A General Introduction to the Order of Operations Copyright Scott Storla 2015.

Copyright Scott Storla 2015

A General Introduction to theOrder of Operations


Operations and Operators

Operation Operator(s)

Addition +

Subtraction

Multiplication

Division

Power 2

Root

Absolute value

Logarithm log ln

Exponential 10 e


Expressions

12 4 3 12 4 3

12 12

3 5 2

386

2

6 1 5y y

4log 2log 4 2t t

2110 24 4 40 19c c c

0.5 0.52 100 5t te e

2 21 4

22 4

x x

xx x


arentheses

xponents

ultiply

ivide

dd

ubtract

P

E

M

D

A

S

12 4 3 12 4 3

12 12

1 2 3

4log 2log 4 2t t

2(sin 1) sinx x

–8 1 6 1 2k k k

18 16 2 3


eamath.comStudent Resources

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.


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.

2 3 4 0.5 0.52 3 4t te e

2ln( 1) 3 ln( ) 1x x

2 3 4

7 7 7

2 3 3 3 4x x

22 3 (2 )xy y xy



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 3 3 3 4x x


A General Introduction to theOrder of Operations


The Order of Operations

The Basic Four


Operations and Operators

Operation Operator(s)

Addition +

Subtraction

Multiplication

Division

Power 2

Root

Absolute value

Logarithm log ln

Exponential 10 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…}


eamath.comStudent Resources










4 4 2 4 6

1 2 4 6

1 8 6

9 6

3

Count the number of operators, discuss the order of the operations and then simplify.Procedure – Order of Operations








4 4 2 4 6

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.


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.


410 3 1 1

2

10 2 3 1 1

10 6 1 1

4 1 1

3 1

4

Count the number of operators, discuss the order of the operations and then simplify.

410 3 1 1

2

10 2 3 1 1

10 6 1 1

4 1 1

3 1

4










80 10 4 2 2 2

8 4 2 2 2

32 2 2 2

32 2 4

30 4

34

Count the number of operators, discuss the order of the operations and then simplify.Procedure – Order of Operations










The Basic Four



Explicit Grouping


Nested 3 16 2 5 1










3 16 2 5 1

3 16 2 6

3 16 12

3 4

12


3 16 2 5 1

3 16 2 6

3 16 12

3 4

12










36 3 2 2

36 3 4

12 4

48











1 3 3 1 3

1 3 3 4

4 12

48


1 3 3 1 3

1 3 3 4

4 12

48










12 2 8 2 8 6

12 2 8 2 2

12 2 8 4

12 2 4

12 8

4


12 2 8 2 8 6

12 2 8 2 2

12 2 8 4

12 2 4

12 8

4










16 16 16 16 6 6

16 16 16 (10) 6

16 16 [6] 6

16 16

0

16 10 6












Explicit Grouping



Implicit Grouping











26 12 2

26 12 2

14 2

26 24

28

2

14


26 12 2

26 12 2

14 2

26 24

28

2

14










2 2 3

8 4 2

2 6

2 2

8

4

2


2 2 3

8 4 2

2 6

2 2

8

4

2










17 2 5 2

17 2 5 2

17 10 2

17 2 7

7 2

17 14

9

3

3

17 2 5 2

17 2 5 2

17 10 2

17 2 7

7 2

17 14

9

3

3











12 4 3 12 4 3

12 12

12 12 12 12

12 12

24 0

12 12

2 0

2

12 4 3 12 4 3

12 12

12 12 12 12

12 12

24 0

12 12

2 0

2


12 4 3 12 4 3

12 12

12 12 12 12

12 12

24 0

12 12

2 0

2










(6 2)(6 2)

6 2(15 2)

(8)(4)

6 2(13)

32

6 26

32

32

1

(6 2)(6 2)

6 2(15 2)

(8)(4)

6 2(13)

32

6 26

32

32

1












Implicit Grouping


Practicing Some VocabularyFor Expressions


Vocabulary

Term

Sum

Factor

Product

Difference

Quotient


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 = 62 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.



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.

12 4 3 12 4 3

12 12

2 4

2

b b ac

a

Using the words term, factor, sum, product, difference or quotient describe each expression.


Before carrying out any operations on 7 45 2

the 2 is a ______,

the 5 2 is a _______, the 7 4 is a _________and 7 45 2

is a

________.

Before carrying out any operations on (3 4)(11 8)

the 4 is a ______,11 8 is a _________, (11 8) is

a________ and (3 4)(11 8) is a _________.

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

______, the 3 is a _____, and 3 4 is a _________.

Before carrying out any operations on 2(4) 3(4) the 4 is a

________, the 2(4) is both a ________, and a _____, and

2(4) 3(4) is a _________.

sumterm term

factor product term

sum

term difference

factor product

term

sum difference

quotient


Given 2(15 1) 3(6 4) the 4 is a ______,

6 4 is a _____, (15 1) is a________

15 1 is a _________ 2(15 1) is both a

_______ and a _______

and 2(15 1) 3(6 4) is a _________.


sum

term

factor

product

difference

term

sum


Given 6 2(3) 10 2(3)4 4

the 6 is a ______,

10 2(3) is a ___________, 6 2(3) is a________

10 2(3)

4

is a_________, 10 2(3)

4

is a _______

and 6 2(3) 10 2(3)4 4

is a _________.


sum

term

factor

product

difference

quotient


Practicing Some VocabularyFor Expressions


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.



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A General Introduction to the Order of Operations Copyright Scott Storla 2015.

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