Objectives (5 - 10 minutes)

1. Conceptual Understanding: Ensure that students understand the concept of complex numbers and their representation in trigonometric form, emphasizing the importance and applicability of this topic in Mathematics.
2. Operational Skills: Develop students' ability to solve multiplications and divisions of complex numbers in trigonometric form, using practical examples like dividing cis 2π by cis π.
3. Critical Thinking Development: Promote logical reasoning and problem-solving skills among students when dealing with complex numbers, encouraging them to solve proposed problems and challenges.

Secondary Objectives:

• Encourage active participation of students in the class, promoting debate and exchange of ideas on the topic.
• Develop students' self-learning skills, encouraging them to seek more information about the subject beyond what will be presented in the classroom.
• Stimulate students' confidence in solving complex problems, reinforcing the importance of practice and continuous effort in learning Mathematics.

Introduction (15 - 20 minutes)

1. Review of Previous Content:

   • The teacher starts the class with a brief review of the concepts of complex numbers, highlighting the real and imaginary parts, and the graphical representation in the complex plane.
   • Next, the teacher reviews the trigonometric representation of complex numbers, the formula cis θ = cos θ + i sin θ, and how this formula is used to represent complex numbers.

2. Problem-Solving Scenarios:

   • The teacher presents two problem-solving scenarios to the students to draw attention to the need to understand the topic of the class and encourage critical thinking. Examples:
     1. "Suppose you have a complex number z = cis 2π and want to divide it by another complex number w = cis π. How would you do that?"
     2. "How can you use the trigonometric formula to simplify the product of two complex numbers?"

3. Contextualization:

   • The teacher contextualizes the importance of the topic, explaining that the study of complex numbers and their operations is fundamental in various areas of science and engineering. They are particularly useful in describing wave phenomena, such as light, sound, and water waves.

4. Engaging Students' Attention:

   • The teacher may share curiosities or stories to spark students' interest in the topic. For example:
     1. "Did you know that complex numbers were initially considered 'imaginary' and 'useless' by the mathematical community, but are now fundamental in many areas of science and technology?"
     2. "Euler's formula, e^ix = cos x + i sin x, is considered one of the most beautiful and profound in mathematics, connecting five of the most important numbers: 0, 1, e, i, and π."

5. Topic Introduction:

   • The teacher concludes the introduction by presenting the topic of the class: "Today, we will learn how to multiply and divide complex numbers in trigonometric form. This will allow us to solve problems like the ones we just discussed and many others. Shall we begin?"

Development (20 - 25 minutes)

1. Group Activity - 'Complexopoly':

   • Objective: The 'Complexopoly' activity, inspired by the famous Monopoly game, aims to teach students how to perform multiplication and division of complex numbers in a playful way.
   • Preparation: The teacher prepares a game board with several spaces, each representing an operation with complex numbers in trigonometric form. The operations include multiplication and division of complex numbers. Additionally, the teacher prepares cards with different complex numbers (in trigonometric form).
   • Instructions: Students are divided into groups of 4 or 5. Each group receives a set of cards with complex numbers. The groups roll a die and move their game piece on the board. When a group lands on a space, they must draw a number of cards equal to the number indicated on the space and perform the indicated operation (multiplication or division) using their complex numbers. If the group successfully performs the operation, they can advance a number of spaces equal to the result of the operation (considering only the real part). The goal is to reach the end of the board first.
   • Feedback: During the activity, the teacher circulates around the room, guiding the groups, clarifying doubts, and correcting errors. Additionally, the teacher encourages discussion among group members to find the best strategy to advance in the game.

2. Group Discussion - 'Exploring the Application of Complex Numbers':

   • Objective: This activity aims to deepen students' understanding of the application of complex numbers in real life.
   • Preparation: The teacher prepares some examples of real-world problems that can be solved with the help of complex numbers. Some examples may include the analysis of electrical circuits, the representation of 3D rotations, among others.
   • Instructions: Students are divided into groups, and each group receives a problem to discuss and present how complex numbers can be used to solve it. Students have some time to research (using textbooks, the internet, etc.) and prepare a presentation for the class.
   • Feedback: The teacher provides feedback after each presentation, highlighting the correct parts and suggesting improvements if necessary. Additionally, the teacher encourages other students to ask questions and make comments to promote discussion in the classroom.

