Introduction

Relevance of the Topic

Understanding Mixed Numbers and applying Operations with them is a cornerstone in the study of Mathematics. These operations provide powerful tools for manipulating quantities expressed in fractions, enabling the resolution of a wide range of everyday problems, as well as mastering more advanced concepts in the discipline.

Contextualization

In our mathematical curriculum, Mixed Numbers are a starting point for more complex concepts such as comparing and ordering fractions, adding and subtracting fractions with different denominators, calculating fractions of quantities, among others.

We will first get to know the structure and meaning of Mixed Numbers and then explore addition and subtraction operations with these numbers. These operations will help consolidate the understanding of Mixed Numbers and provide a solid foundation for future studies in fractions and mixed operations involving fractions.

Theoretical Development

Components

• Whole Numbers and Fractions: Mixed Numbers combine the concept of a whole number with that of a fraction. They express whole(s) and a fraction in a single notation. For example, 3 ½ represents 3 wholes and ½, a fraction.

• Form and Meaning: The structure of a Mixed Number representation is "whole number, space, fraction". The whole number indicates how many whole parts there are and the fraction represents the part that is not whole. The fraction is always less than one unit (denominator smaller than the numerator).

• Addition Operation: When adding two or more Mixed Numbers, the sum of the wholes and the fractions are performed separately and then added together. If the sum of the fractions is an improper fraction (numerator greater than denominator), it is converted into a Mixed Number.

• Subtraction Operation: In the subtraction of Mixed Numbers, the subtraction of the wholes and the subtraction of the fractions are done separately. If the subtraction of the fractions results in a negative number, the whole is adjusted.

Key Terms

• Mixed Numbers: Represent a combination of a whole number with a fraction. They are expressed in the form "whole and fraction". For example, 2 3/4 is a Mixed Number.

• Improper Fraction: It is a fraction where the numerator is greater than or equal to the denominator. Examples include 5/4, 9/2, etc. Improper fractions can be rewritten as Mixed Numbers.

• Conversion of Fraction to Mixed Number: It is a process of expressing an improper fraction as a mixed number. The quotient of the division of the numerator by the denominator becomes the whole number and the remainder of the division is placed over the denominator to become the fraction.

Examples and Cases

• Example of a Mixed Number: 3 ¼ is a Mixed Number, representing 3 wholes and ¼, a fraction.

• Example of Addition in Mixed Numbers: 2 ½ + 1 ¾ = 3 ¼. The fractions are added first (2/4 + 3/4 = 5/4), which generates an improper fraction. This fraction is converted into a mixed number (5 divided by 4 is 1 and remainder 1, which becomes the fraction), resulting in 3 ¼.

• Example of Subtraction in Mixed Numbers: 4 ¾ - 2 ¼ = 2 ½. The fractions are subtracted first (3/4 - 2/4 = 1/4), resulting in 2 1/4.

Detailed Summary

Relevant Points

• Definition of Mixed Numbers: Are ways to represent quantities that combine wholes and fractions. Their representation is in the form "whole and fraction". For example, 1 ½ represents one whole and half.

• Structure of Mixed Numbers: They are expressed in the form "whole number, space, fraction", where the whole number indicates the whole parts and the fraction indicates the part that is not whole. The fraction is always less than one unit, that is, the denominator is always greater than the numerator.

• Composition of Mixed Numbers: The fraction in a Mixed Number always has the same denominator as the whole. This is a consequence of the fraction representing the part without the whole.

• Operations with Mixed Numbers: The operations of addition and subtraction with Mixed Numbers are performed in two stages: first, the addition or subtraction of the wholes, and then, the addition or subtraction of the fractions. The sums or differences are expressed in the form of Mixed Numbers.

• Improper Fractions and Mixed Numbers: Improper fractions are those in which the numerator is greater than or equal to the denominator. They can be converted into Mixed Numbers, where the quotient of the division of the numerator by the denominator becomes the whole and the remainder of the division becomes the fraction.

Conclusions

• Understanding Mixed Numbers and mixed operations with them is a fundamental step for understanding more complex concepts in Mathematics, especially in relation to working with fractions and their operations.

• Mixed Numbers provide an efficient way to represent quantities that include wholes and fractions and allow us to better understand the relationship between the two.

• The process of converting improper fractions into Mixed Numbers and vice versa is a valuable tool for working with fractions and solving problems that involve mixed quantities.

Exercises

1. Addition Exercise with Mixed Numbers: Use the Mixed Numbers 2 ¾ and 1 ½ to perform a sum. Check if the result can be expressed as a Mixed Number.

2. Subtraction Exercise with Mixed Numbers: Perform the subtraction 3 ½ - 1 ¼ using Mixed Numbers. Check if the answer can be expressed as a Mixed Number.

3. Conversion Exercise of Improper Fraction to Mixed Number: Convert the improper fraction 5/3 into a Mixed Number. Check if the conversion is correct by performing the reverse operation, i.e., convert the resulting Mixed Number back into an improper fraction.