Telling The Truth About Adding Fractions

Good Morning!

When I ask my students how they feel about fractions, I don't really need to listen to what they say. I just need to pay attention to their faces. They're generally not fans. When I ask them why they don't like fractions, the responses are not very diverse; they hated adding and subtracting fractions and they hated having to find the lowest common denominator. And I get it. If the fractions have a common denominator, you can just add the numerators. But if they do not have a common denominator, you have to find one first and then rewrite the original fractions as new, equivalent fractions with the new, common denominator. Perhaps you may remember your teacher saying something along the lines of, "You need to find the lowest common denominator first. If you can't find it you can multiply the two denominators." We will come back to that statement later.

First, I want to discuss the topics of Greatest Common Factor (GCF) and Lowest Common Multiple (LCM) and I will tie these to fraction addition later. When I was in elementary school, the process for finding the GCF was to list all the factors of each number, look for the ones they have in common, and choose the largest one. The process for finding the LCM wasn't much different. List multiples of each number in order and one-by-one until you find one they have in common. The first one is the LCM. These methods are fine when kids are first learning about the GCF and LCM. But as they get older, they also learn about prime numbers, prime factorization, and factor trees. These mathematical tools become even more useful in high school mathematics. More on that in a later post.

To find the GCF of two (or more) numbers using prime factorization, complete the factor tree for each number, identify all the prime factors the numbers have in common, and then multiply only one set of them together. Recall that a positive integer is prime if it has only two factors, 1 and the number itself. The number 1 is not considered prime. Here's an example. 

Find GCF(12, 30). Before we begin, I first want to point out that GCF(12, 30) = 6. This will make checking our work easier. 

The prime factorization of 12 is 2, 2, 3 and the prime factorization of 30 is 2, 3, 5. They both have a 2 and 3 in common, so GCF(12, 30) = 2(3) = 6.

To find the LCM of two (or more) numbers using prime factorization, complete the factor tree for each number, find their GCF, and then multiply all of the other prime factors, including the ones they have in common but were not used to determine the GCF.

Find LCM(12, 30), which happens to be 60. From above, the prime factorization of 12 is 2, 2, 3 and 30 is 2, 3, 5. The GCF was constructed from one 2 and one 3. Multiply all of the other factors together, 2(2)(3)(5) = 60.

If you pay attention to what is happening with the prime factors, you may have noticed that LCM(12, 30) = (12)(30) / GCF(12, 30). That is, LCM(12, 30) = 360 / 6 = 60. This is generally true. For two positive integers a and b,

In addition, two numbers a and b for which GCF(a, b) = 1 are called relatively prime. This is why it is sometimes necessary to multiply two denominators together to get the lowest common denominator. The numbers 6 and 7 are relatively prime so LCM(6, 7) = 6(7) / GCF(6, 7) = 6(7) / 1 = 42.

Here's why this matters. The LCM is the lowest common denominator. But if LCM(a, b) = ab / GCF(a, b), then ab = LCM(a, b) * GCF(a, b). The product of the two numbers is a multiple of their lowest common denominator. This implies that ab is a common denominator for a and b, it just might not be the lowest one. Recall that in order to add or subtract fractions we must have a common denominator. We do not need the lowest common denominator, we've only been told since we were young that we needed it. Only we don't. Which means we don't need to do all the work required to find the lowest common denominator. Neither do our students.

Let's come back to, "You need to find the lowest common denominator first. If you can't find it you can multiply the two denominators." The first part of that statement is just a lie. There's no way to put it nicely. We do not need to find the lowest common denominator. We only need a common denominator. And as we have seen, we can get a common denominator by multiplying the two denominators, which will always produce a common denominator. Always.

Once we have a common denominator, we have to rewrite the numerators as equivalent fractions using the common denominator. This typically meant answering questions about what number you need to multiply each denominator by in order to get the common denominator. But if we are just multiplying the two denominators together to begin with, we already know the answers to those questions; the other denominator. To generalize, 

Here are a few examples to show how this works in practice.

Using this method is much less frustrating than finding the lowest, and unnecessary, common denominator. Believe me, I've seen the "why didn't they teach us this the first time" look on my students faces too many times. I would much rather my students understood the connection between common denominators, prime factorization, GCFs, and LCMs since these last three are very rich mathematical concepts with a lot of payoff. I also don't want my students to shut down when they see fractions. So if telling them the truth about adding fractions keeps them from tuning out, I'll take that every time.

[This is the first post in what will be a series of posts addressing topics in mathematics which are often not treated honestly.]