Introduction

The first-degree function, also known as a linear function, is a topic of great importance in mathematics, being one of the fundamental concepts in the study of functions. It is characterized by a dependency relationship between two quantities, where the degree of the function is equal to one, presenting itself in the form f(x) = ax + b, where a and b are real numbers and a ≠ 0.

The study of first-degree functions is essential, as these functions are present in various branches of science, such as physics and economics, enabling the analysis of various phenomena, from the growth of a plant to the calculation of simple interest.

First-degree functions can be represented in two main ways: in a table or in a graph. The table represents the points of the domain and the codomain of the function, while the graph visually shows us the dependency relationship between these points. Both representations allow for a better understanding of the function and its characteristics.

Contextualization

The graphical representation of first-degree functions is a powerful tool that allows us to visualize how variables relate to each other. For any first-degree function, the graph will always be a straight line, which simplifies the understanding of this relationship. In the real world, the graphical representation of first-degree functions can help understand anything from the relationship between the number of hours worked and the salary received to the relationship between a car's speed and time.

On the other hand, the tabular representation of first-degree functions allows us to visualize a series of ordered pairs that belong to the function. This representation is very useful in situations where one wants to calculate the value of the function for certain values or compare values between different first-degree functions. In the real world, the tabular representation can be useful in various situations, such as comparing the performance of different investments over time.

Practical Activity

Activity Title: Graph vs Table - A Clash of Representations

Project Objective

The objective of the activity is to understand how the graphical representation and the tabular representation of a first-degree function relate to each other. For this, students will need to create a first-degree function, express it both graphically and in a table form, and finally, analyze, compare, and contrast both representations.

Detailed Project Description

Students will be divided into groups of 3 to 5 people, and each group will be responsible for creating a first-degree function based on a real-world situation that the students themselves will create. Additionally, they will be responsible for drawing the graph of this function, creating a table with at least 10 points of this function, and finally, preparing a report on the findings and learnings.

This activity requires approximately five to ten hours per student to complete and must be submitted at the end of one month.

Required Materials

1. Paper and pencil for sketches and initial calculations.
2. Computer with internet access for research, use of graphic software (such as GeoGebra), and preparation of the final report.
3. Google Docs account (or similar) for team report preparation.

Detailed Step-by-Step

1. Students should research and choose a real-world situation that can be represented by a first-degree function.

2. Students should then create the first-degree function that models the chosen situation, identifying the coefficients a and b of this function.

3. Next, they should create a table for the function with at least 10 points.

4. After creating the table, they should draw the graph of the function on the Cartesian plane.

5. Students should then compare the representations in the table and graph, identifying the similarities and differences between both.

6. Finally, they should prepare a comprehensive report on the activity, explaining the chosen situation, the function created, the methodology used, and the results obtained.

Project Deliverables Description

Students must submit the following:

1. A written report with the following sections:

   • Introduction: Should contextualize the first-degree function, its relevance and application in the real world, as well as the objective of this project.
   • Development: Should present the theory of first-degree functions, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusions: Should summarize the main points of the work, explain the learnings obtained, and draw conclusions about the project.
   • Bibliography: Should indicate the sources used for research and project development.

2. Attachments with the table and graph created for the first-degree function.

Students will be evaluated on both the technical content presented in the report and the socio-emotional skills exercised during the project, such as time management, teamwork, communication, and problem-solving.