Butterfly Fraction Addition and Subtraction

As I write this, the back-to-school season is upon us.  As students and their families scramble to be ready with all the supplies and schedules, many are feeling stressed for a different reason entirely: they, or their children, have struggled in school before.

Of all the usual school subjects, probably none provokes more anxiety than mathematics.  One reason for this is that math is a subject in which each year’s work builds on previous years’ work in obvious and necessary ways.  Compare this to a subject like history; if a high school history textbook fell into the hands of a smart junior high student, you would expect them to understand much of it, but even a very smart prealgebra student would not be expected to understand more than the barest scraps and pieces of a precalculus book.

Fortunately, the problem suggests a solution: if students struggle and are stressed in math class because they don’t understand or have forgotten topics from the year before, they can do better if they learn those topics, and when dreaded math topics are discussed, fractions always come up high on the list.  One reason for this is that fraction algorithms are sometimes memorized without understanding — sometimes as a result of being taught with minimal understanding by an elementary school teacher who doesn’t like fractions!  (Another reason is that students may not have thoroughly learned their multiplication facts, which are used in most calculations in fraction problems as well as making it easier to recognize patterns in fractions.)

The “butterfly method,” a creative algorithm for adding and subtracting fractions, is illustrated below:

While some may be tempted to dismiss this algorithm as cutesy, it is in fact quite mathematically sound, and can be taught for understanding by following these steps, discussing each one as much as is needed.

• Before trying this method, students should already know about equivalent fractions, and they should have added fractions that already had the same denominator.
• Students should first draw the wings, lower body, and antenna over the problem as illustrated.
• Next, the denominators (3 and 4 in the illustration) should be multiplied and written into the butterfly’s lower body.  Do not worry at all about getting the least common denominator.  That can be taught when students have learned to simplify fractions and have some interest in shortening the process of simplifying.  Even if they have learned to simplify already, give this process some time to soak in before expecting them to find least common denominators.
• After this, the numbers in each wing (in the illustration, 2 and 4 in one, and 1 and 3 in the other) should be multiplied and written with the antenna that is attached to their respective wing.  Point out that the number above each fraction and the number below the fractions (8 and 12 for 2/3, 3 and 12 for 3/4, in the illustration) makes an equivalent fraction.  For example, 8/12 is equivalent to 2/3 because both the 2 and the 3 got multiplied by 4.
• Then write the lower body number in as the answer’s denominator, and add the antenna numbers together for the answer’s numerator.  (If the problem is subtraction, this step changes to subtracting the second antenna number from the first.)  Mention that this is like what the student did before, when fractions had the same denominator to begin with.
• If the student has already learned about simplifying and the fraction does not happen to be in simplest terms, this would be the time to have him or her simplify.  However, that is not a necessity while they are first getting the idea of adding fractions.  My advice is to include this step if they do quite well with the rest of the process and seem to be ready for a little more.

With practice, students can become proficient at this and other strategies for fractions.  By combining practice with thoughtful consideration, they can also understand fractions well and be better prepared for topics involving proportionality, as well as for algebra.

Hat tip and resource (note that the resource gives steps in a slightly different order; the order shown in this article is designed to make it easier for students to understand that adding uses a common denominator and equivalent fractions)