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Unformatted text preview: m707_c01_002_003 1/19/06 8:26 PM Page 2 Algebraic Reasoning 1A Patterns and Relationships 1-1 1-2 1-3 1-4 LAB Numbers and Patterns Exponents Metric Measurements Applying Exponents Scientific Notation with a Calculator Order of Operations Explore Order of Operations Properties 1-5 LAB 1-6 1B Algebraic Thinking 1-7 Variables and Algebraic Expressions 1-8 Translate Words into Math 1-9 Simplifying Algebraic Expressions 1-10 Equations and Their Solutions LAB Model Solving Equations 1-11 Addition and Subtraction Equations 1-12 Multiplication and Division Equations KEYWORD: MS7 Ch1 2 Chapter 1 Astronomical Distances Object Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Nearest star Distance from the Sun (km)* 5.80  10 7 1.082  10 8 1.495  10 8 2.279  10 8 7.780  10 8 1.43  10 9 2.90  10 9 4.40  10 9 5.80  10 9 3.973  10 13 *Distances of planets from the Sun are average distances. Cosmologist Dr. Stephen Hawking is a cosmologist. Cosmologists study the universe as a whole. They are interested in the origins, the structure, and the interaction of space and time. The invention of the telescope has extended the vision of scientists far beyond nearby stars and planets. It has enabled them to view distant galaxies and structures that at one time were only theorized by astrophysicists such as Dr. Hawking. Astronomical distances are so great that we use scientific notation to represent them. m707_c01_002_003 1/19/06 8:27 PM Page 3 Vocabulary Choose the best term from the list to complete each sentence. 1. The operation that gives the quotient of two numbers is __?__. division multiplication 2. The __?__ of the digit 3 in 4,903,672 is thousands. place value 3. The operation that gives the product of two numbers is __?__. product quotient 4. In the equation 15  3  5, the __?__ is 5. Complete these exercises to review skills you will need for this chapter. Find Place Value Give the place value of the digit 4 in each number. 5. 4,092 9. 3,408,289 6. 608,241 10. 34,506,123 7. 7,040,000 8. 4,556,890,100 11. 500,986,402 12. 3,540,277,009 Use Repeated Multiplication Find each product. 13. 2  2  2 14. 9  9  9  9 15. 14  14  14 16. 10  10  10  10 17. 3  3  5  5 18. 2  2  5  7 19. 3  3  11  11 20. 5  10  10  10 Division Facts Find each quotient. 21. 49  7 22. 54  9 23. 96  12 24. 88  8 25. 42  6 26. 65  5 27. 39  3 28. 121  11 Whole Number Operations Add, subtract, multiply, or divide. 29. 425  12  30. 619  254  31. 33. 62  42  34. 122  15  35. 76 2 3  62  47  32. 373  86  36. 241 4 9  Algebraic Reasoning 3 m707_c01_004_004 12/29/05 11:37 AM Page 4 Previously, you algebraic expression expresión algebraica Associative Property propiedad asociativa Commutative Property used multiplication and division to solve problems involving whole numbers. propiedad conmutativa Distributive Property propiedad distributiva equation ecuación • converted measures within the same measurement system. exponent exponente • wrote large numbers in standard form. numerical expression expresión numérica order of operations orden de las operaciones term término variable variable • Study Guide: Preview • used order of operations to simplify whole number expressions without exponents. You will study • simplifying numerical expressions involving order of operations and exponents. • using concrete models to solve equations. • finding solutions to application problems involving related measurement units. • writing large numbers in scientific notation. You can use the skills learned in this chapter 4 Key Vocabulary/Vocabulario • to express distances and sizes of objects in scientific fields such as astronomy and biology. • to solve problems in math and science classes such as Algebra and Physics. Chapter 1 Algebraic Reasoning Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The words equation, equal, and equator all begin with the Latin root equa-, meaning “level.” How can the Latin root word help you define equation ? 