Introduction

Relevance of the Topic

Sorting rational numbers is a fundamental concept in Mathematics. It is the foundation for many more advanced areas of study, including functions, calculus, and algebra. Moreover, the ability to sort rational numbers is applied in various real-life situations, from sorting different-sized fractions in a recipe to comparing prices at the supermarket.

Contextualization

The sorting of rationals is part of the 7th-grade Mathematics curriculum, being a natural extension of the sorting of integers and natural numbers, which are taught in previous years. Understanding and mastering the sorting of rationals is a crucial step in the progression of learning Mathematics, moving from the concrete (natural numbers, integers) to the abstract (rationals, irrationals, reals).

This topic not only enhances students' understanding of the numerical system but also leads them to explore concepts such as equivalence, inequality, decimal numeration, among others. Therefore, the ability to sort rationals serves as a bridge to a deeper understanding of Mathematics.

Theoretical Development

Components

• Rational Numbers: These are all numbers that can be expressed in the form a/b, where "a" and "b" are integers and b ≠ 0. The fraction 3/5, for example, is a rational number. However, not every rational number is a fraction. For example, 0.75 is a rational number, but it is not a fraction.

• Module in R: The module (or absolute value) in R is the positive value of a real number. For example, the module of -4 is 4, and the module of 4 is also 4. The module of a rational number is the module of the numerator divided by the module of the denominator.

• Rational Number Criterion: In the case of two rational numbers having the same sign, the number with the larger module is considered the larger number. If they have opposite signs, then the number with the smaller module is the larger number. For example, -3/4 < -2/3 < -1/2.

Key Terms:

• Ordering: In Mathematics, the term "ordering" refers to classifying numbers in a specific order, usually from smallest to largest or from largest to smallest. For rationals, this is done considering their modules and signs.

• Positional Value: It is the position that a digit occupies in a number. For example, in the number 486, the "8" represents the positional value of 80.

• Decimal Numerical System: The numerical system we use in everyday life is the decimal system, which is base 10. This means that each position in the number (units, tens, hundreds, etc.) represents a multiple of 10. Therefore, the value of each digit depends on its position.

Examples and Cases

• Example 1: Order the fractions 2/3, 4/5, and 1/2. For this, we must consider the modules of the numerators and denominators. Their modules are 2/3, 4/5, and 1/2. Comparing modules, we have the order: 1/2, 2/3, 4/5.

• Example 2: Order the decimal numbers 0.78, 0.9, and 0.45. In this case, it is important to keep in mind the concept of positional value. These numbers are equivalent to the fractions, 78/100, 9/10, and 45/100. Comparing these fractions, we have the order: 0.45, 0.78, 0.9.

• Example 3: Order the rational numbers -9/4, -4/3, and -1/2. In this case, we are dealing with negative numbers. Applying the rational number criterion, the order is: -4/3, -9/4, -1/2.