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Real Numbers ? MODULE 1 LESSON 1.1 ESSENTIAL QUESTION Rational and Irrational Numbers How can you use real numbers to solve real-world problems? 8.NS.1, 8.NS.2, 8.EE.2 LESSON 1.2 Sets of Real Numbers 8.NS.1 LESSON 1.3 Ordering Real Numbers © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Daniel Hershman/Getty Images 8.NS.2 Real-World Video my.hrw.com my.hrw.com Living creatures can be classified into groups. The sea otter belongs to the kingdom Animalia and class Mammalia. Numbers can also be classified into groups such as rational numbers and integers. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 3 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this module. Find the Square of a Number EXAMPLE my.hrw.com Online Practice and Help Find the square of _23. 2 _ 2 × 2 _ × 23 = ____ 3 3 × 3 = _49 Multiply the number by itself. Simplify. Find the square of each number. 1. 7 2. 21 3. -3 4. _45 5. 2.7 6. _14 7. -5.7 8. 1_25 Exponents EXAMPLE 53 = 5 × 5 × 5 = 25 × 5 = 125 Use the base, 5, as a factor 3 times. Multiply from left to right. Simplify each exponential expression. 9. 92 13. 43 ( _13 ) 10. 24 11. 14. (-1)5 15. 4.52 2 12. (-7)2 16. 105 EXAMPLE 2_25 = 2 + _25 10 _ + 25 = __ 5 12 = __ 5 Write the mixed number as a sum of a whole number and a fraction. Write the whole number as an equivalent fraction with the same denominator as the fraction in the mixed number. Add the numerators. Write each mixed number as an improper fraction. 17. 3_13 4 Unit 1 18. 1_58 19. 2_37 20. 5_56 © Houghton Mifflin Harcourt Publishing Company Write a Mixed Number as an Improper Fraction Reading Start-Up Vocabulary Review Words Visualize Vocabulary integers (enteros) ✔ negative numbers (números negativos) ✔ positive numbers (números positivos) ✔ whole number (número entero) Use the ✔ words to complete the graphic. You can put more than one word in each section of the triangle. Integers 1, 45, 192 Preview Words 0, 83, 308 -21, -78, -93 Understand Vocabulary Complete the sentences using the preview words. 1. One of the two equal factors of a number is a 2. A © Houghton Mifflin Harcourt Publishing Company 3. The of a number. . has integers as its square roots. cube root (raiz cúbica) irrational numbers (número irracional) perfect cube (cubo perfecto) perfect square (cuadrado perfecto) principal square root (raíz cuadrada principal) rational number (número racional) real numbers (número real) repeating decimal (decimal periódico) square root (raíz cuadrada) terminating decimal (decimal finito) is the nonnegative square root Active Reading Layered Book Before beginning the lessons in this module, create a layered book to help you learn the concepts in this module. Label the flaps “Rational Numbers,” “Irrational Numbers,” “Square Roots,” and “Real Numbers.” As you study each lesson, write important ideas such as vocabulary, models, and sample problems under the appropriate flap. Module 1 5 GETTING READY FOR Real Numbers Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Key Vocabulary rational number (número racional) A number that can be expressed as a ratio of two integers. irrational number (número irracional) A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal. 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). Visit my.hrw.com to see all CA Common Core Standards explained. my.hrw.com 6 Unit 1 What It Means to You You will recognize a number as rational or irrational by looking at its fraction or decimal form. EXAMPLE 8.NS.1 Classify each number as rational or irrational. _ 0.3 = _13 0.25 = _41 These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals. _ √ 5 ≈ 2.236067977… π ≈ 3.141592654… These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals. What It Means to You You will learn to estimate the values of irrational numbers. EXAMPLE 8.NS.2 _ Estimate the value of √8. 8 is not a perfect square. Find the two perfect squares closest to 8. 8 is between the perfect squares 4 and 9. _ _ _ So √_8 is between √4 and √9. √ 8 is between 2 and 3. _ 8 is closer to 9, so √8 is closer to 3. 