Open Sentences
Objectives To introduce vocabulary and notation for open
sentences;
and to provide practice solving open sentences.
s
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Teaching the Lesson
Family
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Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Using a Map Scale
• Add, subtract, multiply, and divide to solve
open sentences. Math Journal 1, p. 75
ruler
Students use a map scale to convert
measurements to actual distances.
[Operations and Computation Goals 1–4]
• Use a “guess-and-check” strategy to make
reasonable estimates for open sentences. [Operations and Computation Goal 6]
• Identify the solution of an open sentence. [Patterns, Functions, and Algebra Goal 2]
• Determine whether number sentences are
true or false. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Math Boxes 3 11
Math Journal 1, p. 76
Students practice and maintain skills
through Math Box problems.
Study Link 3 11
Math Masters, p. 99
Students practice and maintain skills
through Study Link activities.
Students learn about open sentences and
their solutions. They participate in the Broken
Calculator activity to reinforce the concept of
open sentences and to practice estimation.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Using Fact Triangles to Solve
Open Sentences
Math Masters, p. 100
º, / Fact Triangles
Students explore the concept of open
number sentences.
ENRICHMENT
Solving Open Sentences
Math Masters, p. 101
Students find missing values for letters.
EXTRA PRACTICE
Solving Broken-Calculator Problems
Math Masters, p. 424
Students practice solving open sentences.
Ongoing Assessment:
Informing Instruction See page 215.
Ongoing Assessment:
Recognizing Student Achievement
Use a Math Log or Exit Slip. [Patterns, Functions, and Algebra Goal 2]
Key Vocabulary
variable open sentence solve solution
Materials
Math Journal 1, pp. 73 and 74
Study Link 310
Math Masters, p. 388 or 389; p. 424
transparency of Math Masters, p. 425
(optional) slate calculator overhead
calculator (optional)
Additional Information
An open sentence is a number sentence that contains one or more variables, such as 3 + x = 5. When the variable x is
replaced by a number in 3 + x = 5, the sentence becomes either true or false: 3 + 2 = 5 is true, but 3 + 4 = 5 is false.
Teacher’s Reference Manual, Grades 4–6 pp. 284–297
214
Unit 3
Multiplication and Division; Number Sentences and Algebra
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Getting Started
Mental Math and Reflexes
Math Message
Students solve addition and subtraction problems
mentally and share their strategies. Suggestions:
Is this sentence true or false? The sum of 10
and some number is 15. Be ready to explain
your thinking.
16 + 5 = 21
8 + 14 = 22
25 - 8 = 17
28 - 9 = 19
70 + 40 = 110
180 + 50 = 230
190 - 60 = 130
210 - 30 = 180
92 + 59 = 151
76 + 25 = 101
92 - 48 = 44
184 - 126 = 58
Study Link 3 10 Follow-Up
Partners compare answers. Ask students to rewrite
Problem 14 so that it is a false number sentence
and Problem 13 so that it is a true number sentence.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
The Math Message is likely to cause some confusion. Students
should conclude that they can’t tell because some information
is missing, but some students may make good arguments for
other conclusions.
Tell students that in this lesson they will explore number
sentences with missing information and learn to solve them.
Exploring the Meaning of
WHOLE-CLASS
DISCUSSION
Open Sentences
Algebraic Thinking Now write the same sentence with
math symbols:
10 + x = 15
In this sentence, the letter x stands for the missing number. A
different letter could also be used; for example, 10 + n = 15. Any
letter or other symbol that is not a number will do. A letter or
symbol that stands for a missing number is called a variable.
Now ask students what number they would write in place of x
to change 10 + x = 15 into a true number sentence. 5, because
10 + 5 is equal to 15.
A sentence that has a variable in it, such as 10 + x = 15, is
called an open sentence. To solve an open sentence, replace
the variable with a number that makes the sentence true. The
number that makes the number sentence true is called the
solution. The solution of 10 + x = 15 is the number 5.
Ongoing Assessment:
Informing Instruction
Watch for students who have difficulty with
variables when they are positioned in
different places. For example, a student may
have little difficulty with a problem such as
15 - x = 9 but struggle with a problem such
as x - 6 = 9. Suggest that students write the
number sentence with their solution to see if
it makes sense.
