Contextualization

Mathematics is an incredible science that allows us to understand the world in innovative and surprising ways. One of these ways is through the factorial of a number. The factorial is a mathematical operation symbolized by an exclamation point (!) after a number. To calculate the factorial of a number, we multiply that number by all the natural numbers that precede it until reaching one. For example, the factorial of 5 (symbolized by 5!) is 120, because 54321 = 120.

Although it may seem simple, the factorial has profound meanings and mathematical applications that extend beyond simple multiplication. Factorials are essential in calculating combinations and permutations, which are fundamental in statistics. Furthermore, they also play a critical role in combinatorial analysis, graph theory, and even quantum physics!

If you think that the factorial has nothing to do with real life, you are completely mistaken. The applications of factorials in everyday life include, among other things, calculating the various ways things can be organized or combined. For example, if you are trying to figure out how many possible pizza combinations you can make with 5 different ingredients, or how many different ways a set of books can be organized on a shelf, or even calculating probabilities in lottery games, the factorial is extremely useful.

This task will not only provide a deep understanding of the factorial but also challenge you to apply it in various interdisciplinary areas, such as statistics, physics, and computer programming. Additionally, it will develop important skills such as teamwork, time management, and problem-solving.

To start your journey, here are some reliable sources in Portuguese that you can consult to better understand the factorial:

1. Britannica School: Factorial
2. Mathway: Factorial
3. Educamaisbrasil: What is factorial in mathematics?
4. Mundo Educação: Factorial of a number

Practical Activity

Activity Title: The Factorial in Real Life and Virtual

Project Objective:

Groups of 3 to 5 students will explore the application of the factorial in real life and the virtual universe of computer programming to increase understanding of the factorial, its properties, and applications.

Detailed Project Description:

This project is divided into two parts:

1. In the first part, students will explore the application of the factorial in real life. They will identify a situation where the factorial can be applied and will calculate the factorial manually.

2. In the second part, students will venture into the world of computer programming and create a Python program to calculate the factorial of a number.

Required Materials:

• Paper, pens, and calculator for part 1
• Computers with internet access and a Python development environment for part 2

Detailed Step-by-Step for Activity Execution:

Part 1:

1. Form a group of 3 to 5 members.
2. Identify a real-life situation where the factorial can be applied. It can be organizing a bookshelf, calculating combinations in games, or any other situation involving permutations or combinations.
3. See how the factorial can be used to solve this problem.
4. Calculate the factorial manually and record the entire process.

Part 2:

1. Learn the basics of Python through free online resources, such as Codecademy or the official Python documentation. Focus on loops and functions, as they are necessary for this project.
2. Write a Python program that takes a number from the user and returns the factorial of that number.
3. Test the program with various numbers to ensure it is working correctly.

Project Deliverables:

Students will deliver, in addition to the Python program, a written report containing:

• Introduction: Describe the chosen real-life situation where the factorial was applied, the relevance of this concept, and the relationship with programming.
• Development: Explain the theory of the factorial, the manual calculation process of the factorial, the programming learning methodology, and the creation of the Python program. Include the obtained results.
• Conclusions: Summarize what was learned during the project, how the understanding of the factorial was expanded, and the programming skills acquired.
• Bibliography: List all sources used in the project, from factorial study materials to resources for learning Python.

Remember: The report is a fundamental piece to demonstrate what you learned during the activity. Therefore, be clear, be honest, and be detailed. Make sure to review your work to ensure it is complete and professional.