Objectives (5 - 7 minutes)

1. Understand what a linear system is: The first objective of the lesson is to ensure that students understand the concept of a linear system and how it is formed. This understanding includes the identification of linear equations and variables.

2. Develop skills to discuss a linear system: The second objective is for students to gain skills to discuss a linear system. This involves understanding how variables relate to each other and how the system's equations can be manipulated to obtain a unique solution, multiple solutions, or no solution.

3. Apply the acquired knowledge to practical problems: Finally, students should be able to apply the acquired knowledge to solve practical problems. This means they should be able to identify real-world situations that can be modeled by linear systems and use the learned techniques to solve these problems.

Secondary Objectives:

• Promote active student participation: A secondary objective is to ensure that students are actively involved in the lesson, asking questions, discussing concepts, and solving problems.

• Foster critical thinking and problem-solving: Additionally, the lesson should foster the development of students' critical thinking and problem-solving skills. This involves encouraging careful analysis of problems, formulating effective strategies to solve them, and critically evaluating the solutions obtained.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the lesson by reminding students of previous concepts that are fundamental to understanding the current topic. These concepts include: linear equations, systems of equations, variables, and coefficients. This review can be done through direct questions to students or through a brief theoretical summary.

2. Problem situations: The teacher presents two problem situations involving linear systems. The first one can be a mixing problem, where students need to determine the quantities of different substances that should be mixed to obtain a mixture with certain properties. The second one can be a financial planning problem, where students need to determine how to allocate resources among different projects to maximize the return. These situations are designed to arouse students' interest and demonstrate the relevance of the topic.

3. Contextualization: After presenting the problem situations, the teacher explains that these are just two of the many real-world situations that can be modeled and solved using linear systems. He may mention additional examples, such as linear programming, analysis of electrical circuits, and weather forecasting. The idea is to show students that the topic is not just theoretical but has practical applications in various areas.

4. Capturing students' attention: To capture students' attention, the teacher can share some curiosities or interesting facts about linear systems. For example, he may mention that the technique of solving linear systems known as Gaussian elimination was first used by Chinese astronomers in the 3rd century BC to solve problems of celestial observation. Another interesting fact is that the theory of linear systems is one of the foundations of artificial intelligence and machine learning.

5. Presentation of the topic: Finally, the teacher introduces the topic of the lesson - 'Linear Systems: Discussion of the System'. He explains that in this lesson, students will learn not only to solve linear systems but also to discuss them, that is, to determine whether they have a unique solution, multiple solutions, or no solution. He may also mention that to achieve this goal, students will learn new techniques, such as Cramer's rule and the inverse matrix method.

Development (20 - 25 minutes)

1. Theoretical Discussion (10 - 12 minutes): The teacher should start the lesson with a theoretical review of linear systems, focusing on the identification of variables, linear expressions, and the difference between equations and systems of equations. Some points that can be addressed include:

   • Definition of a linear system: a set of linear equations that share the same variables.
   • Types of linear systems: homogeneous equations (when the constant part of each equation is zero) and non-homogeneous equations (when the constant part of at least one equation is different from zero).
   • Discussion on the number of solutions of a linear system: a system can have a unique solution, multiple solutions, or no solution.
   • Introduction to Cramer's rule and the inverse matrix method: two techniques that can be used to solve and discuss linear systems.

2. Presentation of the Inverse Matrix Method (5 - 7 minutes): The teacher should then introduce the Inverse Matrix Method, explaining how it can be used to solve and discuss linear systems. Some points that can be addressed include:

   • What is an inverse matrix: a matrix that, when multiplied by the original matrix, produces the identity matrix.
   • How to find the inverse matrix of a square matrix: the teacher should demonstrate the process of finding the inverse matrix, step by step, using examples.
   • How to use the inverse matrix to solve a linear system: the teacher should demonstrate how the Inverse Matrix Method can be applied to a linear system to obtain the solution.

