Objectives (5 - 7 minutes)

1. Understand the concept of binomial probability: Students should be able to understand the definition of binomial probability, recognizing the characteristics that differentiate it from other probability distributions.

2. Apply the binomial theorem: Students should be able to solve problems involving the binomial theorem, using the formula and calculator to facilitate calculations.

3. Solve binomial probability problems: Students should be able to identify if a problem fits into a binomial distribution, and if so, to use the formula and the binomial theorem to solve it effectively.

Secondary Objectives

• Promote critical thinking skills: When working with binomial probability problems, students will be encouraged to think critically, analyzing the situation presented and applying the knowledge acquired to find the solution.

• Foster teamwork skills: Group activities will allow students to work together, discussing strategies and sharing ideas to solve the proposed problems.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by reviewing the concepts of probability and combination, which are fundamental for understanding binomial probability. This review can be done through a quick interactive quiz or by solving a problem involving these concepts. (3 - 5 minutes)

2. Problem situations: The teacher should present two or three problem situations involving binomial probability to arouse the students' interest. For example, the teacher can ask: 'What is the probability of a student getting exactly 3 questions right on a multiple-choice test with 5 answer options, if he guesses all the questions?' Or 'What is the probability of a coin landing on the same face every time it is flipped 1000 times?' (5 - 7 minutes)

3. Contextualization: The teacher should contextualize the importance of the subject, explaining that binomial probability is widely used in various areas, such as statistics, quantum physics, genetics, among others. Additionally, the teacher can mention that the study of binomial probability can help better understand everyday phenomena, such as the outcome of an election, the probability of a medication working, among others. (2 - 3 minutes)

4. Introduction to the topic: To capture the students' attention, the teacher can introduce the topic with some curiosities or interesting applications. For example, the teacher can mention that binomial probability was developed by Pierre-Simon Laplace, a French mathematician from the 18th century, and that it was used to solve the famous St. Petersburg gambler's problem. Another curiosity is that binomial probability is the basis for the creation of many machine learning algorithms, which are widely used today. (5 - 7 minutes)

Development (20 - 25 minutes)

1. Activity 'Coin Toss' - (10 - 12 minutes): In this activity, students will be divided into groups of 3 to 4 students. Each group will receive a coin and a table to record the results of the tosses. The objective of the activity is for students to understand the concept of binomial probability in practice. The steps of the activity are:

   • Step 1: The teacher should explain that the coin represents a random event, where heads is a success and tails is a failure.

   • Step 2: Each group must decide on a number of tosses (between 10 and 20 tosses) and record the results (heads or tails) in the table.

   • Step 3: Next, students must calculate the probability of success (heads) and failure (tails) for each toss.

   • Step 4: Students should discuss in their groups the relationship between the number of tosses, the probability of success, and the probability of failure.

   • Step 5: Finally, each group must present their results and conclusions to the class.

2. Activity 'Urn Simulation' - (10 - 12 minutes): In this activity, students will continue in their groups. Each group will receive an urn (a paper bag, for example) containing balls of two different colors (for example, 10 red balls and 10 blue balls). The objective of this activity is for students to understand the application of the binomial theorem. The steps of the activity are:

   • Step 1: The teacher should explain that the urn represents a random experiment, where drawing a red ball is a success and drawing a blue ball is a failure.

   • Step 2: Each group must decide on a number of draws (between 10 and 20 draws) and record the results (red or blue) in the table.

   • Step 3: Next, students must calculate the probability of success (drawing a red ball) and failure (drawing a blue ball) for each draw.

   • Step 4: Students should discuss in their groups the relationship between the number of draws, the probability of success, and the probability of failure.

   • Step 5: Finally, each group must present their results and conclusions to the class.

3. Activity 'Real-world Problems' - (5 - 7 minutes): After the practical activities, the teacher should challenge the students to think about real-world problems that can be solved using binomial probability. Some examples could be: 'What is the probability of a vaccine being effective against a disease, knowing that it has 90% efficacy?' or 'What is the probability of a football team winning a championship, knowing that it has a 60% chance of winning each match?' Students should discuss in their groups and present their proposals to the class. The teacher should guide them in applying the binomial theorem to solve these problems.

Feedback (8 - 10 minutes)

1. Group discussion (3 - 5 minutes): After the conclusion of the activities, the teacher should lead a group discussion about the solutions found by each team. In this discussion, it is important to highlight the strategies used, the difficulties encountered, and the conclusions drawn. The teacher should guide the discussion so that students realize the applicability of binomial probability in different contexts and the importance of the binomial theorem for solving these problems.

2. Connection to theory (2 - 3 minutes): The teacher should take advantage of the discussion to make the connection between the practice carried out and the theory taught. For example, he can review the formula of binomial probability and the binomial theorem and show how they were applied in the activities. Additionally, the teacher can reinforce the concepts of probability and combination, which are the basis for understanding binomial probability.

3. Individual reflection (2 - 3 minutes): The teacher should propose that students reflect individually on what they have learned. For this, he can ask the following questions:

   • What was the most important concept learned today?
   • What questions have not been answered yet?
   • How can you apply what you learned today in everyday situations?

   Students should write down their answers and, if they feel comfortable, share them with the class. The teacher should remind students that there are no right or wrong answers, the important thing is to reflect on the learning and think about how to apply it in the future.

4. Teacher feedback (1 minute): Finally, the teacher should give general feedback to the class, highlighting the positive aspects of the work done and pointing out possible areas for improvement. He should end the lesson by reinforcing the importance of studying binomial probability and the binomial theorem, and encouraging students to continue practicing these concepts.

Conclusion (5 - 7 minutes)

1. Topic summary (2 - 3 minutes): The teacher should start the Conclusion by recalling the main points covered during the lesson. He should emphasize the concept of binomial probability, the binomial theorem, and the formula of binomial probability. Additionally, he should reinforce the importance of understanding the difference between a binomial distribution and other probability distributions, and of being able to identify if a problem fits into a binomial distribution.

2. Linking theory to practice (1 - 2 minutes): The teacher should then explain how the lesson connected theory to practice, referring to the activities carried out. He should highlight how binomial probability was applied in practice, and how the binomial theorem assisted in solving the proposed problems. Additionally, he should emphasize that practice is essential to consolidate the understanding of the theory.

3. Extra materials (1 minute): The teacher should suggest some extra materials for students who wish to deepen their knowledge on the subject. These materials may include explanatory videos, extra exercises, simulation websites, among others. The teacher should encourage students to explore these materials in their own time to strengthen their learning.

4. Real-world applications (1 - 2 minutes): Finally, the teacher should explain some real-world applications of binomial probability. He can mention, for example, how binomial probability is used in genetics to predict the probability of a certain genetic trait occurring in a population, or how it is used in statistics to analyze opinion poll results or drug test results. The teacher should reinforce that the study of binomial probability is not just an academic exercise, but a powerful tool to understand and predict real-world phenomena.