Renaming Fractions by Multiplying by 1

Instructor: Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

In this lesson, you will learn how to rename fractions. We will look at a visual example. Then, we will work out several examples involving multiplying a fraction by a specific version of 1.

Review of the Properties of 1 and Fractions

Does everyone call you by the same name in all situations? Or do some people call you by a nickname? Let's take a professor by the name of Robert Smith, for example. His students call him Dr. Smith. His friends call him Rob. His wife calls him Honey and his mother calls him Bobbie. Robert Smith knows that each of those names refers to him. Fractions are a bit like Robert Smith, in the sense that every fraction can have many different names that we may need to use in different situations.

The number 1 can also go by many different names. Remember that when you divide any number (except 0) by itself, the result is 1. Other names for 1 could be 2/2, 6/6, 93/93, etc. Another feature of the number 1 is that when you multiply any number by 1, the original number stays the same. We can use these two properties of the number 1 to rename fractions.

A fraction represents part of a whole. For example, 5/6 means you cut something whole into six equal pieces and are now dealing with five of those six pieces. The top number in a fraction is called the numerator and the bottom number is called the denominator

Visual Representation of Renaming Fractions

Let's take a look at why you might want to rename a fraction and how it works. We will start with an easy fraction like 1/2. In Diagram 1, I have drawn a representation of 1/2, a sheet of paper creased in the middle with the left half colored blue.

A rectangle split into two sections, one half blue and one half white.

Now, let's say you wanted to represent 1/2 of 6 sections. You could take the sheet from Diagram 1 and fold it twice so that you have 6 sections. To do that, as shown in Diagram 2, we draw two horizontal lines. Now, we have 3 shaded pieces out of 6 total pieces or, to write it as a fraction, 3/6. As you can see, 3/6 is still half of the entire sheet, we just changed how many pieces we have. 3/6 is another name for the fraction 1/2.

A rectangle split into 6 sections, 3 of which are white and 3 of which are blue.

Mathematically Renaming Fractions

So, how do you represent that paper-folding mathematically? We started with a sheet split into 2 sections. However, we then decided we wanted to look at it split into 6 sections. To increase from 2 sections to 6 sections, we need to multiply the denominator by 3.

However, we can't just multiply denominators of fractions by whatever we want. Just saying that you have 1/2 of the sheet shaded and now you have 1/6 shaded would not be true. To make sure that you have a correctly renamed fraction, you have to multiply the numerator by the same number as the number you decided to multiply the denominator by.

Fraction Example 1

That's where those properties of the number 1 that we talked about earlier are applied. You multiply the fraction 1/2 by 3/3 to rename it in terms of 6 parts and get 3/6. Three parts of 6 is the same as 1 part of 2. You are allowed to multiply by 3/3 because that is just another name for 1 and multiplying by 1 does not change a number. Let's look at a couple more examples!

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