2. The word numerical means “of numbers.” How might a numerical expression differ from an expression such as “the sum of two and five”? 3. When something is variable, it has the ability to change. In mathematics, a variable is an algebraic symbol. What special property do you think this type of symbol has? m707_c01_005_005 12/29/05 11:37 AM Page 5 Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. As you read through an example problem, pay attention to the margin notes , such as Helpful Hints, Reading Math notes, and Caution notes. These notes will help you understand concepts and avoid common mistakes. The index is located at the end of your textbook. Use it to find the page where a particular concept is taught. Reading and Writing Math The glossary is found in the back of your textbook. Use it to find definitions and examples of unfamiliar words or properties. The Skills Bank is found in the back of your textbook. These pages review concepts from previous math courses. Try This Use your textbook for the following problems. 1. Use the index to find the page where exponent is defined. 2. In Lesson 1-9, what does the Remember box, located in the margin of page 43, remind you about the perimeter of a figure? 3. Use the glossary to find the definition of each term: order of operations, numerical expression, equation. 4. Where can you review how to multiply whole numbers? Algebraic Reasoning 5 m707_c01_006_009 12/29/05 11:37 AM 1-1 Learn to identify and extend patterns. Page 6 Numbers and Patterns Each year, football teams battle for the state championship. The table shows the number of teams in each round of a division’s football playoffs. You can look for a pattern to find out how many teams are in rounds 5 and 6. Football Playoffs EXAMPLE 1 Round 1 2 3 4 Number of Teams 64 32 16 8 5 6 Identifying and Extending Number Patterns Identify a possible pattern. Use the pattern to write the next three numbers. 64, 32, 16, 8, , 64 , ,... 32 16 8 2 2 2 2 2 2 A pattern is to divide each number by 2 to get the next number. 824 422 221 The next three numbers will be 4, 2, and 1. 51, 44, 37, 30, , 51 , 44 ,... 37 30 7 7 7 7 7 7 A pattern is to subtract 7 from each number to get the next number. 30  7  23 23  7  16 16  7  9 The next three numbers will be 23, 16, and 9. 2, 3, 5, 8, 12, 2 , , 3 ,... 5 8 12 1 2 3 4 5 6 7 A pattern is to add one more than you did the time before. 12  5  17 17  6  23 23  7  30 The next three numbers will be 17, 23, and 30. 6 Chapter 1 Algebraic Reasoning m707_c01_006_009 12/29/05 11:37 AM EXAMPLE Page 7 2 Identifying and Extending Geometric Patterns Identify a possible pattern. Use the pattern to draw the next three figures. The pattern is alternating squares and circles with triangles between them. The next three figures will be . The pattern is to shade every other triangle in a clockwise direction. The next three figures will be EXAMPLE 3 . Using Tables to Identify and Extend Patterns Make a table that shows the number of triangles in each figure. Then tell how Figure 1 Figure 2 Figure 3 many triangles are in the fifth figure of the pattern. Use drawings to justify your answer. The table shows the number of triangles in each figure. Figure 1 2 3 4 5 Number of Triangles 2 4 6 8 10 2 2 2 Figure 4 has 6  2  8 triangles. Figure 4 The pattern is to add 2 triangles each time. 2 Figure 5 has 8  2  10 triangles. Figure 5 Think and Discuss 1. Describe two different number patterns that begin with 3, 6, . . . 2. Tell when it would be useful to make a table to help you identify and extend a pattern. 1-1 Numbers and Patterns 7 m707_c01_006_009 12/29/05 1-1 11:37 AM Page 8 Exercises KEYWORD: MS7 1-1 KEYWORD: MS7 Parent GUIDED PRACTICE See Example 1 Identify a possible pattern. Use the pattern to write the next three numbers. 1. 6, 14, 22, 30, 3. 59, 50, 41, 32, See Example 2 , , , 2. 