Download Lesson 1 - Triumph Learning

Buckle Down South Carolina
PASS 6 Mathematics
Number and Operation
Lesson 1: Number Sense
Lesson 2: Fractions and Mixed Numbers
Lesson 3: Decimals
Lesson 4: Ratio, Proportion, and Rate
Lesson 5: Number Relationships
Unit 2
Algebra
Lesson 6: Expressions, Equations, and Inequalities
Lesson 7: Patterns
Unit 3
Geometry
Lesson 8: Geometric Concepts
Unit 4
Measurement
Lesson 9: Perimeter, Area, and Surface Area
Unit 5
Data Analysis and Probability
Lesson 10: Data Analysis
Lesson 11: Probability
PASS
6
Mathematics
6 MATHEMATICS
Go to www.BuckleDown.com to review our complete line of PASS materials for grades 3–8
ENGLISH LANGUAGE ARTS • MATHEMATICS
South Carolina
South Carolina PASS
The cover image depicts rulers.
These important measurement tools
help students learn about length in
both inches and centimeters
through hands-on activities in realworld situations.
Unit 1
Student Set SC02064S2
P.O. Box 1270
Littleton, MA 01460-4270
Includes: Student Workbook, Form A
Practice Test, Form B Practice Test
PHONE: 800-776-3454
FAX: 877-365-0111
Individual Products:
www.BuckleDown.com
2BDSC06MM01.indd 1
Student Workbook SC02064W2
Form A Practice Test SC02064A2
Form B Practice Test SC02064B2
11/9/09 4:21:44 PM
TABLE OF CONTENTS
Introduction ........................................................................................ 1
Testwise Strategies ................................................................ 2
Unit 1 – Number and Operations ...................................................... 3
Lesson 1: Number Sense ...................................................... 4
Indicators: 6-2.2, 6-2.3, 6-2.7, 6-2.8, 6-2.9, 6-3.2
Lesson 2: Fractions and Mixed Numbers ............................ 17
Indicators: 6-2.3, 6-2.4, 6-2.5
Lesson 3: Decimals .............................................................. 31
Indicators: 6-2.3, 6-2.5
Lesson 4: Ratio, Proportion, and Rate ................................. 36
Indicators: 6-2.6, 6-5.6, 6-5.7
Lesson 5: Number Relationships ......................................... 49
Indicators: 6-2.1, 6-2.3
Unit 2 – Algebra ............................................................................... 59
Lesson 6: Expressions, Equations, and Inequalities ............ 60
Indicators: 6-3.3, 6-3.4, 6-3.5
Lesson 7: Patterns ............................................................... 78
Indicator: 6-3.1
Unit 3 – Geometry ............................................................................ 91
Lesson 8: Geometric Concepts ............................................ 92
Indicators: 6-4.1, 6-4.2, 6-4.3, 6-4.4, 6-4.5, 6-4.6, 6-4.7,
6-4.8, 6-4.9
Unit 4 – Measurement .................................................................... 113
Duplicating any part of this book is prohibited by law.
Lesson 9: Perimeter, Area, and Surface Area ................... 114
Indicators: 6-5.1, 6-5.2, 6-5.3, 6-5.4, 6-5.5
Unit 5 – Data Analysis and Probability ....................................... 131
Lesson 10: Data Analysis .................................................. 132
Indicators: 6-6.1, 6-6.2, 6-6.3
Lesson 11: Probability ........................................................ 149
Indicators: 6-6.4, 6-6.5
iii
2BDSC06MM01_FM_i-iv.indd iii
10/27/09 3:12:17 PM
P rinter P DF
Unit 1 – Number and Operations
Indicators: 6-2.2, 6-2.3
Lesson 1: Number Sense
In this lesson, you will compare and order integers. You will review the rules of
exponents. You will also classify whole numbers as prime or composite and then
use this in prime factorization. Finally, you will use the order of operations to
simplify expressions.
Integers
Integers are whole numbers and their opposites (positive numbers, zero, and negative
numbers). Negative numbers are the numbers less than zero. 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. For example, 4 and ⫺4 are opposites.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
neither positive
nor negative
Integers in Real-Life Situations
Integers can be used in real-life situations. Some keywords that indicate positive
integers are gained, increased, rose, above, more, and up. Some keywords that indicate
negative integers are lost, decreased, dropped, below, less, and down.
