Objectives (5 - 7 minutes)

1. Introduce the concept of Factorial and its applicability in real-world problems, developing students' ability to recognize and calculate the factorial of a number.

2. Provide students with the opportunity to solve Combinatorial Analysis problems that involve calculating the factorial of a number.

3. Stimulate critical thinking and problem-solving skills by presenting situations where factorial calculation is necessary.

Secondary Objectives:

• Promote collaboration and teamwork through practical activities that encourage discussion and exchange of ideas among students.

• Reinforce the use of mathematical language and logical reasoning through problem-solving and interpretation of results.

• Develop research skills and self-learning by encouraging students to seek additional information on the topic covered.

Introduction (10 - 15 minutes)

1. Review of essential concepts: The teacher starts the lesson by reviewing the concepts of multiplication, factorization, and permutation, which were previously studied. This review is important for students to understand the logic behind the factorial.

2. Problem Situation 1 - The mystery of anagrams: The teacher proposes the following situation: 'Suppose you are trying to solve a puzzle that consists of forming all possible words with the letters 'A', 'B', and 'C'. How many different words can you form?' Then, the teacher asks the students to try to solve the problem without using the factorial.

3. Problem Situation 2 - The logistics of the chessboard: The teacher proposes a second situation: 'Imagine you have a chessboard and need to arrange 8 queens in a way that none of them attacks each other. How many arrangement possibilities do you have?' Again, the teacher asks the students to try to solve the problem without using the factorial.

4. Contextualization: The teacher explains that the factorial is a fundamental tool in Combinatorial Analysis, which is the area of Mathematics that studies the possibilities of arrangement, combination, and permutation of elements. He emphasizes that the factorial is very useful in practical situations, such as solving puzzles, organizing events, programming computers, among others.

5. Introduction to the topic - What is factorial? The teacher then introduces the concept of factorial. He explains that the factorial of a number is the product of that number by all its predecessors down to 1. For example, the factorial of 5 is 5x4x3x2x1 = 120. The teacher uses practical examples, such as solving puzzles and programming computers, to illustrate the importance of the factorial.

6. Curiosity - The largest factorial ever calculated: To spark students' interest, the teacher shares the curiosity that the largest factorial ever calculated is the factorial of 170, which has no less than 309 digits! He explains that calculating this factorial took 29 minutes and was performed by a supercomputer.

With this Introduction, students should be prepared to understand and engage with the topic of factorial in Combinatorial Analysis.

Development (20 - 25 minutes)

1. Activity 1 - 'Factorial Pyramid' (10 - 12 minutes): The teacher divides the class into groups of up to 5 students. Each group receives a sheet of paper, colored pens, and toothpicks. The objective of the activity is to build a pyramid with the number of floors corresponding to the factorial of a number drawn by the teacher. Students must calculate the factorial of the drawn number and then build the pyramid, placing the number of toothpicks corresponding to each floor. For example, if the drawn number is 4, the group must calculate 4! = 4x3x2x1 = 24 and build a pyramid with 24 floors. The group that finishes first must verify the factorial calculation answer and the pyramid construction.

   • Step by step:
     • The teacher draws a number for each group.
     • Students calculate the factorial of the drawn number.
     • Based on the result, students build the pyramid with toothpicks and glue.
     • The group that finishes first must verify if the pyramid construction is correct and if the factorial calculation was done correctly.

2. Activity 2 - 'Real World Problems' (10 - 13 minutes): The teacher presents the following situation to the groups: 'Imagine you are event organizers and need to distribute 10 different prizes to the first 5 participants who arrive. How many different distribution forms exist?' Each group must solve the problem using the factorial concept. The teacher circulates around the room, assisting groups that encounter difficulties.

   • Step by step:
     • The teacher presents the problem and explains that each prize must be given to a different participant.
     • Students calculate the factorial of 10 (number of prizes) and divide by the factorial of 5 (number of participants).
     • Students find the result, which represents the number of different distribution forms.

