Context

Mathematics is a science that surrounds us on all sides and is present in our daily lives in subtle and evident ways, from the moment we wake up until the time we prepare to sleep. One of the topics that intrigues students the most is the Change of Base of a numerical system, a concept frequently used in areas such as computer science and engineering. In this project, we will delve into this intriguing theorem, exploring its applications and developing our own practical knowledge.

Our usual counting system is based on base 10, which makes sense, since we have 10 fingers on our hands. However, base 2, or the binary system, is of great importance in computer science, as all data within a computer is stored in this format. Other bases, such as base 8 (octal system) and base 16 (hexadecimal system), are also used in computing and mathematics. It is interesting how we can count and perform operations in various bases, but base 2 holds a special place in our digitized world, and it is precisely this base that will be the focus of our project.

The concept of Base Change is, therefore, an essential topic for any student interested in pursuing a career in STEM (Science, Technology, Engineering, Mathematics). Furthermore, this practical skill has various applications, whether in solving everyday problems or in a deeper understanding of how our digital devices work.

For the development of this project, students can base and deepen their knowledge on the topic using the following resources:

1. Math Is Fun - Binary, Decimal and Hexadecimal Numbers (in English, the use of a translator is suggested)
2. Essential Mathematics - Number Systems and Base Change

In addition, explanatory videos can also be useful to complement the study:

3. Mathematics Rio with Prof Rafael Procopio - How does the BINARY SYSTEM work
4. Physics Show - Number Systems

We hope that this project can open students' eyes to the beauty and usefulness of Mathematics in our daily lives!

Practical Activity

Activity Title: "Decoding Binary Language"

Project Objective

To deepen the understanding of the concept of Base Change (from decimal to binary system), develop collaboration skills among students, and promote the practical application of Mathematics in real-world situations.

Detailed Project Description

Students will be challenged to create a "Human Binary Translator." This project consists of converting messages from our alphabet to binary language and vice versa. Each group will receive a secret message in binary code that must be translated into common language, and they must also create their own message in common language to be converted to binary.

Required Materials

1. Paper and pen
2. Calculator (optional)
3. Conversion tables of characters to binary and vice versa (provided by the teacher)

Detailed Step-by-Step for Activity Execution

1. Each group will receive a secret message in binary code, the conversion table of characters to binary and vice versa, and a blank sheet of paper.

2. The group must use the provided table to decode the secret message, writing the decoded message on the paper.

3. After decoding the message, each student in the group must create a secret message in our common language, convert this message to binary, and record the binary message on the paper.

4. In the end, the groups must take turns to present the messages created (in common language and binary), explain the translation process, and discuss the difficulties encountered and the strategies used to overcome them.

Project Deliverables

Upon completing the practical part of the project, students must prepare a detailed collaborative report containing the following sections:

1. Introduction: The group must contextualize the concept of Base Change and its relevance in the real world, explaining the motivations for the project development and presenting the secret binary message received.

2. Development: The group must explain in detail the process of decoding the secret message and encoding the messages created by the students. This process must be explained from both a theoretical perspective (using Base Change 2) and a practical one (application for message translation). At the end of this section, the group must present the messages in common language and binary.

3. Conclusions: The group must discuss the learnings obtained during the project, highlighting the difficulties encountered, the strategies used to overcome them, and the level of collaboration among group members. Additionally, they must reflect on the application of the Base Change concept in various contexts, including daily life and technology.

4. Bibliography: The group must indicate the sources used for the project construction, including books, web pages, videos, among others.

By the end of the project, students will have not only acquired knowledge about the Base Change of a numerical system but also developed important socio-emotional skills such as time management, communication, collaboration, problem-solving skills, and creative thinking.