Objectives (5 - 7 minutes)

1. Understanding the Concept of Complex Numbers: The teacher must ensure that students understand what complex numbers are, their characteristics, and how they differ from real numbers. This includes the definition of complex numbers, representation in algebraic and geometric form, and the idea of the real and imaginary part.

2. Understanding the Geometric Representation of Complex Numbers: Students should be able to represent a complex number on an Argand-Gauss plane and identify its real and imaginary part. The teacher should emphasize the importance of this representation and how it helps visualize operations with complex numbers.

3. Introduction to the Module and Argument of a Complex Number: Students should begin to understand the concept of module and argument of a complex number. The teacher should explain what these concepts are and how they are related to geometric representation.

Secondary Objectives

1. Encouragement of Critical Thinking: The teacher should encourage students to question and reflect on what they are learning, promoting discussion in the classroom.

2. Development of Problem-Solving Skills: The teacher should include examples and practical exercises to help students apply what they are learning to solve problems.

Introduction (10 - 12 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing the concepts of real, imaginary, and complex numbers. It is important for students to understand the difference between these three types of numbers before moving on to the Introduction of complex numbers. It may be helpful to briefly review imaginary numbers, focusing on the idea that they are multiplied by 'i' and that 'i' squared is equal to -1.

2. Presentation of Problem Situations: To spark students' interest, the teacher can present two situations involving complex numbers. For example, the first situation could be solving an equation of the type x² + 1 = 0, which has no real solution but has complex solutions. The second situation could be the representation of a complex number on an Argand-Gauss plane and the identification of its real and imaginary part.

3. Contextualization of the Subject's Importance: The teacher can then explain the importance of complex numbers, mentioning that they are widely used in various areas of science and engineering, particularly in physics and electronics. For example, the theory of electromagnetic waves is based on complex numbers, and electrical engineering uses complex numbers to model circuits.

4. Introduction of the Topic in an Engaging Way: To capture students' attention, the teacher can introduce the topic of complex numbers in an intriguing way. For example, he can mention that the discovery of complex numbers was initially considered useless and even impossible, but it turned out to be one of the most important discoveries in the history of mathematics. Another curiosity is that the German mathematician Carl Friedrich Gauss, whose name is associated with the Argand-Gauss plane, a fundamental tool for the study of complex numbers, was a strong advocate of complex numbers and called them 'imaginary numbers' to emphasize their importance.

Development (20 - 25 minutes)

1. Theory of Complex Numbers (7 - 10 minutes): The teacher should start the theoretical part by presenting the definition of complex numbers, their notation (a + bi), where a is the real part and bi is the imaginary part. It is important to explain that 'i' is the imaginary unit, where i² = -1. The teacher should emphasize that complex numbers are an extension of real numbers and that all real numbers can be considered as complex numbers with the imaginary part equal to zero.

   • At this point, the teacher can use the blackboard or a math software to illustrate the notation of complex numbers and provide practical examples, such as the representation of the numbers 3 + 2i and -1 - 4i on the Argand-Gauss plane.

2. Argand-Gauss Plane and Geometric Representation (5 - 7 minutes): The teacher should introduce the Argand-Gauss plane, which is a geometric representation of complex numbers. The teacher should explain that the real part of a complex number is represented on the x-axis and the imaginary part on the y-axis.

   • The teacher can use the blackboard or a math software to draw the Argand-Gauss plane and show the representation of the complex numbers 3 + 2i and -1 - 4i.

3. Module and Argument (5 - 7 minutes): The teacher should then introduce the concept of module and argument of a complex number. The module of a complex number is its distance from the origin on the Argand-Gauss plane, and the argument is the angle that the line connecting the complex number to the origin forms with the x-axis counterclockwise.

   • The teacher can use the blackboard or a math software to show the geometric representation of the module and argument and provide examples with the complex numbers 3 + 2i and -1 - 4i.

4. Operations with Complex Numbers (3 - 5 minutes): Finally, the teacher should explain the basic operations with complex numbers: addition, subtraction, multiplication, and division. It is important to emphasize that addition and subtraction are performed component by component (real part with real part and imaginary part with imaginary part) and that multiplication and division are performed according to the rules of multiplication by 'i' (i² = -1).

   • The teacher can use the blackboard or a math software to demonstrate how to perform these operations and provide practical examples.

The teacher must ensure that students understand each of these concepts before moving on to the next stage. This can be done through questions to the class, group discussions, and problem-solving. The teacher should also encourage students to ask questions and clarify any doubts they may have.

Return (8 - 10 minutes)

1. Connection to the Real World (3 - 4 minutes): The teacher should guide students to reflect on how complex numbers are applied in the real world. Some examples may include:

   • The theory of electromagnetic waves, which is based on complex numbers.
   • Electrical engineering, which uses complex numbers to model circuits.
   • Quantum physics, which uses complex numbers to describe the behavior of subatomic particles.

2. Content Review (3 - 4 minutes): The teacher should briefly review the main points covered in the lesson, recalling the definition of complex numbers, the representation on the Argand-Gauss plane, the concept of module and argument, and the basic operations. The teacher can do this through a summary on the blackboard or in presentation slides.

3. Reflection on Learning (2 - 3 minutes): The teacher should ask students to reflect on what they learned in the lesson. Some questions that can be asked include:

   • What was the most important concept you learned today?
   • What questions have not been answered yet?

   The teacher should encourage students to share their reflections and ask questions, if any. This can be done through a brief classroom discussion or in writing, if students prefer.

4. Teacher's Feedback (Optional): If time allows, the teacher can provide feedback on students' participation, highlighting positive points and areas that may need more attention. The teacher can also take the opportunity to clarify any remaining doubts.

This Return moment is essential to consolidate students' learning and for them to realize the relevance of the content studied. In addition, reflection on learning and teacher feedback can help identify areas that may need more practice or review in future classes.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should summarize the main points covered during the lesson. This includes the definition of complex numbers, their representation on the Argand-Gauss plane, the concept of module and argument, and the basic operations. The teacher can use the blackboard or presentation slides to highlight these points and reinforce students' understanding.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should highlight how the lesson connected the theory of complex numbers with practice, through examples and exercises, and with applications in the real world. The teacher can recall the examples of applications of complex numbers that were discussed during the lesson, such as the theory of electromagnetic waves and electrical engineering.

3. Additional Materials (1 - 2 minutes): The teacher should suggest some additional study materials for students who wish to deepen their knowledge of complex numbers. This may include math books, educational websites, online videos, and math apps. Some examples of materials may be:

   • 'Complex Numbers from A to... Z' by Titu Andreescu and Dorin Andrica.
   • 'Complex Numbers and Geometry' by Liang-shin Hahn.
   • 'Complex Numbers: Introduction and Basic Operations' from the Khan Academy math channel on YouTube.

4. Importance of the Subject (1 minute): Finally, the teacher should emphasize the importance of complex numbers, highlighting that they are fundamental to various areas of science and engineering, and that understanding them can open doors to many study and career opportunities. The teacher can encourage students to continue exploring this subject and to look for ways to apply what they have learned in other contexts.