Contextualization

Mathematics is an abstract discipline, but sometimes we forget how its principles and concepts can be directly applied to the real world. An example of this is permutations, an area of study in mathematics that deals with the different ways of organizing a set of objects. Although it may seem like an abstract topic, it is actually applied in various everyday situations.

Theoretical Introduction

Permutations are an area of study in mathematics focused on calculating the number of possible arrangements of a set of elements, where the order of these elements matters. This is closely related to the calculation of probability, as it allows us to understand how many possible outcomes exist.

To understand the concept, imagine you have three friends and want to invite them to a party. How many different ways are there to invite them? You can invite friend A first, then friend B, and finally friend C. Or, you could invite friend B first, then friend A, and finally friend C. Each of these possibilities is a different permutation.

However, permutations are not limited to these simple scenarios. They can also encompass more complex situations, such as calculating the possible number of genetic sequences of an organism, or the quantity of possible arrangements of a 4-digit numerical password.

Contextualization

Permutations are used in a wide range of applications. In software engineering, for example, they are used to test all possible states of a program. In cryptography, permutations are used to encode and decode messages. In strategy games, permutations can help calculate the number of possible moves. And, of course, in mathematics and statistics, permutations are fundamental to understanding the principle of counting and probability.

Practical Activity: "Cracking Codes: An Introduction to Permutations"

Project Objective

Engage students in mathematics through the study of permutations and their practical applications, specifically in code breaking and password creation.

Project Description

Students will work in groups of 3 to 5 people and, imagining themselves as cryptographers, will create and then try to "crack" codes/passwords based on the principle of permutations. In this way, they will learn about the concept of permutations in a practical and applied manner.

Necessary Materials

1. Paper and pencil
2. Calculator
3. Computer with internet access for research.

Step by Step

Step 1: Research and Study

Students should start by researching and studying the concept of permutations. They should use the provided resources and any other reliable sources they find. Each group member should understand both the theoretical concept of permutations and their practical applications.

Step 2: Creation of Codes/Passwords

Each group should then create a set of 5 different codes or passwords, each with 4 characters. The characters can be letters, numbers, or both. They should write down the created codes.

Step 3: Cracking the Codes

After each group has their codes, they should use the concept of permutations to calculate how many attempts would be necessary to "crack" or guess each of the codes.

Step 4: Reflection

Finally, each group should reflect on what they learned through this activity. They should discuss how permutations affect password security, how the number of attempts needed to crack a code changes with the number of characters, and so on.

Project Delivery

Each group should produce a report containing:

1. Introduction: A brief description of permutations and their application in the real world.

2. Development: A detailed explanation of how they developed their codes, how they calculated the number of attempts to crack the codes, and what they learned in the process.

3. Conclusion: A reflection on the group's findings and their significance. They should discuss what they learned about permutations, counting, and how it applies to password security.

4. Bibliography: A list of all sources used by the group during the project, including books, websites, videos, and so on.

The report should be well-structured, with clarity in ideas and language used. It will be used to assess how well students understood the concept of permutations and the application of mathematics in the real world.