Solutions Ke
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Table of Contents UNIT 1 Real Numbers, Eponents, and Scientific Notation Module 1 Lesson 1.1... 1 Lesson 1.... Lesson 1.... Module Lesson.1.... 8 Lesson.....11 Lesson..... 1 Lesson..... 1 UNIT Proportional and Nonproportional Relationships and Functions Module Lesson.1.... 19 Lesson..... 0 Lesson..... Module Lesson.1.... Lesson..... Lesson..... 7 Lesson..... 9 Module Lesson.1.... Lesson..... Lesson..... UNIT Solving Equations and Sstems of Equations Module 7 Lesson 7.1.... Lesson 7..... Lesson 7..... 8 Lesson 7..... 1 Module 8 Lesson 8.1.... Lesson 8..... 8 Lesson 8..... Lesson 8..... Lesson 8..... 70 UNIT Transformational Geometr Module 9 Lesson 9.1.... 7 Lesson 9..... 7 Lesson 9..... 78 Lesson 9..... 79 Lesson 9..... 80 Module 10 Lesson 10.1... 8 Lesson 10.... 8 Lesson 10.... 8 Module Lesson.1.... 8 Lesson..... 9 Lesson..... 1 Lesson..... iii
Table of Contents UNIT Measurement Geometr Module 11 Lesson 11.1.... 88 Lesson 11..... 89 Lesson 11..... 91 Module 1 Lesson 1.1... 9 Lesson 1.... 9 Lesson 1.... 97 UNIT Statistics Module 1 Lesson 1.1... 107 Lesson 1.... 108 Module 1 Lesson 1.1...110 Lesson 1.... 11 Module 1 Lesson 1.1... 100 Lesson 1.... 10 Lesson 1.... 10 iv
UNIT 1 MODULE 1 Real Numbers Are You Read? 1. 7 7 = 9. 1 1 = 1. ( ) ( ) = 9. = 1. (.7 ) (.7 ) = 7.9. ( - 1 ) ( - 1 ) = 1 1 7. (.7) (.7) =.9 8. 1 = + = 7 7 7 = 9 = 1 or 19. 9. 9 = 9 9 = 81 10. = = 1 11. ( ) 1 = 1 1 = 1 9 1. ( 7) = ( 7) ( 7) = 9 1. = = 1. ( 1) = ( 1) ( 1) ( 1) ( 1) ( 1) = 1 1. (. ) = (. ) (. ) = 0. 1. 10 = 10 10 10 10 10 = 100,000 17. 1 + 1 9 + 1 10 18. 1 8 1 + 8 8 8 + 8 1 8 19. 7 + 7 1 7 + 7 17 7 Solutions Ke Real Numbers, Eponents, and Scientific Notation 0. + 0 + LESSON 1.1 Your Turn 0. 1. 11.00 _- 0 _- Because the number repeats during the division process, the answer is a repeating decimal: 0. _. _ 0.1. 8 1.000 _-8 0 _-1 0 _-0 0 0.1. 1 = 7. 7.0 _- 10 _-9 1 Because the number 1 repeats during the division process, the answer is a repeating decimal:. _.. Write the decimal 0.1 as a fraction. 0.1 = 1 100 Simplif using the same numerator and denominator. 1 100 = 1
. = 0. _ 7 ( 100 ) = 100 ( 0. _ 7 ) 100 = 7. _ 7 Because = 0. _ 7, subtract from one side and 0. _ 7 from the other. 100 = 7. _ 7 _- -0. _ 7 99 = 7 99 99 = 7 99 = 7 99, or 19. Write the decimal 1. as a fraction. 1. = 1 10 Simplif using the same numerator and denominator. 1 10 = 7 7, or 1 7. = 19 _ X = _ 19 X = _ 19 X = ±1 The solutions are 1 and -1. 8. = 9 X = 9 _ = 9 = 1 The solutions are 1 and - 1. 9. 1 = 1 = 1 = 8 = The solution is 8. 10. = = = 7 The solution is 7. Guided Practice 1. 0..0 _-0 0 0.. 8 9 0. 8 9 8.0 _-7 8 Because the number 8 repeats during the division process, the answer is a repeating decimal: 0. 8 _.. can also be written as 1, so _.7 1.00 _-1 0 _-8 0 _-0 0.7. 7 10 _ 0.7 10 7.0 _-70 0 0.7. can also be written as 19 8 8 _.7 8 19.000 _-1 0 _- 0 _- 0 _-0 0.7. 0.8.00 _-8 0 _-18 0 Because the number 0 repeats during the division process, the answer is a repeating decimal: 0.8 _ 7. Write the decimal 0.7 as a fraction. 0.7 = 7 1000 Simplif using the same numerator and denominator. 7 1000 = 7 0 8. The decimal. is the can be written as +, or 10.
9. Write the decimal 0. as a fraction. 0. = 100 Simplif using the same numerator and denominator. 100 = 11 10. 10 =. - - 0. _ 9 = = 9 11. 100 =. - - 0. _ 99 = = 99 1. 1000 =. - - 0. _ 999 = = 999 1. = 17 = _ 17 ±.1 1. = 89 _ = 89 = ± 17 1. = 1 = 1 = 1. _. 17. _ 1.7 18. _ 10. 19. Rational numbers can be written in the form a b, where a and b are integers and b 0. Irrational numbers cannot be written in this form. Independent Practice 0. 7 1 _ 0.7 1 7.0000 _- 0 _-8 10 _-11 80 _-80 0 0.7 in. 1. 1 0.1 1.00 _- 0 _- 0 Because the number 0 repeats during the division process, the answer is a repeating decimal: 0.1 _.. can also be written as 1, so _.8 1.0 _-10 0 _-0 0 The distance is.8 km.. 98 can also be written as 9, so 98. 9.00 _-7 _- 0 _-18 0 Because the number 0 repeats during the division process, the answer is a repeating decimal: 98. _ innings.. Write the decimal 0.8 as a fraction. 0.8 = 8 10 Simplif using the same numerator and denominator. 8 10 = A heartbeat takes second.. Separate the decimal from. so that: 0. = 10 Simplif using the same numerator and denominator. 10 = 1 Therefore,. mi = 1 mi.. Separate the repeating digit and let = 0. _ 1 = 0. 1 _ ( 100 ) = 100 ( 0. 1 _ ) 100 = 11. 1 - -0. _ 1 99 = 11 = 11 99 = 1 9 Therefore, 7. 1 _ = 7 1 9. 7. Write the decimal 0.0 as a fraction. 0.0 = 0 1000 Simplif using the same numerator and denominator. 0 1000 = 101 00 A metal penn is worth 101 00 cent.
