Objectives

(5 - 7 minutes)

1. Introduce the concept of Factorial: Students should be able to understand what the factorial of a number is and how it is calculated.

   • Secondary objectives: Students should be able to recognise the factorial notation (n!) and understand its meaning.

2. Apply the concept of Factorial in Practical Problems: Students should be able to use the concept of factorial to solve practical problems, such as the permutation of elements in a set.

3. Identify the properties of Factorial: Students should be able to identify and apply the properties of factorial, such as the decomposition property of factorial and the factorial property of a prime number.

   • Secondary objectives: Students should be able to explain why the factorial property of a prime number is true.

Complementary Objectives:

• Promote critical thinking skills: In addition to learning the concept and properties of factorial, students should be encouraged to think critically about how and when to apply these concepts in practical problems.

• Stimulate group collaboration: Through group activities, students should be encouraged to collaborate and discuss their ideas and solutions, promoting active learning and enhancing their communication skills.

Introduction

(10 - 15 minutes)

1. Review of Previous Content: The teacher should start the class by briefly reviewing the concepts of factorisation and permutation, which are fundamental to understanding factorial. This can be done through a quick quiz or interactive activity to check students' level of understanding. (3 - 5 minutes)

2. Problem Situations: Next, the teacher should present two problem situations that will be solved throughout the class, but which will serve as a starting point for the introduction of the concept of factorial:

   • How many different ways can a group of 5 people arrange themselves to take a photo?
   • How many different anagrams can be formed with the letters of the word "MATHEMATICS"? (3 - 5 minutes)

3. Contextualisation: The teacher should explain that factorial is a fundamental mathematical concept in various areas of knowledge, such as statistics, game theory, and computer science. In addition, examples of everyday situations where factorial can be applied can be cited, such as in the analysis of possible combinations on a restaurant menu or in determining the number of possible sequences in a card game. (2 - 3 minutes)

4. Introduction to the Topic: To gain the attention of students, the teacher can present some curiosities or interesting applications of factorial:

   • The factorial of a number n, represented by n!, is equal to the product of all natural numbers less than or equal to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
   • The factorial of 0, 0!, is always equal to 1. This is because, by definition, the product of no numbers is 1.
   • Factorial is used in statistics to calculate the number of ways an event can occur in a sample. (2 - 3 minutes)

Development

(20 - 25 minutes)

1. Activity "Factorial in the Cube" (10 - 12 minutes):

   • The teacher should divide the class into groups of up to 5 students and provide each group with a box of wooden cubes of the same size. Each cube should have one face marked with a number from 1 to 6.
   • The students' task is to create all possible arrangements of the cubes so that the sum of the numbers on each face is equal to a factorial number previously drawn by the teacher (for example, 6! = 720).
   • Students should record all possible arrangements and, at the end of the activity, present the solution to the class.
   • This activity allows students to visualise the concept of factorial in a fun and interactive way, as well as reinforcing the permutation skill, which is fundamental to calculating the factorial.

2. Group Discussion (5 - 7 minutes):

   • After the activity, the teacher should promote a group discussion, where each team will present their solution and explain the reasoning used.
   • The teacher should encourage students to identify patterns and regularities in the arrangements and relate them to the concept of factorial.
   • This discussion will help students to consolidate their understanding of factorial and develop communication and critical thinking skills.

3. Activity "Factorial Decomposition" (5 - 6 minutes):

   • Next, the teacher should distribute an activity sheet containing a series of integers.
   • The students' task is to decompose each number into prime factors and then calculate the factorial of each of the factors.
   • At the end of the activity, the students should add up all the factorials to obtain the factorial of the original number.
   • This activity aims to deepen students' understanding of the decomposition property of factorial and the relationship between factorial and factorisation.

4. Activity "Factorial of a Prime" (5 - 6 minutes):

   • Finally, the teacher should propose a final activity, where the students must calculate the factorial of a prime number (for example, 7) and of a non-prime number (for example, 8).
   • The students must compare the results and try to identify possible patterns.
   • This activity aims to deepen students' understanding of the factorial property of a prime number and the importance of prime numbers in mathematics.

