Objectives (5 - 10 minutes)

1. Understand the concept of the surface area of the sphere and how to calculate it.

   • Identify the elements that compose a sphere and the total area of its surface.
   • Use the correct formula to calculate the surface area of a sphere.

2. Apply the acquired knowledge to solve practical problems.

   • Develop critical and logical thinking skills to apply theory in problem-solving.
   • Use mathematics in a practical and contextualized way, relating it to everyday situations.

3. Stimulate curiosity and interest in mathematics.

   • Present examples and real situations involving the calculation of the surface area of the sphere.
   • Promote discussions and reflections on the importance of mathematics in understanding the world around us.

Secondary Objectives:

• Develop teamwork and communication skills during problem-solving activities.
• Encourage autonomy and responsibility for their own learning, as the inverted classroom methodology requires students to study the content beforehand.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher starts the lesson by reviewing the spatial geometry concepts already studied, such as sphere, radius, and diameter. He can ask students questions to assess prior knowledge and clarify any doubts that may arise. (3 - 5 minutes)

2. Problem Situations: The teacher presents two problem situations involving the calculation of the surface area of a sphere. For example:

   • "If we had to cover a soccer ball with paint that perfectly covered its entire surface, how many square centimeters of paint would be needed?"
   • "Imagine you have an orange and want to completely wrap it with aluminum foil. How much aluminum foil will be needed?" (4 - 5 minutes)

3. Subject Contextualization: The teacher explains the importance of calculating the surface area of the sphere in various areas of science and engineering. For example, in satellite construction, the study of planetary motion physics, the manufacturing of glass bulbs, among others. He can mention some curiosities, such as the fact that the sphere is the most efficient shape for storing volume with the least amount of surface area. (3 - 4 minutes)

4. Gaining Attention: To spark students' interest, the teacher can share curiosities and real-world applications of calculating the surface area of the sphere. For example:

   • "Did you know that the formula for calculating the surface area of the sphere is used in the manufacturing of golf balls? It helps determine how much force is needed to hit the ball and make it travel the desired distance."
   • "And in space, astronauts need to understand the concept of the surface area of the sphere to calculate the amount of insulating material that will be necessary to protect them from the extreme temperatures of the sun and the vacuum of space." (3 - 4 minutes)

Development (20 - 25 minutes)

1. "Styrofoam Ball" Activity:

   • The teacher divides the class into groups of 3 to 4 students and gives each group a medium-sized styrofoam ball and a roll of string.
   • The teacher explains that the challenge is to completely cover the styrofoam ball with the string, leaving no empty spaces.
   • Each group must measure the length of string needed to cover the ball and record the value.
   • After the activity, the teacher instructs the students to measure the diameter of the styrofoam ball and calculate the value of π (pi) based on the formula C = 2πr, where C is the length of the string and r is the radius of the ball. Then, they must calculate the surface area of the sphere with the formula A = 4πr², where A is the surface area and r is the radius of the sphere.
   • The teacher circulates around the room, assisting the groups as needed and clarifying doubts. (10 - 12 minutes)

2. "Planet Earth" Activity:

   • The teacher prepares in advance a globe or a soccer ball decorated as planet Earth.
   • The students, still in their groups, are tasked with calculating the surface area of the sphere representing Earth.
   • The teacher guides the students to measure the diameter of the sphere and calculate the surface area based on the formula A = 4πr².
   • After the activity, the teacher leads a classroom discussion on how calculating the surface area of the sphere is important in cartography and weather forecasting, for example. (7 - 10 minutes)

3. "Magic Sphere" Activity:

   • The teacher proposes a final challenge: each group must create a "Magic Sphere" using cardboard, scissors, and glue.
   • The Magic Sphere should be decorated with images and words related to the calculation of the surface area of the sphere.
   • The teacher provides a template of a cardboard sphere and instructs the students to cut and glue the parts to form the sphere.
   • During the activity, the teacher circulates around the room, assisting the groups and promoting discussions about the importance of calculating the surface area of the sphere.
   • At the end, each group presents their Magic Sphere, explaining the meaning of the chosen images and words. (3 - 5 minutes)

These playful and practical activities allow students to understand the concept of the surface area of the sphere in a more concrete and meaningful way. By working in groups, they also develop collaboration and communication skills, as well as stimulate curiosity and interest in mathematics.