8. a. You can set up the equation = 00 to find the length of a side. = 00 _ = _ 00 = ± 0 The solutions are = ±0; the equation has solutions. b. The solution = 0 makes sense, but the solution = -0 doesn t make sense, because a painting can t have a side length of -0 inches. c. The length of the wood trim needed is 0 = 80 inches. 9. = 1 _ = _ 1 ±.7 0. = 11 = 11 = 11 1. = 1 _ = _ 1 = ±1. = 9 _ = _ 9 ±.. His estimate is low because 1 is much closer to 1 than it is to 9. So, a better estimate would be higher, such as.8 or.9.. Sample answer: A good estimate is., because = and = 1. Since 9 is about half wa between and 1, 9 is probabl closer to. than to or.. V = r = r = r 7 = r 7 = r = r The radius of the sphere is feet. Focus on Higher Order Thinking. Yes; the cube root of a negative number is alwas negative, because a negative number cubed is alwas negative, and a nonnegative number cubed is alwas nonnegative. 7. =, and _ 1 81 = 9, and 1 _ 81 9 = 7, and _ 9 = = 9 = 7 Because the epressions ield the same answer, ou can see that a b = a _. Therefore, ou can b make a conjecture about the multiplication rule for square roots that _ a _ b = _ a b. 8. The value of a is, because the solutions are = ± 1, and 1 - ( -1 ) = 0. LESSON 1. Your Turn 1. 1 is a rational number because it can be represented as the ratio 8. It is a real number because all rational numbers are real numbers.. The length of the side is _ 10 d. _ 10 is an irrational number because 10 is a whole number that is not a perfect square. It is a real number because all irrational numbers are real numbers.. False. Ever integer is a rational number, but not ever rational number is an integer. For eample, rational numbers such as and - are not integers.. False. Real numbers are either rational numbers or irrational numbers. Integers are rational numbers, so no integers are irrational numbers.. The set of real numbers best describes the situation. The amount can be an number greater than 0.. The set of rational numbers best describes the situation. A person s weight can be a decimal such as 8. pounds. Guided Practice 1. 7 is a rational number because it is the ratio of two 8 integers: 7 and 8. It is a real number because all rational numbers are real numbers.. _ is a whole number because it is equal to, which is a positive number with no fractional or decimal part. Ever whole number is also an integer, a rational number, and a real number.. _ is an irrational number because is a whole number that is not a perfect square. It is a real number because all irrational numbers are real numbers.. 0.7 is a rational number because it is a terminating decimal. It is a real number because all rational numbers are real numbers.. 0 is a whole number because it is a number with no fractional or decimal part. Ever whole number is also an integer, a rational number, and a real number.. - _ 100 is an integer because it is equal to -10, which is a number with no fractional or decimal part. Ever integer is also a rational number and a real number.
7.. _ is a rational number because it is a repeating decimal. It is a real number because all rational numbers are real numbers. 8. - 18 is an integer because it is equal to -, which is a number with no fractional or decimal part. Ever integer is also a rational number and a real number. 9. True. Whole numbers are a subset of the set of rational numbers and can be written as a fraction with a denominator of 1. 10. True. Whole numbers are rational numbers. 11. The set of integers best describes the situation. The change can be a whole dollar amount and can be positive, negative, or 0. 1. The set of rational numbers best describes the situation. The ruler is marked ever 1 1 inch. 1. Sample answer: Describe one set as being a subset of another, or show their relationships in a Venn diagram. Independent Practice 1. - _ 9 is an integer because it is equal to -. Ever integer is also a rational number and a real number. 1. 7 is a whole number because it is a positive number with no fractional or decimal part. Ever whole number is also an integer, a rational number, and a real number. 1. _ 0 is an irrational number because 0 is a whole number that is not a perfect square. It is a real number because all irrational numbers are real numbers. 17. 8 1 is a rational number because it can be represented as the ratio 17. It is a real number because all rational numbers are real numbers. 18. 1. is a rational number because it is a terminating decimal. It is a real number because all rational numbers are real numbers. 19. _ 1 is a whole number because it is equal to, which is a positive number with no fractional or decimal part. Ever whole number is also an integer, a rational number, and a real number. 1. Rational Numbers Integers Whole Numbers 1 7 Real Numbers 9 1 8 Irrational Numbers 0 1. The set of integers best describes the situation. The scores are counting numbers, their opposites, and 0.. Nathaniel is correct. A rational number is a number that can be written as a fraction, and 1 is a fraction. 11. A whole number. The diameter is π π mi, or 1 mi.. It can be a rational number that is not an integer, or an irrational number.. The total number of gallons of milk is either a whole number or a mied number in which the fractional part is 1. Therefore, the number is a rational number. Focus on Higher Order Thinking. The set of negative numbers also includes non-integer rational numbers and irrational numbers. 7. Sample answer: If the calculator shows a decimal that terminates in fewer digits than what the calculator screen allows, then ou can tell that the number is rational. If not, ou cannot tell from the calculator displa whether the number terminates because ou see a limited number of digits. It ma be a repeating decimal (rational) or a non-terminating non- repeating decimal (irrational). 8. It is a whole number. 0. _ = 1 = 1. Since 0. _ is equal to 0. 9 _, then 0. 9 _ is equal to 1, which is a whole number. 9. Sample answer: In decimal form, irrational numbers never terminate and never repeat. Therefore, no matter how man decimal places ou include, the number will never be precisel represented. There will alwas be more digits. LESSON 1. Your Turn. _ is between 1 and, so _ 1.. _ + 1. + =. _ = + _ = + = Since. >, _ + > + _.. _ 1 is between and, so _ 1.. _ 1 +. + = 9. _ is between and, so _.. 1 + _ 1 + 1. = 1. Since 9. < 1., _ 1 + < 1 + _.. _ is between and, but is closer to. So _ <.. _ is between 1 and, so _ 1.. _, _,. 0 0. 1 1... 0. The set of real numbers best describes the situation. The height can be an number greater than 0.
. An approimate value of π is.1. So, π 9.89. _ 7 is between 8 and 9, so _ 7 8.. _ 7, π, 10 7 π 8 8. 9 9. 10 10. 11 11. 1 7. 10 =. _ ; 1 =. _ 10 is between and, but is ver close to. So _ 10 <. _. 1 mi,. _ mi, 10 mi, _ 10 mi Guided Practice 1. _ is between 1 and, so _ 1.. _ + 1. +. _ + 1. +. Since. <., _ + < _ +.. _ 8 is between and but ver close to. Use.8. _ 8 + 17.8 + 17 = 19.8 _ 11 is between and, so _ 11.. _ 11 + 1. + 1 = 18. Since 19.8 > 18., _ 8 + 17 > _ 11 + 1.. _ is between and, so _.. _ +. + = 7. _ is also between and, but will be a bit less than.. Use.. + _ +. = 8. Since 7. < 8., _ + < + _.. _ 9 = _ 9 + = + = _ is between 1 and, so _ 1.. 9 + _ 9 + 1. = 10. Since < 10., _ 9 + < 9 + _.. _ 17 is between and, but ver close to. Use.1. _ 17 -.1 - = 1.1 _ is between and, but ver close to. Use.. - + _ - +. = 0. Since 1.1 > 0., _ 17 - > - + _.. _ is between 1 and, so _ 1.. 1 - _ 1-1. = 10. _ 8 is between and, so _ 8. 1 - _ 8 1 -. = 11. Since 10. < 11., 1 - _ < 1 - _ 8. 7. _ 7 is between and, so _ 7.. _ 7 +. + =. _ 10 is between and, so _ 10.. _ 10-1. - 1 =. Since. >., _ 7 + > _ 10-1. 8. _ 17 is between and, but ver close to. Use.1. _ 17 + 10 -. = 7. _ 11 is between and, so _ 11.. + _ 11 +. =. Since 7. >., _ 17 + > + _ 11. 9. _ is between 1.7 and 1.8, so _ 1.7. π.1, so π.8 1., _, π π 0 1 7 10. _ 17 is between and but ver close to, so _ 17.1, and _ 17 -.1. π.1, so π π 1.7, and 1 +.7. 1 =. 1 + π km,. km, 1 km, _ 17 - km 11. Sample answer: Convert each number to a decimal equivalent, using estimation to find equivalents for irrational numbers. Graph each number on a number line. Read the numbers from left to right to order the numbers from least to greatest. Read the numbers from right to left to order the numbers from greatest to least. Independent Practice 1. _ 7 is between and, so _ 7.. _ 8 is between and, so _ 8. and _ 8 1.7. 8,, _ 7 1. _ 10 is between.1 and., so _ 10.1. π.1. π, _ 10,. 1. _ 0 is between 1 and 1, so _ 0 1.. _ 100 = 10-10, _ 100, 11., _ 0 1. _ 8 is between and, so _ 8.. 9 =. -.7, 9, _ 8, 1. a. A =. = 1. m b. C = π = π m π.1 = 1. m c. The circle would give her more space to plant because it has a greater area. 17. a. _ 0 is between 7.7 and 7.8, so _ 0 7.7. 8 8 = 7. 7. _ 7. 7 = 7.0 7.7 + 7. + 7. + 7.0 = 9.9 = 7.8 The average is 7.8 km. b. _ 7.8, which is slightl greater than, but ver close to, Winnie s estimate. 18. Sample answer:.7. 19. Sample answer: _ 1. _ 0. 11 is between 10.7 and 10.8, so _ 11 10.7. 11 11 = 10. _ Neither student is correct. The answer should be: 11 11, 10., _ 11.
