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Benchmark Exam 7 Mathematics
Lesson 1: Number Concepts
Lesson 2: Comparing and Ordering Rational Numbers
Lesson 3: Computation
Lesson 4: Ratio, Proportion, and Percent
Lesson 5: Estimation and Problem Solving
Unit 2
Algebra
Lesson 6: Expressions, Equations, and Inequalities
Lesson 7: Graphing Linear Equations and Inequalities
Lesson 8: Patterns and Functions
Unit 3
Geometry
Lesson 9: Geometric Figures
Lesson 10: Geometric Concepts
Unit 4
Measurement
Lesson 11: Measurement Systems
Lesson 12: Geometric Measurement
Unit 5
Data Analysis and Probability
Lesson 13: Data Analysis
Lesson 14: Probability
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Catalog # 1BDAR07MM01
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Arkansas Benchmark
EXAM
7 MATHEMATICS
Number and Operations
Arkansas Benchmark Exam
The cover image depicts a game
of dominos. These game pieces
can also be utilized as a classroom
manipulative to illustrate probability.
Unit 1
7
Mathematics
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Page iii
TABLE OF CONTENTS
Introduction .................................................................................................... 1
Test-Taking Tips................................................................................ 2
Unit 1 – Number and Operations ................................................................. 3
Lesson 1: Number Concepts ............................................................. 4
ACTAAP Coverage: NO.1.7.2, NO.1.7.3, NO.1.7.6, NO.3.7.4, NO.3.7.5
Lesson 2: Comparing and Ordering Rational Numbers ................. 17
ACTAAP Coverage: NO.1.7.4, NO.1.7.5
Lesson 3: Computation ................................................................... 32
ACTAAP Coverage: NO.2.7.1, NO.2.7.2, NO.2.7.3, NO.2.7.4, NO.3.7.1
Lesson 4: Ratio, Proportion, and Percent ....................................... 50
ACTAAP Coverage: NO.1.7.1, NO.3.7.6
Lesson 5: Estimation and Problem Solving.................................... 66
ACTAAP Coverage: NO.3.7.1, NO.3.7.2, NO.3.7.3
Unit 2 – Algebra ........................................................................................... 79
Lesson 6: Expressions, Equations, and Inequalities ....................... 80
ACTAAP Coverage: A.5.7.1, A.5.7.3, A.5.7.4
Lesson 7: Graphing Linear Equations and Inequalities.................. 97
ACTAAP Coverage: A.5.7.1, A.5.7.2, A.6.7.1, A.6.7.2, A.6.7.3
Lesson 8: Patterns and Functions.................................................. 110
ACTAAP Coverage: A.4.7.1, A.4.7.2, A.4.7.3, A.7.7.1
Unit 3 – Geometry...................................................................................... 121
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Lesson 9: Geometric Figures ........................................................ 122
ACTAAP Coverage: G.8.7.1, G.8.7.2, G.8.7.3, G.8.7.4, G.11.7.1,
G.11.7.2
Lesson 10: Geometric Concepts ................................................... 139
ACTAAP Coverage: G.8.7.6, G.9.7.1, G.9.7.2, G.10.7.1, G.10.7.2,
M.13.7.5
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Table of Contents
Unit 4 – Measurement ............................................................................... 155
Lesson 11: Measurement Systems ................................................ 156
ACTAAP Coverage: M.12.7.1, M.12.7.2, M.13.7.1
Lesson 12: Geometric Measurement ............................................ 169
ACTAAP Coverage: M.12.7.3, M.13.7.2, M.13.7.3, M.13.7.4,
M.13.7.6, M.13.7.7, G.8.7.5
Unit 5 – Data Analysis and Probability ................................................... 189
Lesson 13: Data Analysis.............................................................. 190
ACTAAP Coverage: DAP.14.7.1, DAP.14.7.2, DAP.14.7.3,
DAP.15.7.1, DAP.15.7.2, DAP.16.7.1
Lesson 14: Probability .................................................................. 213
iv
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ACTAAP Coverage: DAP.17.7.1, DAP.17.7.2
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Page 4
Unit 1 – Number and Operations
ACTAAP Coverage: NO.1.7.6
Lesson 1: Number Concepts
In this lesson, you will review basic number concepts such as rational and irrational numbers,
absolute value, multiples and factors, exponents, square roots, and scientific notation.
