Explaining Order of Operations

Date: 09/06/2001 at 18:25:28
From: Carrie
Subject: The order of operations

How would you explain the order of operations to a 5th or 6th grader 
who has not yet studied the subject?

Date: 09/06/2001 at 22:49:02
From: Doctor Peterson
Subject: Re: The order of operations

Hi, Carrie.

You can use the Dr. Math search page to find all the answers we've 
given on this topic, including our FAQ:

   http://mathforum.org/dr.math/faq/faq.order.operations.html   

Some of our answers are at a reasonably basic level, such as

   Why Rules?
   http://mathforum.org/dr.math/problems/ramila.08.15.01.html   

I'm not sure I've ever written up how I would introduce the concept to 
someone who hasn't seen it before at all, so I'll give it a try.

We can start by thinking of a simple calculation we want to have 
someone work out for us; say, I buy 3 items at $5 each, and 4 items 
at $6 each. How much do I spend? It's easy to describe in words what 
to do; multiply 3 by 5 and 4 by 6, and add the products. That gives 
15 + 24, or 39. But rather than do this ourselves, we want to tell 
someone else what to do, and we want to use symbols rather than words. 
How do we write this down for him? The natural thing to do is to write

    3 * 5 + 4 * 6

(I'm using "*" for the multiplication sign.)

Now suppose we give this to someone and tell him to do the 
calculation, without telling him what it's for. What will he do? He 
may just type it into a calculator just as he sees it:

    3 * 5 = 15; add 4 and we get 19; that times 6 is 114.

What happened? He didn't do what we wanted him to do, and he got a 
different answer! And he could do it in several other ways as well, 
such as

    5 + 4 = 9; 3 * 9 * 6 = 162.

What went wrong is that we haven't said what "grammar" we want our 
symbolic language to have. We know what all the "words" (numbers and 
symbols) mean, but you can put them together in different ways. We 
need rules for the order in which we do the operations we've written 
down.

At this point, you might want to take some time to think about how you 
can solve the problem yourself; kids can often devise strategies that 
are very similar to what mathematicians developed among themselves.

One way to solve this is to indicate explicitly what order to do it 
in, by putting parentheses around parts of our expression that are to 
be done first:

    (3 * 5) + (4 * 6)

This says exactly what to do, so that it can be done only one way. The 
parentheses "package" the operations: you have to treat (3 * 5) as a 
single number, so you can't do anything with it until you have turned 
it into 15; once you've processed both packages, you have nothing left 
to do but add 15 and 24.

That would be a fine solution, but we're lazy. Writing parentheses 
everywhere would get boring and confusing; we'd like some rules to let 
us avoid them when we can. Sometimes such rules come naturally; for 
instance, there's no need for parentheses in 1 + 2 + 3, because the 
addition

    (1 + 2) + 3 = 1 + (2 + 3)

works out the same regardless of the order. So we can drop parentheses 
in such cases, where we do a string of additions or multiplications. 
But some similar cases are not so easy:

    (3 - 2) - 1 = 1 - 1 = 0

    3 - (2 - 1) = 3 - 1 = 2

We can't just drop parentheses in a string of subtractions or 
divisions.

It turns out that we very often want to do something much like my 
original example expression, a sum of products. So we've agreed on a 
rule:

    Do all multiplications before all additions.

Because divisions are closely related to multiplication, and 
subtraction is backward addition, we can expland this rule:

    Do all multiplications and divisions;
    before all additions and subtractions.

This lets us write our expression just the way we did at first, and it 
will always be interpreted the way we wanted:

    3 * 5 + 4 * 6 = 15 + 24 = 39

But what about that string of subtractions? We'd like a rule to let us 
write

    3 - 2 - 1

without confusion. For that, we just agree to do everything from left 
to right, as if it said, Start with 3, subtract 2, then subtract 1. So 
now the rule is

    Do all multiplications and divisions
    (left to right, in the order they appear), and then
    do all additions and subtractions
    (left to right, in the order they appear).

Of course, you can still use parentheses to change the order when you 
want to say something different:

    3 * (5 + 4) * 6 = 3 * 9 * 6 = 162

There are other details to be added later, such as exponents; but this 
is the core of the concept: the multiplication family goes first, and 
then the addition family cleans up.

That's how I'd explain this: no "PEMDAS" or arbitrary decrees, just a 
development of reasonable rules, in the same way kids inventing a game 
together might agree on rules that satisfy them. Of course, we want to 
make sure we come up with the accepted rules, and not some new ones, 
since we have to communicate with others; but it's good to see that it 
all makes sense, and arises from simple principles.

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