Contextualization

Introduction

Polynomials are algebraic expressions made up of terms, each of which consists of a coefficient multiplied by a variable raised to some power. They are a foundational concept in algebra, and understanding their properties is essential to mastering this branch of mathematics.

In this project, we will focus on four primary properties of polynomials: degree, leading coefficient, constant term, and standard form. The degree of a polynomial is the highest power of the variable in any of its terms. The leading coefficient is the coefficient of the term with the highest power of the variable, and the constant term is a term that does not have a variable. The standard form of a polynomial is when the terms are arranged from highest to lowest degree.

Polynomials have a significant application in various fields, including physics, engineering, economics, computer graphics, and many more. They are used to model real-world situations and solve complex problems. For instance, in physics, they are used to describe the motion of objects under the influence of forces. In economics, they are used to model market demand and supply curves.

To understand these properties, you will need to have a good grasp of basic algebraic operations such as addition, subtraction, multiplication, and exponentiation. You should also be familiar with the concept of variables and constants and how they are used in algebraic expressions.

Resources

The following resources will help you understand the properties of polynomials in more depth:

1. Khan Academy: Polynomials
2. Math is Fun: Algebra - Polynomials
3. Purplemath: Introduction to Polynomials
4. Book: "Algebra and Trigonometry" by Michael Sullivan and Michael Sullivan III. Chapter 5: Polynomials and Polynomial Functions.
5. YouTube: Polynomials Playlist by The Organic Chemistry Tutor

These resources provide a mix of text, video, and practice exercises, allowing you to learn the material in a way that suits your learning style. Use them as a guide and feel free to explore other resources that you find helpful.

Practical Activity

Activity Title: "Exploring Polynomials: Properties and Applications"

Objective of the Project:

To understand and apply the properties of polynomials (degree, leading coefficient, constant term, and standard form) in a real-world context. Students will also learn to model and solve problems using polynomials.

Detailed Description of the Project:

In this project, each group of students will create four polynomial equations, each highlighting a different property of polynomials. These polynomial equations should be created based on real-world scenarios, such as a business's profit model or a car's fuel efficiency.

The students will then solve these polynomial equations and present their findings in a report. The report should detail the process of creating the polynomial equations, solving them, and interpreting the results in the context of the real-world scenario.

Necessary Materials:

• Pen and paper for brainstorming and problem-solving
• A computer with internet access for research and report writing
• A mathematical software program (like Wolfram Alpha or Geogebra) for solving the polynomial equations

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

1. Form Groups and Assign Roles: Form groups of 3-5 students. Assign each group member a role, such as researcher, problem solver, writer, and presenter.

2. Research and Real-world Scenario Selection: As a group, research different real-world scenarios that can be modeled using polynomials. Choose four scenarios, each highlighting a different property of polynomials (degree, leading coefficient, constant term, and standard form).

3. Polynomial Equation Creation: For each scenario, create a polynomial equation that accurately models the situation and showcases the chosen property.

4. Solving the Polynomials: Use a mathematical software program to solve the polynomial equations. Make note of the steps taken and the results obtained.

5. Interpreting the Results: Interpret the solutions in the context of the real-world scenario. What do the solutions tell us about the scenario?

6. Report Writing: Based on the findings, write a report detailing the entire process.

7. Presentation: Each group will present their findings to the class, explaining their real-world scenarios, the polynomial equations they created, how they solved them, and what they learned from the exercise.

Project Deliverables:

At the end of the project, each group will submit a report and deliver a presentation.

The report should be structured as follows:

• Introduction: Contextualize the theme, its relevance, and real-world application. State the objective of the project.

• Development: Detail the theory behind the properties of polynomials and their real-world applications. Explain the activity in detail, including the steps taken, the methodology used, and the results obtained. Discuss the process of creating the polynomial equations, solving them, and interpreting the results.

• Conclusion: Revisit the main points of the project, state the learnings obtained, and draw conclusions about the project.

• Bibliography: List all the resources used in the project.

The presentation should cover the same points as the report, but in a more condensed and visual format. Each group will have 10-15 minutes to present, with time for questions and discussion afterwards.

The duration of the project is one week, and the report and presentation are due at the end of that week.

Remember, the goal of this project is not just to understand the properties of polynomials but also to apply them in real-world scenarios. Be creative and have fun with it!