1, 3, 9, 27, ,... , 4. 8, 9, 11, 14, ,... , , ,... , ,... Identify a possible pattern. Use the pattern to draw the next three figures. 5. See Example 3 , 6. 7. Make a table that shows the number of green triangles in each figure. Then tell how many green triangles are in the fifth figure of the pattern. Use drawings to justify your answer. Figure 1 Figure 2 Figure 3 INDEPENDENT PRACTICE See Example 1 See Example 2 Identify a possible pattern. Use the pattern to write the next three numbers. 8. 27, 24, 21, 18, , , ,... 10. 1, 3, 7, 13, 21, , , ,... 11. 14, 37, 60, 83, , , , , ,... 13. 14. Make a table that shows the number of dots in each figure. Then tell how many dots are in the sixth figure of the pattern. Use drawings to justify your answer. Figure 1 Figure 2 Figure 3 Figure 4 PRACTICE AND PROBLEM SOLVING Extra Practice See page 724. Use the rule to write the first five numbers in each pattern. 15. Start with 7; add 16 to each number to get the next number. 16. Start with 96; divide each number by 2 to get the next number. 17. Start with 50; subtract 2, then 4, then 6, and so on, to get the next number. 18. Critical Thinking Suppose the pattern 3, 6, 9, 12, 15 . . . is continued forever. Will the number 100 appear in the pattern? Why or why not? 8 ,... Identify a possible pattern. Use the pattern to draw the next three figures. 12. See Example 3 9. 4,096, 1,024, 256, 64, Chapter 1 Algebraic Reasoning m707_c01_006_009 12/29/05 11:37 AM Page 9 Identify a possible pattern. Use the pattern to find the missing numbers. 19. 3, 12, 21. , , 192, 768, , , 19, 27, 35, 20. 61, 55, , ... 22. 2, , 51, . . . , 43, , 8, , 32, 64, 23. Health The table shows the target heart rate during exercise for athletes of different ages. Assuming the pattern continues, what is the target heart rate for a 40-year-old athlete? a 65-year-old athlete? 25. 1 , 4 , , 5 13 9 5 , 7 , , 21 17 , 10 , 14 , , 19 , , . . . 25 , 25, . . . , ... Target Heart Rate Draw the next three figures in each pattern. 24. , Age Heart Rate (beats per minute) 20 150 25 146 30 142 35 138 , . . . 26. Social Studies In the ancient Mayan civilization, people used a number system based on bars and dots. Several numbers are shown below. Look for a pattern and write the number 18 in the Mayan system. 3 5 8 10 13 15 27. What’s the Error? A student was asked to write the next three numbers in the pattern 96, 48, 24, 12, . . . . The student’s response was 6, 2, 1. Describe and correct the student’s error. 28. Write About It A school chess club meets every Tuesday during the month of March. March 1 falls on a Sunday. Explain how to use a number pattern to find all the dates when the club meets. 29. Challenge Find the 83rd number in the pattern 5, 10, 15, 20, 25, . . . . 30. Multiple Choice Which is the missing number in the pattern 2, 6, , 54, 162, . . . ? A 10 B 18 C 30 D 48 31. Gridded Response Find the next number in the pattern 9, 11, 15, 21, 29, 39, . . . . Round each number to the nearest ten. (Previous course) 32. 61 33. 88 34. 105 35. 2,019 36. 11,403 Round each number to the nearest hundred. (Previous course) 37. 91 38. 543 39. 952 40. 4,050 41. 23,093 1-1 Numbers and Patterns 9 m707_c01_010_013 12/29/05 11:38 AM 1-2 Learn to represent numbers by using exponents. Vocabulary power exponent Page 10 Exponents A DNA molecule makes a copy of itself by splitting in half. Each half becomes a molecule that is identical to the original. The molecules continue to split so that the two become four, the four become eight, and so on. Each time DNA copies itself, the number of molecules doubles. After four copies, the number of molecules is 2  2  2  2  16. base This multiplication can also be written as a power , using a base and an exponent. The exponent tells how many times to use the base as a factor. Read 24 as “the fourth power of 2” or “2 to the fourth power.” The structure of DNA can be compared to a twisted ladder. Exponent Base EXAMPLE 1 Evaluating Powers Find each value. 