Order of Operations
Objectives To review the order of operations; and to evaluate
expressions
containing parentheses.
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Teaching the Lesson
Key Concepts and Skills
• Evaluate numerical expressions involving
parentheses and exponents. [Number and Numeration Goal 4]
• Add, subtract, multiply, and divide
signed numbers. [Operations and Computation Goals 1 and 2]
• Apply the order of operations to
evaluate expressions. [Patterns, Functions, and Algebra Goal 3]
Key Activities
Students evaluate expressions according
to the order of operations.
Ongoing Assessment:
Informing Instruction See page 563.
Ongoing Assessment:
Recognizing Student Achievement
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Differentiation Options
Ongoing Learning & Practice
1 2
4 3
Playing Name That Number
Student Reference Book, p. 329
Math Masters, p. 462
per partnership: complete deck of
number cards (the Everything Math
Deck, if available)
Students practice applying order
of operations.
Math Boxes 6 6
Math Journal 2, p. 224
Students practice and maintain skills
through Math Box problems.
ENRICHMENT
Exploring a Grouping Symbol
Math Masters, p. 197
Students explore the use of the fraction
bar as a grouping symbol.
EXTRA PRACTICE
Evaluating Expressions
Student Reference Book, p. 247 (optional)
Math Masters, p. 198
Students practice using the order of
operations to evaluate expressions.
Study Link 6 6
Math Masters, p. 196
Students practice and maintain skills
through Study Link activities.
Use journal page 222. [Patterns, Functions, and Algebra Goal 3]
Key Vocabulary
order of operations nested parentheses
Materials
Math Journal 2, pp. 222 and 223
Student Reference Book, p. 247
Study Link 6 5
calculator slate
Advance Preparation
Review pages 269–270 of the Student Reference Book. If students do not use TI-15 or Casio fx-55
calculators, check that their calculators are programmed for order of operations.
Teacher’s Reference Manual, Grades 4–6 p. 102
560
Unit 6
Number Systems and Algebra Concepts
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP2, SMP3, SMP4, SMP5, SMP6
Content Standards
Getting Started
6.NS.5, 6.EE.1, 6.EE.2c
Mental Math and Reflexes
Math Message
Students write a number to model each situation.
Evaluate the following expressions.
In golf, par is the number of strokes a player should need
to complete a hole or round. In Jeff’s first round of golf,
he shot 3 over par. 3
In Jeff’s second round of golf, he shot 5 under par. -5
1
Karina’s math test score was 3 _
2 points above the class
1
mean. 3 _
1. 12 - 8 / 4 10
3. 6 ∗ 5 + 7 37
Anastasia’s math test score was 5.75 points below the class
mean. -5.75
Ask students to share the patterns they
found in the tables. Make sure that the
following patterns are mentioned:
2. 24 / 3 + 6 14
4. 17 - 9 ∗ 3 -10
6. 23 / 8 + 3 4
5. 15 + 62 / 3 27
Study Link 6 5 Follow-Up
2
On Monday, the stock market dropped 63.2 points. -63.2
2
2
_
On Tuesday, the stock market rose 58 _
7 points. 58 7
x+y=y+x
x∗y=y∗x
y - x is the opposite of x - y.
y ÷ x is the reciprocal of x ÷ y.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Algebraic Thinking Record students’ answers on the board. If
students disagree, record all proposed answers. Then have students
use a scientific calculator to resolve discrepancies. If they do not
have scientific calculators, skip ahead to the next section.
NOTE Scientific calculators have built-in
order of operations. Many four-function
calculators do not. Please check students’
calculators to be sure.
Order of operations was formally introduced in Fifth Grade
Everyday Mathematics. Prior to this lesson, Everyday Mathematics
has stressed the use of parentheses to evaluate expressions. If
students did not use order of operations in this activity, their
answers for expressions 1, 4, and 5 will differ from the answers
they obtained with their calculators.
Ask students to insert parentheses in each expression to show
which operation the calculator did first. Record these expressions
on the board.
Expression
Number
Expression
Left to
Right
1.
12 - 8 / 4
1
10
12 - (8 / 4)
2.
24 / 3 + 6
14
14
(24 / 3) + 6
3.
6∗5+7
37
37
(6 ∗ 5) + 7
4.