3. Activity 3 - 'Factorial Challenge' (optional, if time allows) (5 - 7 minutes): The teacher proposes a challenge to the groups that have already finished the previous activities. He presents the following situation: 'Can you calculate the factorial of a large number, like 20, without using a calculator?' The groups that accept the challenge have the remaining time to try to solve the problem. The teacher rewards the group that manages to calculate the factorial correctly.

   • Step by step:
     • The teacher presents the challenge of calculating the factorial of 20 without using a calculator.
     • The groups that accept the challenge have the remaining time to try to solve the problem.
     • The teacher rewards the group that manages to calculate the factorial correctly.

With these practical activities, students will have the opportunity to apply the factorial concept and develop skills in logical reasoning, problem-solving, and teamwork. Additionally, contextualized activities help make learning more meaningful and interesting.

Feedback (8 - 10 minutes)

1. Group discussion (3 - 5 minutes): The teacher gathers all students and promotes a group discussion about the solutions found by each group. He asks a representative from each group to share how they arrived at the solutions to the problems presented in the activities. The teacher also encourages students to ask each other questions, promoting the exchange of ideas and collaboration among groups.

   • Step by step:
     • The teacher gathers all students.
     • He asks a representative from each group to share the solutions to the problems.
     • The teacher encourages students to ask questions and comment on the solutions presented.

2. Connection with theory (2 - 3 minutes): The teacher then makes the connection between the activities carried out and the theory studied. He briefly reviews the concept of factorial and how it was applied to solve the proposed problems. The teacher also highlights the importance of the factorial in Combinatorial Analysis and in practical everyday situations.

   • Step by step:
     • The teacher reviews the concept of factorial and how it was applied in the activities.
     • He highlights the importance of the factorial in Combinatorial Analysis and in practical everyday situations.

3. Individual reflection (2 - 3 minutes): Finally, the teacher proposes that students reflect individually on what they learned in the lesson. He asks the following questions:

   1. 'What was the most important concept you learned today?'
   2. 'What questions have not been answered yet?'

   Students have a minute to think about the questions. Then, they can share their answers with the class if they wish.

   • Step by step:
     • The teacher proposes that students reflect individually on what they learned in the lesson.
     • He asks the questions and gives a minute for students to think about them.
     • Students can share their answers if they want.

With the Feedback, the teacher ensures that students understand the concept of factorial and how it can be applied to problem-solving. Additionally, he identifies possible doubts or difficulties that students may have to plan the next lessons more effectively.

Conclusion (5 - 7 minutes)

1. Summary of key points (2 - 3 minutes): The teacher starts the Conclusion by summarizing the key points covered in the lesson. He reinforces the concept of factorial, its notation, and its applicability in Combinatorial Analysis. The teacher emphasizes the importance of the factorial in solving practical problems and in developing logical thinking.

2. Connection between theory, practice, and applications (1 - 2 minutes): Next, the teacher highlights how the lesson connected the factorial theory with practice through playful and contextualized activities. He reinforces that the factorial is not just an abstract concept but a powerful tool for solving real-world problems. The teacher also mentions the applications of the factorial in various areas, such as computer programming, statistics, and game theory.

3. Additional materials (1 minute): The teacher suggests complementary study materials for students who wish to deepen their understanding of the factorial. He can recommend math books, educational websites, explanatory videos, and online exercises. The teacher also encourages students to explore the topic on their own through research and experimentation.

4. Importance of factorial in everyday life (1 - 2 minutes): Finally, the teacher emphasizes the importance of the factorial in everyday life. He mentions that the factorial is used in various everyday situations, such as solving puzzles, organizing events, programming computers, among others. The teacher concludes the lesson by highlighting that understanding and mastering the factorial are essential not only for Mathematics but also for developing critical thinking and problem-solving skills.

With the Conclusion, students should have a clear and comprehensive view of the factorial topic in Combinatorial Analysis. Additionally, they will be encouraged to continue exploring the subject and applying it in their daily lives.