Contextualization

Newton's Binomial Theorem is a fundamental concept in mathematics, with established relevance in algebra, as well as being a cornerstone in understanding more advanced concepts in mathematics, physics, and even biology.

It is a formula that expresses the expansion of powers of a binomial. Named after Isaac Newton, this formula has a wide range of applications, ranging from simple calculations of binomial powers to solving complex problems in differential calculus and statistics.

Introduction

Newton's Binomial Theorem is an expression that presents the expansion of a binomial (a + b) raised to a natural number n. It is reduced to a series of terms, written in the form of the combination of binomial coefficients, which are found in Pascal's Triangle. This expansion develops from the repeated application of the distribution formula.

The application of Newton's Binomial Theorem is not limited to the field of mathematics alone. In physics, it is often used in the analysis of problems involving Taylor series. In the field of biology, Newton's Binomial Theorem also plays an important role, where it can be applied, for example, in the analysis of genetic probabilities.

Now that we understand what Newton's Binomial Theorem is and its relevance, it is essential that we delve deeper into this concept, further exploring its applications, the reason why it works, and its relationship with other fields of mathematics.

Practical Activity

Activity Title: Unraveling Newton's Binomial Theorem through Problem Solving

Project Objective

The objective of this project is to enhance students' understanding and skills with Newton's Binomial Theorem. They will learn the theoretical concept, practical application of Newton's Binomial Theorem, and the connection with other fields of Mathematics through problem solving, research, and presentation.

Detailed Project Description

Students will work in groups of 3 to 5. Each group will be responsible for exploring and presenting in detail Newton's Binomial Theorem and its practical application through problem solving, totaling up to 10 hours per student.

The project will be divided into three parts:

1. Research and theoretical study (3 hours)

   • Groups will research in books and on the internet about Newton's Binomial Theorem and its applications.
   • After completing the research, groups will study Newton's Binomial Theorem and the applications found.

2. Problem solving (5 hours)

   • Each group will select problems that use Newton's Binomial Theorem and solve them.
   • The problems should not be too simple; they need to have a certain level of difficulty that requires the application of Newton's Binomial Theorem.

3. Preparation of the presentation (2 hours)

   • Each group will prepare a presentation on Newton's Binomial Theorem, the problems solved, and their results.

Required Materials

• Notebook for research
• Paper and pencil for problem solving
• Presentation software (Google Slides, PowerPoint, etc.)

Detailed Step-by-Step Guide for the Activity

1. Form groups of 3 to 5 students.
2. Research Newton's Binomial Theorem, follow the recommended sources, and look for more.
3. Study the concept, how to calculate the expansion of a binomial, calculate the independent term of x of a binomial, calculate the sum of the coefficients of the expansion of a binomial, find the value of the coefficient of a specific term of a binomial.
4. Select and solve problems that use Newton's Binomial Theorem. Try to pick problems from different areas (e.g., physics, statistics, and genetics).
5. Prepare a presentation together, including the theory of Newton's Binomial Theorem, the problems solved, and the results.

Submission and Written Document

After completing the practical activities, students must prepare a report based on their findings, problem solutions, and presentations. The report should include:

1. Introduction: Contextualization of the theme, its relevance and real-world application as well as the objective of this project.
2. Development: The theory behind Newton's Binomial Theorem, detailed explanation of the activity in detail, the methodology used, and finally presentation and discussion of the results obtained.
3. Conclusions: The main points, explaining the learnings obtained and the conclusions drawn from the project.
4. Bibliography: The sources used to work on the project such as books, web pages, videos, etc.

In addition to the written document, students must present the theory, problem solutions, and results discussion in a 15-20 minute presentation.

At the end of this project, it is expected that students have acquired not only technical skills but also socio-emotional skills such as time management, communication, problem solving, creative thinking, proactivity, etc.