17 - 9 ∗ 3
24
-10
17 - (9 ∗ 3)
2
Calculator
Parentheses
2
5.
15 + 6 / 3
17
27
15 + (6 / 3)
6.
23 / 8 + 3
4
4
(23 / 8) + 3
For expressions 1, 4, and 5, the scientific calculator did not
evaluate the expressions in the order in which the numbers and
operations were entered. In each of these expressions, the calculator
carried out the addition or subtraction last. A scientific calculator
will follow order of operations, regardless of the order in which an
expression is entered.
Lesson 6 6
561
Student Page
Algebra
Order of Operations
▶ Reviewing Order of Operations
In many everyday situations, the order in which things are done
is important. When you bake a cake, for example, you crack the
eggs before adding them to the batter. In mathematics, too,
many operations should be done in a certain order.
Nancy fed Tom the big gray cat.
3. Multiply and divide in order from left to right.
4. Add and subtract in order from left to right.
Some people remember the order of operations by memorizing
this sentence:
P
E
E
M
D
A
M
D
S
A
S
Evaluate. 5 * 4 ⫺ 6 * 3 ⫹ 2 ⫽ ?
5*4⫺6*3⫹2
20 ⫺ 18 ⫹ 2
2⫹2
4
Multiply first.
Subtract next.
Then add.
5*4⫺6*3⫹2⫽4
Evaluate 52 ⫹ (3 * 4 ⫺ 2) / 5.
52 ⫹ (3 * 4 ⫺ 2) / 5
52 ⫹ 10 / 5
25 ⫹ 10 / 5
25 ⫹ 2
27
Clear parentheses first.
Calculate exponents next.
Divide.
Then add.
52 ⫹ (3 * 4 ⫺ 2) / 5 ⫽ 27
Evaluate each expression.
1. 28 ⫺ 15 / 3 ⫹ 8
3. 10 * 6 / 2 ⫺ 30
(Student Reference Book, p. 247)
Algebraic Thinking Write the following sentence on the board:
Rules for the Order of Operations
1. Do the operations inside parentheses.
Follow rules 2–4 when you are working inside parentheses.
2. Calculate all expressions with exponents.
P
WHOLE-CLASS
DISCUSSION
2. 1 ⫹ (5 * 10) / 4
4. 10 * (12 / 6 ⫹ 4) / 12 ⫹ 1
Check your answers on page 423.
Student Reference Book, p. 247
Ask students to punctuate the sentence so it is clear that Nancy
is not giving Tom a cat to eat. Nancy fed Tom, the big gray cat.
Explain that the use of parentheses in an expression and the
order of operations make up a mathematical “punctuation
system.” As a class, read and work through the examples on
Student Reference Book, page 247.
Discuss why, in the absence of parentheses, it is necessary to
adhere to the order of operations. Sample answer: Everyone can
agree on the same value for an expression.
Use the mnemonic device Please Excuse My Dear Aunt Sally to
identify the order in which operations should be performed. Point
out that in expressions with only addition and subtraction or only
multiplication and division, neither operation has priority over the
other. They are done in order from left to right. For example, the
answer to 4 - 3 + 1 is 2, not 0, and the answer to 24 / 3 ∗ 4 is 32,
not 2.
Adjusting the Activity
ELL
Have students create original mnemonic devices to remember the
order of operations, for example, Please Exile My Devious Annoying Sibling.
Post students’ devices in the classroom.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Have students evaluate the Check Your Understanding
expressions and explain how they got their results. Suggest that
they follow the format of the examples in the Student Reference
Book and write each step on a separate line. This makes it easy to
keep track of the order in which operations are done. Have
students use a calculator to check their results. If the calculator
is scientific, students can enter the expression from left to right.
Pose additional expressions if students need more practice.
562
Unit 6
Number Systems and Algebra Concepts
Student Page
▶ Evaluating Expressions
Date
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
(Math Journal 2, pp. 222 and 223)
INDEPENDENT
ACTIVITY
ELL
Order of Operations
6 6
䉬
Evaluate each expression. Show your work. Then compare your results to those
of your partner.
4ⴱ6⫹3⫽
夹
夹
1.
