Fractions - Equivalent Fractions

Understanding Equivalence: Equivalent fractions are different fractions that represent the same part of a whole. Visually, if you have two identical rectangular cakes, and you cut one into 2 equal parts and take 1 ($\frac{1}{2}$), and cut the other into 4 equal parts and take 2 ($\frac{2}{4}$), you have the same amount of cake in both cases.

Finding Equivalents by Multiplication: To find an equivalent fraction, you can multiply both the numerator and the denominator by the same non-zero whole number. This is like taking the same area and dividing it into even smaller, equal-sized pieces without changing the total shaded amount.

Finding Equivalents by Division: You can also find an equivalent fraction by dividing both the numerator and the denominator by a common factor. This process, often called simplification, reduces the number of parts while keeping the total value the same.

Simplest Form (Lowest Terms): A fraction is said to be in its simplest form if its numerator and denominator have no common factor other than 1. Visually, this represents the fraction using the fewest possible number of parts to describe the same whole.

The Cross-Multiplication Rule: To check if two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, you can multiply the numerator of the first by the denominator of the second ($a \times d$) and the denominator of the first by the numerator of the second ($b \times c$). If the products are equal, the fractions are equivalent.

Number Line Representation: Equivalent fractions occupy the exact same spot on a number line. For example, if you draw a number line from 0 to 1, the point exactly in the middle represents $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, and $\frac{50}{100}$ all at once.

Property of 1: Multiplying a fraction by $\frac{n}{n}$ (where $n \neq 0$) is the same as multiplying by 1. Since any number multiplied by 1 remains the same, the value of the fraction does not change, even though its 'appearance' does.