O.K., I’m ready to share my amazing approach to dividing a fraction by another fraction. Well, maybe not breathtaking … like Andrew Wiles’ proof of Fermat’s Last Theorem … but at least interesting. And best of all, fun and student-friendly!
Last week I asked if anyone had any tricks up their sleeves that make it easier for students to divide fractions. And I said that I would share a trick after I heard from you.
I got a nice response from Michelle, who said that she has used the mnemonic “KFC” (like the fried chicken), which in her class stands for Keep-Change-Flip. The idea being that you KEEP the first fraction, and next you CHANGE the sign from multiplication to division. Finally you FLIP the second fraction, the fraction on the right. We have similar mnemonic where I live, which goes by the phrase: Copy-Dot-Flip, with the “dot” meaning the dot of multiplication.
But what I want to share with you is a completely different approach to dividing one fraction by another, an approach that saves time, and makes it both easier and more fun — in my humble opinion — than the standard approach.
The approach I’m going to show you works for any complex fraction situation you might encounter, such as these:
For this blog post, I’m going to limit my chat to complex fractions of the arithmetic type, meaning those with numbers only, and no variables. And if it seems important, I’ll do another post later on using this very same process for algebraic fractions.
So what is this amazing approach, anyway? Well, it’s based on something I discovered on day when I was just messing around with fractions divided by fractions. I realized that after you do the KFC or the Copy-Dot-Flip, what you get — in general — is actually something really easy to grasp, as this next image will show you, along with a Quick Proof:
If you take a moment to think about it, the terms in the numerator of the result — terms a and d — have something in common; they were on the outside of the original complex fraction, so I call these terms the “outers.” In the same way, the terms in the denominator of the result — terms b and c — were both on the inside of the complex fraction, so I call them the “inners.”
So when you divide fractions in this vertical format, the answer is simply the outers, multiplying each other divided by the inners, multiplying each other.
I find that students find this easy to remember and a cinch to do. This next sheet summarizes the idea, and also provides a fun way of remembering the concept, thinking about the stack of terms as a fraction “sandwich.”
So, to put this in words, the four-level complex fraction that you start out with can be thought of as a sandwich, with two pieces of bread at top and bottom, and slices of bologna and cheese in the middle.
The main point is that to simplify the fraction sandwich, all you need to do is put the two slices of bread together in the numerator and multiply them, And then put the bologna and cheese together in the denominator, and multiply them.
Using this idea it becomes a lot easier to simplify these complex fractions. Here’s an image that shows how it is done, and how this approach saves time over the way we were taught to do it, using reciprocals.
And there’s more good news. This new way of looking at complex fractions also gives students a cool, new way to simplify the fractions before they get the answer. And when you do simplify fully, the answer you get will be a fraction that’s already completely reduced, so you won’t have to stress about that part.
The next two pages show you this fun and easy new way to simplify:
or, or what? … Here’s what …
So now you might like to see the whole process from start to finish, so you can decide for yourself if this technique is for you. Well that’s exactly what we’re showing next. As you can see I consistently highlight the outers with pink, the inners with yellow.
And finally, a “harder” problem, you might say. But check it out. Is it really any harder than the one we’ve just done? You decide.
In my next blog I’ll give you a few problems like these, so you can get used to this trick, and start shaving precious seconds and nano-seconds off the time it take you to do your homework, so you spend more time doing all of those things that you want to do more: texting, watching You-Tube, taking hikes, skating (roller and ice), etc. etc. , etc. You know better than me.
Happy Teaching and Learning!
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!
Comments on: "How to Divide Fractions: from annoying to FUN!" (5)
Have you tried getting a common denominator and the dividing across
3/4 ÷ 1/5
3/4 • 5/5 ÷ 1/5 • 4/4
15/20 ÷ 4/20 now, divide num & then divide the denom
(15÷4)/1 =15÷ 4 = 3 and 3/4
One more time…
2/3•2/2 ÷ 5/6 • 1/1
4/6 ÷ 5/6 now, divide num & then divide the denom
(4÷5)/1 = 4/5
Hi Teri! Thanks for the comment. What you’ve described is a nice approach for students who know how to convert fractions by changing the denominators. It also helps students understand the logic in these kinds of problems. I do, however, have two concerns about this approach. 1) There are many students who have trouble converting fractions to make the denominators the same, and your approach will make kids walk through this trouble zone. And 2) the approach would be rather time-consuming for many students. Otherwise, though, it is a good alternative approach for dividing fractions, and I appreciate your sharing it.
Josh, I am now designing a fraction activity for the Family Multiplication Study. Because of the nature of the study, we need two things:
1 – models for fractions that support multiplication and division
2 – ways for each student to create something of their own as they engage in the activity
I am thinking of a “tree of models” design where you take different models of division (splitting, round robin sharing, partitive and so on) and extend (“grow”) them into rational number operations.
Do you have anything on the topic, from those angles? Your trick is great at the formal stage, and I am asking about pre-formal, conceptual stages – making sense of fraction division.
I’d like to introduce you to the FMS people anyway 🙂
this helpet me alot thank u so much i am in 6 th grade and we are learning this right now and thank u so so so so so so so so so so so so so so so so so so so much and i think u for this so so so so so soo sooooo much because yea and so much
Hi, Thanks for your feedback.
I find in tutoring that students can use this shortcut and use it successfully.
Perhaps it is more difficult to have students retain it in the classroom, though.
I wonder why they call it “the ear.” I just call it the bologna-cheese sandwich,
or “the sandwich,” for short.
Thanks for your feedback. Interesting to get this perspective.