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • The teacher gathers all students and initiates a group discussion about the solutions or conclusions found by each team during the "Styrofoam Ball" and "Planet Earth" activities.
   • Each group will have up to 3 minutes to present their findings, explaining how they arrived at the calculations of the surface area of the sphere and what difficulties they encountered.
   • During the presentations, the teacher encourages other students to ask questions, comment on the solutions presented, and compare them with their own. This stimulates reflection and the exchange of ideas, enriching everyone's learning.

2. Connection to Theory (3 - 4 minutes):

   • After all presentations, the teacher gives a brief review of the theory, reinforcing the concepts of sphere, radius, diameter, and the formula for calculating the surface area of the sphere.
   • The teacher highlights the connections between theory and the practical activities carried out, emphasizing how the acquired knowledge was applied to solve the proposed problems.
   • The teacher also takes the opportunity to clarify any remaining doubts and reinforce the important points of the content.

3. Individual Reflection (2 - 3 minutes):

   • The teacher suggests that students make a brief reflection on what they learned in the lesson.
   • He can ask questions like: "What was the most important concept you learned today?" and "What questions have not been answered yet?".
   • Students will have a minute to think about their answers. Then, those who wish to can share their reflections with the class.
   • This step is important for students to consolidate what they have learned and identify any gaps in their understanding, which can be addressed in future lessons.

4. Feedback and Closure (1 minute):

   • Finally, the teacher thanks everyone for their participation, praises the students' effort and teamwork, and reinforces the importance of calculating the surface area of the sphere in various areas of science and engineering.
   • The teacher also asks for quick feedback from the students about the lesson, inquiring if they enjoyed the activities, felt they learned, and have suggestions to improve future classes.

This Feedback is essential to consolidate learning, promote reflection and metacognition among students, and provide feedback to the teacher on the effectiveness of the lesson and possible improvements for the future.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes):

   • The teacher gives a brief summary of the main concepts covered in the lesson, recalling the definition of sphere, radius, diameter, and the formula for calculating the surface area of the sphere (A = 4πr²).
   • He reinforces the practical applications of calculating the surface area of the sphere, citing examples of areas in science and engineering where these concepts are used.
   • The teacher also highlights the skills developed during the practical activities, such as problem-solving, teamwork, and communication.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes):

   • The teacher explains how the lesson connected theory, practice, and applications, emphasizing that theoretical knowledge was applied to solve practical problems and understand real-world applications.
   • He highlights how the activities with the styrofoam ball and the Earth globe allowed students to visualize and manipulate spheres, making the concept more concrete and understandable.

3. Additional Materials (1 - 2 minutes):

   • The teacher suggests additional materials for students to deepen their understanding of the surface area of the sphere. This may include explanatory videos, interactive websites, online math games, among others.
   • For example, he may recommend a video showing how the formula for the surface area of the sphere was developed or a website that allows students to explore the properties of spheres interactively.

4. Relevance of the Content (1 - 2 minutes):

   • Finally, the teacher reinforces the importance of calculating the surface area of the sphere in daily life and in various areas of knowledge.
   • He may cite additional examples of applications, such as calculating the area of human skin to determine the radiation dose in a radiotherapy treatment, or calculating the area of an eyeball to determine the amount of eye fluid in cases of glaucoma.
   • The teacher concludes the lesson by emphasizing the relevance of studying mathematics to understand the world around us and encouraging students to continue exploring and questioning the concepts learned.