Review:
Number Sentences
Objectives To review number sentences; and to translate word
sentences
into number sentences.
s
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Teaching the Lesson
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Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Solving Challenging Area Problems
• Translate word sentences into
number sentences. Math Journal 2, p. 227
calculator
Students solve challenging problems
involving areas of rectangles.
[Patterns, Functions, and Algebra Goal 1]
• Identify relation symbols in
number sentences. [Patterns, Functions, and Algebra Goal 2]
• Determine whether equalities and
inequalities are true or false. Math Boxes 6 7
Math Journal 2, p. 225
Students practice and maintain skills
through Math Box problems.
[Patterns, Functions, and Algebra Goal 2]
• Apply the order of operations to evaluate
number sentences. [Patterns, Functions, and Algebra Goal 3]
Study Link 6 7
Math Masters, p. 199
Students practice and maintain skills
through Study Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Ordering Operations
Math Masters, p. 200
Students review and apply the order
of operations.
ENRICHMENT
Translating Algebraic Expressions
Math Masters, p. 201
Students list and then use various
word phrases that can be used to refer
to an operation.
EXTRA PRACTICE
Key Activities
Solving Custom-Made Math Boxes
Students review the terms, ideas, and word
translations of number sentences. They
determine whether a number sentence is
true or false.
Math Masters, p. 405
Students complete teacher-generated
Math Boxes.
Ongoing Assessment:
Recognizing Student Achievement
Use Mental Math and Reflexes. [Patterns, Functions, and Algebra Goal 3]
Key Vocabulary
relation symbol equation inequality operation symbol
Materials
Math Journal 2, pp. 226 and 227
Student Reference Book, p. 241
Study Link 6 6
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 284–294
566
Unit 6
Number Systems and Algebra Concepts
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP6, SMP8
Content Standards
Getting Started
6.NS.1, 6.EE.2c, 6.G.1
Mental Math
and Reflexes Math Message
Students evaluate expressions. Suggestions:
1. ≠ not equal to
Identify the following symbols.
3. > greater than
5. ≤ less than or equal to
9(15 ÷ 3) 45
84 ÷ 3 ÷ 4 7
14 + 7 ∗ 8 70
6 ∗ 15 + 9 99
45 ÷ 3 - 4 ∗ 2 7
9 ∗ 4 ÷ 12 ∗ 7 - 18 3
5. < less than
4. = equal
6. ≥ greater than or equal to
Study Link 6 6 Follow-Up
Briefly go over the answers with the class.
Ongoing Assessment:
Recognizing Student Achievement
Mental Math
and
Reflexes
Use Mental Math and Reflexes to assess students’ ability to apply the order
of operations in evaluating numerical expressions. Students are making
adequate progress if they are able to solve the suggested problems.
[Patterns, Functions, and Algebra Goal 3]
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 241)
Student Page
Algebraic Thinking Students of Everyday Mathematics have
worked with the relation symbols =, <, and > since first grade.
They are probably less familiar with the symbols ≠, ≤, and ≥.
Present several examples to clarify the meanings of these
symbols. Suggestions:
6 + 4 ≠ 7 is a true inequality because the sum of
6 and 4 is 10, not 7.
2 ∗ 5 ≠ 10 is a false inequality because the product of
2 and 5 is 10.
7 ≤ 14 is a true inequality because 7 is less than 14.
7 + 7 ≤ 14 is a true inequality because 7 + 7 is equal
to 14.
Algebra
Number Sentences
Number sentences are made up of mathematical symbols.
Mathematical Symbols
Operation Symbols
Relation Symbols
Digits
Variables
0, 1, 2, 3, 4,
n x y z
plus
is equal to
( ) parentheses
5, 6, 7, 8, 9
a b c d
minus
is not equal to
[ ] brackets
or *
⁄ or times
is less than
divided by
is greater than
is less than or equal to
is greater than or equal to
C M P ?ppp
Grouping Symbols
A number sentence must contain numbers (or variables) and a
relation symbol. It may or may not contain operation symbols
and grouping symbols.
Number sentences that contain the symbol are called
equations. Number sentences that contain any one of the
symbols , , , , or are called inequalities.
If a number sentence does not contain variables, then it is
always possible to tell whether it is true or false.
Equations
3 3 8 False
(24 3) / 9 3 True
100 92 9 False
Inequalities 27 * 4 42 False
4
5
2
1
3 2 True
27 72 True
19 19 False
16 * 4 80 3 True
8 ≥ 2 + 10 is a false inequality because 2 + 10 is not
less than 8 nor is it equal to 8.
