Objectives (5 - 7 minutes)

1. Understand the concept of factorial and its application in mathematical calculations, focusing on the idea of permutation.

2. Learn to calculate factorials of integers, developing calculation skills and logical reasoning.

3. Solve practical problems involving the calculation of factorial, in order to apply the concept effectively and in a contextualized manner.

Secondary Objectives:

• Stimulate analytical thinking and problem-solving, encouraging students to find different ways to apply the concept of factorial.

• Promote interaction in the classroom through group work to solve problems involving the calculation of factorial.

• Develop communication and argumentation skills through discussion and presentation of solutions found.

Introduction (10 - 15 minutes)

1. Review of previous contents:

   • The teacher starts the lesson by briefly reviewing the concepts of multiplication and permutation, which are fundamental for understanding the factorial.
   • Simple and practical examples can be used to illustrate these concepts, such as arranging chairs in a room or organizing books on a shelf.

2. Problem situations:

   • The teacher proposes two problem situations to arouse students' interest and contextualize the theme:
     1. 'How many different ways can we arrange the letters in the word 'MATHEMATICS'?'
     2. 'Suppose a football team has 11 players. How many different ways can we choose 7 players to form the starting lineup?'

3. Contextualization:

   • The teacher explains that the factorial is a concept widely used in probability studies, statistics, and combinatorial analysis, being essential for solving various mathematical problems.
   • In addition, it can be mentioned that the factorial is applied in various areas of knowledge, such as in computing, quantum physics, and even in music, in the theory of tones.

4. Introduction to the topic:

   • The teacher presents the topic of the lesson: factorial. Explains that the factorial of a number is the product of that number by all its predecessors down to 1.
   • To arouse students' interest, the teacher can mention the curiosity that the factorial of 0 is 1, which may seem strange at first, but has a logical explanation.
   • Another curiosity that can be mentioned is the notation of the factorial, which is represented with an exclamation point (!) after the number, as in 5!, which is read as 'factorial of 5'.

With this Introduction, students will have a solid foundation to start studying the factorial, understanding the importance and application of this concept.

Development (20 - 25 minutes)

1. Factorial Theory (10 - 12 minutes)

   • The teacher begins the explanation of the Factorial concept, explaining that the factorial of a non-negative number n, denoted by n!, is the product of all positive integers less than or equal to n.
   • Next, the teacher reinforces the importance of the order of factors in multiplication, and how this order affects the result of the factorial.
   • The teacher can illustrate this idea with a practical example, such as the factorial of 5, which is the product of 5 x 4 x 3 x 2 x 1. If the order of these factors were changed, the result would be completely different. For example, if it were 1 x 5 x 3 x 2 x 4, the result would be 120, which is different from the factorial of 5, which is 120.
   • Next, the teacher should emphasize that the factorial of 0 is equal to 1, which can be a confusing concept for students. One way to explain this is through the idea that 0! represents the number of ways to arrange no objects, which is only one way, that is, 1.
   • Finally, the teacher should explain the notation of the factorial, which is an exclamation point (!) after the number, and how it should be read, for example, 5! is read as 'factorial of 5'.

2. Calculation of Factorial (5 - 7 minutes)

   • The teacher should explain how to calculate the factorial of a number. For this, the teacher can present a simple algorithm:
     1. Start with the number for which the factorial is to be calculated (n).
     2. Multiply n by the number immediately smaller than it.
     3. Repeat step 2 until reaching 1.
   • The teacher should demonstrate this algorithm with a practical example, such as calculating the factorial of 5:
     • 5! = 5 x 4 x 3 x 2 x 1 = 120
   • The teacher can then ask students to calculate the factorial of other numbers, following this algorithm.

