*Note: Dividing fractions by fractions is a 6th grade standard, and the underlying skills are learned in progression, starting in 3rd grade.*
Many students (and adults) are intimidated by fractions. I get it, I’ve been there, too.
Chances are, if you were taught the traditional method for dividing fractions as a student, one of the following phrases would ring a bell: Keep, change, flip; invert and multiply; or multiply by the reciprocal. If you were anything like me as a student, you jumped through the hoop, solved the problem, and moved on because you could produce the right answer. Ask many adults why flipping/inverting/multiplying by the reciprocal works, and you’ll likely find out that they are just as confused now as they were when they were young students. What’s a reciprocal, anyway?
The good news is, dividing fractions can be made clear by revisiting what division is and what fractions really are. In this post, we’re going to revisit the basic idea of division and how it works, discuss what fractions are, and finally, demonstrate why the shortcut works when dividing fractions.
Simply put, division is taking a quantity and distributing it into an equal number of sets or groups; it is the inverse of multiplication, which is combining a number of equal sets or groups to create a new, single quantity. If multiplication is repeated addition, then division is repeated subtraction. In division, the number being divided is referred to as the dividend, the number or size of sets/groups the dividend is being divided by or into is called the divisor, and the term used for the answer to a division problem is the quotient.
Let’s take a look at this simple division expression with whole numbers: 24÷6. In this expression, we are being asked to determine the number of sets or groups of 6 (the divisor) that can be removed from (or are contained within) 24 (the dividend). If we repeatedly subtract 6 from 24, we will find that we can subtract 6 a total of 4 times from 24. Four sets/groups of 6 is equal to 24, so the expression would simplify to 4 (our quotient).
Perhaps you’ve heard of a broken bone as a fracture. I think that’s a great way to think about fractions; they’re just whole numbers broken into equal-sized pieces. Fractions are used to represent equal-sized parts of whole numbers. Commonly, we represent fractions with two different numbers, separated by a horizontal line (although in this post, due to formatting constraints, the line with be a diagonal). The top number is called the numerator, and it represents the number of equal-sized pieces that are grouped together. Underneath the horizontal line, the number on the bottom is referred to as the denominator, which is the number of equal-sized pieces needed to create one whole.
Let’s take a look at the fraction 3/4 (pronounced “three fourths”). Here, the numerator is 3 and the denominator is 4. The fraction 3/4 means that there are three equal-sized pieces of something that you would need four pieces of to make a whole. An example of this would be my family of four: my wife and I have two children, so if we had 3/4 of our family present, that would mean that one of us was missing.
To divide fractions, it might be best to look at an example problem, such as 1/2 ÷ 1/4. Remembering that division is taking one quantity and distributing it into equal-sized sets or groups, we can read this expression as asking the question, How many 1/4 are in 1/2? There are a few methods we can use to solve this question, and in this post, we’ll look a few that will help us understand the shortcut of multiplying by the reciprocal.
Dividing Fractions with Repeated Subtraction
In third grade, students learn about equivalent fractions. Fourth grade students learn to create equivalent fractions by multiplying a fraction by a fraction equal to one (because of the multiplicative identity of one). By fifth grade, students are adding and subtracting fractions with unlike denominators.
By repeatedly subtracting 1/4 from 1/2, we can see our answer, but to subtract fractions with unlike denominators, we must first find a common denominator, which in this case, is 4. Any fraction in which the numerator is equal to the denominator is worth one whole, such as 4/4=1. This is why students “do the same thing to the numerator as the denominator” when creating common denominators or equivalent fractions.
To subtract 1/4 from 1/2, we create a common denominator of 4 by multiplying 1/2 by 2/2, creating an equivalent fraction of 2/4. We can then subtract 1/4 two times from 2/4, so the answer to 1/2 ÷ 1/4 is 2.
Dividing Fractions by Creating a Common Denominator
By creating a common denominator, each unit fraction (single part of the whole) is the same size. When 1/2 is changed into 2/4, the fourths in 2/4 are the same size as the fourth in 1/4. Having the same sized unit fractions allows us to divide the dividend’s numerator by the divisor’s numerator, and the dividend’s denominator by the divisor’s denominator. To make this clearer, let’s take a look at how this plays out for our example of 1/2 ÷ 1/4.
Dividing Fractions by Multiplying by the Reciprocal
Before we multiply by the reciprocal, we should give a quick, simplistic definition of a reciprocal. A reciprocal is a number, that when multiplied by another number, will result in 1. When we “flip” the second fraction (divisor) and then multiply by the first fraction (dividend), we are multiplying by the reciprocal of the original divisor. To see what we mean, look at the example with 1/2.
1/2 represents one half of one whole. To create one whole, you would need two one halves. 1/2 x 2 is the same as 1/2 + 1/2, which equals 2/2, which equals 1. The number 2 can be represented as the fraction 2/1 (which also happens to be the inverse of 1/2). As you can see, the inverse of a fraction is also the reciprocal, because when you multiply a fraction by a fraction where the numerator and denominator have been switched, the product is 1.
When we multiply by the reciprocal of the divisor, we are really just shortcutting the method where we find a common denominator before dividing. If we multiply by the reciprocal of 1/4 (4/1), we would have an expression that looks like this: 1/2 x 4/1. This expression simplifies to 4/2, which is equivalent to 2/1, or 2 wholes. When we multiply by the reciprocal, we are doing the same thing as finding a common denominator and dividing the numerators. The only difference is that it might be faster.
Why Teach Other Methods?
Perhaps you’re thinking, If we can get the same answer when we multiply by the reciprocal, then why bother teaching other methods? That’s a fair question. The problem isn’t with students learning that you can divide fractions by fractions when you multiply by the reciprocal. The problem is that many times, the conceptual understanding behind why this shortcut works is skipped over, either because the teacher feels rushed (there’s a lot to cover in a school year) or because the well-meaning adult/teacher doesn’t understand the concept well enough themselves to explain it any other way.
I actually like the idea of teaching students to “multiply by the reciprocal” because it makes sense to do so (after they understand why they are doing that). This phrase really captures what’s going on when we want to quickly arrive at an answer, unlike tricks such as “keep, change, flip” or “invert and multiply.” Phrases such as these don’t rely on any reasoning or require a student to make sense of what they’re doing. By having students model repeated subtraction and then advancing to creating common denominators (which could again be modeled), students build conceptual understanding of what is going on when a fraction is divided by another fraction. Once understanding has been developed and students are able to explain the process, shortcuts such as multiplying by the reciprocal are helpful rather than harmful.