Introduction

Relevance of the Theme

"Variables and Unknowns" constitutes a central pillar in Mathematics, being a crucial tool in numerous scenarios of everyday life and in advanced fields of study. Understanding variables and unknowns is fundamental to unravel patterns, formulate equations, solve problems, and explore the very nature of mathematics. Without mastering these concepts, the doors to a deeper understanding of the discipline will remain closed.

Contextualization

In the Mathematics curriculum, the thematic block of "Operations and Equations" naturally emerges after the study of numbers and arithmetic operations. Here, variables and unknowns play a prominent role: they are the key to introducing notions of the unknown and equation, skills that will pave the way for a wide range of mathematical competencies. Understanding these concepts prepares the student for situations where the solution is not immediately apparent, prompting them to explore different possibilities. Furthermore, they will be the foundations for the understanding of subsequent topics, such as proportions, functions, and more advanced algebra.

Theoretical Development

Components

• Variable: It is a symbol that represents an unknown value. In Mathematics, we use letters (x, y, z) to denote variables. A variable represents an "empty box" that can be filled with a number or an expression. The value of the variable can change depending on the context.

  • For example, in the expression 2x + 3, x is a variable. If we let x = 2, then the expression becomes 2 * 2 + 3 = 7. If we let x = -1, the expression becomes 2 * -1 + 3 = 1.

• Unknown: It is a value that is being sought in a problem. Normally, we use variables to represent unknowns. When we solve equations, we are actually looking for the value of the variable (the unknown) that satisfies the equality.

  • For example, in the equation 2x + 3 = 7, x is the unknown. By solving the equation, we find that x = 2, meaning if x is 2, then the equality is true.

Key Terms

• Equation: An equation is a mathematical expression that contains an equality between two expressions. The equality is tested for different values of its unknowns. The goal is to determine which values of the unknowns satisfy the equality.

  • For example, in the equation 2x + 3 = 7, the left side (2x + 3) is equal to the right side (7). For which values of x is this equality true?

• Simplification: It is the process of performing mathematical operations (addition, subtraction, multiplication, division, exponentiation, etc.) on an expression to make it shorter and more understandable, without altering its value. It is a crucial step in solving equations, as it transforms the original equation into simpler equivalent equations.

  • For example, in the expression 2x + 3, we can simplify by adding 2x with 3, obtaining a simpler expression: 2x + 3 = 7. Here, the left side (2x + 3) was simplified.

Examples and Cases

• Case of candies: Let there be a package of candies with an unknown number of candies. If every two candies you eat, three will remain. How many candies are in the package?

  • In this case, we can use the variable x to represent the number of candies in the package. The phrase "every two candies you eat, three will remain" can be translated into the equation 2x + 3 = x. Solving this equation, we find x = 3, which means there are three candies in the package.

• Case of triple a number: If triple a number added to seven results in 19, what is that number?

  • In this case, we can use the variable x to represent the unknown number. The phrase "triple a number added to seven results in 19" can be translated into the equation 3x + 7 = 19. Solving this equation, we find x = 4, meaning the number is 4.

• Case of the square of a number: If the square of a number minus three is equal to 10, what is that number?

  • Here we also use the variable x to represent the unknown number. The phrase "the square of a number minus three is equal to 10" can be translated into the equation x^2 - 3 = 10. Solving this equation, we find x = -3 or x = 4. Therefore, there are two possible solutions: -3 and 4.

These examples demonstrate the direct applicability of the concepts of variables and unknowns, highlighting why they are so relevant to Mathematics.

Detailed Summary

Key Points

• What is a variable?: A variable is a symbol that represents an unknown value. In mathematics, we use letters (x, y, z) to denote variables. They are essential for constructing equations and expressions with unknown values.

• Incorporating Variables in Equations: Variables are incorporated into equations to represent unknowns. An equation is a mathematical expression that contains an equality between two expressions. Solving an equation involves finding the value of the variable (unknown) that makes the equality true.

• Understanding Unknowns: An unknown is an unknown value in an equation or problem. It is the target of our search when solving an equation or problem.

• Simplification Process: Simplification is a crucial process when dealing with variables and equations. It is the act of performing mathematical operations on an expression to make it easier to manipulate, without altering its value.

• Solving Problems with Variables and Unknowns: The ability to solve everyday problems, such as calculating the number of candies in a package, can be acquired through the application of the concepts of variables and unknowns in equations.

Conclusions

• Mastering Variables and Unknowns: Mastering the concept of variables and unknowns is essential to deepen the study of Mathematics. These concepts pave the way for more complex topics, such as functions and advanced algebra.

• Practical Applicability: Variables and unknowns are not mere theoretical abstractions; they have practical applications in solving everyday problems, as well as in various areas of study and professions.

• Critical and Logical Thinking: The use of variables and unknowns in mathematical equations and problems favors the development of critical and logical thinking, as it requires the analysis of information, the formulation of assumptions, and the elaboration and resolution of strategies.

Suggested Exercises

1. Finding the Value of an Unknown: Solve the equation 2x + 3 = 7 to find the value of x.

2. Building Equations: Create an equation that represents the following situation: The triple of a number added to five is equal to 20.

3. Solving a Practical Problem: In a game, you scored a total of 50 points. Each basket is worth 2 points and each free throw is worth 1 point. How many baskets and free throws did you score if the total number of shots you took was 25? Represent this situation with an equation and solve it to find the number of baskets (x) and the number of free throws (y) you scored.