Objectives (5-10 minutes)

1. Understand the concept of centroid:

   • Recognize the centroid as the point where the medians of a triangle meet.
   • Comprehend that the centroid divides the medians of the triangle in a 2:1 ratio, that is, the distance from the vertex to the centroid is twice the distance from the centroid to the opposite side.

2. Calculate the coordinates of the centroid:

   • Apply the formula to calculate the coordinates of the centroid of a triangle in the Cartesian plane.
   • Solve problems that involve calculating the coordinates of the centroid.

3. Apply the concept of centroid in practical situations:

   • Relate the location of the centroid to the equilibrium of an object.
   • Solve problems that involve the application of the concept of centroid in everyday situations.

Secondary Objectives:

• Develop critical thinking and problem-solving skills by applying the concept of centroid in practical situations.
• Encourage collaboration and communication among students through group activities.
• Reinforce learning in Analytical Geometry and the ability to work with coordinates in the Cartesian plane.

Introduction (10-15 minutes)

1. Review of previous concepts:

   • The teacher begins the lesson by reviewing the concepts of triangle, vertex, and median.
   • Then, they quickly review coordinates in the Cartesian plane and how to calculate the midpoint of two points.
   • These are fundamental concepts for the understanding of the topic of the day, the centroid.

2. Problem situations to introduce the topic:

   • The teacher presents two hypothetical situations to spark students' interest. The first situation could be about an object that needs to be balanced on a specific point. The second situation could be about an architect who needs to calculate the center of gravity of a project to ensure the stability of the structure.
   • These situations serve to contextualize the importance of the centroid and encourage students to understand how to calculate and apply this concept.

3. Contextualization of the importance of the centroid:

   • The teacher explains that the centroid is a fundamental concept in Physics, Engineering, and Architecture, since it is related to the equilibrium and stability of structures.
   • Furthermore, the teacher can mention that the centroid is used in many other areas, such as Biology to calculate the center of mass of an organism, or in Economics to calculate the center of gravity of an industry.

4. Capturing students' attention:

   • The teacher can share the curiosity that the term "centroid" comes from the Greek "baros", which means weight, and "kentron", which means center. That is, the centroid is the center of weight of an object.
   • Then, the teacher can present a more playful application of the centroid, such as calculating the meeting point of the force lines in a magnetic field, or the center of equilibrium of a flat figure.
   • Finally, the teacher should introduce the topic in a clear and direct manner, defining the centroid as the meeting point of the medians of a triangle.

Development (20-25 minutes)

1. Hands-on activity "Building Triangles and Finding the Centroid":

   • The teacher divides the class into groups of 3 or 4 students. Each group receives a sheet of paper, a pencil, and a ruler.
   • The objective of the activity is to construct different types of triangles (equilateral, isosceles, and scalene), and then find the centroid of each one.
   • Students should first draw the triangle on the paper, marking the vertices with letters (A, B, C).
   • Then, they should draw the medians, which should meet at one point. This point is the centroid.
   • Students should then measure the distances from the vertex to this point and from the centroid to the opposite side. They should check that these distances are always in a 2:1 ratio.
   • Finally, students should write down the coordinates of the centroid and the triangle on the Cartesian plane.
   • The teacher should walk around the class, helping the groups and clarifying doubts.

2. Group activity "Real-world Problems":

   • The teacher presents students with a series of real-world problems that can be solved using the concept of centroid. For example: "How could we balance a wooden beam on one end?", "How could we determine the center of gravity of an irregular object?", "How could we calculate the equilibrium point of an engineering project?"
   • Each group should choose one problem to solve. They should discuss among themselves, apply the concept of centroid, and reach a solution.
   • After a certain time, each group should present their solution to the class. The teacher should guide the discussion, clarifying doubts and encouraging students to explain their reasoning.

