Context

The first-degree function is one of the fundamental elements in the study of Mathematics. It is present in everything that involves progression and constant increase. Every time you save money, for example, you are applying the logic of the first-degree function. This is one of the reasons why this is one of the first functions we learn when studying algebra. In summary, a first-degree function follows a linear pattern, which means it progresses at a constant rate.

The graph of the first-degree function is a straight line on the Cartesian plane, which provides a graphical representation of this constant progression. This makes first-degree functions and their graphs valuable tools not only in mathematics but also in various disciplines and fields of study, including physics, economics, engineering, among others.

Understanding the construction and interpretation of this graph is an essential skill for anyone who wants to understand how things progress over time at a constant rate. Furthermore, this function manifests itself in many of the phenomena we observe in our daily lives, from population growth to the economy, and even the behavior of the weather.

Introduction

In algebra, a first-degree function is a polynomial expression of degree 1. Generally, a first-degree function is written in the form y = mx + b, where m and b are constants. The coefficient m is called the angular coefficient, which determines the slope of the line on the graph, and b is called the linear coefficient, which gives the point where the line intersects the y-axis.

This graphical presentation is one of the most visual ways to understand the concept of first-degree functions and their behavior. Furthermore, the graph allows us to glimpse the possible roots of the function, that is, the values that make y equal to zero. In addition, the graph also clearly shows us the increasing or decreasing behavior of the function.

To have a solid foundation on the subject, I suggest the following reliable sources:

• First Degree Function - Brasil Escola
• First Degree Function - Basic Mathematics
• First Degree Function - Só Matemática

Practical Activity

Activity Title:

"Building Graphs - a journey into the world of First Degree Functions"

Project Objective:

To understand the concept and application of first-degree functions and their graphical representations, while developing collaboration, communication, and critical thinking skills.

Detailed Project Description:

In this project, students will be challenged to create an animation that shows the behavior of various first-degree functions, their graphical representations, and the consequences of varying their linear and angular coefficients.

Each group must choose a real-world phenomenon that is modeled by a first-degree function, for example, the revenue from selling lemonade as a function of the number of lemonades sold, the speed of a vehicle as a function of time, among others. Each chosen phenomenon will be the basis for the animation.

This activity requires students to thoroughly investigate first-degree functions: definition, properties, applications, and interpretation of graphs. Additionally, they will also need to learn about the use of some technological tools for the production of the animation.

Required Materials:

• Computers with internet access
• Graphic design software (such as Paint, Adobe Illustrator) or animation software (such as Blender, Adobe Animate)
• Pencils, paper, ruler, calculator for initial calculations
• Books, videos, and web pages on First Degree Function

Detailed Step-by-Step Guide for the Activity:

1. After forming groups of 3 to 5 students, each team must choose a real-world phenomenon that is represented by a first-degree function.

2. Each group must research and study the first-degree function, its definition, properties, applications, and how to interpret its graphs.

3. With the phenomenon in hand, students must create a first-degree equation that models the chosen situation.

4. Students must produce several graphs of the created function, showing the effect of modifying the linear and angular coefficients. They should analyze the slope of the line, intersection points, among other aspects.

5. After studying and creating the graphs, students must develop a script for the animation, showing the variation of the coefficients and their result on the graph.

6. Using the chosen software, students must create the animation based on the script. The animation should show the relationship between the first-degree function and the chosen phenomenon.

7. Students must prepare a presentation of the animation for the class, explaining the first-degree function, the chosen phenomenon, and how this phenomenon is represented by the function and the graph.

8. Finally, students must write a detailed project report. The document should include:

   • Introduction: contextualization of the first-degree function and its importance, the chosen phenomenon, and the project's objective.

   • Development: In-depth discussion of the theory behind the first-degree function, detailed description of the activity, methodology used, and results obtained.

   • Conclusion: Return to the main points, discuss the conclusions drawn from the project, and the lessons learned.

   • Bibliography: References of the information sources used.

The estimated time for the project is approximately 15 to 20 hours, depending on the complexity of the animation and the level of detail of the report.