Fraction Division — A Poem
[Rescued from my old blog.]
Division of fractions is surely one of the most difficult topic in elementary arithmetic. Very few students (or teachers) actually understand how and why it works. Most of us get by with memorized rules, such as:
Ours is not to reason why;
just invert and multiply!
The problem with the “invert and multiply” rule is that students don’t internalize it properly. They may invert one or the other fraction almost at random, or invert them both for good measure — or they get the idea that inverting and multiplying go together, so they carefully invert every fraction that they want to multiply. In my math classes, I have seen all these mistakes and more. Haven’t you?
This fall, my pre-algebra students studied fraction division. I tried to give them an understanding of why it works, but I doubt that got through to many of them. I also tried to communicate the basic principle that dividing by any number is the same as multiplying by that number’s reciprocal. I hope I repeated that phrase often enough for it to sink in, at least for some of the students.
But I didn’t want to trust to hope, so I also wrote a new mnemonic poem, aimed at preventing as many of the mistakes listed above as possible. Perhaps you will find it useful with your students:
When you must divide a fraction,
Do this very simple action:
Flip what you’re dividing BY,
And then it’s easy—multiply!
:: 
:: 
:: 
:: 
:: 
:: 
:: 
:: 
:: 
:: 
:: 
Don’t miss anything! 
Subscribe in a feed reader, or get updates by Email.
Have more fun on Let’s Play Math! blog:
hey! I like this … and it is just in time for back to school with my two boys in grade 6 and 5.
I am thankful for your tip today!
thanks
Kristina
while surfing thru WordPress tagsurfer I came across your blog. I enjoyed your reading your blog. I intend to visit again to make some more relevant comments. Thanks for sharing. Lots of love
Thanks for the mnenoic,
should make fractions less demonic,
with this simple song,
she’ll stop dividing wrong.
Seriously, my daughter would invert whichever of the two fractions happened to hit her fancy. Seems, though, she tended to invert the first fraction rather than the second.
If your student inverts the first fraction, but does the multiplication correctly, she will get an answer that is the exact reciprocal of the true answer. Not much help, but I think it’s cool that it works out that way. I use that fact sometimes when I realize I’ve entered a chemistry problem wrong on my calculator: just hit the 1/x button to make it right.
I just wrote out the poem on an index card and stuck it in my daughter’s math book for ready reference.
An alternative approach to invert-and-multiply is to find a common denominator, then simply divide the numerators by writing them as a fraction. For example: 2/3 divided by 1/4 = 8/12 divided by 3/12. At this point, it doesn’t matter whether you have 12ths or books or hippos — you have 8 things and you want to know how many groups of 3 things you can create. So 8/3 = 2 and 2/3. Many students find this easier to do and easier to understand, especially for fractions for which it’s simple to find a common denominator.
You’re right, Lynn. Finding a common denominator is a great shortcut when working with easy numbers. Unfortunately, I don’t think it translates well to algebra fractions — or have you used it there, too? Or do you use the common denominator approach as a springboard from which to teach the more general invert-and-multiply rule?
I used to know a mnemonic phrase for dividing, had something to do with kings and mushrooms. Can anyone help me?
Sorry, Bev, I don’t know that one. Sounds intriguing. Maybe someone else will comment?
My daughter had a terrible time with invert and multiply until I showed her how to find 3 divided by 1/3, or, how many 1/3’s are there in 3 tangerines? There are 3 thirds in one tangerine, so how many thirds are there in three tangerines? Nine. How did we get 9? By mulitplying 3 x 3. She had an “aha” moment. Ever since then, dividing fractions has been simple for her.
Sally, I am a 23 year old Harvard graduate and just had the exact same aha moment your daughter had.
In trying to teach fractions to a kid I volunteer with I realized that I must have learned division of fractions by a mnemonic memorization method myself even though I much prefer to understand math conceptually and had been teaching it that way too. We got through addition, subtraction, and multiplication just fine and all of a sudden I realized I couldn’t explain conceptually why I flipped the fraction I was dividing by. Your example made so much sense and helped me to see how the same simple problem solving concepts could be applied to more difficult fraction problems. Thanks a bunch! 
hey thanks for the A+ on my math homework all i had to do was copy down ur poem cas that was our home work just to make up a dividing fraction poem THANKS
Glad to help!
Um, Mia, that’s plagiarism. Unless of course you gave Denise credit for her poem when you handed it in.
Oh, you’re right, Mathmom! I didn’t read Mia’s comment carefully enough, so I didn’t realize she turned in the poem as her homework — I just assumed she meant the poem helped her work homework problems.
i love this poem my mom showed it to me and it really helped me in school i wrote it down and showed it to the teacher and she loved it thanks for the poem!!!!!!!!!!!!
*yay*
You’re welcome, Cindy. I’m glad you liked it, and it was very nice of you to write such an encouraging comment!
i am so glad my 5th grade daugthers teacher asked for a fraction poem.THANK YOU LORD!!!!!!!!!!!!!!!
zaria
I’m glad to be a blessing, Zaria, but I hope you didn’t let your daughter pretend that she wrote the poem. That would be stealing!
this iiz a very cool… i likt this
web site alot…
fraction are very kwl…
did you know we use fraction in our lives every day
YER !! it’s true
<3 <3 elouise