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Compare Fractions: Butterfly and XD Methods

How do I teach students to compare fractions?

Most students love that they can multiply the diagonals and instantly compare the fractions.  They think it is genius and don’t understand why they haven’t been doing it all along.  However, it is important for the students to realize that what they are actually doing is getting a common denominator, even if they don’t multiply the denominators together.

Some middle school math teachers that I know require students to ALWAYS multiply the diagonals and the denominators and NEVER use the traditional, “find the least common denominator” method.  Make sure that your students can simplify fractions (even big ones) if you decide to use this method all the time.

I don’t think there is anything wrong with allowing students to choose a method that works best for them, even if it isn’t a method you taught them.  However, it is nice to have an algorithm or two that you plan to model for your students.  After all, some students get very confused when you show them too many methods to complete the same process.

Check out two methods that I use to teach students how to compare fractions, add fractions, and subtract fractions:

1.  Cross Multiply and Smile

This method is fun for students who love the XD emoji (or any emoji).  Students can use this method to compare fractions, add fractions, and subtract fractions.

Do you want the PDF?  Get it below:

Cross Multiply and Smile PDF

2.  Butterfly

The butterfly method is the same as the cross multiply and smile method.  It is just a different name for the same process.  Students can use this method to compare fractions, add fractions, and subtract fractions.

Butterfly Method PDF

Speaking of different names for the same process…

One teacher I knew called this process shooting the ducks.  Her students loved (and REMEMBERED) that!

Compare Fractions

If your students ONLY need to compare fractions and don’t need to add or subtract them, there is no reason to follow through with getting a common denominator.  All you need to do is multiply the diagonals and you are ready to compare the fractions.

Be careful not to confuse your diagonals.  I like to always go from the bottom to the top.  Then, the diagonal’s product is next to the correct fraction.

Why do teachers ditch the LCD?

I think it is to avoid confusing our students.

We try everything to avoid confusing our students, but sometimes we make it even worse.

I recall one year when we decided to NOT teach the prime factorization method for finding the greatest common factor and least common multiple because students were confusing them with just regular prime factorization.  Perhaps this was a mistake? Maybe they would have understood better if they could use a similar method for all three?  Who knows…

The funny thing about teachers who have chosen to forego teaching common denominators?  The XD and butterfly methods force students to get a common denominator, but not always the least common denominator.  It is not necessary to find the LCD to compare fractions, add fractions, and subtract fractions.  Any common denominator will do the trick.

I have noticed that especially with ELL students, the methods shown above are easy to remember and the process is always the same, so it isn’t confusing.  Anytime they see fractions, their eyes light up and they start multiplying the diagonals and denominators, sometimes before reading the multiplication or division problem carefully.

Did I notice any issues when teaching the butterfly or xd?

I’m sure you feel like a broken record when you begin multiplying and dividing fractions, “Do NOT get a common denominator for multiplying and dividing fractions”, “Think about it… does making all the numbers bigger make it easier or harder to solve the problem?”.

Some of the students that love the XD or butterfly method(s) will automatically want to multiply the diagonals and the denominators when they don’t have to.  Before they know it, those poor students will be working with huge fractions and will most likely make a mistake when multiplying, dividing, or trying to simplify the fractional monstrosity.

Regardless of your method, it is a guarantee that at least a few students will get a common denominator (most likely out of habit) when multiplying/dividing fractions.  It’s no big deal.  Just remind them again that they don’t need or want to do that.

Why do students get so confused when finding the least common denominator in the first place?

They have to make too many decisions to make…

  • Can they multiply just ONE of the fractions on the top and bottom to get an equivalent fraction?
  • Do they have to rewrite BOTH fractions to get a common denominator?
  • What is the least common multiple?
  • How about the least common denominator?
  • Do they have to list all the multiples to figure out the LCD?
  • After all of that did they still end up having to multiply the denominators by one another?

I am not saying that one method is better than the other.  I like to show a few examples using various methods, making sure to point out the benefits and drawbacks of each method.

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