Valorie works as a teacher's aid in a 6th grade classroom at a local elementary school.
They've been working on dividing fractions recently, and she spent about two hours yesterday working with one student trying to explain exactly how division of fractions works.
So I figured I'd toss it out to the blogsphere to see what people's answers are. How do you explain to a 6th grader that 1/2 divided by 1/4 is 2?
Please note that it's not sufficient to say: Division is the same as multiplication by the inverse, so when you divide two fractions, you take the second one, invert it, and multiply. That's stating division of fractions as an axiom, and not a reason.
In this case in particular, the teacher wants the students to be able to graphically show how it works.
I can do this with addition and subtraction of numbers (both positive and negative) using positions on a number line. Similarly, I can do multiplication of fractions graphically - you have a whole, divide it into 2 halves. When you multiply the half by a quarter, you are quartering the half, so you take the half, divide it into fours, and one of those fours is the answer.
But how do you do this for division?
My wife had to type this part because we have a bit of, um, discussion, about how simple this part is....
How can you explain to 9-11 year old kids why you multiply by the reciprocal without resorting to the axiom? It's easy to show graphically that 1/2 divided by 1/4 is 2 quarters because the kids can see that there are two quarters in one half. Equally so, the kids can understand that 1/4 divided by 1/2 is 1/2 of a half because the kids can see that only half of the half is covered by the original quarter. The problem comes in when their intuition goes out. They can solve it mathematically, but the teacher is unwilling to have them do the harder problems “on faith“ and the drawing is really confusing the kids. Having tried to draw the 5/8 divided by 3/10, I can assure you, it is quite challenging. And no, the teacher is not willing to keep the problems easy. And no, don't get me started on that aspect of this issue.
I'm a big fan that if one method of instruction isn't working, I try to find another way to explain the concept. I visited my usual math sites and found that most people don't try to graph this stuff until 10th grade or adulthood. Most of the sites have just had this “go on faith“ response (show the kids the easy ones, and let them “go on faith“ that it will hold true for all cases). I really wish I could figure out a way to show successive subtraction, but even that gets difficult on the more complicated examples.
What I am hoping is that someone out there can provide me with the “aha!“ I need to come up with a few more ways to explain this. What this has been teaching me is that I've been doing this “on faith“ most of my life and never stopped to think about why myself.
Any ideas/suggestions would be much appreciated.