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Buckle Down North Carolina
EOG 7 Mathematics
Number Sense
Lesson 1: Number Concepts and Representation
Lesson 2: Computation
Lesson 3: Estimation and Problem Solving
Lesson 4: Ratio, Proportion, and Percent
Unit 2
Algebra
Lesson 5: Expressions, Equations, and Inequalities
Lesson 6: Patterns and Functions
Unit 3
Geometry and Measurement
Lesson 7: Geometric Concepts
Lesson 8: Solids
Lesson 9: Geometric Measurement
Unit 4
Data Analysis and Probability
Lesson 10: Data Analysis and Representation
Lesson 11: Probability
North Carolina
4TH EDITION
North Carolina EOG
The cover image depicts cubes that
feature operational symbols. This
hands-on teaching tool helps students
understand order of operations as
well as probability.
Unit 1
Catalog # 4BDNC07MM01
ISBN 0-7836-3867-1
51495
P.O. Box 2180
Iowa City, Iowa 52244-2180
PHONE: 800-776-3454
FAX: 877-365-0111
www.BuckleDown.com
9 780783 638676
7 MATHEMATICS
Go to www.BuckleDown.com to review our complete line of EOG and EOC materials for Grades 3–12
READING • WRITING • MATHEMATICS • SCIENCE
7
Mathematics
EOG
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Page iii
TABLE OF CONTENTS
Introduction................................................................................................. 1
Testwise StrategiesTM .................................................................... 2
Unit 1 – Number Sense ............................................................................. 3
Lesson 1: Number Concepts and Representation ........................ 4
Lesson 2: Computation................................................................. 22
EOG Standards: 1.02a, 1.02b
Skills to Maintain: number properties
Lesson 3: Estimation and Problem Solving ................................ 43
EOG Standards: 1.02c, 1.02d, 1.03, 5.04
Lesson 4: Ratio, Proportion, and Percent ................................... 54
EOG Standards: 1.01, 5.04
Skills to Maintain: percent
Unit 2 – Algebra......................................................................................... 77
Lesson 5: Expressions, Equations, and Inequalities.................. 78
EOG Standards: 5.02, 5.03
Lesson 6: Patterns and Functions............................................... 94
EOG Standard: 5.01
Unit 3 – Geometry and Measurement ................................................ 111
Lesson 7: Geometric Concepts ................................................... 112
EOG Standards: 2.01, 3.02, 3.03
Skills to Maintain: transformations in the coordinate plane
Lesson 8: Solids .......................................................................... 127
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EOG Standard: 3.01a–c
Lesson 9: Geometric Measurement ........................................... 137
EOG Standards: 2.02, 5.04
Unit 4 – Data Analysis ........................................................................... 151
Lesson 10: Data Analysis and Representation ......................... 152
EOG Standards: 4.01, 4.02, 4.03, 4.04, 4.05
Lesson 11: Probability ................................................................ 177
Skills to Maintain: probability
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Page 4
Unit 1 – Number Sense
Lesson 1: Number Concepts and Representation
In this lesson, you will review basic number concepts such as absolute value,
exponents, scientific notation, multiples, and factors. You will also review how to
convert between different forms of rational numbers.
Absolute Value
The absolute value of a number is the distance that number is from zero 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 units
–7 –6 –5 –4 –3 –2 –1
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. New Bern is 34 miles southeast of Kinston. Goldsboro is 28 miles in the
exact opposite direction. Which city is farther from Kinston?
____________
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?
____________
4
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Directions: For Numbers 1 through 4, determine each value.
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Page 5
Lesson 1: Number Concepts and Representation
Exponents
An exponent shows how many times to multiply a base number by itself.
When working with exponents, remember the following rules:
1. A 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
320 1
(3)2 (3) • (3) 9
7491 749
62 54 5 • 5 • 5 • 5 625
(1)3 (1) • (1) • (1) 1
1
62
1
6•6
1
36
Practice
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Directions: For Numbers 1 through 8, find the value of each expression.
1. 102 __________
5. 170 __________
2. (4)3 __________
6. (12)2 __________
3. 93 __________
7. (2)3 __________
4. 73 __________
8. 848,0001 __________
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Page 6
Unit 1 – Number Sense
Scientific Notation
Scientific notation is used to represent very large or very small numbers.
In scientific notation, a number is written as a number between 1 and 10
multiplied by a positive or negative power of 10.
