Contextualization

Theoretical Introduction

Relationships and equations of quantities are a fundamental theme in Mathematics and are present in many aspects of our daily lives, ranging from simple calculations of proportions in culinary recipes to complex calculations in Physics and Engineering.

To understand relationships and equations of quantities, we need to understand what quantities are. Quantities are everything that can be measured. For example, the distance between two points, the amount of matter in a substance, the energy consumed by a device, among others.

Quantities can be directly proportional, inversely proportional, or not proportional. Two quantities are directly proportional when an increase in one results in an increase in the other, and a decrease in one results in a decrease in the other. On the other hand, two quantities are inversely proportional when an increase in one results in a decrease in the other. Non-proportional quantities are those that do not have a direct or inverse proportional relationship between them.

Contextualization

Relationships and equations of quantities may seem like concepts far from our reality, but that's not quite the case. They are present in various aspects of our daily lives, from preparing a recipe to deciding whether to use a car or public transportation based on the cost of fuel or the ticket, respectively.

For example, if the price of the transportation ticket increases but the fuel price remains the same, it may be more economical to use the car. In other words, the choice of transportation mode can be influenced by the inversely proportional relationship between the cost of the transportation ticket and the cost of fuel.

Therefore, mastering the concept of relationships and equations of quantities is essential for making better and more informed decisions in our daily lives.

Practical Activity

Activity Title: "Exploring Relationships of Quantities in the Real World"

Project Objective

The objective of this project is to use the knowledge of relationships and equations of quantities to analyze and understand everyday situations, reinforcing the idea that Mathematics is present in various aspects of our reality.

Detailed Project Description

Each group must select an everyday situation involving directly proportional, inversely proportional, or non-proportional quantities. For example, analyzing the relationship between the amount of ingredients in a recipe and the number of servings it yields, the relationship between the usage time of an appliance and the consumed electrical energy, the relationship between the distance traveled by a vehicle and the amount of fuel consumed, among others.

After choosing the situation, students must collect data and information to analyze and identify the nature of the relationship between the quantities involved, whether they are directly proportional, inversely proportional, or non-proportional. In addition, students must represent this relationship through algebraic sentences and on the Cartesian plane.

Finally, students must prepare a report detailing the chosen situation, data collection and information, the analysis performed, and the conclusions drawn.

Required Materials

• Notebook or sheets of paper for notes and calculations
• Pens and pencils
• Ruler
• Calculator
• Internet access for research and data collection

Detailed Step-by-Step for Activity Execution

1. Firstly, students must form groups of 3 to 5 members.

2. Next, each group must choose an everyday situation involving proportional or non-proportional quantities. This situation must be relevant and interesting to the group.

3. Once the situation is chosen, students must seek information and collect data about the quantities involved in that situation.

4. With the data in hand, students must analyze the relationship between the quantities involved: are they directly proportional, inversely proportional, or non-proportional?

5. Students must represent this relationship through algebraic sentences and graphical representations on the Cartesian plane.

6. Students must prepare a report detailing the entire work process: the choice of the situation, data collection, analysis of quantities, representation on the Cartesian plane, and conclusions drawn.

7. Finally, the report must include a section with the bibliographic references used by the students.

8. The project must be delivered in one week, with each student spending two to four hours on project execution.

Project Deliverables

Project deliverables include:

• Notes and calculations made for the analysis of quantities.
• Graphical representations on the Cartesian plane.
• Detailed report containing Introduction, Development, Conclusions, and Bibliography:
  • Introduction: The student must contextualize the chosen situation, its relevance and application in the real world, as well as the objective of this project.
  • Development: The student must explain the theory behind the central theme of the project, detail the activity, indicate the methodology used, and finally present and discuss the results obtained.
  • Conclusions: The student must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
  • Bibliography: The student must indicate the sources they relied on to work on the project, such as books, web pages, videos, etc.

Students must ensure that the report complements the practical work done, serving as a way to document and record the learning process, as well as helping to develop writing and communication skills.