Objectives (5 - 7 minutes)

1. Understanding the concept of sample space: Students should be able to define what a sample space is, understanding that it is the set of all possible outcomes of a random experiment.

2. Identification of elements in a sample space: Students need to be able to identify the elements that make up a sample space, such as events, possible outcomes, and the probability of each occurrence.

3. Application of formulas and calculations in sample spaces: Students should learn to apply relevant formulas and calculations to solve problems involving sample spaces, such as the probability of an event occurring.

Secondary Objectives:

• Development of logical-mathematical reasoning: Through the study of sample spaces, students should be able to develop their logical-mathematical reasoning, a fundamental skill for solving mathematical problems.

• Stimulation of critical thinking: When working with sample spaces, students should be encouraged to question and think critically about the proposed problems, thus developing their critical thinking skills.

• Promotion of collaboration and communication: Inverted classroom promotes collaborative learning, and students should be encouraged to work together, discuss ideas, and communicate their solutions to the proposed problems.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by reviewing the concepts of random experiment, events, and probability that have already been studied. This review is essential for students to understand the new content of sample spaces. The teacher can do this through a quick discussion or an interactive activity.

2. Problem-solving situations: The teacher should present two problem-solving situations involving sample spaces. For example:

   • If a die is rolled, what is the sample space and the probability of each face being drawn?
   • If a coin is flipped three times, what is the sample space and the probability of getting two heads and one tail?

3. Contextualization: The teacher should explain the importance of sample spaces in solving everyday problems. For example, one can mention the application in weather forecasting, analyzing results of electoral surveys, among others. It is important for students to understand that mathematics is not just a school subject, but a tool that can be used to understand and solve real-world problems.

4. Introduction of the topic: The teacher should introduce the topic 'Sample Spaces' in a way that sparks the students' interest. For example, one can tell the story of how sample spaces were developed and how they are used in various areas, such as quantum physics, genetics, and economics. Another strategy is to present mathematical curiosities or paradoxes involving sample spaces, such as the birthday paradox or the Monty Hall problem.

Development (20 - 25 minutes)

1. Simulation activity with games of chance (10 - 12 minutes):

   • The teacher should divide the class into groups of 4 to 5 students.
   • Each group will receive a set of games of chance, such as a die, a coin, a roulette, etc.
   • The students' task will be to carry out a series of experiments with each game, recording the results and discussing the sample space of each one.
   • Students should try to predict the results of the next rolls based on previous results, in order to understand the concept of probability.
   • At the end of the activity, each group should present to the class the results of their experiments and their conclusions about the sample spaces.

2. Problem-solving activity (10 - 12 minutes):

   • The teacher should provide each group with a series of problems involving sample spaces and probability, such as the examples presented in the Introduction.
   • The students, in their respective groups, should discuss and solve the problems, applying the concepts of sample space and probability.
   • The teacher should move around the room, guiding the groups, clarifying doubts, and stimulating discussion.
   • After a set time, each group should present their solutions and the reasoning used to reach them to the class.

3. Group discussion (5 - 8 minutes):

   • After all presentations, the teacher should promote a group discussion about the activities carried out.
   • The teacher can ask questions to the class, such as: 'What was the most challenging problem?', 'How did you solve this problem?', 'Was there a strategy that worked well for most groups?'.
   • The objective of this discussion is to reinforce the concepts learned, clarify possible doubts, and promote reflection on the problem-solving process.

Throughout the Development, the teacher should be attentive to correcting conceptual errors, encouraging the participation of all students, and ensuring that learning is taking place effectively.

Return (8 - 10 minutes)

1. Group discussion (3 - 4 minutes):

   • The teacher should promote a group discussion, where each team will have up to 2 minutes to share their solutions or conclusions from the activities carried out. Each team should present the main points of their reasoning, the strategies used, and the challenges encountered.
   • The teacher can encourage the participation of everyone by asking directed questions to each team and requesting justification for their answers.

2. Connection with theory (3 - 4 minutes):

   • After all presentations, the teacher should recap the theoretical concepts covered in the lesson, connecting them with the solutions presented by the students.
   • The teacher should highlight how the theory was applied in practice through the activities, reinforcing the importance of understanding sample spaces for solving probability problems.
   • If there is any concept that was not well understood by the students, the teacher should review it and clarify possible doubts.

3. Final reflection (2 minutes):

   • To conclude the lesson, the teacher should propose that students reflect for a minute on the following questions: 'What was the most important concept learned today?' and 'What questions have not been answered yet?'.
   • After a minute of reflection, the teacher can ask some students to share their answers with the class, thus promoting reflection on the learning process and identifying possible points that need to be reinforced in future lessons.

This Return moment is crucial to consolidate learning, ensure that all students have understood the concepts covered, and identify possible difficulties that need to be addressed. The teacher should be open to listening to the students' contributions, valuing their ideas, and encouraging the participation of everyone.

Conclusion (5 - 7 minutes)

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

   • The teacher should recap the main points covered during the lesson, reinforcing the definition and importance of sample spaces, the identification of elements in a sample space, and the application of formulas and calculations related to them.
   • It should be highlighted how these concepts are interconnected and how they can be applied in solving probability problems.

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

   • The teacher should explain how the lesson connected theory, practice, and real-world applications of sample spaces.
   • It should be emphasized how practical activities, such as simulation with games of chance and problem-solving, allowed students to visualize and apply theoretical concepts in a concrete and meaningful way.
   • The teacher can refer to the examples of real-world applications of sample spaces presented in the Introduction, reinforcing the relevance of the topic.

3. Extra Materials (1 - 2 minutes):

   • The teacher should suggest additional study materials for students who wish to delve deeper into the topic.
   • These materials may include books, articles, videos, and websites that explain sample spaces and probability in a clear and didactic manner.
   • The teacher can also suggest extra exercises for students to practice at home, in order to solidify learning.

4. Importance of the Subject (1 minute):

   • Finally, the teacher should reinforce the importance of understanding sample spaces for everyday life and for solving problems in various fields of knowledge.
   • It should be emphasized that the ability to analyze and calculate probabilities is a valuable skill not only in mathematics but also in fields such as science, economics, and engineering.
   • The teacher can end the lesson by encouraging students to continue exploring and applying the concepts learned, and to realize the presence and relevance of mathematics in their daily lives.