Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 17.1 Facing Your Fear

Most students don't like fractions. This is just a reality. This is unfortunate because fractions are everywhere in mathematical applications. A lot of students really only know fractions as a system of rules for manipulations, and have little intuition for what they represent and how we use them. Over the next few sections, we're going to take a closer look at fractions with the goal of making fractions more understandable.

At its core, fractions are a representation of division. The fraction \(\frac{6}{2}\) can be interpreted as asking the question, "How many groups of 2 can we make if we have 6 objects?" The answer is and we can visualize it with a simple picture.

When this works out evenly, our answer is an integer. When it doesn't, then we can also use fractions to communicate the leftover pieces. The fraction \(\frac{5}{2}\) asks the question "How many groups of 2 can we make if we have 5 objects?" The answer is \(2 \frac{1}{2}\text{.}\) The \(\frac{1}{2}\) part simply means that we have one out of the two pieces needed to create another group. We can visualize the missing piece with an unshaded box.

Numbers like \(2 \frac{1}{2}\) that are a mixture of an integer part and a fraction are called mixed numbers. They are commonly used in applications (such as stock prices and lumber measurements), but they are algebraically cumbersome to use. In fact, the mixed number \(2 \frac{1}{2}\) is really \(2 + \frac{1}{2}\) when it comes to algebraic manipulations. So we often leave fractions involving integers in their "improper" form.

The visualization of fractions above is known as "parts of a whole." The idea is that you have some concept of what a "whole" grouping looks like, and you're trying to fill that in with the number of "parts" that you have. It is often the case that we use circles when talking about parts of a whole because it's more intuitive if the whole is always the same final shape.

Converting between improper fractions and mixed numbers can be done using strict arithmetic methods, but it's also important to develop that base intuition of what fractions are and how they behave. So rather than giving an explanation of converting between improper fractions and mixed numbers in words, we're going to do it in pictures.

Activity 17.1.1. Converting Improper Fractions to Mixed Numbers.

Here is the conversion of an improper fraction to a mixed number:

Try it!

Convert \(\frac{11}{4}\) from an improper fraction to a mixed number using a diagram.

Solution.

Activity 17.1.2. Converting Mixed Numbers to Improper Fractinos.

Here is the conversion of a mixed number to an improper fraction:

Try it!

Convert \(2 \frac{3}{4}\) from a mixed number to an improper fraction using a diagram.

Solution.

The visualization of fractions as wedges of circles is one of two primary visualizations that we use for fractions. The second comes from the number line. A number can be represented by a position on the number line. Here are some examples:

For improper fractions and mixed numbers, we can continue doing this process and extend beyond the interval from 0 to 1.

Activity 17.1.3. Locating Fractions on a Number Line.

When locating numbers on a number line, you do not need to mark all of the subdivisions between the integers, just the subdivisions between the integers you are focused on for your value. Here is how could be represented:

Try it!

Represent \(\frac{11}{3}\) on a number line.

Solution.

Let's take another look at the number lines. Notice specifically that several of the fractions that are in the same position even though they have different denominators.

In fact, we can use the parts of a whole picture to visualize these relationships as well. Here are different fraction representations that result in the same amount shaded in:

What this means is that there are multiple fractions that represent the same number:

This leads us to an important question: When are two fractions equivalent? If we look at the pattern of values, we can see that the fractions are related to each other by multiplying the numerator and the denominator by the same value.

There are many ways to think about this. One way to think about it is that we're taking the parts of a whole diagram and cutting them into extra pieces without changing the shaded area. Here are two examples of this:

This gives us a general pattern that we can follow to generate equivalent fractions. If \(x\) is any non-zero number and is any fraction, then we have

When students are introduced to this, it is usually done with non-zero integers. However, this is true non-zero fraction and decimal values of \(x\text{.}\) In fact, it's even true when is a non-zero variable or variable expression. In other words, the following are all equivalent (as long as the variable quantities are non-zero):

This idea can also be turned around. Sometimes there are complicated fractions with common factors that can be "cancelled out" to leave you with a simpler expression. But this can only be done if the numerator and denominator have the same factor and that factor is being multiplied by the rest of the numerator and denominator.