52 52  5  5  25 Use 5 as a factor 2 times. 26 26  2  2  2  2  2  2  64 Use 2 as a factor 6 times. 251 251  25 Any number to the first power is equal to that number. Any number to the zero power, except zero, is equal to 1. 60  1 100  1 190  1 Zero to the zero power is undefined, meaning that it does not exist. 10 Chapter 1 Algebraic Reasoning m707_c01_010_013 1/7/06 4:09 PM Page 11 To express a whole number as a power, write the number as the product of equal factors. Then write the product using the base and an exponent. For example, 10,000  10  10  10  10  104. EXAMPLE 2 Expressing Whole Numbers as Powers Write each number using an exponent and the given base. EXAMPLE Earth Science 3 49, base 7 49  7  7  72 7 is used as a factor 2 times. 81, base 3 81  3  3  3  3  34 3 is used as a factor 4 times. Earth Science Application The Richter scale measures an earthquake’s strength, or magnitude. Each category in the table is 10 times stronger than the next lower category. For example, a large earthquake is 10 times stronger than a moderate earthquake. How many times stronger is a great earthquake than a moderate one? Earthquake Strength Category Magnitude Moderate 5 Large 6 Major 7 Great 8 An earthquake with a magnitude of 6 is 10 times stronger than one with a magnitude of 5. An earthquake measuring 7.2 on the Richter scale struck Duzce, Turkey, on November 12, 1999. An earthquake with a magnitude of 7 is 10 times stronger than one with a magnitude of 6. An earthquake with a magnitude of 8 is 10 times stronger than one with a magnitude of 7. 10  10  10  103  1,000 A great earthquake is 1,000 times stronger than a moderate one. Think and Discuss 1. Describe a relationship between 35 and 36. 2. Tell which power of 8 is equal to 26. Explain. 3. Explain why any number to the first power is equal to that number. 1-2 Exponents 11 m707_c01_010_013 12/29/05 1-2 11:38 AM Page 12 Exercises KEYWORD: MS7 1-2 KEYWORD: MS7 Parent GUIDED PRACTICE See Example 1 Find each value. 1. 25 See Example 2 2. 33 4. 91 5. 106 Write each number using an exponent and the given base. 6. 25, base 5 See Example 3 3. 62 7. 16, base 4 8. 27, base 3 9. 100, base 10 10. Earth Science On the Richter scale, a great earthquake is 10 times stronger than a major one, and a major one is 10 times stronger than a large one. How many times stronger is a great earthquake than a large one? INDEPENDENT PRACTICE See Example 1 See Example 2 See Example 3 Find each value. 11. 112 12. 35 13. 83 14. 43 15. 34 16. 25 17. 51 18. 23 19. 53 20. 301 Write each number using an exponent and the given base. 21. 81, base 9 22. 4, base 4 23. 64, base 4 24. 1, base 7 25. 32, base 2 26. 128, base 2 27. 1,600, base 40 28. 2,500, base 50 29. 100,000, base 10 30. In a game, a contestant had a starting score of one point. He tripled his score every turn for four turns. Write his score after four turns as a power. Then find his score. PRACTICE AND PROBLEM SOLVING Extra Practice See page 724. Give two ways to represent each number using powers. 31. 81 32. 16 33. 64 34. 729 35. 625 Compare. Write , , or . 36. 42 15 40. 10,000 105 37. 23 32 38. 64 43 39. 83 74 41. 65 3,000 42. 93 36 43. 54 73 44. To find the volume of a cube, find the third power of the length of an edge of the cube. What is the volume of a cube that is 6 inches long on an edge? 45. Patterns Domingo decided to save $0.03 the first day and to triple the amount he saves each day. How much will he save on the seventh day? 46. Life Science A newborn panda cub weighs an average of 4 ounces. How many ounces might a one-year-old panda weigh if its weight increases by the power of 5 in one year? 12 Chapter 1 Algebraic Reasoning m707_c01_010_013 12/29/05 11:38 AM Page 13 47. Social Studies If the populations of the cities in the table double every 10 years, what will their populations be in 2034? 