Throughout all the activities, the teacher should circulate around the room, monitoring students' progress, clarifying doubts, and providing feedback. In addition, the teacher should encourage the participation of all students and promote discussion and exchange of ideas between the groups.

Review

(8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should gather all the students and promote a group discussion about the solutions or conclusions found by each team during the activities.
   • Each group should be invited to share their main findings, challenges, and strategies used.
   • The teacher should take advantage of this discussion to reinforce the main concepts of factorial, clarify possible doubts, and highlight points of attention.

2. Connection with Theory (2 - 3 minutes):

   • After the discussion, the teacher should go back to the theoretical concepts covered in class and show how they were applied and reinforced during the activities.
   • For example, the teacher can highlight how the "Factorial in the Cube" activity illustrates the concept of permutation and how the "Factorial Decomposition" activity reinforces the decomposition property of factorial.
   • The objective of this moment is to help students make the connection between theory and practice, reinforcing the understanding of the concepts.

3. Individual Reflection (3 - 4 minutes):

   • The teacher should propose that students reflect individually on what they have learned in class.
   • To do this, the teacher can ask questions such as: "What was the most important concept you learned today?" and "What questions have not yet been answered?".
   • Students should have a minute to think about their answers and then some volunteers can be invited to share their reflections with the class.
   • The objective of this activity is to encourage students to take responsibility for their own learning, promote metacognition, and identify possible gaps in students' understanding, which can be addressed in future classes.

4. Teacher Feedback (1 - 2 minutes):

   • Finally, the teacher should provide general feedback on the class's performance, highlighting the strengths and areas that need improvement.
   • The teacher should also reinforce the importance of the concept of factorial in the context of mathematics and other disciplines, and encourage students to continue practicing and exploring the topic outside the classroom.

This Review is a crucial part of the lesson plan, as it allows the teacher to assess students' progress, identify possible difficulties, and adjust the teaching accordingly. In addition, it helps students to consolidate their learning, develop critical thinking skills, and become more self-aware as learners.

Conclusion

(5 - 7 minutes)

1. Content Summary (2 - 3 minutes):

   • The teacher should begin the Conclusion by recalling the main points discussed during the class. This includes the concept of factorial, its notation (n!), the calculation of factorial, the permutation of elements in a set, the properties of factorial (decomposition factorial and factorial of a prime number), and the application of these concepts in practical problems.
   • This summary serves to consolidate the knowledge acquired by the students and reinforce the most important concepts.

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

   • The teacher should then explain how the class connected theory, practice, and applications.
   • For example, the teacher can highlight how practical activities, such as "Factorial in the Cube", helped to illustrate the theory of factorial and the property of permutation.
   • The teacher should also emphasise how factorial is a useful tool for solving practical problems, such as determining the number of different ways a group of people can arrange themselves to take a photo.

3. Supplementary Materials (1 - 2 minutes):

   • The teacher should suggest some additional study materials for students who wish to deepen their understanding of factorial.
   • This could include mathematics books, educational websites, online videos, interactive games, and maths learning apps.
   • For example, the teacher could indicate a video explaining the factorial of a prime number or an online game that challenges students to calculate the factorial of various numbers in a limited time.

4. Importance of Factorial (1 minute):

   • Finally, the teacher should highlight the importance of factorial in the context of mathematics and other disciplines.
   • For example, the teacher could explain how factorial is used in statistics to calculate the probability of an event occurring, or how it is applied in computer science to analyse the complexity of algorithms.
   • In addition, the teacher can highlight that factorial is a valuable tool for developing essential mathematical skills, such as the ability to think logically, solve problems, and understand numerical relationships.

The Conclusion is a fundamental step in the lesson plan, as it allows the teacher to consolidate the knowledge acquired by the students, reinforce the most important concepts, and establish the basis for future learning. In addition, it provides students with resources to continue learning autonomously and concrete applications for what they have learned.