2 2.8 2.92 = 8.41 _ = 7.84 √ 8 is between 2.8 and 2.9 _ A good estimate for √8 is 2.85. © Houghton Mifflin Harcourt Publishing Company 8.NS.1 LESSON 1.1 ? Rational and Irrational Numbers ESSENTIAL QUESTION 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a relation number. Also 8.NS.2, 8.EE.2 How do you rewrite rational numbers and decimals, take square roots and cube roots, and approximate irrational numbers? Expressing Rational Numbers as Decimals A rational number is any number that can be written as a ratio in the form _ba , where a and b are integers and b is not 0. Examples of rational numbers are 6 and 0.5. 6 can be written as _61 . Math On the Spot my.hrw.com 0.5 can be written as _12 . Every rational number can be written as a terminating decimal or a repeating decimal. A terminating decimal, such as 0.5, has a finite number of digits. A repeating decimal has a block of one or more digits that repeat indefinitely. EXAMPL 1 EXAMPLE 8.NS.1 Write each fraction as a decimal. My Notes © Houghton Mifflin Harcourt Publishing Company A _14 0.25 ⎯ 4⟌ 1.00 -8 20 -20 0 1 _ = 0.25 4 Remember that the fraction bar means “divided by.” Divide the numerator by the denominator. Divide until the remainder is zero, adding zeros after the decimal point in the dividend as needed. 1 — = 0.3333333333333... 3 1 B _3 0.333 ⎯ 3⟌ 1.000 −9 10 −9 10 −9 1 _ 1 _ = 0.3 3 Divide until the remainder is zero or until the digits in the quotient begin to repeat. Add zeros after the decimal point in the dividend as needed. When a decimal has one or more digits that repeat indefinitely, write the decimal with a bar over the repeating digit(s). Lesson 1.1 7 YOUR TURN Personal Math Trainer Online Practice and Help Write each fraction as a decimal. 5 1. __ 11 2. _18 3. 2_13 my.hrw.com Expressing Decimals as Rational Numbers You can express terminating and repeating decimals as rational numbers. Math On the Spot my.hrw.com EXAMPLE 2 8.NS.1 Write each decimal as a fraction in simplest form. My Notes A 0.825 The decimal 0.825 means “825 thousandths.” Write this as a fraction. 825 ____ 1000 To write “825 thousandths”, put 825 over 1000. Then simplify the fraction. 825 ÷ 25 33 ________ = __ 1000 ÷ 25 40 Divide both the numerator and the denominator by 25. 33 0.825 = __ 40 _ B 0.37 _ _ Let x = 0.37. The number _ 0.37 has 2 repeating digits, so multiply each side of the equation x = 0.37 by 102, or 100. _ x = 0.37 _ 100x = 37.37 _ _ 100 times 0.37 is 37.37. _ _ Because x = 0.37, you can subtract x from one side and 0.37 from the other. _ 100x = 37.37 −x _ −0.37 99x = 37 _ _ 37.37 minus 0.37 is 37. Now solve the equation for x. Simplify if necessary. 99x __ ___ = 37 99 99 37 x = __ 99 8 Unit 1 Divide both sides of the equation by 99. © Houghton Mifflin Harcourt Publishing Company _ (100)x = 100(0.37) YOUR TURN Write each decimal as a fraction in simplest form. _ 4. 0.12 5. 0.57 Personal Math Trainer 6. 1.4 Online Practice and Help my.hrw.com Finding Square Roots and Cube Roots The square root of a positive number p is x if x2 = p. There are two square roots for every positive number. For example, the square roots of 36 are 1 __ _1 _1 6 and −6 because 62 = 36 and (−6)2 = 36. The square roots _ of 25 are 5 and − 5. 1 1 You can write the square roots of __ as ±_5. The symbol √5 indicates the positive, 25 or principal square root. Math On the Spot my.hrw.com A number that is a perfect square has square roots that are integers. The number 81 is a perfect square because its square roots are 9 and −9. The cube root of a positive number p is x if x3 = p. There is one cube root for every positive number. For example, the cube root of 8 is 2 because 23 = 8. 3 _ 3 1 1 _1 because _1 = __ √ is . The symbol The cube root of __ 1 indicates the 3 3 27 27 cube root. () A number that is a perfect cube has a cube root that is an integer. The number 125 is a perfect cube because its cube root is 5. EXAMPL 3 EXAMPLE 8.EE.2 Solve each equation for x. A x2 = 121 x2 = 121 Solve for x by taking the square root of both sides. © Houghton Mifflin Harcourt Publishing Company _ x = ±√ 121 Apply the definition of square root. x = ±11 Think: What numbers squared equal 121? The solutions are 11 and −11. B Math Talk Mathematical Practices Can you square an integer and get a negative number? What does this indicate about whether negative numbers have square roots? 