How often have you taught fraction division to the students in your middle school math classes, only to find them "flipping" the wrong number? You may have taught them to "skip, flip, flip," "invert and multiply," or "multiply by the reciprocal." You may have listed out the steps, or taught them a nifty song, but somehow they still flip the wrong one or they forget to flip at all.
OR they change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred...and then they don't flip anything. Why does this happen? I'm going to say that it happens because they don't see the sense in it  it doesn't mean anything to them.
So, I started using another way to teach fraction division  perhaps you've heard of it, or you use it. I never learned it this way as a student, but I like it and it makes more sense to some students. I learned this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course I said no.
The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators. These are the steps: Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to doing this  hopefully). Step 2: Create a new fraction with the numerator of the first fraction over the numerator of the second fraction...this is your answer. Done (unless you need to reduce)! I was shocked  it seemed SO simple!
Check out this example  it's a simple one, for starters:
5/6 divided by 2/3. 1) Find the common denominator of 6 and 3, which is 6. This gives you 5/6 divided by 4/6. 2) The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator. Then reduce.
Let's look at another one, with mixed numbers:
1 and 4/7 divided by 1 and 3/4. 1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4. 2) Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28. 2) The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator. No reducing, in this case.
I've shown both methods to my sixthgraders. Some really like it. Others stick to the flipping method  but I don't know if this is because they like it better or because it was the first way they learned it.....most of them had been taught something about fraction division in 5th grade.
As far as teaching multiplying by the reciprocal  if students are going to use it, I think it's important that they understand WHY it works. It may be tough for them to understand, but if they learn the common denominator method first, the proof may then make more sense to them. I found a great article on the NCTM website that uses the common denominator method to prove why multiplying by the reciprocal works  check it out! Recently I made two math wheels, to use to teach both methods of dividing fractions taking notes will be more fun! What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?
Grab this free fraction operations math wheel!
5 Comments
Marian Lemon
11/13/2017 08:58:10 pm
I have taught both methods to my 6th graders for several years. We talk about easy problems to use "same denominator" method versus using "keep, change, rearrange", or multiplying by the reciprocal.
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11/14/2017 05:14:17 am
This is brilliant. I have never seen this but I WILL be using it! I now teach 6th and 7th grade (always looking for ideas) but have taught 8th through PreCalculus also (18 years) and I have never seen it before but I LOVE IT!
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Annette McKee
11/14/2017 09:53:39 am
Definitely going to try this as my students have just struggled with a Fraction operations test.
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Mrs. S
5/25/2019 12:15:51 pm
Neat trick. Will this help them when they get to Algebra and they're working with polynomials, or confuse them?
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1/27/2020 08:35:56 pm
These are some of the best ways to teach fractions. It presents visual objects and solid facts that cannot be forgotten in learner's mind.
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