Context

Linear functions are an important element of Mathematics, and their understanding is fundamental for the comprehension of many advanced concepts. A linear function is a correspondence between two numerical sets, in which each element of the first set, called the domain, corresponds to a single element of the second set, called the image set. Its general expression is f(x) = ax + b, where a and b are real numbers and a ≠ 0.

This function is important for its ability to model various real-world phenomena. For example, it can represent the variation of the speed of an object with time, the progression of the production of a factory in relation to the resources consumed, among many others. Through the study of this function, we are able to analyze and predict behaviors and results in various fields, such as physics, economics, biology, among others.

By learning linear functions, you will be able to transit between different areas of Mathematics with greater ease. In addition, this knowledge opens doors to the understanding of more complex functions in Mathematics, including quadratic, logarithmic, exponential, and many other functions. In practical terms, linear functions are the starting point for modeling real-world problems, a valuable skill in many professional fields.

To deepen your knowledge on the subject, I suggest the following resources:

1. Book "Fundamentals of Elementary Mathematics, Volume: 2", by Gelson Iezzi and Carlos Murakami.
2. YouTube Channel "Matemática Rio", which features video lessons on linear functions.
3. Website "Portal da Matemática", which offers various exercises on linear functions.

Remember, these resources are tools to help you build a solid and practical knowledge about linear functions. The teamwork and dedication of each member of the group are fundamental to the success of this project.

Hands-on Activity

Activity Title:

"Journey to the World of Linear Functions: A Journey of Inputs and Outputs"

Project Objective:

This project aims to deepen students' understanding of linear functions, their inputs and outputs, as well as the practical application of this theoretical concept. It is expected that, through the execution of this project, students will be able to identify and build linear functions, identify inputs and outputs of these functions, and apply them to everyday situations.

Detailed Description of the Project:

In this project, students will be divided into groups of 3 to 5 members. Each group will receive a list of everyday situations that can be represented by a linear function. For example, "The speed of a car that increases at a constant rate" or "The production cost of a factory in relation to the amount of resources used". Each group will have to choose an everyday situation to model mathematically using the linear function.

Materials Required:

• Paper and pen to develop the calculations.
• Computer with internet access for research.
• Software for creating graphs (can be specific software or a spreadsheet, such as Excel).

Detailed Step-by-Step for the Activity:

1. Each group must choose an everyday situation from the list provided.
2. After choosing, the group should discuss and identify the variables of the chosen situation and how they can be represented by a linear function.
3. The next step is to build the equation of the linear function that represents the chosen situation. Remember that the linear function has the form f(x) = ax + b.
4. With the function in hand, students should identify the meaning of the inputs (x values) and the outputs (f(x) values) in the everyday situation.
5. Using graphing software, students should plot the function and interpret what the graph represents in the real situation chosen.
6. Finally, the group should discuss what they learned from the activity and how the knowledge of linear functions, inputs, and outputs can be useful in the real world.

At the end of the project, each group should write a report on the activity. The report should contain four parts: Introduction, Development, Conclusions, and Bibliography.

1. Introduction: In this section, the group should contextualize the chosen situation, explain why it is relevant, and how the linear function can be applied to it.
2. Development: In this part, the group should explain the theory behind the linear function, what the inputs and outputs of the function are, and how the real situation was modeled mathematically. It should also present the graph of the function and explain what it represents.
3. Conclusions: Here, the group should explain what they learned from the activity and how the knowledge acquired can be useful. It should also reflect on the teamwork, the difficulties encountered, and how they were overcome.
4. Bibliography: Finally, the group should list all the resources they used to carry out the activity, such as books, websites, videos, etc.

This project is expected to take two to four hours per student to complete and should be submitted one week after the start date.