Algebraic Thinking Assign journal pages 222 and 223. Have
students evaluate the expressions independently and then
compare results with partners. Write disagreements on the board
and ask students to justify their thinking. Have students use a
scientific calculator to resolve any disputes.
Time
LESSON
3.
27
4 ⴱ 7 ⫺ (3 ⫹ 5) ⫽
夹
5.
Ongoing Assessment: Informing Instruction
Watch for students who may be confused by the omission of the multiplication
symbol in algebraic expressions. Remind students that expressions such as
2 ∗ N and 2N are equivalent. Similarly, 5(4 + 3) and 5 ∗ (4 + 3) have the
same meaning.
Some expressions use two sets of parentheses, one inside the
other. The operation in the inner parentheses is done before the
operation in the outer parentheses. Double parentheses are often
referred to as nested parentheses. To support English language
learners, write nested parentheses on the board and include
some examples.
8 ⴱ 4 ⫹ 49 ⫼ 7 ⫽
39
7 ⫺ 5 ⫹ 13 ⫺ 23 ⫺ 17 ⫽
9.
7 / 7 ⴱ 4 ⫹ 32 ⫽
4.
24 / 6 ⴱ 4 ⫽
16
9 ⴱ 6 ⫼ 3 ⫹ 28 ⫽
46
100 ⫺ 50 ⫼ 2 ⫹ 4 ⴱ 5 ⫽
夹
8.
13
10.
247
29
夹
6.
⫺25
夹
33 ⫺ 16 / 4 ⫽
夹
夹
20
7.
2.
12 ⴱ 22 ⫺ 33 ⫽
95
21
Math Journal 2, p. 222
When most students have finished, bring the class together and
discuss expressions that students found difficult.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 222
Problems 1–8
Use journal page 222, Problems 1–8 to assess students’ ability to apply the
order of operations to evaluate numerical expressions. Students are making
adequate progress if they are able to evaluate expressions in Problems 1–8.
Some students may be able to evaluate expressions containing exponents
(Problems 9 and 10).
[Patterns, Functions, and Algebra Goal 3]
Student Page
Date
Time
LESSON
Order of Operations
6 6
䉬
⫺1
⫹ 16 ⫺ 0.5 ⴱ 12 ⫽
10.1
11.
10
13.
⫺(⫺8) ⫺ (⫺4) ⴱ 6 ⫺ (⫺12) / 4 ⫽
14.
⫺4 ⫹ (⫺18) / 6 ⫹ (⫺3 ⴱ ⫺3 ⫺ 5) ⫽
Try This
((ᎏ2ᎏ ⫼ ᎏ4ᎏ) ⫹ 3) ⴱ 6 ⫺ 33 ⫽
3
b.
⫺x ⴱ (6 ⫹ x) ⫺ 33 / 9 ⫽
5
1
1
⫺3
2
⫺5(⫺6 ⫺ (⫺3)) / 7.5 ⫽
16.
3
1
1
1
1
⫺(ᎏ4ᎏ ⫼ ᎏ2ᎏ) ⫹ ᎏ2ᎏ ⫺ (ᎏ2ᎏ ⴱ (⫺ᎏ2ᎏ)) ⫽
17.
Evaluate the following expressions for x ⫽ ⫺2.
x ⴱ ⫺x ⫹ 14 / 2 ⫽
12.
35
15.
a.
continued
⫺ᎏ34ᎏ
3
Math Journal 2, p. 223
Lesson 6 6
563
Student Page
Date
Time
LESSON
6 6
Solve. Simplify your answers.
1.
3
ᎏᎏ
4
a.
1
2
⫽8⫼
23
1ᎏ50ᎏ
c.
2.
2
10ᎏᎏ
3
Multiply or divide.
a.
5
2ᎏ8ᎏ
5
7
4ᎏᎏ ⫼ 1ᎏᎏ ⫽
b.
b.
3
10
⫽ 7ᎏᎏ ⫼ 5
c.
91–93
4.
M
L
K
O
––
Which side corresponds with JL?
—––
MO
▶ Playing Name That Number
PARTNER
ACTIVITY
(Student Reference Book, p. 329; Math Masters, p. 462)
Label the axes of this
mystery graph and describe
a situation it might represent.
