Objectives (5 - 7 minutes)

1. Provide a clear and detailed Introduction to the concept of 'complementary events' in probability theory. This includes defining complementary events, the relationship between the probability of an event and its complementarity, and how to determine the probability of a complementary event.

2. Present practical and real examples to illustrate the application of the concept of complementary events. The examples should be varied and contextualized so that students can relate them to everyday situations, facilitating the understanding and application of the concept.

3. Develop students' ability to solve problems involving complementary events, using both the theory presented and practical examples. The problems should progressively become more complex, challenging the students and encouraging logical reasoning and the ability to apply the acquired knowledge.

Secondary Objectives:

• Stimulate active participation of students through classroom discussions, questions and answers, and group problem-solving.

• Promote understanding of the concept of probability and its importance in analyzing uncertain situations, contributing to the development of critical thinking and decision-making skills.

• Encourage the use of technological resources, such as calculators or simulation software, to facilitate and enrich learning.

The teacher should start the lesson by briefly reviewing the probability concepts already studied, such as calculating the probability of a simple event and the probability of two independent events. This will help prepare the ground for the introduction of the new concept of complementary events. Additionally, the teacher should emphasize the importance of practice and continuous study for effective learning of mathematics.

Introduction (10 - 15 minutes)

1. Recalling basic concepts: The teacher begins the lesson by recalling the probability concepts already studied, such as calculating the probability of a simple event and the probability of two independent events. This review is crucial to ensure that students have a solid foundation before introducing the concept of complementary events.

2. Problem Situation 1: The teacher proposes the following situation: 'Imagine you are flipping a fair coin. What is the probability of getting heads and, simultaneously, not getting tails?'. This situation serves as a trigger to introduce the concept of complementary events.

3. Problem Definition: The teacher then defines the problem to be solved: 'How can we calculate the probability of two complementary events, that is, the probability of one event occurring and its complement not occurring?'

4. Contextualization: The teacher then contextualizes the importance of the subject, explaining that the probability of complementary events is often used in various everyday situations, such as in gambling, weather forecasting, market studies, among others.

5. Curiosities: To arouse students' interest, the teacher can share some curiosities about probability. For example: 'Did you know that probability theory was initially developed to study gambling games, but ended up being widely applied in various areas, such as quantum physics, genetics, and even artificial intelligence?'.

6. Problem Situation 2: To conclude the Introduction, the teacher proposes one more situation: 'Imagine you have two coins, one fair and one biased, which tends to always fall on the same side. You randomly choose a coin and toss it. What is the probability of getting heads and, simultaneously, the coin tossed being the fair one?'. This situation serves as a transition to a more detailed explanation of the concept of complementary events.

The teacher should ensure that students are engaged during the Introduction, encouraging them to ask questions and share their ideas. Additionally, the teacher should take this opportunity to assess students' understanding of the subject, asking direct questions and observing students' responses.

Development (20 - 25 minutes)

1. Activity 1: 'Card Game' (10 - 12 minutes)

   • The teacher gives each group of students a deck of cards and explains that each card represents an event. For example, the 'Ace of Spades' card can represent the event 'Raining tomorrow' and the 'King of Hearts' card can represent the event 'Not raining tomorrow'.

   • The teacher instructs the students to choose an event and its complementary event, and then calculate the probability of each of them occurring.

   • Students should record their choices and calculations on a piece of paper.

   • After all groups have completed the activity, the teacher asks a representative from each group to share their event and its complementary event, as well as the calculated probabilities.

   • The teacher reinforces the concept of complementary events, correcting any misunderstandings and providing constructive feedback.

2. Activity 2: 'Coin Simulation' (10 - 12 minutes)

   • The teacher distributes coins to each group of students. It can be a fair coin and a biased one (for example, a coin with two heads and one tail).

   • The teacher instructs the students to randomly choose a coin and toss it. They should record the result (heads or tails) and the coin they used.

   • Students should repeat the process several times and record the results.

   • The teacher guides the students to calculate the probability of getting heads and the probability of tossing the fair coin, using the recorded information.

