What is a Fraction?

Fractions have been used by many different societies since 2000 BCE. (See “Did You Know” for a Brief History of Fractions.) In this video we look at several examples to help explain what a fraction is.

What is a fraction?

A fraction is a way to represent a proportion in between whole values. We make fractions by splitting objects, or values, into equal-sized pieces. When we discuss fractions, what we’re really discussing are the proportion or part of the whole that we’re looking at.


To illustrate the idea of a fraction, we begin with a simple question: How much of the pizza is left?

(To clarify, we are not asking for the amount of pizza left, rather we are trying to describe what part of the whole pizza remains.)
In everyday conversation, the answer is usually “two slices,” but this isn’t precise enough for a mathematical conversation. Clearly, depending on the number of slices the pizza was cut into, two slices could represent a different portion of the whole pizza,

so we might instead specify “two of the eight slices are left.” This is much better, but still leaves some potential problems.

Saying that there were eight slices initially is only meaningful if we know something about how the pizza was cut. To make our lives easier, let’s only look at the case where they were all the same size. This has two major benefits:
1.    It doesn’t matter which two are left, because they are all the same.
2.    Just based on the number of slices, we know exactly how the pizza was cut. Once you know how many there were, it is a lot easier if you don’t need to define the exact size of each.

We will soon see that splitting the whole into equal parts also has the advantage of simplifying calculations that we may want to do.

Again, we are not trying to measure the amount of anything or its arrangement or any such thing, only the amount that we have in comparison to the whole.

Fractions can represent a group of related objects

8 dogs, 4 brown, 3 gray, and 1 golden

Fractions can also describe groups of related objects.

For example, let’s look at the group of 8 dogs above. If we separate them by color, we discover that there are 3 gray dogs. To write this as a fraction we use the number of grey dogs, 3, as the numerator, and the number of total dogs, 8, as the the denominator. Therefore, our fraction would be $\tfrac{3}{8}$.

Using the same reasoning, we can look at the number of golden dogs and say there would be 1 out of the 8, or $\tfrac{1}{8}$. Similarly, the number of brown dogs is $\tfrac{4}{8}$, or $\tfrac{1}{2}$ of all the dogs.

A bag of jelly beans

Fractions as Division

I bought 215 jellybeans for my class’s fraction activity. I want to make sure each of my 15 students get the same amount of candy. Therefore, each student should get $\tfrac{215}{15}$ of the jellybeans. Remember, a fraction can also be thought of as a division problem. So, we can write $\tfrac{215}{15}$ as $215 \div 15$ which equals 14 $r5$. Each student gets 14 jellybeans (with 5 extra jellybeans for the teacher– yum!).


What is a Unit Fraction?

In a unit fraction, the numerator is always one.

one sixthon ninthone third

A half plus fourth is three fourths

Fun fact: The Egyptians worked almost exclusively with unit fractions. They would write $\tfrac{3}{4}$ as $\tfrac{1}{4} + \tfrac{1}{2}.$