1. a. Since _ 7. and _ 8.8, е is between _ 7 and _ 8. b. Since _ 9 = and _ 10.1, π is between _ 9 and _ 10. Focus on Higher Order Thinking. a. 7.19.1 π.10.11.1.1 b..19. It is closer to π on the number line. 7 c. 11 =.11 = 11.11 =.0008. ; Rational numbers can have the same location, and irrational numbers can have the same location, but the cannot share a location. 7. She did not consider that 1. _ = 1.. MODULE 1 Read to Go On? _ 0. 1. 0 7.00 _-0 100 _-100 0 0.. = 1. _ 7 ( 100 ) = ( 100 ) 1. _ 7 100 = 17. _ 7 Because = 1. _ 7, subtract from one side and 1. _ 7 from the other. 100 = 17. _ 7 - -1. _ 7 99 = 1 99 99 = 1 99 = 1 11. 1 7 8 = 1 8 _ 1.87 8 1.000 _-8 70 _- 0 _- 0 _-0 0 1.87. _ 81 = 9; - _ 81 = -9 9 and -9. = = = 7 _. 1 100 = 1 10 ; - 1 100 = - 1 10 1 10 and - 1 10 7. Each side measures _ 00 ft. 1.1 = 198.81; 1. = 01. so, _ 00 is between 1.1 and 1.. _ 00 1.1 Each side is approimatel 1.1 feet long. 8. 11 _ is a whole number because 11 11 _ = 11 = 11, and 11 is a positive number with 11 11 no fractional or decimal part. Ever whole number is also an integer, a rational number, and a real number. 9. π is an irrational number because π is an irrational number and dividing π b gives another irrational number. It is a real number because all irrational numbers are real numbers. 10. True; Integers can be written as the quotient of two integers. 11. _ 8 is between and, so _ 8.. _ 8 +. + =. _ is between 1 and, so _ 1.. 8 + _ 8 + 1. = 9. Since. < 9., _ 8 + < 8 + _. 1. _ is between and, so _.. _ + 11. + 11 = 1. _ 11 is between and, so _ 11.. + _ 11 +. = 8. Since 1. > 8., _ + 11 > + _ 11. 1. π 9.87, 9. 8 _ 9.88, and _ 99 9.9. Therefore, the order from least to greatest should be π, 9. _ 8, _ 99. _ 1. 1 = 1 = 0.0 1 = 0. 0. _ = 0. 1, 0. _, 1 1. Sample answer: Real numbers, such as the rational number 1, can describe amounts used in cooking. 7
MODULE Eponents and Scientific Notation Are You Read? 1. 10 10 10 100. 10 10 10 10 1000. 10 10 10 10 10 10 100,000. 10 7 10 10 10 10 10 10 10 10,000,000.. 10. 1000,00. 7.08 10 7.08 100 0.0708 7. 0.00 10 0.00 1,000,000,0 8.,00 10,00 1,000 0. 9. 0. 10 0. 100 0 10. 7.7 10 7.7 100,000 0.00077 11. 0.007 10 0.007 1,000 7 1. 19 10 19 1,000,000 0.00019 LESSON.1 Your Turn 7. ( 11 ) = = 8 8. ( ) = = = 9. - -1 = --1 = - = 1 = 1 10. [ ( - 1 ) ] ( + ) = [ ( ) ] ( ) = ( ) ( ) = 1 = 11. ( ) - ( 10 - ) - = ( ) - - = - - = + (-) - = - - = - 1 1 = 1 1 Guided Practice 1. As the eponent decreases b 1, the value of the power is divided b 8. 8 0 = 1 8 1 = 1 8. As the eponent decreases b 1, the value of the power is divided b. 0 = 1-1 = 1 - = 1. An number raised to the power 0 equals 1. 0 = 1. As the eponent increases b 1, the value of the power is multiplied b 10. 10 0 = 1 10 1 = 10 10 = 100. As the eponent increases b 1, the value of the power is multiplied b. 0 = 1 1 = = = 1 =. As the eponent decreases b 1, the value of the power is divided b. 0 = 1-1 = 1 - = 1 - = 1 8 - = 1 1 - = 1 8
7. As the eponent decreases b 1, the value of the power is divided b. 0 = 1-1 = 1 - = 1 1 - = 1 - = 1 - = 1 10 8. An number raised to the power 0 equals 1. 89 0 = 1 9. As the eponent decreases b 1, the value of the power is divided b 11. 11 0 = 1 11-1 = 1 11 11 - = 1 11 11 - = 1 11 10. = 1 1 1 = 1+1+1 = 11. ( ) ( ) = ( 1 1 ) ( 1 1 1 ) = ( 1+1 ) ( 1+1+1 ) = = 1. 7 = = 1 1 1 1 1 1 1 1 1 1 1 1 = 1+1+1+1+1+1+1 1+1+1+1+1 = 7 = 7- = 1. 8 1 8 9 = 8 1-9 = 8 1. 10 = 10 1 1 = 10+1+1 = 1 1. 7 8 7 = 7 8+ = 7 1 1. ( ) = 8 = 17. ( 8 1 ) = 8 1 18. 9 0-10 = 9+0-10 = -1 = 1 1 = 1 19. 10 10 7 +7- = 10 10 = 10 = 10,000 0. ( 10 - ) + ( 10 + ) = + 1 = + + 1 = +1 = 1,0 + 1 = 1,18 1. ( 1 - ) 7 [ ( + ) ] = 7 7 ( 7 ) = 7 7 7 = 7 7 7 = 7 7- = 7 =. Sample answer: When multipling powers with the same base, ou add the eponents. When dividing powers with the same base, ou subtract the eponents. When raising a product to a power, ou raise each factor to that power. When raising a power to a power, ou multipl the eponents.. -7 1 - = -7+1- = = 1. 8 1 ( 8 7 ) - = 8 1 8 7 ( - ) = 8 1 8-1 = 8 1-1 = 8 - = 1 8 = 1. ( ) = 1. 9 9 = 9 - = 9 - = 1 9 = 1 79 = = 1,1 9
7. 8. ( ) = 8 8 = 10 8 = 10-8 = = 11 10 11 11 = 11 10- ( + ) = 11 10-8 = 11 = 11 9. ( ) + 0 + 7 - = + 0 + 7- = + 0 + = + 1 + = 9 0. - ( 7 - ) + ( 7 + ) = - ( ) + 1 1 = -+ + 1 1 = 0 + 1 1 = 1 + 11 = 1 1. 10 [( 8 + ) ] - = 10 [( 10 ) ] - = 10 10 (-) = 10 10 - = 10 - = 10 - = 1 10 = 1 100. 7 ( + ) ( 8-1 ) = 7 ( 7 ) 7 ( 7 ) 7 = 7 +-7 = 7 = 9. [( + ) ] ( 9 - ) = [ ( ) ] 1 ( ) 1 = 1 = 1 1 = 1-1 = = 1. ( ) ( + 1 ) = = - = 0 = 1 = 10. - ( 9 ) 9 - = - ( 9 ) 9 - Power of a Product Propert = ( - ) ( 9 9 - ) Associative Propert = 0 9 Product of Powers Propert = 1 9 Zero Eponent Propert = 9 Identit Propert of Multiplication = 81 Definition of an eponent. Sample answer: The product of two fractions is the product of the numerators over the product of the denominators. Writing this as a product of fractions lets ou simplif them separatel, using the Quotient of Powers Propert to simplif the second fraction. Independent Practice 7. The eponents cannot be added because the bases are not the same. 8. To epress as a product of powers, the bases should be and the powers should add up to ; Sample answer: 0 ; 1 ; 9. 7 > ; 7 = 7- = = 10,8 The distance from Earth to Neptune is the greater distance. It is about 10,8 times greater than the distance from Earth to the moon. 0. The student is not correct because 8 8 - = 8 + (-) = 8 - = 1 8 = 1, which is less than 1. 1. ( b ) n = b - n = - n = - n = - ( b ) - = b -. m = 9 m + = 9 m + - = 9 - m = = 9. = n - n = - n - = - -n = -19 -n (-1) = -19 (-1) n = 19 = 19
