Contextualization

Mathematics is known as a universal language, used to describe patterns, relationships, and real-world problems. One of these tools is the first-degree function, which can be represented graphically or through tables. These forms of representation are powerful mechanisms for visualizing, understanding, and solving problems in various areas, from Economics to Engineering.

The first-degree function (or linear function) is a simple but fundamental mathematical relationship. It expresses a constant dependency relationship between two variables, typically represented as y = ax + b, where 'a' is the angular coefficient, 'b' is the linear coefficient, and 'x' is the independent variable. This function produces a graph in the form of a straight line on the Cartesian plane, hence its name: 'linear function'.

To understand this dependency relationship, it is useful to represent it in two ways: in the form of a graph and in the form of a table. The graph shows the relationship between the variables visually, making it easier to perceive the general behavior of the function. On the other hand, the table allows us to see the exact and specific values for each point in the relationship, which can be useful in various scenarios.

In everyday life, there are several situations described by first-degree functions. For example: calculating simple interest, quantifying revenue or expenses as a function of time, determining the distance traveled as a function of time in uniform motion, and many others.

The first-degree function is one of the most basic tools of algebra and, therefore, a deep understanding of this subject is essential. It will be the basis for understanding more complex mathematical concepts in the future, as well as being frequently used in solving practical problems.

Activity: 'Graph vs Table: A Fun Approach with First Degree Functions'

Project Objective

The objective of this practical activity is to develop students' ability to represent first-degree functions both graphically and in tables, and to understand the relationship between the two representations. In addition, it aims to stimulate collaboration and teamwork throughout the project execution.

Project Description

Students will be divided into groups of 3 to 5 members. Each group will be assigned a first-degree function y = ax + b, where the coefficients a and b will be provided by the teacher. The first task for the students will be to create a table with corresponding values of x and y, considering an interval also defined by the teacher.

After creating the table, students should draw the corresponding graph of the function on graph paper. Next, they will be challenged to reconstruct the function's equation using only the drawn graph (without access to the value table or the original equation). The goal here is to understand how the graph of a first-degree function relates to its algebraic equation.

Throughout the process, each group must document all stages, from constructing the table, transitioning to the graph, to reconstructing the equation from the graph. This record will be used to write the final project report.

Required Materials

For the activity, each group will need the following items:

• Graph paper for drawing the graphs;
• Colored pencils for highlighting;
• Rules and squares to assist in making the graphs;
• Calculators (optional).

Step by Step

1. Receive the first-degree function equation from the teacher and the interval for the value of x.
2. Create a table of values for x and y.
3. Draw the corresponding graph of the function on graph paper.
4. Try to reconstruct the function's equation using only the graph.
5. Document all stages of the process.

Project Submission

Students must submit a written report that includes the following sections:

1. Introduction: Students should contextualize the first-degree function, its relevance, and application in the real world. Additionally, they should explain the project's objective.

2. Development: Students should explain in detail the theory behind the first-degree function. In this section, they should describe the activity that was carried out, indicating the methodology used for the practice. Finally, they should present and discuss the results obtained.

3. Conclusions: Students should summarize the main points of the project, explaining what they learned from the activity and what conclusions were drawn.

4. Bibliography: They should cite all sources used to base the project's development, including books, web pages, videos, among others.

The written report is a fundamental part of this project as it consolidates and expresses the students' learning, their understanding of the theory of the first-degree function, and their insights into the practical activity. This document is the students' opportunity to demonstrate not only what they have learned but also their skills in written communication, logical reasoning, critical thinking, and collaborative work.