Contextualization
Functions are one of the most fundamental concepts in mathematics, being extremely important for understanding a large number of concepts in more advanced mathematics, as well as many phenomena in the real world. In general, a function is a relation between two variables, so that each value in the first set (domain) corresponds to exactly one value in the second set (codomain). Functions can be represented in various ways, such as numerically through tables, graphically through graphs, or algebraically through equations.
Within the study of functions, there are several types that are commonly studied, including linear, quadratic, polynomial, exponential, logarithmic functions, among others. Each of these functions has its specific characteristics and properties and is used to model different types of situations. To correctly understand these various types of functions, it is essential to understand the basic concept of a function and how it can be represented.
Functions are not only important for mathematics but are also powerful tools used in many different fields, including physics, engineering, economics, biology, among others. For example, in physics, functions are used to describe the motion of objects. In engineering, they are used to model the behavior of different systems, and in economics, to understand the relationships between different variables, such as supply and demand.
Practical Activity: Modeling Real-World Situations with Functions
Project Objective
The objective of this project is to apply the concept of functions to model and interpret real-world situations. Each group should choose a real-world situation that can be modeled by a function, create a graphical, algebraic, and numerical representation of this function, and finally interpret and discuss the meaning of this function in the context of the chosen situation.
Detailed Project Description
Each group will start by choosing a real-world situation that can be modeled by a function. For example, a group could choose to model the relationship between speed and time in an accelerating car. Another option could be to model the relationship between temperature and the amount of ice cream sold in a day.
After this choice, the group will create a graphical representation of the function using the Geogebra platform. To do this, they will define appropriate domain and codomain for the function in question. Then, they will draw the function's graph on Geogebra.
The next step will be to create an algebraic representation of the function. In other words, students must find an equation that describes the function modeling the chosen situation.
Finally, the group should also create a numerical representation of the function, through a table that relates the values of the two variables.
Necessary Materials
- Computers with internet access for using Geogebra and research.
- Material for taking notes.
Step by Step
- Form groups of 3 to 5 students.
- Each group should choose a real-world situation that can be modeled by a function.
- Using Geogebra, create a graphical representation of the function.
- Develop an algebraic representation of the function.
- Build a numerical representation of the function through a table.
- Each group should prepare a presentation discussing and analyzing their results.
Project Delivery
At the end of the project, each group must deliver a written report containing an introduction to the topic, a detailed explanation of the activity, presentation of the results obtained, and conclusions about the project.
In the Introduction, students should contextualize the topic of functions and their importance, as well as describe the project's objective. It is also important for the group to explain the real situation they chose to model and why it can be described by a function.
In the Development, the group should explain their process for creating the three representations (numerical, graphical, and algebraic) of their function. It is also important for the group to discuss how they defined the domain and codomain of the function.
The Results obtained should be presented clearly and organized. Additionally, it is important for students to discuss these results and what they mean in the context of the situation they chose to model.
In the Conclusion, the group should discuss what they learned from the project, as well as their difficulties and successes. They should revisit their main points and reflect on the relevance of studying functions.
Finally, in the Bibliography, students should indicate all the sources they used to support their work.