. Sample answer: Dividing is the same as multipling b the reciprocal. So when dividing powers with the same base, ou add the opposite of the eponent in the denominator. This is the same as subtracting the eponents.. 10 0 10 7 = 10 0 10 7 = 10 0-7 = 10 = 1,000 10 kg, or 1,000 kg. 10 0 = 10+0 = 0 0 btes 7. 7-7+ (-) = = ; = 7 = 7- = Both epressions equal, so 7 - = 7 ; Sample answer: When multipling powers with the same base, ou add eponents: 7 + (-) =. When dividing powers with the same base, ou subtract eponents: 7 - =. In cases like this, n -m = n. m 8. The number of cubes in each row is raised to the row number. 9. Since the number of cubes in each row is raised to the row number, the number of cubes in Row will be raised to the power, and the number of cubes in Row will be raised to the power. = 79; = - = = 7 The number of cubes in Row will be, or 79. There will be, or 7, times the number of cubes in Row as there are in Row. 0. 1 + + + + + = + 9 + 7 + 81 + + 79 = 1,09 The total number of cubes in the triangle is 1,09; Sample answer: I evaluated 1,,,,, and and added these numbers together. Focus on Higher Order Thinking 1. Sample answer: No, I do not agree, because = = = 1 = 1.. - = -9 - = -7 - = -81 - = -; ( - ) = 9 ( - ) = -7 ( - ) = 81 ( - ) = - For -a n, ou get -9, -7, -81, and -. For ( -a ) n, ou get 9, -7, 81, and -. No, it does not appear that - a n = ( -a ) n. When n is even, the two epressions are opposites. When n is odd, the two epressions are equal.. Let the number equal. 1 = 1 9 1-9 = 1 = 1 The cube root of 1 is the number. LESSON. Your Turn. Move the decimal in,00 to the left to get..,00. = 1,000 = 10,00 =. 10. Move the decimal in 70,000,000,000 to the left to get.7. 70,000,000,000.7 = 100,000,000,000 = 10 11 70,000,000,000 =.7 10 11. Move the decimal in 9,1,000,000,000 to the left to get 9.1. 9,1,000,000,000 9.1 = 1,000,000,000,000 = 10 1 9,1,000,000,000 = 9.1 10 1 km 8. To write 7.0 10 9 in standard notation, move the decimal 9 places to the right. 7.0 10 9 = 7,0,000,000 9. To write. 10 in standard notation, move the decimal places to the right.. 10 =,000 10. To write 10 in standard notation, move the decimal places to the right. 10 =,000,000 g Guided Practice 1. Move the decimal in 8,97 to the left to get.897. 8,97.897 = 10,000 = 10 8,97 =.897 10. Move the decimal in 1,0,000,000 to the left to get 1.0. 1,0,000,000 1.0 = 1,000,000,000 = 10 9 1,0,000,000 = 1.0 10 9. Move the decimal in,70,000 to the left to get.7.,70,000.7 = 1,000,000 = 10,70,000 =.7 10. Move the decimal in 1,00 to the left to get 1.. 1,00 1. = 10,000 = 10 1,00 = 1. 10 11
. Move the decimal in 97,700,000,000,000,000,000,000 to the left to get 9.77. 97,700,000,000,000,000,000,000 9.77 = 10 97,700,000,000,000,000,000,000 = 9.77 10. Move the decimal in 8,000 to the left to get.8. 8,000.8 = 100,000 = 10 8,000 =.8 10 7. To write 10 in standard notation, move the decimal places to the right. 10 = 00,000 8. To write 1.899 10 9 in standard notation, move the decimal 9 places to the right. 1.899 10 9 = 1,89,900,000 9. To write.1 10 in standard notation, move the decimal places to the right..1 10 =,10 10. To write 8. 10 7 in standard notation, move the decimal 7 places to the right. 8. 10 7 = 8,0,000 11. To write 8 10 in standard notation, move the decimal places to the right. 8 10 = 800,000 1. To write 9 10 10 in standard notation, move the decimal 10 places to the right. 9 10 10 = 90,000,000,000 1. To write. 10 in standard notation, move the decimal places to the right.. 10 =,000 s 1. To write 7. 10 in standard notation, move the decimal places to the right. 7. 10 = 7,00,000 cans 1. First move the decimal 9 places to the left to find.8, a number that is greater than or equal to 1 and less than 10. Then multipl.8 b 10 9, using an eponent on 10 that equals the number of places ou moved the decimal. Independent Practice 1. Move the decimal in,000 to the left to get..,000. = 10,000 = 10,000 =. 10 lb 17. Move the decimal in 0,000 to the left to get.. 0,000. = 100,000 = 10 0,000 =. 10 lb 18. Move the decimal in 100,000 to the left to get 1. 100,000 1 = 10,000 = 10 100,000 = 1 10 lb 19. Move the decimal in 0,000 to the left to get. 0,000 = 10,000 = 10 0,000 = 10 lb 0. Move the decimal in 19,80 to the left to get 1.98. 19,80 1.98 = 10,000 = 10 19,80 = 1.98 10 lb 1. Move the decimal in 0,000 to the left to get. 0,000 = 10,000 = 10 0,000 = 10 lb. 1,000 10. = 10,00 10,00 1.0 = 10,000 = 10 10,00 = 1.0 10 mosquitoes. 0 words 0 =,00 words 1 minute 0 1 hour. 10 = 0,000 0,000,00 = 108. _ 108 1 hours, or 108 hours and 0 minutes. a. Write 1.18 in standard notation, 1,18, and then multipl b our weight. b. Sample answer: 9,0 lb; 9. 10. 0 0 =,00,00. = 1,000 = 10,00 =. 10 lb. 9.999 10 and 10 1 ; numbers in scientific notation are written as the product of a number greater than or equal to 1 and less than 10, and a power of 10, so 0.1 10 and.8 10 are not written in scientific notation. 7. a. None of the girls have the correct answer. b. Poll and Samantha have the decimal in the wrong places, causing their eponents to be incorrect. Esther has the decimal in the correct place but miscounted the number of places the decimal moved. 8. Sample answer: Scientific notation is a quicker wa to write large numbers. Also, it s easier to read, it s used b scientists everwhere, and it s eas to compare sizes of large numbers written in scientific notation. Focus on Higher Order Thinking 9. The speed of a car because it is likel to be less than 100. 0..1 10 8 is greater because the eponent 8 is greater than the eponent. 1. Is the first factor greater than or equal to 1 and less than 10? Is the second factor a power of 10? LESSON. Your Turn. Move the decimal in 0.000089 to the right places to get 8.9. 0.000089 = 8.9 10 -. Move the decimal in 0.0000000 to the right 7 places to get.0. 0.0000000 =.0 10-7. Move the decimal in 0.000007 to the right places to get 7.0 0.000007 = 7 10 - m 9. Move the decimal in 1.0 to the left places. 1.0 10 - = 0.0000010 10. Move the decimal in 9.9 to the left places. 9.9 10 - = 0.000099 1
