Contextualization

The theory of permutations is an area of mathematics that studies the different ways of organizing objects in a fixed sequence. Permutation is one of the simplest ways to organize objects and is an example of a counting problem.

In mathematics, a permutation of a set is, roughly speaking, one of the possible orderings of its elements. For example, the six permutations of the set {1,2,3} are: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). The study of permutations of a finite set is central in the field of combinatorics.

Permutation has practical applications in various fields, such as software engineering, where it is used to test all possible combinations of a system. In financial mathematics, it is used to calculate the different ways to invest money, and in biology, it is used to understand genetic diversity.

Permutations are used in everyday life in ways you may not even realize. For example, they are used when unlocking your phone, when you need to enter a specific sequence of numbers or patterns. Permutations are also used to create and break codes and passwords.

The theory of permutations is an important part of the mathematics curriculum and is fundamental to a deeper understanding of many mathematical concepts. This project will not only allow you to understand the theoretical concepts behind permutations but also apply this knowledge in a practical project.

Practical Activity

Activity Title: "Permutations in the Real World"

Project Objective:

This project aims to lead students to understand in a practical and creative way the theory of permutations and their applications in the real world.

Project Description:

In this project, students in groups of 3 to 5 will create a real-life representation of situations involving permutations. Students will be encouraged to explore the various ways in which permutations can be applied, such as: creating a new code to open a safe, planning a birthday party with different guest combinations, or even creating a new snack menu for a snack bar with ingredient combinations.

Students should also research how permutations are applied in two different areas of study other than exact sciences, such as biology and engineering, for example, and include this in their presentation.

The project will culminate in the presentation of the findings, followed by a written report detailing the project development process, discoveries, and conclusions reached.

Required Materials:

1. Computers or laptops with internet access for research.
2. Pencils and paper for notes.
3. Presentation software (e.g., Microsoft PowerPoint, Google Slides).
4. Camera to record the presentation (optional).

Step by Step:

Part 1: Research and Planning

1. Form groups of 3-5 students.
2. Choose a real-life situation involving permutations.
3. Research permutations in two other disciplines besides mathematics. For example, in biology and engineering.
4. Develop a plan on how you will explore the concept of permutations in this situation.

Part 2: Development

1. Develop your chosen situation, making permutation calculations if necessary.
2. Prepare a detailed presentation demonstrating the use of permutations in the chosen situation.
3. Record a video of your presentation (optional).

Part 3: Documentation and Presentation

1. Write a detailed report of your project following the structure: Introduction, Development, Conclusions, and Bibliography.
2. Schedule a date to present your results to the class.

Project Deliverables:

Students must deliver the following items at the end of the project:

1. A detailed project presentation, through slides or video.
2. A written report following the structure: Introduction, Development, Conclusions, and Bibliography.

In the report, in the Introduction section, students should contextualize the choice of their situation, its relevance, and real-world application. In Development, students are expected to discuss the theory of permutations, explaining in detail the activity, the methodology used, and presenting and discussing the results. In Conclusions, students should recap their main points, discuss the learnings obtained, and draw conclusions about the project. Finally, in Bibliography, students should indicate the sources they used to support their work.