Objectives (5 - 10 minutes)

Main Objectives

1. Understand the concept of independent events and how they relate to the calculation of probabilities.
2. Develop the ability to calculate the probability of two or more independent events occurring simultaneously.
3. Apply the concepts learned in solving practical problems involving independent events.

Secondary Objectives

1. Stimulate students' critical and analytical thinking in solving problems involving probabilities.
2. Develop logical reasoning skills in applying the concepts of independent events.
3. Promote teamwork and collaboration among students through discussion and joint problem solving.

Introduction (10 - 15 minutes)

1. Review of Previous Content

   • The teacher should remind students about the concept of probability and the difference between independent and dependent events. This can be done through a brief discussion or oral quiz to assess students' prior knowledge.
   • It is important that students have a solid foundation in these concepts before moving on to the new material.

2. Presentation of Problem Situations

   • The teacher can present two problem situations:
     • The first one could be about the probability of flipping a fair coin and getting "heads" and at the same time, rolling a die and getting the number "3".
     • The second one could be about the probability of drawing a card from a standard deck of cards, without replacement, and the second card being from a different suit than the first one.
   • These situations should be presented in a way that encourages students to think about how to calculate the probability of both events occurring simultaneously.

3. Contextualization of the Importance of the Subject

   • The teacher should explain to students how the probability of independent events is applied in various everyday areas, such as gambling, weather forecasting, statistical analysis, among others.
   • It is important that students realize the relevance of the subject beyond the classroom, which can motivate them to engage more in the lesson.

4. Introduction of the Topic

   • To introduce the topic, the teacher can present curiosities or interesting facts about the probability of independent events. For example, he can mention the "Birthday Paradox", which shows how, in a group of just 23 people, the probability of two people having the same birthday is higher than one might imagine.
   • Another curiosity could be about the "Monty Hall Paradox", which is a famous mathematical problem involving the probability of choosing the correct door in a game of "Let's Make a Deal".
   • These curiosities can help spark students' interest in the subject and motivate them to learn more.

Development (20 - 25 minutes)

1. Activity 1: "Probability Bingo" (10 - 15 minutes)

   • The teacher should distribute bingo cards to each student. Each card should have a combination of numbers from 1 to 75.
   • Next, the teacher should draw numbered balls from 1 to 75 from an urn, without replacement.
   • The objective of the activity is for students to calculate the probability of a specific number being drawn at a given time and the probability of two or more specific numbers being drawn consecutively.
   • Students should record their predictions and calculations on their cards. The one with the most accurate prediction at the end of the activity will be the winner.
   • During the activity, the teacher should walk around the room, helping students solve the problems and clarifying doubts.

2. Activity 2: "Coin and Dice Toss" (10 - 15 minutes)

   • In this activity, the teacher should distribute coins and dice to each group of students.
   • Students should flip the coin and roll the die, recording the results in a table. They should repeat this procedure several times.
   • Next, students should calculate the probability of the coin landing on "heads" and the die showing the number "3" simultaneously, and the probability of the coin landing on "tails" and the die showing the number "3" simultaneously.
   • Students should compare the results obtained with the predictions made initially and discuss the possible reasons for differences, if any.
   • During the activity, the teacher should walk around the room, observing the groups' work, clarifying doubts, and guiding the discussion.

3. Activity 3: "Probability Deck" (5 - 10 minutes)

   • The teacher should distribute a deck of cards to each group of students.
   • Students should draw one card at a time, without replacement, and record the card's suit in a table.
   • Next, the students should calculate the probability of the second card being from a different suit than the first one.
   • Students should compare the results obtained with the predictions made initially and discuss the possible reasons for differences, if any.
   • During the activity, the teacher should walk around the room, observing the groups' work, clarifying doubts, and guiding the discussion.

These fun and practical activities will allow students to better understand the concept of independent events and the application of probability in solving problems. In addition, they promote interaction between students and development of teamwork skills. The teacher should ensure that all students are involved and actively participating in the activities.

Feedback (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 reached by each team.
   • During this discussion, the teacher should encourage students to explain the reasoning behind their solutions and to share their probability calculation strategies.
   • The teacher should also take the opportunity to correct any misunderstandings that may have arisen during the activities.

2. Connection to Theory (3 - 5 minutes)

   • After the discussion, the teacher should resume the theoretical concepts presented in the Introduction, and explain how they apply to the practical situations experienced by students during the activities.
   • The teacher should emphasize the importance of considering the independence of events when calculating the probability of them occurring simultaneously.
   • It is at this point that the teacher should clarify any doubts that may still exist on the subject.

3. Individual Reflection (2 - 3 minutes)

   • To end the lesson, the teacher should suggest that students individually reflect for a minute on what they have learned.
   • The teacher can ask questions like: "What was the most important concept learned today?" and "What questions have not yet been answered?"
   • After the reflection, students can share their answers with the class or write them down for future discussion.

This Feedback moment is essential for consolidating student learning and for the teacher to evaluate the effectiveness of the lesson. It allows students to connect theory to practice, reflecting on what they have learned and identifying possible gaps in their understanding. In addition, the group discussion enables the exchange of ideas and the clarification of doubts, enriching the learning process.

Conclusion (5 - 10 minutes)

1. Summary of Main Points (2 - 3 minutes)

   • The teacher should summarize the main points of the lesson, recapping the concept of independent events and the importance of considering this independence when calculating the probability of events occurring simultaneously.
   • They should reinforce the strategies and methods learned for calculating the probability of independent events.
   • The teacher can use practical examples, such as those from the activities carried out, to illustrate these points and reinforce students' understanding.

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

   • The teacher should explain how the lesson connected the theory, practice, and applications of the concepts of independent events and probability.
   • They can highlight how the practical activities helped illustrate the application of these concepts in everyday situations.
   • The teacher should emphasize that the ability to calculate the probability of independent events is useful in many contexts, from gambling to risk analysis and forecasting.

3. Extra Materials (1 - 2 minutes)

   • To deepen students' learning, the teacher can suggest extra materials for study, such as readings, videos, or websites that explain the calculation of probabilities of independent events in more detail.
   • They can recommend, for example, videos from math education channels on YouTube, such as "Math Antics" or "Khan Academy", which usually have excellent explanations on the subject.
   • In addition, the teacher can indicate exercises from textbooks or math websites so that students can practice what they have learned.

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

   • Finally, the teacher should emphasize the importance of the subject presented for students' daily lives.
   • They can mention again examples of real situations in which the ability to calculate the probability of independent events is useful, such as in gambling, weather forecasting, among others.
   • The teacher should encourage students to continue exploring and applying these concepts, remembering that they are powerful tools for understanding and analyzing many phenomena and situations in the real world.