11. Move the decimal to the left places. 1 10 - = 0.01 m Guided Practice 1. Move the decimal in 0.00087 to the right places to get.87. 0.00087 =.87 10 -. Move the decimal in 0.00008 to the right places to get.8. 0.00008 =.8 10 -. Move the decimal in 0.00009 to the right places to get.9. 0.00009 =.9 10 -. Move the decimal in 0.017 to the right places to get.17. 0.017 =.17 10 -. Move the decimal in 0.0000 to the right places to get. 0.0000 = 10 -. Move the decimal in 0.00001 to the right places to get 1.. 0.00001 = 1. 10-7. Move the decimal to the left places. 10 - = 0.0000 8. Move the decimal in.8 to the left places..8 10 - = 0.000008 9. Move the decimal in 8. to the left places. 8. 10 - = 0.0008 10. Move the decimal in.97 to the left places..97 10 - = 0.097 11. Move the decimal in 9.0 to the left places. 9.0 10 - = 0.000090 1. Move the decimal to the left places. 10 - = 0.0000 1. Move the decimal in 0.0001 to the right places to get 1. 0.0001 = 1 10-1. Move the decimal to the left places. 1.7 10 - = 0.0000000000000000000000017 1. Move the decimal point places right to find.7, a number greater than or equal to 1 and less than 10. Then multipl.7 b 10 -, using a negative eponent on 10 that equals the number of places ou moved the decimal. Independent Practice 1. Move the decimal in 0.0077 to the right places to get.77. 0.0077 =.77 10 - cm 17. Move the decimal in 0.001 to the right places to get 1.. 0.001 = 1. 10 - cm 18. Move the decimal in 0.00 to the right places to get.. 0.00 =. 10 - cm 19. Move the decimal in 0.00 to the right places to get.. 0.00 =. 10 - cm 0. Move the decimal in 0.01 to the right places to get 1.. 0.01 = 1. 10 - cm 1. Move the decimal in 0.0008 to the right places to get 8. 0.0008 = 8 10 - cm. The ounces in a cup of milk; it is more than 1 but less than 10.. 7 cm = 0.07 m, 7 cm = 7 10 0 cm; 0.07 m = 7 10 - m. The first factors are the same; the eponents differ b.. 1.89E 1; the eponent on 10. If the eponent on 10 is nonnegative, the number is greater than or equal to 1.. Move the decimal in 0.00007 to the right places to get.7. 0.00007 =.7 10 - L 7. Negative, because a ladbug would weigh less than 1 ounce. 8. Move the decimal in 1,70,000 to the left to get 1.7. 1,70,000 1.7 = 1,000,000 = 10 1,70,000 = 1.7 10 9. Move the decimal in 1. to the left 10 places. 1. 10-10 = 0.0000000001 0. Move the decimal in 0.008 to the right places to get.8. 0.008 =.8 10-1. To write 7.19 10 7 in standard notation, move the decimal 7 places to the right. 7.19 10 7 = 71,90,000. Move the decimal in 0.00000000018 to the right 10 places to get 1.8. 0.00000000018 = 1.8 10-10. To write.97 10 in standard notation, move the decimal places to the right..97 10 =,97,000. Atom of silver, atom of aluminum, Atlantic wolfish egg, the Moon, Mars, Jupiter Focus on Higher Order Thinking. 1. 10 - = 0.01 1. 10 = 10.8 10 - = 0.008. 10 - = 0.0 9. 10-1 = 0.9.8 10 - m, 1. 10 - m,. 10 - m, 9. 10-1 m, 1. 10 m. Al treated the eponent as if it were positive instead of negative and moved the decimal in the wrong direction. The answer should be 0.000000. 7. The result will be greater than the number with the positive eponent because the divisor is less than 1. 1
LESSON. Your Turn 1. ( 1.1 10 8 ) - (.8 10 7 ) = ( 1.1 10 8 ) - ( 0.8 10 8 ) = ( 1.1-0.88 ) 10 8 = 0.7 10 8 = 7. 10 7 7. 10 7 more people live in Meico than in Canada.. ( 1.8 10 ) (.8 10 ) = ( 1.8.8 ) ( 10 10 ) = ( 1.8.8 ) 10 + = 8.98 10 8 The approimate distance from the Sun to Saturn is 8.98 10 8 miles..,70,000,000 1.17 10 7 9 =.7 10 1.17 10 7 =.7 1.17 10 9 10 7 =.7 9-7 10 1.17 =.1 10 On average, it takes sunlight.1 10 minutes to reach Pluto.. The letter E takes the place of 10. 7. 10 = 7.E. The letter E takes the place of 10. 10-7 = E-7. The letter E takes the place of 10..7 10 1 =.7E1 7. 10 takes the place of the letter E..E-1 =. 10-1 8. 10 takes the place of the letter E..E1 =. 10 1 9. 10 takes the place of the letter E..98E-8 =.98 10-8 Guided Practice 1.. 10 +. 10 +.8 10 =. 10 + 0. 10 +.8 10 = (. + 0. +.8 ) 10 = 7. 10. 8. 10 -. 10-1.0 10 = 8. 10 -. 10-0.10 10 = ( 8. -. - 0.10 ) 10 =.1 10. 1. 10 + 0.0 10 +. 10 = ( 1. + 0.0 +. ) 10 = 10.. 10 -. 10-1.9 10 =. 10-0. 10-0.0019 10 = (. - 0. - 0.0019 ) 10 =.981 10. ( 1.8 10 9 ) (.7 10 1 ) = ( 1.8.7 ) ( 10 9 10 1 ) = ( 1.8.7 ) 10 9+1 = 1.0 10 1 = 1.0 10.. 10 17 10 9 =. 17 10 10 9 =. 17-9 10 = 1.7 10 8 7. ( 10 1 )(.8 10 ) = (.8 ) ( 10 1 10 ) = (.8 ) 10 1+ = 1.9 10 18 = 1.9 10 19 8. 8. 10 1. 10 1 = 8.. 1 10 10 1 = 8. 1-1 10. = 10 7 9. The letter E takes the place of 10.. 10 11 =.E11 10. The letter E takes the place of 10. 7. 10 - = 7.E- 11. The letter E takes the place of 10. 8 10-1 = 8E-1 1. 10 takes the place of the letter E. 7.E- = 7. 10-1. 10 takes the place of the letter E. 1.E1 = 1. 10 1 1. 10 takes the place of the letter E. 9E1 = 9 10 1 1. Sample answer: To add or subtract, rewrite the numbers to the same power of 10, add or subtract the multipliers, and rewrite the answer is scientific notation. To multipl or divide, multipl or divide the multipliers, use the rules of eponents to multipl or divide the powers of 10, and rewrite the answer in scientific notation. Independent Practice 1. (.0 10 7 ) (. 10 ) = (.0. ) ( 10 7 10 ) = (.0. ) 10 7+ = 1. 10 9 = 1. 10 10 An adult blue whale can eat 1. 10 10 krill in. 10 das. 1
17..9 10 1,000,000,000 =.9 10 1. 10 10 =.9. 10 1 10 10 =.9. 10 1-10 = 1.9 10 An adult has about 1900 times as man cells as a newborn. 18. ( 7.11 10 7 ) + ( 1.1 10 7 ) + (.10 10 7 ) = ( 7.11 + 1.1 +.10 ) 10 7 = 11.88 10 7 = 1.188 10 8 The total amount of paper, glass, and plastic waste generated is 1.188 10 8 tons. 19. (.7 10 7 ) + ( 0.1 10 7 ) + ( 0. 10 7 ) = (.7 + 0.1 + 0. ) 10 7 =.0 10 7 The total amount of paper, glass, and plastic waste recovered is.0 10 7 tons. 0. ( 1.188 10 8 ) - (.0 10 7 ) = ( 1.188 10 8 ) - ( 0.0 10 8 ) = ( 1.188-0.0 ) 10 8 = 0. 10 8 =. 10 7 The total amount of paper, glass, and plastic waste not recovered is. 10 7 tons. 1. Paper:.7 10 7 7.17 10 7 =.7 7.11 10 7-7 = 0. 10 0 = 0. 1 = 0. Glass: 0.1 10 7 1.1 10 7 = 0.1 1.1 10 7-7 = 0.71 10 0 = 0.71 1 = 0.71 Plastics: 0. 10 7.10 10 7 = 0..10 10 7-7 = 0.08 10 0 = 0.08 1 = 0.08 Plastics have the lowest recover ratio.. (.8 10 7 ) - (.1 10 7 ) = (.8 -.1 ) 10 7 =. 10 7. 10 7 more people live in France than Australia...1 10 7.9 10 =.1.9 10 7 10 =.1.9 10 7- = 0.7 10 1 = 0.7 10 = 7 The approimate average number of people per square mile in Australia is 7.. 1. 10 9.8 10 7 = 1..8 10 9 10 7 = 1..8 10 9-7 = 0.01 10 =.01 10 1 = 0.1 The population of China is about 0.1 times as great as the population of France.. First, convert 7.018 1 0 minutes to hours: 7.018 10 min 1 h 0 min = 7.018 10 10 1 = 7.018 10 10 1 = 1.198 10 hours Net, convert Mia s age, in hours, to das. 1.198 10 hours 1 da hours = 1.198 10. 10 1 = 1.198. 10 10 1 = 0.87 10-1 = 0.87 10 =.87 10 = 87 das If each ear has si 0-da months and si 1-da months, the average number of das in a month is 0.. Convert 87 das to ears. 87 das 1 month 0. das 1 ear 1 months = 1.117 ears Convert the fraction of ears into months: 0.117 ears 1 months =.77709 months 1 ear Convert the fraction of months into das: 0. das 0.77709 months 1 month =. _ 9 das Mia is 1 ears, months, and. das old. 1
