Parentheses in
Number Sentences
Objective To review the use of parentheses.
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Identify and write sentences that model
number stories. [Patterns, Functions, and Algebra Goal 2]
• Solve problems involving parentheses and
nested parentheses. [Patterns, Functions, and Algebra Goal 3]
• Insert parentheses in order to make true
number sentences. [Patterns, Functions, and Algebra Goal 3]
Key Activities
Students use parentheses in number
sentences involving more than one
operation. They translate number stories into
number sentences, inserting parentheses in
the proper places, and determine whether
number sentences containing parentheses
are true or false.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Name That Number
Student Reference Book, p. 325
per partnership: 1 complete deck of
number cards (from the Everything
Math Deck, if available)
Students apply number properties,
equivalent names, arithmetic
operations, and basic facts.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Reviewing Parentheses in Number
Sentences
Math Masters, p. 198
Students insert parentheses to make
number sentences true.
ENRICHMENT
Math Boxes 7 4
Describing Dot Patterns
with Number Models
Math Journal 2, p. 221
Students practice and maintain skills
through Math Box problems.
Math Masters, p. 199
Students partition dot grids and use the
patterns to write number models.
Study Link 7 4
Math Masters, p. 197
Students practice and maintain skills
through Study Link activities.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, pp. 77 and 78
Students practice using grouping symbols.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters,
page 414). [Patterns, Functions, and Algebra Goal 3]
Key Vocabulary
expression ambiguous nested
parentheses
Materials
Math Journal 2, pp. 219 and 220
Study Link 73
Math Masters, p. 414
slate
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 98, 99, 102, 103, 288, 289
Lesson 7 4
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Getting Started
Mental Math and Reflexes
Math
Message
Use your slate procedures for problems such as the following:
47 ∗ 104 = 470,000
4.7 ∗ 103 = 4,700
0.47 ∗ 102 = 47
0.047 ∗ 10 = 0.47
356 ∗ 103 = 356,000
42.6 ∗ 102 = 4,260
0.862 ∗ 102 = 86.2
0.009 ∗ 103 = 9
0.109 ∗ 103 = 109
7.08 ∗ 104 = 70,800
0.084 ∗ 102 = 8.4
79.04 ∗ 103 = 79,040
Complete Problems 1 and 2 at the
top of journal page 219.
Study Link 7 3
Follow-Up
Have partners compare answers
and resolve differences.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Journal 2, p. 219)
Algebraic Thinking Discuss students’ answers. Ask: What do
parentheses mean in number sentences? Operations inside
parentheses are done first.
Write the statements from Problem 1 on the board with their
correct expressions, but without parentheses. Guide students to see
that without the mathematical punctuation of parentheses,
number expressions can take on different values depending on the
order in which the operations are performed. Without parentheses,
the expression is said to be ambiguous because it has more
than one possible meaning.
Student Page
Date
Time
LESSON
74
NOTE An expression is a group of mathematical symbols (numbers, operation
Parentheses and Number Stories
signs, variables, grouping symbols) that represents a number—or can represent
a number if values are assigned to any variables it contains. A number sentence
is made up of at least two numbers or expressions separated by a relation
symbol such as =, >, or <.
Math Message
1.
Write each statement as an expression.
Add 8 and 3, and then multiply by 5.
b.
Add 8 to the product of 3 and 5.
c.
d.
2.
(8 + 3) ∗ 5
8 + (3 ∗ 5)
∗ (8 - 7)
10 times the difference of 7 subtracted from 8. 10
∗
(10
8)
7
The product of 10 and 8, minus 7.
a.
Example: The number sentence 6 ∗ 4 - 2 / 2 = n includes the expression
6 ∗ 4 - 2 / 2, the variable n, and the equal symbol =.
Insert parentheses to rewrite the following problem to make four different
true sentences.
6 ∗ 4 – 2/2 = ?
Ask volunteers to share their answers to Problem 2 and explain
the steps they used to solve the problem. As the students explain
the steps, list them on the board. For example:
Sample answers:
((6 ∗ 4)) – 2) / 2 = 11
6 ∗ (4 – 2) / 2 = 6
(6 ∗ 4) – (2 / 2) = 23
6 ∗ (4 – (2 / 2)) = 18
Draw a line to match each number story with the expression that fits it.
3.
Story 1
Tom’s Total Number of Bottles of Juice
Tom had 4 bottles of juice.
He went shopping and bought
3 six-packs of bottles of juice.
