The butterfly method in fractions and the danger of overemphasizing tricks

I first got to know about the “Butterfly Method” in fractions from one of friend’s son. My friend, knowing that I have been teaching Math for many years in Singapore, commented to me, “My son’s school just adopted Singapore Math, and I don’t understand the math!”. Curious, I asked the Mom to elaborate and she started talking about the “Butterfly Method”. The son went on to explain this to me.

• First, you multiply the numerators and denominators diagonally and add –  that becomes your new numerator
• Then, you multiply the denominators – that becomes your new denominator

I became very curious. I understand how the math works, but how about the student? I went on to ask him, “Do you know why you are doing this?”. He replied matter-of-factly, “Well, you just do it!” Determined to show him the conceptual way, I told him that we need to first make sure the pieces are the same size and we do so by changing the fractions to equivalent fractions with the same denominator.

At this point in time, the student was getting a little impatient with me! “Why are we doing this? This is too much work. I like my trick and it works!”, he says. I knew this is coming, and I asked him to add three fractions using the butterfly method. He was dumbfounded.

In our opinion, tricks like the butterfly method should be avoided when students are first introduced to fractions. There are several reasons, e.g.

• There is no conceptual understanding in the instruction.
• It reinforces the belief that fractions is just a bunch of tricks.
• What happens if you add three or more fractions?

The session with the young boy left a deep impression on me. Other than learning a new method that I have not heard of, it made me realized the difficulties of starting a new curriculum. Singapore Math, with its focus on conceptual understanding and visualization, has become very popular in the United States over the years, and many private and public schools have replaced their curriculum with Singapore Math. However, it is not easy.

Many teachers teach the way that they were taught, and in this case, the teacher might be teaching the way that she was taught when she was a student; never mind that she is using the Singapore Math curriculum, and never mind that the Singapore Math curriculum has no mention of the “Butterfly Method” at all.

This was the turning point for me. That was when I started professional development for teachers using the Singapore Math curriculum, and through my interactions with teachers, started realizing the support that teachers need, not just in a two-day workshop, but on a on-going basis, throughout the school year. This is also the reason why we started our online Singapore Math membership site, so that we can serve the community of teachers on a regular basis.

In the opening example, although the final answer is correct, students failed to see directly that 2/5 + 1/5 is simply 3/5 and resorted to using algorithms or tricks without thinking. Have you seen similar behavior in your students?

More Fraction Resources

For more fraction resources, refer to our main fractions page.

13 thoughts on “The butterfly method in fractions and the danger of overemphasizing tricks”

1. Even the Ohio Department of Education in their standards for math page reminds teachers not to use tricks to with elementary students. Sadly they haven’t read the info on the website and don’t reference it when making lesson plans.

2. The problem is not only for the examples you mentioned, but also students will tend to do the same when they do rational expression exercises like x/(x -2) – (x2+4)/(x2-4) which takes a day while it can be done in several techniques in a very short time.

3. Nice post! I avoided teaching this “trick” with my students this year. Sadly, an administrator said we should use it last year. I only show this now, when students have a solid method to adding and subtracting fractions. What was interesting this year however, is that I was just beginning a unit on adding and subtracting fractions this year with one of my classes. A student came to school the next day, proficient with this “trick!” She showed me, and I said it will always work, and had her try to add 3 fractions with unlike denominators as you did above to prove my point. She knew enough to use the butterfly method with two fractions, and then repeat the butterfly method until she had a common denominator for all three. I was somewhat impressed (this is a student with an IEP). I told her there was a more efficient way of arriving at a common denominator, and a reason why the “trick” worked, and I wanted her to learn that way and why the “trick” worked too. She was reluctant, but said okay. I told her that I would let her share the method with the class after we finished the unit. This cheered her up! We have since finished the unit, and she uses both ways, but the “trick” is her go-to way. She learned it by watching YouTube videos at home and attempting to do her homework at home on her own! Again, impressed that she took learning how to do something in her own hands! Sadly though, many students are using this “trick,” and it’s mainly my struggling learners.

4. I teach high school math and I had some students show me this method. Another problem with this trick is when the first number is negative. Students always signs wrong.

5. I came here looking for reasons to support my belief that “butterfly” is harmful. I found them!
You can find more alternatives to “tricks” at https://nixthetricks.com/ (by teachers, for teachers). Lots of good stuff there.

Mostly I feel like these things can be useful once you understand why they work, as a shorthand way of keeping track of things or aiding memory for the steps of a procedure. But then, the really shorthand way is to use a calculator, so it seems like these kinds of procedures are largely obsolete. Things that show the WHY much more visibly are what we need more of, and things that get correct pencil-and-paper answers quickly are just not that useful any more.

6. I agree. I am a teacher and honestly I had never heard of this method until recently (I usually teach older grades) but teaching 7th grade and 7th grade RTI. We have a math strategist. When I mentioned teaching students about lcm’s and lcd’s I was told that was a bad way to teach this skill. And I was told to teach this using BAM method (butterfly). I thought, so the students will have no conceptual knowledge. Not good.

7. It’s a trick if you perceive it as such. Technically, you ARE CREATING equivalent fractions, except expressing both numerators on one denominator.

It is mathematically sound, and that’s precisely why it works.

You don’t need to know why things work to know how to do it effectively. Just like when you divide a fraction and you flip the second fraction and multiply.

8. I created this “trick” myself when I was teaching middle school math back in 2008. Granted, other folks in the world may have created the same concept themselves independently, but I was blown away tonight when I saw a YouTube video showing the strategy and then did a Google search to discover how all over the place it has become. Holy cow!

On my own end, this was a graphical strategy that was taught to remind students of the approach needed when faced with two unlike denominators in an addition or subtraction problem, and was taught to 6th-8th grade students after having already been introduced to operations with fractions and the need to find common denominators. It was never intended to be an introductory or one-strategy-fits-all-problems method. I’m a bit saddened that it appears to have turned into that when it was never my intent when sharing it with students or colleagues.

Regardless, the principles behind it are sound and do work with both addition and subtraction, providing students are aware of negative numbers, which is why a conscious and deliberate application is important when it is appropriate.

9. I use this method with students when comparing fractions but have not used it with addition and subtraction of fractions. It has been a good strategy for my students to use when comparing fractions. This strategy is taught after we have done diagrams showing equivalent fractions so my students do understand why it works.

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