Objectives (5 - 7 minutes)

1. Understand Cavalieri's Principle: Students should be able to explain the concept of Cavalieri's Principle, both in terms of volumes and areas. They should understand that if two figures have sections that have the same area (or volume, in the case of solids), then the figures have the same area (or volume).

2. Apply Cavalieri's Principle: Students should be able to apply Cavalieri's Principle to solve practical problems. They should be able to identify when the principle is relevant and how to use it to find unknown areas or volumes.

3. Recognize the importance of Cavalieri's Principle: Students should be able to understand the relevance and applicability of Cavalieri's Principle in various contexts. They should be able to identify everyday situations or fields of study where the principle is useful and necessary.

   Secondary Objectives:

   • Develop logical reasoning skills: Through the study and application of Cavalieri's Principle, students should be able to develop their logical reasoning skills, which are crucial for the discipline of mathematics and for other areas of knowledge.

   • Promote active student participation: Students should be encouraged to actively participate in the class by asking questions, sharing their ideas and solutions, and collaborating with their peers. This will help promote a collaborative learning environment and increase students' understanding and engagement.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing the basic concepts of spatial geometry that were covered in previous classes and that are fundamental for understanding Cavalieri's Principle. This may include the definition of geometric solids (cubes, spheres, prisms, etc.), the difference between area and volume, and how to calculate the area and volume of some simple solids. In addition, the teacher may briefly review the concept of a solid's cross-section.

2. Problem Situations: The teacher can present two problem situations that will serve as a starting point for the Introduction to Cavalieri's Principle. For example:

   • "If we cut a cube with a plane parallel to one of its faces, what will be the area of the resulting cross-section?"
   • "If we cut a sphere with a plane parallel to its diameter, what will be the area of the resulting cross-section?"

3. Contextualization: The teacher can contextualize the importance of Cavalieri's Principle by explaining that this principle is widely used in various areas, such as engineering, architecture, physics, and chemistry. For example, in civil engineering, the principle is used to calculate earthwork volumes in construction projects. In physics, the principle is used to calculate moments of inertia of objects in rotation.

4. Capturing Students' Attention: To spark students' interest, the teacher can share some curiosities or practical applications of Cavalieri's Principle. For example:

   • "Did you know that the famous Italian mathematician Bonaventura Cavalieri, who gave his name to this principle, used his discoveries to calculate the volume of the sphere?"
   • "Cavalieri's Principle is also used in medicine! For example, in computed tomography, which is a diagnostic imaging technique, the principle is applied to calculate tissue density in the human body."

By the end of this stage, students should be motivated to learn more about Cavalieri's Principle and understand its importance and application.

Development (20 - 25 minutes)

1. Activity 1 - The Magic Cube: The teacher should divide the class into groups of 3 to 4 students and provide each group with a toy magic cube. The students' task will be to observe the changes in the cube's cross-sections through the manipulation of the cube. They should take notes of their observations and try to find a pattern.

   • Step 1: The teacher should explain to the students that the magic cube has an internal mechanism that allows it to be manipulated in a way that changes the size of its cross-sections without altering the total area of the cube.
   • Step 2: The students, in their groups, should manipulate the cube, observing the changes in the cross-sections and taking notes of their observations.
   • Step 3: After a period of exploration, each group should present their observations to the class. The teacher should facilitate the discussion, helping students relate their observations to Cavalieri's Principle.

2. Activity 2 - The Birthday Cake: In this activity, students will be challenged to apply Cavalieri's Principle to solve a practical problem. The teacher should provide the groups with an image of a birthday cake, which is a cylinder-shaped solid. The students' task will be to determine the amount of filling in the cake, which will be represented by the area of a cross-section of the cake.

   • Step 1: The teacher should explain the problem situation, showing the image of the cake and explaining that the filling is concentrated in a thin layer inside the cake.
   • Step 2: The teacher should guide the students to identify the information needed to solve the problem, such as the diameter of the cake and the thickness of the filling.
   • Step 3: The students, in their groups, should discuss and plan the strategy to solve the problem, which will involve the application of Cavalieri's Principle.
   • Step 4: Each group should present their solution to the class. The teacher should guide the discussion, checking if the students correctly applied Cavalieri's Principle.

3. Activity 3 - The Prism Challenge: In this activity, students will be challenged to apply Cavalieri's Principle to solve a problem in spatial geometry. The teacher should provide the groups with an image of an irregular prism, and the students' task will be to determine the volume of the prism.

