A General Introduction to the
Order of Operations
Copyright Scott Storla 2015
Operations and Operators
Operation
Addition
Operator(s)
+

Subtraction
Multiplication
 
 

Division
2
Power
Root
Absolute value
Logarithm
Exponential
log ln
10
Copyright Scott Storla 2015
e
Expressions
8
110  24c  4c 2  40  19c
 63
2
3  52

12  4  3  12  4  3 

12
12

2 e0.5t  100  5e0.5t
x 1
x2  2

x
x2  4

4
x 2
 6y  1 y  5 
4log  t   2log  t   4  2
Copyright Scott Storla 2015
1 2  3
12  4  3  12  4  3 

12
12
P arentheses
E xponents
 18  16  2  3
Multiply
D ivide
–8
A dd

 
k 1  6

k 1  2 k
4log  t   2log  t   4  2
S ubtract
2(sin x  1)  sin x
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
eamath.com
Student Resources
Copyright Scott Storla 2015
2 3 4
 
7 7 7
2  34
2 3x  3 3x  4
2e0.5t  3  4e0.5t
2ln( x  1)  3  ln( x )  1
2 xy 2  3 y (2 xy )
Think like an expert
When novices first view an expression they tend to
focus on the numbers and letters. When experts first
view an expression they notice the numbers and letters
but put the majority of their attention on the operators
and grouping symbols. Start thinking like an expert by
consciously analyzing operations and their order.
Copyright Scott Storla 2015
Think like an expert
Automaticity is the ability to do something without
thought. For example most adults are automatic at
reading. Experts are automatic at correctly
processing each step of the order of operations.
Novices often rely on a calculator. You need to
practice order of operations problems using your
brain, not a calculator.
2 3x  3 3x  4
Copyright Scott Storla 2015
A General Introduction to the
Order of Operations
Copyright Scott Storla 2015
The Order of Operations
The Basic Four
Copyright Scott Storla 2015
Operations and Operators
Operation
Addition
Operator(s)
+

Subtraction
Multiplication
 
 

Division
2
Power
Root
Absolute value
Logarithm
Exponential
log ln
10
Copyright Scott Storla 2015
e
Definition – Natural Numbers
The set of numbers {1,2…}
The natural numbers are {1,2,3…}
The whole numbers are {0,1,2,3…}
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
eamath.com
Student Resources
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
4  4  2 4  6
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
Copyright Scott Storla 2015
1 2  4  6
1 8  6
96
3
Plan/Proceed Pairs
Order of Operations
1. Pens/pencils down.
2. Together plan a strategy for simplifying the expression.
a) Count the number of operators.
b) Discuss the order for the operations.
3. Pens/pencils up. Individually finish the problem.
4. Compare your answers.
5. One pair will be asked to share their process.
Copyright Scott Storla 2015
Plan/Proceed Pairs
Order of Operations
1. Pens/pencils down.
2. Together plan a strategy for simplifying the expression.
a) Count the number of operators.
b) Discuss the order for the operations.
3. Pens/pencils up. Individually finish the problem.
4. Compare your answers.
5. One pair will be asked to share their process.
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
4
10   3  1  1
2
10  2  3  1  1
10  6  1  1
4  1 1
3 1
4
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
80  10  4  2  2  2
8 4  2  2 2
32  2  2  2
32  2  4
30  4
34
Copyright Scott Storla 2015
The Order of Operations
The Basic Four
Copyright Scott Storla 2015
The Order of Operations
Explicit Grouping
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
Nested 3 16  2  5  1 
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
3 16  2  5  1 
3 16  2  6  
3 16  12
3  4
12
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
36  3  2  2 
36  3  4 
12  4 
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
48
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
1  3  3 1  3 
1  3  3  4 
 4  12
48
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
12  2 8  2  8  6 
12  2 8  2  2 
12  2 8  4
12  2 4
12  8
4
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.


