Objectives (5 - 7 minutes)

1. Understand the concept of complementary events in probability theory.

   • Define complementary events.
   • Identify examples of complementary events in everyday life.

2. Apply the probability formula for complementary events.

   • Explain the probability formula for complementary events.
   • Solve probability problems involving complementary events.

3. Develop critical thinking skills and problem-solving abilities.

   • Propose problem situations involving the concept of complementary events.
   • Encourage discussion and logical reasoning for problem-solving.

Secondary Objectives:

• Stimulate active student participation in the class, promoting a collaborative and interactive learning environment.
• Spark students' interest in the subject by relating it to practical and everyday situations.

Introduction (10 - 15 minutes)

1. Review of previous contents:

   • The teacher starts the class with a brief review of the probability concepts already studied, such as the sample space and the calculation of the probability of a simple event. This review can be done through a quick discussion with the students, asking them to recall the concepts and provide practical examples.

2. Presentation of problem situations:

   • The teacher proposes two problem situations related to the lesson's theme. For example, 'What is the probability of raining or not raining today?' and 'What is the probability of winning or not winning the lottery?'. These situations will serve as a starting point for introducing the concept of complementary events.

3. Contextualization of the theme:

   • The teacher explains that probability theory, besides being an important tool for mathematics, is also widely used in everyday life in various situations. For example, when predicting the probability of an event, such as the weather, or when calculating the chances of winning in games of chance, like the lottery.

4. Introduction of the topic:

   • The teacher presents the lesson's topic, 'Probability of Complementary Events,' explaining that it refers to events that cannot occur at the same time. Additionally, shows how this concept can be useful for solving the problem situations presented.

5. Curiosities and practical applications:

   • The teacher shares some curiosities about the subject to spark students' interest. For example, the history of how probability theory was developed by mathematicians to solve problems in games of chance.
   • Next, the teacher presents some practical applications of the concept of complementary events. For example, in medicine, when calculating the probability of a patient having or not having a disease based on their symptoms, or in engineering, when calculating the probability of a component failing or not in a system.

Development (20 - 25 minutes)

1. Theory presentation (7 - 10 minutes)

   • Definition of Complementary Events: The teacher starts the theory presentation by explaining that complementary events are those that cannot occur simultaneously. In other words, if one occurs, the other cannot. For example, 'winning' and 'not winning' in the lottery, 'raining' and 'not raining' in a day.
   • Sample Space: Next, the teacher reinforces the concept of sample space, which is the set of all possible outcomes of a random experiment. For example, in the case of rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.
   • Complementary Events and Probability: The teacher explains how the probability of an event and its complement are related, using the formula P(A) + P(A') = 1, where A is an event and A' is its complement. In other words, the probability of an event happening added to the probability of it not happening is always equal to 1.

2. Practical examples (5 - 7 minutes)

   • The teacher presents some practical examples to illustrate the concept of complementary events. For example, when flipping a coin, the events 'heads' and 'tails' are complementary. Another example is with a die, where the events 'rolling an even number' and 'rolling an odd number' are complementary.
   • The teacher can also use examples from students' daily lives, such as the probability of passing or failing a test, of raining or not raining in a day, etc. This helps make the concept more concrete and applicable.

3. Exercise resolution (8 - 10 minutes)

   • The teacher proposes some exercises for the class to solve, using the probability formula for complementary events. The exercises can involve both calculating probabilities and identifying complementary events in practical situations.
   • During the exercise resolution, the teacher should move around the classroom, assisting students who encounter difficulties and verifying if they are applying the concept correctly.

4. Discussion and clarification of doubts (5 - 8 minutes)

   • After the exercise resolution, the teacher promotes a discussion with the class, asking them how they applied the concept of complementary events in solving the problems.
   • The teacher also takes this opportunity to clarify any doubts the students may have and reinforce the concepts presented. This is an important moment to ensure that all students have understood the lesson content.

Return (8 - 10 minutes)

1. Content review (3 - 4 minutes)

   • The teacher starts this stage by giving a brief summary of the main points covered in the lesson. He can review the definition of complementary events, the probability formula for complementary events and how to apply it, and the practical examples that were discussed.
   • This review serves to consolidate students' learning and ensure they have understood the concepts presented. Additionally, it helps prepare them for the next stage, where they will be challenged to apply what they have learned in practical situations.

2. Connection with the real world (2 - 3 minutes)

   • The teacher then asks the students to reflect on how the concept of complementary events can be applied in real situations. For example, he can ask if they can think of any other examples of complementary events beyond those discussed in class.
   • This stage is important for students to realize the relevance of the content learned and how it can be useful in their daily lives. Additionally, it helps stimulate critical thinking and creativity among students.

3. Learning assessment (2 - 3 minutes)

   • The teacher proposes a final challenge to the students: solve a problem involving the calculation of the probability of complementary events in a practical situation. For example, he can ask students to calculate the probability of a student passing or failing a test, based on their previous grades and the class average.
   • Students have a few minutes to reflect on the problem and try to solve it. Then, the teacher asks some students to share their solutions and explanations with the class. He can also present his own solution, highlighting the key points and the strategy used.
   • This activity serves to verify if students were able to correctly apply the concept of complementary events and the probability formula, and if they can solve problems involving this concept. Additionally, it allows the teacher to identify any difficulties students may have and adjust his planning for future lessons, if necessary.

4. Final reflection (1 minute)

   • To conclude the lesson, the teacher suggests that students reflect for a minute on what they have learned. He can ask questions like 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'. Students can write down their answers in a notebook or share them orally with the class.
   • This final reflection helps students consolidate what they have learned and identify any remaining doubts. Additionally, it allows the teacher to assess the effectiveness of his lesson and plan activities for the next class based on student feedback.

Conclusion (5 - 7 minutes)

1. Recapitulation of contents (2 - 3 minutes)

   • The teacher starts the Conclusion of the lesson by recalling the key concepts covered: the definition of complementary events, the probability formula for complementary events and how to apply it, and the examples of complementary events in everyday life.
   • He can give a quick summary of each of these topics and ask students to share what they remember about them. This allows the teacher to assess students' level of understanding and clarify any remaining doubts.

2. Connection between theory, practice, and applications (1 - 2 minutes)

   • The teacher then highlights how the lesson was structured to connect theory, practice, and applications. He can review the problem situations and exercises solved, showing how they required the application of the discussed theoretical concepts.
   • Additionally, the teacher can reinforce the relevance of the subject, showing how the understanding of complementary events and probability has practical applications in various areas, such as weather forecasting, gambling, and many other everyday situations.

3. Additional materials (1 - 2 minutes)

   • The teacher suggests some additional study materials for students who wish to deepen their knowledge on the subject. These materials may include math books, educational websites, explanatory videos, educational games, among others.
   • He can also recommend some extra exercises for students to practice at home, in order to consolidate what they have learned. These exercises can vary in difficulty, from simple probability calculation problems to more complex problems that require logical reasoning.

4. Importance of the subject for everyday life (1 minute)

   • Finally, the teacher reinforces the importance of the subject for students' daily lives. He can cite examples of everyday situations where the understanding of complementary events and probability can be useful, such as predicting the probability of an event, making risk-based decisions, or understanding news involving probabilities, like election polls.
   • This connection between the lesson's subject and students' practical lives helps motivate them to continue studying and applying what they have learned in their lives.