Objectives (5 - 7 minutes)

1. Understand the Cartesian coordinate system: Students should be able to understand and describe the Cartesian coordinate system, including the notion of X and Y axes, and how they are used to locate points on the plane.

2. Identify the quadrants on the Cartesian plane: Students should be able to identify and name the four quadrants on the Cartesian plane, and understand how they are used to represent different combinations of positive and negative values.

3. Plot points on the Cartesian plane: Students should be able to plot points on the Cartesian plane, given a coordinate (x, y), and also identify the coordinates of a plotted point.

   Secondary objectives:

   • Recognize the utility of the Cartesian plane in different contexts: Students should be able to recognize the importance of the Cartesian coordinate system in various disciplines and fields of study, including mathematics, physics, engineering, among others.

   • Apply the acquired knowledge to solve problems: Students should be able to apply the acquired knowledge about the Cartesian plane to solve problems, such as determining the distance between two points or the coordinate of a point given a distance and the coordinate of another point.

Introduction (10 - 15 minutes)

1. Review of previous content:

   • The teacher should start the lesson by reviewing the concepts of X and Y axes, and how they relate to the location of points on the plane. (3 minutes)
   • Next, the definition of coordinates and how they are used to identify the position of a point on the plane should be reviewed. (2 minutes)
   • Finally, it is important to remind students about the idea of positive and negative numbers, and how they are represented on the X and Y axes. (2 minutes)

2. Problem situation:

   • The teacher can propose two problem situations to start the discussion:
     1. 'Imagine you are in a city that has a street system following the pattern of a Cartesian plane. You receive the following instruction: 'Turn right at the next intersection and walk 3 blocks forward.' How would you identify the arrival point on the city's Cartesian plane?'
     2. 'Suppose you are playing a battle game where the battlefield is represented by a Cartesian plane. You are informed that the enemy is located at point (4, -2). How could you locate this point on the plane and plan your attack strategy?' (5 minutes)

3. Contextualization of the importance of the subject:

   • The teacher should then explain to the students how the Cartesian coordinate system is used in various areas of everyday life and study. This may include examples such as GPS navigation, geometry, physics, engineering, among others. (3 minutes)

4. Introduction of the topic:

   • To introduce the topic and capture the students' attention, the teacher can share two curiosities:
     1. 'Did you know that the Cartesian coordinate system was developed by the mathematician René Descartes in the 17th century? He created this system to solve geometry problems, but his invention ended up becoming one of the most important and widely used mathematical tools worldwide.'
     2. 'Have you heard of the 'emotional Cartesian plane'? It is a creative way to use the Cartesian coordinate system to represent different emotions. For example, sadness can be represented in the third quadrant, while joy can be represented in the first quadrant.' (5 minutes)

5. Lesson objectives:

   • Finally, the teacher should present the learning objectives of the lesson, explaining that students will learn to plot points on the Cartesian plane, identify the quadrants, and apply these skills to solve problems. (1 minute)

Development (20 - 25 minutes)

1. Role-Playing Activity: 'Adventure on the Cartesian Plane' (10 - 15 minutes)

   • In this activity, students will be divided into groups of 4 to 5 people and will receive a fictional adventure scenario. In this scenario, they will have to use the Cartesian coordinate system to navigate through a 'maze' and find the 'treasure'.
   • Necessary materials: Grid paper sheets, cards with pre-determined coordinates (x, y), colored markers.
   • Development:
     1. The teacher should prepare the 'maze' on the grid paper, marking points of interest such as obstacles, alternative paths, and the 'treasure'.
     2. Each group receives a set of cards with different coordinates (x, y) associated with different parts of the 'maze'.
     3. The goal of the game is for the groups, using the coordinate cards, to identify the location of the obstacles, find the path to the 'treasure', and 'mark' it on the grid paper.
     4. The groups should discuss and plan their actions, deciding in which order they will use their coordinate cards.
     5. After each round, the groups should present their actions and justify their decisions based on the coordinates used.
     6. The game continues until all groups find the 'treasure' or the activity time runs out.

