Why Order of Operations?

Date: 17 May 1995 07:21:59 -0400
From: Anonymous
Subject: A question

     Recently some 7th-grade math students were introduced to the concept of
"order of operations."  Several of the students had questions about WHY we
use the order of operations at all... why not just set up problems to read
from left-to-right?  (One student even came in with a demonstration of
"reverse Polish notation"!)

     Can you suggest some "real-life" examples that show the "necessity" of
using the order of operations?  Thank you.

Jerry Taylor
Educational Technology Specialist
Arcadia Middle School
Greece, NY
JerryTaylr@aol.com

Date: 17 May 1995 11:23:21 -0400
From: Dr. Sydney
Subject: Re: A question

Dear Jerry,

Good question!  It certainly helps people understand mathematical concepts
better when they can relate them more closely to their own lives, so this is
great!

On a simple level, I'll give you a few examples.  You could perhaps ask the
students to come up with more complex examples for fun!

Suppose two classes are going on a field trip to the zoo.  There
are 28 people in one class and 22 people in the other class.  The teachers
want to order lunch for all of the students, and in each lunch, they
want there to be 2 packages of crackers.  How many packages of crackers
should the teachers order?  Well, here is where order of operations comes in:
The teachers want to order 2*(28+22) packages of graham crackers.  If the
teachers didn't use order of operations, then instead of ending up with 100
packages of graham crackers, the teachers would end up with 78 packages of
graham crackers, and some of the kids would be very unhappy. 

The above example demonstrates one kind of "order of operations."  Here is 
another example which uses what perhaps you really mean when you say "order 
of operations."

Suppose on that same bus trip each teacher also wants one package of
crackers.  Then, the teachers write this down mathematically as:

2 + 2*(28+22) = 2 + 2*(50) 

Using correct "order of operations" the teachers will figure out that they
should order 102 packages of crackers.  If instead the teachers were to not
use "order of operations," they would order 200 crackers, and that would
just be too much!!!!!!!!!

I hope these examples help, and if you have any other questions, feel free
to write back!  

--Sydney, "Dr. Math"

Date: 18 May 1995 19:10:15 -0400
From: Anonymous
Subject: Re: A question

Actually, if I did NOT use the "order of operations," I would come up with an
answer of ** 134 ** , rather than your 200, right?  ( 2 + 2 x 28 + 22 = 134 ,
not 200 ... there would be NO parentheses used if I truly did NOT follow
order of op.)

Your examples are interesting.  Let me share with you a couple of excerpts
from some of the comments I received from the NCTM listserv:

    real life problem
    The perimeter of a rectangle is L+L+W+W
    if l is 2 and w is 2 then  2 + 2 + 2 +2 = 8
    It is also 2(l + w) = 2 * (2 + 2) = 2 * 4 = 8  or
                          2 * 2 + 2  = 4 + 2 = 6

Not so "real life." The only reason that the example 2(l+w) "works" is
BECAUSE of the order of operations.  Let me suggest this approach: l+w*2.  It
"works" the simplest way...left-to-right!  All we have to do is all agree to
DO it that way, right?

P.S. Not trying to be argumentative...we're just trying to understand the
"why" behind the decision.  ;-)

    A man finds three wallets on the gound, each containing $4.  The next day,
    he finds five wallets, each containing $2.  (Let's assume he's: 1) lucky, 
    and 2) dishonest.)  In all, how much did he find?
         3 X 4 + 5 X 2 = 22  (order of operations)

    But, let me propose a problem like this:

    A man finds three wallets on the ground, each containing $4.  The next 
    day, he finds a five-dollar bill lying on the ground.  He bets ALL this 
    money with his bookie, and DOUBLES his cash.  How much does he have now?
         3 X 4 + 5 X 2 = 34 (left to right)

    Order of operations?  Or "common sense"?  Just wondering!    ;-)

Date: 20 May 1995 09:51:39 -0400
From: Dr. Ken
Subject: Re: A question

Hello there!

Let me offer a couple of comments on the order of operations issue:

To be sure, our way of notating arithmetic is not the only way of doing
things, as is shown by the reverse Polish notation that the student brought
in.  The main thing that must be true in a notational system is consistency,
i.e. when you use conventional notation you don't just start from the left
and chug through, but you use the traditional order of operations, and when
you use another notational system you stick to it as strictly.  Most of the
examples given to you on why we use the standard order of operations have
been examples of using the wrong order for that particular system.

Here's one reason we might think that our normal system is so neat.  You
know that the operations of addition and multiplication are _commutative_,
i.e. they give you the same result no matter what order you write them in:
2*3 = 3*2.  In conventional notation, that property is clearly reflected, and
you get lots of options for how you want to write your expressions:  
2*3+5 = 3*2+5 = 5+2*3 = 5+3*2 = 11.  In calculator-order notation, for
example, you only get the first two, 2*3+5 and 3*2+5.  In calculator-order
notation the third evaluates to 21, and the fourth evaluates to 16.  So the
main reason for using conventional order of operations is the flexibility it
gives you in writing down mathematical expressions.  It's important to
remember, though, that it really is just a "conventional" system, i.e. it's
a convention that we use it (albeit a pretty good one in my opinion).

I hope this gives you folks something to think about!

-Ken