16  16  16  16  6    6
16  16  16  (10)  6
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
16  16  [6]  6
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
16  10  6
16  16
0
Copyright Scott Storla 2015
The Order of Operations
Explicit Grouping
Copyright Scott Storla 2015
The Order of Operations
Implicit Grouping
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
 26  12  2 
26  12  2 
14  2 
26  24
28
2
14
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
2  23
8  42
26
22
8
4
2
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
17  2  5   2
17  2  5  2 
17  10   2
17  2  7 
72
17  14
9
3
3
Copyright Scott Storla 2015
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
12  4  3 
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
12

12  4  3 
12  12 12  12

12
12
24 0

12 12
20
2
Copyright Scott Storla 2015
12
Count the number of operators,
discuss the order of the
operations and then simplify.
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations;
in the numerator or denominators of fractions.
inside absolute value bars.
in radicands or exponents.
1. Start to the left and work right simplifying each operation
beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each
multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each
addition or subtraction as you come to them.
(6  2)(6  2)
6  2(15  2)
(8)(4)
6  2(13)
32
6  26
32
32
1
Copyright Scott Storla 2015
The Order of Operations
Implicit Grouping
Copyright Scott Storla 2015
Practicing Some Vocabulary
For Expressions
Copyright Scott Storla 2015
Vocabulary
Term
Sum
Factor
Product
Difference
Quotient
Copyright Scott Storla 2015
We add terms to get a sum.
For example 2 + 3 = 5.
2 is a term. 3 is a term. 5 is the sum.
We multiply factors to get a product.
For example 2 x 3 = 6
2 is a factor. 3 is a factor. 6 is the product.
When we subtract we have a difference.
When we divide we have a quotient.
We often name an expression by the last
operation we would carry out.
Copyright Scott Storla 2015
Using the words term, factor, sum, product, difference
or quotient describe each expression.
12  4  3  12  4  3 

12
12
b  b2  4ac
2a
Think like an expert
When novices first view an expression they tend to
focus on the numbers and letters. When experts first
view an expression they notice the numbers and letters
but put the majority of their attention on the operators
and grouping symbols. Start thinking like an expert by
consciously analyzing operations and their order.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference or quotient.
Before carrying out any operations on 3  4 the 4 is a
term the 3 is a _____,
sum
term and 3  4 is a _________.
______,
Before carrying out any operations on 2(4)  3(4) the 4 is a
product and a _____,
factor the 2(4) is both a ________,
term and
________,
sum
2(4)  3(4) is a _________.
Before carrying out any operations on (3  4)(11 8)
term 11 8 is a _________,
difference (11 8) is
the 4 is a ______,
factor and (3  4)(118) is a _________.
product
a________
term
Before carrying out any operations on 7  4 the 2 is a ______,
52
7  4 is a
sum
difference
the 5  2 is a _______,
the 7  4 is a _________and
52
quotient
________.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference or quotient.
term
Given 2(15  1)  3(6  4) the 4 is a ______,
sum (15  1) is a________
factor
6  4 is a _____,
difference 2(15  1) is both a
15  1 is a _________
term
product and a _______
_______
sum
and 2(15  1)  3(6  4) is a _________.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference or quotient.



term
Given  6  2(3)  10  2(3)  the 6 is a ______,

4

4

difference
sum
10  2(3) is a ___________,
6  2(3) is a________


10  2(3)
quotient
 10  2(3)  is a _______
factor
is a_________,


4
4





product
and  6  2(3)  10  2(3)  is a _________.

4

4

Copyright Scott Storla 2015
Practicing Some Vocabulary
For Expressions
Copyright Scott Storla 2015
You understand the order of operations when you’re
able to correctly ;
a) Attend to the operators.
b) Consciously order the operations.
c) Simplify the expression.
d) Use the proper vocabulary when you
describe the process to yourself and others.
Copyright Scott Storla 2015
The Order of Operations
Copyright Scott Storla 2015