What is the Order of Operations?

My very educated mother just served us nachos.

Every good boy deserves fudge.

Roy G. Biv

Mnemonics. Be it the planets in our solar system, the lines of the treble clef, or the order of the colors in a rainbow, most of us have relied on mnemonics to remember the information at some point. We use them because they are so effective. A mnemonic device used in Math-U-See is “Parachute expert, my dear Aunt Sally” (or “Please excuse my dear Aunt Sally,” as it often appears in other curricula). This is often referred to by the acronym PEMDAS. What does PEMDAS mean, and what does Aunt Sally have to do with math?

The acronym PEMDAS is often used to remember the order of operations in mathematics. The order of operations is a convention that really began to take shape during the sixteenth century as algebraic notation developed and has continued to evolve ever since. It provides a uniform method of evaluating mathematical expressions without the overuse of parentheses (or brackets). The use of the acronym PEMDAS is a more recent development. Each letter of the acronym represents an operation:

P – parentheses
E – exponents
M – multiplication
D – division
A – addition
S – subtraction

Regardless of how many values you start within an expression, you can only work with two at a time. Without applying the order of operations, one person might work an expression from left to right, and another might work random groupings that he creates. This would result in different answers. Consider this example:

42 + 3 X 5 =

Working this expression left to right, I would get 42 + 3 = 45, and then multiplying 45 X 5 would result in 225.

42 + 3 X 5 =

Applying the order of operations (PEMDAS), I would complete the multiplication of 3 X 5 = 15. Adding the 15 to 42 would result in 57.

If we were having a conversation regarding these numbers, the intent would be clear. It is dealing with mathematical ideas in written notation that makes the order of operations so valuable. Therefore, having a clear understanding of this concept is vital.

However, is remembering PEMDAS enough? Unfortunately, no. Many a student has dutifully worked through an expression, applying each operation in the order set forth by PEMDAS, only to find their answer incorrect. The Math-U-See program clearly teaches the student that multiplication and division are completed simultaneously, left to right, as these are related operations. The same is true for addition and subtraction. To expand on the previous example:

42 + 3 X 5 – 20 ÷ 10 X 2 + 3 =

Working this example in the order of each letter of PEMDAS would yield the following:

42 + 15 – 20 ÷ 20 + 3 = Completing the multiplication first as there are no parentheses or exponents
42 + 15 – 1 + 3 = Completing the division
57 – 4 = Completing the addition
53 Completing the subtraction

This time, applying the understanding that multiplication and division are completed simultaneously, as are addition and subtraction, I would get the following:

42 + 3 X 5 – 20 ÷ 10 X 2 + 3 =
42 + 15 – 20 ÷ 10 X 2 + 3 = Completing first multiplication
42 + 15 – 2 X 2 + 3 = Continuing to the right to division
42 + 15 – 4 + 3 = Continuing right to the last multiplication
57 – 4 + 3 = Completing first addition
53 + 3 = Continuing right to subtraction
56 Continuing right to the last addition

(Note: Steps can be combined in practice. They are broken down here for clarity.)

Although this important concept of order of operations seems simple, it can actually be quite challenging to apply. A fun activity to practice the order of operations is Four 4’s. To play, use exactly four 4’s to make every integer from 0-10 using any mathematical operation, decimal points, and parentheses. You must use all four 4’s each time and cannot use any other numbers.

Have fun exploring the order of operations, and say “hi” to Aunt Sally while you do!

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