Contextualization

The concept of function is one of the cornerstones of mathematics. It is so fundamental that it permeates almost all areas of knowledge, from physics to economics, through biology, chemistry, and even seemingly disconnected areas such as sociology and psychology.

Functions help us describe, in a simplified way, how one quantity varies in relation to another. For example, in physics, we may want to understand the relationship between time and the position of an object. In economics, we may be interested in the relationship between income and consumption. In all these situations, the idea of function allows us to model and understand these relationships in a precise and rigorous way.

Introduction

The word function comes from Latin, 'functio', which means execution or performance of something. And in the mathematical context, a function is a correspondence law between two sets, so that each element of the first set (domain) corresponds to a unique element of the second set (codomain).

To have a function, it is necessary and sufficient that, for each input value, there is exactly one output value. This mathematical concept is extremely versatile and is the basis for many fields of study and real-world applications.

From a practical point of view, a function can be visualized as a machine: you input a value, the machine processes that value according to a specific rule (the function), and then produces an output.

Consequently, it is imperative to understand and master this concept, as it is fundamental to many of the disciplines that students will encounter throughout their academic journey.

Practical Activity: 'The Role of Real in the World'

Project Objective

Connect the mathematical theory of function with its applicability in the real world, through the creation of a 'Function Guide', which is a collection of mathematical functions that model phenomena of the real world, along with a detailed analysis of each of these functions.

Detailed Project Description

Students, in groups of 3 to 5, will research and select five real-world situations that can be modeled through mathematical functions and their respective application. For each situation, students will:

1. Describe the real-world situation.
2. Identify and explain the mathematical function that models the situation.
3. Explore the concepts of domain and codomain in the chosen situation.
4. Draw the graph of the function.
5. Explain how the function allows predicting or understanding the real-world situation.

Students should select at least one situation that is modeled by a linear function, a quadratic function, an exponential function, a logarithmic function, and a trigonometric function.

Necessary Materials

• Computers with Internet access for research and report writing.
• Graphic design software. It can be dedicated software like GeoGebra or simply a spreadsheet software like Microsoft Excel.

Detailed Step-by-Step

1. Form groups of 3 to 5 students.
2. The groups will research and select five real-world situations that can be modeled by mathematical functions.
3. For each situation, students will identify and explain the mathematical function that models the situation.
4. Students will explore the concepts of domain and codomain in the chosen situation.
5. Students will draw the graph of the function using graphic design software.
6. Students will write a detailed report for each situation, explaining how the function allows predicting or understanding the real-world situation.
7. The teams will compile their reports into a single document, the 'Function Guide'.

It is expected that students will spend more than twelve hours on this activity, considering the time for research, discussion, drawing graphs, and report writing.

Project Deliverables

Groups should deliver the 'Function Guide', which is a compilation of the five elaborated reports, along with an introduction contextualizing the importance and application of functions in the real world.

The structure of each function report should follow the format:

1. Introduction: Description of the real-world situation and the mathematical function that models it.
2. Development: Detailed explanation of the chosen function, explaining its key concepts such as domain, codomain, and graph.
3. Conclusion: Discussion on how the function allows predicting or understanding the real-world situation.
4. Bibliography: References used for the study and elaboration of the report.

The final report should be formatted clearly and organized, being easily understandable for a reader not familiar with the topic. The use of graphics and illustrations is encouraged to facilitate the understanding of the concepts.

In addition to the report, students should also deliver a PowerPoint presentation, where each group will present one of their situations and explain how mathematics helps to understand the real world.

Students will be evaluated both on the technical content of their report and on their ability to present their ideas clearly and convincingly, and on their ability to work effectively in a group.