Objectives (5-10 minutes)

1. Familiarize students with the concept of translation in the Cartesian plane, understanding that it is a movement in which the original figure is moved in a certain direction and distance without changing its shape or orientation.

2. Develop students' ability to perform translations in the Cartesian plane, understanding how the coordinates of a point change during this process and how to identify the coordinates of a point after the translation.

3. Apply the knowledge acquired about translations in the Cartesian plane to solve practical problems, such as determining the final position of an object after a series of translations.

   Secondary objectives:

   • Stimulate students' logical reasoning and spatial visualization skills.

   • Promote students' self-confidence in their mathematical abilities, encouraging active participation and problem solving independently.

Introduction (10-15 minutes)

1. Review of Previous Concepts: The teacher should begin the lesson by reviewing mathematical concepts that are fundamental to understanding the topic of the lesson. In this case, it is important to review the concept of the Cartesian plane, Cartesian coordinates, x and y axes, and how to locate points on the plane. This review can be done through a brief classroom discussion or with the help of visual aids, such as a graph of the Cartesian plane projected on the board or screen. (3-5 minutes)

2. Problem Situations: To pique students' interest and contextualize the subject, the teacher can present two problem situations. The first may involve locating an object on a map after it has been translated in a certain way. The second could be the task of drawing the figure of an animal on a Cartesian plane after a series of translations. These problem situations should serve as a hook for introducing the theory of translations. (3-5 minutes)

3. Contextualization: The teacher should then explain how translations in the Cartesian plane are applied in real-world situations. Examples can be given of how translation is used in fields such as engineering, architecture, graphic design, and even GPS navigation. The idea is to show students that the mathematics they are learning has practical, real-world applications. (2-3 minutes)

4. Introduction to the Topic: To introduce the topic of translations in the Cartesian plane, the teacher can tell a brief story about how the concept was developed and who the mathematicians involved were. It can be mentioned that the idea of translation was introduced by Euclid, the "father of geometry,” and that over the centuries, many other mathematicians have contributed to the Development of this concept. This historical introduction can help arouse students' curiosity and show the relevance of the subject that will be covered in the lesson. (2-3 minutes)

Development (20-25 minutes)

1. Theory of Translations in the Cartesian Plane (10-12 minutes)

   1.1. The teacher should begin by explaining that a translation is a movement of a figure on the plane without changing its shape or orientation.

   1.2. Then, the mathematical process of translation should be described. The teacher should explain that to translate a figure, each point of the original figure is moved by a certain distance in a certain direction.

   1.3. It is important to emphasize that during translation, all the points of the original figure move in the same way. In other words, if a point is translated 3 units to the right and 2 units up, all other points in the figure will also be translated 3 units to the right and 2 units up.

   1.4. The teacher should demonstrate the translation process in the Cartesian plane using a practical example. For instance, it can be started with a square with vertices at (1,1), (1,2), (2,2), and (2,1). Then, the teacher can show how the figure is translated 3 units to the right and 2 units up, resulting in a new square with vertices at (4,3), (4,4), (5,4), and (5,3).

   1.5. The teacher should conclude the explanation of the theory of translations in the Cartesian plane by emphasizing that the general formula for translating a point (x,y) in a Cartesian plane is (x + a, y + b), where (a,b) is the translation vector.

2. Practice of Translations in the Cartesian Plane (5-7 minutes)

   2.1. After explaining the theory, the teacher should ask students to perform some translations in the Cartesian plane. To do this, the teacher can provide a series of simple figures (such as squares, triangles, etc.) and ask students to translate them according to the teacher's instructions.

   2.2. During this practical activity, the teacher should circulate around the room, observing students' work and providing feedback and help when necessary. It is important that the teacher emphasize that all points in the figure must be translated in the same way.

   2.3. To increase student engagement, the teacher can turn this activity into a game. For example, teams of students can be formed and each team can be given a figure to translate. The team that translates the figure correctly in the shortest amount of time wins the game.

3. Application of Translations to Practical Problems (5-6 minutes)

   3.1. Finally, the teacher should apply the knowledge acquired about translations in the Cartesian plane to solve practical problems. The teacher can present students with a series of problems that involve determining the final position of an object after a series of translations.

