Order of Operations vs. Associative Property

Date: 10/10/2002 at 12:39:38
From: Jeff Horn
Subject: Order of Operations vs. Associative Property

I volunteer to teach a weekly extra-credit math course in my 
daughter's 5th grade class. Today I taught a lesson on the correct 
"Order of Operations" in mathematical expressions (i.e.: parenthesis, 
exponents, multiplication/division from left to right, and lastly 
addition/subtraction from left to right).  

That very day, her regular math teacher taught a lesson on the 
associative property, in which operations in expressions only 
consisting of addition or multiplication can basically be performed 
in any order, with the same result. In fact, her homework assignment 
included some mental math problems in which she was to rearrange the 
order of operations in multiplication problems in order to more easily 
solve the equation in her head. For example, 4 x (6 x 25) might be 
rearranged to (4 x 25) x 6, which is perfectly okay under the 
associative property.

My question is: does this make the associative property an exception 
to the order of operations, a corollary, or something totally 
unrelated?  

Thanks for whatever help you can give me and thanks for the invaluable 
help your site gives me every week as I prepare my lesson plan (and 
try to keep one step ahead of the students!).

Date: 10/10/2002 at 15:28:23
From: Doctor Peterson
Subject: Re: Order of Operations vs. Associative Property

Hi, Jeff.

The ideas of Order of Operations, and properties of those operations 
such as commutativity and associativity, are distinct concepts. The 
former deals with the mere syntax of how we write algebraic 
expressions; the latter deals with the semantics - what the operations 
mean. It's as if I taught you that you should say "I ate a pizza" 
rather than "Me eat pizza" (teaching grammar), and the next day told 
you that you could change that sentence to "The pizza was consumed by 
me" (restating the same fact using different words and word order). 
These are not contradictory, but two different levels of language 
understanding.

So when we write

    4 * (6 * 25)

that means that we multiply 6 times 25, then multiply 4 by that value. 
It's just a set of instructions for calculating. But when we say that

    4 * (6 * 25) = (4 * 6) * 25

that says that if we make two different calculations, the results will 
be the same. This is an equation, asserting that this fact is true. 
And if we say that FOR ALL a, b, and c,

    a * (b * c) = (a * b) * c

this is a theorem, presenting a fact that we can then make use of 
whenever we want. It says that because of the ways numbers work, when 
we see this particular arrangement of operations, we can safely 
rearrange the order in which we do things, and will still get the 
same answer. It doesn't mean that these expressions don't mean 
different things (in terms of the operations they say to do), only 
that their values will always be the same when we carry out those 
operations.

So, combining the two concepts, if we see an expression like

    4*6*25

without parentheses, although we know that the rules of grammar say 
we should multiply from left to right, first doing 4*6 and then 
24*25, the rules of algebra say we can change the order to make the 
work easier. We can use the commutative property (swapping the order 
of the 4 and 6) and say that the expression has the same value as

    6*4*25

and then using the associative property we can do this by first 
multiplying 4*25 to get 100, then multiplying by 6 to get 600.

The Order of Operations rules told us what this expression _meant_; 
the properties of multiplication told us tricks we could use to do the 
calculation (or, later, to rearrange an equation in algebra).

I assume you have found discussions in our archives of order of 
operations, such as

   Explaining Order of Operations
   http://mathforum.org/library/drmath/view/57199.html 

and also discussions of properties, like

   Properties Glossary
   http://mathforum.org/dr.math/faq/faq.property.glossary.html 

Thanks for letting me know you find our site useful! I am always 
especially happy to be able to assist teachers (of any sort), which 
multiplies the usefulness of our efforts. If you have any further 
questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/10/2002 at 15:29:14
From: Doctor Roy
Subject: Re: Order of Operations vs. Associative Property

Thanks for writing to Dr. Math.

The left to right rule really only applies to division and 
subtraction. This is because they don't satisfy the associative
property of equality. To be really precise, it is because we are
really "loose" about the definition of subtraction and division. If we
define subtraction properly, it is adding by an inverse. For example,
(3 - 5) + 7, is really 3 + (-5) + 7.  By writing it this way, we can
rewrite to get 3 + ((-5)+ 7), since addition is associative. We get
the same result as before. 

The same applies to division. By dividing, we are really multiplying 
by a multiplicative inverse. For example 5/3 is really 5 * (1/3).  
Written this way, we can use the associative property. 

The "left to right" rule in the order of operations really exists
because elementary school/middle school students are generally too
young to understand the abstractions that would make the explanation
make sense. They are also generally too young to understand the
concept of an additive inverse and a multiplicative inverse.  So the
"left to right" rule allows us to keep subtraction and division as
they are without changing the basic rules of math that students are
taught (i.e. that subtraction and division are really addition and
multiplication with different names).  Even now, whenever I see
something like (3-5)-2, I have no problems associating 3 + [(-5) - 2]
to get the same answer, or even switching the order to get (-5) - 2 +
3. It's all really the same, even for multiplication/division.

So, really, by introducing of the associative property (and all the
other field axioms), we are introducing concepts like additive
identity (i.e. negative numbers) and multiplicative identity
(multiplying by 1/3 instead of dividing by 3). This is the point when
the somewhat artificial "left to right" rule can be abandoned for a
more formal explanation.

I suppose I have given a rather long-winded response, but the question
deserved a very full answer. Simply put, you can think of the
associative property of multiplication and addition to be an exception
to the order of operations, if it makes arithmetic easier. But in a
more real sense, the order of operations going from left to right was
already more an artificial structure that isn't properly needed except
when we want to use the very familiar constructs of division and
subtraction.

I hope this helps.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/