Contextualization

Theoretical Introduction

First-degree functions are the backbone of mathematics, appearing in various real-world applications. They are defined as functions where the independent variable x is multiplied by a constant, and another constant is added or subtracted from the result. The general form is f(x) = ax + b, where a and b are real constants and a ≠ 0.

Furthermore, the graph of a first-degree function is a straight line, which can be easily represented on a Cartesian plane. The coefficient a is the slope of the line, while the coefficient b is where the line crosses the y-axis (called the intercept).

In addition, the table of a first-degree function is an alternative way to represent the function. Each row of the table corresponds to a point on the function's graph, and the complete table usually provides an overview of the function. Tables are particularly useful for understanding the function's behavior at specific values of x.

Importance and Real-World Applications

First-degree functions are extremely relevant and applicable in a wide variety of real-world situations. They are often used in fields such as physics, economics, biology, among others, to model and predict linear behaviors. For example, the rate of acceleration of a car in constant motion or the relationship between cost and the quantity of units produced in a company.

Furthermore, understanding how to represent first-degree functions in graphs and tables is a fundamental skill in mathematics and many other disciplines. It is a basic tool that allows for easily visualizing the behavior of a function, facilitating the analysis and understanding of its characteristics.

Activity

Activity Title: Crime Scene Investigation: The Mystery of the First Degree Function

Project Objective

This project aims to have students apply the concepts of first-degree functions, tables, and graphs in an integrated and contextualized manner, through the simulation of a criminal investigation.

The idea is for students to divide into groups of 3 to 5 people, with each student taking on a specific role in the investigation team. Roles may include lead investigator, data analyst, graph specialist, etc.

The project should last around 12 to 15 hours and will include aspects such as teamwork, problem-solving, time management, and creative thinking.

Detailed Project Description

In a fictional crime scene, students will receive information in the form of tabular data representing different types of evidence (such as times and locations of crimes, for example). This data carries a first-degree function relationship.

The students must then analyze the data, identify patterns, generate hypotheses, build graphs, and use mathematics to solve the mystery.

This will be done in conjunction with the Forensic Science discipline, allowing students to learn about the different aspects of criminal investigations and how mathematics is used in practice.

Required Materials

• Crime scene data (provided by the teacher)
• Paper and pencil
• Computer with spreadsheet software (Microsoft Excel, Google Sheets, etc)
• Colored markers for graph construction

Step-by-Step Guide for the Activity:

1. Formation of Investigation Teams: Students should divide into groups of 3 to 5 people.

2. Initial Study: Each group should study the provided material, which includes a detailed description of the crimes and the corresponding tabular data.

3. Data Analysis: Students should analyze the tabular data, looking for patterns and relationships that may be linked to the occurrence of the crimes. They should formulate conjectures, which will be tested later.

4. Graph Construction: After analyzing the data, groups should build graphs corresponding to the first-degree function present in the data.

5. Interpretation of Graphs: Students should interpret the created graphs, based on the knowledge acquired about first-degree functions.

6. Creation of the Investigation Report: Each group should prepare a detailed report of their findings. This report should include an introduction (with a presentation of the problem and the project's relevance), development (with the theoretical explanation and description of the activities carried out), conclusions (with the main learnings and results obtained), and bibliography used.

Project Deliverables

Each group must deliver:

• The solution to the mystery, explaining step by step how they arrived at the resolution;
• Unique and original graphs, representing the first-degree function identified in the data;
• A detailed investigation report, describing the applied theory, the methodology used, the results obtained, and the conclusions drawn. This document should be written clearly and objectively, containing all the elements of a scientific report (introduction, development, conclusions, bibliography).

Remember, the goal of this project is not only to apply mathematical concepts but also to develop skills in collaboration, investigation, critical thinking, and communication.