Parts and Wholes - Comparing and Ordering Fractions

Understanding Fractions: A fraction represents a part of a whole and is written as $\frac{Numerator}{Denominator}$. The denominator tells us the total number of equal parts the whole is divided into, while the numerator tells us how many of those parts are being considered. Imagine a circular pizza cut into 8 equal slices; if you eat 3 slices, you have consumed $\frac{3}{8}$ of the pizza.

Like Fractions: Fractions that have the exact same denominator are called like fractions, such as $\frac{2}{9}$ and $\frac{5}{9}$. Visually, this is like comparing two identical chocolate bars where both are divided into 9 equal pieces. Because the size of each piece is the same, we can compare them easily by just looking at the number of pieces (the numerator).

Unlike Fractions: Fractions with different denominators are called unlike fractions, such as $\frac{1}{3}$ and $\frac{1}{4}$. If you visualize two identical rectangles, the one divided into 3 parts will have larger individual pieces than the one divided into 4 parts. Therefore, unlike fractions involve parts of different sizes.

Equivalent Fractions: These are fractions that look different but represent the same value or the same part of a whole. For example, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{5}{10}$ are all equivalent. On a number line, these fractions would all land on the exact same spot, halfway between $0$ and $1$.

Comparing Like Fractions: When denominators are the same, the fraction with the larger numerator is the greater fraction. For example, $\frac{5}{6} > \frac{1}{6}$. If you have two strips of paper of the same length divided into 6 parts, shading 5 parts covers more area than shading only 1 part.

Comparing Fractions with the Same Numerator: If the numerators are the same, the fraction with the smaller denominator is actually the larger fraction. For example, $\frac{2}{3} > \frac{2}{5}$. Think of sharing 2 apples among 3 people versus 2 apples among 5 people; the group of 3 people will get larger portions.

Comparing Unlike Fractions using Cross-Multiplication: To compare two unlike fractions like $\frac{a}{b}$ and $\frac{c}{d}$, we cross-multiply. We calculate $a \times d$ and $b \times c$. If $a \times d$ is greater than $b \times c$, then $\frac{a}{b} > \frac{c}{d}$. This is a quick way to check which fraction covers more of the 'whole' without drawing them.

Ordering Fractions using LCM: To arrange unlike fractions in ascending (smallest to largest) or descending (largest to smallest) order, we find the Least Common Multiple (LCM) of the denominators. We then convert each fraction into an equivalent fraction with that common denominator. This turns them into like fractions, making them easy to order by comparing their new numerators.