Contextualization

Pascal's Triangle is an arrangement of numbers arranged in a triangular shape that generates fascinating patterns and has various applications in mathematics. The importance of Pascal's Triangle is notable in various aspects and concepts of mathematics, such as the combination of elements, in Newton's binomials, and even in probabilities.

Theoretical Introduction

Pascal's Triangle is a triangular figure formed by numbers in a series of lines that form numerical patterns. It starts with “1” at the top, and the subsequent numbers are obtained by adding the two numbers directly above. For example, the third row is 1 2 1, where 2 is the sum of 1 and 1; in the next row, we have 1 3 3 1, where 3 is the sum of 1 and 2, and also of 2 and 1.

Pascal's Triangle can also be used to expand (a+b)^n, where n is a non-negative integer. In other words, we can simplify the expression (a+b)^n with the help of Pascal's Triangle. Additionally, there are a number of surprises and hidden patterns in the design of Pascal's Triangle, such as triangular numbers, squares, and Fibonacci.

Importance of Pascal's Triangle

Pascal's Triangle has great relevance in various areas of mathematics, but also in nature and art. It can be used to calculate permutations and combinations - a common topic in probability and statistics. It also has applications in binomial series, geometry, and fractals.

Pascal's Triangle is deeply connected to many different concepts and has practical applications in real life, such as: combinatorial analysis, game theory, algorithms, computer science, statistics, and many others. It is a topic with great potential to arouse curiosity and exploration, allowing mathematics to be seen as a dynamic and surprising field.

Practical Activity

Activity Title: Exploring Pascal's Triangle

Project Objective

Students should apply the concept of Pascal's Triangle to:

1. Calculate binomial coefficients.
2. Identify patterns in Pascal's Triangle.
3. Apply the binomial theorem to expand polynomials.
4. Discuss applications in everyday situations and in various areas of knowledge.

Detailed Project Description

Students will work in groups of 3 to 5 members, dividing tasks of research, calculations, practical experiments, and report writing.

The project will be divided into two main parts: the practical part, where students will perform some activities involving Pascal's Triangle, and the theoretical part, where students will research the subject and write a document containing the four topics: Introduction, Development, Conclusions, and Bibliography used.

Required Materials

For this project, you will need:

• Pencil and paper or spreadsheet software (excel, google sheets, etc) for constructing Pascal's Triangle.
• Internet access for research and studies.

Detailed Step-by-Step for Activity Execution

1. Group Formation: Form groups of 3 to 5 students.
2. Theory Study: Students should study the properties, applications, and history of Pascal's Triangle.
3. Construction of Pascal's Triangle: Using pencil and paper or a spreadsheet software (such as excel or google sheets), students should construct Pascal's Triangle up to the 10th line.
4. Identify Patterns: Students should identify the different patterns in Pascal's Triangle that were discussed during the theory study.
5. Application of the Binomial Theorem: Students should expand some polynomials using Pascal's Triangle and compare the results with direct expansion using the Newton binomial formula.
6. Discussion of Applications: Students should research and discuss the applications of Pascal's Triangle in real life, in different fields of knowledge.
7. Preparation of the Final Report: Next, students should prepare a detailed report of the activity containing the results of the activities, the answers to the questions, and their conclusions.

Project Deliverables

Students should deliver the following items:

1. Final Report: The report should include an introduction, with brief explanations about Pascal's Triangle and its uses in mathematics. The development should explain step by step the activities carried out and the observations made. The conclusion should address the achievements and learnings of the project. Finally, a bibliography should be included, listing all materials consulted during the project preparation.
2. Pattern Analysis: A section of the report should be dedicated to the analysis of the patterns observed in Pascal's Triangle.
3. Expanding Polynomials: A section of the report should describe the application of the binomial theorem to expand polynomials, with complete examples.

When developing the report, students should fit the discussions and activities carried out in the project, explaining how they contribute to a deep understanding of Pascal's Triangle. Additionally, students should use the report to express their learnings and insights about Pascal's Triangle and its multiple uses.