Contextualization

In this project, we will deal with an important mathematical concept: permutation with repetition.

Permutations are a branch of Mathematics that is dedicated to the study of the arrangement of elements of a set in a certain order. In particular, permutation with repetition occurs when one or more elements of a set are repeated, as in the word BANANA where the letter 'A' is repeated three times and 'N' twice. Correctly manipulating these permutations is crucial in various fields, from combinatorial analysis in mathematics to genetics in biology.

In combinatorial analysis, permutations with repetition are essential for solving complex questions about the number of distinct possibilities to organize elements of a set, even when they are repeated. For example, when we want to calculate how many password combinations we can have with a certain number of digits, we resort to this concept.

In the field of genetics, permutation with repetition is fundamental in the analysis of genetic sequences. For example, the human DNA is composed of 3 billion base pairs whose combinations, despite being only four possible (A, T, C, and G), can repeat along the chain, making the study of combinations a very complex task without the knowledge of permutation with repetition.

In everyday life, permutation with repetition can be found in tasks as simple as creating a password for an email or setting up a code to open a padlock. For these reasons, the ability to understand and apply permutations with repetition is invaluable in many academic, professional, and everyday scenarios.

Practical Activity

Activity Title: "Unraveling Permutations: A Game of Words and Numbers"

Project Objective

This project aims to put into practice the knowledge acquired about permutations with repetition, developing technical skills for problem-solving with permutations and socio-emotional skills such as time management, communication, creative thinking, and proactivity.

Detailed Project Description

In groups of three to five students, you will create a "Permutation with Repetition Game." The game will consist of a list of words and expressions (3 to 10 characters), where players must calculate the number of possible permutations considering the repetitions of characters.

Additionally, you will create a table that indicates the score for each word or expression in the game, based on the difficulty (number of possible permutations). Finally, you will organize a "battle" where each group will present their game to the others, and everyone will have to solve the proposed challenges.

Required Materials

• Paper and pen for notes and calculations
• Computer with internet access for research

Step by Step

1. Form a group of 3 to 5 students.
2. Research and study permutation with repetition using the recommended resources.
3. Create a list of words and expressions (3 to 10 characters) that will be used in the game. Make sure to use words with character repetitions.
4. Calculate the number of possible permutations for each word or expression and assign a score (a tip is to assign a higher score to words with a higher number of permutations).
5. Create a table with the words/expressions, the number of permutations, and the corresponding score.
6. Organize a "battle" where each group will present their game to the others, and everyone will have to solve the proposed challenges.
7. At the end of the "battle," have a discussion in the classroom about the difficulties, strategies used, and interesting facts learned.
8. Conclude the project with a written report, following the guidelines below.

Project Deliverables

After completing the practical activities, the groups must produce a written report containing:

1. Introduction: Provide context on the theme of permutation with repetition, discussing its relevance and real-world applications, and explain the project's objective.
2. Development: Explain the theory of permutations with repetition, describe in detail the activity carried out, the methodology used, and present the results. Include in this section a detailed description of the game created, including the chosen words/expressions, the calculations performed, the scoring table, and how the "battle" experience was.
3. Conclusion: Reflect on the learnings obtained, revisit the main points of the project and the conclusions drawn. Also, report on the socio-emotional skills developed.
4. Bibliography: Indicate the sources used to work on the project.

The final report must be submitted one week after the project introduction.