   3.2. The teacher should guide students in solving these problems, showing them how to apply the translation formula to determine the new position of a point. The teacher should encourage students to work in groups to solve these problems, promoting collaboration and discussion among students.

   3.3. It is important that the teacher provide continuous feedback during this activity, correcting errors and praising good work. The teacher should also reinforce the importance of checking that all points in the figure have been translated in the same way.

Feedback (10-15 minutes)

1. Group Discussion (5-7 minutes)

   1.1. The teacher should start a group discussion by asking students to share the solutions or conclusions they found during the practical activity and problem-solving.

   1.2. The teacher should encourage students to explain how they arrived at their answers, what strategies they used, and what difficulties they encountered.

   1.3. During this discussion, the teacher should ask questions to stimulate critical thinking and deepen students' understanding of the topic of the lesson. For example, the teacher could ask: "Why must all points in a figure be translated in the same way?" or "How is the translation formula (x + a, y + b) applied to determine the new position of a point?".

   1.4. The teacher should also take this opportunity to correct any misunderstandings and reinforce the most important concepts.

2. Connection with Theory (3-5 minutes)

   2.1. After the group discussion, the teacher should make the connection between the practical activity and the theory of translations in the Cartesian plane.

   2.2. The teacher should explain how the practical activity of translating figures in the Cartesian plane helps visualize and understand the concept of translation.

   2.3. Additionally, the teacher should reinforce that the translation formula (x + a, y + b) is the key to performing translations in the Cartesian plane.

3. Individual Reflection (2-3 minutes)

   3.1. Finally, the teacher should ask students to reflect individually on what they have learned in class.

   3.2. The teacher can do this by asking questions such as: "What was the most important concept you learned today?" and "What questions do you still have about the topic of the lesson?"

   3.3. Students should have a minute to think about these questions and then be asked to share their answers with the class. The teacher should listen carefully to students' responses and take notes, which can be used to adjust planning for future lessons.

   3.4. The teacher should also encourage students to continue studying the topic of the lesson at home, reviewing the concepts learned and solving additional exercises if possible.

4. Teacher Feedback (1-2 minutes)

   4.1. To conclude the lesson, the teacher should give general feedback on the class's performance.

   4.2. The teacher can praise the students' effort and participation, highlighting individual and collective achievements. In addition, the teacher should reinforce the importance of the topic of the lesson and how it connects to other mathematical concepts and practical applications.

   4.3. The teacher should also take the opportunity to remind students about the next lesson and what topics will be discussed.

Conclusion (5-7 minutes)

1. Content Summary (2-3 minutes)

   1.1. The teacher should begin the conclusion by summarizing the main points covered during the lesson. This includes the definition of translation in the Cartesian plane, the mathematical process of translation, the general translation formula (x + a, y + b), and how to apply it to determine the new position of a point.

   1.2. The teacher should reinforce that during a translation, all points in a figure move in the same way, resulting in a new position for the figure but without changing its shape or orientation.

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

   2.1. Next, the teacher should explain how the lesson connected the theory of translations in the Cartesian plane with the practice of performing translations and the application of this knowledge to solving practical problems.

   2.2. The teacher should emphasize that the practical activity of translating figures in the Cartesian plane helped visualize and understand the concept of translation and that solving practical problems allowed students to apply the knowledge acquired in a relevant and meaningful way.

3. Complementary Materials (1 minute)

   3.1. The teacher should suggest some complementary materials so that students can deepen their understanding of translations in the Cartesian plane. These materials may include math books, educational websites, explanatory videos, and online exercises.

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

   4.1. Finally, the teacher should highlight the importance of the subject for everyday life, explaining that translations in the Cartesian plane are used in various fields, such as engineering, architecture, graphic design, GPS navigation, among others.

   4.2. The teacher should encourage students to seek out examples of translations in their daily lives, to reinforce the relevance and applicability of what has been learned.

   4.3. The teacher should end the lesson by reinforcing the importance of continuous study and practice for consolidating knowledge of translations in the Cartesian plane and encouraging students to continue exploring the topic at home.