Example
What integer is represented in the following sentence?
Between 2:00 P.M. and 10:00 P.M., the temperature dropped 12 degrees.
Practice
Directions: For Numbers 1 through 3, write the integer that is represented in
each sentence.
1. The temperature decreased 18°F from 1:00 P.M. to 7:00 P.M. __________
2. Erik’s dog gained 6 pounds. __________
3. Katie’s checking account dropped $127. _________
Duplicating any part of this book is prohibited by law.
The integer that represents dropped 12 degrees is ⴚ12.
4
2BDSC06MM01_L01_03-16.indd 4
10/27/09 3:13:29 PM
P rinter P DF
Lesson 1: Number Sense
Indicators: 6-2.2, 6-2.3
Comparing and Ordering Integers
Integers can be compared using a number line. As you move from left to right on the
number line, the numbers are ordered from least to greatest. The number line on page 4
shows that ⫺5 ⬍ ⫺2 because ⫺5 is to the left of ⫺2.
Example
Compare the following integers using ⬍ or ⬎.
12 _______ ⫺13
⫺35
_______ 25
⫺13
12 _______ 25
_______ ⫺35
First, find the integers on a number line.
⫺35
⫺40
⫺13
⫺30
⫺20
12
⫺10
0
10
25
20
30
40
Then, compare the integers by looking at where they are located in relation to
each other on the number line.
12 ⬎ ⫺13
⫺35
⬍ 25
12 ⬍ 25
⫺13
⬎ ⫺35
Practice
Directions: For Numbers 1 through 6, refer to the number line and use ⬍, ⬎, or ⫽ to
compare the integers.
Duplicating any part of this book is prohibited by law.
⫺30
⫺25
⫺20
⫺15
⫺10
⫺5
0
5
10
15
20
1. ⫺12 _______ 8
4. ⫺2 _______ 2
2. ⫺4 _______ ⫺16
5. ⫺18 _______ ⫺13
3. ⫺30 _______ ⫺30
6. 27 _______ 9
25
30
7. Write the following integers in order from least to greatest.
⫺25,
25, 7, 0, ⫺10, ⫺16, 5, 1, ⫺2, ⫺1, 4, ⫺5
8. Is the following list in order from greatest to least? _____________________________
⫺10, ⫺9, ⫺8, ⫺6, ⫺5
5
2BDSC06MM01_L01_03-16.indd 5
10/27/09 3:13:30 PM
P rinter P DF
Unit 1 – Number and Operations
Indicators: 6-2.7, 6-2.9
Exponents
An exponent shows how many times a base number occurs as a factor. Using a base
number and an exponent is sometimes called repeated multiplication.
When working with exponents, remember the following rules:
1. Any base number (except zero) with zero as the exponent equals 1.
100 ⫽ 1
2. Any base number with 1 as the exponent equals the base number.
101 ⫽ 10
Example
What is the value of 105?
The exponent (5) shows that the base number (10) occurs as a factor
5 times.
exponent
‘
105 ⫽ 10 • 10 • 10 • 10 • 10 ⫽ 100,000
“
base number
Therefore, 105 ⫽ 100,000.
Notice that when the base number is 10, the number of zeros in the value is
equal to the exponent.
Example
Write 8 in exponential form.
In exponential form, 8 is written as 23.
TIP: Exponents are called “powers.” When you read them, 64 is read “six to
the fourth power.” An exponent of 2 can be read as “to the second power ” or
“squared,” and an exponent of 3 can be read as “to the third power” or “cubed.”
Duplicating any part of this book is prohibited by law.
8 ⫽ 2 • 2 • 2 ⫽ 23
6
2BDSC06MM01_L01_03-16.indd 6
10/27/09 3:13:30 PM
P rinter P DF
Lesson 1: Number Sense
Indicators: 6-2.7, 6-2.9
Practice
Directions: For Numbers 1 through 7, write the expression as the factors of the base
number and then evaluate.
