Contextualization

Mathematics is a science full of applicabilities that transcend the limits of the classroom and are present in our daily lives in ways we may not always perceive. The ability to solve practical problems using mathematical concepts is one of the most valuable skills we can acquire. Within this spectrum, the compound rule of three is an exceptionally powerful tool that, once understood and mastered, provides students with the ability to make predictions, solve challenges, and better understand the world around them.

Our focus in this project is to explore the concept of compound rule of three and how it can be applied to solve problems involving multiple quantities directly or inversely proportional to each other. This statistical technique is used to simplify and solve problems that may seem complex at first glance, being essential in fields ranging from economics to engineering, passing through biology, cooking, and many others.

Importance of Compound Rule of Three in the Real World

In the real world, we rarely encounter situations involving direct and simple relationships between two variables. More often, we face situations where multiple variables interact in a more complex manner. The ability to use the compound rule of three to understand these relationships and make precise calculations is essential in various professional contexts, whether for an urban planner estimating the impact of a new public transportation system on traffic reduction or a chef adapting recipes for different quantities of servings.

Furthermore, the compound rule of three is a fundamental skill at the core of logical reasoning to make informed decisions in daily life. Whether calculating the amount of paint needed to paint a house or adjusting the quantity of ingredients in a recipe for a specific number of guests, the compound rule of three emerges as a daily tool for efficiency and practicality.

Reliable Resources for In-Depth Study

To deepen their knowledge and prepare for this project, we suggest the following resources:

• Khan Academy (link) - Offers a wide range of video lessons covering different mathematical topics, including proportions, ratios, and the rule of three.
• Brasil Escola (link) - Presents articles that explain clearly and accessibly the concept of compound rule of three, along with practical examples.
• Só Matemática (link) - A website with detailed lessons on various mathematical concepts, including the compound rule of three, with application exercises.
• Book 'Fundamentos de Matemática Elementar' - Vol. 1 - Conjuntos e Funções: A renowned collection, available in school libraries, that provides a solid foundation on the mathematical fundamentals relevant to high school education.

These resources serve as a starting point for study and preparation for the activities, as well as providing additional context and examples that can be discussed in a group.

Practical Activity

Activity Title

'The Great Challenge of Proportionality: A Journey through the Compound Rule of Three'

Project Objective

The objective of this project is to provide 2nd-year high school students with the understanding and application of the compound rule of three in real and complex problem situations, developing technical and socio-emotional skills through collaborative work.

Detailed Project Description

In this project, students will be challenged to create and solve problems involving multiple proportional quantities, in a kind of 'mathematical treasure hunt.' They will use knowledge of direct and inverse proportion to interpret and correlate data from fictional scenarios, yet inspired by the real world.

The student groups, consisting of 3 to 5 members, will need to collaborate to design a set of interconnected mathematical puzzles that, when solved, will lead to a final 'discovery.' Each puzzle will be based on a situation that requires the application of the compound rule of three. In the end, they must document the process and present their solutions in a comprehensive report.

Required Materials

• Internet access for research and consultation of additional resources
• Computers or tablets for typing and formatting the final report
• Paper, pencil, eraser, and ruler for sketching graphs and tables
• Calculator to facilitate calculations

Detailed Activity Steps

1. Group Formation and Initial Research (1-2 hours): Each group should consist of 3 to 5 students. Initially, they should dedicate themselves to studying and discussing the recommended resources to understand the theory of the compound rule of three.

2. Theme Definition and Problem Creation (2-3 hours): Each group must choose three real-world themes (such as cooking, economics, engineering, etc.) and develop a problem related to each theme involving the compound rule of three. The complexity should be sufficient to require a calculation that includes at least three proportional quantities.

3. Problem Solving (2-3 hours): After creating the problems, the groups should exchange them for resolution. Each group will solve the problems created by another group and verify the accuracy of the proposed solutions.

4. Compilation of Results and Analysis (1-2 hours): Students must compile the solved problems, showing the calculation steps and commenting on the difficulties encountered and the strategies used to overcome them.

5. Report Writing (2-3 hours): Each group must write a detailed project report, which will include the creation of the problems, the resolution methodology, and reflections on the activities carried out.

Project Duration

The estimated duration for project execution is 5 to 10 hours per student, with the deadline set at one month from the project start date.

Project Deliverables

The project culminates in the delivery of a written document, which must be structured as follows:

1. Introduction (1-2 pages): It should contextualize the compound rule of three, describe its relevance and practical application, as well as the project's objective.

2. Development (4-6 pages): This section should detail the theoretical content of the compound rule of three, present the problems created and solved, and the methodology used. The development should also discuss the results obtained and the difficulties faced during the resolutions.

3. Conclusions (1-2 pages): The groups will summarize the main points of the work, highlighting the learnings, skills acquired, and personal reflections on the experience.

4. Bibliography (1 page): Indication of all sources consulted during the project - books, articles, websites, videos, among others.

The report must be formatted according to ABNT standards and submitted in digital form through the learning management system used by the school (e.g., Google Classroom, Moodle, etc.). The evaluation will take into account not only mathematical accuracy but also the clarity of communication, the originality of the problems, teamwork, and students' critical reflection.