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.

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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

Divisibility Rules Foldable

How did I create my divisibility rules foldable?

First, I decided how to word the divisibility rules.  I based my rules off of a poster that I have for free here.  The only thing that I changed was adding a “or 0” to the divisibility rules for 2.  Eventually, I will go back in and change the free posters to show that.

HERE IS WHAT I CAME UP WITH FOR MY RULES:

  • A number is divisible by 2 if the last digit is even [2, 4, 6, 8] or 0.
  • A number is divisible by 3 if the (sum of the digits) is divisible by 3.
  • A number is divisible by 4 if the (last 2 digits) are divisible by 4.
  • A number is divisible by 5 is the last digit is a 5 or a 0.
  • A number is divisible by 6 if it is divisible by BOTH 2 and 3.
  • NOTE:  I skipped 7
  • A number is divisible by 8 if the (last 3 digits) are divisible by 8.
  • A number is divisible by 9 if (the sum of the digits) is divisible by 9.
  • A number is divisible by 10 if the last digit is 0.

After I listed the rules, it is clear that I need at least 8 sections so that I can show the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10. Right away, I knew that I wanted to create a window/flap template, so I started by folding the paper to get sixteen equal sections. Then, I cut the flaps to complete the foldable.

SEE IT STEP BY STEP:

Foldable Directions

If you are having your students create a foldable with this template, I suggest having them completely fill out the foldable with the information BEFORE cutting.  I typically have students wait to cut until right before they glue their foldable in their notebook.

BACK TO THE DIVISIBILITY RULES FOLDABLE:

HAND WRITTEN ROUGH DRAFT:

NOTE:  I did not cut the rough draft of my foldable until I finished filling it out.  As you can see, I did not cut very carefully.

Back/Cover:Divisibility Rules

Some Flaps Open / Some Flaps Closed:

I put the rules on the back of the flap and the examples underneath the flap.  I typically put the examples on the inside and the rules/steps on the back of the flap.  There is nothing wrong with flip-flopping the examples and the rules, but I think it is important to keep the entire foldable consistent.

Divisibility Rules

Inside:  All Flaps Open:

I made up the examples as I went, making sure to have at least one non-example for each section.

Divisibility Rules

HAND WRITTEN & COLORED FINAL COPY:

Front Cover Flaps:

I thought it would be fun to add some color to my hand written final copy.

Divisibility Rules

Some Flaps Open / Some Flaps Closed:

I used colored pencils and flair pens to create this.  I was so nervous that the pens would bleed through.  To my surprise, they didn’t bleed much even though I was just using THIN copy paper.

Divisibility Rules

Inside:  All Flaps Open:

Feel free to use my pictures to help your students put together their foldable.

Divisibility Rules


WOULD YOU RATHER PURCHASE THE COMPUTER VERSION?

GET IT HERE:

Divisibility Rules Foldable Pictures

Divisibility Rules

Divisibility RulesDivisibility RulesDivisibility Rules

NOTE:  The examples in the paid version are different than the examples in the  handwritten version so you can use both.

AmyHarrisonBWSquare

Creating a Foldable on Any Topic

How do I create a foldable on any topic?


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SECTIONS:

First, I decide how many sections I will need based on the topic I plan to use.

For example, if I want to create a foldable on fraction operations, I will need four sections: adding fractions, subtracting fractions, multiplying fractions, and dividing fractions.  If I split each section approximately in half, I can separate the rules from the examples.

Once I know the number of sections needed,  I must decide which style of foldable I want to use.


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STYLE:

A few ideas come to mind right away:

Flaps/Windows

Benefits of Flaps/Windows:  

  • They help students focus on one topic at a time by giving them the ability to open just one section at a time.
  • The flaps are great for quizzing/memorization.

Disadvantages of Flaps/Windows:

  • Inevitably some students will accidentally rip off a flap.  Don’t worry, you can always tape it back on.
  • Some cutting is required to create the flaps/windows.

    Check Out Some Examples:

        4 flaps all facing the same direction

FractionsFoldable4FlapsAmyHarrisonOutsideFractionsFoldable4FlapsAmyHarrisonInside

If you chose to use this template, I suggest putting it in your math notebook so that the folds on the left hand side face the center binding.  The foldable goes on the right hand side page in your composition book to accomplish this.

Now, when the students open and close their books, they don’t have to worry about making sure that all of the flaps on the foldable are closed.  That is a great benefit of this foldable over the next option.  However, the flaps on this foldable are longer and can accidentally get torn off.

4 flaps with 2 on each side

FractionsFoldable4FlapsAmyHarrison2OnEachSideOutside

FractionsFoldable4FlapsAmyHarrison2OnEachSideInsideAmyHarrison

If you chose to use this template, it doesn’t matter if you put the foldable on the left hand side or the right hand side of your notebook.

If you put it on the right hand side of your notebook, the flaps on the right hand side of the foldable might misbehave when your students open/close their notebook.  Make sure that students close the flaps on the foldable before closing their notebook.  This will prevent the windows from getting smashed or torn off.

I enjoy the flap/window style because it allows for two sections for each topic:  one on the back of the cover, and the other inside the foldable.  When using this style, I like to keep it consistent:  all of the steps are on the back of the cover, and all of the examples are inside the foldable.

If you are having trouble deciding which flap/window template to use take a look at the shape of the sections and decide which one would be better for what you plan to write in it.

Both foldables have flaps/windows, and they take up the same amount of space.  However, I think the second option is better for writing the fraction operations steps, so I am leaning towards that one right now.