(4 + 3) ∗ 6
Steps
Story 2
Tom had 4 six-packs of bottles of juice.
He went shopping and bought 3 more
six-packs of bottles of juice.
Numerical Expression
Multiply 6 times 4.
6∗4
Subtract 2 from the result.
(6 ∗ 4) - 2
Divide that result by 2.
((6 ∗ 4) - 2) / 2
4 + (3 ∗ 6)
Math Journal 2, p. 219
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Unit 7
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Exponents and Negative Numbers
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Student Page
Point out that when two or more sets of parentheses are used in
the same expression, the operation inside the inner parentheses is
done first. In ((6 ∗ 4) - 2) / 2, 6 times 4 is the operation in the
inner parentheses. The operation in the outer parentheses is done
next, followed by all the remaining operations. Parentheses inside
parentheses are referred to as nested parentheses.
Date
Time
LESSON
Parentheses and Number Stories
74
䉬
4.
Story 1
continued
Number of Cookies Alice Ate
3 ⫹ (45 / 5)
Alice ate 3 cookies before going to a
party. At the party, Alice and 4 friends
ate equal shares of 45 cookies.
Story 2
(45 ⫹ 3) / 5
There was a full bag with 45 cookies
and an opened bag with 3 cookies.
Alice and 4 friends ate equal shares
of all these cookies.
5.
Links to the Future
Story 1
Number of Cookies Baked
Mr. Chung baked 5 batches of cookies.
Each of the first 4 batches contained
15 cookies. The final batch contained
only 5 cookies.
15 ⴱ (4 ⫹ 5)
Story 2
This lesson stresses the use of parentheses to create unambiguous expressions.
Lesson 7-5 will introduce the conventional order of operations, which often
reduces or eliminates the need for grouping symbols in many number sentences.
(4 ⴱ 15) ⫹ 5
In the morning, Mr. Chung baked
4 batches of 15 cookies each. In the
afternoon, he baked 5 more batches
of 15 cookies each.
6.
A grocery store received a shipment of 120 cases of apple juice. Each
case contained 4 six-packs of cans. After inspection, the store found that
9 cans were damaged.
夹
Write an expression that represents the number of undamaged cans.
▶ Matching Number Stories
to Appropriate Expressions
Sample answer: (120 ⴱ (4 ⴱ 6)) ⫺ 9
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
(Math Journal 2, pp. 219 and 220)
Ask students to read the two number stories in Problem 3 on
journal page 219 and match each story with an expression. Ask
volunteers to explain their choices. Story 1 goes with the second
expression since 3 must be multiplied by 6 to find the total
number of cans in 3 six-packs. Story 2 goes with the first
expression since 4 + 3 stands for the total number of six-packs.
Math Journal 2, p. 220
Assign journal page 220. When most students are done, bring the
class together and go over the answers. Have students explain
their reasons for matching a given number story with a
particular expression.
Ongoing Assessment:
Recognizing Student Achievement
Exit Slip
Student Page
Use an Exit Slip (Math Masters, page 414) to assess students’ facility with
writing expressions containing parentheses to represent a number story. Have
students explain how they used parentheses in Problem 6 on journal page 220
to write the expression for the total number of undamaged cans. Students
are making adequate progress if they refer to the use of nested parentheses
to identify the total number of undamaged cans.
[Patterns, Functions, and Algebra Goal 3]
Games
Name That Number
Materials 䊐 1 complete deck of number cards
Players
2 or 3
Skill
Naming numbers with expressions
Object of the game To collect the most cards.
Directions
1. Shuffle the deck and deal 5 cards to each player. Place the
remaining cards number-side down on the table between the
players. Turn over the top card and place it beside the deck.
This is the target number for the round.
2. Players try to match the target number by adding,
subtracting, multiplying, or dividing the numbers on as many
of their cards as possible. A card may only be used once.
3. Players write their solutions on a sheet of paper. When
players have written their best solutions:
2 Ongoing Learning & Practice
♦ Each player sets aside the cards they used to match the
target number.
♦ Each player replaces the cards they set aside by
drawing new cards from the top of the deck.
♦ The old target number is placed on the bottom of the deck.
♦ A new target number is turned over, and another round
is played.
Target number: 16
5
8
5
7
Students practice applying number properties, equivalent names,
arithmetic operations, and basic facts by playing Name That
7
2
8
Player 1’s cards:
10
2
(Student Reference Book, p. 325)
4. Play continues until there are not enough cards left to
replace all of the players’ cards. The player who has set
aside the most cards wins the game.
