Lesson Plan | Technical Methodology | Complex Numbers: Basic Operations

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to clearly present the main and secondary objectives of the lesson, highlighting the importance of developing practical skills through the manipulation of complex numbers. This approach not only facilitates theoretical understanding but also demonstrates the relevance of this knowledge in the job market, preparing students for practical situations and challenges they may encounter in their future careers.

Main Objectives

1. Understand and perform operations of addition, subtraction, multiplication, division, and exponentiation of complex numbers in algebraic form.

2. Develop practical skills through maker activities involving complex numbers.

3. Establish the relationship between mathematical concepts and their practical applications in the job market.

Side Objectives

1. Encourage critical thinking and problem-solving through mini practical challenges.
2. Promote collaboration and teamwork during maker activities.

Introduction

Duration: 15 - 20 minutes

The purpose of this stage is to capture students' attention and establish the relevance of complex numbers, connecting them to practical situations and the job market. This prepares students for active and engaged learning throughout the lesson.

Contextualization

Complex numbers are fundamental in various areas of mathematics and engineering. They are used to solve problems involving the roots of negative numbers, something that real numbers cannot do. For example, in electronics, complex numbers are essential for analyzing alternating current circuits, allowing for the calculation of impedances and the understanding of phenomena like resonance.

Curiosities and Market Connection

A curiosity about complex numbers is that they were initially rejected for being 'non-real', but today they are indispensable in various technological and scientific fields. In the job market, electrical and telecommunications engineers routinely use complex numbers to design and optimize systems. Moreover, in computer graphics, algorithms that render 3D images often employ complex numbers to perform transformations and rotations of objects.

Initial Activity

Initial Activity: Provocative Question: Write the following question on the board: "How would you solve the equation x² + 1 = 0 using only real numbers?" Brief Discussion: Allow students to discuss the question and try to find a solution. Short Video: Show a 3 to 5-minute video introducing complex numbers and demonstrating their application in a real-world context, such as in electrical engineering.

Development

Duration: 45 - 55 minutes

The purpose of this stage is to deepen students' understanding of basic operations with complex numbers, providing a practical experience that simulates real-world situations in the job market. This facilitates the application of theoretical concepts to real-world problems, promoting problem-solving skills and collaborative work.

Covered Topics

1. Addition of complex numbers
2. Subtraction of complex numbers
3. Multiplication of complex numbers
4. Division of complex numbers
5. Exponentiation of complex numbers

Reflections on the Theme

Guide students to reflect on how operations with complex numbers can be applied in real situations, such as in the development of communication technologies and electrical engineering. Encourage them to think about the importance of mastering these operations to solve complex problems and innovate in their future careers.

Mini Challenge

Mini Challenge: Building a Simulated Circuit

Students will be divided into groups and receive a kit of simulated electronic components (cards with impedance values represented by complex numbers). They will have to assemble a simple alternating current circuit and calculate the total impedance of the circuit using operations with complex numbers.

Instructions

1. Divide the class into groups of 4 to 5 students.
2. Distribute the simulated electronic component kits to each group.
3. Explain that each component has an impedance value represented by a complex number.
4. Ask them to assemble a simple circuit (series or parallel) with the provided components.
5. Guide the groups to calculate the total impedance of the circuit using the operations of addition, subtraction, multiplication, and division of complex numbers.
6. After assembly and calculations, ask each group to present their circuit and explain the process of calculating the total impedance.

Objective: Apply operations with complex numbers in a practical and relevant context, developing teamwork and problem-solving skills.

Duration: 35 - 40 minutes

Evaluation Exercises

1. Calculate (3 + 4i) + (2 - 3i).
2. Subtract (5 + 2i) from (7 - 4i).
3. Multiply (1 + i) by (2 - 3i).
4. Divide (6 + 2i) by (3 - i).
5. Find the square of (1 + 2i).

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to consolidate students' learning, ensuring they understand the connection between theory and practice and the applicability of complex numbers in real contexts. The final discussion and recap of content aim to reinforce understanding and the importance of the topic, preparing students for future applications in the job market.

Discussion

Promote an open discussion with students about the main operations with complex numbers that were covered in the lesson. Guide them to reflect on the practical mini challenges, such as building the simulated circuit, and how these activities helped solidify their understanding of the concepts. Encourage students to share their experiences and difficulties, highlighting the importance of teamwork and the practical application of acquired knowledge.

Summary

Summarize the main content presented, including the operations of addition, subtraction, multiplication, division, and exponentiation of complex numbers in algebraic form. Highlight the relevance of these concepts through practical activities and their application in the job market, such as in electrical engineering and computer graphics.

Closing

Explain how the lesson connected theory to practice through maker activities and the proposed challenges. Reinforce the importance of understanding and applying complex numbers in real situations, underscoring their numerous applications in the job market. Conclude the lesson by emphasizing how mastering these concepts can open doors to various technological and scientific careers.