2. Practical Activity: 'Creating an Emotional Cartesian Plane' (10 - 15 minutes)

   • In this activity, students will be encouraged to use their creativity and critical thinking to create their own 'emotional Cartesian plane'. They will have to identify different emotions and assign them to different parts of the plane, justifying their choices.
   • Necessary materials: Grid paper sheets, colored markers.
   • Development:
     1. The teacher should briefly explain the concept of emotions and how they can vary in intensity and polarity (positive or negative).
     2. Next, students, individually or in pairs, should think of various emotions they know and assign them to different parts of the Cartesian plane, justifying their choices based on the intensity and polarity of the emotion.
     3. After finishing, students should share their creations with the class, explaining their choices and listening to their classmates' opinions and justifications.
     4. The teacher should guide the discussion, highlighting the importance of empathy and understanding emotions in the development of socioemotional skills.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should gather all students and promote a group discussion, asking each group to share their solutions or conclusions from the activities.
   • Each group will have a maximum of 3 minutes to present, ensuring that all students have the opportunity to speak.
   • While the groups present, the teacher should ask questions to stimulate critical thinking and deepen students' understanding, such as: 'How did you arrive at this solution?', 'What were the challenges you faced and how did you overcome them?' and 'What did you learn from this activity?'.
   • The teacher should take advantage of this discussion to reinforce the key concepts of the lesson, such as identifying coordinates, locating points on the plane, and the importance of the Cartesian coordinate system.

2. Connection with Theory (2 - 3 minutes)

   • After the presentations, the teacher should revisit the theoretical concepts discussed at the beginning of the lesson and make connections with the solutions presented by the students.
   • For example, the teacher can point out how the 'Adventure on the Cartesian Plane' activity illustrated the practical application of the Cartesian coordinate system in solving navigation problems.
   • The teacher can also highlight how the 'Creating an Emotional Cartesian Plane' activity demonstrated the versatility of the Cartesian plane as a tool to represent and understand different aspects of life, not only in mathematics but also in social and human sciences.

3. Final Reflection (2 - 3 minutes)

   • To conclude the lesson, the teacher should propose a moment of reflection, where students will have a minute to think about the answers to the following questions:
     1. 'What was the most important concept you learned today?'
     2. 'What questions have not been answered yet?'
   • After a minute of reflection, the teacher should invite students to share their answers.
   • The teacher should listen carefully to the students' responses, take note of any unanswered questions, and consider these answers and questions when planning future lessons.

4. Feedback and Closure (1 minute)

   • Finally, the teacher should thank the students for their participation and effort during the lesson, and remind them that learning is a continuous process that requires practice and effort.
   • The teacher should also encourage students to review the lesson material at home and prepare any doubts or questions for the next lesson.
   • Lastly, the teacher should collect feedback from students about the lesson, asking what they liked, what they found challenging, and what could be improved. This will help the teacher adjust and improve future lessons.

Conclusion (5 - 7 minutes)

1. Recap of Contents (2 - 3 minutes)

   • The teacher should start the Conclusion by recalling the main points covered in the lesson. This includes the definition and function of the Cartesian coordinate system, the identification of quadrants, plotting points on the plane, and the application of these skills in problem-solving.
   • The teacher can use visual aids, such as charts and diagrams, to reinforce the concepts and facilitate recapitulation.
   • It is important for the teacher to provide a clear and concise summary, ensuring that all students have understood the fundamental concepts.

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

   • Next, the teacher should highlight how the lesson connected theory, practice, and applications.
   • The teacher can mention how practical activities, such as 'Adventure on the Cartesian Plane' and 'Creating an Emotional Cartesian Plane', allowed students to apply theoretical concepts in real-world situations and understand the relevance of these concepts in different contexts.
   • Additionally, the teacher should reinforce how the Cartesian coordinate system is an essential tool in various disciplines and professions, from mathematics and physics to engineering, architecture, and GPS navigation.

3. Additional Materials (1 minute)

   • The teacher should suggest additional study materials for students who wish to deepen their knowledge on the topic.
   • This may include math books, educational websites with interactive tutorials, explanatory videos on YouTube, and learning apps that allow students to explore the Cartesian coordinate system in a gaming environment.
   • Additionally, the teacher can recommend additional practice exercises for students to consolidate what they learned in the lesson.

4. Relevance of the Subject (1 - 2 minutes)

   • Finally, the teacher should emphasize the importance of the topic for students' daily lives.
   • The teacher can mention examples of how the Cartesian coordinate system is used in everyday situations, such as GPS navigation, locating addresses on a map, planning travel routes, representing data in graphs, and many others.
   • Furthermore, the teacher should reinforce that the development of spatial thinking skills, such as the ability to visualize and manipulate objects on a plane, is fundamental not only for mathematics but also for many other areas of knowledge and life.