Lesson Plan | Technical Methodology | Linear Systems: System Discussion

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear understanding of the classification of linear systems concerning the existence and number of solutions. It is crucial that students develop practical and analytical skills, which are highly valued in the job market, particularly in areas that require critical thinking and solving complex mathematical problems.

Main Objectives

1. Identify and classify linear systems regarding the existence and number of solutions.

2. Apply resolution methods to determine the nature of the solutions of linear systems.

3. Develop analytical skills to discuss the compatibility and indeterminacy of linear systems.

Side Objectives

1. Introduce basic concepts of linear algebra applicable in the job market.
2. Encourage logical reasoning and problem-solving in practical contexts.

Introduction

Duration: (15 - 20 minutes)

The purpose of this stage is to engage students, contextualize the theme of the class, and show the relevance of linear systems in the job market. This will help spark students' interest and motivation to learn and apply the concepts that will be developed throughout the lesson.

Contextualization

Linear systems are present in various situations of our daily lives, from a company's financial organization to the optimization of industrial processes. The ability to discuss and solve linear systems is essential for understanding and modeling complex problems, facilitating more assertive and efficient decision-making.

Curiosities and Market Connection

• Linear systems are fundamental in areas such as engineering, economics, computer science, and even business administration.
• In engineering, for example, they are used to analyze electrical circuits and mechanical structures.
• In economics, they help solve optimization problems, such as maximizing profits or minimizing costs.
• In the financial market, they are used to model and predict investment behaviors.

Initial Activity

• Short Video: Start the class with a 3-5 minute video showcasing the application of linear systems in a real context, such as optimizing delivery routes for a logistics company.
• Provocative Question: Ask the students: 'How would you solve a problem where you need to find the best combination of products to maximize profits with limited resources?'

Development

Duration: (60 - 65 minutes)

The purpose of this stage is to deepen students' understanding of linear systems through practical activities. By solving real problems, students develop analytical and problem-solving skills that are essential in the job market. This phase also encourages critical reflection and application of the learned concepts, ensuring a solid and practical understanding of the content.

Covered Topics

1. Classification of linear systems regarding the existence and number of solutions
2. Methods for solving linear systems (substitution, elimination, augmented matrix)
3. Discussion criteria for systems (Rouché-Capelli Theorem)
4. Practical applications of linear systems in the job market

Reflections on the Theme

Guide students to reflect on how the ability to solve linear systems can be applied in various professions. Encourage them to think of everyday examples where this skill is useful, such as in financial data analysis, production engineering, or even algorithm development in computer science. Prompt reflection on the importance of understanding the conditions that determine the existence and number of solutions of a system, and how this analysis can influence critical decisions in projects and business operations.

Mini Challenge

Maker Challenge: Building an Optimization Model

Students will construct a practical optimization model using linear systems to solve a real problem. Divide the class into groups and propose a problem where they need to determine the best combination of resources to maximize the profits of a fictional company.

Instructions

1. Divide the class into groups of 3-4 students.
2. Provide the groups with a problem description: 'A factory produces two products, A and B. The profit per unit of A is $40 and the profit per unit of B is $30. The production of A requires 2 hours of labor and 1 kg of material, while the production of B requires 1 hour of labor and 2 kg of material. The factory has a total of 100 hours of labor and 80 kg of material available per month. Determine the quantity of each product that the factory should produce to maximize profits.'
3. Each group must formulate the system of linear equations representing the problem, identify the constraints, and solve the system to find the optimal solution.
4. Ask each group to present their model and the solution found, explaining the reasoning used to reach the answer.
5. Encourage discussion among the groups about different approaches and solutions.

Objective: Apply the concepts of linear systems to solve a real optimization problem, developing mathematical modeling and decision-making skills.

Duration: (40 - 45 minutes)

Evaluation Exercises

1. Determine if the following linear system is possible and determined, impossible, or possible and indeterminate: 2x + 3y = 5 4x + 6y = 10
2. Solve the linear system using the substitution method: 3x - y = 7 2x + 3y = 1
3. Check if the linear system below has a solution using the Rouché-Capelli Theorem: x + y + z = 6 2x - y + 3z = 14 -x + 2y - z = -2
4. Explain how linear systems can be used for optimization in a business context, giving practical examples.

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to consolidate the knowledge acquired by students throughout the lesson, reinforcing the importance of linear systems both in theory and practice. By promoting discussion and reflection on the activities carried out, students can better internalize the concepts and understand their relevance in real contexts, thus preparing to apply them in the job market.

Discussion

Facilitate a discussion on the main points covered in the lesson. Ask students about their perceptions regarding the maker challenge, how they applied the concepts of linear systems to solve the optimization problem, and what difficulties they encountered. Encourage them to share their reflections on the importance of linear system resolution methods and practical applications in the job market, highlighting concrete examples discussed during the lesson.

Summary

Summarize and recap the main content presented about linear systems: classification concerning existence and number of solutions, resolution methods (substitution, elimination, augmented matrix), discussion criteria for systems (such as the Rouché-Capelli Theorem), and practical applications in the job market. Highlight how each of these topics was addressed in the practical activities and discussions.

Closing

Explain how the lesson connected theory and practice, allowing students to apply the concepts learned in a realistic context. Reinforce the importance of linear systems in everyday life, especially in areas such as engineering, economics, computer science, and administration. Conclude by highlighting how the ability to solve linear systems contributes to more effective decision-making and the resolution of complex problems in the job market.