Fractions and Decimals - Comparing and Ordering Fractions

Understanding the Parts: A fraction $\frac{a}{b}$ consists of a numerator $a$ (how many parts we have) and a denominator $b$ (how many equal parts make a whole). Imagine a chocolate bar divided into $8$ equal squares; the denominator is $8$, and if you eat $3$ squares, the numerator is $3$, represented as $\frac{3}{8}$.

Comparing Fractions with Like Denominators: When two fractions have the same denominator, the pieces are the same size. Therefore, the fraction with the larger numerator is greater. For example, $\frac{5}{6} > \frac{2}{6}$ because $5$ slices of a $6$-slice pizza are more than $2$ slices. Visually, imagine two identical circles both divided into $6$ wedges; the one with more shaded wedges represents the larger fraction.

Comparing Fractions with Like Numerators: When numerators are the same, the fraction with the smaller denominator is larger because the whole is divided into fewer, larger pieces. Imagine two same-sized sandwiches: one cut into $2$ halves and one cut into $8$ tiny squares. A piece from the sandwich cut into $2$ (represented as $\frac{1}{2}$) is much bigger than a piece from the sandwich cut into $8$ (represented as $\frac{1}{8}$). Thus, $\frac{1}{2} > \frac{1}{8}$.

Using Benchmark Fractions: We can use $0$, $\frac{1}{2}$, and $1$ as 'benchmarks' to compare fractions. If you are comparing $\frac{1}{8}$ and $\frac{3}{4}$, you can see that $\frac{1}{8}$ is close to $0$, while $\frac{3}{4}$ is more than $\frac{1}{2}$ and closer to $1$. On a number line, $\frac{1}{8}$ sits far to the left, and $\frac{3}{4}$ sits far to the right.

Equivalent Fractions for Comparison: To compare fractions with different numerators and denominators, we can find equivalent fractions to make the denominators the same. For example, to compare $\frac{1}{2}$ and $\frac{3}{6}$, we multiply the numerator and denominator of $\frac{1}{2}$ by $3$ to get $\frac{3}{6}$. Visually, this is like taking a large piece of cake and cutting it into smaller pieces without changing the total amount of cake you have.

Ordering Fractions: To order a set of fractions from least to greatest, we compare them in pairs or find a common denominator for all of them. On a horizontal number line, the smallest fractions are placed on the left, and as values increase toward the whole number $1$, they move toward the right.

Decimals and Fractions: Fractions with denominators of $10$ or $100$ can be easily compared by looking at their decimal equivalents. For example, $\frac{7}{10}$ is $0.7$ and $\frac{70}{100}$ is $0.70$. Visually, a grid of $100$ small squares with $70$ shaded is exactly the same area as a grid of $10$ columns with $7$ columns shaded.