Objectives (5 - 7 minutes)

1. Understand Cavalieri's Principle: Students should be able to understand Cavalieri's principle and its application in spatial geometry. They should be able to explain the concept and how it differs from the principle of adding and subtracting volumes.

2. Identify Figures with the Same Volume: Students should be able to identify which figures have the same volume, applying Cavalieri's principle. They should be able to solve problems involving the comparison of volumes of different figures.

3. Apply Cavalieri's Principle in Practical Problems: Students should be able to apply Cavalieri's principle in real situations, such as calculating the volume of a solid obtained by rotating a figure around an axis.

Secondary Objectives

1. Develop Critical Thinking: In addition to learning the content, students should be encouraged to develop their critical thinking by analyzing and questioning the concepts presented.

2. Stimulate Autonomous Learning: The inverted classroom methodology promotes autonomous learning. Students should be encouraged to research and study the content before class so they can actively participate in the discussions and activities proposed.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing concepts of spatial geometry, such as volume, area, and the notion of solids. It is important to ensure that students have a solid understanding of these concepts, as they will be fundamental to understanding Cavalieri's Principle. The teacher can do this through a brief theoretical review or a practical activity, such as comparing the volume of different cubes.

2. Problem Situation: Next, the teacher should present two problem situations that will be discussed and solved during the lesson. The first situation may involve comparing two solids of different shapes but with the same volume. The second situation could be as follows: 'If a triangle is rotated around one of its sides, what will be the resulting solid and how can we calculate its volume?' These situations should serve as a hook for the Introduction to Cavalieri's Principle.

3. Contextualization: The teacher should then contextualize the importance of Cavalieri's Principle, explaining that this is a fundamental concept in geometry, with applications in various areas such as physics and engineering. For example, it can mention that this principle is used to calculate the volume of solids obtained by rotation, a calculation used in engineering and architectural projects.

4. Introduction to the Topic: Finally, the teacher should introduce the topic of Cavalieri's Principle. It can start with the following question: 'If two figures have the same base and height, do they have the same volume?' Then, it should reveal that the answer to this question is not so simple, and that this is where Cavalieri's Principle comes in. It can also arouse students' curiosity by telling the story of Bonaventura Cavalieri, an Italian mathematician from the 17th century who was the first to enunciate this principle.

The Introduction should be planned to arouse students' interest in the topic, showing its relevance and applicability, and preparing the ground for the deeper exploration that will take place in the Development of the lesson.

Development (20 - 25 minutes)

1. Modeling Activity (10 - 12 minutes)

   • Description: Students will be divided into groups of five and will receive a variety of three-dimensional objects (cubes, cylinders, pyramids, etc.) of different sizes. Each group will also receive a sheet of paper and a pencil. The task will be to identify which pairs of objects have the same volume, using Cavalieri's Principle.
   • Step by Step:
     1. The groups must choose two objects to compare.
     2. They must draw the cross-sections of the objects on their sheet of paper.
     3. Then, they must count the squares in each cross-section and multiply this number by the height of the object.
     4. If the two numbers are equal, this means that the objects have the same volume.
     5. The groups must record their findings and share them with the class.

2. Rotation Activity (10 - 12 minutes)

   • Description: In this activity, students will explore how Cavalieri's Principle applies when a flat figure is rotated around an axis.
   • Step by Step:
     1. Each group will receive a piece of modeling clay and a ruler.
     2. They must shape the modeling clay into a triangle and a cylinder.
     3. Then, they must measure the base and height of the triangle and the cylinder.
     4. Using the appropriate formulas, students must calculate the volume of the triangle and the cylinder.
     5. Finally, they must compare the two volumes and verify if they agree with Cavalieri's Principle.

3. Group Discussion (5 - 7 minutes)

   • Description: After the conclusion of the activities, there will be a group discussion where each group will have the opportunity to share their findings and difficulties. The teacher should take this opportunity to clarify any misunderstood concepts and reinforce the application of Cavalieri's Principle.

These practical activities and group discussions will allow students to explore Cavalieri's Principle in a concrete and meaningful way, reinforcing their understanding of the topic. Additionally, they will also be developing important skills such as teamwork, problem-solving, and mathematical communication.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • Description: The teacher should promote a group discussion with all students. Each group will have a maximum of 3 minutes to share the solutions or conclusions they reached during the modeling and rotation activities. The teacher should encourage students to explain the reasoning behind their answers, not just present the final result. Additionally, the teacher should take this opportunity to clarify any misunderstood concepts and reinforce the application of Cavalieri's Principle.

2. Connection to Theory (2 - 3 minutes)

   • Description: After the discussion, the teacher should make the connection between the practical activities carried out and the theory of Cavalieri's Principle. The teacher should highlight how the activities demonstrated the principle in practice and reinforce the theoretical concepts presented in the Introduction of the lesson. This will help students consolidate their understanding of the topic and see the relevance and applicability of Cavalieri's Principle.

3. Individual Reflection (2 - 3 minutes)

   • Description: To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. The teacher can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?' Students should write down their answers, which can be used as a starting point for the next lesson or as a basis for formative assessment. The goal of this activity is to encourage students to reflect on their own learning process and identify any areas they still do not fully understand.

This Return is a crucial part of the lesson plan, as it allows the teacher to assess students' progress, clarify any misunderstandings, and reinforce key concepts. Additionally, it promotes students' metacognition, helping them become more autonomous and reflective learners.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes)

   • Description: The teacher should review the main points covered during the lesson, reinforcing Cavalieri's Principle, its application in spatial geometry, and the difference between this principle and the principle of adding and subtracting volumes. It should also highlight the main discoveries and conclusions obtained during the practical activities, reinforcing the importance of critical thinking and the application of theoretical concepts in solving practical problems.

2. Connection of Theory with Practice (1 - 2 minutes)

   • Description: The teacher should highlight how the lesson connected the theory of Cavalieri's Principle with practice, through the modeling and rotation activities. It should emphasize how these activities allowed students to explore and understand the principle in a concrete and meaningful way, reinforcing its applicability and the importance of spatial geometry in everyday situations.

3. Extra Materials (1 minute)

   • Description: The teacher should suggest extra materials so that students can deepen their knowledge of Cavalieri's Principle. These materials may include textbooks, educational videos, websites specialized in mathematics, among others. Additionally, the teacher can suggest additional exercises so that students can practice what they learned in the lesson.

4. Importance of the Subject for Everyday Life (1 - 2 minutes)

   • Description: To conclude, the teacher should emphasize the importance of Cavalieri's Principle for everyday life. It should explain that this principle is used in various areas, such as physics and engineering, to calculate the volume of solids obtained by rotation, a calculation that is applied in engineering and architectural projects. Furthermore, the teacher can mention that the ability to visualize and manipulate three-dimensional objects is an essential skill in many professions and daily activities, and that spatial geometry, including Cavalieri's Principle, can help develop this skill.

The Conclusion is a fundamental part of the lesson plan, as it allows the teacher to consolidate students' learning, reinforce the connection between theory and practice, and highlight the relevance and applicability of the topic. Additionally, by suggesting extra materials and explaining the importance of the subject for everyday life, the teacher encourages students to continue studying the topic and applying what they have learned in their lives.