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!


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38 comments on “Fraction Division — A Poem

  1. 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

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  2. 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.

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  3. 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.

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  4. 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.

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  5. 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?

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  6. 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.

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  7. 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! ;)

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  8. 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

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  9. 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.

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  10. 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*

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  11. 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

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  12. I was looking at a home
    When i noticed a distortion
    I am sure I am not alone
    The front door was out of proportion.

    This is a cool poem my mom and i made up.

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  13. Incidentally, this reminds me of a scene from a Japanese anime, where a young girl gets her elder sister to explain why 1/2 divided by 1/4 equals 2. The elder girl replies without skipping a heartbeat: you simply invert the 1/4 to become 4/1 and hence 1/2 times 4 equals 2.

    The young one isn’t convinced, and asks how on earth it is possible to divide something by a quarter-she reasons you can cut a pie into 4 pieces, but how do you cut a pie into one quarter pieces? The elder one was at a loss, and simply told her to “accept it” and move on.

    How would you explain the above in a manner which makes sense?

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  14. You can cut a pie into 4 pieces and ask what size the pieces will be. But you can also cut a pie into pieces of size 1/4 and ask how many pieces you will get. Either way, it’s division — just as with whole number division, you can either name the number of groups you are making or name the size of the groups. Whichever you name, the answer of the division gives you the other one.

    If you cut 1/2 pie into pieces of size 1/4, you will get two pieces. Or to put it in other words: “How many 1/4’s are there in one half?”

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  15. Sounds workable, but somehow kids are just troubled by the 1/4 thingy because it is logical
    to cut up stuff in whole quantities but not in fractions. Thanks for sharing.

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  16. You tutor high school students, right? Do they do any cooking or crafts? You might be able to explain it as measurement. For example: I have 1/2 pound of cheese, and I need to grate 1/4 pound to top each pizza. How many pizzas can I make? Again, you need to find out how many fourths in a half, but perhaps the mental image would be easier.

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  17. Pingback: How to Understand Fraction Division « Let's Play Math!

  18. Yes I do, though in Singapore high school is called junior college (for 17 and 18 year olds) and we handle much higher order stuff like calculus and complex number. Fractions is a concept taught at the primary school level here (for children around the age of 7-9), but ever since I taught maths I became pretty fascinated with getting ideas across regardless of their difficulty level. I believe that teaching is somewhat a body of work which must be constantly updated and re-learnt constantly. Your blog is greatly inspirational :)

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  19. Fraction division is usually taught (though unfortunately not mastered) in middle school in the U.S. — around ages 11-14. Most students don’t master it until algebra, when they just memorize the rule through repeated use. But I hope that if students can learn to think of division as the inverse of multiplication, they should be able to see the sense in how it works.

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  20. Digesting stuff without even attempting to thoroughly understand the underlying mechanics of various maths concepts is also a grave problem in Singaporean schools, where attaining good grades supersedes everything else. So the child simply memorises stuff with the sole intent of vomiting (for lack of a better word lol) it out in its entirety during examinations, because history has proven that you can score well with such a strategy over here.Our system is too deeply embedded in rote learning and hasn’t been very successful in tearing apart from these shackles so far, despite the tenuous efforts of the education authorities to change learning mindsets.

    Maths demands substantial agility and flexibility to succeed, however these skillsets can only be truly acquired if the student sits down and starts seeking out the Whys behind the formulas taught.

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  21. Sounds very much like the U.S. — except that in Singapore, students mostly succeed in learning the rules (judging by performance on international tests). Ours don’t, in part because our culture doesn’t put the same stress on good grades. In fact, having too high of grades is culturally discouraged. :(

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  22. I don’t know how to articulate this, but all I can say is the picture looks better when viewed from the outside As a Singaporean tutor, my personal opinion of Singaporean maths students is that most of them aren’t that capable as teachers from other countries made them out to be.

    Not too long ago I wrote a guest post on wild about maths that discusses the Singapore maths education system, the URL is here: http://wildaboutmath.com/2011/08/13/taking-a-deep-hard-look-at-singapore-mathematics/#more-532 ;you can take a look if you are interested, Peace.

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  23. I love your poem! I plan to use it for a presentation I have to do for my English 345 class and when I graduate and have a class of my own! I will also be including a short bio of you, the author. Thank you, it was precisely what I wanted for my math poem :)

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  24. Sure, Margo, go ahead! But also, I’d recommend emphasizing the real principle: dividing by any number is the same as multiplying by that number’s reciprocal. This is true of whole numbers, which is how I usually start — that dividing by 2 is the same as multiplying by 1/2, and dividing by three is the same as 1/3 of the number, and so on up the number line. But of course, the principle applies to any kind of number, so it gets used a ton in algebra.

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