On Wed, Jul 13, 2011 at 3:38 AM, <[email protected]> wrote: > > > > Special bonus points for anyone who can come up with an example of > > division with fractions (ex: 1/3 divided by 1/2) > > 1/2 goes into 1 twice.
When the divisor OR the quotient are whole, people do use fraction division in life. Many people will conceptualize "How many halves are there in three pizzas?" or even "How many quarters are there in three halves?" (the last one is a stretch) as division. However, after messing with fraction division for about a year (see http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction ) I believe people who don't have PURE MATH purposes avoid conceptualizing division of a fraction by a fraction, when it's not immediately clear the result is a whole number. Instead, they conceptualize it as TWO operations (multiplication and division) where at least one number is whole. The pure math purposes have to do with extensions of operations. In mathematics, figuring out how operations work for all types of numbers and even non-number entities is a very strong value. As such, we want to subtract greater numbers from smaller ones, take square roots of negatives, and multiply anything whatsoever (zeros, ordered arrays, transformations, etc.) This extension value definitely tramps any muggle values such as cognitive accessibility or ease of calculation. There are strong mathematical reasons for holding the extension value dear. We just have to realize these reasons don't necessarily apply to eating pizzas, or even to math-rich professional practices such as nursing (let me know if you want "Proportional Reasoning in Nursing Practice" study). > In fact it goes into any whole number N by dividing > N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into > 1/3 twice 1/3 There - you conceptualize it through whole-number steps. These steps are entirely sufficient for dividing pizzas. You only need to re-conceptualize these steps (at a significant cognitive cost, as my teaching experiments indicate, if you go beyond the example of 1/2) as division by a fraction if you are going for the mathematical value of figuring how fraction division works. There are no utilitarian or artistic purposes, that I could find in more than a year of looking for them, in conceptualizing the separate steps as division by a fraction. In practice, nurses, pizza cooks, carpenters and so on don't "really" divide by fractions - they work with numerators and denominators separately. I would suggest exploring reasons behind the math value of stretching operations, for example, talking about how inefficient it would be to program operations separately for different types of variables. Cheers, Maria Droujkova 919-388-1721 Make math your own, to make your own math. > , or 2/3. If you divide a circle into sixths, you can easily > see that a third of the circle (two pieces) is two-thirds of half the > circle (three pieces), in just the same way that, for example, two beads > is 1/4 of eight beads. > > It has been done in detail, and is available on various OER sites, some of > which are given at > > http://wiki.sugarlabs.org/go/Open_Education_Resources > >
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