Sample answers:
Time (seconds)
x-axis
Speed (mph)
y-axis
Situation A car begins to
move after the traffic
light turns green.
N
J
⫺10(⫺14.35) ⫽ 143.5
4 ⴱ 3 ⴱ (⫺5) ⫽ ⫺60
⫺70 ⫽ ᎏ2⫺8ᎏ40
97
Triangles JKL and MNO are congruent.
3.
2 Ongoing Learning & Practice
Math Boxes
䉬
178
Students use the order of operations to match a target number. For
each round, students write their answers as a single number
sentence on the game record sheet. Ask students to organize their
number sentences to avoid using parentheses whenever possible.
140
Two dice are tossed. Some possible outcomes appear in the table below.
5.
Complete the table.
(1,1)
(1,2)
(2,1) (2,2)
(3,1) (3,2)
(4,1)
(4,2)
(5,1) (5,2)
(6,1) (6,2)
(1,3)
(1,4)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(1,5)
36
a.
How many equally likely outcomes are there?
b.
What is the probability of tossing a multiple of 2 on both dice?
c.
6
ᎏᎏ,
36
What is the probability of tossing a composite number
on the first die and a prime number on the second die?
▶ Math Boxes 6 6
(1,6)
(2,4) (2,5)
(3,4)
(3,5)
(4,4) (4,5)
(5,4) (5,5)
(6,4) (6,5)
9
ᎏᎏ ,
36
or
INDEPENDENT
ACTIVITY
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
(Math Journal 2, p. 224)
or
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-8. The skill in Problem 5 previews
Unit 7 content.
1
ᎏᎏ
4
1
ᎏᎏ
6
148–153
Math Journal 2, p. 224
Writing/Reasoning Have students write a response to the
following: Explain how you know that ∠ L and ∠O in
Problem 3 are congruent. Sample answer: The two
triangles are congruent and ∠ L and ∠O are corresponding,
so ∠ L and ∠O are congruent.
▶ Study Link 6 6
INDEPENDENT
ACTIVITY
(Math Masters, p. 196)
Home Connection Students use order of operations to
evaluate expressions.
Game Master
Study Link Master
Name
Date
STUDY LINK
Name
Time
Name That Number Record Sheet
Using Order of Operations
6 6
Date
Time
329
1 2
4 3
Round 1
Please Excuse My Dear Aunt Sally
247
Target Number:
Parentheses Exponents Multiplication Division Addition Subtraction
My Cards:
My Solution (number sentence):
Evaluate each expression.
1.
5
6º3
2
21
2
21
32
2.
4º9/2
( 4
6)
20
Number of cards used:
Round 2
1
3.
2
5
8
º
1
2
4.
(2.3
7.8) º 4
3
43.4
Target Number:
My Cards:
My Solution (number sentence):
5.
42
7(3
72
( 5))
Evaluate the following expressions for m
7.
9.
m
m
6
m2
1
4
( (m3))
6.
28
3) º 6 / 2
33
Number of cards used:
3.
8.
8
((2 º 4)
10.
((4
1
2
11) º
ºm
5
4
3) / 9 º ( m)
3
5
1
10
-15
7
10
Name
Date
Name That Number Record Sheet
Round 1
Practice
Units of Capacity
Find each missing number.
11.
3 gal 7 qt
12.
5 gal 3 qt
13.
13 pt
14.
10 c
15.
18 qt
4 gal
23
6
2
4
3
2 cups (c)
1 pint (pt)
Target Number:
My Cards:
My Solution (number sentence):
qt
2 pints
qt
4 quarts
1 quart (qt)
Number of cards used:
1 gallon (gal)
qt 1 pt
qt 1 pt
gal 4 pt
Round 2
Target Number:
My Cards:
My Solution (number sentence):
Number of cards used:
Math Masters, p. 196
564
Unit 6
Number Systems and Algebra Concepts
Math Masters, p. 462
Time
329
1 2
4 3
Teaching Master
Name
3 Differentiation Options
LESSON
6 6
䉬
Date
Time
Another Grouping Symbol
A fraction bar (sometimes referred to as the vinculum) indicates division.
