Contextualization

Mathematics is a subject taught throughout all years of schooling, where gradually the topics become more complex. One of these complex topics is the 'change of base' in numeration. When hearing this term, most students may feel intimidated, but the reality is that this subject is fascinating and very useful for understanding the world around us.

The numeration we use in our daily lives is the decimal system - a counting method that uses 10 as its base. Therefore, we have 10 possible digits: from 0 to 9. But why 10? Why not 8, or 16? In fact, in various occasions and areas of science and technology, we use other base systems. For example, computers operate based on the binary system, where the base is 2 and the possible digits are only 0 and 1.

So, why is studying the change of base so important? One of the reasons is that it allows us to better understand how computers and digital technology, which we use so much in our daily lives, work. Additionally, the change of base helps us to better understand our own decimal system and to recognize similarities and differences with other numeration systems.

Theoretical Introduction

The study of numerical bases allows us to explore and understand the structure of numeration systems. In the decimal system, which is the system we use in our daily lives, the base is 10. However, there are other numerical systems that use different bases. The most well-known is probably the binary system, used in computing, which is base 2. But we also have the octal system (base 8) and the hexadecimal system (base 16), commonly used in computer programming.

But what does this 'base' mean? The base of a numerical system is the amount of different numbers that can be used before we need to add an additional digit. For example, in our decimal system with a base of 10, we have numbers from 0 to 9. When we reach 10, we need to add a new digit to represent that number, in other words, a 'ten'. Similarly, in the binary system, we have only the numbers 0 and 1. When we reach 2, we need to add a new digit to represent that number.

The change of base is the process by which a number expressed in a certain base is rewritten in another base. For this, it is necessary to understand well the structure of the original numerical system and the destination numerical system. This is a fundamental concept in mathematics and computer science.

Practical Activity: 'Journey through Bases'

Project Objective:

The activity aims to familiarize students with the change of numerical bases, specifically between the decimal, binary, octal, and hexadecimal systems. The work will consist of creating a 'guide' or 'map' to transition between these bases, with examples and explanation of the techniques used. Additionally, the project will promote socio-emotional skills, such as time management, teamwork, critical thinking, and proactivity.

Project Description:

The project will be carried out by groups of 3 to 5 students, with a total workload of approximately 12 hours per student. It will be a challenge that will require research, study, group discussions, and creativity.

The 'Travel Guide Between Bases' will be created in the form of a written report, describing the main characteristics of each numerical system (decimal, binary, octal, and hexadecimal), explaining how to convert a number from one base to another, and providing practical examples of these conversions. Additionally, the work should present a discussion on the importance and application of these different numerical systems in our society, especially in the field of computing.

Necessary Materials:

1. Paper, pens, pencils, and erasers.
2. Computer with internet access.
3. Books and reference materials suggested in the project's resources section.
4. Text editing programs (Microsoft Word, Google Docs, etc) for the elaboration of the final report.

Detailed Step-by-Step for Carrying Out the Activity:

1. Research and Study: Start by studying the different numerical systems mentioned. Use the indicated resources and any others you find. Take note of the main characteristics of each system and the techniques for converting numbers between the different bases.

2. Group Discussion: Hold meetings to discuss what you have learned. Each group member can bring different examples and propose problems to be solved by the group.

3. Guide Development: Write, in a clear and detailed manner, how to convert numbers between the different bases. Use practical examples to illustrate each step.

4. Discussion on Applications: Research and discuss where these numerical systems are used in the real world. Why do computer programmers use the hexadecimal system? Why is the binary system so important for computing?

5. Report Development: Write the final report, organizing all the collected and discussed information. The report should include: Introduction, Development (with explanation of theory and techniques), Discussion on practical applications, and Conclusions. Remember to cite all sources of information in the Bibliography.

6. Review and Correction: Review the report, correct any errors, and improve the writing, if necessary. Make sure the report is clear and understandable.

7. Presentation: Prepare a presentation of the report for the class. The presentation should be creative and interactive, allowing classmates to also learn from the group's work.

Project Deliverables:

1. Written Report: The report should be submitted in electronic document format (Word, Google Docs, etc). It should be well-organized and clearly present all the requested items: Introduction, Development, Discussion on practical applications, Conclusions, and Bibliography.

2. Oral Presentation: An oral presentation to the class should be made, lasting 15 to 20 minutes, explaining the main points of the report in a creative and interactive way.