Context

The world around us is full of patterns and relationships that can be mathematically described through equations. One of these equations is the Equation of a Line, which describes how two variables are linearly related. That is, if one variable changes, the other also changes in a specific way that can be described by a straight line on a graph. All of this, in a simple and precise manner.

There are many everyday situations that can be modeled through line equations. Have you ever stopped to think about how weather forecasting is done? Or how it is possible to calculate the average speed of a car? Or even how to determine if it is more advantageous to take a taxi or an Uber? All of these answers can be found through the study of the equation of a line. This is a powerful mathematical tool and quite useful in various areas, from Physics to Economics.

Introduction

The Equation of a Line is one of the most fundamental concepts in Mathematics and provides the basis for more advanced areas, such as Calculus and Linear Algebra. In general, a line in the plane is determined by an equation of the form y = mx + n, where m is the slope of the line (which indicates the inclination of the line with respect to the x-axis) and n is the y-intercept (which represents the point where the line intersects the y-axis).

The applications of this theme are vast and range from pure Mathematics to the understanding of natural and social phenomena. Therefore, understanding the equation of a line is of fundamental importance to see and understand the world from a mathematical perspective.

The equation of a line is also a cornerstone for the study of more complex forms. Understanding the basics is the first step to deepen the teaching of mathematics and solve more complex problems. Therefore, in the following project, we will delve into the study of this fascinating theme.

Practical Activity - Modeling the Real World with the Equation of a Line

Project Objective

The objective of this project is to put into practice the understanding of the equation of a line, through the modeling of practical situations from the real world. Students will identify a situation where the equation of a line can be applied, collect data, formulate the equation, and infer conclusions.

Detailed Project Description

Students should work in groups of 3 to 5 people. Each group must choose a situation in which the relationship between two variables can be modeled by a line. Some examples could be: the cost of a taxi in relation to the distance traveled, the growth of a plant over time, or the amount of waste produced by a household in relation to the number of inhabitants.

Each group must collect enough data to formulate the equation of the line that represents the chosen situation. After collecting the data, the groups must use them to calculate the slope (m) and the y-intercept (n) of the equation.

Required Materials

• Paper and pen for notes
• Calculator
• Internet access for data research

Detailed Step-by-Step

1. Group Formation and Topic Selection: Form groups of 3 to 5 students and choose a real-world situation where the relationship between two variables can be modeled by a straight line.

2. Data Collection: Conduct research (online, experiments, interviews, etc.) to collect data that represent the chosen situation. It is recommended to collect at least 10 pairs of data.

3. Calculation of Equation Coefficients: Using the collected data, calculate the slope (m) and the y-intercept (n) of the equation of the line that represents the chosen situation.

4. Report Preparation: Write a detailed report including the introduction, development, conclusions, and bibliography used.

   • In the introduction, contextualize the chosen theme and the relevance of the equation of a line to model the situation.

   • In the development, explain how the data was collected, how the coefficients were calculated, and how the equation of the line was formulated. Discuss the results obtained.

   • In the conclusion, summarize the main points of the work and explain what was learned from the project.

   • In the bibliography, present all sources of information used during the work.

5. Project Presentation: Prepare a short presentation to explain the chosen situation, the data collection process, and the obtained equation of the line. Show what was learned and how the equation of a line can be used to model everyday situations.

Project Deliverables

Students must deliver the project in two parts.

1. Written Report: The writing of the report is a crucial part of the project. The report should follow the format indicated in the step-by-step, being well-organized into well-defined sections of introduction, development, conclusions, and bibliography. This document will serve as a detailed record of the work, the calculations performed, and the conclusions drawn.

2. Project Presentation: In addition to the report, groups must prepare a presentation (can be in PowerPoint or another preferred format of the group) to be held in class, so they can share their learnings, the obtained equation of the line, and their interpretations with the class.

Remember, the objective of this project is not only to understand the equation of a line itself, but also to develop important skills such as time management, communication, problem-solving, creative thinking, and proactivity.