Contextualization

Introduction

Functions are one of the fundamental concepts in mathematics. They are a way to describe the relationship between two variables, often represented by x and y. A linear function, also known as a linear equation, is a function where the rate of change is constant. It takes the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis).

Linear functions are incredibly useful in many areas, both in mathematics and in the real world. They can be used to model and predict various phenomena, such as population growth, speed and distance in physics, and even the price of goods and services in economics. Understanding linear functions can provide a solid foundation for more advanced concepts in mathematics, such as quadratic and exponential functions.

Real-World Relevance

Linear functions have numerous applications in the real world. One of the most common uses is in the field of economics, where they are used to model and predict supply and demand. For example, the price of a product might decrease by a certain amount for each additional unit produced (this is the slope of the line). The initial price, on the other hand, represents the y-intercept.

Linear functions are also used in physics to describe the relationship between two variables. For instance, in the formula for speed (speed = distance/time), distance and time have a linear relationship. The speed is the slope of the line, while the starting point (when time is 0) represents the y-intercept.

Resources

Students can use the following resources to deepen their understanding of linear equations and functions:

1. Khan Academy: Linear Equations
2. Math is Fun: Linear Functions
3. BBC Bitesize: Linear Graphs
4. Purplemath: Linear Equations

Practical Activity

Activity Title: "Linear Equations: From Paper to Practice"

Objective of the Project

The objective of this project is to allow students to apply their knowledge of linear equations and functions in a practical setting. By the end of the project, students should be able to understand and create linear equations, calculate and interpret the slope and y-intercept, and analyze real-world situations using linear functions.

Detailed Description of the Project

In this project, each group will be tasked with selecting a real-world scenario that can be modeled by a linear equation. They will then create a graph of the linear function and use this graph to make predictions and solve problems related to their chosen scenario. Along with their final report, the students must also present their work in a short video format.

The project duration is one week, and it's designed to be carried out by groups of 3 to 5 students.

Necessary Materials

• Graphing paper or any graphing tool
• Computer with internet access for research and report writing
• Presentation software for creating the video presentation

Detailed Step-by-Step Guide

1. Form Groups (Estimated Time: 10 minutes): Form groups of 3 to 5 students. Each group should be diverse, encouraging collaboration and teamwork.

2. Select a Scenario (Estimated Time: 20 minutes): Each group should choose a real-world scenario that can be modeled by a linear equation. This could be anything from the growth of a population, the depreciation of a car's value over time, or the cost of a phone plan.

3. Research (Estimated Time: 1 hour): Research the chosen scenario. Gather data and information that can be used to create a linear equation to model the scenario.

4. Create the Equation and Graph (Estimated Time: 1 hour): From the research, create a linear equation that models the scenario. Plot the graph of the function on the graphing paper or using a digital tool.

5. Solve Problems (Estimated Time: 1 hour): Using the graph, solve problems related to the scenario. This could involve predicting future values, finding the rate of change, or determining when certain conditions will be met.

6. Write the Report (Estimated Time: 2 hours): Write a report in the following structure:

   • Introduction: Discuss the chosen scenario and its relevance. Also, explain the objective of the project.

   • Development: Detail the theory behind linear functions, how they are used to model the chosen scenario, and how the graph and equation were created. Include the problems you solved and explain the solutions using your graph and equation.

   • Conclusion: Summarize the main points of the project, the learnings obtained, and the conclusions drawn about the chosen scenario.

   • Bibliography: Indicate the sources you used for your research, such as books, web pages, videos, etc.

7. Create the Video (Estimated Time: 1 hour): Create a short video presentation (3-5 minutes) summarizing your project. The video should be engaging and informative, presenting the main points of your report.

8. Submit the Project (Estimated Time: 1 hour): Submit your report and video in the format specified by your teacher. Make sure to meet the deadline.

Project Deliverables

At the end of the project, each group must submit:

1. Written Report: This should be a detailed account of your project, following the structure indicated above. The report should be submitted in a digital format (e.g., PDF or Word Document).

2. Video Presentation: This should be a short video (3-5 minutes) summarizing your project. The video should be engaging and informative, presenting the main points of your report. The video should be submitted in a format specified by your teacher (e.g., YouTube link or a digital file).

Remember that both the report and the video should reflect your understanding of linear functions and how they can be used to model real-world scenarios. It should also demonstrate your ability to work collaboratively in a group and manage your time effectively.