Objectives (5 - 7 minutes)

1. Introduce students to the concept of Analytical Geometry and its importance in the field of mathematics and in other areas of knowledge and real life.

2. Develop students' ability to understand and apply the formula of the equation of a line, as well as to interpret its components and the meaning of its solutions.

3. Provide students with the opportunity to solve practical problems involving the application of the equation of a line, stimulating logical thinking and problem-solving skills.

Secondary objectives:

• Encourage active participation of students throughout the class, through practical activities and group discussions.

• Foster students' autonomy, promoting research and individual study as a complement to the content presented in the classroom.

Introduction (10 - 15 minutes)

1. Review of previous contents: The teacher should start the class by reviewing the concepts of geometry and algebra, with emphasis on the definition of point, line, plane, first-degree equations, and linear functions. This review can be done through questions directed at the students or through a brief summary presented by the teacher. (3 - 5 minutes)

2. Problem-solving situations: After the review, the teacher can present two problem-solving situations that involve the need to find the equation of a line. For example:

   • 'An airplane departs from point A at time t = 0 and arrives at point B at time t = 1. How can we represent the trajectory of this airplane in space-time?'
   • 'If we know two points in a plane, how can we determine the line that passes through these points?' (3 - 5 minutes)

3. Contextualization: The teacher should then contextualize the importance of analytical geometry, explaining that it is essential in various areas such as physics, engineering, architecture, geography, and even in computer games. For example, analytical geometry is used to model the trajectory of a rocket, to design a building, to determine the shortest route between two points on a map, and to create graphics in computer games. (2 - 3 minutes)

4. Gaining attention: To draw students' attention to the subject, the teacher can share some curiosities or interesting applications of analytical geometry. For example:

   • 'Did you know that analytical geometry was developed by René Descartes, the same philosopher who said 'I think, therefore I am'?'
   • 'Did you know that analytical geometry was one of the main tools used by Isaac Newton to develop the theory of universal gravitation?' (2 - 3 minutes)

Development (20 - 25 minutes)

1. Coordinate Drawing Activity: The teacher should propose a practical activity where students will draw simple geometric figures (such as a square, a triangle, a circle) on a Cartesian plane. Each group of students will receive a sheet of paper with a Cartesian plane drawn. They will have to choose three non-collinear points (not on the same line) and draw the geometric figures, respecting the condition that all points belong to the figure. After completing the drawing, students should identify the equation of the line passing through two of the drawn points. (10 - 12 minutes)

   • Activity steps:
     1. Divide the class into groups of 4 to 5 students.
     2. Hand out a sheet of paper with a Cartesian plane to each group.
     3. Students must choose three non-collinear points and draw the geometric figures.
     4. After completing the drawing, students must identify the equation of the line passing through two of the drawn points.
     5. Groups present their drawings and equations to the class.

2. Tangram Activity on the Cartesian Plane: The teacher should propose a second practical activity, using the Tangram game. Each group of students will receive a set of Tangram pieces and a sheet of paper with a Cartesian plane drawn. They will have to create a figure on the Cartesian plane using the Tangram pieces and then find the equation of the line passing through the extreme points of each piece. (10 - 12 minutes)

   • Activity steps:
     1. Each group receives a set of Tangram pieces and a sheet of paper with a Cartesian plane.
     2. Students must create a figure on the Cartesian plane using the Tangram pieces.
     3. Then, they must find the equation of the line passing through the extreme points of each piece.
     4. Groups present their figures and equations to the class.

3. Discussion and Analysis: After the practical activities, the teacher should promote a discussion in the classroom about the solutions found by the groups. The goal is for students to understand the relationship between the position of points on the Cartesian plane and the equation of the line passing through them. Additionally, the teacher can propose solving a more complex problem that involves applying the equation of the line in a real-world context. (5 - 8 minutes)

   • Discussion steps:
     1. The teacher reviews the solutions presented by the groups, highlighting the main points and difficulties encountered.
     2. Students are encouraged to ask questions and share their own problem-solving strategies.
     3. The teacher proposes solving a more complex problem, which can be an exercise from a textbook or a real-world situation (for example, determining the trajectory of a rocket based on speed and time data).
     4. Students are guided to solve the problem in groups, with the teacher's assistance.
     5. After solving the problem, groups present their solutions to the class, explaining the reasoning used.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should gather all students and promote a group discussion about the solutions found by each team in the practical activities. Students are encouraged to share their problem-solving strategies, the challenges they faced, and how they overcame them. The teacher should ask questions to stimulate critical thinking and deepen students' understanding of the subject. For example:

   • 'How did you decide which points to use to draw the figure on the Cartesian plane?'
   • 'How did you find the equation of the line passing through those points?'
   • 'Why is the equation of the line important in analytical geometry?'

2. Connection with Theory (2 - 3 minutes): The teacher should then make the connection between the practical activities and the theory presented at the beginning of the class. The goal is for students to see how the equation of the line can be applied in real situations and how it facilitates the understanding and representation of geometric phenomena. The teacher can ask questions to verify if students can make this connection. For example:

   • 'How did the equation of the line help you draw the figures on the Cartesian plane?'
   • 'How would the equation of the line help determine the trajectory of an airplane or a rocket?'

3. Final Reflection (2 - 3 minutes): To conclude the class, the teacher should propose that students reflect individually on what they have learned. The teacher can ask questions to guide students' reflection. For example:

   • 'What was the most important concept you learned today?'
   • 'What questions have not been answered yet?'
   • 'How can you apply what you learned today in everyday situations or in other disciplines?'

4. Teacher's Feedback (1 minute): The teacher should then give overall feedback on the class, praising students' effort and participation, highlighting the class's strengths, and suggesting areas for improvement. The teacher should also encourage students to continue studying the subject and to ask questions in the next class.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion of the class by summarizing the main topics covered. This includes the definition of Analytical Geometry, the formula of the equation of a line, and how to apply it to solve problems. The teacher should emphasize that the equation of a line is a powerful tool that allows the representation and analysis of geometric phenomena in a precise and efficient way.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should explain how the class connected theory, practice, and applications. The teacher can recall the practical activities carried out, highlighting how they helped students understand and apply the theory. The teacher should also mention the applications of analytical geometry in the real world, reinforcing the relevance of the subject beyond the school environment.

3. Additional Materials (1 minute): The teacher should suggest some additional study materials for students. This may include textbooks, math websites, explanatory videos, and additional exercises. The teacher should encourage students to explore these materials on their own, as a way to reinforce learning and clarify any doubts.

4. Importance of the Subject (1 - 2 minutes): Finally, the teacher should emphasize the importance of the subject in everyday life. The teacher can explain that analytical geometry is a widely used tool in various areas of knowledge and practical life. For example, it is essential in physics and engineering to model and understand the movement of objects in space. Additionally, analytical geometry can be used to solve everyday problems, such as determining the shortest route between two points on a map. The teacher should motivate students to continue studying the subject, reminding them that mathematics is a discipline that, besides being useful, can be very fun and challenging.