Objectives (5 minutes)

1. Understand the concept of volume of a sphere: The teacher must ensure that students understand the concept of volume and how it applies to spatial geometry. This includes understanding what a sphere is and how it is differentiated from other solids.

2. Develop the formula for calculating the volume of a sphere: Students should be able to derive the formula for calculating the volume of a sphere, with the teacher guiding and clarifying doubts throughout the process. This involves understanding the use of the radius in the formula and how it affects the volume.

3. Apply the formula to solve sphere volume problems: Students should be able to apply the formula they have developed to solve sphere volume problems. This includes the ability to work with different units of measurement and interpret the result in a relevant context.

Secondary Objectives:

• Promote critical thinking and problem-solving skills: By solving sphere volume problems, students will be encouraged to think critically and develop their problem-solving skills.

• Foster collaboration and communication: The teacher should encourage students to work together, discuss their solutions, and communicate their ideas clearly and effectively.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by briefly reviewing previously learned spatial geometry concepts, such as the volume of prisms and cylinders. This will serve as a foundation for the new content to be presented. (3 - 5 minutes)

2. Presentation of problem situations: The teacher can present two or three problems involving the calculation of sphere volumes. For example:

   • "Imagine you need to fill a soccer ball with air. How could you calculate the amount of air needed to completely fill the ball?"
   • "If you have a box that can perfectly accommodate a marble, what would be the volume of the marble?" These situations will help contextualize the importance of the topic and spark students' interest. (5 - 7 minutes)

3. Contextualization of the subject: The teacher should explain how the calculation of sphere volumes is applied in everyday situations, industries, and other disciplines. For example, in the manufacturing of sports balls, in architecture (calculating sphere volumes for the construction of domes, for instance), in physics (calculating sphere volumes to understand the distribution of electric charges in an atom), etc. (2 - 3 minutes)

4. Engage students' attention: To pique students' interest, the teacher can share some curiosities about spheres. For example:

   • "Did you know that the most efficient way to pack spheres is in the form of a pyramid? This is because a pyramid can be placed inside a box with less wasted space than if the spheres were placed directly in the box."
   • "Did you know that a sphere is the only solid that has no edges or vertices? This makes the sphere a very interesting object to study in geometry." (2 - 3 minutes)

This Introduction should establish the relevance of the topic, spark students' curiosity, and prepare them for the content to be presented.

Development (20 - 25 minutes)

1. Theory presentation (10 - 12 minutes): The teacher should present the theory necessary for calculating the volume of a sphere. This includes:

   • Definition of a sphere and its properties: The teacher should remind that a sphere is a geometric solid formed by all points in space that are at a distance r (radius) from a fixed point (center). Additionally, a sphere has no edges or vertices and has a single surface.
   • Derivation of the formula for the volume of a sphere: The teacher should guide students in deriving the formula for the volume of a sphere. One way to do this is by considering a sphere inscribed in a cube and then dividing the sphere into a series of thin sections resembling orange slices. The teacher should show how, by summing the volume of these sections, we obtain the formula for the volume of the sphere: 4/3 * π * r³.
   • Explanation of the formula: The teacher should explain the meaning of each term in the formula: π is a constant representing the relationship between the circumference of a circle and its diameter, r is the radius of the sphere.

2. Practical demonstration (5 - 7 minutes): The teacher should perform a practical demonstration to show how the formula for the volume of a sphere works in practice. For example, the teacher can use a styrofoam sphere and a graduated container with water. The teacher should ask students to predict the volume of the sphere and then demonstrate how the formula for the volume of the sphere can be used to calculate the volume of the styrofoam sphere. This will help students see the relevance of the concept and the practical application of the formula.

3. Problem solving (5 - 6 minutes): After presenting the theory and the practical demonstration, the teacher should provide students with problems to solve in groups. The problems should vary in difficulty and context (for example, one problem may involve the Earth's sphere, while another may involve a bowling ball). The teacher should circulate around the room, monitoring students' progress, clarifying doubts, and encouraging discussion among group members.

4. Feedback and discussion (3 - 5 minutes): After students have solved the problems, the teacher should facilitate a classroom discussion, asking groups to share their solutions and strategies. The teacher should provide feedback on students' solutions, clarify any misunderstandings, and highlight important aspects of calculating the volume of a sphere.

The Development of the lesson should ensure that students understand the theory behind calculating the volume of a sphere, can apply this theory to solve problems, and see the relevance and practical application of the concept.

Return (10 minutes)

1. Group discussion (3 - 4 minutes): The teacher should ask each group to share their solutions or conclusions with the class. Each group should have a maximum of 3 minutes to present. This allows students to learn from others' approaches and see different ways of solving the same problems.

2. Connection with theory (3 - 4 minutes): After all presentations, the teacher should provide a general review of the solutions presented, highlighting how each connects with the theory presented at the beginning of the lesson. This helps reinforce theoretical concepts and show students how they apply in practice.

3. Teacher feedback (2 - 3 minutes): The teacher should provide feedback on the groups' presentations, praising creative solutions, pointing out any errors or misunderstandings, and highlighting important aspects of calculating the volume of a sphere. The teacher can also take this opportunity to clarify any remaining doubts and reinforce key concepts of the lesson.

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

   • "What was the most important concept you learned today?"
   • "What questions have not been answered yet?"
   • "How can you apply what you learned today in everyday situations or in other disciplines?" This reflection helps students consolidate what they have learned, identify any areas where they may still have doubts, and see the relevance of what they have learned.

The Return is a crucial part of the lesson, as it allows the teacher to assess students' progress, provide feedback, clarify any remaining doubts, and help students consolidate what they have learned and see the relevance of the lesson content.

Conclusion (5 - 7 minutes)

1. Content summary (2 - 3 minutes): The teacher should recap the key points of the lesson, recalling the concept of a sphere, the formula for calculating its volume (V = 4/3 * π * r³), and the importance of spatial geometry in solving real-world problems. The teacher should emphasize the derivation of the formula from dividing a sphere into thin sections to facilitate students' understanding of the formula.

2. Connection between theory, practice, and applications (1 - 2 minutes): The teacher should explain how the lesson connected the theory of calculating the volume of a sphere with practice, through the demonstration of calculating the volume of a styrofoam sphere, and real-world applications, through the discussion of examples of using sphere volume calculations in different contexts, such as in the manufacturing of sports balls, in architecture, and in physics.

3. Extra materials (1 minute): The teacher should suggest additional study materials for students who wish to deepen their knowledge on the subject. This may include math books, educational websites, explanatory videos on YouTube, among others. For example, the teacher may suggest using an online sphere simulator to visualize the concept of sphere volume interactively.

4. Relevance of the topic (1 - 2 minutes): Finally, the teacher should emphasize the importance of the topic studied for everyday life and other disciplines. Calculating the volume of a sphere, although it may seem like an abstract concept, has practical applications in many areas, from product manufacturing to space exploration. Additionally, the ability to think geometrically and solve complex problems is a valuable skill that can be applied in various life situations. The teacher can encourage students to reflect on how what they learned in the lesson can be useful in their daily lives and in their learning in other disciplines.