3. Hands-on activity "Centroid Game":

   • To conclude the hands-on portion of the lesson, the teacher proposes a board game called "Centroid Game".
   • Each group receives a board with the figure of a triangle and a point in the center. The objective is to move the point to different positions on the board, in order to keep the triangle balanced.
   • Students should calculate the new coordinates of the centroid after each move and check if they are correct.
   • This game serves to consolidate the understanding of the concept of centroid and to reinforce the skill of calculating the coordinates of the centroid.

Review (10-15 minutes)

1. Group Discussion (5-7 minutes):

   • The teacher should promote a group discussion with all the students, where each group has a maximum of 3 minutes to share their solutions or conclusions about the activities carried out.
   • During the discussion, the teacher should encourage students to explain their reasoning, to argue about their solutions, and to question the solutions of other groups.
   • The teacher should ask questions that encourage reflection and deepen the understanding of the concept of centroid, such as "Why does the centroid divide the medians in a 2:1 ratio?", "How is the centroid related to the equilibrium of an object?" or "How did you apply the concept of centroid to solve problem X?".

2. Connection with the Theory (2-3 minutes):

   • After the discussion, the teacher should go back to the theoretical concepts presented at the beginning of the lesson and connect them to the hands-on activities carried out.
   • The teacher could, for example, reinforce the definition of centroid as the meeting point of the medians of a triangle, and explain how this was evidenced in the constructions of triangles carried out by the students.
   • The teacher should also highlight the importance of the centroid for the equilibrium and stability of structures, and how this was applied in the solution of the proposed real-world problems.

3. Individual Reflection (3-5 minutes):

   • To conclude the lesson, the teacher proposes that students make an individual reflection about what they have learned.
   • The teacher can guide this reflection by asking questions such as "What was the most important concept you learned today?", "What questions have not yet been answered?" or "How can you apply what you learned today in other situations?".
   • Students should write down their answers in a notebook or on a piece of paper, which will be collected by the teacher at the end of the lesson.
   • This individual reflection serves to consolidate learning, to identify possible gaps in students' understanding, and to establish connections between the content of the lesson and the real world.

4. Closing (1 minute):

   • To conclude the lesson, the teacher should thank everyone for their participation, reinforce the main concepts learned, and make a brief Introduction to the topic of the next lesson.
   • The teacher should also remind students to study the lesson content at home, and to seek the teacher or their classmates if they have any doubts.

Conclusion (5-10 minutes)

1. Recapitulation of the main points (2-3 minutes):

   • The teacher should start the Conclusion by recapitulating the main points covered during the lesson. This includes the definition of centroid, how the centroid divides the medians of a triangle, and how to calculate the coordinates of the centroid.
   • The teacher should also review the hands-on activities carried out, highlighting the insights and learnings that students obtained from them.

2. Connection between theory, practice, and applications (1-2 minutes):

   • The teacher should then reinforce how the lesson connected the theory, practice, and applications of the concept of centroid.
   • The teacher could, for example, mention how the activity of building triangles and finding the centroid helped to visualize and understand the theoretical concept.
   • Furthermore, the teacher should stress how the practical applications of the centroid, discussed during the lesson, helped to contextualize and give meaning to the concept.

3. Extra materials for study (1-2 minutes):

   • The teacher should then suggest some extra materials for students to deepen their study of the centroid. This could include textbooks, educational websites, explanatory videos, and additional exercises.
   • The teacher could, for example, recommend the use of dynamic geometry software, such as GeoGebra, to explore the concept of centroid in an interactive environment.
   • Furthermore, the teacher should remind students to review their notes and to reflect on the questions proposed during the lesson.

4. Importance of the centroid in everyday life (1-2 minutes):

   • Finally, the teacher should summarize the importance of the centroid in everyday life. This could include the application of the centroid in various areas, such as Physics, Engineering, Architecture, Biology, and Economics.
   • The teacher could also highlight how the ability to solve problems involving the centroid can be useful in everyday situations, such as when balancing an object or calculating the center of gravity of a figure.
   • This connection between theory and practice, and between mathematics and the real world, is essential to motivate students to continue studying and applying what they have learned.