3 3 6, not 8
27 / 9 3
92 9 90, and 90 is not equal to 100
27 * 4 108, which is not greater than 42.
2
2
4
2
1
, and is less than .
15
15
5
3
2
27 is not equal to 72.
19 is not less than itself.
2
64 is greater than or equal to 263.
True or false?
5
5
1. 32 14 18
2. 4 * 7 30
3. 0 4. 25 5 5 * 6
5. 50 12 7 * 22
6. 84 84
Check your answers on page 423.
Student Reference Book, p. 241
Lesson 6 7
567
Student Page
Date
Time
LESSON
Discuss and write on the board the following terms and ideas:
Number Sentences
6 7
䉬
A number sentence must contain a relation symbol (=, ≠, <,
>, ≤, or ≥). A number sentence that contains an equal sign (=)
is called an equation. A number sentence that contains any
of the other relation symbols is called an inequality. Point out
that an expression, such as 15 + 7, is not a number sentence
because it does not contain a relation symbol. Ask students to
write an equation and an inequality. Select some volunteers to
write examples on the board.
Translate the word sentences below into number sentences. Study the first one.
240 241
3 ⴱ 5 15
1.
Three times five is equal to fifteen.
2.
Nine increased by seven is less than twenty-nine.
3.
Thirteen is not equal to nine more than twenty.
4.
The product of eight and six is less than or equal to the sum of twenty and thirty.
5.
Thirty-seven increased by twelve is greater than fifty decreased by ten.
6.
Nineteen is less than or equal to nineteen.
9 7 29
13 20 9
8 ⴱ 6 20 30
37 12 50 10
19 19
Tell whether each number sentence is true or false.
7.
3 ⴱ 21 63
9.
42 12 / 6 5
11.
24 / 4 2 8
13.
21 (7 ⴱ 3) 5
15.
63 / 7 8
true
true
true
false
true
8.
true
true
true
true
false
(3 ⴱ 4) 7 19
10.
871
12.
9 / (8 5) 3
14.
8 ⴱ 7 72
16.
35 5 ⴱ 8 320
A number sentence may contain one or more operation
symbols (+, -, ×, ∗, ÷, or /). Some number sentences do not,
for example, 14 ≠ 20, 17 < 22, and x = 5.
If a number sentence contains only numbers (no variables), it is
always possible to tell whether the sentence is true or false.
Insert parentheses so each number sentence is true.
(5 ⴱ 8)(4 2) 42
10 (2 ⴱ 6) 24
21. 33 (24 / 3) 25
23. 3 ⴱ(4 3)(5 ⴱ 3) 3
( )
( )
22. (36 /(7 2))ⴱ 3 12
24. 48 /(8 4) 100 / 10
17.
18.
7 ⴱ 9 6 21
19.
20.
97/78
A common misconception is that a number sentence must be
true—that if it is false, the statement is not a number sentence.
Explain that number sentences, like word sentences, may be
true or false, but they are still sentences.
Math Journal 2, p. 226
Adjusting the Activity
ELL
Record an example of each symbol, idea, and term on the board. Leave
these examples posted so students can refer to them throughout the lesson.
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
Ask students to solve the Check Your Understanding problems
on page 241 of the Student Reference Book. Briefly
review answers.
▶ Solving Problems Involving
Student Page
Date
Number Sentences
Time
LESSON
Number Sentences
6 7
䉬
Number Sentence
240–241
True or false?
Answers vary.
The word HOPE is printed in shaded
block letters inside a 15 ft by 5 ft
rectangular billboard. What is the area of
the unshaded portion of the billboard?
3'
1,368 cm2
28.
Pennies tossed onto the gameboard at
the right have an equal chance of landing
anywhere on the board. If 60% of the
pennies land inside the smaller square,
what is the length of a side s of the
smaller square to the nearest inch?
3'
3'
1'
2'
Adjusting the Activity
1'
Have students highlight or underline the inequality symbols before
working on Problems 7–16. Ask students to write word sentences for number
sentences involving division (Problems 9, 11, 12, 15).
6 cm
6 cm
Square corners, 6 centimeters on a side,
are removed from a 36 cm by 42 cm
piece of cardboard. The cardboard is then
folded to form an open box. What is the
surface area of the inside of the box?
