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 emoji:  XD (or any emoji).  Students can use this method to compare fractions, add fractions, and subtract fractions.

compare fractionscompare 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 cross multiply and smile 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.

comparing fractions

How You Can Make a W-Shaped (or an M-Shaped) Accordion Style Foldable

Feel free to use the images and/or video to help your students create their w-shaped accordion foldable.




You need enough copy paper and writing utensils for your class.

It is important to create your foldable ahead of time, that way you aren’t trying to come up with something on the spot.

I find it very helpful to have multiple foldables with various states of completion. That way, you can show a little at a time, but don’t have to worry about writing everything down while your students are.

Now that you don’t have to spend all of your time in the same spot writing, you can monitor your class better by moving around the room.  Take this opportunity to make sure everyone is on task and that no one needs help.

Do you want to print the above pictures?  Get a free PDF here!


Check out some foldables that I made using this template:




Square Roots Reference Card

It can be very overwhelming to work with square roots of non-perfect squares.  Use this FREE square roots reference card to help your students as they begin working with square roots.  Later, have students memorize their perfect squares from the square root of 1 to at least the square root of 225.

Get the FREE .pdf here: SquareRootsReferenceCardFrom1to400

4×5 Square Roots Reference Card:


5×4 Square Roots Reference Card:


Square Roots Reference Card – 5 to a page


Square Roots Reference Card – 10 to a page


Get the FREE .pdf here: SquareRootsReferenceCardFrom1to400

Try some related activities:

ApproximatingSquareRootsMaze1 ApproximatingSquareRootsPuzzles1 RationalAndIrrational1


FREE 6th Grade Ratios and Proportional Relationships Common Core Math Posters

Are you looking for common core math posters for the 6th grade ratios and proportional relationships standards?

Try these FREE posters:

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Download the .pdf here:  6thGradeCCSS_RP_Posters

or FREE from my TpT Store:  6thGradeCCSS_RP_Posters

Try Some Related Activities:



Divisibility Rules Poster Options

I am working on converting old anchor charts and notes to typed posters.

Divisibility Rules Poster Options:



Black and White:


Use these photos as a poster to display in your classroom, or print for use in student notebooks/binders.  Here is the FREE .pdf:  DivisibilityRulesPoster.

So, what is this trick for the 7’s divisibility rules?  

Check it out:

  • Double the last digit and subtract it from the remaining part of the number.  If the answer is either 0 or it is divisible by 7, then the original number is divisible by 7.


  • Start with the number:  1015.
  • Isolate the last digit:  1015 .
  1. Double the last digit.  5 x 2 = 10.
  2. Subtract the result (10) from the remaining part of the number (101):  101 – 10 = 91
  3. Determine whether your answer is equal to 0 or divisible by 7.  If it is, the number is divisible by 7.  If it is not, the number is not divisible by 7.
  • 91 divided by 7 equals 13, which is a whole number.
  • Since 91 is divisible by 7, that means 1015 is also divisible by 7.
  • If you are curious, and don’t feel like working it out… 1015 divided by 7 equals 145.

It is probably easier just to do long division in most cases, but it is always fun to learn a new trick 😉

Check out my post on making a divisibility rules foldable.


Math is a Language

Understanding “Mathlish”:

Have you ever tried to learn another language?

Can you speak English? Spanish? Both? Another Language? Learning about Math is just like learning another language!

Once you can speak Mathlish, doing math problems is easier!  Note:  We started calling math vocabulary “mathlish” last year. The students loved it!  

Here are a few FREE pages from the Mathlish Dictionary:





Get the free PDF


First Week of School Ideas and Resources 4

A unique back to school activity for middle school.

The Cost of Coloring in South America:

FREE Coloring Activity Involving Basic Math

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• Cover Page: Cost of Coloring South America Project
• Goal/Directions Page
• South America Coloring Project Worksheet
• South America Coloring Project Answer Key
• Thank You & Author Information

How does this activity work? Directions:
• Color the map of South America. Each color is worth a different dollar amount.
• Make the cheapest map possible.
• WARNING! Before you get started, there is a catch… no colors can touch.
• Your map has to cost ≤ $2,800 or LESS!
• How LOW Can You Go?
• When you finish, explain how you got your answer. Did you have a strategy? Explain.
• Make sure you show all your math (adding up the numbers and state your answer.

Download this FREE product from my TPT Store.

Read more about back to school resources:

First Week of School Ideas and Resources Part 1

First Week of School Ideas and Resources Part 2

First Week of School Ideas and Resources Part 3