Contextualization

Introduction

In the field of mathematics, a function can be thought of as a machine that takes an input, does something to it, and gives back an output. Functions are one of the most fundamental concepts in mathematics and are used to describe many real-world phenomena. They are like the building blocks of mathematical models.

In this project, we will be focusing on the types of functions - specifically, the Linear Function, the Quadratic Function, and the Exponential Function. These are some of the most common functions that we encounter in our daily lives.

Real-world Application

Functions are not just abstract mathematical concepts; they have real-world applications. Linear functions, for example, can be used to model how the cost of a product changes with the number of items produced. If the cost increases by a fixed amount per item, then the relationship between the cost and the number of items is a linear function.

Quadratic functions can be used to model the motion of a projectile. The path of a projectile, such as a ball thrown into the air, can be described by a parabola, which is a type of quadratic function.

Exponential functions are used to model exponential growth or decay. For example, the growth of a population over time, the decay of a radioactive substance, or the increase in the value of an investment with compound interest can all be described by exponential functions.

Resources

To delve deeper into these concepts, you may use the following resources:

1. Khan Academy - Functions: This is an excellent resource that provides detailed explanations and examples of various types of functions.

2. Math is Fun - Functions: This resource provides a simple and intuitive introduction to functions, along with interactive examples.

3. Purplemath - Function Notation: This resource focuses on a specific aspect of functions - function notation. Understanding function notation is crucial for working with functions.

4. Wolfram MathWorld - Linear Function, Wolfram MathWorld - Quadratic Function, Wolfram MathWorld - Exponential Function: These resources provide more in-depth information about each type of function, including their properties and graphing.

Remember, the goal of this project is not just to understand these concepts, but also to apply them in a real-world context. So, let's get started!

Practical Activity

Activity Title: "Exploring Functions: Linear, Quadratic, and Exponential"

Objective of the Project:

The main objective is to understand and differentiate the characteristics of three main types of functions: Linear, Quadratic, and Exponential. The project aims to foster a deeper understanding of mathematical functions and their real-world applications.

Detailed Description of the Project:

In groups of 3 to 5 students, you will create Function Fact Sheets for each of the three types of functions. Each fact sheet should include the following details:

1. Definition of the function type.
2. Explanation of the function's form.
3. The graph of the function.
4. Real-world examples of the function.

To complete this project, you will need to research each function type, understand its definition, learn how to recognize it, and find real-world examples.

Necessary Materials:

1. Access to the internet for research.
2. Graphing paper or a graphing calculator to plot the functions.
3. Stationery for note-taking and drafting the fact sheets.
4. A computer with word processing or graphic design software for finalizing the fact sheets.

Detailed Step-by-Step for Carrying Out the Activity:

1. Form Groups and Assign Roles: Divide yourselves into groups of 3 to 5. Within each group, assign roles to each member. Suggested roles could be: Researcher, Writer, Graphic Designer, and Editor.

2. Research: Use the provided resources and any other reliable sources to research each of the three types of functions: Linear, Quadratic, and Exponential. As you conduct your research, take notes on the key points for each function.

3. Discuss and Plan: As a group, discuss your findings and start planning the content and structure of your Function Fact Sheets. Each fact sheet should be concise, and the information should be presented clearly and logically.

4. Draft Fact Sheets: Begin drafting your Fact Sheets. The researcher should provide the information, the writer should articulate it clearly, the graphic designer should create a visually appealing layout, and the editor should review and polish the content.

5. Review and Revise: After each Fact Sheet is drafted, review it as a group. Make sure the content is accurate, the language is clear, and the layout is visually appealing. Make any necessary revisions.

6. Finalize and Compile: When all three Fact Sheets are ready, compile them into a single document. This document should include a cover page with the title of your project, the names of your group members, and the date.

7. Write the Report: Write a report detailing your project following the structure: Introduction, Development, Conclusion, and Bibliography. The Introduction should contextualize the theme, its relevance, and real-world application. The Development should detail the theory behind the functions, explain the activity in detail, indicate the methodology, and finally present and discuss the obtained results. The Conclusion should revisit the main points of the project, state the learnings obtained, and the conclusions drawn about the project.

Project Deliverables:

The final deliverable should include:

1. Function Fact Sheets: A document containing three Function Fact Sheets - one for each of the three types of functions.

2. Project Report: A report detailing your project.

Remember, the goal is not just to deliver the final products but also to learn and apply your knowledge of functions. Therefore, focus on understanding the concepts and making clear connections to their real-world applications. Good luck!