Rational and Irrational Numbers
The real number system consists of rational and irrational numbers.
Rational numbers
Rational numbers can be expressed in fractional form,
a
,
b
where a (the numerator) and b (the
denominator) are both integers and b 0. The decimal form of the number either terminates or
repeats. Counting numbers, whole numbers, integers, and non-integers are all rational numbers.
Counting numbers are the natural numbers.
{1, 2, 3, 4, 5, 6, . . .}
Whole numbers consist of the counting numbers and zero.
{0, 1, 2, 3, 4, 5, . . .}
Integers consist of the counting numbers, their opposites, and zero.
{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
3
7
4
5.25
3
4 11
9.261
0.6
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Non-integers consist of fractions that can be written as terminating or repeating decimals. A
terminating decimal comes to a complete stop, whereas a repeating decimal continues the
same digit or block of digits forever. Here are some examples of non-integers.
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Lesson 1: Number Concepts
ACTAAP Coverage: NO.1.7.6
Irrational numbers
Irrational numbers are numbers that cannot be written as a ratio of two integers. The decimal
form of the number never terminates and never repeats. The most common irrational number is
pi (). The value of is 3.141592654. . . but is usually approximated as 3.14.
The following tree diagram shows the subsets of the real number system.
Real Numbers
Rational Numbers
Irrational Numbers
Integers
Whole
Numbers
Non-Integers
Negative
Integers
Practice
Directions: For Numbers 1 through 5, write whether each real number is rational or irrational.
1. 23.75 _________________________
2. 4.750918362. . . _________________________
3.
5
9
_________________________
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4. 1,469,000 _________________________
5.
15
_________________________
6. Circle all the real numbers below that are rational.
81
9
0.251896857. . .
2
3.6
7. Circle all the real numbers below that are irrational.
15
16
8.493870015. . .
4 1
2
18
5
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Unit 1 – Number and Operations
ACTAAP Coverage: NO.3.7.5
Absolute Value
The absolute value of a number is that number’s distance from 0 on a number line. When you
write the absolute value of a number n, use the notation n.
Example
The absolute value of 7 7 7.
The absolute value of 7 7 7.
7 units
–10 –9 –8
–7 –6 –5 –4 –3 –2 –1
7 units
0
1
2
3
4
5
6
7
8
9 10
The absolute value of every number will be either positive or 0. Negative signs on the
outside of absolute value signs act as factors of 1. You need to evaluate what’s
inside the first, then multiply by 1.
50
1(50) 50
Practice
1. 2 __________
3. 75 __________
2. 26 __________
4. 31 __________
Directions: For Numbers 5 and 6, represent each distance or depth using absolute value signs.
Then determine each value and answer the question.
5. Benton is 25 miles southwest of Little Rock. Jacksonville is 16 miles in the opposite
direction from Little Rock. Which city is farther from Little Rock?
_________________
6. The lowest point in the United States, Death Valley, is at 282 feet. If sea level represents
0 feet, how far is Death Valley from sea level?
_________________
6
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Directions: For Numbers 1 through 4, determine each value.
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Lesson 1: Number Concepts
ACTAAP Coverage: NO.3.7.4
Multiples and Factors
Multiples of a number are the products that result from multiplying the number by each of the
whole numbers (0, 1, 2, 3, 4, . . .).
Example
What are the first eight multiples of 3?
Multiply 3 by each of the first eight whole numbers.
3•00
3•13
3•26
3•39
3 • 4 12
3 • 5 15
3 • 6 18
3 • 7 21
The first eight multiples of 3 are 0, 3, 6, 9, 12, 15, 18, and 21.
A number that is a multiple of two or more numbers is a common multiple of those numbers.
The smallest common multiple of two or more numbers is called their least common multiple
(LCM). Zero is not considered a common multiple.
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Example
What are some common multiples and the least common multiple of 3, 4, and 6?
multiples of 3: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, . . .
multiples of 4: 0, 4, 8, 12, 16, 20, 24, 28, 32, . . .
multiples of 6: 0, 6, 12, 18, 24, 30, 36, 42, . . .
The numbers 12 and 24 are common multiples of 3, 4, and 6.
The least common multiple of 3, 4, and 6 is 12.
