Are you teaching that PEMDAS is flexible? Are your students equipped to handle that flexibility?
You remember PEMDAS, right? It’s the acronym we use to help kids understand, or better yet remember, the order of operations. We’ve even added the catchy phrase Please Excuse My Dear Aunt Sally to assist with recalling the order. There's even a song! And everyone is still wondering, who is Aunt Sally? But of course, the name isn’t important. What's important is understanding the correct order in which to perform operations. A couple of years ago, I even wrote a blog explaining the chaos that would occur without having order in our lives. While I stand behind that idea, I feel the need to go deeper. Can we ignore PEMDAS?
Believe it or not, yes, sometimes we CAN ignore PEMDAS! But allow me to clarify that statement. As a high school math teacher who mainly taught Algebra 2, a subject I absolutely love by the way, I honestly didn’t always follow the rigidness of PEMDAS. Knowing that my students understood the order of operations and the basic properties of numbers, I would routinely step outside of the PEMDAS lane, and I encouraged my students to do the same. Why? Because we can!
Consider this problem: 81 + 42 – 1
Not too difficult, right? According to PEMDAS, we should add before subtracting because the addition and subtraction should be handled from left to right. Let's use Simplifying Expressions Using Order of Operations to work this problem.
So the answer is 122. But is that the only way to simplify this expression? Dare we ignore Aunt Sally? Yes, let’s do it. Here’s another way to simplify the same expression.
But how can we subtract before we add? Recall these basic properties of real numbers:
- Commutative Property of Addition
- Commutative Property of Multiplication
- Associative Property of Addition
- Associative Property of Multiplication
- Distributive Property
The Commutative Property of Addition allows you to change the order of numbers being added. By doing so, we can perform the subtraction before the addition since subtraction is equivalent to adding a negative value.
Let’s try another one: 10 + 52 + 6(7 – 3)
First, we’ll strictly follow PEMDAS.
And now, let's try another approach.
One extra step, but notice the flexibility in the order. The Distributive Property allows you to multiply first rather than subtract the values within the parentheses! And notice that the exponent wasn’t evaluated until the end!
But, be careful! While there’s flexibility, there are still rules that must be followed. Don’t confuse flexibility with improper mathematics.
Consider one last expression: 11 + 2(3 – 1)3
In this case, the exponent prevents you from distributing the 2 to each term first. But the learning opportunity here is to notice when the order can be modified and when it cannot.
So, what am I trying to say? First, the order of operations shouldn’t be taught in isolation. Instead, teach the order along with the properties of numbers. Make sure students thoroughly understand the Commutative, Associative, and Distributive Properties as they simplify expressions. Second, don’t confuse a different approach with chaos. The instruction and order still exist; we just have more wiggle room!
Need more practice with the order of operations? Check out Simplifying Expressions Using Order of Operations.