Order of Operations Dispute
Date: 09/26/2000 at 16:37:55 From: Terre and Robert Reidy Subject: Order of Operations Dispute The problem reads: N divided by (division sign)ml where n=12, m=6, and l=3. I believe the correct answer should be .6666, as 12 divided by 18 equals this. My husband agrees with me. My son came home very upset from school, with a note from his teacher that the answer was wrong. She indicated that I should have divided the 6 (m) into 12 (n) before I divided the 3 (l) into the equation. Her answer was 6. My son is very upset with me; his teacher told him I was doing "old fashioned math." Do I need to go back to school?
Date: 09/26/2000 at 21:03:38 From: Doctor Peterson Subject: Re: Order of Operations Dispute Hi, Terre and Robert. Let's write the problem as n / ml, with n=12, m=6, l=3 I can give you some good news and some bad news. First, the bad news: according to the usual order of operations rules now taught, your answer is wrong. You can read about the rules in the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.order.operations.html Basically, you do all multiplications and divisions before all additions and subtractions; and when you have a string of multiplications and divisions to do, you go from left to right in the order they come. In your case, that means you divide n by m giving 2, then multiply by l to get 6. BUT... You are not alone in your opinion. This part of the rule - doing multiplication and division together - is probably the last rule to have stabilized; I know that in the 1920's, at least, there was no agreement. It seems that an agreement developed, but it is unraveling now, as I hear from many students whose texts answer questions like this the way you did. It appears that they are adding an unstated rule, which seems entirely reasonable in this context, that an implied multiplication (indicated by simply placing two variables or expressions together, as in "ml") should be done first. It certainly looks as if it should mean that. The problem is that, although I've heard of this rule being followed frequently, I've hardly ever heard of it being taught, so these texts are not following their own stated rules. Since this type of expression is so ambiguous, with people disagreeing on the rules, and the rules being easy to overlook, my own opinion is that neither your answer nor the teacher's is right: the question is wrong. No responsible mathematician would write such an expression; we would just say n --- m l so there would be no question about its meaning. After all, the purpose of rules is to allow us to communicate clearly, not to help us trick students and start fights among families. So you may in fact be "old-fashioned"; or you may be on the cutting edge. In any case, I'm afraid you'll just have to learn how they are doing it in class, and follow along. There shouldn't be many more issues like this to worry about. You can read more about these issues in the Dr. Math archives: Order of Operations http://mathforum.org/dr.math/problems/wuandheil.05.19.99.html More on Order of Operations http://mathforum.org/dr.math/problems/breitenbach.2.13.00.html I hope this helps at least a little. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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