Contextualization

Theoretical Introduction

Polynomials are essential in all areas of mathematics. They are mathematical expressions that involve numbers and variables, and can be as simple as x+2 or as complex as 5x^4 - 3x^3 + 2x^2 - x +1. To effectively work with polynomials, we need to understand their essential properties. The properties of polynomials include equality properties, basic operations such as addition and subtraction, multiplication and division, and how to solve polynomials.

The first property to be understood is the property of equality. Two polynomials are considered equal if and only if the corresponding coefficients of the same degree terms are equal. The degree of a polynomial is the highest degree of its terms. This property is essential for solving polynomials.

Basic operations on polynomials, such as addition, subtraction, and multiplication, follow the same rules as operations on real numbers. Division, however, is a bit more complex and involves the concept of synthetic division. Effective operations with polynomials and understanding the properties of polynomials allow us to solve complex problems.

Contextualization

Polynomials and their properties have various applications in many areas such as physics, engineering, economics, and biology. In physics, for example, polynomial equations are used to describe the motion of objects. In engineering, they are used to calculate material resistances, while in economics, they are used to model growth rates. In other words, the study of polynomials is an essential tool for solving complex and real-world problems.

The ability to work with polynomials opens doors to other areas of mathematics and sciences, such as calculus. Furthermore, familiarity with polynomials and their properties also makes you a more critical and logical thinker, essential skills in any career.

Sources for Research and Further Study:

• Khan Academy: Polynomials
• S.O.S Mathematics: Polynomials
• Brazil School: Polynomials
• Fundamentals of Elementary Mathematics Collection – Vol. 5: Polynomials and Algebraic Equations – Gelson Iezzi, Carlos Murakami

Practical Activity

Activity Title: "Polynomials in Practice: From Theory to Application"

Project Objective:

This project aims to provide students with a deeper understanding of the properties of polynomials through their application in real-world scenarios. Students will be encouraged to apply their theoretical math skills, work in teams, and, at the same time, develop their research and communication skills.

Detailed Project Description:

Based on their research and studies on the properties of polynomials, each group should choose an area of science or engineering (such as physics, economics, electrical engineering, etc.) and develop a mathematical model that predicts some important phenomenon or process. This activity involves creating a detailed report and presenting the findings.

Required Materials:

• Mathematics books and/or access to online resources for research.
• Mathematics software such as Maple, Mathematica, or GeoGebra (optional)
• Paper and pen for sketches and calculations
• Presentation software such as PowerPoint, Google Slides, or Prezi

Detailed Step-by-Step:

1. Research and topic selection (Estimated duration: 3 hours): Each group should research and choose a topic of interest that involves the direct or indirect application of polynomial properties.

2. Development of the mathematical model (Estimated duration: 5 hours): The groups should then develop a mathematical model using polynomials that describes or explains the chosen phenomenon.

3. Interpretation and validation of the model (Estimated duration: 2 hours): Students should then interpret the model results and validate its accuracy and applicability. This step involves performing calculations, data analysis, and possibly conducting small experiments (if applicable).

4. Writing the report (Estimated duration: 2 hours): Each group should then prepare a detailed report that describes the entire process of model development, their conclusions, and their learnings. The report should follow the structure mentioned above (Introduction, Development, Conclusions, Bibliography).

5. Preparing the presentation (Estimated duration: 2 hours): Finally, the groups should prepare a visual presentation of their project, which will be presented to the class. The presentation should be creative, engaging, and clear, exposing the chosen topic, the process of model development, the results obtained, and the conclusions drawn.

Project Deliverables:

The project has two main deliverables:

1. Written report: Including all the topics mentioned above. The report should clearly demonstrate that students understand the properties of polynomials and are able to apply them to real-world problems. Additionally, the report should show that students are able to work in teams and communicate their ideas clearly and logically.

2. Visual presentation: Which is an exhibition of the work done by the students. The presentation should be creative, engaging, and should demonstrate the students' ability to communicate their ideas effectively.

By the end of the project, students will have had the opportunity to deepen their technical skills and knowledge about polynomials and their properties, as well as develop important socio-emotional skills such as time management, communication, problem-solving, creative thinking, and proactivity.

Note that the requirement for this project to be interdisciplinary is achieved through the need to integrate mathematics knowledge with other areas of science and engineering, such as physics, economics, etc., in the development of the mathematical model.