Fractions - Concept of a Fraction

A fraction is a number representing a part of a whole. The 'whole' can be a single object (like a circle) or a collection of objects (like a box of pencils). For example, if a circular cake is cut into 4 equal slices and you take 1 slice, you have $\frac{1}{4}$ of the cake. Visually, this is represented by shading one quadrant of a circle.

A fraction consists of two parts separated by a horizontal line: the Numerator (top) and the Denominator (bottom). The denominator tells us the total number of equal parts the whole is divided into, while the numerator tells us how many parts are being considered. In the fraction $\frac{3}{5}$, 3 is the numerator and 5 is the denominator.

Proper Fractions are fractions where the numerator is smaller than the denominator ($N < D$). They represent a value that is always less than 1. On a number line, proper fractions like $\frac{1}{2}$ or $\frac{2}{3}$ always lie between 0 and 1.

Improper Fractions have a numerator that is equal to or greater than the denominator ($N \ge D$). These represent values that are 1 or greater. Visually, $\frac{5}{4}$ would be shown as one complete square (divided into 4 parts) plus one additional part from a second identical square.

Mixed Fractions are a combination of a whole number and a proper fraction, such as $2 \frac{1}{3}$. This can be visualized as 2 full circles and $\frac{1}{3}$ of a third circle. Every improper fraction can be written as a mixed fraction and vice versa.

Fractions on a Number Line: To represent a fraction like $\frac{3}{8}$, we divide the distance between 0 and 1 into 8 equal parts or segments. Starting from 0, the third marking point represents the fraction $\frac{3}{8}$.

Equivalent Fractions represent the same portion of a whole even though they have different numbers. For instance, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ are all equivalent. Visually, if you shade half of a rectangle, it remains the same area whether you divide the rectangle into 2 large pieces or 6 smaller pieces.

Simplest Form: A fraction is in its simplest (or lowest) form if its numerator and denominator have no common factor other than 1. For example, $\frac{2}{3}$ is in simplest form, but $\frac{4}{6}$ is not because both numbers can be divided by 2 to get $\frac{2}{3}$.