Lesson 3 11
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Finding Solutions of
WHOLE-CLASS
ACTIVITY
Open Sentences
Algebraic Thinking Use a procedure like the following. (See margin.)
10 + x= 15
Write an open sentence on the board.
Students solve the open sentence.
On their slates, students write the number sentence with the
solution in place of the variable. They circle the solution.
If students disagree on the solution, they check their solutions
on their calculators.
Begin with problems like the following:
12 + x = 55
Example:
12 + 43 = 55
36 / p = 9
36 / 4 = 9
Teacher: 10 + x = 15
Student: 10 + 5 = 15
2 ∗ x = 18
17 = z - 8
2 ∗ 9 = 18
17 = 25 - 8
21 - 8 = n
k / 6 = 10
21 - 8 = 13
14 = t - 9
60 / 6 = 10
m / 25 = 4
14 = 23 - 9
100 / 25 = 4
Adjusting the Activity
Have students restate the open sentences in words. For example,
for 12 + x = 55, ask: What number added to 12 will equal 55? For 2 ∗ x = 18,
ask: 18 is 2 times as many as what number?
A U D I T O R Y
NOTE The Broken Calculator activity is
a good way to reinforce the idea that the
solution of an open sentence is a number
that makes the sentence true. It is an activity
you can do with students from time to time
to remind them of this basic idea. Broken
Calculator is also an excellent routine for
practicing estimation.
K I N E S T H E T I C
Introducing the Broken
T A C T I L E
V I S U A L
WHOLE-CLASS
ACTIVITY
Calculator Activity
(Math Masters, pp. 424 and 425)
Algebraic Thinking Ask students to pretend that the minus key
on their calculator is broken. Write the following open sentence on
the board, and ask students to solve it using their calculators but
without using the minus key:
452 + x = 735
216
Unit 3 Multiplication and Division; Number Sentences and Algebra
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Broken Key:
Have the class share solution strategies. Use an overhead
calculator, if available. Students who are very skilled in mental
computation may have subtracted 452 from 735 in their heads.
Others probably replaced the variable x with various numbers
until they found a true number sentence. This guess-and-check
strategy can be organized in a table like the one shown in
the margin.
–
To Solve: 452 + x = 735
452 + 300 = 752
too much
452 + 250 = 702
too little
452 + 280 = 732
very close
452 + 283 = 735
Got it!
The solution of 452 + x = 735 is 283, because 452 + 283 = 735
is true.
Pose a few more problems like the following on a transparency
of Math Masters, page 425. Have students record their work on
Math Masters, page 424.
Open Sentence
Broken Key
Solution
75 + x = 415
340
y + 128 = 563
435
r - 156 = 954
1,110
p / 34 = 27
918
y / 29 = 52
1,508
19 ∗ t = 1,330
70
Solving Broken Calculator
Problems
PARTNER
ACTIVITY
PROBLEM
PR
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IN
NG
G
(Math Journal 1, p. 73)
The journal page contains five Broken Calculator problems and
a blank table on which students write problems for their partners
to solve.
Student Page
Date
LESSON
3 11
䉬
Ongoing Assessment:
Recognizing Student Achievement
Math Log or
Exit Slip
Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess
students’ ability to use and explain a strategy for solving open number sentences.
Have students explain the strategy they used to solve Problem 1, 2, 3, or 4 on
journal page 73. Students are making adequate progress if their strategy involves
using estimation to close in on the solution to the open sentence. Some students
may be able to explain how they solved Problem 5, which involves estimating the
product of two 2-digit numbers.
Time
Broken Calculator
Solve each open sentence on your calculator without using the “broken” key.
Only one key is broken in each problem. Record your steps. Sample answers:
1.
Broken Key:
2.
–
3.
Broken Key:
–
To Solve: z ⴙ 643 ⴝ 1,210
To Solve: 68 ⴙ x ⴝ 413
68 350 418
too much
600 643 1,243
too much
68 345 413
Got it!
550 643 1,193
too little
Broken Key:
4.
+
560 643 1,203
closer
567 643 1,210
Got it!
Broken Key:
Æ
To Solve: d ⴚ 574 ⴝ 1,437
To Solve: w / 15 ⴝ 8
2,000 574 1,426 too little
100 15 6.667
too little
2,010 574 1,436 closer
120 15 8
Got it!
2,011 574 1,437 Got it!