Accordion

Benefits of Accordion:  

  • If you are starting with copy paper, there is ABSOLUTELY NO CUTTING necessary. However, if you start with a template from the computer, you may want to cut off the excess margin.
  • The foldable takes up less space than the flap/window style, so there is more room to write on the notebook page next to the foldable.
  • It is less likely that a student will rip off part of the foldable because it is all one piece of paper with no cuts.

Disadvantages of Accordion:

  • It is harder to only show one section at a time.
    • In this example, adding and subtracting are shown at the same time, and multiplying and dividing are shown at the same time.
    • If students need to “block out” the other sections, consider having them use a piece of cardstock (so they can’t see through the paper) that is cut in thirds to cover the other section.
    • Another option:  have them put a folder over the section they need to cover.

SEE AN EXAMPLE HERE:

I chose this option because it is similar to some other foldables that I made recently, so I was determined to make something with this template.

Another reason I picked this one?  The other flap/window options are the FIRST that came to mind, so I thought that this one might be more unique.

Need more reasons?  I decided that it might be helpful to have both addition/subtraction showing at the same time since they have nearly all of the same steps.  Then, I realized that multiplication/division can benefit from being next to each other as well.  After all, division becomes multiplication after flipping the second fraction!


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CREATE YOUR TEACHER FOLDABLE

  • Fill out your cover/titles depending on which style you chose.
  • Decide the steps/rules you want to use.  Write them out on your foldable.  Make sure there is enough space for students to write everything comfortably.
  • Determine how many examples you have room for.  Write them out.
  • Solve all of your examples.  Make sure that you have enough room to fit all of the necessary work.  If you need more room, consider taking out one or more examples.

Here is what my accordion style fraction operations foldable looked like after I typed up my steps and wrote in some examples.

After I took these photos, I decided to add some lines to separate each of the problems.

I started with my student version that had numbers for the steps and all of the questions typed.   Then, I wrote out the steps and solved all of the problems.  I put a box around my answers.

Later I typed up all of the steps and answers to get a completed “Answer Key” foldable.

MultiplyingDividingFractionsSideComplete_AmyHarrisonAddingSubtractingFractionsSideCompleteAmyHarrison

 

 

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CONSIDER MAKING MULTIPLE VERSIONS

Now that you have completed your teacher foldable, YOU ARE FINISHED!  Make sure that you have your complete version (a.k.a. answer key) for your reference and partially completed versions for each class.  All of these can be made by hand, but you may want to consider making versions on the computer for easy printing and copying.

Check out my typed student versions and answer key:

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If you are going to go to the computer, make one version that is the same as what the students will fill out.  Then, you can print it and make two-sided copies.

You just saved your class some time since they won’t have to write every single problem. PLUS, there is something to be said about having a typed worksheet-like paper with all of the questions already on it.  It is impossible for students to write down the problems incorrectly and they are easily distinguished from the work and answers that the students will write in.

Typed Student Version 1:

MultiplyingDividingFractionsSideStudentVersion1_AmyHarrisonAddingSubtractingSideStudentVersion1_AmyHarrison

 

INCLUDED IN FIRST PRINTABLE STUDENT VERSION:

  • All Titles (Adding Fractions, Subtracting Fractions, Multiplying Fractions and Dividing Fractions)
  • Text on the Cover (FRACTION OPERATIONS) and Operation Symbols
  • All Questions

Students will start with the printed foldable above, write out the steps, work out the problems, and write the answers to get the completed version to glue in their math notebook.

Completed Student Version:

MultiplyingDividingSideStudentVersion1COMPLETE_AmyHarrisonAddingSubtractingSideStudentVersion1_COMPLETE_AmyHarrison

Creating even more versions can be helpful if you want to show only one part at a time. However, this can be accomplished by covering most of the foldable with paper.

I created a second student version that has all of the steps already typed out.  This will be great for accommodations or if I am looking to save some time.

Typed Student Version 2:

MultiplyingDividingFractionsSideStudentVersion2_AmyHarrisonAddingSubtractingSideStudentVersion2_AmyHarrison

INCLUDED IN SECOND PRINTABLE STUDENT VERSION:

  • All Titles (Adding Fractions, Subtracting Fractions, Multiplying Fractions and Dividing Fractions)
  • Text on the Cover (FRACTION OPERATIONS) and Operation Symbols
  • All Questions
  • Steps Typed

NOT INCLUDED IN EITHER PRINTABLE VERSION

  • Steps
  • Worked out problems

REMEMBER:

By taking the time to type up the questions, you will save your students and yourself some time.  Simply make copies instead of having to create everything by hand.  Now you can use the same foldable template next year, unless you are teaching a new grade 😦

WOULD YOU PREFER TO PURCHASE THE COMPLETED FRACTIONS FOLDABLE AND TEMPLATE INSTEAD OF MAKING YOUR OWN?  GET IT HERE!

FractionOperationsAccordionStyleFoldable_COVER.png

CHECK OUT ANOTHER FOLDABLE I MADE WITH THIS TEMPLATE FOR CONVERTING BETWEEN FRACTIONS, DECICMALS, AND PERCENTAGES:

FractionsDecimalsPercentsFoldable_PIN [Autosaved] [Autosaved]

 

 

AmyHarrisonBWSquare

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.

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ANY SUGGESTIONS?

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:

FractionsFoldablePicture2

SlopeFoldablePin3

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