Fractions - Fraction on the Number Line

A number line is a horizontal line where numbers are marked at equal intervals. To represent fractions, we divide the unit length (the distance between $0$ and $1$, $1$ and $2$, etc.) into equal parts based on the denominator. Visual: Imagine a straight line starting at $0$ on the left, with points marked with ticks to show equal spacing.

Proper fractions, where the numerator is less than the denominator ($n < d$), always lie between $0$ and $1$ on the number line. Visual: If you are plotting $\frac{1}{2}$, the point will be exactly in the middle of the segment starting at $0$ and ending at $1$.

The denominator of a fraction tells us the total number of equal divisions we must make in each unit length. For example, if the denominator is $4$, the space between $0$ and $1$ must be divided into $4$ equal segments using $3$ marks between them.

The numerator tells us how many parts to count from the zero mark. To plot $\frac{3}{5}$, you would divide the space between $0$ and $1$ into $5$ equal parts and then count $3$ marks to the right of $0$.

On any number line divided into $n$ equal parts between $0$ and $1$, the point $0$ can be written as $\frac{0}{n}$ and the point $1$ can be written as $\frac{n}{n}$. Visual: On a line divided into $7$ parts, the point $1$ sits exactly at the $7^{th}$ mark, which represents $\frac{7}{7}$.

Improper fractions, where the numerator is greater than the denominator ($n > d$), represent values greater than $1$. To plot these, the number line must extend beyond $1$. Visual: To plot $\frac{5}{4}$, you continue the sequence of equal divisions past $1$ to the next mark, which represents $1 \frac{1}{4}$ or $\frac{5}{4}$.

Mixed fractions are plotted by first identifying the whole number part and then the fractional part. For $2 \frac{1}{3}$, you move to the whole number $2$ and then move $\frac{1}{3}$ of the way toward the next whole number $3$. Visual: This point will be located between the marks for $2$ and $3$, specifically at the first of three equal divisions.