. (. 10 ) 810 = (. 10 ) ( 8.1 10 ) = (. 8.1 ) ( 10 10 ) = (. 8.1 ) 10 + = 19. 10 = 1.9 10 7 There are 1,000,000 millimeters in a kilometer, so to convert millimeters to kilometers, divide the number of millimeters b 1,000,000. 1.9 10 7 1,000,000 7. = 1.9 10 7 1 10 = 1.9 1 10 7 10 = 1.9 10 1 = 1.9 10 = 19. Sample answer: Courtne covered 1.9 10 7 mm or 19. km during her run. 9.0 10 1.08 10 8 9 1 10 10 8 9 1-8 10 10 0,000 The average US public debt per American was about $ 10, or $0,000 per person in October 010. Focus on Higher Order Thinking 8. Sample answer: You can add or subtract numbers written in scientific notation onl if their powers of 10 are the same. You can multipl and divide numbers written in scientific notation that have different powers. The laws of eponents are used to combine the powers. 9. ( 8 10 ) ( 10 9 ) = ( 8 ) ( 10 10 9 ) = ( 8 ) 10 +9 = 0 10 1 = 10 1 The student was off b a power of 10. The correct product is 10 1. 0. (.87 10 1 ) - ( 7 10 10 ) ( 10 7 ) + (.1 10 8 ) = ( 87 10 10 ) - ( 7 10 10 ) ( 10 7 ) + (.1 10 8 ) = = = = = ( 87-7 ) 10 10 ( 10 7 ) + (.1 10 8 ) 80 10 10 ( 10 7 ) + (.1 10 8 ).8 10 1 ( 10 7 ) + (.1 10 8 ).8 10 1 ( 10 7 ) + ( 1 10 7 ).8 10 1 ( + 1 ) 10 7 =.8 10 1 10 7 =.8. 1 10 10 8 =.8 1-8 10. = 0.7 10 = 7. 10 Sample answer: First, simplif the numerator b rewriting both numbers to the same power of 10 ( 10 10 ) and subtracting to get 80 10 10 or.8 10 1. Then simplif the denominator b rewriting both numbers to the same power of 10 ( 10 7 ) and adding to get 10 7 or. 10 8. Finall, divide the multipliers (.8. ) to get 0.7, use the 1 division rule for eponents ( 10 ) 10 to get 10, and 8 rewrite 0.7 10 in scientific notation as 7. 10. MODULE Read to Go On? 1. As the eponent decreases b 1, the value of the power is divided b. 0 = 1-1 = 1 - = 1 9 - = 1 7 - = 1 81. An number raised to the power 0 equals 1. 0 = 1. As the eponent increases b 1, the value of the power is multiplied b. 0 = 1 1 = = 1 = =. 8 8 7 = 8 +7 = 8 10. 1 1 = 1 - = 1. ( 10 ) = 10 = 10 1 7.,000 Move the decimal point in,000 to the left places to get.,000 = 10 8. 91,007,00 Move the decimal point in 91,007,00 to the left 7 places to get 9.1007. 91,007,00 = 9.1007 10 7 1
9. To write 1.09 10 9 in standard notation, move the decimal point 9 places to the right. 1.09 10 9 = 1,09,00,000 10. To write 10 in standard notation, move the decimal point places to the right. 10 = 00 11. Move the decimal point in 0.0 to the right places to get. 0.0 = 10-1. Move the decimal point in 0.000701 to the right places to get 7.01. 0.000701 = 7.01 10-1. To write 8.9 10 - in standard notation, move the decimal point places to the left. 8.9 10 - = 0.000089 1. To write.1 10 - in standard notation, move the decimal point places to the left..1 10 - = 0.01 1. ( 7 10 ) - (. 10 ) = ( 7 -. ) 10 = 1.7 10 1. (. 10 ) + ( 7.1 10 ) = ( 0. 10 ) + ( 7.1 10 ) = ( 0. 7.1 ) 10 = 7. 10 17. ( 10 ) (. 10 ) = (. ) ( 10 10 ) = ( 10.8 ) ( 10 + ) = ( 10.8 ) ( 10 10 ) = 1.08 10 11 18. 7.8 10 9 10 = 7.8 10 9 10 =. 10 9 19..0 10.791 10 7 =.0.791 10 9 10 7 =.0 9-7 10.791 = 0.777 10 = 7.77 10 1 0. Sample answer: Ver large numbers, such as distances in space, and ver small numbers, such as the sizes of atomic particles, can be written in scientific notation. 17
UNIT MODULE Proportional Relationships Are You Read? _ 0.7 1. 8.000 _- 00 00 _-0 0 _-0 0 0.7. 0. 0. = 0. 10 0. 10 = 0.7.00 _- 80 0 _- 0 0 0.7. 0.1 0. = 0.1 100 0. 100 = 1 0 _ 0. 0 1.00 _-1.00 100 _-100 0 0.. 0.9 0.7 = 9 7 _ 0. 7 9.00 _-7 0 10 _-10 0 0. _ 0.8..0 _-0 0 0.8 Solutions Ke Proportional and Nonproportional Relationships and Functions = 0.9 100 0.7 100 18 _ 0.0. 0.10 _-10 0 0.0 _ 0. 7. 1.0 _-8 70 _-70 0 0. _ 0. 8. 1 7.0 _-70 0 0. _ 0.0 9. 10 0.0 _-0 0 0.0 10. 0 18 = 10 11. 0 18 = 10 10 9 = 10 = 9 1 = 0 7 1 = 0 7 7 = 0 7 = 0 = 1. = 1 = 1 1 = 1 = = 1 1. 11 = 1 10 11 1 1 = 1 10 1 1 = 1 10 1 = 10 = 10
1. 8 = 1 8 1 = = = 1. 9 = 1 7 9 = 1 7 7 = 1 7 = 1 = 7 1. 1 = 17. 8 1 8 = = = 0 1 = 0 1 = = = 18. = 18 = 18 = = LESSON.1 Your Turn. Number of hours Number of biccles 1 1 = 1, 0 1 1 0 0 = 1, = 1, 0 = 1 Let represent the number of hours. Let represent the number of biccles. = 1. The point (, ) indicates that in hours, the hiker hiked miles.. Time (h) 10 1 Distance (mi) 1 18, 1 10 =, 18 1 = Let represent the number of hours. Let represent the number of miles. = 19 Guided Practice 1. A proportional relationship is a relationship between two quantities in which the ratio of one quantit to the other quantit is constant.. When writing an equation of a proportional relationship in the form = k, k represents the constant of proportionalit.. a. Time (weeks) 1 8 10 Time (das) 7 1 8 70 b. 7 1 = 7, 1 = 7, 8 = 7, 8 = 7, 70 10 = 7 Let represent the time in weeks. Let represent the time in das. The equation that describes the relationship is = 7.. Ogen atoms 17 10 Hdrogen atoms 10 0 Let represent the number of ogen atoms. Let represent the number of hdrogen atoms. =. =, 10 =, 17 =, 0 10 =. Distance (in.) 1 Actual distance (mi) 0 1 = 0, 0 0 0 90 = 0, 90 = 0 Let represent the distance in inches. Let represent the actual distance in miles. = 0.. Sample answer: Use the equation to make a table with -values and -values. Then graph the points (, ) and draw a line through the points. Independent Practice 7. 0 =, 8 10 0 = 17 No; the ratios of the numbers in each column are not equal. 8. Degrees Fahrenheit 00 10 10 80 0 O 0 0 0 80 100 Degrees Celsius Sample answer: The graph is a line starting at (0, ) and slanting upward to the right.
