Objectives (5 - 7 minutes)

1. Understand the fundamental definitions and properties of point, plane, and line in the context of geometry.
2. Identify and distinguish between points, planes, and lines in practical problems and everyday situations.
3. Develop skills to represent points, planes, and lines in diagrams and coordinates.

Secondary Objectives:

• Apply the acquired knowledge in solving geometric problems involving points, planes, and lines.
• Stimulate critical thinking and analytical skills when dealing with abstract concepts of geometry.
• Promote collaboration and group discussion to deepen the understanding of the concepts.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the lesson by recalling the previous concepts that are essential for understanding the topic of the lesson. He may briefly mention flat geometry, spatial geometry, and Cartesian coordinates, reinforcing the importance of visualization and representation of objects in space. (3 - 5 minutes)

2. Problem situation: The teacher presents two situations involving points, planes, and lines to arouse students' interest.

   • First situation: 'Imagine you are an architect and need to design a new building. How would you use points, planes, and lines to create an efficient and safe project?'
   • Second situation: 'Imagine you are in an unknown city and need to plot the shortest path between two points. How would you use points, planes, and lines to solve this problem?' (3 - 5 minutes)

3. Contextualization: The teacher explains how the concepts of point, plane, and line are applied in various areas of knowledge, such as architecture, engineering, physics, and even in everyday situations, such as GPS navigation. This serves to show students the relevance and practical utility of what will be learned. (2 - 3 minutes)

4. Topic introduction: The teacher introduces the topic of point, plane, and line, explaining that these are the most basic and fundamental elements of geometry. He may mention that, although they seem simple, these concepts are the basis for understanding more complex structures in geometry and other areas of mathematics. To further arouse students' curiosity, the teacher can share some curiosities, such as the fact that, in modern mathematics, a point is considered a dimensionless object, or that the concept of a line is one of the oldest in mathematics, being studied since ancient Greek times. (3 - 5 minutes)

At the end of the Introduction, students should have a clear understanding of what will be covered in the lesson and be motivated to explore more about the topic. The teacher should encourage active student participation by asking questions and facilitating discussion.

Development (20 - 25 minutes)

1. Activity 1: The Architect (10 - 12 minutes)

   • The teacher divides the class into small groups and provides each group with a large sheet of paper, ruler, pencil, and a list of instructions. Each group is designated as an 'architect team' and must work together to complete the task.
   • The task consists of designing a simple but functional building using points, planes, and lines. They must first identify the main points of the building (for example, the vertices of each floor), then draw the planes representing the walls, ceilings, and floors, and finally draw the lines indicating the corridors and connections between spaces.
   • To make the activity more challenging, the teacher can introduce some constraints, such as limiting the number of points that can be used or requiring that the planes and lines be drawn at specific angles.
   • During the activity, the teacher should move around the room, observing the progress of the groups and providing guidance and clarifications as needed. In the end, each group must present their project to the class, explaining how they used points, planes, and lines in their design.

2. Activity 2: The Unknown City (10 - 12 minutes)

   • Still in their groups, students receive a new task: they are now 'explorers' in an unknown city and must find the shortest path between two reference points using only points, planes, and lines.
   • The teacher provides each group with a simple map of the city, marking the reference points and challenging the students to trace the shortest path between them, using only points (the reference points), planes (the streets), and lines (the paths).
   • To make the activity more interesting, the teacher can add obstacles to the map (for example, buildings or rivers) that the students will have to navigate around when tracing their routes.
   • During the activity, the teacher should encourage students to discuss in their groups, test different strategies, and justify their choices. In the end, each group must present their solution to the class, explaining the reasoning behind their path.

3. Discussion and Reflection (5 - 7 minutes)

   • After the conclusion of the activities, the teacher should lead a class discussion, addressing questions such as: What were the challenges faced? How did you use points, planes, and lines to solve the problems? What did you learn from these activities? How do these concepts apply in the real world?
   • The teacher should encourage students to reflect on what they have learned and make connections with the theory presented at the beginning of the lesson. He may also ask questions to test students' understanding and clarify any misunderstandings that may have arisen during the activities.

At the end of the Development, students should have a clear and practical understanding of the concepts of point, plane, and line, and have been able to apply them to solve geometry problems. The teacher should reinforce the key concepts and highlight the importance of teamwork, discussion, and reflection for learning.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should lead a brief group discussion, where each team will share their solutions or conclusions from the 'The Architect' and 'The Unknown City' activities.
   • Each group will have a maximum of 3 minutes to present, and during this time, the other students should pay attention and be prepared to ask questions or make comments.
   • The teacher should ensure that all presentations are made in an environment of respect and collaboration, where all contributions are valued.

2. Connection with Theory (2 - 3 minutes)

   • After all presentations, the teacher should make the connection between the activities carried out and the theory presented at the beginning of the lesson.
   • He can highlight how the groups used the concepts of point, plane, and line to solve practical problems and reinforce the importance of these concepts in geometry and other areas of mathematics and the real world.
   • The teacher should also clarify any misunderstandings that may have arisen during the presentations, reinforcing the key concepts and answering any questions not addressed by the students themselves.

3. Individual Reflection (2 - 3 minutes)

   • To consolidate learning, the teacher should propose that students reflect individually on the lesson. He can ask questions like:
     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 ask some students to share their answers with the class. This can help identify any remaining misunderstandings and provide valuable feedback to the teacher on the effectiveness of the lesson.

At the end of the Return, students should have a clear understanding of the concepts of point, plane, and line, and how they are applied in practice. They should also have had the opportunity to reflect on their own learning and express any doubts or concerns. The teacher should use this information to plan future lessons and to adapt his teaching to the individual needs of the students.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes)

   • The teacher should start the Conclusion by summarizing the main points discussed in the lesson. He should reaffirm the definition and characteristics of points, planes, and lines, and the importance of these concepts in geometry and other areas of knowledge.
   • He can use examples from the activities carried out during the lesson to illustrate and reinforce the theoretical concepts. For example, when reviewing the concept of a line, he can refer to the paths traced by the groups in the 'The Unknown City' activity.
   • The teacher should also recall the relevance and applicability of these concepts in the real world, referring to the problem situations presented at the beginning of the lesson.

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

   • The teacher should emphasize how the lesson connected theory, practice, and applications. He can explain that by designing a building or plotting a path in the city, students put into practice the concepts of point, plane, and line, which were initially discussed theoretically.
   • He should also emphasize that these activities reflect real situations in which these concepts are used, reinforcing the importance of mathematics and geometry in everyday life.

3. Additional Materials (1 - 2 minutes)

   • The teacher should suggest additional study materials for students who wish to deepen their knowledge on the topic. This may include textbooks, educational videos online, interactive math websites, among others.
   • For example, he may recommend a video that visually explains the concepts of point, plane, and line, or a website where students can practice representing these elements on a Cartesian plane.

4. Subject Importance (1 minute)

   • Finally, the teacher should summarize the relevance of the subject to students' daily lives, highlighting that geometry is not just a set of abstract rules, but a powerful tool for understanding and solving problems in many fields, from architecture and engineering to physics and biology.
   • He can encourage students to look for examples of these concepts in their everyday environment, reinforcing the idea that mathematics is present in all aspects of our lives.

At the end of the Conclusion, students should have a clear and comprehensive view of the lesson topic, its relevance, and how they can continue learning and exploring the subject on their own. The teacher should reinforce the importance of continuous practice and independent study for mastering mathematics and other fields of knowledge.