16 x2 = ___ 169 16 x2 = ___ 169 _ 16 x = ±√ ___ 169 4 x = ±__ 13 Solve for x by taking the square root of both sides. Apply the definition of square root. 16 Think: What numbers squared equal ____ ? 169 4 4 The solutions are __ and −__ . 13 13 Lesson 1.1 9 C 729 = x3 _ _ 3 3 √ 729 = √x3 Solve for x by taking the cube root of both sides. _ 3 √ 729 = x Apply the definition of cube root. 9=x Think: What number cubed equals 729? The solution is 9. 8 x3 = ___ 125 D _ _ 3 3 8 3 ___ √ x = √ 125 Solve for x by taking the cube root of both sides. _ 8 3 ___ x = √ 125 Apply the definition of cube root. x = _25 8 Think: What number cubed equals ____ ? 125 The solution is _25. YOUR TURN Solve each equation for x. Personal Math Trainer Online Practice and Help my.hrw.com 7. x2 = 196 9 8. x2 = ___ 256 9. x3 = 512 64 10. x3 = ___ 343 EXPLORE ACTIVITY 8.NS.2, 8.EE.2 Estimating Irrational Numbers _ x2 = 2 © Houghton Mifflin Harcourt Publishing Company Irrational numbers are numbers that are not rational. In other words, they cannot be written in the form _ba , where a and b are integers and b is not 0. Square roots of perfect squares are rational numbers. Square roots of numbers that are not perfect squares are irrational. Some equations like those in Example 3 involve square roots of numbers that are not perfect squares. _ √ 2 is irrational. x = ± √2 _ Estimate the value of √2. A Find two consecutive perfect squares that 2 is between. Complete the inequality by writing these perfect squares in the boxes. B Now take the square root of each number. C Simplify the square roots of perfect squares. _ √ 2 is between 10 Unit 1 and . _ √ < 2 < < _ √2 _ < < √2 < _ √ D √2 ≈ 1.5 _ Estimate that √2 ≈ 1.5. 0 1 2 3 4 E To find a better estimate, first choose some numbers between 1 and 2 and square them. For example, choose 1.3, 1.4, and 1.5. 1.32 = _ 1.42 = 1.52 = Is √2 between 1.3 and 1.4? How do you know? _ Is √2 between 1.4 and 1.5? How do you know? 2 is closer to than to _ , so √ 2 ≈ . F Locate and label this value on the number line. 1.1 1.2 1.3 1.4 1.5 Reflect _ 11. How could you find an even better estimate of √ 2? _ © Houghton Mifflin Harcourt Publishing Company 12. Find a better estimate of √2. 1.412 = 1.422 = 2 is closer to than to 1.432 = _ , so √ 2 ≈ . Draw a number line and locate and label your estimate. 13. Solve x2 = 7. Write your answer as a radical expression. Then estimate to one decimal place. Lesson 1.1 11 Guided Practice Write each fraction or mixed number as a decimal. (Example 1) 1. _25 2. _89 3. 3 _34 7 4. __ 10 5. 2 _38 6. _56 Write each decimal as a fraction or mixed number in simplest form. (Example 2) 7. 0.675 8. 5.6 9. 0.44 _ _ 10. 0.4 _ 11. 0.26 10x = -x 12. 0.325 100x = -x - - 1000x = -x - _______________ ___________________ _______________________ x= x= x= x= x= x= Solve each equation for x. (Example 3 and Explore Activity) 25 14. x2 = ___ 289 13. x2 = 17 __ x= ± √ ≈± √ __ 15. x3 = 216 __ x= x = ± __________ = ± _____ √ 3 = 16. ? _ √5 ≈ 17. _ √3 ≈ ESSENTIAL QUESTION CHECK-IN 19. What is the difference between rational and irrational numbers? 12 Unit 1 18. _ √ 10 ≈ © Houghton Mifflin Harcourt Publishing Company Approximate each irrational number to one decimal place without a calculator. (Explore Activity) Name Class Date 1.1 Independent Practice Personal Math Trainer 8.NS.1, 8.NS.2, 8.EE.2 my.hrw.com Online Practice and Help 7 -inch-long bolt is used in a machine. 20. A __ 16 What is this length written as a decimal? 21. The weight of an object on the moon is _16 its weight on Earth. Write _61 as a decimal. 22. The distance to the nearest gas station is 2 _45 kilometers. What is this distance written as a decimal? 23. A baseball pitcher has pitched 98 _32 innings. What is the number of innings written as a decimal? 