Objectives (5 - 10 minutes)

1. Understanding the concept of Sample Space: Students should be able to understand and define what a sample space is in probability. This includes the notion that the sample space is the set of all possible outcomes of a random experiment.

2. Identification of Sample Space in practical situations: Students should be able to identify the sample space in different contexts and practical situations. This may include coin tosses, dice rolls, card selections from a deck, among others.

3. Calculation of Probabilities: Students should be able to calculate the probability of an event occurring, given the sample space. This includes understanding that the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

Secondary Objectives:

• Application of Sample Space and Probabilities in everyday situations: Students should be able to apply what they have learned about sample space and probabilities in everyday situations. This may include making decisions based on probabilities, such as the probability of rain on a given day, or the probability of winning a game of chance.

• Development of critical thinking: Through the study of sample space and probabilities, students should also develop critical thinking skills, such as the ability to analyze and interpret information, make predictions, and make informed decisions.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by briefly reviewing the concepts of random events, possible outcomes, and events that were covered in previous classes. This will serve as a basis for understanding the new topic. (3 - 5 minutes)

2. Problem situation 1: The teacher can then present the following problem situation to the students: 'If we toss a coin, what are the possible results we can get?'. The teacher should encourage students to think about all possible outcomes - heads or tails - and write them down. (2 - 3 minutes)

3. Problem situation 2: Next, the teacher can present a second problem situation: 'If we roll a die, what are the possible results we can get?'. Again, students should be encouraged to think about all possible outcomes - from 1 to 6 - and write them down. (2 - 3 minutes)

4. Contextualization: The teacher should then explain that these situations are examples of random experiments, and that the list of all possible outcomes of a random experiment is called the sample space. The teacher can then introduce the term 'sample space' and explain that it will be the focus of the lesson. (2 - 3 minutes)

5. Capturing students' attention: To spark students' interest, the teacher can share two curiosities about probability:

   • Curiosity 1: The teacher can ask students if they know the probability of tossing a coin and it landing on its edge. After receiving answers, the teacher can reveal that the probability is very low, about 1 in 6000 tosses. (1 - 2 minutes)

   • Curiosity 2: The teacher can then ask students if they know the probability of winning the lottery. After receiving answers, the teacher can explain that the probability is very low, making the lottery a game of chance. (1 - 2 minutes)

Development (20 - 25 minutes)

1. Activity 1: Sample Space of a Deck of Cards (10 - 12 minutes)

   • Activity description: The teacher presents students with a deck of 52 cards and asks them to identify the sample space, that is, all possible outcomes, if a card is drawn from the deck.

   • Activity steps:

     1. The teacher distributes a deck of cards to each group of students.
     2. Students inspect the deck of cards and discuss among themselves all possible outcomes (the 52 cards) that can occur if a card is drawn from the deck.
     3. Each group should record their sample space, listing all 52 cards.
     4. The groups share their answers and the teacher verifies if all possible outcomes were identified.

   • Discussion and Conclusion: The teacher should then lead a discussion about the activity, emphasizing that the sample space is the set of all possible outcomes of a random experiment. The teacher can also discuss the importance of correctly identifying the sample space for calculating probabilities.

2. Activity 2: Sample Space and Probabilities in Monopoly (10 - 12 minutes)

   • Activity description: The teacher presents students with the Monopoly board game and proposes the following situation: 'If we roll a die, what is the probability of getting a certain number?'. Students must then identify the sample space and calculate the probability.

   • Activity steps:

     1. The teacher distributes the Monopoly boards and dice to each group.
     2. Students roll the die and record the number obtained.
     3. Students repeat this process several times (at least 20 rolls) and record the results.
     4. Students identify the sample space (the possible outcomes of rolling the die, from 1 to 6) and calculate the probability of getting each number, by dividing the number of times each number was obtained by the total number of rolls.
     5. The groups share their answers and the teacher verifies if all possible outcomes were identified and if the probability calculations are correct.

   • Discussion and Conclusion: The teacher should then lead a discussion about the activity, highlighting how the identification of the sample space and the calculation of probabilities can be applied in real-life situations, such as in board games. The teacher can also discuss the importance of repeating the experiment several times to obtain a more accurate estimate of probability.

