Introduction

Relevance of the Theme

"Variables" are essential components of mathematics, being a crucial connection between elementary arithmetic and advanced algebra. They are fundamental in solving mathematical problems, as well as in other areas of knowledge, such as physics, economics, software engineering, among many others. Understanding variables is a turning point, moving beyond mere numerical operations to a realm of more abstract ideas and concepts, establishing the basis for the understanding of more complex mathematical concepts.

Contextualization

In the 6th-grade mathematics curriculum, students are first becoming familiar with symbolic and algebraic representation, which are the basis for future algebra. The study of variables, therefore, prepares the ground for the construction of more advanced concepts, such as algebraic expressions, equations, and inequalities. Furthermore, this knowledge also contributes to solving everyday problems, as variable concepts are frequently used in practical contexts. A solid understanding of variables now will drive success in understanding subsequent math subjects.

Theoretical Development

Components

• What is a Variable? A 'variable' is a symbol that represents an unknown value. It derives its name from 'vary,' as the value it represents can change depending on the context. In mathematics, variables are often represented by letters of the alphabet, such as 'x,' 'y,' or 'z'.

• Value of a Variable: The value of a variable can vary depending on the context. This unknown value is what we seek to find or express in an equation. In a mathematical expression, a variable can be replaced by any number, leading to different results.

• How are Variables Used? Variables are used to represent a range of quantities, from unknown numbers to changing quantities. They are used to form algebraic expressions, equations, and inequalities, which are fundamental tools for modeling real-world problems in mathematics.

Key Terms

• Independent and Dependent Variables: In a mathematical relationship, the variable whose value is known or defined is called the 'independent variable.' It 'influences' or 'determines' the dependent variable. The dependent variable, in turn, 'depends' on the value of the independent variable. For example, in the formula for calculating the area of a circle 'A = π r^2,' 'r' is the independent variable (the radius of the circle) and 'A' is the dependent variable (the area of the circle).

• Constants: In a mathematical expression, constants are fixed values that do not change. For example, in the formula for the area of a circle, 'π' is a constant, as its value is always the same, regardless of the radius value.

• Concept of 'Solving an Equation': Solving an equation means finding the value of the variable that makes the equation true. This is done by applying equivalent mathematical operations, both on each side of the equation and for the equation as a whole, until the variable is isolated.

Examples and Cases

• Example 1: Algebraic Expression If we have a box with 'x' apples and add 'y' more apples to it, the total number of apples in the box can be represented by the expression 'x + y'. Here, 'x' and 'y' are the variables representing the quantity of apples in the box.

• Example 2: Equation The equation '3x + 5 = 20' represents the situation where we have an unknown number multiplied by 3, added to 5, and the total results in 20. In this case, we are trying to find the value of 'x' that satisfies the equality. Solving the equation results in 'x = 5'.

• Example 3: Independent and Dependent Variables If the cost to buy 'x' books is 'y' reais, then 'x' is the independent variable and 'y' is the dependent variable. For each value of 'x' we choose (independent variable), there will be a corresponding value of 'y' (dependent variable) that will determine the total cost of the books.

Detailed Summary

Key Points

• Meaning and Use of Variables: Variables are symbols that represent unknown values. They are essential in mathematics to model and solve problems. Variables can be replaced by any number, which can lead to different results.
• Independent and Dependent Variables: In mathematics, a variable that is known or defined is called the 'independent variable.' It influences or determines the value of another variable called the 'dependent variable'.
• Importance of Constants: Constants are fixed values that do not change in a mathematical expression, they accompany or mix with the variables. For example, in 'A = π r^2,' π is a constant.

Conclusions

• Application of Variables: Variables are used to represent unknown or changing quantities and are fundamental for mathematical expressions, equations, and inequalities. Understanding their applicability opens doors to solving mathematical and real-world problems.
• Solving Equations: The concept of 'solving an equation' has been elucidated, highlighting the importance of equivalent mathematical operations to isolate the variable and find its solution.

Exercises

1. Write an algebraic expression using the variables 'a' and 'b' that represent the sum of 'a' apples and 'b' oranges.
2. Consider the equation '2x + 3 = 11'. Solve it, finding the value of 'x'.
3. In a situation, a car salesman earns a fixed amount per month (R$ 500.00) plus a commission (R$ 100.00) for each car sold. Write this situation in the form of an equation, with 'v' representing the total amount the salesman earns in a month and 'c' representing the number of cars sold. How much does the salesman earn if he sells 5 cars in the month?