Contextualization

The proportionality constant is a fundamental concept in mathematics that has applications in various everyday situations. It is a number that directly relates two proportional quantities. When one quantity grows or decreases, the other follows the same rhythm, multiplied by the proportionality constant.

To understand the 'Proportionality Constant,' we need to understand what 'Proportionality' is. In simple terms, two values are proportional if the ratio between them is always constant. For example, if each pastry costs R$ 2.00, then the cost of 3 pastries will be R$ 6.00, of 4 pastries will be R$ 8.00, and so on. The ratio between the number of pastries and the cost is always the same, 2, and that is our proportionality constant.

Imagine you are planning a trip and need to understand the car's fuel consumption. Let's say the car travels 12km with 1 liter of gasoline. Thus, this ratio, 12km/liter, is the proportionality constant that allows you to calculate the amount of gasoline needed to travel a certain distance. Now, imagine you are cooking and the recipe calls for 3 cups of flour for every 2 eggs. The proportion between flour and eggs is also a proportionality constant.

The proportionality constant is essential for solving everyday problems, understanding natural phenomena, and developing strategies for decision-making. Understanding it means being able to see and apply mathematical relationships in a wide variety of contexts.

For a deeper dive into the concepts discussed here, you can refer to the book 'Mathematics and its Technologies,' by FTD publisher, where there is a chapter dedicated to proportionality and the proportionality constant. Additionally, the website 'Só Matemática' (https://www.somatematica.com.br/emedio/) has a series of exercises and explanations on the topic. Another option is the YouTube channel 'Matemática Rio with Prof. Rafael Procopio,' which has several explanatory videos on the subject.

Practical Activity: 'The Proportions of Our City'

Project Objective

In this activity, teams of 9th-grade students will explore the use of the proportionality constant in an urban context, analyzing the proportional relationships in the planning and growth of a city. This means not only working with Mathematics but also having a vision of disciplines such as Geography, Arts, and Social Sciences.

The interdisciplinary project aims to:

1. Teach students to apply proportionality concepts in the real world.
2. Encourage teamwork, research, communication, and problem-solving.
3. Develop a deeper understanding of how cities are planned and grow.

Detailed Project Description

Students will be divided into groups of 3 to 5 members. Each group will receive a different amount of cardboard building blocks (representing buildings), a scale-printed map of a part of the city (which they will have to enlarge using the proportionality constant), and the estimated number of inhabitants for the upcoming years.

From there, students will have to plan the city's growth, taking into account the increasing number of inhabitants, the need for green areas, the layout of the buildings, among other factors. They will have to calculate the necessary proportions to correctly distribute the buildings and other areas on their map, so that they fit the predicted population growth.

Additionally, each group should research about their city (or another of their choice), seeking data that can be correlated through the proportionality constant.

Required Materials

• Cardboard blocks of different sizes
• Scale rules
• Printed maps
• Calculators
• Internet access for research

Detailed Step-by-Step

1. Formation of groups and delivery of materials.
2. Presentation of the activity and its requirements.
3. Groups plan how they will develop the activity and divide tasks among members.
4. Students research about the city and collect relevant data.
5. Perform calculations to understand the proportionality constant between buildings, green areas, and population.
6. Plan and execute the construction of the city to scale.
7. Prepare the final project presentation, including the report.
8. Present the project to the class.

Project Deliverables

Groups must present the project of their city with an explanation of how they used the proportionality constant in their planning. Along with this, each group must submit a written document, in the form of a report.

Report: It should follow the format with introduction, development, conclusions, and bibliography used.

• Introduction: Contextualize the importance of the proportionality constant and how it is present in the planning and growth of cities. Explain the project's motivation and objective.

• Development: Explain the process of carrying out the activity, the theory that permeates the proportionality constant and how it was applied in the city's construction. Present the methodology used to organize group work, the data researched and collected, the calculations performed, and the challenges faced.

• Conclusions: Bring the learnings obtained from the activity, the conclusions about city planning, the application of the proportionality constant, and the relevant aspects of the teamwork experience.

• Bibliography: Indicate all sources of information used for the project.

At the end of this activity, students will have experienced a practical application of a proportionality constant and will have developed important skills such as teamwork, problem-solving, communication, and time management.