Introduction: Connecting Your Learning

People need a common set of rules for performing basic calculations. What does 3 + 5 ⋅ 2 equal? Is it 16 or 13? Your answer depends on the order in which you perform the operations. Mathematicians have developed a standard order of operations that tells you which calculations to make first in an expression with more than one operation. Without a standard procedure for making calculations, two people could get two different answers to the same problem.

The order of operations is defined as the set of rules that tells you the order in which addition, subtraction, multiplication, and division are performed in any calculation.

The order of operations is very important when trying to solve a programming problem or find a GPS location. If the order of operations is done incorrectly, a GPS location could potentially put someone out in the middle of nowhere when he or she was trying to find a store across town, or it could cause a checkbook balancing error in the mobile application that you developed. By paying close attention to the order of operations, a user can successfully navigate to their desired location or calculate the balance for a checking account.

Focusing Your Learning

Key Terms

Order of Operations

The order of operations is the order that must be followed to simplify math expressions. The universally agreed-upon and accepted order of operations for evaluating a mathematical expression is as follows:

Please	Parentheses	( )	grouping symbols from the inside out
Excuse	Exponents	x3
My	Multiplication	∙	as they appear from left to right
Dear	Division	÷
Aunt	Addition	+	as they appear from left to right
Sally	Subtraction	-

Grouping Symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations should be grouped together and considered as one number.

1. In a computation with more than one operation involved, grouping symbols indicate which operation to perform first. If possible, you should perform operations inside grouping symbols first.

(5 · 5) + 20 = 45   5 times 5 is 25. Adding 20 you get 45.

whereas

5 · (5 + 20) = 125   Here 5 + 20 is performed first as that sum is inside the parentheses. 5 + 20 is 25. Then multiplying 5 and 25, you get 125.

2. If there are no parentheses, you should always complete multiplications and divisions first, as they appear left to right, followed by additions and subtractions, as they appear left to right. You can always put your own parentheses into equations using this rule to make things easier for yourself, for example:

a ÷ b + c · d = (a ÷ b) + (c · d)

20 ÷ 4 + 5 · 5 = (20 ÷ 4) + (5 · 5)

Take a look at the following grouping symbols examples.

Example: 9 + (3 · 8)

Since 3 and 8 are within parentheses, they are to be combined first:

= 9 + 24

Then add the terms:

= 33

Thus, 9 + (3 · 8) = 33.

Example: (10 ÷ 0) · 6

Since (10 ÷ 0) is undefined, this operation is meaningless, and no value is attached to it. You write "undefined" as the answer.

Order of Operations in Action

Now that you know about grouping symbols and their importance, you will apply what you learned in the following examples.

Example 1: 21 + 3 ⋅ 12
Step 1:	21 + 3 ∙ 12	=	3 ∙ 12	Multiplication
Step 2:	21 + 36	=	21 + 36	Addition
		=	57

Example 2: (15 − 8) + 5(6 + 4)
Step 1:	(15 − 8) + 5(6+4)	=	(15 − 8) + 5(6+4)	Simplify inside parentheses first.
Step 2:	7 + 5 ⋅ 10	=	7 + 5 ⋅ 10	Multiply
Step 3:	7 + 50	=	7 + 50	Add
		=	57

Example 3: 63 − (4 + 6 ⋅ 3) + 76−4
Step 1:	63 – (4 + 6 ⋅ 3) + 76 – 4	=	63 – (4 + 6 ⋅ 3) + 76 – 4	Simplify inside parentheses (multiplication first then addition)
Step 2:	63 − (4+18) + 76 − 4	=	63 − (4+18) + 76 − 4
Step 3:	63 – 22 + 76 − 4	=	63 – 22 + 76 – 4	addition and subtraction left to right
Step 4:	41+76 − 4	=	41+76−4
Step 5:	117 − 4	=	117 − 4
Step 6:		=	113

Here is a different way to see the process of using the order of operations to solve a problem.

Example 1: 7 ⋅ 6 − 4² + 1⁵

Evaluate the exponential forms, moving from left to right.

= 7 ⋅ 6 − 16 + 1

Multiply 7 · 6.

= 42 − 16 + 1

Subtract 16 from 42.

= 26 + 1

Add 26 and 1.

= 27

Example 2: 4² + 6 ⋅ (3² + 2²)

Evaluate the exponential forms in the parentheses.

= 4² + 6 ⋅ (9 + 4)

Add 9 and 4 in the parentheses.

= 4² + 6 ⋅ (13)

Evaluate the exponential form 4².

= 16 + 6 ⋅ (13)

Multiply 6 and 13.

= 16 + 78

Add 16 and 78.

= 94

Practice Exercises: Order of Operations

Now that you have gained some knowledge, it is time to practice what you have learned. Check your answers after completing the activity to see what you have learned.

For additional practice with Order of Operations select the following link: Order of Operations Additional Practice Once you complete the additional practice, check to see how well you did by selecting the following link: Order of Operations Additional Practice Solutions

Watch the following Khan Academy video to see additional examples that can help you gain a better understanding of these new concepts.

Math Video Toolkit: Introduction to the Order of Operations More Complicated Order of Operations Examples

Summarizing Your Learning

In this lesson, you discovered the importance of the order of operations in mathematics. You learned how one simple problem like 3 + 5 ∙ 2 could lead to two different answers if a standard order is not followed.

Take a few minutes to research how the order of operations relates to various fields. For example, imagine if the banking industry calculated your bank account by subtracting all items that you purchased with your debit card on a particular day before they calculated your paycheck deposit. This would cause a lot of people to overdraw their accounts. The order in which many things are done truly makes a difference to the outcome.

Assessing Your Learning

Now that you have read over the lesson carefully and attempted the practice exercises, it is time for a knowledge check. Please note that this is a graded part of this lesson so be sure you have prepared yourself before starting.

1. Complete the Arithmetic Review: Order of Operations.

Resource:

“Exponents, Roots, Factorization of Whole Numbers: Grouping Symbols and the Order of Operations” by Ellis, W., & Burzynski, D. © 2010 retrieved from http://cnx.org/content/m34872/1.2/ is used under a Creative Commons Attribution http://creativecommons.org/licenses/by/3.0/. This adaption of the lesson, “Order of Operations,” by the National Information Security and Geospatial Technologies Consortium (NISGTC) is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0.

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