Objectives (5 - 7 minutes)

• Understand the representation of linear systems through matrices: The main objective of this lesson is for students to understand how linear systems can be represented through matrices. This implies understanding the relationship between the system's unknowns and the matrix elements.

• Develop matrix manipulation skills to solve linear systems: Once students have understood the representation of linear systems by matrices, the next step is for them to acquire skills to manipulate these matrices. This includes applying elementary row operations, which are essential for solving linear systems.

• Solve linear systems using the row-echelon form technique: Finally, the objective is for students to solve linear systems using the row-echelon form technique. This involves applying elementary row operations to transform the system's matrix into an echelon form.

Secondary Objectives:

• Stimulate critical thinking and problem-solving: In addition to the main objectives, the lesson also aims to develop critical thinking and problem-solving skills in students. Solving linear systems through matrices is a process that requires analysis and reasoning, and therefore can be an opportunity to work on these skills.

Introduction (10 - 15 minutes)

• Review of previous concepts: The teacher should start by reviewing the concepts of linear equations systems and matrices. It may be helpful to present some examples of linear systems and their solutions, reinforcing the idea that solving linear systems involves determining values for the unknowns that satisfy all the system's equations.

• Presentation of problem situations: Next, the teacher can present two problem situations involving linear systems. For example, a system of equations representing a real problem, such as determining the prices of different products in a store, or solving a complex mathematical problem involving systems of equations. These problem situations should be challenging and interesting to capture the students' attention.

• Contextualization of the subject's importance: The teacher should then explain the importance of solving linear systems in various areas of knowledge and practical life. It can be mentioned, for example, that solving linear systems is fundamental in areas such as physics, engineering, economics, and computing. Furthermore, it can be emphasized that solving linear systems is a skill that can be useful in daily life, for example, for solving optimization problems.

• Engaging topic introduction: To capture students' attention, the teacher can present two curiosities related to the topic. For example, it can be mentioned that the study of linear systems dates back to antiquity, with the Chinese mathematician Sun Zi being one of the first to study the subject. Additionally, it can be mentioned that solving linear systems is a very challenging problem in computing, and that there are quite complex algorithms for efficiently solving large-scale linear systems.

Development (20 - 25 minutes)

• Theory of Representation of Linear Systems by Matrices (5 - 7 minutes): The teacher should start by explaining the theory of representing linear systems by matrices. It should be emphasized that each linear system can be represented by a matrix, called the system matrix, so that solving the system is equivalent to finding the solution matrix. The teacher can use the whiteboard to illustrate this representation, writing a linear system and the corresponding matrix.

• Application of Theory (5 - 7 minutes): Next, the teacher should show how to apply the theory in practice. For this, some examples of linear systems can be presented, and students can be asked to represent them by matrices. The teacher should move around the classroom, assisting students and clarifying any doubts that arise.

• Row-Echelon Form Theory (5 - 7 minutes): After the practice, the teacher should introduce the row-echelon form theory. It should be explained that the row-echelon form technique consists of applying a sequence of elementary row operations to the system matrix, in order to obtain an echelon form matrix. The teacher should emphasize that the echelon form matrix facilitates the system's solution, as it allows for the immediate identification of solutions.

• Application of Theory (5 - 7 minutes): Finally, the teacher should show how to apply the row-echelon form technique in solving linear systems. A linear system can be presented, and students can be asked to solve it using row-echelon form. The teacher should once again move around the classroom, assisting students and clarifying any doubts that arise.

• Discussion and Reflection (3 - 5 minutes): To conclude the Development of the lesson, the teacher should promote a discussion and reflection on the content presented. Students should be asked if they can perceive the usefulness of representing linear systems by matrices and the row-echelon form technique. Additionally, the teacher should ask students if they have any questions or difficulties regarding the content. The teacher should be prepared to clarify any doubts that arise and to propose reinforcement activities, if necessary.

Return (8 - 10 minutes)

• Review and Connections (3 - 4 minutes): The teacher should start the Return stage by briefly reviewing the concepts and techniques presented in the lesson. This can be done through a quick recap of the main points, or by asking students to summarize what they have learned. Then, the teacher should make connections between the presented theory and practice. For example, students can be asked how they could apply what they have learned to solve a real problem or to better understand a concept from another discipline.

• Reflection on Learning (3 - 4 minutes): The teacher should then propose that students reflect on what they have learned. This can be done through questions such as:

  • What was the most important concept you learned today?
  • What questions do you still have about the subject?
  • How can you apply what you learned today in other situations?
  • What were the easiest and most difficult parts of the lesson?
  • What would you do differently if you had to solve the same problem again?

• Student Feedback (2 - 3 minutes): Finally, the teacher should ask for feedback from students about the lesson. This can be done through a quick survey, oral questions, or a short questionnaire. Student feedback is a valuable tool for the teacher to assess the effectiveness of the lesson and make improvements for future classes. Additionally, student feedback can help identify individual or group difficulties, which can be addressed in future classes or reinforcement activities.

• Lesson Closure (1 minute): To conclude the lesson, the teacher should thank the students for their participation and reinforce the importance of the subject. The teacher can also suggest additional study materials, such as books, videos, or websites, for students who wish to deepen their knowledge on the subject.

Conclusion (7 - 10 minutes)

• Summary of Key Points (2 - 3 minutes): The teacher should start the Conclusion stage by summarizing the main points of the lesson. This may include the representation of linear systems through matrices, matrix manipulation, and the row-echelon form technique for solving linear systems. The teacher should emphasize that solving linear systems through matrices is a powerful tool that can be applied in various areas of knowledge and practical life.

• Connection between Theory and Practice (2 - 3 minutes): Next, the teacher should make a connection between the presented theory and the performed practice. The teacher should reinforce that the theory provides a set of tools for solving linear systems, while practice allows students to apply these tools to solve concrete problems. The teacher can revisit the examples of linear systems presented during the lesson and explain how theory and practice relate in those cases.

• Additional Materials (1 - 2 minutes): The teacher should then suggest some additional study materials for students who wish to deepen their knowledge on the subject. This may include textbooks, explanatory videos, math websites, and online exercises. The teacher can also suggest some extra problems for students to solve at home, in order to practice what they have learned during the lesson.

• Importance of the Subject (1 - 2 minutes): Finally, the teacher should emphasize the importance of the subject presented for daily life and other disciplines. The teacher can mention, for example, that solving linear systems is a useful skill in various areas, such as physics, engineering, economics, and computing. Furthermore, the teacher can explain that solving linear systems can help develop critical thinking and problem-solving skills, which are essential for success in many areas of life.

• Lesson Closure (1 minute): To conclude the lesson, the teacher should thank the students for their participation and reinforce the importance of the subject. The teacher can also remind students of the date and time of the next lesson, and suggest that they review the lesson content at home to prepare for the next one.