3. Application of Factorial (5 - 6 minutes)

   • The teacher should show how the factorial is applied in practical situations, such as in combinatorial analysis and probability problems.
   • For example, the teacher can use a practical problem, such as the formation of the starting lineup of a football team, which was presented in the Introduction. The teacher should explain how the factorial is used to calculate the number of different ways to choose the players.
   • The teacher can then propose other practical problems and ask students to apply the concept of factorial to solve these problems.

At the end of this stage, students will have acquired the necessary knowledge to understand and calculate the factorial of a number, as well as how to apply this concept in practical situations.

Return (8 - 10 minutes)

1. Concepts Review (3 - 4 minutes)

   • The teacher should start the Return phase by reviewing the main concepts covered in the lesson, reinforcing the definition of factorial, the algorithm for calculating the factorial, and the application of the factorial in practical situations.
   • The teacher can use the whiteboard or a slide presentation to summarize the concepts, highlighting the most important points.
   • During this review, the teacher should check students' understanding by asking questions and encouraging active participation from the class.

2. Connection to Practice (2 - 3 minutes)

   • The teacher should then make a connection between the theory presented and its application in practice, recalling the practical problems that were proposed during the lesson and how the concept of factorial was used to solve them.
   • The teacher can also mention other everyday situations or other disciplines where the factorial can be applied, reinforcing the importance of this concept.
   • The objective of this stage is to show students that mathematics is not just a theoretical and abstract discipline, but has practical and useful applications.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should then propose that students reflect on what they learned during the lesson.
   • The teacher can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • The teacher should give students time to think and write down their answers.
   • This reflection is important for students to internalize what they have learned and identify possible doubts or difficulties they still have.

4. Sharing (1 - 2 minutes)

   • Finally, the teacher should invite some students to share their reflections.
   • The teacher should listen attentively to the students' contributions, reinforcing the positive points and clarifying possible doubts.
   • This stage is important for the teacher to have immediate feedback on what the students have learned and what difficulties still need to be overcome.

At the end of this stage, students will have had the opportunity to consolidate what they have learned, reflect on their learning process, and clarify possible doubts. In addition, the teacher will have obtained valuable feedback on the effectiveness of the lesson and will be able to adapt the next lessons according to the students' needs.

Conclusion (5 - 7 minutes)

1. Summary of Contents (1 - 2 minutes)

   • The teacher should start the Conclusion of the lesson by summarizing the main contents covered.
   • Should reinforce the definition of factorial, the algorithm for calculating the factorial, and the application of the factorial in practical situations, such as in combinatorial analysis and probability problems.
   • Can use the whiteboard or a slide presentation to visualize the most important concepts.

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

   • The teacher should highlight how the lesson connected the theory of the factorial with its application in practice.
   • Should recall the practical problems that were proposed during the lesson and how the concept of factorial was used to solve them, showing the importance and usefulness of this concept.
   • Can also mention other everyday situations or other disciplines where the factorial can be applied, reinforcing the relevance of this content.

3. Extra Materials (1 - 2 minutes)

   • The teacher should suggest extra materials for students who wish to deepen their understanding of the factorial.
   • Can indicate books, websites, videos, or math apps that offer additional explanations and exercises on the factorial.
   • These materials can be made available on the school's virtual learning environment or indicated to students on a printed list.

4. Importance of the Factorial (1 - 2 minutes)

   • Finally, the teacher should emphasize the importance of the factorial for daily life and other disciplines, highlighting that, despite seeming like an abstract concept, the factorial has practical and useful applications.
   • Can reinforce its importance for combinatorial analysis and probability, but also mention other areas of knowledge that use the factorial, such as computing, quantum physics, and music.
   • For example, can mention that the factorial is used to calculate the number of different ways to organize elements in a data structure in a computer program, to calculate the number of possible states in a quantum system, or to calculate the number of possible arrangements of musical notes in a melody.

With this Conclusion, students will have had the opportunity to review the main contents of the lesson, reflect on what they have learned, and understand the importance of the factorial. In addition, they will have received guidance on how they can continue learning about this subject.