[Patterns, Functions, and Algebra Goal 2]
Try This
6.
Make up one for your partner to solve.
5.
Solving Open Sentences
Broken Key:
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 74)
Have students solve open sentences and rewrite each sentence with
the solution in place of the variable.
Broken Key:
÷
To Solve: s ⴱ 48 ⴝ 2,928
To Solve:
50 ⴱ 48 2,400
too little
60 ⴱ 48 2,880
closer
65 ⴱ 48 3,120
too much
61 ⴱ 48 2,928
Got it!
Answers vary.
Answers vary.
Math Journal 1, p. 73
Lesson 3 11
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Student Page
Date
Time
LESSON
Open Sentences
3 11
2 Ongoing Learning & Practice
148
Solve each open sentence. Copy the entire sentence with the solution
in place of the variable. Circle the solution.
48 + d = 70
1.
2.
51 = n + 29
4.
32 = 76 - p
6.
b - 7 = 12
48 + 22 = 70
3.
34 - x = 7
5.
h-6=9
34 - 27 = 7
32 = 76 - 44
15 - 6 = 9
8.
(Math Journal 1, p. 75)
Social Studies Link Students measure the distances
between locations on a map of Egypt. They use the map
scale to convert the measurements to actual distances.
5 ∗ m = 35
5 ∗ 7 = 35
40 - 30 = 10
y=3∗8
9.
10.
24 = 3 ∗ 8
21 / x = 7
21 / 3 = 7
11. x = 32 / 8
12. 5 = w / 10
4 = 32 / 8
5 = 50 / 10
Math Boxes 3 11
Try This
13.
INDEPENDENT
ACTIVITY
19 - 7 = 12
u - 30 = 10
7.
Using a Map Scale
51 = 22 + 29
INDEPENDENT
ACTIVITY
Mr. O’Connor wrote two open sentences on the board.
45 + x = 71
45 + y = 71
(Math Journal 1, p. 76)
Isabel says the two open sentences must have different solutions because
the variables are different.
a.
Do you agree with Isabel?
b.
Explain your answer.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 3-10. The skill in Problem 5
previews Unit 4 content.
no
Sample answer: In both sentences the variable equals 26.
You can use any variable in a number sentence—different
variables do not necessarily mean different values.
Math Journal 1, p. 74
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Writing/Reasoning Have students write a response to the
following: In Problem 1b, you wrote the factor pairs of 16. Is 16
a prime number or a composite number? Explain how you know.
Sample answer: Composite. Composite numbers have more than
one factor pair and prime numbers have only one factor pair. The
number 16 has 3 factor pairs.
Student Page
Student Page
Date
Date
Time
Time
LESSON
LESSON
Estimating Distances
3 11
Math Boxes
3 11
145
1.
Alexandria
Name all the factors of 12.
b.
100
200 mi
EGYPT
State
Name the factor pairs of 16.
1
2
4
Nile
1 inch represents 200 miles
Luxor
Lake
Nasser
The areas of which two states differ
by 944 square miles?
Rhode Island
1 , 2 , 3 , 4 , 6 , 12
Giza Cairo
0
2.
Complete.
a.
Suez
Canal
and
and
and
16
8
4
and
Delaware
Total Area
Connecticut
5,543 square miles
Rhode Island
1,545 square miles
Delaware
2,489 square miles
New Jersey
8,721 square miles
Aswan
7
Abu Simbel
You want to take a trip to Egypt and see the following sights:
Cairo, the capital, on the Nile River, near the Pyramids at Giza
Use the bar
graph to answer
the questions.
a.
Alexandria, a busy modern city and port on the Mediterranean
How many
students slept
8 hours?
The Aswan High Dam across the Nile River, completed in 1970, and Lake Nasser,
7
which formed behind the dam
b.
The temples at Abu Simbel, built more than 3,000 years ago and moved to their
present location in the 1960s to escape the rising water of Lake Nasser
Number of Hours Students
Slept Last Night
Number of Students
3.
10
8
6
4
2
0
6
7
8
4.
Which of the angles below have a
measure of more than 90 degrees?
Circle them.
9 10
Hours Slept
What is the mode for the number
of hours slept?
9
73
93
You want to know how far it is between locations.
1.
That represents about
2.
600
400
2
miles.
The distance between Abu Simbel and Aswan is about
That represents about
100
5. a.
inch(es) on the map.
