Context

The study of functions is one of the most important parts of mathematics and finds applications in numerous areas, from physics to economics. Among the various categories of functions, we find modular functions, also known as absolute value functions. These functions, characterized by their structure |x|, have the peculiarity of always providing a positive value or zero as a result, regardless of the value of the variable x. In other words, |x| = x if x >= 0 and |x| = -x if x < 0.

Modular functions are often used to describe real-world situations where the quantity of interest is always positive or zero, regardless of the conditions. For example, the distance between two points on a map, or the time difference between two events. That is why modular functions are so important and why we are studying them.

Theoretical Introduction

In any function, we can identify two main parts: the input (x) and the output (y), where the output is determined by the input through a specific rule, which is the function itself. In the case of modular functions, this rule involves obtaining the absolute value of the input. If we take, for example, the function y = |x|, we see that for any value of x we introduce, we will obtain its absolute value as output.

The geometry of modular functions is equally interesting and important. If we graphically represent a modular function, we will obtain a V-shaped figure, with the vertex located at the point (0, 0). This shape is a direct consequence of the definition of the modular function: no matter how negative the input is, the output will always be positive or zero.

To master modular functions, it is necessary to understand not only the theory but also how to calculate inputs and outputs, draw graphs, and interpret them correctly. And that is exactly what we will learn in this project.

Practical Activity

Activity Title: Unraveling the Modular Function

Project Objective:

The main goal of this activity is to enable students to acquire competence in calculating inputs and outputs of modular functions, understand their behavior, and know how to represent them graphically. Additionally, we want to encourage teamwork, time management, and effective communication.

Detailed Project Description:

You and your group will be the mathematicians in charge of exploring the intriguing world of modular functions! You will have to shed light on the mysteries of the inputs and outputs of these functions and, finally, master the art of representing them graphically.

Required Materials:

• Paper and pens.
• Mathematics books or internet access for research.
• Calculator.

Step-by-Step Guide for the Activity:

1. Preparation and Research: Gather as a group and study the theory of modular functions using the recommended resources. Discuss the theory and solve practical examples together.

2. Input and Output Exercises: Within the group, each member should create three different modular functions and calculate the outputs for five distinct inputs, noting all the details. Then, exchange the exercises among group members and solve the problems created by your colleagues. Check each other's answers.

3. Graphical Drawing: For each modular function solved in the previous step, draw the corresponding graph. Analyze how the change in the input (the value of x) affects the output (the value of y) and how this is reflected in the graph.

4. Discussion and Reflection: Discuss your observations and conclusions about the behavior of modular functions. Try to link the theory learned with what you observed when drawing the graphs.

5. Final Report: As a group, write a report documenting your activities and findings. The report should include the following topics:

   • Introduction: Provide context for the modular function, its relevance and application in the real world, as well as the project's objective.
   • Development: Describe the theory of modular functions. Explain in detail the activity carried out, indicate the methodology used in the calculations, and present the results obtained, including the modular functions created and their graphs. Discussions and observations about the behavior of modular functions should also be included in this section.
   • Conclusion: Summarize the main points of the work, present the learnings obtained, and draw conclusions about the project.
   • Bibliography: Indicate the sources you relied on to work on the project.

This project is designed to be carried out in groups of 3 to 5 students and should take each participating student 2 to 4 hours. The project deadline is one week.