Contextualization

Introduction to Function: Concept

The concept of a 'function' is a fundamental one in mathematics and plays a significant role in various disciplines such as physics, engineering, economics, and computer science. Simply put, a function is a rule that takes an input and produces an output. It is like a machine that takes an object, does something to it, and gives you a result.

In mathematics, we use functions to express the relationship between two or more variables. These variables can be numbers, like in the simple function f(x) = x + 2, where 'x' is the variable and the rule of the function is to add 2 to the input. Or, they can be more complex, like in the function that describes the trajectory of a projectile in physics, where the variables are time, velocity, and acceleration.

A function has a domain, which is the set of all possible inputs, and a range, which is the set of all possible outputs. For example, in the function f(x) = x^2, the domain can be any number, but the range is always greater than or equal to zero.

Why is it important?

The concept of a function is one of the building blocks of mathematics. It helps us understand and explain the world around us by providing a way to model and analyze real-world phenomena. For example, functions are used in physics to describe the motion of objects, in economics to model supply and demand, and in computer science to write algorithms.

Moreover, understanding functions is key to understanding more advanced mathematical concepts. It forms the basis for calculus, which is used in many fields including engineering, physics, and economics. Functions also play a central role in algebra and are used extensively in geometry, statistics, and probability.

Resources

1. Khan Academy: Functions
2. Math is Fun: What is a Function?
3. Purplemath: Introduction to Functions
4. Book: "Algebra and Trigonometry" by Michael Sullivan. Chapters 2 and 3 specifically touch upon the concept of functions in detail.

Practical Activity

Activity Title: Exploring Functions

Objective of the Project

The main objective of this project is to understand and apply the concept of functions in a practical manner. Students will create and manipulate simple functions, explore their domains and ranges, and learn about different types of functions.

Detailed Description of the Project

In this project, students will work in groups of 3 to 5 and create a "Function Maker" machine. This machine will take a number as an input, apply a pre-defined mathematical function to it, and produce an output. The function can be as simple as addition or subtraction, or it can be more complex like multiplication or division.

The students will then use their "Function Maker" to explore different functions. They will change the input and observe how the output changes. They will also discuss the concepts of domain and range and try to predict the output for different inputs.

Necessary Materials

• Cardboard or foam board
• Rulers
• Markers
• Scissors
• Glue
• Small objects (like beads or buttons) for input and output

Detailed Step-by-step for Carrying out the Activity

1. Prepare the Function Maker: Each group will design and build their own "Function Maker" machine using the materials provided. The machine should have a place to put the input, a place to display the output, and a mechanism to apply the function.

2. Define the Function: The group will then choose a mathematical function to use in their machine. This can be a simple function like addition, or it can be more complex like multiplication or division. The function should be written on a card and placed in the machine.

3. Test the Function: The group will test their machine by putting different inputs and observing the outputs. They should record their observations in a table.

4. Explore Different Functions: After understanding how their machine works, the group will explore different functions. They can change the function card and repeat the test, or they can change the input and predict the output. They should record their observations for each function in the table.

5. Discuss the Findings: The group will then discuss their findings. They should try to identify patterns and relationships between the inputs and outputs, and relate these back to the concept of functions. They should also discuss the concepts of domain and range and try to predict the output for different inputs based on their observations.

Project Deliverables

At the end of the project, each group will submit a written report and a presentation. The report should contain the following sections:

1. Introduction: This should provide context for the project, explain the concept of functions, and state the objectives of the project.

2. Development: This section should detail the theory behind functions, describe the "Function Maker" machine and the functions used, and explain the methodology followed.

3. Results: This section should present the observations made, the patterns and relationships identified, and the understanding of the concepts of domain and range.

4. Conclusion: The conclusion should revisit the main points of the project, state the learnings obtained, and draw conclusions about the project.

5. Bibliography: The report should list all the sources of information used in the project.

The presentation should summarize the main points of the report and should include a demonstration of the "Function Maker" machine.

The total duration of the project is one week, with an estimated workload of 3 to 5 hours per participating student.