(8 ⫹ 4)
(5 ⫺ 2)
ENRICHMENT
▶ Exploring a Grouping Symbol
INDEPENDENT
ACTIVITY
5–15 Min
Copyright © Wright Group/McGraw-Hill
An expression such as (8 ⫹ 4) ⫼ (5 ⫺ 2) can be written as ᎏᎏ.
(Math Masters, p. 197)
A fraction bar also acts as a grouping symbol. Both the numerator and the denominator
can be treated as if there were parentheses around them. Any operations in the numerator
or the denominator must be performed before the division.
冢 249 º⫹37 冣
Example: Evaluate 72 ⫼ ᎏᎏ .
72 ⫼
冢
(29 ⫹ 7)
36
ᎏ ⫽ 72 ⫼ ᎏᎏ ⫽ 72 ⫼ 3 ⫽ 24
冣 ⫽ 72 ⫼ ᎏ
(4 º 3)
12
29 ⫹ 7
ᎏᎏ
4º3
Evaluate each expression.
1.
4⫹5
3
20 ⫺ ᎏᎏ ⫽
17
7(8 ⫺ 1) ⫹ (42 ⫼ 3)
3. ᎏᎏᎏ
(10 ⫺ 7)3
To further explore order of operations, students use the fraction
bar as a grouping symbol. They evaluate expressions that involve
fractions and nested parentheses.
⫽
4(5 ⫹ 7)
2. ᎏᎏ
6(6 ⫺ 4)
7
Name
LESSON
66
䉬
⫽
6[24 ⫺ 2(7 ⫺ 3)]
4. ᎏᎏ
27 ⫼ 9
4
⫽
32
Date
Time
Another Grouping Symbol
PARTNER
ACTIVITY
EXTRA PRACTICE
▶ Evaluating Expressions
15–30 Min
(Student Reference Book, p. 247; Math Masters, p. 198)
Copyright © Wright Group/McGraw-Hill
A fraction bar (sometimes referred to as the vinculum) indicates division.
(8 ⫹ 4)
(5 ⫺ 2)
An expression such as (8 ⫹ 4) ⫼ (5 ⫺ 2) can be written as ᎏᎏ.
A fraction bar also acts as a grouping symbol. Both the numerator and the denominator
can be treated as if there were parentheses around them. Any operations in the numerator
or the denominator must be performed before the division.
冢 249 º⫹37 冣
Example: Evaluate 72 ⫼ ᎏᎏ .
冢 249 º⫹37 冣
36
12
Evaluate each expression.
1.
Students practice evaluating numerical expressions involving
the order of operations. This page provides practice with order
of operations without the distraction of difficult computations.
(29 ⫹ 7)
(4 º 3)
72 ⫼ ᎏᎏ ⫽ 72 ⫼ ᎏᎏ ⫽ 72 ⫼ ᎏᎏ ⫽ 72 ⫼ 3 ⫽ 24
4(5 ⫹ 7)
2. ᎏᎏ
6(6 ⫺ 4)
4⫹5
3
20 ⫺ ᎏᎏ ⫽
7(8 ⫺ 1) ⫹ (42 ⫼ 3)
3. ᎏᎏᎏ
(10 ⫺ 7)3
⫽
⫽
6[24 ⫺ 2(7 ⫺ 3)]
4. ᎏᎏ
27 ⫼ 9
⫽
197
Math Masters, p. 197
Teaching Master
Name
LESSON
6 6
䉬
Date
Time
Order of Operations
Please Excuse My Dear Aunt Sally
Parentheses Exponents Multiplication Division Addition Subtraction
Evaluate each expression. Compare your result to a partner’s. If you don’t agree,
discuss how you evaluated the expression to decide which result is correct.
50
1.
26 ⫹ 15 º 2 ⫺ 6 ⫽
3.
50 ⫹ 70 / 2 ⫽
5.
18 / 3 ⫹ (37 ⫹ 13) ⫽
7.
42 ⫹ 6 / 6 ⫺ 8 ⫽
85
56
35
113
2.
18 ⫺ 5 ⫹ 102 ⫽
4.
39 ⫹ 1 ⫺ 24 / 6 ⫽
6.
10 ⫹ 28 ⫼ 14 ⫺ 5 ⫽
8.
5 ⫹ 32 º 4 / 2 ⫽
36
7
23
Math Masters, p. 198
Lesson 6 6
565