Objectives (5 - 7 minutes)

1. Understand the concept of a 3x3 matrix determinant and its importance in solving linear systems.
2. Calculate the determinant of a 3x3 matrix using the Sarrus rule and the triangle rule.
3. Apply Cramer's rule to solve linear systems using the coefficient matrix and the independent terms matrix.

Secondary Objectives

1. Develop the skill of identifying and manipulating 3x3 matrices.
2. Reinforce knowledge of matrix operations, such as multiplication and subtraction.
3. Stimulate logical and analytical thinking in solving applied mathematical problems.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher begins the lesson by reviewing the concepts of matrices, their properties and basic operations (addition, subtraction and multiplication by a scalar). It is important to highlight that the 3x3 matrix is a square matrix and, therefore, has a determinant. The teacher should also review Cramer's rule to solve linear systems.

2. Problem Situations: The teacher proposes two problem situations to arouse students' interest and contextualize the subject. The first situation could be solving a linear system using Cramer's rule, where students are challenged to identify the coefficient matrix and the independent terms matrix. The second situation could be the need to calculate the area of a parallelogram on the plane, where the sides of the parallelogram are defined by vectors.

3. Contextualization: The teacher explains that the determinant of a 3x3 matrix is a fundamental concept in mathematics and has several practical applications. For example, in physics, the inertia matrix of an object is a 3x3 symmetric matrix and its determinant is related to the amount of resistance the object has to change its state of motion.

4. Introduction of the Topic: The teacher introduces the topic of the determinant of a 3x3 matrix, explaining that it is a number that can be calculated in different ways. He should mention that, for this lesson, two rules will be presented to calculate the determinant: the Sarrus rule and the triangle rule.

5. Curiosities: To capture students' attention, the teacher can share some curiosities about the determinant of a 3x3 matrix. For example, he can mention that the determinant of a three-dimensional rotation matrix is always 1 (which means that the rotation does not change the volume of space) and that the determinant of a reflection matrix is always -1 (which means that the reflection inverts the volume of space). Another curiosity is that if the determinant of a matrix is zero, it means that the matrix has no inverse.

Development (20 - 25 minutes)

1. Sarrus Rule (10 - 12 minutes)

   1.1. The teacher should start by explaining the Sarrus rule, which is a formula for calculating the determinant of a 3x3 matrix.

   1.2. Next, the teacher should draw a 3x3 matrix on the board and explain how the Sarrus rule works.

   1.3. The teacher should then instruct students to multiply the elements of the main diagonal (from left to right) and the elements of the secondary diagonals (from right to left).

   1.4. After the explanation, the teacher should demonstrate the Sarrus rule with a practical example.

   1.5. Finally, the teacher should instruct students to try to solve some 3x3 determinant exercises using the Sarrus rule.

2. Triangle Rule (5 - 7 minutes)

   2.1. Next, the teacher should introduce the triangle rule to calculate the determinant of a 3x3 matrix.

   2.2. The teacher should explain that, in the triangle rule, the elements of the matrix are arranged in the form of a triangle, with the elements of the main diagonal at the base of the triangle.

   2.3. The teacher should then instruct students to multiply the elements of the main diagonal (from left to right) and the elements of the secondary diagonals (from right to left).

   2.4. After the explanation, the teacher should demonstrate the triangle rule with a practical example.

   2.5. Finally, the teacher should instruct students to try to solve some 3x3 determinant exercises using the triangle rule.

3. Cramer's Rule (5 - 6 minutes)

   3.1. The teacher should revisit Cramer's rule, which was introduced in the concept review step.

   3.2. The teacher should explain that Cramer's rule allows solving a linear system using the determinant of a 3x3 matrix.

   3.3. The teacher should then show how the coefficient matrix and the independent terms matrix are formed from the linear system.

   3.4. The teacher should demonstrate the application of Cramer's rule to solve a linear system with a practical example.

   3.5. Finally, the teacher should instruct students to try to solve some linear systems using Cramer's rule.

The teacher should be attentive throughout the Development process, clarifying doubts, reinforcing concepts and correcting possible calculation errors. It is also important to encourage teamwork and exchange of ideas among students.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   1.1. The teacher should form small groups of up to 5 students and ask them to discuss with each other the solutions or conclusions they reached during the Development of the lesson.

   1.2. Each group should be encouraged to share their solutions with the class, thus promoting the exchange of ideas and collaborative learning.

   1.3. The teacher should intervene during the discussions to clarify doubts, correct possible errors and reinforce important concepts.

2. Connection with the Theory (2 - 3 minutes)

   2.1. The teacher should then return to the theoretical concepts presented at the beginning of the lesson and make the connection with the practical activities carried out.

   2.2. The teacher can, for example, ask students how they applied the Sarrus rule, the triangle rule and Cramer's rule to solve the proposed exercises.

   2.3. The teacher should encourage students to explain in their own words the concepts and techniques used, thus verifying whether they really understood the subject.

3. Individual Reflection (2 - 3 minutes)

   3.1. To conclude, the teacher should suggest that students reflect individually on what they have learned in class.

   3.2. The teacher can ask questions such as: "What was the most important concept you learned today?" and "What questions have not yet been answered?"

   3.3. Students should be encouraged to write down their reflections in a notebook or a digital document, so that they can consult them later and review the content of the lesson.

4. Feedback and Closing (1 - 2 minutes)

   4.1. The teacher should end the lesson by thanking everyone for their participation and highlighting the main points of the content covered.

   4.2. The teacher can also ask students for feedback on the lesson, asking, for example, if they found the content clear and if they had any difficulties at any time.

   4.3. The teacher should remind students of the importance of reviewing the content at home and of clarifying any doubts that may arise.

   4.4. The teacher should also inform students about the next topic to be covered in the subject and if any prior preparation is necessary.

Conclusion (5 - 7 minutes)

1. Summary of the Content (2 - 3 minutes)

   1.1. The teacher should begin the Conclusion by recapitulating the main points covered during the lesson, recalling the definition of the determinant of a 3x3 matrix, the Sarrus and triangle rules to calculate the determinant, and Cramer's rule to solve linear systems.

   1.2. The teacher should emphasize the importance of understanding and mastering these concepts, as they are fundamental for solving various mathematical problems and have applications in several areas of knowledge.

2. Connection between Theory, Practice and Applications (1 - 2 minutes)

   2.1. Next, the teacher should highlight how the lesson connected the theory, practice and applications of the determinant of a 3x3 matrix.

   2.2. The teacher should reinforce that the theory was presented in a clear and didactic way, allowing students to understand the fundamental concepts.

   2.3. The teacher should also emphasize that the practice, through examples and exercises, allowed students to apply and assimilate the theoretical concepts.

   2.4. Finally, the teacher should reiterate the applications of the determinant of a 3x3 matrix in several fields of knowledge, such as in solving linear systems and in the analysis of geometric transformations.

3. Extra Materials (1 - 2 minutes)

   3.1. The teacher should suggest extra materials for students who wish to delve deeper into the study of the determinant of a 3x3 matrix.

   3.2. These materials may include books, websites, videos and online exercises that address the topic in a broader and more detailed way.

   3.3. The teacher should instruct students to use these materials as a complement to the content studied in the classroom, as a way to reinforce the concepts learned and to improve their calculation and problem-solving skills.

4. Importance of the Subject and Closing (1 minute)

   4.1. Finally, the teacher should emphasize the importance of the determinant of a 3x3 matrix for the students' mathematical education and for its application in several areas of knowledge.

   4.2. The teacher should end the lesson by reinforcing the need to review the content at home, to solve the proposed exercises and to clarify any doubts that may have arisen during the lesson.