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Buckle Down North Carolina
EOG 6 Mathematics
Number and Operations
Lesson 1: Number Representation
Lesson 2: Fractions, Decimals, and Percents
Lesson 3: Computation
Lesson 4: Estimation and Problem Solving
Unit 2
Algebra
Lesson 5: Expressions, Equations, and Inequalities
Lesson 6: Graphs and Tables
Unit 3
Geometry
Lesson 7: Plane Figures
Lesson 8: Coordinate Geometry
Unit 4
Measurement
Lesson 9: Measurement Systems
Lesson 10: Geometric Measurement
Unit 5
Data Analysis and Probability
Lesson 11: Data Collection and Analysis
Lesson 12: Probability
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Catalog # 4BDNC06MM01
4TH EDITION
6 MATHEMATICS
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READING • WRITING • MATHEMATICS
North Carolina
North Carolina EOG
The cover image depicts a protractor.
This important tool for measuring
angles is also a useful instrument for
drafting and plotting.
Unit 1
6
Mathematics
EOG
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Page iii
TABLE OF CONTENTS
Introduction................................................................................................. 1
Testwise StrategiesTM .................................................................... 2
Unit 1 – Number and Operations .......................................................... 3
Lesson 1: Number Representation ................................................ 4
EOG Standards: 1.01a, 1.01b, 1.05, 1.06
Lesson 2: Fractions, Decimals, and Percents ............................. 17
EOG Standards: 1.01a, 1.01b, 1.02a, 1.03
Lesson 3: Computation................................................................. 35
EOG Standards: 1.04a, 1.04b
Skills to Maintain: addition and subtraction of non-negative
rational numbers
Lesson 4: Estimation and Problem Solving ................................ 49
EOG Standards: 1.01c, 1.02b, 1.04c, 1.04d, 1.07
Unit 2 – Algebra......................................................................................... 63
Lesson 5: Expressions, Equations, and Inequalities.................. 64
EOG Standards: 5.01a–e, 5.02, 5.03
Skill to Maintain: number properties
Lesson 6: Graphs and Tables....................................................... 91
EOG Standard: 5.04
Unit 3 – Geometry................................................................................... 109
Lesson 7: Plane Figures ............................................................. 110
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EOG Standards: 3.01, 3.02
Skills to Maintain: symmetry and congruency
Lesson 8: Coordinate Geometry................................................. 125
EOG Standards: 3.03, 3.04
Skills to Maintain: transformations, coordinate grid
Unit 4 – Measurement ........................................................................... 137
Lesson 9: Measurement Systems .............................................. 138
EOG Standard: 2.01
Lesson 10: Geometric Measurement ......................................... 143
EOG Standards: 2.01, 2.02
Skills to Maintain: perimeter and area
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Table of Contents
Unit 5 – Data Analysis and Probability............................................. 153
Lesson 11: Data Collection and Analysis .................................. 154
EOG Standard: 4.06
Skills to Maintain: median, mode, and range; bar graphs
and leaf plots
Lesson 12: Probability................................................................ 171
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EOG Standards: 4.01. 4.02, 4.03, 4.04, 4.05
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Unit 1 – Number and Operations
Lesson 1: Number Representation
In this lesson, you will learn how to use extremely small and large numbers—
including numbers less than zero.
Whole Numbers
Whole numbers are the counting numbers and zero. Here is the set of whole
numbers.
{0, 1, 2, 3, 4, 5, . . .}
Integers
Integers are whole numbers and their opposites (positive numbers, zero, and
negative numbers). Negative numbers are the numbers less than zero. They
can be shown on a number line.
5 4 3 2 1
0
1
2
3
4
5
neither positive
nor negative
The opposite of a number is the number that is the same distance from 0 on a
number line, but on the opposite side of 0. The opposite of a positive number is
a negative number, and the opposite of a negative number is a positive number.
4
4
is 4 units from 0.
4 is 4 units from 0.
and 4 are opposites.
The sum of two opposites is 0.
Examples
40
4 (4) 0
125 0
125 (125) 0
4
125
245 (245) 0
245
4
245 0
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Example
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Lesson 1: Number Representation
Practice
1. Write the following numbers in order from least to greatest.
25,
25, 7, 0, 10, 16, 5, 1, 2, 1, 4, 5
2. Is the following list in order from greatest to least? ______________________
10, 9, 8, 6, 5
Directions: In Numbers 3 through 6, use the following information to write the
integer that is represented by the bold-faced words.
At 6:00 A.M., the temperature in Greensboro was 63°.
Temperature
at 6:00 A.M.
70
°F
70
°F
70
65
65
°F
70
°F
65
60
60
55
55
Temperature
at 10:00 P.M.
65
60
60
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Temperature
at 12 noon
Temperature
at 8:00 A.M.