Powers of 10
Negative
Positive
101 10
10 1 0.1
102 100
10 2 0.01
103 1,000
10 3 0.001
104 10,000
10 4 0.0001
105 100,000
10 5 0.00001
–
–
–
–
–
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 or right 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 or right and use that number as the positive or negative
power of 10.
Examples
Write 5,860,000 in scientific
notation.
Write 0.0000319 in scientific
notation.
Move the decimal point 6 places
to the left and multiply by 106.
Move the decimal point 5 places
to the right and multiply by 105.
5.860000.
0.00003.19
0.0000319 3.19 • 105
5,860,000 5.86 • 106
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Step 3: Multiply the decimal (in Step 1) by the power of 10 (in Step 2).
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Page 7
Lesson 1: Number Concepts and Representation
Changing from scientific notation to standard form
To change a number written in scientific notation to standard form, move the
decimal point to the right for a positive power of 10 and to the left for a
negative power of 10. The exponent tells you the number of places to move
the decimal point. Remember to add zeros as placeholders when necessary.
Examples
Write 9.437 • 107 in
standard form.
Write 2.5 • 104 in
standard form.
Move the decimal point
7 places to the right.
Move the decimal point
4 places to the left.
9.4370000.
0.0002.5
9.437 • 107 94,370,000
2.5 • 104 0.00025
Practice
Directions: For Numbers 1 through 4, write each number in scientific notation.
1. 5,209 _________________________
2. 652,000 _________________________
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3. 0.000058 _________________________
4. 0.00000000214 _________________________
Directions: For Numbers 5 through 7, write each number in standard form.
5. 2.4 • 108 _________________________
6. 9.05 • 106 _________________________
7. 5.284 • 105 _________________________
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Page 8
Unit 1 – Number Sense
Multiples and Factors
Multiples of a number are the products that result from multiplying the
number by each of the integers (0, 1, 2, 3, 4, . . .). In this lesson, only
positive multiples will be found.
Example
What are the first ten multiples of 3?
Multiply 3 by each of the first ten positive integers.
3•13
3•26
3•39
3 • 4 12
3 • 5 15
3 • 6 18
3 • 7 21
3 • 8 24
3 • 9 27
3 • 10 30
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).
Example
What are some common multiples and the least common multiple
of 3 and 4?
multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, . . .
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, . . .
The numbers 12 and 24 are common multiples of 3 and 4.
The least common multiple of 3 and 4 is 12.
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The first ten multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30.
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Page 9
Lesson 1: Number Concepts and Representation
Factors of a number divide that number exactly (no remainder). A number is
divisible by each of its factors. In this lesson, only positive factors will be found.
Example
What are the factors of 18?
Find the positive integers 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.
Notice in the previous example that all factors come in pairs. When you divide
18 by 3, you see that both 3 and 6 are factors of 18. These are called factor
pairs. The factor pairs of 18 are 1 and 18, 2 and 9, and 3 and 6.
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
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What are the common factors and the greatest common factor of
18 and 30?
factors of 18: 1, 2, 3, 6, 9, 18
factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The numbers 1, 2, 3, and 6 are common factors of 18 and 30.
The greatest common factor of 18 and 30 is 6.
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Unit 1 – Number Sense
Practice
Directions: For Numbers 1 through 4, fill in the missing multiples.
1. Multiples of 6: 6, ________, 18, ________, ________, . . .
2. Multiples of 8: 8, 16, 24, ________, 40, ________, 56, ________, 72, . . .
3. Multiples of 9: 9, 18, ________, 36, 45, ________, 63, ________, . . .
4. Multiples of 15: 15, ________, ________, 60, 75, 90, ________, 120, 135, . . .
5. What is the LCM of 6 and 8? ________
6. What is the LCM of 8 and 9? ________
7. What is the LCM of 9 and 15? ________
Directions: For Numbers 8 through 11, list all the factors.
8. Factors of 9: ______________________________
10. Factors of 28: ______________________________
11. Factors of 36: ______________________________
12. What is the GCF of 9 and 27? ________
13. What is the GCF of 27 and 28? ________
14. What is the GCF of 28 and 36? ________
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9. Factors of 27: ______________________________
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Lesson 1: Number Concepts and Representation
Prime and Composite Numbers
Prime numbers have only two factors: 1 and the number. Composite
numbers have at least three factors.
Example
2, 3, and 5 have only two factors. They are prime numbers.
Factors of 2: 1 and 2
3: 1 and 3
5: 1 and 5
4 has three factors. It is a composite number.
