Objectives (5 - 10 minutes)

Main Objectives

1. To introduce students to the concept of point, plane and line in geometry, explaining their definitions and fundamental characteristics.
2. To develop students' ability to identify and differentiate between points, planes and lines in different geometric contexts.
3. To provide students with the opportunity to apply the acquired knowledge in solving practical problems and everyday situations that involve the use of points, planes and lines.

Secondary Objectives

1. To stimulate students' logical thinking and capacity for abstraction through the exploration of geometric concepts.
2. To encourage students' active participation in the class, promoting discussion and questioning about the topics covered.
3. To encourage the development of problem-solving and critical thinking skills through the application of mathematical knowledge in practical situations.

The Objectives should be clearly communicated to the students at the beginning of the class, in order to guide their expectations and promote the understanding of the purpose of the content to be learned.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher begins the class by briefly reviewing fundamental concepts of geometry, such as points, lines, planes and their intersections. The goal is to ensure that all students are on the same page and ready to move on to the new content. (3-5 minutes)

2. Problem Situations: The teacher presents two situations that awaken the problem to be solved.

   • Situation 1: Imagine that you are on the beach and you see a boat on the horizon. How would you describe the position of that boat?
   • Situation 2: Suppose you are building a model house. How would you represent the walls (planes) and the beams (lines) in your construction? (3-5 minutes)

3. Contextualization of the importance of the subject: The teacher discusses how the understanding of points, planes and lines is fundamental in various fields, such as architecture, engineering, astronomy, visual arts and even in our daily lives. For example, on a map, points can represent cities, lines the roads, and planes the areas of land or water. (2-3 minutes)

4. Introduction of the topic:

   • Curiosity 1: The teacher can mention that, in mathematics, a point is considered an entity without dimension, that is, it has no length, width or height. However, it can be used to represent any object in a space, from a subatomic particle to the most distant star.
   • Curiosity 2: Another curiosity is the origin of the term "plane". It comes from the Greek "plános", which means "flat space, surface". A plane is a flat surface that extends infinitely in all directions.
   • Curiosity 3: Finally, the teacher can mention that the concept of line is one of the most basic and intuitive ideas in geometry. In fact, the idea of a line is so fundamental that it cannot be defined, only understood intuitively. (2-3 minutes)

This Introduction seeks to awaken students' interest in the subject, showing its relevance and applicability. In addition, curiosities and problem situations stimulate curiosity and active student participation.

Development (20 - 25 minutes)

1. Definition and Characteristics: Point, Plane and Line (5-7 minutes)

   • The teacher explains that, in geometry, a point is an abstract entity that has no dimensions, that is, it has no length, width or height. It is represented by a capital letter.
   • Then, the teacher introduces the concept of a plane as a two-dimensional flat surface that extends infinitely in all directions. It is represented by a capital letter.
   • Finally, the teacher presents the definition of a line as a line that extends infinitely in both directions. It is represented by a lowercase letter.
   • The teacher emphasizes that, in geometry, a point, a plane and a line are considered primitive concepts, that is, they are not defined in terms of other concepts. They are accepted as basic entities from which other geometric concepts are constructed.
   • To reinforce the definition and characteristics of point, plane and line, the teacher can use visual illustrations, diagrams and practical examples.

2. Identification of Point, Plane and Line (5-7 minutes)

   • The teacher presents students with a series of geometric figures and asks them to identify the points, planes and lines present in each figure.
   • To make this activity more interactive, the teacher can divide the class into small groups and assign each group a set of figures to analyze. Then, the groups can share their observations with the class.
   • The teacher circulates through the room, providing guidance and clarifying doubts as necessary.

3. Application of Point, Plane and Line in Practical Situations (5-7 minutes)

   • The teacher proposes some everyday situations or practical problems that involve the use of points, planes and lines. For example, how to represent the trajectory of a plane in the sky, the construction of a building or the organization of a classroom.
   • The teacher guides students to apply the acquired knowledge to solve these situations, encouraging them to think critically and creatively.
   • The teacher can provide continuous feedback and guidance as students work on the activities, promoting active and autonomous learning.

