Getting Started

Theoretical Introduction

A First Degree Function is a fundamental topic in Mathematics that serves as a foundation for more advanced studies such as Calculus and Linear Algebra. Initially, we must understand that a function is a correspondence between two numerical sets, where each number in the first set (called the domain) corresponds to a unique number in the second set (called the image). First degree functions are mainly characterized by being linear, meaning that their graphs are straight lines.

First degree functions are represented by an equation in the form y = ax + b, where 'a' and 'b' are real numbers and 'a' cannot be equal to zero. In this equation, 'a' is the angular coefficient (representing the slope of the line relative to the x axis) and 'b' is the linear coefficient (representing the point where the line intercepts the y axis). Therefore, by manipulating these coefficients, it is possible to control the slope and position of the line, respectively.

It is important to emphasize that first degree functions are the basis for the formation of many more complex functions, giving the student the necessary foundation for understanding more advanced topics. Furthermore, the concept of function is extremely relevant to virtually all areas of science, from physics to economics, including biology, geography, engineering, among others.

Relevance and Applications

A First Degree Function has a wide range of practical applications. It can be used to understand direct relationships between variables in various fields, such as physics, economics, biology, and many others. For example, a biologist can use a first degree function to predict the growth of a population of organisms over time. An economist can use it to analyze the relationship between the price of a product and the quantity demanded by consumers.

In addition, the First Degree Function can be applied in our own daily lives. For example, you have probably heard that the distance a car travels is directly proportional to the car's speed and travel time. This is a direct application of the first degree function. Therefore, mastering this concept not only broadens our academic horizons, but also helps us to understand and analyze real-life situations.

Hands-On Activity

Activity Title: "Creating and Analyzing Linear Models: A First Degree Function Adventure"

Project's Aim:

• Encourage collaboration among students in the elaboration of a project.
• Develop a theoretical understanding of first degree functions through practical application.
• Apply the knowledge about first degree functions in an interdisciplinary context, involving Mathematics and Economics.

Project's Detailed Description:

The main purpose of this project is to put into practice the theory and skills acquired about first degree functions in a practical and relevant scenario. Students will gather in groups of 3 to 5 people to create a simplified business model for a fictitious small company. They will have to relate the sales of a specific product (quantity) to its price.

The groups will have to base themselves on a first degree function to model this relationship, considering a reasonable price range and predicting the quantity that would be sold at these prices. Students will have to take into account variables such as production cost, market demand, and competition.

Required Materials:

• Notebook or paper for notes and calculations.
• Computer with spreadsheet software (such as Excel or Google Sheets) to create the model and graph.
• Internet access for research.

Step-by-Step Instructions:

1. Research and Development of the Business Model: Each group must choose a product and research its production costs, market price, competition, and demand. This can be done through online research or by using the personal knowledge of the group's members.
2. Elaboration of the Function: Based on the data collected, the students will create a first degree function that relates the product's price (x) to the quantity sold (y). They will have to justify the choice of the angular and linear coefficients, demonstrating how these values were influenced by the information gathered in the research.
3. Analysis of the Model: Next, the students will analyze the graph of the obtained function, discussing its meaning in relation to the business model. They will need to identify the profitability of the product in different price ranges, which are the most advantageous prices, and at what point the company would start to make a loss.
4. Final Report: Finally, the groups will write a comprehensive report explaining the process of developing and analyzing the model.

Project Deliverables:

• Business Model and Function: The first degree function that represents the sales as a function of the product's price, along with the data and justifications used to arrive at this model.
• Analysis of the Model: Discussion on the interpretation of the function's graph, demonstrating the understanding of the product's profitability in different price ranges.
• Final Report: This document must be written according to the following guidelines:
  • Introduction: The student must contextualize the chosen product and business model, highlighting the relevance of first degree functions in the real world.
  • Development: Here, the student must discuss the theory of first degree functions, justify the choice of the function's coefficients, and detail how the function was used to model the product's sales as a function of its price.
  • Conclusion: The group must summarize the main points of the report, make explicit the lessons learned, and discuss the conclusions drawn from the project.
  • Bibliography: Cite the sources of information used in the research and development of the project.

This project should be carried out in approximately 12 hours per student, distributed among research, model creation, analysis, and report writing. Argumentation, teamwork, time management, problem solving, and critical thinking are skills that will be essential for the completion of the project.