Measure the line segment to the nearest centimeter.
L
miles.
The distance between Cairo and Aswan is about
That represents about
3.
3
The distance between Alexandria and Abu Simbel is about
About
inch(es) on the map.
cm
b.
__
Draw a line segment that is half the length of L P .
c.
How long is the line segment you drew?
1
_
2
inch(es) on the map.
About
5.5
cm
128
miles.
Math Journal 1, p. 76
Math Journal 1, p. 75
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1/7/11 1:22 PM
Unit 3 Multiplication and Division; Number Sentences and Algebra
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Study Link Master
Name
Study Link 3 11
Date
STUDY LINK
INDEPENDENT
ACTIVITY
(Math Masters, p. 99)
Home Connection Students tell whether number
sentences are true or false, make true number sentences
by filling in missing numbers and inserting parentheses,
and find solutions for open sentences.
3 11
䉬
Time
Open Sentences
Write T if the number sentence is true and F if the number sentence is false.
1.
35 7 º 5
3.
25 25 50
T
F
2.
43 34
4.
49 (7 7) 0
148
T
T
Make a true number sentence by filling in the missing number.
2
5.
12 / (3 3)
4
(3 8) 6 7.
6.
(60 28) / 4 8.
30 (4 6) 8
20
Make a true number sentence by inserting parentheses.
(4 º 2) 10 18
(27 / 9)/ 3 1
9.
11.
(
)
( )
10.
16 16 8 º 2
12.
27 / 9 / 3 9
Find the solution of each open sentence below. Write a number sentence with the
solution in place of the variable. Check to see whether the number sentence is true.
Example: 6 x 14
3 Differentiation Options
13.
12 x 32
14.
s 200 3
15.
5 º y 40
7x /4
16.
PARTNER
ACTIVITY
READINESS
Using Fact Triangles to
Solution: 8
Open sentence
5–15 Min
Number sentence: 6 + 8 = 14
Solution
Number sentence
20
197
8
28
12 20 32
197 200 3
5 ⴱ 8 40
7 28 / 4
Practice
17.
366 7,565 19.
9,325 756 7,931
8,569
18.
3,238 9,784 20.
4,805 2,927 13,022
1,878
Solve Open Sentences
Math Masters, p. 99
(Math Masters, p. 100)
To explore the concept of open number sentences, have students
use Multiplication/Division Fact Triangles to write and solve
open sentences. For example:
Cole picked up a Fact Triangle and asked, “3 times what
number equals 15?”
He wrote 3 ∗ ? = 15; ? = 5
NOTE For practice solving
simple inequalities, see
www.everydaymathonline.com.
15
ⴱ, 3
ENRICHMENT
Solving Open Sentences
INDEPENDENT
ACTIVITY
3 11
䉬
15–30 Min
To apply students’ understanding of open sentences, have them
determine the unknown values of letters in animal names.
Solving Broken-Calculator
Date
LESSON
(Math Masters, p. 101)
EXTRA PRACTICE
Teaching Master
Name
Time
Solve Open Sentences
Each letter in the animal names on this page has a value.
C
E
I
L
M
W
Y
8
17
2
12
9
10
4
Some of the values of the letters are known.
A
D
K
N
O
P
13 21 3 16 5 20
Some of the values of the letters are unknown.
Use the information below to find the unknown values.
COW is worth 23.
KOALA is worth 46.
DONKEY is worth 66.
MONKEY is worth 54.
LION is worth 35.
PANDA is worth 83.
INDEPENDENT
ACTIVITY
5–15 Min
Problems
(Math Masters, p. 424)
To provide practice solving open sentences, have students
complete Broken Calculator problems. Use Math Masters,
page 424 to create problems to meet the needs of individual
students, or have students create and solve their own problems.
Name
Date
LESSON
3 11
䉬
Time
Solve Open Sentences
Each letter in the animal names on this page has a value.
C
E
I
L
M
W
Y
8
17
2
12
9
10
4
Some of the values of the letters are known.
A
D
K
N
O
P
Some of the values of the letters are unknown.
Use the information below to find the unknown values.
COW is worth 23.
KOALA is worth 46.
DONKEY is worth 66.
MONKEY is worth 54.
LION is worth 35.
PANDA is worth 83.
Math Masters, p. 101
Lesson 3 11
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