Visual Representations of Fractions

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

To understand fractions, it is helpful to represent them visually. In this lesson, learn about three ways to visually represent fractions: linear models, area models, and set models.

Why Represent Fractions Visually?

How long do you think it will take to read this lesson? Maybe a quarter of an hour? Well, how long is that?

You probably know that a quarter of an hour is fifteen minutes, but you may not have realized that you just used a fraction to figure that out. If there are 60 minutes in an hour, 1/4 of 60 minutes is 15 minutes. No matter what you are doing, you never have to look too far to find a fraction, so it is important to understand what they are and how to represent them.

A fraction is a mathematical way to represent a part of a whole. The number on the bottom of the fraction, known as the denominator, tells you how many parts there are total, and the number on the top (the numerator) tells you how many parts you actually have.

It's often easiest to understand fractions by seeing them as parts of a whole object or group of objects, rather than simply as abstract numbers. To understand fractions, there are several visual models that you can use.

Linear Models of Fractions

One way to represent a fraction visually is to use a linear model. A linear model uses a number line to show the size of a fraction. In a linear model, the total length of a line is divided into equally spaced intervals. A fraction can then be shown by another line that only covers part of the total length. The ratio of the length of the line to the total length is the fraction!

In this linear model, the green line is five units long and the whole number line is 8 units long, so the fraction represented here is 5/8.
linear model of fractions

One advantage of using linear models is that, with a linear model, you can easily identify fractions that are equivalent (like 1/2, 2/4, and 4/8).

linear model showing equivalent fractions

Area Models of Fractions

Another common way to represent a fraction visually is to use an area model. In an area model, a shape (often a circle or rectangle) is divided into a number of equal sections. The total number of sections is equal to the the denominator of the fraction. The numerator of the fraction tells you how many of the sections should be shaded.

Area models are a great way to visualize fractions, and they also make it easy to see which fractions are equivalent.

area model of fractions

In addition to just visualizing fractions, area models can be used to help you add fractions. Let's look at an example of how to use an area model to add two fractions, 1/3 and 1/4.

To add 1/3, and 1/4, first, draw two rectangles that represent each fraction. The rectangle representing 1/3 will be divided into three equal sections and one of them will be shaded. The rectangle representing 1/4 will be divided into four equal sections and one of them will be shaded.

adding fractions_part 1

Next, imagine that you put one of the rectangles on top of the other. Now, each rectangle would be divided into twelve equal boxes. This means that 12 is the common denominator between 3 and 4.

adding fractions_part 2

Finally, draw a new rectangle and divide it into twelfths as well. Shade the total number of boxes from the TWO shapes you had before (there are 7 in this case).

adding fractions_part 3

As you can see, there are 7 boxes shaded and 12 boxes total, so 1/3 + 1/4 = 7/12. This is a great way to understand how fractions are added together!

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