3. Activity 3: Sample Space and Probabilities in Real Life (5 - 10 minutes)

   • Activity description: The teacher presents students with some real-life situations and asks them to identify the sample space and calculate the probability of a certain event occurring.

   • Activity steps:

     1. The teacher presents students with situations, which may include: the probability of rain on a given day, the probability of winning the lottery, the probability of a soccer team winning a game, among others.
     2. Students, in their groups, must identify the sample space and calculate the probability of each event occurring.
     3. The groups share their answers and the teacher verifies if the sample spaces were correctly identified and if the probability calculations are correct.

   • Discussion and Conclusion: The teacher should then lead a discussion about the presented situations, highlighting how probability can be used to make informed decisions. The teacher can also discuss the importance of considering other factors, in addition to probability, when making decisions, such as risk and benefits.

Return (10 - 15 minutes)

1. Group discussion (5 - 7 minutes)

   • The teacher should gather all students and promote a group discussion about the solutions or conclusions found by each group in the activities carried out.
   • Each group should briefly share their answers, explaining how they identified the sample space and how they calculated the probability.
   • The teacher should encourage students to ask each other questions and provide constructive feedback. The goal is for students to learn from each other and develop communication and collaboration skills.
   • The teacher should also take the opportunity to clarify any doubts that may have arisen during the activities.

2. Connection with theory (3 - 5 minutes)

   • After the group discussion, the teacher should revisit the theoretical concepts presented in the Introduction and make the connection with the practical activities carried out.
   • For example, the teacher can explain how the identification of the sample space and the calculation of probabilities are applied in real situations, such as in board games and weather forecasting.
   • The teacher should also reinforce the importance of considering other factors, in addition to probability, when making decisions, and how this relates to the development of critical thinking.

3. Individual reflection (2 - 3 minutes)

   • To conclude the lesson, the teacher should propose that students reflect individually on what they have learned.
   • The teacher can ask the following questions to guide students' reflection:
     1. What was the most important concept you learned today?
     2. What questions have not been answered yet?
   • The teacher should encourage students to write down their answers and bring their questions to the next class, if any.

4. Teacher feedback (1 - 2 minutes)

   • The teacher should then provide overall feedback to the class, highlighting strengths and areas that need improvement.
   • For example, the teacher can praise the active participation of students during the activities and discussion, and suggest that they continue to engage in this way in the upcoming classes.
   • The teacher should also reinforce the most important concepts covered in the lesson and remind students to study these concepts for the next class.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes)

   • The teacher should start the Conclusion by recapping the main points covered in the lesson. This includes the definition of sample space, the importance of correctly identifying the sample space, and the way to calculate the probability of an event occurring.
   • The teacher can provide a brief but comprehensive summary of each activity carried out, reinforcing how they are connected to the concept of sample space and probabilities.

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

   • The teacher should then highlight the connection between theory, practice, and applications. This can be done by emphasizing how the practical activities helped students better understand the theory, and how the understanding of the theory can be applied in real situations.
   • For example, the teacher can mention how the deck of cards activity helped illustrate the concept of sample space, and how the Monopoly activity allowed students to apply the concept of sample space and probabilities in a familiar context.
   • The teacher can also mention the real-life situations discussed in activity 3, and how the understanding of sample space and probabilities can help people make informed decisions in their daily lives.

3. Extra Materials (1 - 2 minutes)

   • The teacher can then suggest some extra materials for students to deepen their knowledge on the subject. This may include math books, educational websites, YouTube videos, among others.
   • For example, the teacher can suggest that students watch a YouTube video that explains the concept of sample space and probabilities in a different way, or read a chapter from a math book that addresses the subject in more detail.

4. Relevance of the Subject (1 - 2 minutes)

   • Finally, the teacher should emphasize the importance of the subject studied for the students' lives. This can be done by explaining how probability is used in various areas, from weather forecasting to finance and gambling.
   • The teacher can also mention how the understanding of sample space and probabilities can help students develop valuable skills, such as critical thinking, the ability to analyze and interpret information, and make informed decisions.