Why Does the Order of Operations Work?

Why does the order of operations help us arrive at the correct calculation?  How does it work, why is it PEMDAS?  Why not addition first, then multiplication then groups, or something else?

I took it upon myself to get to the bottom of this question because I realized that I am so familiar with manipulating Algebra and mathematical expressions that I know how to navigate the pitfalls.  That instills a sense of conceptual knowledge, but that was a false sense.  I did not understand why we followed PEMDAS, until now.

By fully understanding why we follow the order of operations we can gain insight into the way math is written and have better abilities to understand and explain ourselves to others.  So, if you’re a teacher, this is powerful because you’ll have a foundation that can be shared with others without them having to trip in all of the holes.  If you’re a student, this is a great piece of information because it will empower you to see mathematics more clearly.

Let’s start with the basics, addition and subtraction.  First off, subtraction is addition of negative integers.  We are taught “take-away,” but that’s not the whole story.  Addition and subtraction are the same operation.  We do them from left to right as a matter of convention, because we read from left to right.

But what is addition?  In order to unpack why the order of operations works we must understand this most basic question.  Well, addition, is repeated counting, nothing more.  Suppose you have 3 vials of zombie vaccine and someone donates another 6 to your cause.  Instead of laying them out and counting from the beginning, we can combine six and three to get the count of nine.

And what is 9?  Nine is | | | | | | | | |.

What about multiplication?  That’s just skip counting.  For example, say you now have four baskets, each with 7 vials of this zombie vaccine.  Four groups of seven is twenty-eight.  We could lay them all out and count them, we could lay them all out and add them, or we could multiply.

Now suppose someone gave you another 6 vials.  To calculate how many vials you had, you could not add the six to one of the baskets, and then multiply because not all of the baskets would have the same amount.  When we are multiplying we are skip counting by the same amount, however many times is appropriate.

For example:

4 baskets of 7 vials each

4 × 7 = 28


7 + 7 + 7 + 7 = 28


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Consider the 4 × 7 method of calculation.  We are repeatedly counting by 7.  If we considered the scenario where we received an additional six vials, it would look like this:

4 × 7 + 6

If you add first, you end up with 4 baskets of 13 vials each, which is not the case.  We have four baskets of 7 vials each, plus six more vials.

Addition compacts the counting.  Multiplication compacts the addition of same sized groups of things.  If you add before you multiply, you are changing the size of the groups, or the number of the groups, when in fact, addition really only changes single pieces that belong in those groups, but not the groups themselves.

Division is multiplication by the reciprocal.  In one respect it can be thought of as skip subtracting, going backwards, but that doesn’t change how it fits with respect to the order of operations.  It is the same level of compacting counting as multiplication.

Exponents are repeated multiplication, of the same thing!


3 + 6

4 × 7 = 7 + 7 + 7 + 7

74 = 7 × 7 × 7 × 7

This is one layer of further complexity.  Look at 7 × 7.  That is seven trucks each with seven boxes.  The next × 7 is like seven baskets per box.  The last × 7 is seven vials in each basket.

The 4 × 7 is just four baskets of seven vials.

The 3 + 9 is like having 3 vials and someone giving you six more.

So, let’s look at:

9 + 4 × 74

Remember that the 74 is seven trucks of seven boxes of seven baskets, each with seven vials!  Multiplication is skip counting and we are skip counting repeatedly by 7, four times.

Now, the 4 × 74 means that we have four groups of seven trucks, each with seven boxes, each with seven baskets containing seven vials of zombie vaccine each.

The 9 in the front means we have nine more vials of zombie vaccine.  To add the nine to the four first, would increase the number of trucks of vaccine we have!

Now parentheses don’t have a mathematical reason to go first, not any more than why we do math from left to right.  It’s convention.  We all agree that we group things with the highest priority with parentheses, so we do them first.

(9 + 4) × 74

The above expression means we have 9 and then four more groups of seven trucks with seven boxes with …  and so on.

Exponents are compacted multiplication, but the multiplication is of the same number.  The multiplication is compacting the addition.  The addition is compacting the counting.  Each layer kind of nests or layers groups of like sized things, allowing us to skip over repeated calculations that are the same.

We write mathematics with these operations because it is clean and clear.  If we tried to write out 35, we would have a page-long monstrosity.  We can perform this calculation readily, but often write the expression instead of the calculation because it is cleaner.

The order of operations respects level of nesting, or compaction, of exponents, multiplication and addition, as they related to counting individual things.  The higher the order of arithmetic we go, up to exponents, the higher the level of compaction.

Exponents are repeated multiplication, and multiplication is repeated addition, and addition is repeated, or skip counting.  We group things together with all of these operations, but how that grouping is done must be done in order when perform the calculations.

The Problem with PEMDAS

The problem with PEMDAS

This problem has really stirred a lot of interest and created a buzz on the internet. I can see why, it’s an easy one to miss.  And yet, PEMDAS is such an easy thing to remember, the mnemonic devices offered make for a strong memory.  So people passionately defend their answers.

6 ÷ 2(2 + 1)

I am going to tell you the answer in just a moment, but before I do, please listen to why I think this is a worthy problem to explore.

There are two fundamental misconceptions with math that make math into a monster for so many people, and this problem touches on both.  In a sense, neither has anything to do with the order of operations specifically.

The first issue is understanding that spatial arrangements in math mean something.  The way we write the numbers and symbols has a meaning, very specific at that.  In this video by Mind Your Decisions, https://youtu.be/URcUvFIUIhQ, he shares where there was a moment in time when we used different conventions to write math.

And while math may or may not be a human invention, the symbols and arrangements and their meanings certainly are.  Just like the letter A is only a letter and with a specific sound because we all agree.  Just like a red light means stop, a green light means go and a yellow light means HURRY HURRY HURRY!

The second, and more over-arching issue here, is the misconception that addition and subtraction are different.  They are fundamentally the same thing.  Subtraction is really addition of opposite numbers.  Perhaps to shore this misconception negative numbers should be introduced instead of subtraction.

Now you might argue and say, Wait, addition has properties that subtraction lacks, like the commutative property.

You’re correct, 5 + 3 = 3 + 5, while 5 – 3 does not equal 3 – 5.  However, 5 – 3 is really five plus the opposite of three, like written below.

5 + - 3

And that is the same as this expression below.

-3 + 5

So the AS at the end of PEMDAS is really just A, or S, whichever leads to the better nursey rhyme type device to improve recall.

Since we believe that addition and subtraction are different, we also come away with the belief that multiplication and division are different.  Sorry, they’re not.  Division is multiplication of the reciprocal.  Remember that whole phrase from your school days? (How was that for a mnemonic device?)

And while division does not have the commutative property, that again is a consequence of the way we write math.  If we only wrote division as multiplication of the reciprocal, we would see that multiplication and division are in fact the same.

So, back to the problem.  The most common wrong answer is 1.  The correct answer is 9.  Here’s a great video on the order of operations, super catchy and articulates the importance of left to right as written for multiplication and addition.

Last thing:  Now, in creative writing the intent of the author must be considered, should it also be considered here?

Let me know what about this you like, dislike or disagree with.  Let me know what is helpful.  I really want to promote success through making math transparent.  It’s my mission.  You can help support my mission by just sharing and liking this.  Subscribe to my blog if you’re a teacher as I will be populating it with lots of teacher advice, not all math related.

Thank you again for reading.