3. Individual Activity - 'Complex Challenge':

   • Objective: This activity aims to reinforce students' individual learning about the multiplication and division of complex numbers.
   • Preparation: The teacher prepares a list of exercises on multiplication and division of complex numbers in trigonometric form.
   • Instructions: Each student receives a list of exercises to solve individually. Students must use the knowledge acquired in class and apply it to solve the problems.
   • Feedback: The teacher collects the exercise lists at the end of the class and corrects them. In the next class, the teacher provides individual feedback to each student, highlighting strengths and areas that need improvement.

Feedback (10 - 15 minutes)

1. Review of Learned Concepts:

   • The teacher asks students to share what they have learned during the class. This can be done through group discussions or individual presentations. Students should be encouraged to talk about the concept of complex numbers, the trigonometric formula, and how to solve multiplications and divisions of complex numbers in this form.
   • The teacher listens attentively to students' responses and makes relevant observations to reinforce learning or correct possible misunderstandings.

2. Connection with Theory:

   • After the discussions, the teacher reviews the theoretical concepts covered in the class, connecting them with the practical activities carried out. They emphasize the importance of understanding the theory to solve problems, and how practice helps consolidate theoretical learning.

3. Student Feedback:

   • The teacher requests feedback from students about the class. This can be done through a brief survey or open discussion. Student feedback is important for the teacher to assess the effectiveness of the lesson plan and make necessary adjustments for future classes.

4. Individual Reflection:

   • The teacher suggests that students reflect for a minute on some questions, such as:
     1. "What was the most important concept learned today?"
     2. "What questions have not been answered yet?"
     3. "How can you apply what you learned today in real-life situations?"
   • After reflection, the teacher may ask some students to share their answers with the class. This can help reinforce learning and promote self-assessment.

5. Lesson Closure:

   • Finally, the teacher concludes the class by summarizing the main points covered and highlighting the importance of the studied topic. They may also suggest some additional material for home study and introduce the topic of the next class.
   • The teacher thanks the students for their participation and effort, encouraging them to continue studying and striving to improve their mathematical skills.

Conclusion (5 - 10 minutes)

1. Summary of Key Concepts:

   • The teacher revisits the main concepts covered during the class, reinforcing students' understanding of the trigonometric representation of complex numbers and the operations of multiplication and division within this context.
   • They emphasize the importance of understanding the cis θ = cos θ + i sin θ formula and how it is used to perform such operations, highlighting practical examples worked on during the class.

2. Connection between Theory, Practice, and Applications:

   • The teacher recaps the teaching methods used, connecting the theory presented in the class introduction with the practical activities carried out by students.
   • They highlight how practice helps in consolidating theoretical knowledge and how the learned concepts can be applied in real-world situations.
   • They may also use this moment to discuss student feedback on the effectiveness of practical activities in improving the understanding of theoretical concepts.

3. Additional Material:

   • The teacher suggests additional study materials, such as textbooks, educational videos, websites, and math apps that address complex numbers and their operations.
   • They may also provide a list of additional exercises with solutions for students to practice at home.

4. Everyday Applications:

   • The teacher concludes by contextualizing the importance of complex numbers in everyday life. They may cite examples of applications in various areas such as electrical engineering, physics, signal processing, among others.
   • The idea is for students to understand that mathematics, although complex at times, is an extremely useful and applicable tool in various everyday situations.

5. Class Closure:

   • The teacher thanks the students for their participation and effort, encouraging them to continue studying and deepening their knowledge on the subject.
   • They also introduce the topic that will be covered in the next class, creating a sense of anticipation and motivating students to prepare for the next meeting.