Background

Introduction

Studying mathematics is a fundamental part of our educational system, not only because of its direct application in many areas of our lives, but also because of the development of logical reasoning and understanding of the world that it provides. Among the various areas of mathematics, we have Algebra, which is the part of mathematics that studies mathematical operations in a generalized way.

A key topic in algebra is the study of functions. Put simply, a function is a relation between two sets, in which each element of one set is associated with a single element of another set. There are several types of functions, one of which is the linear function, which is the focus of our project. The linear function is represented by the equation f(x) = ax + b, where "a" is the slope, which indicates the inclination of the line when the function is represented graphically; and "b" is the y-intercept, which corresponds to the point where the line intercepts the y-axis (or the value of f(x) when x = 0).

Contextualization of the theme

The linear function plays an important role in several practical applications, for example, in economics it is used to determine the total cost of production based on fixed and variable costs. In physics, the linear function is used to calculate the speed of an object in constant motion.

We will also explore how the graphical representation of a linear function can help us to better understand its properties and applications. By creating a graph of a linear function, you will be able to visualize the relationship between the variables, the slope of the line, and the intercept, and how they impact real-world situations.

Practical Activity

Activity Title: "Building and Understanding Linear Function Graphs: A Practical Approach"

Project Objective:

The objectives of this project are:

1. To comprehensively understand the concept of linear function, both theoretically and practically.
2. Converting the algebraic representation of linear function to graphical representation.
3. To understand the practical application of the linear function.

Step-by-Step Project Description:

In this project, students will build linear function graphs to represent real-world situations, so as to better understand how these functions work and where they are applied. To this end, each group of students, consisting of 3 to 5 participants, will choose a real-world situation that can be modeled by a linear function, and will build - both on paper and using the software of their choice - the graph that represents that function.

Materials Required:

• Graph or millimeter paper.
• Pencils, erasers, and colored pens.
• Computer with internet access for the use of dynamic geometry and algebra software or applications.

Step by Step:

1. Selection of Situation: The groups must choose a real-world situation that can be modeled by a linear function. For example, the total cost of production of a factory in relation to the quantity produced; the cost of a taxi ride in relation to the distance traveled, among others.

2. Function Construction: Based on the chosen situation, students must construct the linear function that represents it. They must clearly identify the slope (indicating the rate of change) and the y-intercept (indicating the starting value) of the function.

3. Graphical Representation: Next, students must construct the graph of this function, both manually on graph paper and digitally using specific software.

4. Graph Analysis: After constructing the graph, students must perform a complete analysis of it. This should include interpreting the slope and y-intercept in the context of the situation, and discussions on the behavior of the function.

5. Report Preparation: Finally, the students must write a report containing all the processes carried out and the conclusions reached. The report should be structured in the following sections: Introduction, Development, Conclusions, and Bibliography.

Project Delivery:

The final project should consist of:

• The chosen linear function, with a clear explanation of its meaning in the chosen context.
• Graphical representation of the function on graph paper (which must be scanned for inclusion in the report) and screenshot of the digital graphical representation.
• Final report, containing:
  • Introduction: The student should contextualize the theme, discussing the chosen situation and its relevance, the application of the linear function in it, and the objective of this project.
  • Development: The student should explain the theory behind the linear function, detail the activity carried out, the methodology used, and present and discuss the results obtained.
  • Conclusion: The student should revisit the main points of the work, outlining the lessons learned and the conclusions drawn about the project.
  • Bibliography: The student should indicate the sources used to work on the project, such as books, web pages, videos, etc.