Order of Operations (PEMDAS)

So many of us have been there.  We’re given a giant, ridiculously long, multi-step equation that we need to solve.  We solve it.  Phew!  A friend solves it, they got a different answer….but it seems we’ve both done the computations correctly. 

What is going on!?

Take this scenario for example:

$$9+3\times11(3+6)=?$$

Student 1 solved the problem like this:

$$\style{font-size:16px}{\begin{array}{l}9+3=12\\12\times11=132\\132\times3=396\\396+6=402\\402\end{array}}$$

This student solved in order from left to right.  Each computation was done correctly and an answer of 402 was found.

Student 2 did something quite different: 

$$\begin{array}{l}9+3\times11(3+6)=?\\9+3\times11(9)=?\\9+3\times99=?\\9+297=?\\306=?\end{array}$$

This student followed the order of operations and did all of the computations correctly.  In this case, the answer they found was 306.

Two answers.  All correct computations, but both answers are NOT correct.  Which one is right and how do we know?!

Enter the order of operations.

Try to solve

What is the order of operations?

Many years ago mathematicians found themselves in this exact situation!  They realized that there were many ways to solve these complex equations and even if all computations were done correctly a variety of seemingly correct answers could be found.

They decided this wouldn’t do. Somehow, likely even before algebraic notation existed, mathematicians came to a common understanding that there was an order in which equations should be completed so as to find only one right answer.

Today, this is what we call the order of operations.

The order of operations applies to all equations and is a math skill that you’ll use for the rest of your life once you learn it so pay close attention.

The order of operations tells us that: (Perhaps this could be made more graphically interesting?)

1. First, you deal with the parentheses.
2. THEN you factor in the exponents.
3. After that, you multiply OR divide from left to right.
4. Finally, you add OR subtract from left to right.

Following these rules for the order in which you do the operations will help you find the right answer each and every time.

PEMDAS

Everyone knows that long lists of instructions can be hard to remember so math teachers have adopted an acronym to help students remember the order of operations.  PEMDAS or, if you prefer a sentence, Please Excuse My Dear Aunt Sally.

In this memory device, each word or letter reminds us of an operation and the operations should be completed in that order.

Let’s take a closer look:

Stands
S	for subtraction
A	for addition
D	for division
M	for multiplication
E	for exponent
P	for parentheses

• Please or P stands for parentheses:  any operations that are inside of parentheses should be done first of all.
• Excuse or E stands for exponents:  after any parentheses are complete, apply the exponents.
• My or M stands for multiplication:  this step can be a bit confusing because it is done in conjunction with the next step.
• Dear or D stands for division:  remember, multiplication and division are done IN ORDER from left to right.
• Aunt or A stands for addition:  addition is also done in conjunction with subtraction from left to right.
• Sally or S stands for subtraction:  the last step, but remember it should be done at the same time as addition and always from left to right.

This silly sentence, Please Excuse My Dear Aunt Sally. Is simply a helpful way to remember that these are the steps we need to do in this specific order to find the correct solution:  parentheses, exponents, multiply/divide, add/subtract.

Let’s Practice

Let’s tackle this equation keeping PEMDAS in mind.

$$44^2+12\times(11-6)=?$$

PEMDAS tells us to handle the parentheses first so we can simplify the equation like this.

$$44^2+12\times(5)=?$$

We look back at PEMDAS and see that exponents are next on our list.  We can further simply the equation using that information.

$$1936+12\times(5)=?$$

Again, we look at PEMDAS.  Multiplication and division are the next in line so the next step of simplification looks like this.

$$1936+60=?$$

And finally, we’ve made it to the end of PEMDAS and it’s time to add and subtract.  We’ll finish solving the equation like so.

$$1936+60=1996$$

We’ve done it!  We followed the order of operations and found our answer of 1,996.  Nicely done.

Now that you’re getting the hang of it, let’s try one more practice problem.