24. A heartbeat takes 0.8 second. How many seconds is this written as a fraction? 25. There are 26.2 miles in a marathon. Write the number of miles using a fraction. 26. The average score on a biology test _ was 72.1. Write the average score using a fraction. 27. The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction? © Houghton Mifflin Harcourt Publishing Company • ©Photodisc/Getty Images 28. Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting. a. If x is the length of one side of the painting, what equation can you set up to find the length of a side? How many solutions does the equation have? b. Do all of the solutions that you found make sense in the context of the problem? Explain. c. What is the length of the wood trim needed to go around the painting? Solve each equation for x. Write your answers as radical expressions. Then estimate to one decimal place, if necessary. 29. x2 = 14 30. x3 = 1331 31. x2 = 144 32. x2 = 29 Lesson 1.1 13 _ 2 2 33. Analyze Relationships To find √15, Beau _found 3 = 9 and 4 = 16. He said that since 15 is between_ 9 and 16, √15 must be between 3 and 4. He 3+4 thinks a good estimate for √15 is ____ = 3.5. Is Beau’s estimate high, low, 2 or correct? Explain. 34. Justify Reasoning What is a good estimate for the solution to the equation x3 = 95? How did you come up with your estimate? 35. The volume of a sphere is 36π ft3. What is the radius of the sphere? Use the formula V = _43 πr3 to find your answer. FOCUS ON HIGHER ORDER THINKING Work Area 37. Make a Conjecture Evaluate and compare the following expressions. _ √ _ √4 4 __ _ and ____ 25 √ 25 _ √ _ √ 16 16 __ _ and ____ 81 √ 81 _ √ _ √ 36 36 __ _ and ____ 49 √ 49 Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots. 38. Persevere in Problem Solving The difference between the solutions to the equation x2 = a is 30. What is a? Show that your answer is correct. 14 Unit 1 © Houghton Mifflin Harcourt Publishing Company • © Ilene MacDonald/Alamy Images 36. Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning. Sets of Real Numbers LESSON 1.2 ? 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a relation number. ESSENTIAL QUESTION How can you describe relationships between sets of real numbers? Classifying Real Numbers Animals Biologists classify animals based on shared characteristics. A cardinal is an animal, a vertebrate, a bird, and a passerine. Vertebrates Birds Math On the Spot Passerines You already know that the set of rational numbers consists of whole numbers, integers, and fractions. The set of real numbers consists of the set of rational numbers and the set of irrational numbers. my.hrw.com Real Numbers Rational Numbers 27 4 0.3 -2 Whole Numbers -1 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Wikimedia Commons 7 Integers -3 0 1 √4 Irrational Numbers -6 √17 Passerines, such as the cardinal, are also called “perching birds.” - √11 √2 3 π 4.5 EXAMPL 1 EXAMPLE 8.NS.1 Write all names that apply to each number. A _ √5 5 is a whole number that is not a perfect square. irrational, real B –17.84 rational, real _ √ 81 C ____ 9 Animated Math my.hrw.com –17.84 is a terminating decimal. _ √ 81 9 _____ = __ =1 9 9 whole, integer, rational, real Math Talk Mathematical Practices What types of numbers are between 3.1 and 3.9 on a number line? Lesson 1.2 15 YOUR TURN Personal Math Trainer Write all names that apply to each number. 1. A baseball pitcher has pitched 12_23 innings. Online Practice and Help my.hrw.com 2. The length of the side of a square that has an area of 10 square yards. Understanding Sets and Subsets of Real Numbers By understanding which sets are subsets of types of numbers, you can verify whether statements about the relationships between sets are true or false. Math On the Spot my.hrw.com EXAMPLE 2 8.NS.1 Tell whether the given statement is true or false. Explain your choice. A All irrational numbers are real numbers. B No rational numbers are whole numbers. Math Talk Mathematical Practices Give an example of a rational number that is a whole number. Show that the number is both whole and rational. False. A whole number can be written as a fraction with a denominator of 1, so every whole number is included in the set of rational numbers. The whole numbers are a subset of the rational numbers. YOUR TURN Tell whether the given statement is true or false. Explain your choice. 