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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