10
▶ Playing Name That Number
PARTNER
ACTIVITY
Some possible solutions:
10 ⫹ 8 ⫺ 2 ⫽ 16 (3 cards used)
7 * 2 ⫹ 10 ⫺ 8 ⫽ 16 (4 cards used)
8 / 2 ⫹ 10 ⫹ 7 ⫺ 5 ⫽ 16 (all 5 cards used)
The player sets aside the cards used to make a solution and draws
the same number of cards from the top of the deck.
Student Reference Book, p. 325
Lesson 7 4
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Student Page
Date
Time
LESSON
Number. Encourage students to extend the game by using the
cards in their hands to form exponents and/or fractions.
Math Boxes
74
䉬
Measure the length and width of each of the following
objects to the nearest half inch.
1.
Answers vary for b–d.
journal cover
a.
length
10 ᎏ78ᎏ in.
width
8ᎏ12ᎏ
b.
in.
length
index card
c.
length
width
in.
b.
Stems
(100s and 10s)
15
16
17
18
in.
width
Find the following landmarks for the data.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lessons 7-2 and 7-6. The skill in
Problem 4 previews Unit 8 content.
478
3555
0059
5
117–119
4.
M
167 ⬚.
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 221)
in.
165
154
Range: 31
Mode(s): 165,170
Measure ⬔M to the nearest degree.
⬔M measures about
Minimum:
Leaves
(1s)
4
2
0
2
▶ Math Boxes 7 4
in.
Median:
163, 179, 170, 165, 182, 157,
154, 165, 170, 175, 162, 185,
158, 170, 165, 154
3.
width
(your choice)
length
Make a stem-and-leaf plot of the
hand-span measures in Ms. Grip’s
fifth-grade class.
2. a.
in.
d.
in.
183
desktop
Calculate the sale price.
Regular
Price
Discount
$8.99
20%
$11.99
25%
$89.00
20%
$9.99
20%
Sale
Price
$7.19
$8.99
$71.20
$7.99
204
Writing/Reasoning Have students write a response to the
following: Use your solution for Problem 2 to explain how
to read a stem-and-leaf plot. Sample answer: In this
problem, the stems are the hundreds and tens digits for each
number, and the leaves are the ones digits. The first number on
this stem-and-leaf plot has 1 in the hundreds place, 5 in the tens
place, and 4 in the ones place. It is read one hundred fifty-four.
51
▶ Study Link 7 4
Math Journal 2, p. 221
INDEPENDENT
ACTIVITY
(Math Masters, p. 197)
Home Connection Students insert parentheses to make
number sentences true. In several cases, students will
need to insert nested parentheses.
3 Differentiation Options
READINESS
▶ Reviewing Parentheses in
Study Link Master
Name
Date
STUDY LINK
Using Parentheses
74
䉬
4.
2 =(3 ⴱ 2)⫺(4 / 1)
2.
3 =(4 ⫹ 3 ⫺ 1) / 2
Write seven names for 8. Use only numbers
less than 10, and use at least three different
operations in each name. Use parentheses.
Follow the directions in Problem 7 to fill in
the last two rows.
Reminder: When you have a pair of parentheses
inside another pair, the parentheses are called
nested parentheses.
Example: 8 ⫽ ((5 ⴱ 6) ⫹ 2) / 4
5.
7.
1 ⫽((4 ⫹ 1) ⫺ 3) / 2
6.
Sample answers for
Problems 1–6:
3. 4 ⫽(3 ⫺ 1) ⫹(4 / 2)
8
Make each sentence true by inserting parentheses.
15–30 Min
Number Sentences
222 223
Make each sentence true by inserting parentheses.
1.
Time
PARTNER
ACTIVITY
(9 ⫻ 1) ⫺ (1 / 1)
(9 ⫺ 5) ⫹ (2 ⫻ 2)
(0.5 / 1) ⫹ (2.5 ⫻ 3)
(6.3 ⫹ 9.7) / (6 ⫺ 4)
(6.2 / 2) ⫹ (7 ⫺ 2.1)
1
(ᎏ2ᎏ ⫻ 8) ⫹ (9 ⫺ 5)
(72 ⫺ 9) ⫼ 5
(42 ⫺ (3 ⴱ 3)) ⫹ 1
((2 ⫹ 1)4 ⫼ 9) ⫺ 1
(Math Masters, p. 198)
To explore the use of parentheses in number sentences, have
students insert parentheses to make true sentences. Students
compare the use of commas in text sentences to the use of
parentheses in number sentences. When students have finished
the page, have them share why they think parentheses are
important.