3. All rational numbers are integers. Personal Math Trainer Online Practice and Help my.hrw.com 16 Unit 1 4. Some irrational numbers are integers. © Houghton Mifflin Harcourt Publishing Company • Image Credits: Digital Image copyright ©2004 Eyewire True. Every irrational number is included in the set of real numbers. The irrational numbers are a subset of the real numbers. Identifying Sets for Real-World Situations Real numbers can be used to represent real-world quantities. Highways have posted speed limit signs that are represented by natural numbers such as 55 mph. Integers appear on thermometers. Rational numbers are used in many daily activities, including cooking. For example, ingredients in a recipe are often given in fractional amounts such as _23 cup flour. EXAMPL 3 EXAMPLE Math On the Spot my.hrw.com 8.NS.1 Identify the set of numbers that best describes each situation. Explain your choice. My Notes A the number of people wearing glasses in a room The set of whole numbers best describes the situation. The number of people wearing glasses may be 0 or a counting number. B the circumference of a flying disk has a diameter of 8, 9, 10, 11, or 14 inches The set of irrational numbers best describes the situation. Each circumference would be a product of π and the diameter, and any multiple of π is irrational. YOUR TURN © Houghton Mifflin Harcourt Publishing Company Identify the set of numbers that best describes the situation. Explain your choice. 5. the amount of water in a glass as it evaporates 6. the weight of a person in pounds Personal Math Trainer Online Practice and Help my.hrw.com Lesson 1.2 17 Guided Practice Write all names that apply to each number. (Example 1) 1. _78 3. 2. _ √ 24 _ √ 36 4. 0.75 _ 6. - √ 100 5. 0 _ 7. 5.45 18 8. - __ 6 Tell whether the given statement is true or false. Explain your choice. (Example 2) 9. All whole numbers are rational numbers. 10. No irrational numbers are whole numbers. Identify the set of numbers that best describes each situation. Explain your choice. (Example 3) 11. the change in the value of an account when given to the nearest dollar 12. the markings on a standard ruler IN. ? ESSENTIAL QUESTION CHECK-IN 13. What are some ways to describe the relationships between sets of numbers? 18 Unit 1 1 © Houghton Mifflin Harcourt Publishing Company 1 inch 16 Name Class Date 1.2 Independent Practice Personal Math Trainer 8.NS.1 my.hrw.com Online Practice and Help Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram. _ 14. -√9 15. 257 16. 17. 8 _12 _ √ 50 18. 16.6 19. _ √ 16 Real Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Identify the set of numbers that best describes each situation. Explain your choice. © Houghton Mifflin Harcourt Publishing Company 20. the height of an airplane as it descends to an airport runway 21. the score with respect to par of several golfers: 2, – 3, 5, 0, – 1 1 22. Critique Reasoning Ronald states that the number __ 11 is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain. Lesson 1.2 19 23. Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain π mi your answer. 24. Critical Thinking A number is not an integer. What type of number can it be? 25. A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons? FOCUS ON HIGHER ORDER THINKING Work Area 26. Explain the Error Katie said, “Negative numbers are integers.” What was her error? _ _1 28. Draw Conclusions _ The decimal _0.3 represents 3 . What type of number best describes 0.9, which is 3 · 0.3? Explain. 29. Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this? 20 Unit 1 © Houghton Mifflin Harcourt Publishing Company 27. Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain. Ordering Real Numbers LESSON 1.3 ? 