 minutes):

   • Finally, the teacher should emphasize the importance of the Theorem of Thales to everyday life and to other disciplines. For example, the teacher could mention how the theorem is used in architecture and engineering to design and build structures, and in sciences such as physics and astronomy to measure inaccessible distances.
   • The teacher could also emphasize that the ability to apply the theorem to real-world problems is a valuable skill that can be used in many life situations, not just in mathematics. For example, the ability to solve problems of proportionality can be useful when shopping, cooking, or planning a trip.