Today in my 6th-grade classes we were talking about dividing fractions, and we were modeling them with rectangles, á la Fawn Nguyen. My second period is full of awesome students, and they were doing such good work and were keyed into the discussion today. I was so proud of their efforts. As we got to the last example (dividing a unit fraction by a whole number), a student piped up and said, “Hey (discovery moment) I have a shortcut, you can just multiply the numbers together and get the denominator of the answer.” The class was split on whether or not this would hold true for other problems. So we tried a few more. A happy cry went out from across the room, “Oh, my gosh, this works!” The bell rang.
For me, student discovery of the algorithm of dividing fractions is a rare thing. I’m not sure I’ve heard a student discover this particular algorithm on their own. Usually, if I have students who “know the easy way” have been taught the algorithm explicitly and are remembering that way. I heard the “KFC,” “flip it, change it, what was it?” discussed between students in another class about five different ways today. When I had students direct those comments to me today, I said we would be talking about that later, but for today we were going to look at models to help us understand why our quotients made sense.
So, 2nd-period ended right as the discussion was getting good, but I used that A-ha moment to challenge the next two classes. When we got to that problem, I said, “Hey, class, a student in a previous class came up with this rule, what do you think?”. We would work through a couple of examples and then I challenged them to find an example where the rule didn’t work. I’d just recommended this very type of problem to them on Khan Academy, so I know they’ll have a steady stream of problems to check it against. So, the classes are figuring out if we now have a shortcut called “Student’s Name Rule” or “Student Name’s idea that works sometimes”.
This was a high point today. Motivation to keep moving forward.