Fraction Division in Middle School Math
Students may have been taught to “skip, flip, flip,” “invert and multiply,” or “multiply by the reciprocal.”
They may have “learned” the steps or learned a fun fraction division song to help them remember; but somehow they still flip the wrong one or they forget to flip at all.
OR students 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 students don’t understand why they’re flipping anything – it doesn’t mean anything to them.
Fraction Division Using Common Denominators
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 simplify)!
I was shocked – it seemed SO simple!
Fraction Division Example
5/6 divided by 2/3.
- Find the common denominator of 6 and 3, which is 6. This gives you 5/6 divided by 4/6.
- The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator.
Fraction Division Example 2
1_4/7 divided by 1_3/4.
- Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4.
- Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28.
- The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator.
- No simplifying, in this case.
Why Does Using Common Denominators Work?
When we look at 5/6 ÷ 4/6, there are 5 pieces and 4 pieces that are the same size (our numerators).
So, we’re really looking at 5 ÷ 4, or how many times 4 fits into 5.
Student Response to Dividing Fractions with Common Denominators
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.
Fraction Division Using the Reciprocal
It may be tough for them to understand in 5th or 6th grade, 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, if you have time:-)
Fraction Division Math Wheels
Students can save their wheel to use as a reference throughout the year.
What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?
Resources to teach and practice fraction division
Check out the course, Fractions: From Foundations to Operations.