Contextualization

The First Degree Function is a fundamental concept in mathematics and plays an important role in modeling real-world situations. This function, also known as a linear function, is widely used in various fields such as economics, engineering, physics, among others, to describe phenomena that vary in a constant manner.

Each first degree function is defined by a mathematical formula that relates two variables: the independent (or input) and the dependent (or output). The independent variable is the one you have control over and can change. The dependent variable is the one that changes depending on the independent variable, meaning it depends on the independent variable to have any value.

For example, in a car trip, the travel time (dependent variable) is directly proportional to the distance traveled (independent variable) at a constant speed. This can be described by a linear function, where the distance is the input and the time is the output.

In this sense, the first degree function is essential for representing and solving many practical problems. When you understand this function, you acquire a powerful tool that helps you understand and model various everyday situations.

Practical Activity: Modeling the Real World with First Degree Functions

Project Objective

Practice the theory learned about first degree functions by designing and simulating a real-world scenario where this function is applicable, understanding its inputs and outputs, and presenting this information in the form of a report.

Detailed Project Description

Students, divided into groups of 3 to 5, should choose a daily life topic where a first degree function can be applied, and create a mathematical model that represents this situation. Throughout the project, students should identify the input and output variables, calculate the slope and intercept, and draw a graph to represent the function.

Required Materials

1. Paper and pen for notes.
2. Internet access for research.
3. Calculator for mathematical calculations.
4. Spreadsheet software for creating graphs (e.g. MS Excel, Google Sheets)

Step-by-Step for Carrying Out the Activity

1. Form groups of 3 to 5 students.

2. Choose a real scenario where a first degree function could be applied. Examples of scenarios could be: the production cost of a company versus the quantity produced, the distance traveled according to speed, among others.

3. After defining the scenario, identify the input and output variables. The input is usually what we control and the output is the result we obtain.

4. Build the equation of the first degree function that best describes the chosen scenario. Remember that the equation of the first degree function is in the form y = ax + b where 'a' is the slope and 'b' is the intercept.

5. Calculate the slope (a) and intercept (b) according to the scenario you chose.

6. With the help of a spreadsheet software, draw a graph that represents the function. Plot the independent variable on the X-axis (input) and the dependent variable on the Y-axis (output).

7. Prepare a report detailing each step above, including the choice of scenario, the function equation, and its graph. The report should follow the format specified in the project description (Introduction, Development, Conclusions, and Bibliography used).

8. Present the project to the group explaining the scenario, the function, the graph, and the conclusions.

Project Deliverables

Students should deliver:

1. The choice of scenario and a detailed description of it.

2. The equation of the first degree function that models the chosen phenomenon, as well as the identification of the inputs and outputs of this function.

3. The calculation of the slope and intercept of the function.

4. The graph of the function using spreadsheet software.

5. A written report, where they will:

   • In the introduction, provide an overview of the chosen scenario and its relevance, explaining why the first degree function was chosen to model it.

   • In the development, describe in detail the modeling process, including the identification of input and output variables, the construction of the function equation, and the graph elaboration.

   • In the conclusion, describe what they learned throughout the project, how the first degree function applies to the chosen scenario, and what insights they were able to obtain from the graph analysis.

   • In the bibliography, list the sources consulted throughout the project.