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### 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/
```
Associated Topics:
Elementary Division
Elementary Multiplication
Elementary Number Sense/About Numbers
Elementary Subtraction
Middle School Division
Middle School Number Sense/About Numbers

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