Take this equation:   

$$23+\frac{(14\times2)}3+3^4$$

PEMDAS tells us to start with parentheses so we’ll simplify the equation like this to begin.

$$23+\frac{(28)}3+3^4$$

Next, PEMDAS says to tackle the exponents.  With that in mind the next simplification will look this way.

$$23+\frac{(28)}3+81$$

Now that parentheses and exponents are taken care of, we’ll move on to the next step.  Referencing PEMDAS will show us that it’s multiplication and division. 

Remember, we handle these in order from left to right.

We can go ahead and simplify like so.

$$23+7+81$$

And our final step according to PEMDAS is to add and subtract, again, in order from left to right.  We’ll finish off our equation like this.

$$\begin{array}{l}30+81\\111\end{array}$$

Our answer, after following the order of operations aka PEMDAS is 111.  Well done!

You’re ready for action!

Now that you’ve got some practice under your belt and you understand just how to follow the order of operations you’re ready to tackle any multi-step equation that comes you’re way.  Good luck and remember, Please Excuse My Dear Aunt Sally!!!!

Order of Operation Lecture

Watch the video below on PEMDAS where we teach you everything you need to know! You will find the transcript below as well👇🏼 

Let’s go ahead and get right to it. So more than likely, you’re already familiar with the notion of order of operations, but let’s go ahead and talk about this.

So the order of operations is a set of rules in mathematics that guides us on how to evaluate a given mathematical expression. And the best way to think about the order of operations is to look at this acronym here, PEMDAS.

P 👉🏻 stands for parentheses, which is the highest in order.

E 👉🏻 stands for exponents.

M 👉🏻 stands for multiplication.

D 👉🏻 stands for division.

A 👉🏻 stands for addition.

S 👉🏻 stands for subtraction.

P 👉🏻 for parentheses is the highest-ranking order followed by E and then the rest over here.

So what does that mean? So if I were to provide you with a very simple example, let’s say five plus parentheses three plus two. And let’s just go ahead and add times two over here.

What would you do first? Well, if we go ahead and look at our PEMDAS, it states that you must follow parentheses first. So we have to evaluate what’s inside the parentheses first.

So three plus two is five, and now I can rewrite what’s on the left side and right side. So I have five plus five times two, and now I can refer back to my acronym PEMDAS to see what I have to do next.

And you’ll notice that multiplication comes before addition. So I must multiply first. So five times two is 10 and now very simply five plus 10 is equal to 15.

So as you can see, I have a mathematical expression over here. And with my order of operations PEMDAS, I was able to evaluate this very nicely and efficiently.

One of the things that I want to clear up is that some students have a minor misconception with the concept of PEMDAS when it comes to multiplication division and addition and subtraction. What do I mean by that?

So I want to just very quickly review multiplication, division, addition, and subtraction. Notice that I’ve colored both multiplication and division the same color green, meaning that multiplication and division are ranked exactly the same.

So let me go ahead and give you first an example and how students make this common mistake. So let’s just say five divided by five times three.

Okay. Now, many students will say, “hey, division, right?” I have division here first and I have multiplication here. Second, I’m going to go ahead and multiply first because M comes before division and that is not true.

Multiplication and division are actually in the same rank, meaning that if you have a scenario where you just have division and multiplication, you have to go ahead and approach the problem from left to right. So left to right is here, right?

What I’ve drawn in blue. So I’m going to divide first followed by multiplication. So five divided by five is one, and one time three is three.

The same thing goes for addition and subtraction. If I have five minus five plus three, don’t go ahead and add first, we’re going to first subtract, right?

We’re going to go from left to right because addition and subtraction still have the same rank as each other. So five minus five is zero and zero. Plus three is three. So that is a common misconception from some students.

If you don’t know this acronym, or if you weren’t aware of this common misconception, please write this down in your notebook.

So let’s move on to the next slide. So PEMDAS is something you must memorize. You can use the saying “Please Excuse My Dear Aunt Sally” to remember PEMDAS. 

… continued in the video!

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Practice Questions