   • Students should compare their estimates with the theoretical probabilities and discuss possible sources of error.

3. Activity 3: 'Probability Problems' (5 - 6 minutes)

   • The teacher distributes an activity sheet with probability problems involving complementary events.

   • Students work in their groups to solve the problems, applying the knowledge acquired in the previous activities.

   • After a set time, the teacher asks a representative from each group to share their solutions and strategies. The teacher provides feedback and clarifies any remaining doubts.

These activities were designed to be interactive, engaging, and contextualized. They allow students to explore the concept of complementary events in a practical way, developing their problem-solving skills, critical thinking, and collaboration. The use of games, simulations, and real problems makes learning more meaningful and fun, helping students visualize the importance and applicability of probability in their daily lives.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher gathers all students and promotes a group discussion, where each group of students shares their solutions and conclusions from the activities carried out.

   • The teacher encourages students to explain their problem-solving strategies, how they arrived at their answers, and what difficulties they encountered.

   • During the discussion, the teacher asks questions to verify students' understanding of the concept of complementary events and its application in solving the proposed problems.

   • The teacher also takes the opportunity to correct any misunderstandings and provide constructive feedback.

2. Connection with Theory (2 - 3 minutes)

   • After the discussion, the teacher reviews the theoretical content, highlighting how the practical activities and real examples connect with the theory.

   • The teacher reinforces the concept of complementary events and the relationship between the probability of an event occurring and the probability of its complement occurring.

   • The teacher revisits the practical examples used in the activities and explains how the theory was applied to solve the problems.

3. Individual Reflection (2 - 3 minutes)

   • The teacher suggests that students reflect individually on what they learned in the lesson.

   • The teacher formulates guiding questions, such as: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.

   • Students are encouraged to write down their answers, which can be shared with the class or handed to the teacher for later feedback.

This Return stage is crucial to consolidate students' learning, clarify any remaining doubts, and assess the effectiveness of the lesson. Through group discussion, students have the opportunity to learn from each other, reflect on their problem-solving strategies, and receive immediate feedback from the teacher. The connection with theory helps reinforce the concepts and practical application of the probability of complementary events. Individual reflection allows students to evaluate their own progress and identify areas that need more practice or study.

The teacher should facilitate the discussion, ensuring that all students have the opportunity to share their ideas and express their doubts. Additionally, the teacher should provide constructive feedback and reinforce the importance of critical thinking, collaboration, and continuous practice for effective learning of mathematics.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes)

   • The teacher starts the Conclusion by summarizing the main points covered during the lesson. This includes defining complementary events, the relationship between the probability of an event and its complementarity, and how to determine the probability of a complementary event.
   • The teacher uses practical and real examples to reinforce students' understanding of the content and clarify any remaining doubts.
   • The teacher also reinforces the importance of practice and continuous study for effective learning of mathematics, especially when it comes to probability.

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

   • The teacher highlights how the lesson connected the theory, practice, and applications of the concept of complementary events.
   • The teacher emphasizes that the practical activities with the deck of cards and coins helped students visualize and better understand the theory.
   • The teacher also reiterates the applications of the concept of complementary events in real situations, such as in gambling, weather forecasting, market studies, among others.

3. Extra Materials (1 - 2 minutes)

   • The teacher suggests extra materials for students to deepen their understanding of the subject. These may include educational videos, probability simulation websites, math books, among others.
   • The teacher can share links to these resources on an online learning platform, such as Google Classroom, so that students can easily access them.

4. Importance of the Subject (1 minute)

   • To conclude, the teacher highlights the importance of the concept of complementary events in everyday life.
   • The teacher reinforces that the probability of complementary events is often used in various situations in daily life, such as in gambling, weather forecasting, market studies, among others.
   • The teacher encourages students to continue exploring the subject and applying it in practical situations, to further strengthen their understanding and skills.

The Conclusion is an essential part of the lesson, as it helps consolidate students' learning, reinforce the importance of the content, and guide future study. The teacher should ensure that the Conclusion is clear, concise, and informative, and that students have the opportunity to ask questions or express any remaining doubts.