Order of Operations
Sometimes there may be more than one way to go about solving a mathematical expression. The
answer will depend on how you look at the problem. But, mathematics cannot work if the same
problem produces a different answer. To eliminate confusion, an “order of operations” was developed.
The order simply states what operations take precedence over other operations. A common technique
that is used to help students remember this order is called “PEMDAS”, which is the phrase, “Please
Excuse My Dear Aunt Sally”. It stands for parentheses, exponents, multiplication, division, addition and
subtraction. Most students are able to memorize this acronym; however it is more difficult for them to
remember how to apply this in simplifying various expressions.
Students who lack the skills to simplify problems using the correct order of operations:



Work problems from left to right, ignoring the correct order of operations
Do not recognize various grouping symbols other than parentheses. Examples of
grouping symbols are parentheses, brackets, braces, absolute value bars, fraction bars
and radical symbols.
Work with parentheses first regardless of the purpose of the parentheses. For example,
when a student sees the problem 25  5(3) , they will multiply first because they see some
parentheses; therefore they think the multiplication comes first in the order.
The Pellissippi State team envisions a maze game as a way to show fluency with order of
operations. At each turning point the student will either choose the next operation to be
performed or the correct answer to the next operation until they successfully get through the
maze.
An Order of Operations game may progress as follows.

Expressions involving addition and subtraction only
o Example: 5  16  8  23
o Example: 12  (8)  (3)  49

Expressions include addition, subtraction, multiplication and division
o Example: 8  7*9
o Example: 4*3  8*6
o Example: (14)  7 *5
o Example: 3*5  (15)  4  6

Expressions include all of the previous operations plus exponents and radicals
o Example: 12  42
o Example: 33   2*5 
o Example:
2
49  42
o Example: 16  (5)  36*33  92

Expressions include all previous operations plus grouping symbols. Problems should become
increasingly more difficult as the student progresses.
o Example: 12(2  7)  24  12
o Example: 6  5  3
2
o Example:  82  52    24  6   32
o Example: 6 
o Example:
16  4
2
22  2
9  4(2)(3)  10
NOTE: All currently designed games require students to work quickly through the problems to
demonstrate their math fluency. The examples for order of operations represent problems that
students should be able to simplify quickly. If speed is not an important factor for
demonstrating fluency in this area, more difficult problems will need to be included in the
game.
Test Questions
MATH 0801
11. Simplify. Show each step on your paper.
862
12. Simplify. Show each step on your paper.
4  32   7  12 
13. Simplify. Show each step on your paper.
4  4( 7)
3  4 2 
2