Objectives (5 - 7 minutes)

1. Understanding Spheres:

   • Students should be able to define and identify the characteristics of a sphere, such as radius, diameter, and chord.
   • They should understand the relationship between the radius, diameter, and chord of a sphere.

2. Metric Relations in Spheres:

   • Students should be able to understand and apply the metric relations between spheres and tangent planes.
   • They should understand how to calculate the distance from a point to a sphere and the distance from a point to a tangent plane to a sphere.

3. Practical Skills:

   • Students should be able to apply the concepts and formulas learned to solve practical problems involving spheres and their metric relations.

Secondary Objectives:

• Critical Thinking: Encourage students to think critically and analyze the proposed problems before applying the formulas.
• Teamwork: Promote collaboration in the classroom by encouraging students to work together to solve problems.
• Connection to the Real World: Show students practical examples of how spheres and their metric relations are used in real-world applications, such as the geometry of three-dimensional objects and physics.

Introduction (10 - 12 minutes)

1. Review of Previous Content:

   • The teacher should begin the lesson by reviewing the concepts of sphere, radius, diameter, and chord, which were previously studied. This review can be done through direct questions to the students or through a brief crossword puzzle or puzzle, where students must associate the correct terms with their respective definitions.

2. Problem Situations:

   • Next, the teacher should present two problem situations to arouse students' interest and contextualize the subject:
     • Situation 1: "Imagine that you are in a mini-golf game and need to calculate the distance between the ball and the hole, which is represented by two spheres. How would you do that?"
     • Situation 2: "In a billiard game, a player needs to calculate the distance between the cue ball and the target ball to make a successful shot. How can this be done considering that the balls are represented by spheres?"

3. Contextualization:

   • The teacher should then explain how spheres and their metric relations are used not only in games, but also in various real-world applications, such as architecture (calculations of volumes for the construction of domes, for example), engineering (calculations for the construction of tanks and reservoirs), and physics (calculations of forces and movements in spherical bodies).

4. Gaining Students' Attention:

   • To arouse students' curiosity, the teacher can share two curiosities:
     • Curiosity 1: "Did you know that the sphere is the geometric figure that has the largest number of points on its surface?"
     • Curiosity 2: "Did you know that the sphere is one of the solids most used in works of art and architecture, due to its unique properties of symmetry and balance?"

At the end of the Introduction, students should have a clear understanding of what will be studied in the lesson, with an overview of the concepts to be learned and an awareness of the importance of these concepts in various real-world applications.

Development (20 - 25 minutes)

1. Modeling Activity with Plasticine (10 - 12 minutes):

   • The teacher should divide the class into groups of up to 5 students and provide each group with plasticine of different colors and sizes.
   • The activity consists of modeling three spheres with plasticine of different colors, each representing an element of the problem: the planet Earth (larger sphere), the Moon (medium sphere), and an artificial satellite (smaller sphere).
   • Students should calculate and mark, on each sphere, the midpoint of any diameter (chord) and any point on the surface of the sphere.
   • The students should then measure the distance between the midpoint of the diameter (chord) and the point on the surface in each of the spheres.
   • Finally, students should calculate the distance from the surface point to the chord in each of the spheres, using the formulas learned.
   • Students should record their calculations and conclusions and present their findings to the class.

2. Practical Activity with Billiard Balls (8 - 10 minutes):

   • The teacher should provide each group with a set of numbered billiard balls, representing the spheres in the problem.
   • The activity consists of calculating the distance between two billiard balls, using the distance formula between two points in three-dimensional space.
   • Students should measure the diameter of the balls with a ruler and choose two points on each ball to calculate the distance.
   • After the calculations, students should check if the calculated distance corresponds to the actual distance between the balls and discuss the possible sources of error.

3. Research and Presentation Activity (5 - 8 minutes):

   • The teacher should assign each group an everyday object that has a spherical shape (such as a tennis ball, an orange, a pump, etc.).
   • Students should research the object, including its function, its dimensions, and how spherical geometry is applied to the manufacture and use of the object.
   • Each group should prepare a short presentation (2 to 3 minutes) to share their findings with the class.
   • The presentations should be followed by a classroom discussion on the application of spherical geometry in the real world.

