Date: Fri, 01 Nov 1996 20:38:15 -0800
Sender: wallace 
Subject: middle school math question
I am a middle school teacher with a question about the order of operations. A fraction bar implies a grouping above and below the bar, however the old division symbol does not. What is the answer to this problem? Let x=-2 and y=3. Evaluate 12x / 2y (if the old division sign is used). One author says that 2y is an implied grouping therefore the answer is -4. Another author just says to substitute in the numbers and follow the order of operations- which is multiplication and division in order from left to right. The answer then is -36 because 12 times -2 is -24 divided by 2 is 12 times 3. (Please understand that the division symbol was not a fraction bar, but a line with a dot above and below.) I am wondering if this area has not been taught consistently in math textbooks. Dolciani is the only author I find that says that a number and a variable is an implied grouping. Other authors don't mention it. Thank you.
The problem arises not only in middle-school textbooks. Every so often while reading mathematics -- even written by professionals -- one comes across ambiguous notation involving division. That means the reader has to try all possible meanings until discovering the one the author had intended.

Rule: Either use the multiplication symbol or use parentheses. I prefer parentheses. 12x/(2y) is clear as is 12(x/2)y. The official rule says that 12*x/2*y means the latter (and all calculators that I'm familiar with would do the computation that way), but one shouldn't assume that everybody would agree with that rule, particularly when it is written 12x / 2y with a space on either side of the division symbol. I believe that the author who gave the contrary advice is thinking of expressions written by hand, where the grouping is implied by the spacing. On the other hand, I would bet that 1 / 2sin x would mean (1/2)*sin x.

Chris

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