3'
5'
29 ft2
27.
42 cm
36 cm
A U D I T O R Y
9 in.
s
s 9 in.
7 in.
Try the Penny Toss!
Math Journal 2, p. 227
568
Algebraic Thinking Assign Problems 1–25 on journal pages 226
and 227. Remind students to apply the order of operations.
When most students have completed the problems, bring the
class together to review the answers. Have volunteers share
their translated number sentences.
Try This
26.
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
N
G
(Math Journal 2, pp. 226 and 227)
continued
Write three true and three false number sentences. Trade journals with your
partner and determine which sentences are true and which are false.
25.
PARTNER
ACTIVITY
Unit 6
Number Systems and Algebra Concepts
K I N E S T H E T I C
T A C T I L E
V I S U A L
Student Page
Date
2 Ongoing Learning & Practice
Time
LESSON
Math Boxes
6 7
䉬
1.
Simplify.
4
a. 7
▶ Solving Challenging Area Problems
1
b. 20
PARTNER
ACTIVITY
(Math Journal 2, p. 227)
2.
of 84
of 35
8
c. 3
of 9
1
d. 3
of 3
4
48
3
14
26
11
7
a. 12
1
4
51
68
Divide. Simplify if possible.
124
2
5
7
8
12
25
5
b.
28 3 c.
3 6 d.
4 27 1
4
1
2
2
10
13
87–89
Algebraic Thinking Assign Problems 26–28 on journal page 227 to
pairs of students. While some students may not be able to solve all
the problems, encourage everyone to try. Have students share
solution strategies.
Problem 26: Some students may calculate the area of the
word HOPE and subtract it from the area of the rectangle.
75 ft2 - 46 ft2, or 29 ft2 Others may add the areas that
are not shaded. (1 ∗ 2) + (1 ∗ 2) + (1 ∗ 5) + (1 ∗ 1) + (1 ∗ 5)
+ (1 ∗ 1) + (3 ∗ 2) + (1 ∗ 3) + (2 ∗ 1) + (2 ∗ 1), or 29 ft2
Problem 27: The surface area of the open box is the same as the
area of the original rectangular piece of cardboard (36 cm ∗ 42 cm,
or 1,512 cm2) minus the area of the four square corners that are
removed (4 ∗ (6 cm ∗ 6 cm), or 144 cm2).
1,512 cm2 - 144 cm2 = 1,368 cm2
3.
93
Give a ballpark estimate for each quotient.
4.
Sample estimates:
a.
137.8 15
b.
248.19 / 12
c.
4,507.08 89.76
d.
0.6 / 14.7
10
20
50
0.04
Complete each sentence using an
algebraic expression.
a.
If each bag of potatoes weighs at least
p pounds, then 6 bags weigh at least
b.
Jack is 6 inches taller than Michael.
If Jack is h inches tall, then Michael is
6p
pounds.
h6
inches tall.
261
5.
240
Which fraction is equivalent to 2.015?
Choose the best answer.
6.
2,015
10,000
4,030
20,000
403
200
2,015
100
You draw one card at random from a
regular deck of 52 playing cards (no
jokers). What is the chance of drawing:
1
13
a. a 4?
4
13
b. a card with a prime number?
c.
a face card
(jack, queen, or king)?
d.
an even-numbered
5
26
black card?
59 60
3
13
148–153
Math Journal 2, p. 225
Problem 28: The area of the entire gameboard is 81 in2. The area
of the smaller square is 60 percent of the area of the entire board,
or 48.6 in2. This is almost 49 in2. Therefore, the length of a side of
the smaller square is almost 7 in., because 7 ∗ 7 = 49. Some
students may notice that they can find the solution quickly by using
the square root key on their calculators.
▶ Math Boxes 6 7
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 225)
Study Link Master
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lessons 6-4a and 6-5. The skills in
Problems 5 and 6 preview Unit 7 content.
Name
Date
STUDY LINK
Number Sentences
6 7
䉬
1. a.
Time
Draw a circle around each number sentence.
241–243
Writing/Reasoning Have students write a response to the
following: Write a number story that can be modeled by
the number sentence in Problem 2b. Sample answer:
5 feet of yarn and
Vanessa uses yarn to make bracelets. She has 2 _
8
wants to make 3 bracelets. How much yarn can she use for each
bracelet?
b.
2.
