Contextualization

First degree functions are fundamental concepts in mathematics and are widely used in various areas of knowledge, such as physics, economics, engineering, biology, among others. This is because they represent a direct relationship between two variables, which is extremely common in natural and social phenomena. Their graphical representations facilitate the understanding of this relationship and are a powerful tool in problem solving.

Understanding the first degree function is opening a door to a range of possibilities that mathematics can offer. Whether in predicting the behavior of a commodity, estimating the growth of a population, decoding a coding system, or even in the calculations involved in the construction of a building, the first degree function is there to simplify the resolution and make the understanding of the problem more accessible.

Introduction

The first degree function, also known as a linear function, is a function of the type f(x)=ax+b, where a and b are real numbers and a is not equal to 0. The number a is called the angular coefficient, it indicates the slope of the line on the function graph. The number b is called the linear coefficient, corresponding to the point where the line touches the y-axis on the graph.

When represented on a graph, the first degree function is a line that can have a positive slope (when a>0) or negative slope (when a<0). Understanding this slope is fundamental, as it indicates how the function behaves, that is, whether it increases or decreases as the value of x also increases.

The graphical representation of a function, besides being an intuitive visualization of the function, is an important tool for problem solving. Through the graph, it is possible to identify points of intersection between functions, maximums and minimums, among other properties that can be useful in different contexts.

For the development and deepening of the topic, students can use the following resources:

• Khan Academy - First Degree Functions: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:linear-functions/v/what-is-a-function
• Mundo Educação UOL - First Degree Function: https://mundoeducacao.uol.com.br/matematica/funcao-1-grau.htm
• Só Matemática - First Degree Function: https://www.somatematica.com.br/fundam/funcao1.php
• YouTube – Matemática Rio with Prof. Rafael Procópio - First degree function - Graph and equation: https://www.youtube.com/watch?v=Z5dN6wamu8k

Practical Activity: "Building and Analyzing Graphs of First Degree Functions"

Project Objective

Develop the ability to build and analyze graphs of first degree functions through the use of graphic representation software and logical-mathematical reasoning. In addition, foster teamwork and problem solving.

Project Description

Students, divided into groups of 3 to 5 people, should create graphs of first degree functions, analyze them, and later write a report. The activity will be divided into 3 main parts:

1. Creating graphs of first degree functions using graphic representation software.
2. Analysis of the created functions, focusing on the function properties and the visual aspects of the graph.
3. Writing a report detailing the entire process.

Required Materials

• Computer with internet access
• Graphic representation software (we suggest Desmos or GeoGebra)
• Paper and pencil for notes

Step-by-Step for Activity Execution

1. Students group together and choose 3 different first degree functions to work on.
2. Each group should create the graphs of the chosen functions using the graphic representation software.
3. With the created graphs, students should analyze each function, observing characteristics such as angular coefficient, linear coefficient, slope of the line, and the point where the line touches the y-axis.
4. The team should discuss the results together, sharing observations and conclusions.
5. Finally, students should write a report detailing the entire process. The report should include the introduction, development (with a detailed description of the chosen functions, the created graphs, the analysis of the function properties and visual aspects of the graphs), conclusions, and bibliography.

Project Deliverables and Connection with Suggested Activities

The project is completed with the delivery of the written report. This report should contain the following sections:

1. Introduction: A clear description of the project's purpose, the relevance of studying first degree functions, and the learning objective.
2. Development: A summary of the practical activities carried out. Here, students should specify which first degree functions they chose, how they created their graphs, and the discussion and analysis carried out on each function and its graphs.
3. Conclusion: A recap of the results obtained, as well as the lessons learned throughout the project. It is important for students to mention here what the angular and linear coefficients mean in practice, when analyzing the graphs.
4. Bibliography: The research resources used by students to carry out the project should be listed here, including books, web pages, videos, etc.

The report must be delivered within a week from the start of the project. During this week, students should manage their time efficiently to carry out all parts of the project effectively. The report will be evaluated based on the understanding of the topic, the analysis performed, the presentation of results, and the quality of writing.