7
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Unit 1 – Number and Operations
ACTAAP Coverage: NO.3.7.4
Factors of a number divide that number exactly (no remainder). A number is divisible by each
of its factors.
Example
What are the factors of 18?
Find the whole numbers that divide 18 evenly.
18 1 18
18 2 9
18 3 6
18 6 3
18 9 2
18 18 1
The factors of 18 are 1, 2, 3, 6, 9, and 18.
A number that is a factor of two or more numbers is a common factor of those numbers. The
largest common factor of two or more numbers is called their greatest common factor (GCF).
Example
What are the common factors and the greatest common factor of 18, 24, and 30?
factors of 18: 1, 2, 3, 6, 9, 18
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The numbers 1, 2, 3, and 6 are common factors of 18, 24, and 30.
The greatest common factor of 18, 24, and 30 is 6.
8
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factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
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Lesson 1: Number Concepts
ACTAAP Coverage: NO.3.7.4
Practice
Directions: For Numbers 1 through 3, fill in the missing multiples.
1. Multiples of 4: 0, 4, ________, 12, 16, ________, 24, 28, ________, 36, . . .
2. Multiples of 6: 0, 6, 12, 18, ________, 30, ________, 42, ________, 54, . . .
3. Multiples of 9: 0, 9, 18, ________, 36, 45, ________, 63, ________, . . .
4. What is the least common multiple of 6 and 9? ____________
5. What is the least common multiple of 4, 6, and 9? ____________
6. Dakota has art class every 4 school days and music class every 5 school days. If he has
both classes today, how many school days will pass before he takes both classes again on
the same day?
____________
Directions: For Numbers 7 through 9, list all the factors.
7. Factors of 9: ______________________________
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8. Factors of 27: ______________________________
9. Factors of 36: ______________________________
10. What is the greatest common factor of 9 and 27? ____________
11. What is the greatest common factor of 9, 27, and 36? ____________
12. The organizers of a school concert set up chairs in two sections. The orchestra section has
84 chairs, while the audience section has 312 chairs. If all rows have the same number of
chairs, what is the greatest number of chairs that each row can have?
____________
9
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Unit 1 – Number and Operations
ACTAAP Coverage: NO.3.7.5
Exponents
An exponent shows how many times to multiply a base number by itself. When working with
exponents, remember the following rules:
1. Any base number with an exponent of 1 equals the same number.
2. Any base number (except zero) with zero as the exponent equals 1. (00 is undefined.)
3. Any base number with a negative exponent is written as its reciprocal with a positive
exponent.
The following model shows how to find the value of 25.
exponent
‘
25 2 • 2 • 2 • 2 • 2 32
“
base
Examples
Each of the following examples shows the same number written in exponential form
and standard form.
320 1
7491 749
216 6 • 6 • 6 63
112 11 • 11 121
62 1
62
1
6•6
1
36
2,401 7 • 7 • 7 • 7 74
(1)3 (1) • (1) • (1) 1
10
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(3)2 (3) • (3) 9
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Lesson 1: Number Concepts
ACTAAP Coverage: NO.3.7.5
Practice
Directions: For Numbers 1 through 5, write each expression in standard form.
1. 102 ____________________
2. (4)3 ____________________
3. 93 ____________________
4. 73 ____________________
5. (12)2 ____________________
Directions: For Numbers 6 through 10, write each number in exponential notation using an
exponent other than 1.
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6. 625 ____________________
7. 27 ____________________
8. 49 ____________________
9. 256 ____________________
10. 1,000 ____________________
11
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Page 12
Unit 1 – Number and Operations
ACTAAP Coverage: NO.3.7.5
Square Roots
The square root of a number is indicated by the radical sign (). To find the square root of a
number, find the number that, when multiplied by itself, is equal to the number under the radical
sign (the radicand).
Example
. (It can also be used to model 42.)
The following square can be used to model 16
The square has been divided into 16 smaller, equal squares. To find 16
, simply
count the number of smaller squares that make up each side of the larger square.
Each side is made up of 4 smaller squares.
16
4
(42 16)
Whole numbers such as 1, 4, 9, 16, and 25 are called perfect squares because their square roots
are integers.
Practice
1.
100
__________
5.
49
__________
2.
225
__________
6.
144
__________
3.
9 __________
7.
64
__________
4.
81
__________
8.
36
__________
12
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Directions: For Numbers 1 through 8, evaluate each square root.
