Contextualization

The study of 'First Degree Functions' is a fundamental part of mathematics. A first degree function, or linear function, is a function that can be written in the form f(x) = ax + b, where a and b are given real numbers and a ≠ 0.

Theoretical Introduction

First degree functions are known for their simplicity and are frequently encountered in everyday life. The concept of function is crucial in the field of mathematics and is used in various areas such as physics, economics, and engineering. Furthermore, it serves as the basis for understanding other more complex mathematical concepts.

A first degree function is represented by a line in the xy Cartesian coordinate system. The coefficient 'a' is responsible for the slope of the line, and the independent term 'b' is the point where the line intersects the y-axis. Additionally, the first degree function presents a constant variation, called the angular coefficient.

First degree functions have a very interesting characteristic: their rate of change is constant. This means that for each unit added to the independent variable (x), the dependent variable (f(x)) will increase or decrease by a fixed amount. This is the concept of slope, which is essential in understanding key mathematical concepts and their application.

Contextualization

First degree functions appear in many practical situations and everyday problems. Examples include converting temperatures between Celsius and Fahrenheit scales, calculating the distance traveled on a trip as a function of time, or understanding how the manufacturing cost of a product varies with the quantity produced.

Furthermore, understanding first degree functions allows students to develop a range of important skills, including logical reasoning, problem-solving, and critical thinking. With a solid understanding of first degree functions, students will be well-prepared to study more advanced mathematical concepts.

Reference Materials

We recommend the following reference sources to deepen the study of first degree functions:

• Khan Academy: Khan Academy has a rich selection of videos on first degree functions. Available at: Khan Academy - First Degree Functions
• Brasil Escola: Brasil Escola has a comprehensive article on first degree functions. Available at: Brasil Escola - First Degree Functions
• Só Matemática: Só Matemática offers solved and commented exercises on first degree functions. Available at: Só Matemática - First Degree Functions

Practical Activity

Activity Title: 'Building and Applying First Degree Functions'

Project Objective

This project aims to encourage students to investigate and apply first degree functions in everyday situations, while developing teamwork, problem-solving, and critical thinking skills.

Detailed Project Description:

Students will be divided into groups of 3 to 5 members. Each group must choose a practical situation that can be modeled using first degree functions. Examples of situations include, but are not limited to, the relationship between distance and time on a trip, the production cost of items versus the number of items produced, and currency conversion.

Once the situation is chosen, the group must gather data, create a first degree function model that fits these data, and analyze the resulting model. This project should take about five to ten hours for each participating student and will have a one-month deadline.

Required Materials

• Paper and pencil for notes
• Drawings or photographs related to the chosen situation
• Calculator or computer software to assist in calculating the first degree function
• Web resources, books, or other reference materials to aid in understanding first degree functions

Detailed Step-by-Step:

1. Formation of groups and definition of the situation to be studied
2. Research of data and information relevant to modeling the chosen situation
3. Creation of the mathematical model using a first degree function
4. Analysis and interpretation of the created model. Discussion on how the variation in input x affects the output y
5. Preparation of a detailed report on the project

Project Deliverables

At the end of the project, each group must prepare a written report following the criteria below:

1. Introduction: The group must contextualize the theme, its relevance and real-world application, as well as the objective of this project.
2. Development: The group must explain the theory behind the central theme of the project, detail the activity, indicate the methodology used, and finally present and discuss the results obtained.
3. Conclusions: The group must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
4. Bibliography: The group must indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

The report will be the main deliverable of the project and must be presented clearly and concisely. The report should link the theory of first degree functions with the practice of the chosen situation and demonstrate how the concepts were applied to solve the proposed situation.