1. 103 ⫽ ______________________________ ⫽ ____________
2. 100 ⫽ ______________________________ ⫽ ____________
3. 102 ⫽ ______________________________ ⫽ ____________
4. 104 ⫽ ______________________________ ⫽ ____________
5. 101 ⫽ ______________________________ ⫽ ____________
6. 106 ⫽ ______________________________ ⫽ ____________
7. 105 ⫽ ______________________________ ⫽ ____________
Directions: For Numbers 8 through 12, write each number in exponential form.
8. 32 ⫽ ______________________________
Duplicating any part of this book is prohibited by law.
9. 49 ⫽ ______________________________
10. 125 ⫽ ______________________________
11. 64 ⫽ ______________________________
12. 1,000 ⫽ ______________________________
7
2BDSC06MM01_L01_03-16.indd 7
10/27/09 3:13:30 PM
P rinter P DF
Unit 1 – Number and Operations
Indicators: 6-2.8, 6-2.9
Prime and Composite Numbers
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.
Example
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? __________________________________
4. List all the prime 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
Duplicating any part of this book is prohibited by law.
5. List all the composite numbers between 20 and 30.
8
2BDSC06MM01_L01_03-16.indd 8
10/27/09 3:13:30 PM
P rinter P DF
Lesson 1: Number Sense
Indicators: 6-2.8, 6-2.9
Prime Factorization
Prime factorization is a way of expressing a composite number as the product of
prime numbers. The fundamental theorem of arithmetic states that every counting
number is either prime or can be decomposed (broken down) into its prime factorization.
You can use a factor tree to find the prime factorization of a 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
63
2
21
3
3
7
Duplicating any part of this book is prohibited by law.
The prime factorization of 504 is 2 • 2 • 2 • 3 • 3 • 7 or 23 • 32 • 7.
Note: There is more than one way to make a factor tree. In the first step of
this example, you could have divided by 3 or 7 instead of by 2. The order in
which you find the prime factors does not matter. However, when you list the
prime factors in your answer, list them in order from least to greatest.
9
2BDSC06MM01_L01_03-16.indd 9
10/27/09 3:13:30 PM
P rinter P DF
Unit 1 – Number and Operations
Indicators: 6-2.8, 6-2.9
Practice
1. Draw a factor tree for 45.
What is the prime factorization of 45? _________________________
What is the prime factorization of 120? _________________________
Duplicating any part of this book is prohibited by law.
2. Draw a factor tree for 120.
10
2BDSC06MM01_L01_03-16.indd 10
10/27/09 3:13:31 PM
P rinter P DF
Lesson 1: Number Sense
Indicators: 6-2.8, 6-2.9
3. Draw a factor tree for 1,260.
What is the prime factorization of 1,260? _________________________
Duplicating any part of this book is prohibited by law.
4. Draw a factor tree for 800.
What is the prime factorization of 800? _________________________
11
2BDSC06MM01_L01_03-16.indd 11
10/27/09 3:13:31 PM
P rinter P DF
Unit 1 – Number and Operations
Indicators: 6-3.2
Order of Operations
When simplifying an expression, you must follow the correct order of operations.
The following example shows the steps needed to simplify an expression.
Example
Simplify the following expression.
20 ⫹ 7 • (4 ⫼ 2) ⫺ 52
Step 1: Simplify all expressions in parentheses.
(4 ⫼ 2) ⫽ 2
20 ⫹ 7 • 2 ⫺ 52
Step 2: Simplify exponents.
52 ⫽ 25
20 ⫹ 7 • 2 ⫺ 25
Step 3: Perform all multiplication and division in order from left to right.
7 • 2 ⫽ 14
20 ⫹ 14 ⫺ 25
Step 4: Perform all addition and subtraction in order from left to right.
20 ⫹ 14 ⫽ 34
34 ⫺ 25 ⫽ 9
Duplicating any part of this book is prohibited by law.
The expression simplifies to 9.
12
2BDSC06MM01_L01_03-16.indd 12
10/27/09 3:13:31 PM
P rinter P DF
Lesson 1: Number Sense
Indicators: 6-3.2
Practice
Directions: For Numbers 1 through 6, use the correct order of operations to simplify
each expression.
