Contextualization

Linear functions are fundamental mathematical tools that have pivotal roles in various disciplines and real-life situations. They represent a direct, unchanging relationship between two variables. In other words, a linear function can be visualized as a straight line in a graph, where each point on that line corresponds to a specific set of input and output values. As such, they help us understand and predict the behavior of various phenomena.

In this project, we will delve into the world of linear functions, exploring their properties, their various forms, and their nature. We will also try to understand the concept of slope and y-intercept and how they help in formulating the equation of a line. We will learn about the constant rate of change and how it signifies the consistent relationship between variables in a linear function.

Linear functions are ubiquitous. They serve as the core concept for various essential areas of study such as physics, economics, computer science, engineering, and more. With their ability to depict a constant rate of change, they enable us to comprehend the rate of growth or decrease, evaluate trends, and make predictions. For instance, in economics, linear functions aid in understanding the relationship between supply and demand, while in physics, they enable us to understand the concept of speed.

To help you get started with this exciting topic, some valuable resources you can use include:

1. Khan Academy - This website offers free lessons on a wide range of topics in mathematics, including linear functions. You can find video tutorials, practice exercises, and quizzes that will greatly aid your understanding.
2. IXL Learning - This provides interactive questions and explanations on linear functions, useful for practice and strengthening comprehension.
3. Math is Fun - This site offers easy-to-understand explanations on linear equations and their application.
4. Wolfram Mathworld - Here, you can deep dive into the technicalities and intricacies of linear functions.
5. Linear Functions and Slope - Intercepts Form - This book by Mary Jane Sterling provides an in-depth study of linear functions, the slope, and the y-intercept. It is available in many libraries and online stores.

Amidst this exploration, always remember: Linear functions are not just abstract concepts. They mirror the world around us, providing us with tools to model, understand, and predict a wide range of phenomena. By the end of this project, you will have gained not only a deep understanding of linear functions but also an appreciation for their practicality and relevance in our lives.

Practical Activity

Activity Title: "Exploring the World of Linear Functions Through Real-Life Scenarios"

Objective of the Project

The goal is to apply theoretical knowledge to real-life situations in order to uncover the practical applications of linear functions. Each student group will create a physical model using everyday objects to simulate real-life linear function scenarios and demonstrate their understanding of the subject matter.

Detailed Description of the Project

This project involves two main endeavors: the first is a research portion, and the second is a practical creation of a model demonstrating a real-world linear function. Each group would be required to conduct research on linear functions, their real-life applications, and how they impact our daily lives. The students should then select a real-life scenario to showcase the practical application of linear functions.

The group would need to create a physical model that would allow them to demonstrate their chosen real-life scenario. With this model, they would show the linear relationship between two variables over time using everyday objects. The linear function's slope (rate of change) and y-intercept should also be clearly demonstrated and explained with reference to the model.

Necessary Materials

• Cardboard or any other sturdy base for model construction.
• Everyday objects such as toys, game pieces, kitchen utensils, etc. to represent elements within the scenario.
• Art supplies (like markers, papers, stickers, etc.) to decorate and enhance the model.
• A camera or smartphone to document the process and the final model.
• Internet access for research.

Detailed Step-by-Step for Carrying Out the Activity

1. Begin by researching linear functions, their properties, and real-world applications. This will aid your understanding of the concept and its significance.
2. Choose an everyday real-life scenario where a linear function could be applied. For instance, the relationship between time spent studying and improvement in grades, the depreciation of a car's value with years of use, etc.
3. Design a plan for your physical model that demonstrates the chosen scenario.
4. Start creating the model. Use everyday objects to represent the elements in your scenario and use the cardboard as your base. Make sure the linear function's slope (rate of change) and y-intercept are clearly represented in the model.
5. After the model is complete, document the model and the process you went through to build it. You can use a camera or smartphone for this.
6. Analyze your model, understanding and explaining how the chosen scenario illustrates the linear function, its slope, and y-intercept.
7. Prepare a presentation where each group member explains a different aspect of the model and how it relates to linear functions.

Project Deliverables

The final deliverables of this project will be a recorded presentation detailing the linear function model and a written report. This report should detail the thought process, methodology, the theory behind the real-life application of the linear function chosen, and the results obtained from the project.

The report must contain four main parts:

• Introduction: Contextualize the theme, its relevance, real-world application, and the objective of this project.
• Development: Detail the theory behind linear functions, explain the activity in detail, indicate the methodology used, and finally present and discuss the obtained results.
• Conclusions: Revisit the main points of the project, explicitly stating the learnings obtained, and the conclusions drawn from the project.
• Bibliography: Indicate the sources relied upon to gather information and to work on the project.

This project aims to stimulate collaboration and understanding among students, providing a fun and engaging way to learn about linear functions. By the completion of the project, students should be able to determine a function that can be represented by a straight line, with a constant rate of change between input and output values. Moreover, they should acquire socio-emotional skills such as problem-solving, creative thinking, and proactivity.

The project duration would be one month, with an estimation of five to ten hours of work per student.