55
55
50
50
50
50
40
40
40
40
3. By 8:00 A.M., the temperature dropped 3°. ____________
4. Between 8:00 A.M. and noon, the temperature rose 7°. ____________
5. Between noon and 10:00 P.M., the temperature dropped 13°. ____________
6. What was the temperature at 10:00 P.M.? ____________
5
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Unit 1 – Number and Operations
Multiples
Multiples of a number are the products that result from multiplying the
number by each of the whole numbers (0, 1, 2, 3, 4, and so on).
Example
What are the first five multiples of 6?
Multiply 6 by each of the first five whole numbers.
6•00
6•16
6 • 2 12
6 • 3 18
6 • 4 24
The first five multiples of 6 are 0, 6, 12, 18, and 24.
A number that is a multiple of two or more numbers is a common multiple
of those numbers. (Zero is not considered a common multiple.) The least
number that is a common multiple of two or more numbers is called their
least common multiple (LCM).
Example
What is the least common multiple of 6 and 8?
multiples of 8: 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, . . .
Of the multiples shown, 24 and 48 are common multiples of 6 and 8.
The least common multiple of 6 and 8 is 24.
6
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multiples of 6: 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, . . .
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Lesson 1: Number Representation
Practice
Directions: For Numbers 1 through 5, list the first 10 multiples.
1. multiples of 4: _________________________________________________________
_
2. multiples of 7: _____________________________________________________
_____
3. multiples of 9: _________________________________________________________
4. multiples of 12: ________________________________________________________
5. multiples of 16: ________________________________________________________
6. What is the least common multiple of 4 and 7? __________
7. What is the least common multiple of 7 and 9? __________
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8. What is the least common multiple of 9 and 12? __________
9. What is the least common multiple of 12 and 16? __________
10. What is the least common
multiple of 10 and 15?
11. What is the least common
multiple of 3 and 13?
A 30
A 13
B 50
B 26
C 60
C 39
D 90
D 52
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Unit 1 – Number and Operations
Factors
Factors of a number divide that number evenly (remainder of 0). A number is
divisible by all its factors.
Rules for Divisibility
A number is
divisible by . . .
when . . .
2
its last digit is an even number
3
the total of its digits is divisible by 3
5
its last digit is 0 or 5
6
it is divisible by both 2 and 3 (see rules above)
9
the total of its digits is divisible by 9
10
its last digit is 0
Example
What are the factors of 24?
Find the numbers that divide 24 evenly.
24 1 24
24 6 4
24 2 12
24 8 3
24 3 8
24 12 2
24 4 6
24 24 1
A number that is a factor of two or more numbers is a common factor of those
numbers. The greatest number that is a common factor of two or more numbers
is called their greatest common factor (GCF).
Example
What is the greatest common factor of 24 and 42?
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
The numbers 1, 2, 3, and 6 are the common factors of 24 and 42. The
greatest common factor of 24 and 42 is 6.
8
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The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
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Lesson 1: Number Representation
Practice
Directions: For Numbers 1 through 5, list all the factors.
1. factors of 5: ____________________________________________
_____________
_____
2. factors of 10: ___________________________________________________________
3. factors of 17: ___________________________________________________________
4. factors of 102: __________________________________________________________
5. factors of 110: _________________________________________________________
6. What is the greatest common factor of 5 and 10? __________
7. What is the greatest common factor of 10 and 17? __________
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8. What is the greatest common factor of 17 and 102? __________
9. What is the greatest common factor of 102 and 110? __________
10. What is the greatest common
factor of 44 and 52?
11. What is the greatest common
factor of 39 and 78?
A 1
A
1
B 2
B
3
C 4
C 13
D 6
D 39
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Unit 1 – Number and Operations
Exponents
An exponent shows how many times a base number occurs as a factor.
Exponents show repeated multiplication.
When working with exponents, remember that any base number (except zero)
that has zero as the exponent equals 1. (00 is not defined.) Also, any base
number that has 1 as the exponent equals the base number.
Example
How is 3 • 3 • 3 • 3 • 3 written in exponential notation?
Since the base number (3) occurs as a factor 5 times, the exponent is 5.
exponent
‘
3 • 3 • 3 • 3 • 3 35
“
base number
Therefore, 3 • 3 • 3 • 3 • 3 can be written as 35.
Practice
Directions: For Numbers 1 through 5, write each expression in exponential
notation.
2. 11 • 11 ____________
3. 2 • 2 • 2 • 2 • 2 • 2 • 2 ____________
4. 3 • 3 • 3 • 3 ____________
5. 7 • 7 • 7 ____________
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1. 5 • 5 • 5 • 5 • 5 ____________
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Lesson 1: Number Representation
Primes and Composites
A prime number has only two factors: 1 and the number. A composite
number has at least three factors. Remember, 0 and 1 are neither prime
nor composite numbers.