1. 12 ⫺ 1 • 6 ⫼ 3 • 2 ⫹ 1 ⫽ _______________
2. 12 ⫺ 1 • (6 ⫼ 3) • (2 ⫹ 1) ⫽ _______________
3. (12 • 6) ⫼ (3 • 2) ⫹ 1 ⫺ 1 ⫽ _______________
4. 12 • (6 ⫺ 1) ⫼ 3 • 22 ⫹ 1 ⫽ _______________
Duplicating any part of this book is prohibited by law.
5. 1 ⫹ 12 • 6 ⫼ 3 • 22 ⫺ 1 ⫽ _______________
6. 4 • 33 ⫼ 2 ⫽ _______________
PASS Practice begins on the following page.
2BDSC06MM01_L01_03-16.indd 13
13
10/27/09 3:13:31 PM
P rinter P DF
PASS Practice
Directions: Read each question and choose the best answer.
1.
4.
What is the prime
factorization of 60?
F.
A. 5 • 6
B.
2.
2 • 33
28
G. 45
C. 3 • 4 • 5
H. 64
D. 22 • 3 • 5
I.
What is the value of 103?
F.
5.
83
What is the value of 100?
30
A.
0
100
B.
1
H. 1,000
C.
10
I.
D. 100
G.
3.
Which of the following is
equal to 512?
3,000
What is the prime
factorization of 350?
6.
A. 2 • 52 • 7
Which of the following is a
prime number?
F.
23
22 • 3 • 9
G. 33
C. 2 • 5 • 35
H. 63
D. 5 • 7 • 10
I.
93
Duplicating any part of this book is prohibited by law.
B.
14
2BDSC06MM01_L01_03-16.indd 14
10/27/09 3:13:31 PM
P rinter P DF
7.
Which calculation should you
perform first to simplify the
following expression?
10.
4 • (⫺8 ⫺ 4) ⫼ (9 ⫹ 7)
62 ⫼ 3 • 2 ⫹ (4 ⫺ 3)
A. 3 • 2
B.
Simplify the following expression.
F.
3
G.
1
H. ⫺1
2⫹4
I.
C. 4 ⫺ 3
⫺3
D. 62 ⫼ 3
11.
8.
How many zeros are in the value
of 106?
A. 2 • 3 • 5 • 7
F.
0
B.
G.
3
C. 2 • 5 • 35
H.
6
D. 2 • 5 • 7 • 11
I.
What is 27 in exponential form?
F.
Which of the following is equal
to 100,000?
22
G. 23
A. 106
B.
22 • 3 • 7
60
12.
9.
What is the prime factorization
of 210?
H. 32
105
I.
33
Duplicating any part of this book is prohibited by law.
C. 104
D. 103
15
2BDSC06MM01_L01_03-16.indd 15
10/27/09 3:13:31 PM
P rinter P DF
13.
16.
What is the simplified form of
the expression below?
16 ⫼ 4 ⫹ 4
Which list shows the integers in
order from least to greatest?
F.
⫺20, 20, 0, 50, ⫺50
A. 8
G. 0, ⫺5, 5, 20, 100
B.
H. ⫺187, ⫺178, 0, 36, 143
4
C. 2
I.
200, 198, 73, ⫺10, ⫺44
D. 1
17.
14.
What is the simplified form of
the expression below?
Which list shows the integers in
order from greatest to least?
A. ⫺5, ⫺8, 12, 25, 32
6⫻7⫹3⫼3
B.
48
C. 32, 25, 12, ⫺8, ⫺5
G. 43
D. 32, 25, 12, ⫺5, ⫺8
F.
⫺8, ⫺5, 12, 25, 32
H. 20
I.
15.
15
18.
F.
What is the simplified form of
the expression below?
⬍
G. ⬎
60 ⫼ (5 ⫹ 1)
H. ⫽
A. 10
I.
ⱕ
12
Duplicating any part of this book is prohibited by law.
B.
Which symbol makes the
statement ⫺78 _______ ⫺91 true?
C. 13
D. 15
16
2BDSC06MM01_L01_03-16.indd 16
10/27/09 3:13:31 PM
P rinter P DF