Factors of 4: 1, 2, and 4
Practice
1. Is 6 a prime number or a composite number? ___________________
2. Is 7 a prime number or a composite number? ___________________
3. Is 9 a prime number or a composite number? ___________________
4. List all the prime numbers between 10 and 20.
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5. List all the composite numbers between 10 and 20.
6. Which is a prime number?
7. Which is a composite number?
A 23
A 29
B 33
B 43
C 49
C 73
D 51
D 93
TIP: 0 and 1 are neither prime nor composite. The only even prime number
is 2.
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Page 12
Unit 1 – Number Sense
Prime Factorization
Prime factorization is a way to express a composite number as a product of
prime numbers. Factor trees help you determine the prime factorization of
composite numbers.
Example
What is the prime factorization of 24?
Write the number 24. Put a prime factor under the left branch and
circle it. Put its nonprime factor pair under the right branch. Repeat
the process until you have circled two prime numbers at the bottom of
the tree.
24
12
2
6
2
2
3
The prime factorization of 24 is 2 • 2 • 2 • 3 or 23 • 3.
Practice
The prime factorization of 45 is _____________________________ .
12
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1. Draw a factor tree for 45.
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Lesson 1: Number Concepts and Representation
Prime factorization can be used to find the LCM and the GCF of two or more
numbers.
Example
Use prime factorization to find the LCM of 75 and 90.
Step 1: Find the prime factorizations of 75 and 90.
75 3 • 52
90 2 • 32 • 5
Step 2: Circle the highest power of all the different prime factors of
both numbers.
75 3 • 52
90 2 • 32 • 5
Step 3: Multiply the highest power of all the different prime factors
from Step 2.
2 • 32 • 52 2 • 3 • 3 • 5 • 5 450
The LCM of 75 and 90 is 450.
Example
Use prime factorization to find the GCF of 96 and 180.
Step 1: Find the prime factorization of 96 and 180.
96 25 • 3
180 22 • 32 • 5
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Step 2: Circle the lowest power of the common prime factors of both
numbers.
96 25 • 3
180 22 • 32 • 5
Step 3: Multiply the lowest power of the common prime factors from
Step 2.
22 • 3 2 • 2 • 3 12
The GCF of 96 and 180 is 12.
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Page 14
Unit 1 – Number Sense
Practice
1. Draw a factor tree for 54.
The prime factorization of 54 is ______________________________.
The prime factorization of 420 is ______________________________.
3. What is the LCM of 54 and 420? ____________
4. What is the GCF of 54 and 420? ____________
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2. Draw a factor tree for 420.
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Lesson 1: Number Concepts and Representation
5. Draw a factor tree for 225.
The prime factorization of 225 is ______________________________.
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6. Draw a factor tree for 600.
The prime factorization of 600 is ______________________________.
7. What is the LCM of 225 and 600? ____________
8. What is the GCF of 225 and 600? ____________
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Page 16
Unit 1 – Number Sense
Equivalent Forms of Rational Numbers
Fractions, decimals, and percents are various ways of representing rational
numbers and expressing parts of a whole. The following grid shows 75 of its
100 units shaded.
As a fraction, 75 out of 100 is written
75
100
or
3
.
4
As a decimal, it is written 0.75.
As a percent, it is written 75%. These three numbers,
3
,
4
0.75, and 75%, are
equivalent. They each represent 75 parts of 100. The relationships between
fractions, decimals, and percents allow us to convert from one form to another.
Converting a decimal to a fraction
To convert a decimal to a fraction, write the fraction that you would say if you
read the decimal aloud correctly. Then reduce to lowest terms (when necessary)
by dividing the numerator and the denominator by the same number. Use the
following table of placeholders to help you read the decimals correctly.
‘‘And’’
Tenths
Hundredths
0
.
5
6
2
.
0
4
Thousandths
3
Examples
Write 0.56 as a fraction.
Write 2.043 as a fraction
0.56 is read “fifty-six
hundredths,” so the
fraction is:
2.043 is read “two and
forty-three thousandths,”
so the fraction is:
56 4
100 4
14
25
43
2
1,000
16
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Ones
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Lesson 1: Number Concepts and Representation
Converting a decimal to a percent
To convert a decimal to a percent, move the decimal point two places to the
right and then write the percent sign, %.
Examples
Write 0.085 as a percent.
Write 1.05 as a percent.
0.085 8.5%
1.05 105%
Converting a percent to a decimal
To convert a percent to a decimal, remove the percent sign and then move the
decimal point two places to the left.
Examples
Write 89% as a decimal.
