Introduction to Fractions: What Is a Fraction?

Every fraction has two numbers, written one above the other and separated by a bar:

A fraction is written as parts counted over parts in the whole.

Since words cannot sit inside the fraction bar on this page, look at it with numbers instead: in \( \dfrac{3}{4} \), the top number is called the numerator and the bottom number is called the denominator.

• The denominator (bottom number) tells you how many equal parts the whole has been split into.
• The numerator (top number) tells you how many of those equal parts you are counting or shading.

So in \( \dfrac{3}{4} \), the whole was split into 4 equal parts, and 3 of those parts are being counted.

A fraction only works correctly if the whole is divided into parts that are the same size. If a bar is cut into one big piece and three tiny pieces, you cannot call the big piece "one fourth", even though it is one out of four pieces, because the pieces are not equal. This idea of splitting a whole into same-size pieces first is called equal partitioning, and it is worth practicing on its own at Fractions - Equal Partition before naming fractions.

A fraction can describe more than one kind of whole:

• A region, such as a shaded part of a circle, rectangle, or pizza.
• A set, such as 3 red marbles out of a group of 5 marbles.
• A length, such as a distance on a ruler or number line.

These are all the same idea of a fraction, just applied to different kinds of wholes. You can see many examples of these models drawn out step by step at Fractions - Pictorial Representations.

A fraction can also be shown as a point or a length on a number line. To place \( \dfrac{3}{4} \) between 0 and 1, split the space between 0 and 1 into 4 equal jumps, then count 3 jumps from 0.

This number-line model is especially useful once fractions get larger than 1 or need to be compared, which is covered in more depth at Fractions - Number Lines.

Fractions are usually read by naming the numerator first, then the denominator as an ordinal word: \( \dfrac{1}{2} \) is "one half", \( \dfrac{1}{3} \) is "one third", \( \dfrac{3}{4} \) is "three fourths", and \( \dfrac{5}{8} \) is "five eighths". Once you are comfortable naming fractions, the next natural step is noticing that different-looking fractions, such as \( \dfrac{1}{2} \) and \( \dfrac{2}{4} \), can represent the exact same amount. That idea is explored fully at Equivalent Fractions.

Suppose a garden bed is split into 6 equal sections, and vegetables are planted in 5 of them. What fraction of the garden bed has vegetables?

The whole bed is split into 6 equal sections, so the denominator is 6. Since 5 sections have vegetables, the numerator is 5. The fraction of the garden with vegetables is \( \dfrac{5}{6} \).

Once these basic ideas feel comfortable, try applying them to real situations, including sets, regions, and measurements, at Fractions - Region, Sets & Linear Models, or work through story problems at Fractions - Word Problems.

Related lessons

• Math 3