Contextualization

Mathematics is not just an academic discipline learned in schools, but an indispensable tool used in countless real-world applications. Within mathematics, there is a concept called 'factorial', which plays a vital role in many fields, including science, engineering, economics, and computer science. If you have ever had an algebra class, you may recognize the factorial as an operator represented by the symbol '!', used to multiply a series of decreasing numbers down to one. But do we fully understand the relevance and application of this concept beyond the classroom?

The concept of factorial is a crucial part of mathematics, specifically in combinations and permutations, probability, power series, and is also a fundamental part of calculus. Just to illustrate, let's think about organizing a line of people. How many ways are there to arrange 5 people in a line? The answer is 5 factorial, or 5!, which is 5 * 4 * 3 * 2 * 1 = 120. Thus, we have 120 different ways to arrange these people.

Factorial is also an important tool in algorithm analysis in computer science. The runtime of the 'Bubble Sort' sorting algorithm, for example, has a worst-case time complexity of O(n²), where n represents the number of elements to be sorted. Let's assume a computer can process 1,000,000 operations per second, so to sort 10 elements would take approximately 100 seconds and for 15 elements, this would increase to 225 seconds.

Therefore, a detailed understanding of factorial can open doors to solving complex problems and applying them in a variety of real-life scenarios.

Practical Activity: 'Factorial in Practice'

Project Objective:

The objectives of this project are:

1. Understand the theory of factorial and its relevance in the real world.
2. Apply the theory of factorial in solving problems of combinations and permutations.
3. Develop teamwork skills, time management, and complex problem-solving abilities.

Project Description:

This is a one-month project divided into two main parts. In the first part, you will study the theory of factorial, its properties, and how to calculate it. In the second part, you will apply the factorial to solve problems of combinations and permutations.

Required Materials:

1. Whiteboard or paper sheets and pens.
2. Reference books and online learning resources.
3. Calculators (preferable, but not necessary).
4. Computer with Internet access.

Step by Step:

1. Study of Factorial Theory: In this first step, each group should study the factorial theory, become familiar with its properties, and learn how to calculate the factorial of a number. It is recommended to use the indicated books and online resources for studying.

2. Factorial Exercises: After understanding the theory, the next phase is to practice it. Each group should solve a series of exercises involving factorial calculation.

3. Applying Factorial in Combinations and Permutations Problems: Here, each group should apply the factorial theory in problems of combinations and permutations. This part is crucial as it allows students to understand the practical application of factorial.

4. Report Elaboration: Each group will be responsible for creating a detailed report on the project. This report should include an introduction (contextualization and relevance of factorial), development (factorial theory, how to calculate and its properties, problems solved), conclusions (reflections, learnings, and key points), and bibliography (sources used during the study).

Project Deliverables:

At the end of the project, each group must deliver:

1. Final Report: Detailed document explaining the factorial theory, how to calculate and its properties, application of factorial in problems of combinations and permutations, and their conclusions.

2. Presentation: Each group will make a brief oral presentation of the project to the class, sharing their learnings and experiences.

The report and presentation should reflect the teamwork of the students, their understanding of the factorial theory, their ability to apply the theory in problem-solving, and socio-emotional skills developed during the project.