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Buckle Down Arkansas 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 Go to www.BuckleDown.com to review our complete line of Benchmark Exam materials for Grades 3–8 READING • WRITING • MATHEMATICS • SCIENCE P.O. Box 2180 Iowa City, Iowa 52244-2180 PHONE: 800-776-3454 FAX: 877-365-0111 Catalog # 1BDAR07MM01 ISBN 0-7836-4857-X 5 1 2 9 5 www.BuckleDown.com EMAIL: [email protected] 9 780783 648576 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 1BDAR07MM01 FM 5/31/06 12:22 PM 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 iii 1BDAR07MM01 FM 5/31/06 12:22 PM Page iv 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. ACTAAP Coverage: DAP.17.7.1, DAP.17.7.2 1BDAR07MM01 L01 5/31/06 1:02 PM 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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. 1BDAR07MM01 L01 5/31/06 1:02 PM Page 5 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 _________________________ © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 1BDAR07MM01 L01 5/31/06 1:02 PM Page 6 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Directions: For Numbers 1 through 4, determine each value. 1BDAR07MM01 L01 5/31/06 1:02 PM Page 7 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. © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 1BDAR07MM01 L01 5/31/06 1:02 PM Page 8 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 1BDAR07MM01 L01 5/31/06 1:02 PM Page 9 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: ______________________________ © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 1BDAR07MM01 L01 5/31/06 1:02 PM Page 10 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. (3)2 (3) • (3) 9 1BDAR07MM01 L01 5/31/06 1:02 PM Page 11 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. © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 6. 625 ____________________ 7. 27 ____________________ 8. 49 ____________________ 9. 256 ____________________ 10. 1,000 ____________________ 11 1BDAR07MM01 L01 5/31/06 1:02 PM 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Directions: For Numbers 1 through 8, evaluate each square root. 1BDAR07MM01 L01 5/31/06 1:02 PM Page 13 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 1BDAR07MM01 L01 5/31/06 1:02 PM Page 14 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 4. 830,000,000 _________________________ 1BDAR07MM01 L01 5/31/06 1:02 PM Page 15 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. © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 1BDAR07MM01 L01 5/31/06 1:02 PM Page 16 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 © 2007 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 9.