Introduction

Relevance of the Topic

Fractions and Decimal Numbers: Conversion is a central theme in the study of mathematics. It bridges two fundamental formats of numerical representation, allowing students to understand the relationship between them. The ability to convert numbers between fraction and decimal is crucial for solving a variety of mathematical and real-life problems, such as understanding proportions, making financial calculations, measuring distances, and comprehending statistics.

Contextualization

The conversion between fractions and decimal numbers is a natural extension of the study of fractions and decimal numbers, topics that are routinely explored in the early years of Elementary School. By the 6th grade, students are ready to deepen their understanding of these concepts and expand their mathematical skills.

The theme of conversion is fundamental to the Mathematics curriculum and establishes a solid foundation for more advanced topics. Learning to convert between fractions and decimals is a precursor to the study of percentages, a vital skill in the real world. Furthermore, placing fractional and decimal numbers on the number line enhances the understanding of the number space, a crucial concept in mathematics.

In a broader context, proficiency in converting fractions to decimals and vice versa also contributes to overall proficiency in numbers and operations, one of the key domains of mathematics. Thus, this topic has profound implications for students' learning trajectory.

Theoretical Development

Components

• Fractions: Numerical representation in the form of numerator/denominator that indicates the proportional quantity of a whole. Example: 1/2 represents half of a whole.

  • Numerator: The top part of a fraction that indicates how many parts of the whole are being considered.
  • Denominator: The bottom part of a fraction that represents the total number of parts into which the whole has been divided.
  • Proper Fraction: A fraction whose numerator is smaller than the denominator. Example: 3/4.
  • Improper Fraction: A fraction whose numerator is greater than the denominator. Example: 5/2.
  • Mixed Fraction: Combination of a whole number and a proper fraction. Example: 2 1/2.

• Decimal Numbers: Numerical representation based on the base 10 system. Consists of an integer part and a fractional part divided by a comma or decimal point. Example: 3.25.

  • Integer Part: Number to the left of the decimal point.
  • Decimal Part: Number to the right of the decimal point.

• Conversion from fractions to decimals: Process of transforming a fraction into a decimal number by dividing the numerator by the denominator.

  • Example: The fraction 3/4 is converted to a decimal by dividing 3 by 4, resulting in the decimal 0.75.

• Conversion from decimals to fractions: Process of transforming a decimal number into a fraction. The number to the right of the decimal point becomes the numerator, and the denominator is a power of 10 based on the number of digits to the right of the decimal point.

  • Example: The decimal 0.75 is converted to a fraction with the numerator 75 and denominator 100 (based on 2 digits to the right of the point). Simplifying the fraction, we get 3/4.

Key Terms

• Fraction: Representation of a proportional quantity of a whole.
• Decimal Number: Numerical representation in the base 10 system.
• Conversion: Process of changing from one form of numerical representation to another.
• Mixed Number: Combination of a whole number and a fraction.

Examples and Cases

• Example 1: Converting the fraction 1/2 to a decimal. We divide the numerator by the denominator: 1 ÷ 2 = 0.5. Therefore, 1/2 = 0.5.
• Example 2: Converting the decimal 0.75 to a fraction. The number 75 becomes the numerator and the denominator is 100 (based on 2 digits to the right of the point). Simplifying the fraction, we get 75 ÷ 25 / 100 ÷ 25 = 3/4.
• Example 3: Placing the fraction 3/4 and its equivalent decimal 0.75 on the number line. Both are located between the whole numbers 0 and 1, closer to 1.
• Case: Solving a problem involving the conversion of fractions to decimals. If you have 3/4 of a pizza and want to divide it equally between 2 people, how much does each person get? Converting 3/4 to a decimal, we have 0.75. So, each person receives 0.75 / 2 = 0.375 of the pizza. That is, a little over one-third of the pizza.

Detailed Summary

Key Points

• Understanding Fractions and Decimals: Fractions and decimals are two ways of representing fractional quantities. While fractions are represented as a division of two integers (numerator and denominator), decimal numbers use the decimal point to separate the integer part from the fractional part.
• Conversions: The ability to convert fractions to decimals and vice versa is vital in the study of mathematics. This is done by dividing the numerator by the denominator in the conversion from fractions to decimals and by transforming the number after the decimal point into the numerator and the power of 10 corresponding to the number of digits after the decimal point into the denominator in the conversion from decimals to fractions.
• Number Line: Placing fractions and decimals on the number line helps visualize and better understand the relationships between them, as well as order and size relationships.

Conclusions

• Relevance of Conversion Skill: The ability to convert between fractions and decimals is essential for deepening the understanding of the relationship between these two formats of numerical representation. Additionally, this skill is fundamental for solving everyday problems and real-life situations involving fractions and decimals.
• Comparison and Ordering on the Number Line: Representation on the number line not only aids in visualizing fractions and decimals but also facilitates the comparison and ordering of these numbers.
• Problem Solving: The use of fractions and decimals along with the ability to convert between them becomes a powerful tool in solving problems in various situations and contexts.

Exercises

1. Convert the fraction 7/8 to a decimal. Why is this decimal greater than 0.5?
2. Convert the decimal 0.6 to a fraction. Is it possible to simplify this fraction? If so, how?
3. Represent on the number line the fraction 2/3 and its equivalent decimal. Which whole number is closest to them?