48. Critical Thinking Explain why 63  36. City Population (2004) Yuma, AZ 86,070 Phoenix, AZ 1,421,298 49. Hobbies Malia is making a quilt with a pattern of rings. In the center ring, she uses four stars. In each of the next three rings, she uses three times as many stars as in the one before. How many stars does she use in the fourth ring? Write the answer using a power and find its value. Order each set of numbers from least to greatest. 50. 29, 23, 62, 16, 35 51. 43, 33, 62, 53, 101 52. 72, 24, 80, 102, 18 53. 2, 18, 34, 161, 0 54. 52, 21, 112, 131, 19 55. 25, 33, 9, 52, 81 56. Life Science The cells of some kinds of bacteria divide every 30 minutes. If you begin with a single cell, how many cells will there be after 1 hour? 2 hours? 3 hours? 57. What’s the Error? A student wrote 64 as 8  2. How did the student apply exponents incorrectly? 58. Write About It Is 25 greater than or less than 33? Explain your answer. 59. Challenge What is the length of the edge of a cube if its volume is 1,000 cubic meters? Bacteria divide by pinching in two. This process is called binary fission. 60. Multiple Choice What is the value of 46? A 24 B 1,024 C 4,096 D 16,384 J 82 61. Multiple Choice Which of the following is NOT equal to 64? F 64 G 43 H 26 62. Gridded Response Simplify 23  32. Simplify. (Previous course) 63. 15  27  5  3  11  16  7  4 64. 2  6  5  7  100  1  75 65. 2  9  8  12  6  8  5  6  7 66. 9  30  4  1  4  1  7  5 Identify a possible pattern. Use the pattern to write the next three numbers. (Lesson 1-1) 67. 100, 91, 82, 73, 64, . . . 68. 17, 19, 22, 26, 31, . . . 69. 2, 6, 18, 54, 162, . . . 1-2 Exponents 13 m707_c01_014_017 12/29/05 11:38 AM 1-3 Learn to identify, convert, and compare metric units. Page 14 Metric Measurements The Micro Flying Robot II is the world’s lightest helicopter. Produced in Japan in 2004, the robot is 85 millimeters tall and has a mass of 8.6 grams. You can use the following benchmarks to help you understand millimeters, grams, and other metric units. Metric Unit Length Mass Capacity EXAMPLE 1 Benchmark Millimeter (mm) Thickness of a dime Centimeter (cm) Width of your little finger Meter (m) Width of a doorway Kilometer (km) Length of 10 football fields Milligram (mg) Mass of a grain of sand Gram (g) Mass of a small paperclip Kilogram (kg) Mass of a textbook Milliliter (mL) Amount of liquid in an eyedropper Liter (L) Amount of water in a large water bottle Kiloliter (kL) Capacity of 2 large refrigerators Choosing the Appropriate Metric Unit Choose the most appropriate metric unit for each measurement. Justify your answer. The length of a car Meters—the length of a car is similar to the width of several doorways. The mass of a skateboard Kilograms—the mass of a skateboard is similar to the mass of several textbooks. The recommended dose of a cough syrup Milliliters—one dose of cough syrup is similar to the amount of liquid in several eyedroppers. 14 Chapter 1 Algebraic Reasoning m707_c01_014_017 1/7/06 4:10 PM Page 15 Prefixes: Milli- means “thousandth” Centi- means “hundredth” Kilo- means “thousand” The prefixes of metric units correlate to place values in the base-10 number system. The table shows how metric units are based on powers of 10. 1,000 10 1 Thousands Hundreds Tens Ones Kilo- 100 Hecto- 0.1 Tenths Deca- Base unit Deci- 0.01 0.001 Hundredths Thousandths Centi- Milli- You can convert units within the metric system by multiplying or dividing by powers of 10. To convert to a smaller unit, you must multiply. To convert to a larger unit, you must divide. EXAMPLE 2 Converting Metric Units Convert each measure. 510 cm to meters 510 cm  (510  100) m 100 cm  1 m, so divide by 100.  5.1 m Move the decimal point 2 places left: 510. 2.3 L to milliliters 2.3 L  (2.3  1,000) mL 1 L  1,000 mL, so multiply by 1,000.  2,300 mL Move the decimal point 3 places right: 2.300 EXAMPLE 3 Using Unit Conversion to Make Comparisons Mai and Brian are ...
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