Objectives (5 - 10 minutes)

1. Understanding Thales' Theorem: The teacher must ensure that students fully understand Thales' theorem, which states that if three or more parallel lines are cut by two transversals, then these transversals divide the lines into proportional segments.

2. Practical Application: Students should be able to apply Thales' theorem to practical problems, such as determining unknown measures in similar figures.

3. Development of Reasoning Skills: The teacher should encourage students to develop logical reasoning skills when working with Thales' theorem, in order to strengthen their ability to solve mathematical problems.

   Secondary Objectives:

   • Improving Mathematical Communication: Students should be encouraged to communicate their answers and reasoning clearly and concisely, thus improving their mathematical communication skills.
   • Promoting Teamwork: Through group activities, students should be encouraged to work together to solve problems, promoting teamwork and collaboration.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing essential mathematical concepts that are the basis for the study of Thales' theorem. This may include defining parallel lines, transversals, and proportional segments. It may also briefly review the concept of similarity of figures. (3 - 5 minutes)

2. Problem-Solving Scenarios: Present students with two problem-solving scenarios involving the application of Thales' Theorem, but without immediately providing the solution. For example, "If we have two parallel lines and a transversal that cuts these lines, how can we determine the measures of the resulting segments?" and "If we have two similar figures, how can we use Thales' Theorem to find the measure of an unknown side?" These scenarios will pique students' curiosity and prepare them for the content to be studied. (2 - 4 minutes)

3. Contextualization: Explain the importance of Thales' Theorem in various areas of Mathematics and in practical applications, such as in geometry, physics, and engineering. For example, Thales' theorem is used to determine the distance between inaccessible objects, such as the height of a tree, using the shadow it casts on the ground. (2 - 3 minutes)

4. Topic Introduction: Present Thales' Theorem as a powerful tool for solving complex mathematical problems. The teacher can tell the story of Thales of Miletus, the Greek mathematician credited with the discovery of the theorem. Additionally, present a curiosity about the theorem, such as the fact that it is also known as the "theorem of the bundle of lines" due to the way parallel lines seem to group together when cut by a transversal. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity "Building Thales' Theorem" (10 - 15 minutes)

   • Description: In this activity, students will be divided into groups of 4 to 5 people and will receive materials (ruler, compass, pencil, paper) to draw their own geometric figures. They will be instructed to draw two parallel lines on their papers and then draw a third line (the transversal) that cuts the two parallel lines.

   • Objective: The objective of this activity is to allow students to visualize Thales' theorem by "building" it themselves. They should see that when the transversal cuts the parallel lines, it creates proportional segments.

   • Step by step:

     1. Students receive the necessary materials.
     2. They draw the two parallel lines on their papers.
     3. Then, they draw the third line (the transversal) that cuts the two parallel lines.
     4. They mark the segments created by the parallel lines.
     5. Finally, they measure the segments to confirm that they are proportional, thus validating Thales' theorem.

2. Activity "Applying Thales' Theorem" (10 - 15 minutes)

   • Description: In this activity, students, still in their groups, will receive a set of problems involving the application of Thales' theorem. The problems may include determining unknown measures in similar figures, verifying the proportion of segments in different configurations of parallel lines, among others.

   • Objective: The objective of this activity is to allow students to apply Thales' theorem in real situations. They should work together to solve the problems, discussing their strategies and reasoning.

   • Step by step:

     1. Students receive the problems and discuss in their groups how they can be solved using Thales' theorem.
     2. They start solving the problems, noting their steps and reasoning.
     3. If they encounter difficulties, they can ask the teacher or other groups for help.
     4. Finally, they present their solutions to the class, explaining how they used Thales' theorem to reach them.

3. Group Discussion (5 - 10 minutes)

   • Description: After the conclusion of the activities, the teacher should promote a group discussion so that students can share their solutions and discoveries. This will allow students to learn from each other and reinforce their understanding of Thales' theorem.

   • Objective: The objective of this discussion is to allow students to consolidate their learning, clarify any doubts, and see different approaches to problem-solving.

   • Step by step:

     1. The teacher asks a representative from each group to share the solution to one of the problems they solved.
     2. Students have the opportunity to ask questions and make comments about the solutions presented.
     3. The teacher facilitates the discussion, clarifying any doubts and highlighting the key points of Thales' theorem.
     4. Finally, the teacher reinforces the key concepts of the lesson and prepares the students for the Conclusion stage.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes)

   • Description: In this stage, the teacher should open a discussion for each group to share their conclusions, solutions, and discoveries from the activities carried out. The teacher should ensure that all groups have the opportunity to speak and encourage the participation of all students.