9. a. Sample answer: The account had a balance of $100 to begin with. b. Sample answer: Have Ralph open the account with no initial deposit and then put $0 in ever month. 10. Sample answer: Let represent the number of nickels ou have. Let represent the amount of mone ou have in dollars. = 1 0 11. 8 0 = 8 = 0 8 = 80 8 8 = 80 8 = 10 1. 1 8 = 1 1 1 = 8 1 = 8 1 8 = 8 8 18 = 1. a. Time (min) 1 Distance (in.) 10 0 0 0 0 b. 10 1 = 10, 0 = 10, 0 = 10, 0 Let represent the time in minutes. Let represent the distance in inches. = 10 c. = 10 8 = 10 = 8 10 = 8. It takes 8. minutes. = 10, 0 = 10 Focus on Higher Order Thinking 1. Sample answer: All of the graphs represent real-world data for which both and take on onl nonnegative values. When both coordinates are positive, the corresponding point will be in the first quadrant or on the aes. If either or or both could be negative, then other quadrants would be needed. 1. Length of side 1 of square Perimeter of 8 1 1 0 square Area of 1 9 1 square 0 a. 8 =, 1 Yes. The ratio of the perimeter of a square to its side length is alwas. =, 1 =, 1 =, 0 = b. 1 = 1, 1 = No. The ratio of the area of a square to its side length is not constant. 1. The new constant of proportionalit is the reciprocal of the original constant of proportionalit. LESSON. Your Turn 1. Find the rates of change. 18-0 0. 0 = 18 0. = 1 18 1. 0. = 1 1 = 1 1 1. = 0. = 10 The rates of change are variable.. Use points (, ) and (8, ). rise = + run = + rise run = Slope = Guided Practice 1. - 1 1 - = 9 = 1 9-7 - 1 = 1 = 1-9 7-7 = 1 8 = 1 The rates of change are constant.. 1 - - = = - 1 9 - = 1 The rates of change are variable.. - 1-1 = 1 8. - 1 - = 1 80 The rates of change are variable. 7-8 - = 8 = 19 1-7 7 - = 7 = 19 171-1 = 9-8 7 = 19 The rates of change are constant.. Use points (1, 00) and (, 00). change in distance = 00-00 change in time - 1 = 00 = 00 ft per min 1
. Use points (1, 00) and (, 800). change in distance = 800 00 change in time 1 = 00 = 00 ft per min The rate of change is 00 ft per min. 7. Use points (-1, ) and (1, -). rise = - run = rise run = - The slope is -. 8. Use points (-, -) and (, ). rise = run = rise run = = The slope is. 9. Sample answer: Find the coordinates of two points on the line. Then divide the change in -values (the rise) from one point to the other b the change in -values (the run). Independent Practice 10. a. Slope of _ EF = rise run = - - (-) = - 9 = - 1 Slope of _ EF = - 1 Slope of _ FG = rise run = - - - = - - = Slope of _ FG = Slope of _ GH = rise run = -1 - (-) - - = -9 = - 1 Slope of _ GH = - 1 Slope of _ HE = rise run = -1 - - - (-) = - - = Slope of _ HE = b. The slopes of opposite sides are the same. c. The slopes of adjacent sides are negative reciprocals of each other. 11. The total distance is. miles + 7. miles, or 1 miles. The total amount of time is 8 minutes. 8 minutes = 8 hour or 0.8 hour. 0 The average rate of speed = 1 mi = 1 mph. 0.8 hr 1. Use the first two points to find the slope of the line. slope = rise run = 8 = 1 The slope found using the second two points must also be 1. - - n - 8 = 1 - n - 8 = 1 n - 8 = -1 n = - 1. a. The container lost gallons of water in 10 minutes. gal = 0. gal per min 10 min The water is leaking at a rate of 0. gal per min. b. Solve the proportion gal 10 min = gal min. 10 = = 0 = The container will be empt in minutes. 1. He used the ratio of the change in over the change in instead of the ratio of the change in over the change in. 1. 10-10 - D B - O 10 - C - -10 a. Slope of _ AB = rise run = 1 - - = - = -1 Slope of _ AB = -1 Slope of _ BC = rise run = - - 1 0 - = - - = 1 Slope of _ BC = 1 Slope of _ CD = rise run = 1 - (-) - - 0 = - = -1 Slope of _ CD = -1 Slope of _ DA = rise run = 1 - - - = - - = 1 Slope of _ DA = 1 b. The slopes of opposite sides are the same. c. Yes. The slopes of opposite sides are alwas the same. Focus on Higher Order Thinking 1. Yes. The slope of a line is constant. Therefore, the calculated slope will be the same no matter which two points are chosen. A 1
17. Sample answer: The lines are equall steep, but the one with the positive slope slants upward from left to right and the one with the negative slope slants downward from left to right. 18. Sample answer: The rise along the -ais is zero, while the run along the -ais is not zero. The slope is zero run or zero. LESSON. Your Turn. Distance (mi) 10 O Tomas s Ride 10 Time (min) Slope = rise run =, so the unit rate is. His rate of speed is mi/min.. The slope of the graph of = 7 is 7. Therefore Plane A s rate of speed is 7 mph. The graph for Plane B contains the point (1, ), so the slope of the graph is. Therefore, Plane B s rate of speed is mph. Plane B is fling at a faster rate of speed. Guided Practice 1. Use the points (0, 0) and (1, 10). Slope = rise run = 10 =, so the unit rate is 1 mi/hr.. Use the points (, ) and (8, 10). Slope = rise run =, so the unit rate is mi/hr.. The slope of the graph of = 0. is 0., so Henr s rate is 0. mph. Use the point (, ) to find the slope of the graph given for Clark s hike. Slope = rise run = = = 1. So Clark s rate is 1. mph. Therefore, Clark is faster.. 1 = 0 1 = = 0 = 90 = 1 = 1. 1 = 1 = 18 8 = = 8 = 8. Sample answer: Table of values: The ratio of to gives the unit rate and the slope. Equation: If the equation can be written as = m, then m is the unit rate and the slope. Graph: When the line passes through the origin, the slope of the graph is also the unit rate. Independent Practice 7. a. Time (min) 8 1 1 0 Distance (mi) 9 1 1 b. Distance (mi) 0 10 O Migration Flight 10 0 Time (min) c. Use the point (8, ). Slope = rise run = = 8 The slope is, which means that the unit rate of migration is mi/min. 8. A unit rate is a rate in which the second quantit in the comparison is one unit. 9. a. Machine 1: slope = unit rate = 0. = 0. gal/s 1 For Machine, use the point (8, ). Slope = rise run = = 8 Machine : slope = unit rate = = 0.7 gal/s b. Since 0.7 > 0., Machine is working at a faster rate. 10. In the equation = 1, 1 is the slope, so Patrick s 9 9 rate is 1 kilometer per minute. Jennifer s rate is 9 or 1 kilometer per minute. Since 1 > 1, Jennifer 0 8 8 9 has the faster training rate. Focus on Higher Order Thinking 11. The slope and the unit rate are both.7; If the graph of a proportional relationship passes through the point (1, r), then r equals the slope and the unit rate, which is $.7/min.
