Lesson 1.2: Fractions

Introduction

Fractions are an essential part of mathematics that represent parts of a whole. Understanding fractions is fundamental because it establishes a foundation for more advanced arithmetic and mathematical concepts. In this lesson, students, we will cover various aspects of fractions, including equivalent fractions, simplification, comparison, and operations such as addition, subtraction, multiplication, and division. By the end of this lesson, you will feel more confident in working with fractions and mixed numbers.

Learning Objectives

• Define equivalent fractions, simplify fractions, and understand how to compare and order them.
• Perform operations with fractions: addition, subtraction, multiplication, and division, including handling mixed numbers.
• Convert between improper fractions and mixed numbers and simplify them to their lowest terms.

Understanding Fractions

A fraction consists of two main parts: the numerator and the denominator. The numerator indicates how many parts we have, while the denominator indicates how many parts make up a whole.

Defining Fractions

• Numerator: The top number of a fraction (e.g., in $\frac{3}{4}$, 3 is the numerator).
• Denominator: The bottom number of a fraction (e.g., in $\frac{3}{4}$, 4 is the denominator).

Equivalent Fractions

Two fractions are equivalent if they represent the same part of a whole, even if they look different. For example, $\frac{1}{2}$ is equivalent to $\frac{2}{4}$, as they both represent half of a whole.

Finding Equivalent Fractions

To find equivalent fractions, you can multiply or divide the numerator and denominator by the same non-zero number.

Example 1: Find two equivalent fractions for $\frac{3}{5}$.

1. Multiply by 2:

$$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$$

1. Multiply by 3:

$$\frac{3 \times 3}{5 \times 3} = \frac{9}{15}$$

So, two equivalent fractions for $\frac{3}{5}$ are $\frac{6}{10}$ and $\frac{9}{15}$.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. This means finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.

Steps to Simplify a Fraction:

1. Find the GCD of the numerator and denominator.
2. Divide both the numerator and the denominator by their GCD.

Example 2: Simplify the fraction $\frac{8}{12}$.

1. The GCD of 8 and 12 is 4.
2. Divide both the numerator and the denominator by 4:

$$\frac{8 \div 4}{12 \div 4} = \frac{2}{3}$$

Comparing and Ordering Fractions

To compare fractions, they must have a common denominator or you can convert them to decimals. If two fractions have the same denominator, the one with the larger numerator is greater.

Example 3: Comparing $\frac{2}{5}$ and $\frac{3}{7}$.

1. Find a common denominator. The least common multiple (LCM) of 5 and 7 is 35.
2. Convert both fractions:

$$\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}$$

$$\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}$$

1. Now compare:

Since $\frac{14}{35} < \frac{15}{35}$, we can conclude that $\frac{2}{5} < \frac{3}{7}$.

Operations with Fractions

Adding and Subtracting Fractions

When adding or subtracting fractions, if they have the same denominator, you simply add or subtract the numerators and keep the denominator the same.

Example 4: Add $\frac{2}{5} + \frac{1}{5}$.

$$\frac{2 + 1}{5} = \frac{3}{5}$$

If the fractions have different denominators, you need to find a common denominator first.

Example 5: Subtract $\frac{1}{4} - \frac{1}{6}$.

1. Find the LCM of 4 and 6, which is 12.
2. Convert both fractions:

$$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$$

1. Now subtract:

$$\frac{3}{12} - \frac{2}{12} = \frac{1}{12}$$

Multiplying Fractions

To multiply fractions, you simply multiply the numerators together and the denominators together.

Example 6: Multiply $\frac{2}{3} \times \frac{3}{4}$.

$$\frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$$

Dividing Fractions

To divide by a fraction, multiply by its reciprocal (flip the second fraction).

Example 7: Divide $\frac{1}{2} \div \frac{3}{4}$.

1. Flip the second fraction: $\frac{4}{3}$.
2. Multiply:

$$\frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$$

Mixed Numbers and Improper Fractions

A mixed number consists of a whole number and a fraction (e.g., $1 \frac{2}{3}$). An improper fraction has a numerator that is greater than or equal to the denominator (e.g., $\frac{5}{3}$).

Converting Between Improper Fractions and Mixed Numbers

To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. The result becomes the new numerator, and the denominator remains unchanged.

Example 8: Convert $2 \frac{3}{5}$ to an improper fraction.

1. Multiply the whole number by the denominator: $2 \times 5 = 10$.
2. Add the numerator: $10 + 3 = 13$.
3. Write as an improper fraction: $\frac{13}{5}$.

To convert an improper fraction back to a mixed number, divide the numerator by the denominator.

Example 9: Convert $\frac{9}{4}$ to a mixed number.

1. Divide: $9 \div 4 = 2$ with a remainder of $1$.
2. This means: $2 \frac{1}{4}$.

Conclusion

In this lesson, students, we have explored the essential aspects of fractions, including equivalent fractions, simplification, comparison, and operations. By practicing these skills, you can improve your fluidity and confidence with fractions, laying a solid groundwork for further mathematical learning.

Study Notes

• A fraction consists of a numerator and a denominator.
• Equivalent fractions are different fractions that represent the same value.
• To simplify a fraction, divide both the numerator and denominator by their GCD.
• To compare fractions, convert them to a common denominator or decimals.
• Add and subtract fractions with the same denominator; find a common denominator for different denominators.
• To multiply fractions, multiply the numerators and the denominators.
• To divide fractions, multiply by the reciprocal of the divisor.
• Mixed numbers consist of whole numbers and fractions; improper fractions have numerators greater than the denominators.
• Convert mixed numbers to improper fractions by multiplying and adding; convert improper fractions to mixed numbers by dividing.