This Development aims to consolidate students' understanding of the concepts of point, plane and line, in addition to providing the opportunity to apply these concepts in practical situations. The teacher must encourage active student participation, promoting discussion and questioning to deepen understanding of the content.

Return (10 - 15 minutes)

1. Review of concepts learned (5-7 minutes)

   • The teacher begins this stage by reviewing the concepts of point, plane and line, reinforcing their definitions and characteristics. He can ask students to share their own definitions and perceptions, allowing them to become the "teachers" for a moment.
   • Then, the teacher can propose a question-and-answer game, where students are challenged to answer questions about the concepts learned. This game can be played individually or in teams, and the teacher can reward correct answers to encourage student participation and engagement.
   • The teacher can also ask students to apply what they have learned in new situations, challenging them to think critically and creatively. For example, he could suggest that students draw a map of their homes, using points to represent furniture, planes to represent walls, and lines to represent hallways.

2. Connection with practice (3-5 minutes)

   • The teacher discusses how the concepts of point, plane and line are applied in practice, whether in fields such as architecture and engineering, or in everyday situations. For example, he can mention how engineers use points, planes and lines to design buildings, or how pilots use these concepts to navigate an airplane.
   • The teacher can also ask students to share their own experiences of how they have used or seen these concepts being used in their lives. This helps to reinforce the relevance and applicability of the content learned, and also allows students to make meaningful connections between theory and practice.

3. Final reflection (2-3 minutes)

   • The teacher concludes the class by asking students to reflect for one minute on the following questions:
     1. What was the most important concept you learned today?
     2. What questions have not yet been answered?
   • After the minute of reflection, the teacher can ask some students to share their answers with the class. This not only helps the teacher to assess the effectiveness of the lesson, but also gives students the opportunity to express their doubts and thoughts, promoting active learning and self-assessment.

This Return stage is essential for consolidating students' learning and for assessing the effectiveness of the lesson. The teacher should ensure that students have understood the main concepts and that they can apply them in different situations. In addition, the teacher must encourage students to reflect on what they have learned, to make connections with practice and to express their doubts and thoughts.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2-3 minutes)

   • The teacher reviews the main concepts covered in the class: points, planes and lines. He reaffirms their definitions, characteristics and how they are fundamental elements in geometry.
   • The teacher can use visual diagrams or three-dimensional models to reinforce these concepts and facilitate student understanding.
   • He also recalls the practical applications of these concepts, highlighting how they are used in various areas, from building construction to air navigation.

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

   • The teacher explains how the class connected theory, practice and applications. He can, for example, mention how the definition of points, planes and lines was illustrated through practical examples and how students had the opportunity to apply these concepts in everyday situations.
   • He can also emphasize how the understanding of these concepts is essential in various professions and activities, and how mathematics, in general, plays an important role in our daily lives.

3. Extra Materials (1-2 minutes)

   • The teacher suggests extra materials for students who wish to deepen their knowledge of the subject. This may include books, websites, educational videos, online games, and math learning apps.
   • He can, for example, recommend an animated video that explains the concepts of points, planes and lines in a playful and visually appealing way, or an online game that allows students to explore these concepts interactively.

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

   • Finally, the teacher highlights the importance of the subject for everyday life. He can, for example, mention how the ability to visualize and work with points, planes and lines is useful in various situations, from reading a map to organizing a physical space.
   • He can also highlight that mathematics, in general, helps to develop valuable skills, such as logical thinking, problem solving and abstraction, which are useful in many aspects of our lives.

The Conclusion is an essential part of the class, as it helps to consolidate students' learning and to establish the connection between theory, practice and applications. In addition, by suggesting extra materials and highlighting the relevance of the subject to everyday life, the teacher encourages students to continue exploring the topic outside the classroom and to value the importance of mathematics in their lives.