Incorrect Application of PEMDAS and Order of OperationsDate: 09/14/2006 at 16:13:38 From: Monica Subject: Understanding of the Order of Operations I was working with students on the order of operations today and explained that multiplication and division are done from left to right, as are addition and subtraction. Apparently, they believe they were taught in the past to do all addition and then all subtraction. I tried to show examples of why that wouldn't work, but they simply did the problem their way, obtained a different answer and asked why it was incorrect. Are there any examples or explanations that would clearly explain why they must be done from left to right? Date: 09/14/2006 at 16:37:38 From: Doctor Peterson Subject: Re: Understanding of the Order of Operations Hi, Monica. It's impossible to show that they MUST be done from left to right; that is nothing more than a convention we all agree on. Your class has shown that it makes a difference which order you use; that proves that we MUST make some choice that we can all follow. What that choice is, is not so definite. But it makes a lot of sense to go left to right, for the following reason. We define subtraction this way: a - b = a + -b This allows us to think of any subtraction as an addition; we essentially just attach the negative sign to the number following it, rather than taking it as a different operation. The subtraction requires no extra rules, just the rules we already have for addition. If we do this, then 2 - 3 + 4 = 2 + -3 + 4 = 3 That is the same result we get if we do the operations from left to right (and it doesn't depend on whether we do the ADDITIONS from left to right, since addition is commutative!). If we did the addition first, we would get 2 - 3 + 4 = 2 - (3 + 4) = 2 + -(3 + 4) = 2 + -7 = -5 Note that this time, the negative sign ended up applying to ALL the following numbers, rather than just to the one after it. So doing additions and subtractions from left to right makes it easier to transform an expression into one involving only addition; and since addition is commutative and associative, it is MUCH nicer to work with! The rule, therefore, arises from the wish to make expressions easier to handle. Without it, a lot of algebra would turn out to be a lot harder. So your students should thank whoever first made this choice! Now, your student's misunderstanding of the rule very likely comes from the use of PEMDAS or something equivalent, which is meant to be only a summary of the rules. It sounds as if A comes before S, but that twists the intended meaning of the mnemonic. See this page for another thought: Confusion over Interpretation of PEMDAS http://mathforum.org/library/drmath/view/66614.html That says essentially the same thing I just said, but about multiplication and division, which is an even bigger problem. (Did you know that in other countries they use BODMAS instead of PEMDAS, so students often think division should be done first?) If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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