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). ESSENTIAL QUESTION How do you order a set of real numbers? Comparing Irrational Numbers Between any two real numbers is another real number. To compare and order real numbers, you can approximate irrational numbers as decimals. Math On the Spot EXAMPL 1 EXAMPLE 8.NS.2 _ _ 3 + √5 . Write <, >, or =. Compare √3 + 5 STEP 1 STEP 2 my.hrw.com _ First approximate √3 . _ √ 3 is between 1 and 2. _ Next approximate √5 . _ √ 5 is between 2 and 3. Use perfect squares to estimate square roots. 12 = 1 22 = 4 32 = 9 My Notes Then use your approximations to simplify the expressions. _ √3 + 5 is between 6 and 7. _ 3 + √5 is between 5 and 6. _ _ So, √ 3 + 5 > 3 + √ 5 . © Houghton Mifflin Harcourt Publishing Company Reflect _ _ 1. If 7 + √5 is equal to √5 plus a number, what do you know about the number? Why? 2. What are the closest two integers that √300 is between? _ YOUR TURN Personal Math Trainer Compare. Write <, >, or =. 3. _ √2 + 4 _ 2 + √4 4. _ √ 12 +6 _ 12 + √6 Online Practice and Help my.hrw.com Lesson 1.3 21 Ordering Real Numbers You can compare and order real numbers and list them from least to greatest. Math On the Spot my.hrw.com EXAMPLE 2 8.NS.2 _ Order √22 , π + 1, and 4 _12 from least to greatest. STEP 1 _ First approximate √22 . _ √ 22 is between 4 and 5. Since you don’t know where it falls _ between 4 and 5, you need to find a better estimate for √ 22 so you can compare it to 4 _12 . Since 22 is closer to 25 than 16, use squares _ of numbers between 4.5 and 5 to find a better estimate of √22 . 4.52 = 20.25 4.62 = 21.16 4.72 = 22.09 _ 4.82 = 23.04 Since 4.72 = 22.09, an approximate value for √22 is 4.7. An approximate value of π is 3.14. So an approximate value of π +1 is 4.14. STEP 2 _ Plot √ 22 , π + 1, and 4 _21 on a number line. 1 42 π+1 4 4.2 4.4 √22 4.6 4.8 5 Read the numbers from left to right to place them in order from least to greatest. _ From least to greatest, the numbers are π + 1, 4 _12 , and √22 . Order the numbers from least to greatest. Then graph them on the number line. 5. _ _ √ 5 , 2.5, √ 3 0 0.5 1 1.5 2 2.5 3 3.5 4 Online Practice and Help my.hrw.com 22 Unit 1 6. π2, 10, √75 8 8.5 9 Mathematical Practices If real numbers a, b, and c are in order from least to greatest, what is the order of their opposites from least to greatest? Explain. _ Personal Math Trainer Math Talk 9.5 10 10.5 11 11.5 12 © Houghton Mifflin Harcourt Publishing Company YOUR TURN Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ. It can be important to know not only which are the most accurate but which give the greatest or least values, depending upon the context. EXAMPL 3 EXAMPLE Math On the Spot my.hrw.com 8.NS.2 Four people have found the distance in kilometers across a canyon using different methods. Their results are given in the table. Order the distances from greatest to least. My Notes Distance Across Quarry Canyon (km) Juana Lee Ann Ryne Jackson √ 28 23 __ 4 5.5 5_12 _ STEP 1 _ Write each value as a decimal. _ √28 is between 5.2 and 5.3. Since 5.32 = 28.09, an approximate _ value for √28 is 5.3. 23 __ = 5.75 4 _ _ 5.5 is 5.555…, so 5.5 to the nearest hundredth is 5.56. 5 _12 = 5.5 STEP 2 _ 23 _ , 5.5, and 5 _21 on a number line. Plot √28 , __ 4 √28 © Houghton Mifflin Harcourt Publishing Company 5 5.2 1 5 2 5.5 5.4 5.6 23 4 5.8 6 From greatest to least, the distances are: _ _ 23 km, 5.5 km, _ __ 5 21 km, √28 km. 4 YOUR TURN 7. Four people have found the distance in miles across a crater using different methods. Their results are given below. _ _ 10 √ 10 3_1 Jonathan: __ 3 , Elaine: 3.45, José: 2 , Lashonda: Order the distances from greatest to least. Personal Math Trainer Online Practice and Help my.hrw.com Lesson 1.3 23 Guided Practice Compare. Write <, >, or =. (Example 1) 1. _ √3 +2 _ √3 + 3 2. _ √ 8 + 17 3. _ √6 +5 6+ _ √5 4. _ √9 + 3 5. _ √ 17 - 3 7. _ √7 + 2 -2 + _ √ 11 + 15 9+ _ _ √5 _ 14 - √ 8 6. 12 - √2 _ √ 10 - 1 8. _ √3 _ √ 17 + 3 _ 3 + √ 11 _ 9. Order √ 3 , 2π, and 1.5 from least to greatest. Then graph them on the number line. (Example 2) _ √ 3 is between π ≈ 3.14, so 2π ≈ 0 0.5 1 1.5 _ , so √ 3 ≈ and . . 2 2.5 3 3.5 4 4.5 5 From least to greatest, the numbers are 5.5 6 6.5 , 7 , . ? ESSENTIAL QUESTION CHECK-IN 11. Explain how to order a set of real numbers. 