Contextualization

In mathematics, constantly dealing with numbers and their properties may seem distant from the real world. However, the discipline is full of principles and concepts that play fundamental roles in our daily lives. Today's topic, Factoring: Difference of squares, is an example of this.

Factoring is a powerful mathematical process that allows breaking down a number or an algebraic expression into a product of factors. One of the most commonly used factoring methods is the difference of squares, which refers to factoring an algebraic expression that can be represented as the difference of two squares. It is an important concept in mathematics and is commonly applied in solving quadratic equations.

The application of this concept is present in our lives in ways you may not imagine. For example, it is used in areas such as physics, to calculate speeds and distances, and in engineering, to find solutions to construction problems. In computer science, it is used in decoding and encryption algorithms, and in economics, to calculate profits and losses.

Introduction

Factoring is a mathematical process that involves breaking down a number or an algebraic expression into a product of factors. For example, the number 15 can be factored as 3 * 5. In factoring algebraic expressions, we are interested in breaking down the expression into its constituent factors.

The Difference of Squares is a special case of factoring that deals with the algebraic expression a² - b². This expression can be factored as (a - b)(a + b), where a and b are terms that can be numbers, variables, or algebraic expressions. It is a widely used technique in solving quadratic equations.

This concept is fundamental for understanding second-degree equations and how they are solved. Knowing how to factor a difference of squares is an essential tool for those who want to work with mathematics or physics. A solid understanding of this topic is also crucial for anyone interested in areas such as engineering or economics.

Here are some reliable sources to deepen your knowledge on the subject:

1. Khan Academy: Difference of squares
2. Brasil Escola: Factoring - Difference of Squares
3. Só Matemática: Factoring - Difference of Squares

Practical Activity

Activity Title: Factoring the Real World: The Difference of Squares in our Daily Lives

Project Objective

The objective of this project is to explore in a practical and playful way the concept of factoring, with a special focus on the difference of squares, and understand how this concept is applied in the real world. Students should be able to apply the concepts learned to solve everyday problems related to mathematics and physics.

Detailed Project Description

The project is divided into three parts:

1. Research and learning

2. Practical application of the concept

3. Preparation and presentation of a report on the project

Groups of 3 to 5 students will work collaboratively throughout the project. The project should last approximately two weeks, with an estimated workload of over twelve hours per participating student.

Research and Learning

Students will be tasked with researching and learning about factoring and the difference of squares. They should use the research to understand the concept, its meaning, its importance, and how it is applied in the real world.

Practical Application of the Concept

After researching and understanding the concept, students should identify real-world situations where factoring and the difference of squares are applied. They should formulate and solve problems related to these situations.

Required Materials

The project requires the use of textbooks, internet access, and office supplies for the production of the report.

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

1. Formation of groups and project planning.

2. Research on factoring, with a focus on the difference of squares.

3. Discussion of the concepts learned and how they can be applied in the real world.

4. Identification of real-world situations where the concept can be applied.

5. Formulation of problems based on the identified situations.

6. Resolution of the formulated problems.

7. Preparation of the report on the project.

8. Presentation of the report to the class.

Students should produce a detailed report on the project. The report should include an introduction to the concept, a summary of what was learned during the research, a detailed description of the identified real-world situations and formulated problems, the solutions to the problems, and the conclusions drawn from the project. Citations should be added at the end of the report in the appropriate format to indicate the sources used.

Students should work together to plan, execute the project, and prepare the report. They should use collaboration, effective communication, and critical thinking to solve the problems and compile the necessary information for the report.

The final report should be written in a way that clearly describes the group's trajectory throughout the project, the understanding of factoring concepts, especially the difference of squares, and how they were able to apply the concept in solving real-world problems. The report should demonstrate the students' understanding of the factoring concept, their problem-solving skills, and their ability to work in a team.