(5 4) º 20 20
(4 23) / 9
12 12
Choose one item that you did not circle. Explain why it is not a
number sentence.
9 (6 2) 0.5
24
c. 6
3.
4.
INDEPENDENT
ACTIVITY
33 / 11
true
false
b.
94 49 2 º 2
d.
70 25 45
false
true
Insert parentheses to make each number sentence true.
(28 6) 9 31
(36 / 6)/ 2 12
(
(
)
b.
20 40 9 11
d.
4 º 8 4 16
)
Write a number sentence for each word sentence. Tell whether the number
sentence is true or false.
Word sentence
Number sentence
True or false?
21 increased by 7 is less than 40.
60 14 50
90 3 º 30
21 7 40
false
true
true
The square root of 36 is greater
than half of 10.
兹36
苶
a.
If 14 is subtracted from 60,
the result is 50.
b.
90 is 3 times as much as 30.
c.
d.
(Math Masters, p. 199)
Home Connection Students practice identifying number
sentences, inserting parentheses to make true number
sentences, identifying true and false number sentences,
and translating word sentences into number sentences.
56 / 8
Tell whether each number sentence is true or false.
a.
c.
3 º 15 100
Sample answer: A number sentence must contain a
relation symbol. 56 / 8 does not include one.
a.
▶ Study Link 6 7
17 27
1
2
º 10
true
Practice
5.
1.867 0.947 0.92
6.
6 2.49 3.51
7.
256.3 4.785 251.515
Math Masters, p. 199
Lesson 6 7
569
Teaching Master
Name
Date
LESSON
Time
3 Differentiation Options
Ordering Operations
67
䉬
The order of operations is shown in the diagram below.
in order, left to right
(
an
)
or º or /
Use the diagram to help you label which operation you should perform first, second, third,
and so on when evaluating an expression.
Example: Label the order in which you should perform the operations to evaluate the
expression 9 / (8 5). Then evaluate the expression.
2
1
4
3
5
9 / (8 ⴚ 5) ⴙ 12 º 4 ⴚ 11
Do the operation inside the parentheses.
(8 5) 3
23
Divide and multiply in order from left to right.
9/33
12 º 4 48
45
Add and subtract in order from left to right.
1
▶ Ordering Operations
To provide experience applying the order of operations,
have students label the order in which operations should
be performed to evaluate numeric expressions involving
positive numbers.
3 48 51
51 11 40
For each expression, label the operation you would perform first, second, third,
and so on. Then evaluate the expression.
2 1
3.
6 4 º 42 5.
14 28 / 7 º 2 3
2 1
3
1
2
56
7 º 23 6 0.3 º 10 4.
(9 1) / 2 º 32 6.
1º75/1
3 4 2
1
70
2
1
2.
6
1
3
2
5–15 Min
(Math Masters, p. 200)
9 / (8 5) 12 º 4 11 40
1.
PARTNER
ACTIVITY
READINESS
9
12
SMALL-GROUP
ACTIVITY
ENRICHMENT
45
▶ Translating
5–15 Min
Algebraic Expressions
(Math Masters, p. 201)
Math Masters, p. 200
Name-Collection Box for d - 12
To further explore words used to describe operations,
make a table on the board similar to the one shown. Have
students list phrases that can be used to suggest addition,
subtraction, multiplication, or division.
d – 12
The difference of
d and 12
12 subtracted
from d
Operation
A number
decreased by 12
Addition
12 less than
a number
Teaching Master
LESSON
Name-Collection Boxes
6 7
䉬
Name
Name
Date
Date
Multiplication
Word Phrase
Operation
Word Phrase
the sum of
the difference of
plus
minus
Subtraction
added to
subtracted from
increased by
decreased by
a total of
less than
the product of
the quotient of
times
shared among
Division
multiplied by
divided by
doubled
halved
tripled
split evenly
Then have students use name-collection boxes (Math Masters,
p. 201) to generate word phrases for various algebraic expressions
that you assign. (See margin.) Suggestions:
w
_
4+n
d - 12
5s
5
Name
Name
Date
Date
EXTRA PRACTICE
p
▶ Solving Custom-Made
INDEPENDENT
ACTIVITY
5–15 Min
py g
g
Math Boxes
(Math Masters, p. 405)
Math Masters, p. 201
570
Unit 6
Number Systems and Algebra Concepts
Use Math Masters, page 405 to generate Math Box questions that
focus on a particular concept or skill for which students need
extra practice.