Objectives (5 - 7 minutes)

1. Understand the concept of bisector and perpendicular bisector: Students should be able to define and distinguish the bisector of an angle and the perpendicular bisector of a segment, understanding their functions and importance in geometry.
2. Identify and draw bisectors and perpendicular bisectors: Students should learn to apply the definitions of bisectors and perpendicular bisectors in practice, identifying and drawing them in geometric figures.
3. Solve problems involving the bisector and the perpendicular bisector: Students should be able to apply the acquired knowledge to solve problems involving the bisector and the perpendicular bisector, demonstrating logical reasoning skills and critical thinking.

Secondary objectives:

• Foster teamwork: Through group activities, students should be encouraged to work together, share ideas, and support each other to achieve the lesson's objectives.
• Develop spatial thinking skills: Geometry is a discipline that requires spatial thinking skills, and understanding and applying the concepts of bisector and perpendicular bisector can help enhance these skills in students.

Introduction (10 - 15 minutes)

1. Review of previous contents: The teacher should start the lesson by reviewing the concepts of angle, line, line segment, and parallelism, which are fundamental to understanding the concept of bisector and perpendicular bisector. Questions can be asked to students to verify prior knowledge and clarify any doubts that may arise. (3 - 5 minutes)

2. Problem situations: The teacher should propose two problem situations to introduce the theme:

   • First situation: 'If we have any angle, how can we find the line that divides it in half, that is, the bisector of that angle?'
   • Second situation: 'If we have any line segment, how can we find the line that passes through its midpoint and is perpendicular to it, that is, the perpendicular bisector of that segment?' (3 - 5 minutes)

3. Contextualization: The teacher should explain that the bisector and the perpendicular bisector are important concepts in geometry and have practical applications in various areas, such as in building construction, engineering, architecture, and even in board games, where they are used to determine the trajectory of pieces, for example. (2 - 3 minutes)

4. Engaging students' attention: To arouse students' interest, the teacher can present some curiosities and applications of the concepts of bisector and perpendicular bisector:

   • Curiosity 1: The term 'bisector' comes from the Latin 'bis' (two) and 'secare' (to cut), that is, the bisector cuts an angle in two equal parts.
   • Curiosity 2: In art and nature, there are several examples of bisectors and perpendicular bisectors. For example, in painting, the bisector is used to determine the viewer's point of view, and in nature, the perpendicular bisector is used to determine the center of a flower. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity 'Building Bisectors and Perpendicular Bisectors': The teacher should divide the class into groups of up to 5 students and provide each group with a ruler, a compass, a protractor, and graph paper. Students will be tasked with constructing bisectors in angles and perpendicular bisectors in line segments, measuring and marking the necessary points with the provided instruments. The teacher should circulate around the room, assisting the groups as needed and clarifying doubts. (10 - 12 minutes)

   • Step 1: Students should draw any angle on a paper with the compass and ruler.
   • Step 2: Using the compass, students should draw the bisector of that angle.
   • Step 3: On another paper, students should draw any line segment.
   • Step 4: Using the compass and ruler, students should draw the perpendicular bisector of that segment.

2. Activity 'Solving Problems with Bisectors and Perpendicular Bisectors': After the conclusion of the construction activity, the teacher should propose problems for the groups to solve using the bisectors and perpendicular bisectors they constructed. The problems may involve determining angles or segments, verifying properties, and solving problem situations. The teacher should provide immediate feedback and guidance as the groups progress in solving the problems. (10 - 12 minutes)

   • Example of problem 1: 'If we draw the bisectors of the angles of a quadrilateral, what can we conclude about these bisectors?'
   • Example of problem 2: 'Given a line segment AB and its perpendicular bisector, if we draw a point C on the perpendicular bisector, what can we conclude about the segments AC and BC?'

3. Activity 'Applying in Practice': To consolidate learning, the teacher should propose to the groups to apply the concept of bisector and perpendicular bisector in practical everyday situations. For example, students can observe the bisectors on a map and discuss how they are useful for determining the direction between two points, or they can observe the perpendicular bisector in a drawing and discuss how it is useful for determining the symmetry center of the drawing. (3 - 5 minutes)

Feedback (8 - 10 minutes)

1. Group discussion (3 - 4 minutes): The teacher should promote a group discussion, where each team shares their solutions or conclusions from the activities carried out. At this moment, students will have the opportunity to hear different perspectives and approaches to solving problems, which can enrich their own understanding of the subject. The teacher should encourage students to ask questions and make comments about the presentations of other groups.

2. Connection with theory (3 - 4 minutes): After the group discussion, the teacher should revisit the theoretical concepts of bisector and perpendicular bisector. He should highlight how the practical activities performed relate to the theory presented at the beginning of the lesson. The teacher should reinforce the main theoretical points, clarify any remaining doubts, and provide additional examples if necessary.

3. Individual reflection (2 minutes): To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. He can ask questions like:

   1. 'What was the most important concept you learned today?'
   2. 'What questions have not been answered yet?'

The teacher should give a minute for students to think about these questions. Then, he can ask some students to share their answers with the class. This reflection exercise helps students consolidate their learning and identify any areas that may still need reinforcement.

Conclusion (5 - 7 minutes)

1. Recapitulation: The teacher should give a brief summary of the main points covered in the lesson. He should recall the concept of bisector and perpendicular bisector, the importance of these lines in geometry, and the practical applications of these concepts. The teacher should reinforce the skills developed by students during the lesson, such as the ability to draw bisectors and perpendicular bisectors, and solve problems involving these lines. (2 - 3 minutes)

2. Connection between theory and practice: The teacher should highlight how the lesson connected theory and practice. He should explain that the construction and problem-solving with bisectors and perpendicular bisectors helped students better understand the theoretical concepts. The teacher can mention examples of the activities carried out to illustrate this connection. (1 - 2 minutes)

3. Extra materials: The teacher can suggest additional study materials for students who wish to deepen their knowledge of bisectors and perpendicular bisectors. These materials may include explanatory videos, educational websites, geometry books, and additional exercises. The teacher should emphasize that practice is essential for learning mathematics, and students should make the most of these resources to consolidate their understanding of the subject. (1 - 2 minutes)

4. Importance of the subject: Finally, the teacher should emphasize the importance of the topic addressed for everyday life. He can mention how the concepts of bisector and perpendicular bisector are applied in various areas, such as architecture, engineering, design, art, and even in board games. The teacher should emphasize that mathematics is not an abstract discipline, but a powerful tool to understand and describe the world around us. (1 minute)