Introduction

The 'modular function' is one of the most important topics in the study of mathematics, particularly in the areas of algebra and analytic mathematics. The modular function, expressed as |x|, is the absolute value of x which always results in a positive value or zero; making this concept particularly interesting and challenging for study.

Often, complex mathematical expressions can be simplified using the modular function, allowing for easier analysis of equations and formulas. Furthermore, the modular function has a unique and fascinating graphical representation, as when visualizing the graph of this function, one observes a V-shaped curve, with its vertex located at the origin.

The graph of a modular function is created by applying the modulus (modular function) to each value of x, resulting in a 'reflection' of the negative part of the graph on the x-axis. It is interesting to note that the graph of a modular function will always exist, regardless of the function's domain.

Contextualization

The study and understanding of the modular function are extremely essential in mathematics and its applications, as it paves the way for the interpretation and solution of advanced mathematical problems. Frequently, the modular function is encountered in real-life situations where a negative result is not feasible. For example, the distance between two points on a straight line, the absolute value of a number, the magnitude of a vector in physics, and many others.

Furthermore, the modular function significantly contributes to various areas of real life, such as engineering, physics, economics, computer science, etc. For example, in economics, the modular function is used to calculate the absolute difference between the opening and closing prices of a stock, regardless of the direction in which the price moves.

Practical Activity

Activity Title:

Exploring the Modular Function through Graphs

Project Objective:

To study in-depth the concept of modular functions, learning how to build, interpret, and manipulate them. This activity also aims to develop research skills, teamwork collaboration, and the application of theory to practice.

Detailed Project Description:

Students will be divided into groups of three to five members, with each group being responsible for building, analyzing, and interpreting the graph of a modular function through a real-world application case study.

Each group must identify a real-world situation where the modular function is applicable, for example, the distance between two points in a city or the variation of the absolute value of the price of a stock or commodity.

After identifying the theme, the group should thoroughly investigate and understand how the modular function applies in this situation, from there they will construct the function's graph, indicating how the input (x) and output (y) values were obtained or defined.

The next step is the analysis and interpretation of the constructed graph: identifying the shape, orientation, position relative to the origin, and other relevant details of the graph. It is important to recognize the domain and co-domain of the function and their importance for the real-life situation studied.

Finally, students must present their conclusions, highlighting the importance of the modular function in the analysis and solution of the identified problem.

Note: Although this project is primarily focused on mathematics, it intersects with other disciplines such as geography, economics, physics, etc., depending on the application situation chosen by the students.

Required Materials:

• Graph paper
• Pencils and erasers
• Ruler
• Computer with internet access and programs like Excel or Google Sheets for more precise and sophisticated graphs.
• Mathematics books, math websites, and YouTube videos for research.

Detailed Step-by-Step Guide for the Activity:

1. Formation of groups and initial discussion to select the application theme of the modular function.
2. Detailed research on the chosen theme and on how the modular function can be applied to solve or understand the problem.
3. Collection of necessary data (x and y values) for graph construction.
4. Drawing the graph of the modular function on graph paper or computer.
5. Analysis and interpretation of the graph: identification of the shape, orientation, position relative to the origin, domain, co-domain, etc.
6. Reflection on the experience: Discussion on what was learned during the activity, the challenges encountered, and how they were overcome.
7. Preparation of the report presenting the work done and the conclusions.

Project Delivery:

Students will present their projects in two stages:

1. An oral presentation where they will demonstrate the created graph and explain in detail the entire process, from the theme selection to the construction and interpretation of the graph.
2. Submission of a written report with the following sections:
   • Introduction: Should contain the relevance of studying modular functions and the chosen situation, and the project's objective.
   • Development: Should include the theory of the modular function, a detailed description of the project, the methodology followed, and the results obtained.
   • Conclusions: Summarizing the main points, explaining the learnings obtained, and the conclusions drawn.
   • Bibliography: Indicating the sources of information used to carry out the project.

Students must ensure that the report is clear, well-organized, free of spelling and grammar errors, and clearly demonstrates an understanding of the concept of the modular function and its application to the real-life study situation.

Project Duration: The project should be carried out over four weeks, with weekly meetings to discuss progress and resolve any doubts.