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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 BD NC7M ST FM 6/23/05 2:32 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 iii BD NC7M ST L01 6/23/05 2:35 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Directions: For Numbers 1 through 4, determine each value. BD NC7M ST L01 6/23/05 2:35 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 __________ 5 BD NC7M ST L01 6/23/05 2:35 PM 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 6 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Step 3: Multiply the decimal (in Step 1) by the power of 10 (in Step 2). BD NC7M ST L01 6/23/05 2:35 PM 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 _________________________ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 _________________________ 7 BD NC7M ST L01 6/23/05 2:35 PM 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. 8 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. The first ten multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. BD NC7M ST L01 6/23/05 2:35 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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. 9 BD NC7M ST L01 6/23/05 2:35 PM Page 10 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? ________ 10 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 9. Factors of 27: ______________________________ BD NC7M ST L01 6/23/05 2:35 PM Page 11 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. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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. 11 BD NC7M ST L01 6/23/05 2:35 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 1. Draw a factor tree for 45. BD NC7M ST L01 6/23/05 2:35 PM Page 13 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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. 13 BD NC7M ST L01 6/23/05 2:35 PM 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? ____________ 14 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 2. Draw a factor tree for 420. BD NC7M ST L01 6/23/05 2:35 PM Page 15 Lesson 1: Number Concepts and Representation 5. Draw a factor tree for 225. The prime factorization of 225 is ______________________________. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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? ____________ 15 BD NC7M ST L01 6/23/05 2:35 PM 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Ones BD NC7M ST L01 6/23/05 2:35 PM Page 17 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). © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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 BD NC7M ST L01 6/24/05 3:52 PM Page 18 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Write BD NC7M ST L01 6/28/05 4:14 PM Page 19 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? _______________ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 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? BD NC7M ST L01 6/28/05 4:14 PM Page 20 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 20 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. C 8 BD NC7M ST L01 6/23/05 2:35 PM Page 21 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 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Ellen collected 1 8 of the total. Who collected the greatest number of cans? A Robert B Steven C Marie D Ellen 21