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WHY NOT the Butterfly Method When Adding and Subtracting Fractions?

The “butterfly method” for comparing, adding, and subtracting fractions – WHY are teachers teaching this?

Although it seems like a fun, easy way to add fractions that works every time, the “butterfly method” is NOT helpful to students.

If we want students to understand the meaning of adding and subtracting fractions, we need to stop using shortcuts like this one.

I don’t understand why the “butterfly method” is being used….

I understand that it works and that it can make finding an answer easier for students (for simple fractions).

However, I’ve found definite drawbacks to this method, so I don’t like to see students adding and subtracting fractions this way at all.

In this post, I’ll share what I’ve seen in 6th grade math, with students who were taught the butterfly method in previous grades.

Butterfly Method for Comparing Fractions

In past years, students would come to 6th grade, having learned the butterfly method for comparing fractions. A few observations I’ve made about using the butterfly method for comparing fractions:

Students often had no idea why it worked
They knew the trick better than they knew how to find a common denominator
Students didn’t seem to understand that the products from the cross-multiplying were the actual numerators they would get if they made certain equivalent fractions (shown in figure 1 below)

They just knew it worked.

This bothered me, because I believe students should understand why things work. So, I always made sure to explain why the method worked.

Butterfly Method for Adding and Subtracting Fractions

But this year, I had 6th grade math students tell me they were taught to use the butterfly method to add and subtract fractions (cross multiply and add/subtract those products, then multiply the denominators together, as shown in figures 1 and 2 below).

In this process, students got a correct answer if they followed the steps correctly, but they didn’t have any conceptual understanding of WHY the method works or what adding fractions means.

It seems that many students are being taught this “trick”, to make learning fraction operations “easier and fun.”

In reality, students aren’t learning what it means to add or subtract fractions.

And I have to ask, why wouldn’t we want them to understand that 6 is the LCD in the problem in figure 2?
Why would we want to them to use a larger denominator and then have to do more simplifying to lowest terms??
Why wouldn’t we want students to understand what it means to add or subtract fractions??

Using the Butterfly Method When Adding Larger Fractions

Let’s take a look at what can happen when students don’t understand the concepts and depend on shortcuts like this one.

I recently gave students problem solving that required them to use all fraction operations. It was during this problem solving time that I learned students had been taught the butterfly method for adding fractions.

Since adding and subtracting is in the 5th grade math curriculum (and I teach 6th), we did a very brief review of adding and subtracting fractions before students worked on these problems, and the butterfly method didn’t come up during this time.
It was during their work period that those who had learned it decided it was easier to use the butterfly method than to find common denominators.

In the problem solving, students had to add 5/6, 2/3, 7/12, and 7/10.

And here’s where the butterfly method totally failed the students who had learned to rely on it, not only because they didn’t understand why it works, but also because it became so cumbersome!
They couldn’t use the butterfly method to add 3 or 4 fractions at a time, so they started by adding two fractions.

They took that answer and added a third fraction, again using the butterfly method
And then took that answer and added the 4th fraction, again using the butterfly method

Here’s what that actually looked like:

Students added 5/6 and 2/3, getting a denominator of 18 and an answer of 27/18, as in figure 2.

Then they added 27/18 and 7/12, as in figure 3 below, getting a denominator of 216 and an answer of 450/216.

From there, students added 450/216 and 7/10, shown in figure 4 below. They ended up with a HUGE denominator that they then had to work really hard to simplify!

(AND, we weren’t using calculators, so there were greater chances for errors).

It might seem surprising that they continued this process to get such huge numbers, but some of them did, because:

This was the method they learned and depended on
They had been taught that this was the ‘easy’ way
They didn’t understand what fraction addition meant and couldn’t find common denominators

I was so shocked to see this….and this is the first year I saw this method used in this way.

Needless to say, we’ve spent quite a bit of time working on understanding and ‘unlearning’ the butterfly method.

3 Reasons the Butterfly Method Shortcut Is Not Helpful

1) While this method may seem helpful for students who struggle, teaching students to depend on this method leads to greater struggles for them as they progress in math.

When we as teachers (or parents) find certain tricks that work for simple math problems, we need to look ahead to what our students will experience in future years rather than going for a quick fix for the current year.
We need to try these methods with more complicated problems, to see if they will still be effective. AND if we are to teach shortcuts, they should ONLY be taught AFTER students have mastered the concepts.

2) The butterfly method skips conceptual understanding. Students don’t understand what is actually happening when we add and subtract fractions in this way.

We need to think about whether “tricks” like the butterfly method teach them math concepts, or number sense, or number connections….or do they just teach shortcuts?
If we as teachers don’t understand the meaning ourselves, we need to learn more in order to help our students.

3) The butterfly method avoids the use of common denominators.

Even if students DO understand the concepts of adding and subtracting fractions, using the shortcut means they aren’t practicing how to find LCD, which is a skill they’ll need in middle and high school.

Students are capable of understanding the concepts and we need to have faith that they can “get it” without the tricks. Yes, some students will need more reinforcement and will need to revisit the concepts more frequently (spiral review is helpful here!), but they can get it.

What are your thoughts or experiences with the “butterfly method”?

Interested in more about fractions?
Check out the Fractions: From Foundations to Operations program.