Introduction

Linear functions are symbolized by the expression y = ax + b, where a and b represent the coefficients of the function, with 'a' being the angular coefficient and 'b' the linear coefficient. The linear function is a mathematical tool that can express a wide variety of real-world phenomena in a quantitative language, thus becoming a very powerful resource.

The coefficient 'a' determines the angle at which the line intersects the y-axis, while the coefficient 'b' is the point where the line intersects the y-axis. Understanding the linear function is fundamental because it is the basis for many other concepts in mathematics, such as geometry, calculus, and advanced algebra.

Linear functions are an integral part of mathematics and are used in a variety of contexts, from simple equations to complex mathematical theories. It is also one of the first advanced mathematical concepts that students learn and, therefore, an important topic in mathematical education.

Contextualization

In everyday life, we come across situations that involve direct proportionality, such as the distance traveled by a car in relation to the travel time or the amount of hours worked in relation to the salary received at the end of the month. These situations, among many others, can be mathematically represented by linear functions, and learning to interpret and use them correctly can help us make more informed and efficient decisions.

Consider, for example, a car trip. The distance you can travel depends on the speed you set and the time you have available. If you know your speed and travel time, you can use a linear function to calculate how far you will get. This can be very useful for planning trips or managing your time.

Practical Activity

Activity Title:

"Proportionalities in Professions: An analysis via linear functions"

Project Objective:

1. Understand the concept and application of linear functions.
2. Apply linear functions to solve real-world problems.
3. Develop teamwork and time management skills.

Detailed Project Description:

Students will be divided into groups of 3-5 people. Each group will have to choose two different professions and analyze them based on an aspect that can be modeled by a linear function. For example, a group may choose the profession of a taxi driver and a salesperson. For the taxi driver, they can consider the relationship between working time and income. For the salesperson, they can analyze the relationship between the number of products sold and the commission received.

Required Materials:

1. Internet access for research.
2. Paper, pen, ruler for graphs and calculations.
3. Presentation software or word processor for the final report.

Step-by-Step Guide for Activity Execution:

1. Research and Planning:

• Research on the two chosen professions
• Identification of the aspect that can be modeled by a linear function
• Collection of necessary data to establish the proportional relationship

2. Calculation and Graphs:

• Use the concepts of linear functions to develop equations that model the chosen aspect in each profession.
• Draw graphs representing these functions.

3. Analysis and Interpretation:

• Analyze and interpret the functions and graphs.
• Identify the slope and intercept and their practical implications.

4. Compilation and Presentation:

• Compile the results into a report covering Introduction, Development, Conclusions, and Bibliography.
• Present the work to the class.

Project Deliverables:

Each group will deliver a report following the suggested structure. The report should have the following characteristics:

1. Introduction: Should contextualize the theme, its relevance and application in the real world, as well as the objective of this project.
2. Development: Should explain the theory behind linear functions, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
3. Conclusion: Should conclude the work by summarizing its main points, explaining the learnings obtained, and the conclusions drawn from the project.
4. Bibliography: Should indicate the sources that were used to work on the project such as books, web pages, videos, etc.

In addition, students will present their findings to the class, explaining their equations, graphs, and practical implications. This will allow all students to see the variety of applications of linear functions in different professions.