Contextualization

Mathematics is a science with incredible applicability, becoming a fundamental tool for understanding and solving everyday and complex problems. In this sense, the study of Functions plays a central role within this area of knowledge. Functions are relationships of dependence between two variables, where one of them is always a function of the other. In a simplified way, we can think of a situation where a person's financial gain depends on the number of hours they work. This is a situation of dependence, in which the salary (one variable) is a function of the hours worked (another variable).

For a better understanding, let's segment the concept of functions into three representations: numerical, algebraic, and graphical. The numerical representation is seen in tables that relate two numerical sets, where the rule of the relationship is the function. In the algebraic representation, the function is presented by a mathematical expression, such as the general formula of the second degree. In the graphical representation, the function is seen through a drawing on the Cartesian plane, demonstrating in a more visual way the relationship of dependence between the variables.

With the increasing digitalization of society, understanding functions becomes even more relevant. Almost all current technological applications use some form of function in their algorithm. Google uses functions in search algorithms to offer the most relevant results. Social networks use functions to determine the content that will be shown to each user. Banks use functions in their financial and risk calculations. Thus, understanding what functions are and how they behave is essential for full interaction in the modern world.

An excellent resource platform for students to delve deeper into the subject is Khan Academy, which offers a complete course on Functions, including explanatory videos and practice exercises. In addition, Google Scholar can be used as a research tool to find academic articles on the subject.

• Khan Academy, "Functions" [Online]. Available: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:function-introduction
• Google Scholar [Online]. Available: https://scholar.google.com

Practical Activity: "The Mathematics behind the Roller Coaster"

Activity Title:

Building a Roller Coaster Model

Project Objective:

Through this practical activity, students will develop an understanding of the concept of functions and their numerical, algebraic, and graphical representations, exploring the real world and applying it to a roller coaster model. In addition, they will enhance their socio-emotional skills such as teamwork, time management, problem-solving, and creative thinking.

Detailed Project Description:

Students, in groups of 3 to 5, will be tasked with designing and building a roller coaster model using recyclable materials. They must describe the functions that represent the different sections of the roller coaster track, identifying whether they are linear, quadratic, or exponential functions.

Required Materials:

• Cardboard
• Popsicle sticks
• Tape
• Straws
• Graph paper
• Pencils, erasers, and rulers
• Marble or similar

Detailed Step-by-Step:

1. Study the functions: Start by learning the concept of functions in mathematics. This includes understanding the numerical, algebraic, and graphical representations of a function. Explore online resources, textbooks, and materials on the concept of functions.

2. Plan the roller coaster: Based on your knowledge, plan the design of your roller coaster. This includes determining steep climbs, sharp drops, and loops. Remember that the height and speed of the roller coaster should be representations of the function you studied.

3. Build the model: Using the listed materials, start building the model. Use cardboard as the base, popsicle sticks as supports, and straws as tracks.

4. Test the roller coaster: With the roller coaster built, test it with a marble or something similar. Modify the design as needed to ensure the marble completes the course.

5. Write the functions: Identify the different sections of the roller coaster track and write the functions that represent these sections on graph paper. Identify whether they are linear, quadratic, or exponential functions.

6. Document the project: In a written report, document the entire process from planning to construction and testing of the roller coaster. This should include the identified functions.

Project Deliverables:

Students must deliver two main things:

1. The roller coaster model: A physical model displaying different types of functions.

2. The report: A written document detailing the entire planning, construction, and testing process of the roller coaster, as well as the identification and description of the functions. The document should contain the following parts:

• Introduction: Here, contextualize the theme, its relevance, and applicability in the real world, as well as the objective of this project.

• Development: In this part, students must explain the theory behind the functions, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained, that is, the mathematical models representing the different sections of the roller coaster.

• Conclusion: Summarize the main points of the project, discuss what was learned, and draw conclusions about the project.

• Bibliography: Indicate the sources used to work on the project, such as books, web pages, videos, etc.

Students should dedicate five to ten hours of work to this project, and it must be delivered within a month.