Contextualization

Polynomials are a very versatile form of mathematical expression, used in various areas of knowledge, from physics to economics. They are the basis for many algorithms in computer science and engineering, besides representing real situations in fields such as chemistry and biology. The ability to perform basic operations with polynomials – addition, subtraction, multiplication, and division – is a crucial step in understanding these and other more complex concepts.

In everyday life, when playing a video game, using a cell phone application, or even making a simple internet search, you are using algorithms that often have polynomial operations as their basis. Understanding these operations allows us to have a better comprehension of the world around us and how things work.

Introduction

A relevant characteristic of polynomials is that they are continuous functions, that is, they do not present discontinuities or "jumps" in their graphs. Because of that, polynomials are extremely useful for modeling real world phenomena that are continuous too, such as the trajectory of a projectile, the growth of a population, or the depreciation of an asset among others.

Another important aspect is that, even though polynomials seem simple, polynomial operations involve a range of mathematical skills, such as manipulating algebraic expressions, understanding the properties of operations, and handling abstractions. Practicing these skills is essential to enhance mathematical knowledge.

This project will use everyday objects to represent polynomials and their operations in a way that encourages teamwork, effective communication, and time management. The project also involves other subjects, such as physics and history, making learning more interdisciplinary.

Hands-on Activity: "Polynomials in the Real World"

Project Goal

To understand the structure and function of polynomials through physical representation, and to explore the operations of sum, subtraction, multiplication and division of polynomials, through a creative and engaging hands-on project.

Detailed Description of the Project

Each group will select an everyday object that can be mathematically described by a polynomial. They will use this object to demonstrate the sum, subtraction, multiplication, and division of polynomials.

The group must work to physically represent these operations with the chosen object. To create their representation, the students may use any materials: from colorful papers and drawings to 3D modeling software. They will then create an explanatory video showing the step-by-step process of the operations.

The second part of the work involves research about polynomials in human history. Students will search and discover a time in history in which polynomials were particularly important and will illustrate it through a 5-slide presentation.

Finally, students will reflect on how polynomial operations are applied to solve real world problems, particularly in their area of interest. They must identify a problem, develop a solution using polynomials and present it in a written paper.

Required Materials

• Everyday objects.
• Creative materials (colorful papers, paints, fabrics etc.)
• Digital tools (video editing software, 3D modeling software, etc.).
• Books and online resources for research.

Step-by-Step Instructions for Completing the Activity

1. Form teams of 3 to 5 students and have each team choose an everyday object that can be represented by a polynomial.

2. Each team must use their object to physically demonstrate the operations of addition, subtraction, multiplication and division of polynomials. Students are encouraged to be as creative as possible in their representations.

3. Following the physical representation of the operations, teams will create an explanatory video introducing and explaining the step-by-step process to perform the operations.

4. Once they finish the video, each group should research a time when polynomials have been of great impact on history. Then they will create a 5-slide presentation illustrating this moment in history.

5. Finally, each team will choose a real world problem from an area of interest to them that could be solved using polynomials. They must develop a solution for this problem and present it in a written paper, which must include an introduction, development, conclusion and bibliography.

6. Every project deliverable (along with the written paper) should be submitted to the teacher for grading.

Deliverables of the Project

Students should deliver the project in three parts:

1. The physical representation of the polynomial operations, and the explanatory video.

2. The slide presentation about polynomials in history.

3. The written paper about the use of polynomials to solve a real world problem.

The paper should be written as an extended explanation of the project and its main sections should be:

1. Introduction: Explain the context of the chosen real world problem, indicate why it is relevant to solve it using polynomials and state the objective of the paper.

2. Development: Explain the theory behind the polynomials and their operations, present the methodology used to solve the chosen problem and finally present and discuss the results.

3. Conclusion: Summarize the paper's key points, highlight what was learned, and state the conclusions about the project.

4. Bibliography: List the sources that the team consulted for the project.

The estimated workload for this project is 12+ hours per student, and it approaches both hard skills such as solving polynomial operations, and soft skills such as time management, communication, problem solving and creative thinking.