24 Unit 1 Forest Perimeter (km) Leon _ √ 17 -2 Mika Jason Ashley π 1 + __ 2 12 ___ 2.5 5 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Elena Elisseeva/Alamy Images 10. Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least. (Example 3) Name Class Date 1.3 Independent Practice Personal Math Trainer 8.NS.2 my.hrw.com Online Practice and Help Order the numbers from least to greatest. 12. 14. _ _ √8 √ 7 , 2, ___ 2 _ _ √ 220 , -10, √ 100 , 11.5 13. _ √ 10 , π, 3.5 15. _ 9 √ 8 , -3.75, 3, _ 4 16. Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters. a. Find the area of the square. b. Find the area of the circle. c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant? © Houghton Mifflin Harcourt Publishing Company 17. Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table. Distance Across Father’s Ranch (km) 1 _ √ 60 a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate. 2 58 __ 8 3 _ 7.3 4 7 _35 _ b. Winnie’s father estimated the distance across his ranch to be √ 56 km. How does this distance compare to Winnie’s estimate? Give an example of each type of number. _ _ 18. a real number between √13 and √14 19. an irrational number between 5 and 7 Lesson 1.3 25 20. A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right? _ 115 √ 115 , ___ 11 , and 10.5624 21. Math History There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284. a. Between which two square roots of integers could you find this number? b. Between which two square roots of integers can you find π? FOCUS ON HIGHER ORDER THINKING Work Area 22. Analyze Relationships There are several approximations used for π, 22 including 3.14 and __ 7 . π is approximately 3.14159265358979 . . . 3.140 3.141 3.142 3.143 b. Which of the two approximations is a better estimate for π? Explain. c. Find a whole number x so that the ratio ___ 113 is a better estimate for π x than the two given approximations. 23. Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line, what is the fewest number of distinct points that need to be graphed? Explain. _ 24. Critique Reasoning Jill says that 12.6 is less than 12.63. Explain her error. 26 Unit 1 © Houghton Mifflin Harcourt Publishing Company Image Credits: ©3DStock/iStockPhoto.com a. Label π and the two approximations on the number line. MODULE QUIZ Ready Personal Math Trainer 1.1 Rational and Irrational Numbers Online Practice and Help Write each fraction as a decimal or each decimal as a fraction. ___ 7 1. __ 20 my.hrw.com 2. 1.27 3. 1_78 5. x3 = 343 1 6. x2 = ___ 100 Solve each equation for x. 4. x2 = 81 7. A square patio has an area of 200 square feet. How long is each side of the patio to the nearest tenth? 1.2 Sets of Real Numbers Write all names that apply to each number. 121 ____ 8. ____ √ 121 π 9. __ 2 10. Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice. 1.3 Ordering Real Numbers Compare. Write <, >, or =. __ © Houghton Mifflin Harcourt Publishing Company 11. √8 + 3 __ 8 + √3 __ Order the numbers from least to greatest. ___ __ 13. √99, π2, 9.8 ___ 12. √5 + 11 14. ___ 5 + √11 __ 1 _ , 1, 0.2 √__ 25 4 ESSENTIAL QUESTION 15. How are real numbers used to describe real-world situations? Module 1 27 MODULE 1 MIXED REVIEW Personal Math Trainer Assessment Readiness my.hrw.com Online Practice and Help ___ 1. Look at each number. Is the number between 2π and √52 ? Select Yes or No for expressions A–C. A. 6_23 Yes No 5π B. __ 2 Yes No Yes No __ C. 3√5 11 . 2. Consider the number -__ 15 Choose True or False for each statement. A. The number is rational. B. The number can be written as a repeating decimal. C. The number is less than –0.8. True True False False True False ___ 29 4. A student says that √ 83 is greater than __ . Is the student correct? Justify your 3 reasoning. 28 Unit 1 © Houghton Mifflin Harcourt Publishing Company 3. The volume of a cube is given by V = x3, where x is the length of an edge of the cube. A cube-shaped end table has a volume of 3_38 cubic feet. What is the length of an edge of the end table? Explain how you solved this problem.