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Incorrect Application of PEMDAS and Order of Operations

Date: 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 

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 

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 
Associated Topics:
Elementary Addition
Elementary Division
Elementary Multiplication
Elementary Subtraction
Middle School Algebra
Middle School Division

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