7 ⫽((4 ⴱ 3) / 2) ⫹ 1
Add two names to your name-collection box in Problem 4.
Use nested parentheses.
Practice
Find the number that each variable represents.
4
1
1ᎏ1ᎏ2 , or 1ᎏ3ᎏ 9. (1ᎏ1ᎏ ⫹ p) ⴱ 22 ⫽ 12
5
1
8. 2 ᎏᎏ ⫽ (1ᎏᎏ ⫹ a) a ⫽
12
12
2
2
1
5
15 d ⫽ 2ᎏ8ᎏ, or 2ᎏ4ᎏ
2
10. 6 ᎏᎏ ⫹ d ⫽ 7 ᎏᎏ
11. 6.4 ⫺ y ⫽ 6 ᎏᎏ
8
8
5
1
p ⫽ 1ᎏ2ᎏ
y⫽0
Math Masters, p. 197
560
Unit 7
Exponents and Negative Numbers
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Teaching Master
ENRICHMENT
▶ Describing Dot Patterns
INDEPENDENT
ACTIVITY
15–30 Min
with Number Models
Name
74
Read the following sentence. Mary Grace the lizard ate three crickets.
1.
This sentence could have multiple meanings.
2. The lizard, named Mary Grace, ate three crickets.
3. The speaker is telling someone named Mary that the
lizard, named Grace, ate three crickets.
To apply students’ understanding of parentheses, have
them write number sentences to describe dot patterns.
Students partition a dot grid and write number sentences
to model the indicated number patterns.
Without commas, it’s hard to tell which meaning was intended. Write the
number of the meaning next to each sentence below.
a.
b.
c.
Mary Grace, the lizard, ate three crickets.
Mary Grace, the lizard ate three crickets.
Mary, Grace the lizard, ate three crickets.
Insert parentheses in each sentence to make the sentence true.
SMALL-GROUP
ACTIVITY
5–15 Min
3 (4 ⫹ 7) ⫽ 33
6 ⫹ (9 5) ⫽ 51
27 / 4 ⫹ 5 ⫹ 6 ⫽ 9 27 / (4 ⫹ 5) ⫹ 6 ⫽ 9
a.
3 4 ⫹ 7 ⫽ 33
b.
6 ⫹ 9 5 ⫽ 51
c.
Insert parentheses in the expressions below, and find their solutions.
3.
To offer students more experience with grouping symbols, see
5-Minute Math, pages 77 and 78.
Planning Ahead
2
1
3
By adding commas, the meaning of a sentence becomes clear. In number
sentences, parentheses are used to indicate what to calculate first.
2.
▶ 5-Minute Math
Time
Reviewing Parentheses
1. The speaker is telling someone named Mary Grace
that the lizard ate three crickets.
(Math Masters, p. 199)
EXTRA PRACTICE
Date
LESSON
a.
75⫺4⫽
b.
6⫹9⫼3⫽
(7 5) ⫺ 4 ⫽ 31; 7 (5 ⫺ 4) ⫽ 7
(6 ⫹ 9) ⫼ 3 ⫽ 5; 6 ⫹ (9 ⫼ 3) ⫽ 9
Math Masters, p. 198
In Lesson 7-6, you will need newspapers and magazines that
contain line graphs.
Teaching Master
Name
LESSON
74
Date
Time
Describing Dot Patterns
The total dots in this dot array can be found by using patterns.
Here is one way to find the total:
((3 3) ⫹ (4 3) ⫹ 4)
Use shape outlines or colors to identify a pattern
on this dot array. Write a number model for your
pattern. Then write a number story that
matches your number model.
Sample answers:
Number model:
(11 11) ⫹ (4 (5 4)) ⫹ (12 4) ⫹ (4 4) ⫽ 265
Number story:
Sample answer: In a large city school, the fifth graders fill
the auditorium in colorful choir robes. Students wearing purple
robes stand in 11 rows of 11 students each. Four groups in
gold robes stand in 4 rows of 5 students each. There are 4
groups of 12 students in black robes and 4 groups of 4 students
in green robes. How many students are in the auditorium?
Math Masters, p. 199
Lesson 7 4
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