[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!