Introduction

Linear functions are one of the first introductions to the important field of mathematics called Calculus, and they are extremely useful for understanding how different quantities relate to each other. A linear function is a function between two variables that is proportional to the input and where the output is a linear function of the input. The mathematical expression for this is f(x) = mx + n, where f(x) is the output, x is the input, m is the slope or rate of change (proportionality), and n is the intercept or initial value.

In real-world applications, linear functions can represent things like the relationship between the price of a product and the quantity of products sold, or the relationship between the number of study hours and a student's final grade. Understanding linear functions is also fundamental to entering more advanced fields of mathematics, such as calculus and linear algebra.

Contextualization

Linear functions are everywhere around us. In the business world, for example, the concept of linear functions is used to determine prices and profits. In physics, it is used to express the relationship between speed and time. In engineering, to describe the behavior of electrical circuits. Therefore, learning to understand, express, and solve linear functions means gaining an important tool to understand the world around us.

Mastering the understanding of linear functions and their application allows you to present more efficient and faster solutions to everyday problems. Imagine a business that needs to determine the ideal price for launching a product in relation to the number of units it expects to sell. Or a student who wants to know how much time they need to invest in studying to achieve the desired grade. With linear functions, all these questions can be answered more accurately and with less effort.

Practical Activity: "Real World Problems with Linear Functions"

Project Objective

The objective of this project is for students to apply the concept of linear functions to solve real-world problems, enhancing their teamwork, time management, and problem-solving skills.

Detailed Project Description

Students will be divided into groups of 3 to 5 people. Each group will receive a real-world situation to be solved using linear functions. The situations will be provided by the teacher and may vary between themes such as business, physics, engineering, or any other field where linear functions can be applied.

Students will have to prepare a presentation of their results, which will be made in front of the classroom, and also a written report detailing the process they used to reach their conclusions.

The project must be completed within a one-month timeframe.

Necessary Materials

• Paper and pencil for notes and calculations.
• Internet access for research.
• Presentation software (e.g., Microsoft PowerPoint, Google Slides) for the final presentation.

Detailed Step-by-Step

1. The teacher divides the students into groups of 3 to 5 people.

2. Each group receives a real-world situation that needs to be solved using linear functions.

3. Students have to research the problem, understand how a linear function can be applied to solve it, and finally come to a solution.

4. Students must prepare a presentation of their results for the class, detailing the problem they were assigned, how they solved it, and explaining the theory behind the concepts used.

5. In addition, each group must also write a detailed report of the process they went through to reach their conclusions. This report will have the following sections:

   • Introduction: the student must contextualize the problem, its relevance and application in the real world, as well as the objective of this project.

   • Development: the student must explain the theory behind the central theme of the project, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.

   • Conclusion: the student must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.

   • Bibliography: the student must indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Project Deliverables

Groups must deliver the visual presentation that will be presented to the class, as well as the written report of the project. The report should complement the presentation and vice versa, allowing for a complete understanding of the problem situation, the application of the linear function to solve it, and the results obtained.

With this project, students will not only deepen their understanding of linear functions and their application, but also develop important skills, such as the ability to work in a team, manage time, solve problems, and efficiently communicate their conclusions both orally and in writing.