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Lesson 1: Number Concepts
ACTAAP Coverage: NO.1.7.2, NO.1.7.3
Scientific Notation
Scientific notation is used to represent very large numbers. In scientific notation, a large
number is written as a number between 1 and 10 multiplied by a positive power of 10. The
following table shows the first ten positive powers of 10.
Powers of Ten
1
10 1,000,000
2
10 10,000,000
3
10 100,000,000
4
10 1,000,000,000
10 10
10 100
10 1,000
10 10,000
5
10 100,000
6
7
8
9
10
10
10,000,000,000
Changing from standard form to scientific notation
Follow these steps to change a number from standard form to scientific notation.
Step 1: Move the decimal point to the left until you have a number greater than or equal
to 1 and less than 10.
Step 2: Count the number of places you moved the decimal point to the left and use that
number as the positive power of 10.
Step 3: Multiply the decimal (in Step 1) by the power of 10 (in Step 2).
Example
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Write 5,860,000 in scientific notation.
Move the decimal point 6 places to the left.
5.860000.
Since the decimal point moved 6 places to the left, multiply by 106.
5.86 • 106
Therefore, 5,860,000 5.86 • 106.
13
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Unit 1 – Number and Operations
ACTAAP Coverage: NO.1.7.2, NO.1.7.3
Changing from scientific notation to standard form
To change a number written in scientific notation with a positive power of 10 to standard form,
move the decimal point to the right. The exponent tells you the number of places to move the
decimal point. Remember to add zeros as placeholders when necessary.
Example
Write 9.437 • 107 in standard form.
Since you are multiplying by the positive 7th power of 10, move the decimal point
7 places to the right.
9.4370000.
Therefore, 9.437 • 107 94,370,000.
Practice
Directions: For Numbers 1 through 4, write the number in scientific notation.
1. 5,209 _________________________
2. 652,000 _________________________
3. 2,001,600 _________________________
Directions: For Numbers 5 through 8, write the number in standard form.
5. 2.4 • 108 _________________________
6. 9.05 • 106 _________________________
7. 5.284 • 105 _________________________
8. 3.02 • 108 _________________________
14
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4. 830,000,000 _________________________
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Lesson 1: Benchmark Exam Practice
Benchmark Exam Practice
1.
2.
What is another way of writing 64?
5.
A.
26
B.
6.4 • 102
A.
5.4 • 103
C.
4•4•4•4
B.
5.4 • 104
D.
|64|
C.
5.04 • 103
D.
5.04 • 104
What is the value of 33?
6.
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3.
What is the LCM of 6, 8, and 10?
A.
3
B.
6
A.
40
C.
9
B.
60
D.
27
C.
80
D.
120
Venus circles the sun at an average
distance of 108,000,000 kilometers.
How is this number expressed in
scientific notation?
A.
1.08 • 106 kilometers
B.
1.08 • 107 kilometers
C.
1.08 • 108 kilometers
D.
1.08 • 109 kilometers
7.
8.
4.
How is 50,400 written in scientific
notation?
Evaluate: 256
A.
14
B.
15
C.
16
D.
17
Which statement is true?
A.
|7| 7
B.
|28| 28
C.
|56|
56
D.
|4|
4
In which list are all three numbers
rational?
A.
5
2, , 16
9
B.
0, 3, 4.2
C.
, 10
27, 13
D.
22
, 0.16, 20
5
15
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Unit 1 – Number and Operations
What is the value of (2)4?
A.
16
B.
8
C.
8
D.
16
Use the number line below to answer
question 13.
1
G
13.
10.
3
What is the GCF of 30, 42, and 54?
B.
2
A.
2
C.
2
B.
3
D.
3
C.
6
D.
7
12.
4
What is the value of 2 ?
B.
12
A.
C.
16
D.
24
B.
1
64
C.
D.
64
1
64
15.
Which of the following is a natural
number?
A.
0
B.
1
2
C.
D.
16
64
0.75
1
16.
K
What is the LCM of 4 and 12?
A.
6
J
What is the absolute value of H?
A.
14.
11.
H
0 1
What is the GCF of 576 and 756?
A.
18
B.
36
C.
42
D.
72
How can 1,296 be written in
exponential notation?
A.
64
B.
45
C.
210
D.
123
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9.