Write 13.25% as a decimal.
89% 0.89
13.25% 0.1325
Converting a percent to a fraction
To convert a percent to a fraction, first write the percent as a decimal, then
write the fraction that you would say if you read the decimal aloud correctly.
Finally, reduce to lowest terms (when necessary).
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Examples
Write 31% as a fraction.
Write 10.5% as a fraction.
The decimal form of 31%
is 0.31, so the fraction is:
The decimal form of 10.5%
is 0.105, so the fraction is:
105 5
1,000 5
31
100
17
21
200
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Unit 1 – Number Sense
Converting a fraction to a decimal
To convert a fraction to a decimal, divide the numerator by the denominator.
The decimal will terminate or repeat a digit or block of digits. (Use a bar, —,
to show the repeating digit or block of digits.)
Examples
Write
3
8
as a decimal.
Write
0.375
83
.0
0
0
2 4‘
60
56‘
40
40
0 (terminating)
3
8
1
6
as a decimal.
0.166
61
.0
0
0
6‘
40
36‘
40
36
4 (repeating)
1
6
0.375
0.16
Converting a fraction to a percent
To convert a fraction to a percent, first divide the numerator by the denominator
to find the decimal form. Next, move the decimal point two places to the right
and, finally, write the percent sign, %.
Examples
2
5
as a percent.
The decimal form of
2
5
Write
is
3
4
as a percent.
The decimal form of
0.4, so the percent is:
3
4
is
0.75, so the percent is:
40%
75%
TIP: When converting to a percent, don’t forget to include the percent sign.
The percent sign is always on the right side of the number.
18
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Write
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Lesson 1: Number Concepts and Representation
Practice
Directions: For Numbers 1 through 6, convert to the given forms.
1. Convert 0.23 to a: percent _______________ fraction _______________
2. Convert
2
9
to a: decimal _______________ percent _______________
3. Convert 24.05% to a: decimal _______________ fraction _______________
4. Convert
5
8
to a: decimal _______________ percent _______________
5. Convert 0.92 to a: percent _______________ fraction _______________
6. Convert 37.2% to a: decimal _______________ fraction _______________
7. On the last science test, Mia correctly answered
9
10
of the questions. What
percent of the questions did Mia answer correctly?
_______________
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8. Last baseball season, Ryan got a hit 32.8% of the times he batted. What
decimal represents how often Ryan got a hit?
_______________
9. How is 0.755 written as
a fraction?
A
1
755
B
11
20
C
3
4
D
151
200
10. How is
A 1.2
B 0.83
C 0.65
D 0.56
19
25
30
written as a decimal?
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Unit 1 – Number Sense
EOG Practice
5. What is the greatest common
factor of 42 and 54?
1. How is 0.85 written as a fraction?
A
8
10
B
17
20
C
21
25
D
43
50
A 2
B 3
C 6
D 7
6. Paul makes 34% of the threepoint shots he attempts. What
fraction of three-point shots that
Paul attempts does he make?
2. What is the LCM of 8 and 10?
A 20
B 40
C 60
A
17
25
B
17
50
C
17
75
D
17
500
D 80
3. What is the value of (2)4?
A 16
B 8
7. What is the value of 26?
A
D 16
1
64
B 64
4. Venus circles the sun at
an average distance of
108,000,000 km. How is this
number expressed in scientific
notation?
C 64
1
D 64
A 1.08 • 106 km
B 1.08 • 107 km
C 1.08 • 108 km
D 1.08 • 109 km
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Lesson 1: EOG Practice
11. Which list orders the integers
from least to greatest?
8. What is the least common
multiple of 4 and 12?
4
A 743, 758, 772, 789, 802
B 12
B 599, 627, 726, 642, 531
C 16
C 490, 496, 445, 485, 488
D 24
D 134, 144, 149, 191, 198
A
12. List all of the common factors of
28 and 42.
9. Which statement is true?
A 7 7
B 28 28
A 1, 2, 7
C 56 56
B 2, 4, 7, 14
D 4 4
C 1, 2, 7, 14
D 1, 2, 4, 6, 12
10. Four people worked together
13. What is the prime factorization
of 256?
to collect cans for their school’s
A 24
aluminum can drive. Robert
collected
6
16
B 26
of the total amount
C 28
brought in by the group, Steven
D 212
collected 35% of the total, Marie
collected 0.15 of the total, and
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Ellen collected
1
8
of the total.
Who collected the greatest
number of cans?
A Robert
B Steven
C Marie
D Ellen
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