   • Step 1: The teacher should explain the problem situation, showing the image of the prism and explaining that it can be divided into cross-sections of different shapes.
   • Step 2: The teacher should guide the students to identify the information needed to solve the problem, such as the base area of the prism and the height of the prism.
   • Step 3: The students, in their groups, should discuss and plan the strategy to solve the problem, which will involve the application of Cavalieri's Principle.
   • Step 4: Each group should present their solution to the class. The teacher should guide the discussion, checking if the students correctly applied Cavalieri's Principle.

By the end of this stage, students should have a deeper understanding of Cavalieri's Principle, be able to apply it to solve practical problems, and have developed their logical reasoning and teamwork skills.

Feedback (8 - 10 minutes)

1. Group Discussion: The teacher should gather the class and promote a group discussion. Each group should share their solutions or conclusions from the activities carried out. The goal is for students to learn from each other and see different approaches to the application of Cavalieri's Principle. During the discussion, the teacher should ask questions to stimulate critical thinking and deepen students' understanding.

   • Step 1: The teacher should ask each group to share their solutions or conclusions from the activities carried out.
   • Step 2: The teacher should ask questions to stimulate critical thinking and deepen students' understanding. For example: "Why did you choose this strategy to solve the problem?", "How did you apply Cavalieri's Principle in this situation?", "Can you think of other applications of Cavalieri's Principle?".

2. Connection to Theory: After the group discussion, the teacher should make the connection between the practical activities carried out and the theory of Cavalieri's Principle. The teacher should emphasize how observing the changes in the solids' cross-sections (Activity 1) and applying Cavalieri's Principle to solve practical problems (Activities 2 and 3) illustrate and reinforce the theoretical concept of Cavalieri's Principle.

   • Step 1: The teacher should summarize the main observations and conclusions made by the students during the practical activities.
   • Step 2: The teacher should relate these observations and conclusions to the theory of Cavalieri's Principle. For example, "You noticed that, even by changing the size of the cube's cross-sections, the total area of the cube did not change. This is an example of Cavalieri's Principle, which states that if two figures have sections that have the same area (or volume), then the figures have the same area (or volume)".

3. Individual Reflection: Finally, the teacher should propose that students reflect for a minute on what they learned in the lesson. The teacher should ask questions to guide students' reflection, such as: "What was the most important concept you learned today?", "What questions have not been answered yet?".

   • Step 1: The teacher should propose that students reflect individually for a minute.
   • Step 2: The teacher should ask questions to guide students' reflection.
   • Step 3: The teacher should encourage students to share their reflections with the class if they feel comfortable.

By the end of this stage, students should have consolidated their learning about Cavalieri's Principle, be able to make the connection between theory and practice, and have a clear understanding of any doubts or questions they may have.

Conclusion (5 - 7 minutes)

1. Recapitulation of Contents: The teacher should summarize the main points covered during the lesson, reiterating the definition of Cavalieri's Principle, its application for calculating areas and volumes, and the importance of this principle in spatial geometry. The teacher should also highlight the observations made by students during the practical activities and how they reinforce Cavalieri's Principle.

2. Connection between Theory and Practice: The teacher should emphasize how the lesson was able to connect theory, through the explanation of Cavalieri's Principle, with practice, through the activities carried out with the magic cube, the birthday cake, and the irregular prism. The teacher should reinforce that understanding Cavalieri's Principle is crucial for solving practical problems in spatial geometry.

3. Extra Materials: The teacher should suggest extra materials for students to deepen their understanding of Cavalieri's Principle. This may include mathematics books with chapters dedicated to spatial geometry, explanatory videos available on the internet, math websites with exercises and explanations, among others.

4. Practical Applications: The teacher should recall the practical applications of Cavalieri's Principle discussed during the lesson, such as in civil engineering, physics, and medicine. The teacher may also encourage students to look for other everyday situations where Cavalieri's Principle can be applied, thus fostering critical thinking and the application of acquired knowledge.

5. Importance of the Subject: Finally, the teacher should emphasize the importance of Cavalieri's Principle not only for mathematics but also for other areas of knowledge and for everyday life. The teacher should emphasize that the ability to apply Cavalieri's Principle to solve problems is a valuable skill that can be used in various situations throughout life.

By the end of this stage, students should have consolidated their learning about Cavalieri's Principle, be able to make the connection between theory and practice, and have a clear understanding of the importance and relevance of this principle.