3. Presentation of Cramer's Rule (5 - 7 minutes): The teacher should then present Cramer's Rule, another technique for solving and discussing linear systems. Some points that can be addressed include:

   • What is the determinant of a matrix: the teacher should explain that the determinant of a square matrix is a number that can be used to determine if the matrix has an inverse.
   • How to use Cramer's Rule to solve a linear system: the teacher should demonstrate the process of using Cramer's Rule to solve a linear system, step by step, using examples.
   • Limitations of Cramer's Rule: the teacher should emphasize that Cramer's Rule can only be applied to linear systems with the same number of equations and variables.

4. Guided Practice (5 - 7 minutes): After presenting the two techniques, the teacher should guide students in a practice session where they apply the Inverse Matrix Method and Cramer's Rule to solve and discuss linear systems. The teacher should provide support and feedback during practice, correcting errors and clarifying doubts.

5. Discussion of Practical Examples (3 - 5 minutes): Finally, the teacher should discuss practical examples of real-world situations that can be modeled and solved using linear systems. This will help reinforce the relevance of the topic and motivate students to learn.

Feedback (8 - 10 minutes)

1. Synthesis and Connection to the Real World (3 - 4 minutes): The teacher should summarize the main points of the lesson, reinforcing the concept of a linear system and the techniques for discussing linear systems: Cramer's rule and the inverse matrix method. Then, the teacher should make the connection of these concepts to the real world, recalling the problem situations presented in the Introduction of the lesson and how they can be solved using the learned techniques. The teacher may also present new examples from different contexts to reinforce the applicability of the concepts. For example, he can show how planning a balanced diet, considering the nutrient quantities of different foods, can be modeled as a linear system and solved using the learned techniques.

2. Individual Reflection (2 - 3 minutes): The teacher should propose that students reflect individually for a minute on the following questions:

   1. What was the most important concept learned today?
   2. What questions have not been answered yet?

   After reflection, the teacher should ask some students to share their answers with the class. The aim of this activity is to help students internalize what they have learned and identify any gaps in their understanding that can be addressed in future lessons.

3. Student Feedback (1 - 2 minutes): The teacher should ask students what they thought of the lesson and if they feel they have achieved the proposed objectives. The teacher should be open to constructive feedback and use this information to improve his future lessons. This activity also serves to reinforce the importance of active student participation and engagement with the content.

4. Preparation for the Next Lesson (2 - 3 minutes): Finally, the teacher should inform students about the topic of the next lesson and any preparations, if any, they should make. For example, if the next lesson is about solving linear systems, the teacher may ask students to review the substitution, elimination, and Cramer techniques used in the Development of the current lesson. The teacher may also assign readings or homework to reinforce learning and prepare students for the next topic.

At the end of this stage, the teacher should have a clear idea of what students have learned and what areas still need to be reinforced. This will allow him to adjust his teaching plan to meet the individual and collective needs of the students.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion by recapping the main points covered during the lesson. This includes the definition of a linear system, the techniques for solving linear systems (Cramer's rule and the Inverse Matrix Method), and the discussion of linear systems. The teacher should emphasize the importance of these concepts and how they connect to solve complex problems.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should highlight how the lesson connected theory (mathematical concepts and techniques) to practice (problem-solving) and applications (modeling real-world situations). This helps students understand the relevance of what they have learned and how they can apply this knowledge in their daily lives.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest extra materials for students who wish to deepen their knowledge on the topic. This may include reference books, educational websites, explanatory videos, and additional exercises. The teacher may also provide a brief summary of what these materials cover and how they can complement what was learned in the classroom.

4. Relevance of the Subject (1 minute): To conclude, the teacher should emphasize the importance of linear systems in different areas of knowledge, such as physics, engineering, economics, social sciences, among others. For example, he can mention how the ability to model and solve problems with linear systems is a valuable skill for careers in exact and applied sciences. Additionally, the teacher should encourage students to continue practicing and applying what they have learned, as the ability to solve and discuss linear systems is fundamental for understanding more advanced concepts in mathematics and many other disciplines.