Interesting Fraction Division

After reading Richard Skemp’s Relational Understanding and Instrumental Understanding at the Republic of Mathematics it got me to thinking about something that happened in class this past week.

I teach a class once a week at a local community college close to my school. I like it...it is a nice change, with an amazing mix of students; some, just out of high school, and others being older adults looking to go back to college or to enter college for the first time. I teach a sequence of three courses - Preparatory Mathematics, Introduction to Algebra (part 1), and Introduction to Algebra (part 2) - none of which count towards any of my student's degree plans because they are not considered to be college-level mathematics.


Teaching these classes are very rewarding. I had an older adult who stood outside the classroom door for five minutes on the first day of class last quarter, afraid to come in because she thought she was going to fail the class. She went on to be one of my better students. To see how she (and the rest) grow to realize all the mathematics they know and can do is special. I typically have about 75% of the students from the Prep Math course take all three classes in the sequence with me.

On a side note...it is funny is to see their reaction to how I let them retake parts of quizzes and the like during my office hour before class (actually, it is really funny to see their reaction to WolframAlpha or to GeoGebra). I explain how some students will get it sooner than others, and how I want their grades to be an accurate assessment of their most current understanding of the content. Everyone in the class has gone through high school, they know the game, so this idea is a shock to them. About halfway through the first quarter, they get it, and start stopping in during my office hour. Of course, there are some institutional constraints that are much different than high school, but as long as I can get everything done that is in the syllabus, I am cool.


Something came up this past week when reviewing fraction operations, in particular, division. We discuss the usual finding-of-common-denominators when adding and subtracting, and how you can, but you do not need to, find a common denominator when multiplying or dividing. Many of the students choose to because it is one less thing to remember, and I am ok with that. I am not going to allow fractions to be a deal breaker for them. Show me you can add, subtract, multiply and divide some everyday fractions, that is good. 


When it comes to division, we discuss the normal sorts of mnemonic devices they remember from high school. Same, change, flip. "We do not wonder why, we just flip and multiply." How about THAT for a mnemonic device! The usual question arises: "Why can't you just divide straight across in the same way you multiply straight across?" Since they ask, we talk about it. You can divide fractions by dividing the numerators and denominators straight across, but this method will sometimes leave you with a fraction in the numerator and a fraction in the denominator.

Not a big deal. This just means that you will need to do some additional work to change this fraction into one with integers in the numerator and denominator by rewriting the fractions with common denominators

which of course adds a layer of fraction sophistication, such as multiplying the numerator and denominator by 20.


This past week, someone who liked to find common denominators for multiplication and division problems, noticed that when you found a common denominator first, then divided straight across, you could avoid the messy fraction with fractions in the numerator and denominator.

Thought this was pretty cool. I had never thought about fraction division in this manner, probably because I never found common denominators when I divided. I just always flipped and multiplied!