Contextualization

Mathematics is a universal language used to describe and interpret the world around us. Within this extensive field of study, reason is a fundamental concept. A ratio is a quantitative comparison between two quantities showing the relative amount of one of them in relation to the other. It is extensively used in various areas, whether they are physical, statistical, economic, or financial, for example.

By studying ratios, we become better at making informed decisions. We can improve our understanding of how things relate to each other and how one variable affects another. This knowledge is crucial for solving complex problems and plays a vital role in many fields of everyday life, such as economics, engineering, sciences, and even in cooking and art.

To investigate this concept and its relevance, we will adopt a theoretical and practical approach. In addition to traditional study, we will explore the concept of ratio by creating and solving real-world problems involving ratios. This approach will not only help students understand the theory but also learn how to apply it in practice.

Theoretical Introduction

Ratio is a mathematical concept that expresses the proportional relationship between two quantities. It is the division of the antecedent term (first term) by the consequent term (second term). For example, if there are 5 oranges for every 10 apples in a bag, the ratio of oranges to apples is 1:2.

Ratio is a concept that permeates many areas of mathematics. In geometry, for example, it is used to describe similarity between shapes. In statistics, it is employed to analyze and interpret data. In calculus, the ratio of two infinitesimal quantities results in the notion of the derivative, which is fundamental for analysis.

Learning about ratios is not just about memorizing definitions and applying them to problems. It is about understanding how different quantities relate to each other and how we can use this relationship to solve practical problems. Therefore, it is important to study not only the theory but also apply this knowledge in practice.

We recommend the following resources for a deeper understanding of the topic:

1. Book: 'Elementary Mathematics: Ratio and Proportion' - Authors: Luiz Roberto Dante
2. Khan Academy Video Lesson - Ratios
3. Article on Ratios - Mundo Educação

Practical Activity

Activity Title: 'Reason in Real Life: A Case Study'

Project Objective:

This project proposes the development of a real case study that explores the application of the concept of ratio. Students, working in groups of 3 to 5, should choose an environment, phenomenon, or situation in which the ratio is a key component for analysis and understanding. They should then create a detailed report, investigating the use of the ratio concept in the chosen context and solving encountered problems.

Detailed Project Description:

Each group of students will select a relevant topic in which the concept of ratio is applicable and important to understand the problem at hand. We suggest the following topics as inspiration:

• Economics: Inflation Rate, Interest Rate
• Biology: Population Growth Rate
• Cooking: Recipes and ingredient proportions
• Art: Proportions in artworks

Students should conduct in-depth research on the chosen topic, gather data, calculate ratios, and interpret results. The final product will be a detailed report that should connect theory with practice and show how the concept of ratio is useful for understanding and solving real-world problems.

Required Materials:

• Internet access for research;
• Textbooks or reference materials;
• Word processing software (Microsoft Word, Google Docs, etc.) for report preparation.

Detailed Step-by-Step for Activity Execution:

1. Group Formation: Students should divide into groups of 3 to 5 people.

2. Topic Selection: Groups choose a topic of interest that involves the concept of ratio.

3. Theoretical Research: Investigation on how the concept of ratio is applied to the chosen topic, using references from the school library, the internet, or other sources.

4. Data Collection: Groups should collect relevant data for the chosen topic.

5. Analysis and Interpretation: Students should calculate and interpret the ratios extracted from the collected data, relating them to the chosen topic.

6. Report Preparation: Students should prepare a detailed report, following the structure of Introduction, Development, Conclusions, and Bibliography used, as instructed below:

   • Introduction: Presentation of the chosen topic, its relevance, and the application of the ratio concept in that context.
   • Development: Detailed exposition of the theory behind the central theme, explanation of the activity in detail, presentation of the methodology used, and discussion of the results obtained.
   • Conclusions: Recap of the main points, explaining the learnings obtained and the conclusions drawn from the project.
   • Bibliography: Indication of the sources they relied on to work on the project such as books, web pages, videos, etc.

7. Work Presentation: Each group should present their work to the class, sharing their findings and conclusions.

The estimated duration for the execution of this project is 5 to 10 hours per student, to be divided over a month. Each group will have the freedom to manage their time according to the availability and organization of the members.

By the end of the project, students will have acquired theoretical and practical knowledge about the concept of ratio and how it is applied in the real world. Additionally, they will have enhanced socio-emotional skills such as time management, teamwork, problem-solving, and critical thinking.