Contextualization

Theoretical Introduction

Combinatorics is a branch of mathematics that studies combinations, permutations, arrangements, and groupings of elements within a set. This concept is fundamental not only in mathematics but also in computer science and statistics. For example, if you want to create a basketball team with your friends, how many possible teams can you form? Or in a restaurant, how many different dishes can be made with the options on the menu? In both cases, we use concepts from combinatorics.

The central idea of combination is that the order in which the elements are arranged does not matter, that is, the set {a, b} is the same as the set {b, a}. In this project, we will focus on applying the concept of "combinations", one of the fundamental topics in combinatorics.

In addition to combinations, we will also explore other concepts that are directly related to it, such as the Fundamental Principle of Counting, which determines that if we have two tasks to perform and one of them can be done in n ways and the other in m ways, then there are n * m ways to do both. In addition, we will also explore the concept of Permutation, which is an organization of the elements of a set where order is important.

Contextualization

Combinatorics is present in various areas of our daily lives. When we choose clothes to wear, we are applying the concept of combinations: how many combinations of clothes can we make with the pieces in our wardrobe? Or when we arrange the furniture in our house, we are thinking about different combinations of furniture arrangements.

From a broader perspective, combinatorics is used in various professional fields. Traffic engineers use combinatorics to predict different traffic flow scenarios and optimize the placement of traffic lights, while biologists use it to study the possible combinations of genes and their implications for genetic diversity.

The idea of the project is to show that combinatorics is not just an abstract concept, but a useful tool for solving everyday problems and in various professional fields.

Practical Activity

Activity Title: "Secret Codes and Possible Combinations"

Project Objective

This project aims to apply the knowledge of combination, permutation, and the counting principle in a practical way to solve a real problem: creating and deciphering secret codes. In addition, by carrying out this project, students will develop socio-emotional skills such as teamwork, time management, critical thinking, and problem solving.

Detailed Project Description

In this project, students will be divided into groups of 3 to 5 people and will be challenged to create and decipher secret codes using the knowledge of combination.

The project will have 3 phases: 1) Code Creation, 2) Deciphering Codes, and 3) Formalization and presentation of results.

1. Code Creation: Each group must create a secret code. This code must have the following characteristics:

   • It must consist of 5 digits, which can be numbers from 0 to 9 and/or letters A-Z.
   • It must have at least 3 different characters.
   • It must provide a hint to help decipher the code, which can be a pattern observed in the code or a code formation rule.

2. Deciphering Codes: In this phase, the groups exchange the created codes among themselves, along with the hint provided. Each group receives a code created by another group and must try to decipher it, taking into account the hint provided. The groups must use the knowledge of combinatorics to find the solution.

3. Formalization and presentation of results: The groups must document the entire process in a written report. The report must follow the presentation format described in the "Project Deliverables" section below.

Required Materials

• Notebook
• Pens and pencils
• Calculator

Detailed Step by Step for carrying out the activity

1. Form groups of 3 to 5 students.
2. The groups should study and review the concepts of combinations, permutations, and the counting principle, using the resources suggested in the "Resources for study" section.
3. Each group should create a secret code according to the characteristics described in the "Detailed Project Description" section. Write down this code and the corresponding hint.
4. The groups should exchange the codes among themselves, along with the hint provided.
5. Each group receives a code from another group and must try to decipher it, taking into account the hint provided.
6. Each group must document the entire process, from creating the code to deciphering the code of another group, in a written report.

Project Deliverables

The final product of this project is a written report that documents the entire process of creating and deciphering the secret codes. This report must contain:

1. Introduction: It should portray the contextualization of the project, showing the importance of combinations in mathematics and in real life. It should also describe the objective of the work.

2. Development: The students should explain how they created the code, indicating the methodology used and the hint provided. In addition, they should describe how the process of deciphering the code of another group was, indicating which strategies were used and how the knowledge of combination was applied.

3. Conclusions: The students should conclude the work by summarizing their main points, clarifying the lessons learned and the conclusions drawn about the project. They should discuss how combination was used in practice, and what challenges they encountered in the process.

4. Bibliography: The groups should indicate the sources they used to study and carry out the activities, such as books, websites, videos, etc.

The expectation is that students will work on this project for more than twelve hours, considering the time to study the concepts, create and decipher the codes, and write the report.

The project connects two disciplines - Mathematics and Portuguese Language - because at the same time that students need to apply mathematical knowledge to create and decipher the codes, they also need to use writing skills to write the report, incorporating the correct use of grammar, cohesion, and coherence.