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Textbooks tend to characterize fractions in each of the following three ways, though often without directly saying so. And most adults, including teachers, tend to think of fractions in each of these ways:
Another combination that didn't make sense arose as I was examining the issue: why and how were (1) and (3) equivalent, as in the following: if you take 5 pies and divide them each into 5 equal pieces, then 3/5 of the total amount can either be three out of the five pies or it can be three pieces out of each pie. That makes no difference in terms of amount of pie you have -- three pieces from each of five pies which are divided into fifths will reassemble into three whole, sliced pies. However, in terms of kinds of pie you end up with, this would make a difference if one has different kinds of pies to begin with. If one has three cherry and two apple pies, you don't necessarily get the same thing by taking, say, three of the cherry pies and calling them 3/5 of the pies, as you do when you take 3 pieces out of each pie and reassemble the pieces into (whole) sliced pies. In the former case you have only cherry pie, in the latter case you have a mixture of cherry and apple pie. Since these are not the same thing (in terms of substance), the question is why are they equivalent in terms of amount. It cannot be because they are the same "things", since they aren't.
I think I have finally figured out the answer to all this, which is that the above three things are not the same things, but they are equal things for reasons which may be more complex than you or anyone else needs or cares to know. But the answer is as follows.
Suppose you divide a pie into five parts. One pie DIVIDED by 5 = 5 pieces that are each 1 OUT OF 5 equal parts of the pie and .2 of the pie. Each of these quantities can be represented by 1/5, though the first means "one equal piece out of five" and the second refers to 1 (pie) DIVIDED by 5 in decimal form [which, incidentally but not accidentally, is also the same as 2/10 in fractional form, and which is equivalent to 1/5 by reduction]. Anyway, the DIVISION is of the PIE, and the numerator represents how many of the PIECES you are talking about after you have (figuratively or literally) divided the pie into pieces. So when you are talking about only ONE piece, it turns out that since there was only ONE pie, the numerator of the fraction will be the same as the number of things you divided -- the "1" refers equally to the number of pieces you are referring to and the number of pies; and the "5" refers both to the number of total pieces and the divisor by which you divided the one pie. Hence, these two NUMBERS will be equivalent -- the fraction as quotient and the fraction as portion - even though the "things" you are talking about are not the same. That is one piece out of five pieces ("1 piece/5 pieces") and one pie divided into 5 equal pieces ("1 pie/5 pieces") give you the same numbers in the numerator and denominator even though they don't give you the same units in the fraction or represent the same physical things.
Then, if you are talking about two pieces of one pie, since you are now talking about 2 TIMES one fractional piece of the pie, that will be NUMERICALLY equivalent to 2 TIMES (one PIE divided by 5). And so, 2 out of the 5 pieces [written 2 pieces/5 pieces] will be NUMERICALLY equivalent to 2 times one pie divided into 5 pieces [written 2(1 pie/5 pieces)].
Then if you have more pies to begin with, since NUMERICALLY 2 times (1 PIE/5) [that is, 2 times one pie divided into 5 parts] will be equivalent to (2 pies)/5 [that is, 2 pies divided by 5] -- which means 2 times the fractional portion of one pie is NUMERICALLY EQUIVALENT to one (same total fractional) portion of two pies, even though two fractional pieces of one pie are not the same thing as one (same) fractional piece of each of two pies. (E.g., if you had one blueberry pie and one cherry pie, 1/5 of each of them is not the same thing as two fifths of one of them -- they are just NUMERICALLY equivalent amounts.)
Some of this seems too difficult to point out to little kids learning fractions to begin with, but at the point that texts do start to switch from proportions to divisions, some of it could be useful; or, at the very least, it should be pointed out that the three things are not the same thing even though they turn out to be numerically equal or equivalent things for reasons too difficult to explain. This may save confusion on the part of those kids who might try to figure out WHY fractions of things, fractional portions of groups of things, and division of numerators by denominators are treated the same when they are not really the same things at all.
This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking. But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do. I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account. |