Context

Functions are of great importance in mathematics, and first-degree functions, also known as linear functions, are the most basic but also the most used. A linear function is a function of the type f(x) = mx + n, where m and n are real numbers. Learning this type of function provides us with a fundamental tool to solve a wide range of real-world problems.

Linear functions can be applied in various situations, such as in economics to understand the relationship between the price and demand of a product, in physics to understand changes in speed, or in biology to understand the growth of a population under certain conditions. The versatility and applicability of linear functions make them a fundamental concept in mathematics.

The graph of a linear function on the Cartesian plane is always a straight line, and this line can be ascending or descending depending on the value of 'm' in the formula, which is the slope. In addition, the value of 'n' is called the y-intercept and represents the point where the line touches the y-axis.

Introduction

The idea of this project is to develop activities that allow students to understand the definition, graphical representation, and application of linear functions in a playful and practical way. The activities will be designed to be carried out in groups, encouraging teamwork and collaboration among students, while challenging them to apply their mathematical knowledge in a practical context.

To develop this project, students can use various educational materials available in Portuguese on the internet, such as videos, websites, and digital books. Among the reliable and useful sources for studying linear functions and their applications, the Khan Academy website (https://pt.khanacademy.org/) stands out, offering a series of video lessons and exercises on the topic, as well as the Escola Digital website (https://www.escoladigital.pt/disciplinas/matematica/funcao-linear/), which has a section dedicated to linear functions, with definitions, examples, and exercises.

Practical Activity

Activity Title: Linear Functions in Practice: Building and Analyzing Graphs

Project Objective

The activity aims at the central understanding and practical application of the concept of linear functions. Students will explore the role of linear functions in different contexts, such as economics and physics, through the creation and analysis of graphs. The activity will seek to integrate mathematical concepts with the ability to interpret and apply them in real scenarios.

Detailed Project Description

Students will work in groups of 3 to 5 to develop the project. They will select two different scenarios where linear functions can be applied. For each scenario, they must formulate a linear equation that exemplifies a situation from the chosen scenario and create a graph to represent this equation.

Next, students must prepare a detailed report explaining the chosen scenarios, the derivation of the linear equations from them, the creation of the graphs, and their analysis. The report should connect the theoretical concepts of linear functions with their practical use in solving the scenarios.

Required Materials

• Paper and pens for creating the graphs
• Calculator
• Internet access for research
• Graphing software (such as GeoGebra) - optional

Step-by-Step Guide for the Activity

1. Group Formation: Students should be divided into groups of 3 to 5 members.

2. Scenario Selection: Each group must choose two different scenarios where a linear function can be applied. These scenarios can be related to various areas, such as economics, physics, biology, among others.

3. Formulation of Linear Equations: For each scenario, groups must formulate a linear equation that represents a specific situation from the chosen scenario.

4. Creation of Graphs: For each equation, students must create a graph representing the function. The graphs can be created manually or using graphing software (such as GeoGebra, for example).

5. Analysis of Graphs: Groups must analyze the created graphs, trying to interpret the represented situation and the relationship between the variables.

6. Report Elaboration: Each group must prepare a detailed report, explaining the process of scenario selection, formulation of linear equations, creation of graphs, and their analysis.

7. Project Presentation: Each group must present their work to the class, explaining the process and the conclusions drawn.

Project Deliverables

Groups must deliver the following items:

1. Written Report: The report should have the following sections: Introduction (contextualization on the topic, relevance of linear functions, project objective), Development (explanation of scenario selection, formulation of linear equations, creation and analysis of graphs, methodology used in the project), Conclusions (summary of the main points, learnings, and conclusions about the project), and Bibliography (sources used for the project).

2. Graphs of Linear Functions: The graphs created to represent the linear functions must be delivered along with the report.

3. Presentation: Groups must give a presentation of the project to the class, explaining the process and the conclusions drawn.

The report will be the main means to assess students' ability to connect theory with practice. Through it, it will be possible to evaluate students' understanding of the topic, as well as their research, teamwork, and communication skills.