Examples
The number 3 has only two factors: 1 and 3. Therefore, 3 is a
prime number.
The number 4 has three factors: 1, 2, and 4. Therefore, 4 is a
composite number.
The number 6 has four factors: 1, 2, 3, and 6. Therefore, 6 is a
composite number.
Practice
1. Is 8 a prime number or a composite number? _____________________________
2. Is 11 a prime number or a composite number? ____________________________
3. Is 15 a prime number or a composite number? ____________________________
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4. List all the prime numbers between 20 and 30.
5. List all the composite numbers between 20 and 30.
6. Which is a prime number?
7. Which is a composite number?
A 37
A 43
B 45
B 59
C 51
C 61
D 63
D 77
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Unit 1 – Number and Operations
Prime Factorization
Prime factorization is a way of expressing a composite number as a product
of prime numbers. You can use a factor tree to determine the prime
factorization of a composite number.
Example
What is the prime factorization of 504?
Write the number 504. Write a prime factor under the left branch and
circle it. Write the nonprime factor under the right branch. Repeat
this process under each composite number until you have two prime
numbers at the bottom of the tree. The prime factorization is the
product of all the circled numbers.
504
2
252
126
2
2
63
21
3
3
7
TIP: The order in which you find the prime factors doesn’t matter. In the
first step of this example, you could have divided by 3 or 7 instead of by 2.
When you list the prime factors in your answer, list them in order from
least to greatest. It might help if you always divide by the smallest prime
number, but it is not necessary.
12
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The prime factorization of 504 is 2 • 2 • 2 • 3 • 3 • 7 or 23 • 32 • 7.
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Lesson 1: Number Representation
Practice
1. Draw a factor tree for 45.
3. Draw a factor tree for 1,260.
The prime factorization of 45 is
The prime factorization of 1,260 is
_____________________________.
_____________________________.
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2. Draw a factor tree for 120.
4. Draw a factor tree for 800.
The prime factorization of 120 is
The prime factorization of 800 is
______________________________.
_____________________________.
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Unit 1 – Number and Operations
Scientific Notation
Scientific notation is a way to write very large and very small numbers. It is
written as a number between 1 and 10 multiplied by a 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
The following example shows how to write both a very large number and a very
small number in scientific notation.
Examples
Write 5,860,000 in
scientific notation.
Write 0.0000000029 in
scientific notation.
5.860000.
0.000000002.9
5.86
2.9
Step 2: Count the number of places you moved the decimal point, and
use that number as the power of 10. If you moved it to the left,
it’s a positive exponent; to the right, it’s negative.
The decimal point was
moved 6 places to the left.
The decimal point was
moved 9 places to the right.
109
106
Step 3: Write an expression with the decimal number (from Step 1)
times the power of 10 (from Step 2).
2.9 • 109
5.86 • 106
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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.
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Lesson 1: Number Representation
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. If the power of 10 is
a negative number, move the decimal point to the left. The exponent tells you
the number of places to move the decimal point. Remember to add zeros as
placeholders when necessary.
Examples
9.473 • 107 9.4730000. 94,730,000
4.625 • 105 0.00004.625 0.00004625
Practice
Directions: For Numbers 1 through 4, write the number in scientific notation.
1. 5,800,000 _________________________
2. 960,000 _________________________
3. 245,000,000 _________________________
4. 18,200,000 _________________________
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Directions: For Numbers 5 through 8, write the number in standard form.
5. 9 • 106 _________________________
6. 1.8 • 108 _________________________
7. 2.04 • 105 _________________________
8. 4.55 • 107 _________________________
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Unit 1 – Number and Operations
EOG Practice
5. What is the prime factorization
of 350?
1. Which shows the numbers in
order from greatest to least?
A 5, 8, 12, 25, 32
A 2 • 52 • 7
B 8, 5, 12, 25, 32
B 22 • 3 • 9
C 32, 25, 12, 8, 5
C 2 • 5 • 35
D 32, 25, 12, 5, 8
D 5 • 7 • 10
2. Which number is the opposite
of 23?
6. How is 0.00012 written in
scientific notation?
A 32
A 12 104
B 23
B 1.2 104
C 23
C 12 104
D 32
D 1.2 104
7. What is the greatest common
factor of 32 and 56?
A 41 106
A
4
B 4.1 106
B
8
C 4.1 106
C
16
D 4.1 105
D 224
4. What is the least common
multiple of 5 and 40?
8. How is 2 • 2 • 2 • 2 • 2 • 2
written in exponential notation?
A
5
A 25
B
20
B 26
C
40
C 27
D 120
D 62
16
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3. What is 4,100,000 written in
scientific notation?