Objectives (5 - 7 minutes)

1. Understand the nature of complex numbers: Students should be able to identify complex numbers and understand their nature, including their representation in trigonometric form. They should understand that a complex number is composed of a real part and an imaginary part.

2. Solve multiplications of complex numbers: The main objective of the lesson is for students to be able to solve multiplications of complex numbers in trigonometric form. They should be able to correctly apply the multiplication rules, including the multiplication of complex numbers, such as multiplying cis 2π by cis π.

3. Solve divisions of complex numbers: Another main objective is for students to be able to solve divisions of complex numbers in trigonometric form. They should be able to correctly apply the division rules, such as dividing cis 2π by cis π.

Secondary Objectives:

• Develop problem-solving skills: In addition to learning how to solve multiplications and divisions of complex numbers, students should also develop their problem-solving skills. They should be able to apply what they have learned to solve complex problems in the real world.
• Promote collaboration among students: Through the use of the flipped classroom methodology, students should be encouraged to collaborate with each other, discussing problems and working together to find solutions.

Introduction (7 - 10 minutes)

1. Review of Necessary Content: The teacher starts the lesson by reviewing the basic concepts of complex numbers, including the definition, representation in rectangular and polar form, and Euler's formula. The teacher can do this through targeted questions to the students or through a brief summary on the blackboard. It is important to ensure that all students understand these concepts, as they are essential for understanding the lesson topic.

2. Problem-Solving Situations: Next, the teacher presents two problem-solving situations involving complex numbers. For example, "How can we multiply two complex numbers that are in trigonometric form?" and "How can we divide two complex numbers that are in trigonometric form?" These problem-solving situations serve as initiators for the theory development.

3. Contextualization: The teacher contextualizes the importance of complex numbers, explaining how they are used in various areas of science and engineering. For example, they are used in quantum physics, signal processing, electrical engineering, among others. The teacher can also show practical examples of how the multiplication and division of complex numbers are applied in these areas.

4. Introduction to the Topic: Finally, the teacher introduces the lesson topic in an engaging way. For example, they can tell the story of how complex numbers were discovered, or share a curiosity about them. The teacher can also show a video or animation that visually and intuitively explains what complex numbers are and how they are multiplied and divided.

This introduction should serve to capture the students' attention, arouse their curiosity, and motivate them to learn more about the lesson topic.

Development (20 - 25 minutes)

1. Theory: Multiplication of complex numbers in trigonometric form (8 - 10 minutes): The teacher should discuss the formula for the multiplication of complex numbers. The theory should cover:

   • The basic rule that to multiply two complex numbers in trigonometric form, you multiply the magnitudes and add the arguments.
   • The demonstration of this rule using Euler's formula.
   • Practical examples of how to apply the rule. For example, multiplying cis 2π by cis π.

2. Practice: Multiplication of complex numbers in trigonometric form (5 - 7 minutes): After the theoretical explanation, students should have the opportunity to practice the multiplication of complex numbers. The teacher should prepare a series of problems for the students to solve, preferably in pairs or small groups, to promote collaboration. The teacher should move around the classroom, assisting students who are having difficulties and answering questions.

3. Theory: Division of complex numbers in trigonometric form (8 - 10 minutes): Next, the teacher should present the theory of division of complex numbers. The explanation should include:

   • The basic rule that to divide two complex numbers in trigonometric form, you divide the magnitudes and subtract the arguments.
   • The demonstration of this rule using Euler's formula.
   • Practical examples of how to apply the rule. For example, dividing cis 2π by cis π.

4. Practice: Division of complex numbers in trigonometric form (5 - 7 minutes): Students should then have the opportunity to practice the division of complex numbers. Again, the teacher should prepare a series of problems for the students to solve, preferably in pairs or small groups. The teacher should move around the classroom, assisting students who are having difficulties and answering questions.

Throughout the process, it should be emphasized that understanding the rules of multiplication and division of complex numbers is essential for solving more complex problems in science and engineering. In addition, students should be encouraged to discuss among themselves and ask questions, thus promoting a collaborative learning environment.

Return (8 - 10 minutes)

1. Reflection on Learning (3 - 4 minutes): The teacher should ask students to reflect on what they have learned during the lesson. To facilitate this reflection, the teacher can ask questions such as: "What was the most important concept you learned today?" and "What questions have not been answered yet?" The teacher should give students enough time to think about their answers and then allow some of them to share their reflections with the class.

2. Connection to the Real World (3 - 4 minutes): Next, the teacher should make the connection between what was learned in the lesson and the real world. For this, the teacher can:

   • Present real examples of how the multiplication and division of complex numbers are used in science and engineering. For example, in the analysis of electrical circuits, in solving differential equations, in image and sound compression, among others.
   • Discuss how the problem-solving skills developed in the lesson can be useful in other areas of life. For example, the ability to think logically and analytically, the ability to work in a team, the ability to persist in the face of a difficult problem, among others.
   • Ask students to think about other everyday situations where they could apply what they have learned about complex numbers. For example, in the analysis of complex data, in computer programming, in solving mathematical problems, among others.

3. Feedback for Students (2 minutes): Finally, the teacher should provide feedback to students on their performance during the lesson. The teacher should praise students for their efforts and progress, and offer suggestions on how they can further improve. Additionally, the teacher should encourage students to continue studying the topic at home, and to reach out if they have any questions or difficulties.

4. Planning for the Next Lesson (1 minute): The teacher should inform students about what will be covered in the next lesson and what the expectations are. If there is any reading material or preparatory activity, the teacher should inform the students and explain how it will help in the learning of the next lesson.

This return stage is crucial to ensure that students understand what was learned during the lesson, see the relevance of what they learned to their lives, and feel motivated to continue learning. Additionally, it allows the teacher to assess the effectiveness of their instruction and make adjustments, if necessary, for future lessons.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes): The teacher should summarize the main points discussed during the lesson, reinforcing students' understanding of the multiplication and division of complex numbers in trigonometric form. This is the moment to recall the importance of the multiplication and division rules, the application of Euler's formula, and the need for understanding complex numbers for solving problems in various areas of science and engineering.

2. Theory-Practice-Applications Connection (1 - 2 minutes): The teacher should explain how the lesson united theory, practice, and applications. The theoretical concepts were applied in solving practical problems, and the real-world applications of complex numbers were discussed to show the relevance of the topic. The teacher can reinforce the idea that understanding operations with complex numbers is essential for dealing with real problems in science and engineering.

3. Additional Materials (1 - 2 minutes): The teacher should suggest additional materials for students to deepen their knowledge of complex numbers. These materials may include books, online videos, math websites, among others. The teacher should encourage students to explore these resources and ask questions in the next lesson if they have any doubts.

4. Importance of the Topic (1 minute): Finally, the teacher should emphasize the importance of the lesson topic for students' everyday lives. They can discuss how understanding complex numbers and their manipulation can be useful in various situations, from data analysis in a science project to solving complex problems in their future careers.

This final stage of the lesson serves to consolidate students' learning, encourage them to continue studying the topic on their own, and show the relevance of complex numbers to their lives. By doing this, the teacher helps ensure that students not only understand the concepts presented in the lesson, but also recognize their value and application.