These practical and contextualized activities will help students to better visualize and understand the concepts of spheres and their metric relations, as well as the application of these concepts in the real world. In addition, they will allow students to apply their mathematical skills in a playful and collaborative way.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should gather all the students for a group discussion. Each group will have up to 2 minutes to share the solutions or conclusions they reached in their respective activities.
   • During the presentations, the teacher should encourage students to explain not only the calculations they made, but also the reasoning behind them, thus promoting critical thinking and understanding of the concepts.
   • The teacher should encourage the participation of all students, asking questions to ensure that everyone has understood the presentations of the other groups.

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

   • After the presentations, the teacher should return to the theoretical concepts discussed at the beginning of the lesson and show how they were applied in the practical activities.
   • The teacher should emphasize the importance of understanding the metric relations of spheres and how they can be used to solve real-world problems.
   • The teacher can, for example, ask students how they applied the distance formula between two points in three-dimensional space to calculate the distance between the billiard balls, or how they used the formula for the distance from a point to a sphere to solve the mini-golf problem.

3. Individual Reflection (2 - 3 minutes):

   • To finalize the lesson, the teacher should propose that students reflect individually on what they have learned. The teacher can ask the following question:
     • "What was the most important concept you learned today?"
   • Students should have a minute to think about the question and then share their answers with the class.
   • The teacher should encourage students to explain why they chose that concept as the most important, which will help identify possible gaps in understanding and plan future lessons.

4. Feedback and Closure (1 minute):

   • The teacher should thank the students for their participation, highlight the main points of the lesson, and reinforce that the concepts learned are fundamental to the understanding of spatial geometry.
   • The teacher should also encourage students to continue studying the subject and to seek to apply the concepts learned in other everyday situations.

Feedback is a crucial part of the lesson, as it allows the teacher to assess students' progress, identify and correct possible misunderstandings, and reinforce key concepts. In addition, it helps to promote reflection and students' metacognitive awareness, which are important skills for autonomous and effective learning.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes):

   • The teacher should begin the Conclusion by summarizing the main points covered in the lesson: the definition and characteristics of a sphere, the metric relations between spheres and tangent planes, and how to calculate the distance from a point to a sphere and the distance from a point to a tangent plane to a sphere.
   • The teacher can do this through a diagram on the blackboard, showing the formulas and the steps to solve the problems.
   • It is important that the teacher highlights the relevance of these concepts and how they can be applied in the real world, recalling the problem situations and the practical activities carried out during the lesson.

2. Theory-Practice Connection (1 - 2 minutes):

   • The teacher should explain again how the lesson connected theory, practice, and applications.
   • It should be emphasized how the understanding of theoretical concepts allowed students to solve the proposed practical problems and to understand the applicability of these concepts in the real world.

3. Extra Materials (1 - 2 minutes):

   • The teacher should suggest extra materials so that students can deepen their studies on the subject. These materials may include mathematics books, educational websites, explanatory videos, and online exercises.
   • The teacher can, for example, recommend the book "Euclidean Geometry Plane and Spatial" by Carlos Murakami, the website "Khan Academy" which has a section dedicated to spatial geometry, the video "Learning Spatial Geometry - Spheres" from the channel "Matemática Rio", and the website "Brilliant.org" which offers a variety of math problems and exercises.

4. Importance of the Subject (1 minute):

   • Finally, the teacher should reinforce the importance of the subject studied for everyday life and for other areas of knowledge.
   • The teacher can, for example, explain how spheres and their metric relations are fundamental in various areas, such as physics (for example, in the description of movements of celestial bodies), engineering (for example, in the design of bridges and buildings), and architecture (for example, in the design of domes and vaults).
   • The teacher should also encourage students to reflect on how the understanding of these concepts can benefit them in their lives, whether in solving everyday problems or in developing logical and abstract reasoning skills.

The Conclusion is an essential step to consolidate learning, reinforce the importance of the subject studied, and encourage the continuity of studies. In addition, by connecting theory, practice, and applications, it helps students integrate new knowledge into their cognitive repertoire and perceive the relevance of this knowledge in different contexts.