   • Objective: The objective of this discussion is to allow students to express their opinions, reflect on what they have learned, and hear their peers' perspectives. This helps to consolidate learning and promote mathematical communication.

   • Step by step:

     1. The teacher invites each group to briefly share what they discussed and how they solved the problems.
     2. Other students are encouraged to ask questions and give feedback.
     3. The teacher facilitates the discussion, highlighting key points and correcting any misconceptions.
     4. The teacher reinforces the importance of Thales' theorem and how it can be applied in different situations.

2. Learning Verification (3 - 5 minutes)

   • Description: The teacher should ask questions to verify students' understanding of Thales' theorem and its application. The questions should be open-ended to allow students to express their ideas and use their own words.

   • Objective: The objective of this stage is to allow the teacher to assess the level of understanding of the students and identify any knowledge gaps that may need reinforcement in future classes.

   • Step by step:

     1. The teacher asks questions like "How did you use Thales' theorem to solve problem X?", "How do you know that the segments you found are proportional?", and "Can you think of another way to solve problem Y using Thales' theorem?".
     2. Students respond to the questions, explaining their reasoning and justifying their answers.
     3. The teacher provides immediate feedback, praising correct answers and correcting any misconceptions.
     4. The teacher identifies any knowledge gaps and plans future activities to reinforce these concepts.

3. Individual Reflection (2 - 3 minutes)

   • Description: The teacher asks students to reflect individually on what they learned in the lesson. They should think about Thales' theorem, how it can be applied, and any questions they still have.

   • Objective: The objective of this reflection is to allow students to consolidate their learning, identify any areas of confusion, and prepare for the next lesson.

   • Step by step:

     1. The teacher asks students to close their eyes and reflect on the following questions:
        • "What was the most important concept you learned today?"
        • "What questions do you still have about Thales' theorem?"
     2. After a minute of reflection, the teacher asks students to share their answers.

4. Feedback and Conclusion (1 minute)

   • Description: The teacher asks students to provide feedback on the lesson, including what they liked, what they found challenging, and any suggestions they may have to improve future lessons.

   • Objective: The objective of this stage is to allow the teacher to assess the effectiveness of the lesson and make necessary adjustments for future classes. Additionally, it helps promote students' responsibility for their own learning.

   • Step by step:

     1. The teacher asks students to provide feedback on the lesson, answering questions like "What did you like most about today's lesson?" and "What do you think could be improved?".
     2. The teacher thanks the students for their feedback and concludes the lesson, reinforcing the key concepts of Thales' theorem and reminding students of the work for the next lesson.

Conclusion (5 - 10 minutes)

1. Content Summary (2 - 3 minutes)

   • Description: The teacher should recap the main points covered during the lesson, emphasizing the definition and importance of Thales' theorem, the characteristics of parallel lines and transversals, and how the theorem can be applied to solve practical problems.
   • Objective: The objective of this stage is to reinforce the key concepts of Thales' theorem and ensure that students have understood the content presented.

2. Connection to Practice (2 - 3 minutes)

   • Description: The teacher should explain how Thales' theorem applies in the real world, citing examples of its use in fields such as engineering, architecture, and physics. Additionally, the practical application of the theorem in the activity "Building Thales' Theorem" can be highlighted.
   • Objective: The objective of this stage is to show students the relevance of the learned content, encouraging them to seek connections between mathematics and the real world.

3. Extra Materials (1 - 2 minutes)

   • Description: The teacher should suggest additional study materials for students who wish to deepen their knowledge of Thales' theorem. These materials may include explanatory videos, interactive math websites, textbooks, among others.
   • Objective: The objective of this stage is to encourage students to study autonomously and seek different resources to consolidate their learning.

4. Subject Relevance (1 - 2 minutes)

   • Description: Finally, the teacher should emphasize the importance of Thales' theorem for mathematics and everyday life. For example, how the ability to recognize and apply proportional relationships is fundamental in various areas, from solving practical problems to understanding more advanced concepts in mathematics.
   • Objective: The objective of this stage is to motivate students to value the mathematical knowledge acquired, realizing its applicability and relevance beyond the school environment.