1. Car B is traveling at the faster rate; The slope of the graph is equal to the unit rate of speed. Car A: slope = 7. - 0 0. - 0 = 7. 0. = mph Car B: slope = 0 0 0 = 0 = 0 mph So, Car B is traveling faster. 1. After 1 1 minutes, gallons will have been pumped into the pool; Sample answer: The unit rate is = 18 gal/min. So 1 1 minutes after 1 minutes, an additional 18 1 1 = 7 gallons will be pumped in, so the total is 1 + 7 = gal. MODULE Read to Go On? 1. = 1.,. = 1., = 1., 7. = 1. The constant of proportionalit is 1... = 1., 7. = 1., 0 = 1. The constant of proportionalit k is 1.. The equation is = 1... Use (0, 0) and (1, ). rise run = 1 The slope is.. Use (0, 0) and (1, -). rise run = - 1 The slope is -.. Train A Train B 10 = 70 10 = 7 Train A has a rate of 70 km per hour; Train B has a rate of 7 km per hour. Train B is faster.. Sample answer: The graph of a proportional relationship is a line that passes through the origin. The slope of the line is the unit rate of change.
MODULE Nonproportional Relationships Are You Read? 1. - (-) + 8. - - -9. - 10 -. - - (-) - + -. 8 - (-8) 8 + 8 1. 9-7. - - 9-1 = 1 - = 1 - = 1 ( ) - = 1 ( ) - = = (number of hours) (earnings in dollars) 0. = - + 1 = - + 1 = -(-1) + 1 = -( 0 ) + 1 = = 1 = - + 1 = - + 1 = -( 1 ) + 1 = -( ) + 1 = -1 = - -1 0 1 1-1 - 8. 0 - (-) 0 + 9. 1 - (-9) 1 + 9 1 10. - - (-) - + - - - O - - 11. -7-10 -17 1. - 1-9 1 1. 10 O D (, 7) B (0, ) E (, ) C (8, 0) 10 Guided Practice 1. = + = + = (-) + = (-1) + = 1 = = + = + = ( 0 ) + = ( 1 ) + = = 7 = + = ( ) + = 9 - -1 0 1 1 7 9 LESSON.1 Your Turn 1. Sample answer: = 1 - = 1 - = 1 ( ) - = 1 ( ) - = 0 =
. = 8 - = 8 - = 8 ( -8) - = 8 ( 0 ) - = -8 = - = 8 - = 8 - = 8 ( 8 ) - = 8 ( 1 ) - = - = 1 = 8 - = 8 ( ) - = -8 0 8 1-8 - - 1. Calculate the value of for each pair of values. is undefined. 0 7 =., 11 =.7, 1 =., 19 8 =.7 The relationship is not proportional because the ratio of to is not constant.. The relationship is not proportional because although the graph is a line, it does not pass through the origin.. = - 1 = - 1 = (-) - 1 = (-1) - 1 = - = - = - 1 = - 1 = ( 0 ) - 1 = ( 1 ) - 1 = -1 = 0 = - 1 = ( ) - 1 = 1 - -1 0 1 - - -1 0 1 - - O - -. Sample answer: Choose values that make sense in the contet. For eample, the number of games plaed must be a whole number. Independent Practice 7. The graph is a set of unconnected points; Sample answer: The values of must be whole numbers, since ou cannot bu a fractional part of a lunch. 8. The graph is a solid line; Sample answer: The values of can be an number, since the distance remaining can be measured at an moment in time. 9. a. Sample answer: = 8 + 1 = 8 + 1 = 8 ( 0 ) + 1 = 8 ( 1 ) + 1 = 1 = 0 = 8 + 1 = 8 + 1 = 8 ( ) + 1 = 8 ( ) + 1 = 8 = = 8 + 1 = 8 ( ) + 1 = (number of ears renewed) (total cost in dollars) b. Total cost ($) 8 0 1 8 0 1 1 0 8 Magazine Subscription Costs O 8 10 1 1 Number of ears renewed c. Sample answer: The relationship is not proportional because although the graph is a line, it does not pass through the origin. Also, the ratio of the total cost to the number of ears is not constant. d. No; The graph is a set of unconnected points because the values of represent whole numbers of ears. 10. Sample answer: In a proportional relationship the ratio of each -value to its corresponding -value must be constant. In addition, its graph must be a line that passes through the origin. 11. Sample answer: In a table, the ratios of each -value to its corresponding -value will not be equal. The graph will not pass through the origin. The equation can be written in the form = m + b, with b 0.
Focus on Higher Order Thinking 1. Sample answer: George s observation is true, but his claim is false. There is a constant rate of change. However, the relationship is not proportional because the ratio of to ( 90, 7, 70, 7., ) is not constant. 1. At most one; Sample answer: A line representing a proportional relationship must pass through the origin. A line parallel to it cannot also pass through the origin, so at most one of the lines can represent a proportional relationship. LESSON. Your Turn 1. Use the points (, ) and (, ) to find the slope. m = - - = 10 = The slope m is. The difference between the -values is. The difference between the -values is 10. Working backward, when = 0, = 1. The -intercept b is 1.. Use the points ( 1, 8 ) and (, 1 ) to find the slope. m = 1-8 - 1 = 7 1 = 7 The slope m is 7. The difference between the -values is 1. The difference between the -values is 7. Working backward, when = 0, = 1. The -intercept b is 1. Guided Practice 1. Use the points ( 0, 1 ) and (, -) to find the slope. m = - - 1-0 = - = - The slope is -. The -intercept is 1, since 1 is the -coordinate of the point where the graph intersects the -ais. slope m = - -intercept b = 1. Use the points ( 0, -1) and (, 0 ) to find the slope. m = 0 - (-1) = - 1 0 = The slope is. The -intercept is -1, since -1 is the - coordinate of the point where the graph intersects the -ais. slope m = -intercept b = -1. Use the points ( 0, -) and (, 1 ) to find the slope. m = 1 - (-) - 0 = The slope is. The -intercept is -, since - is the -coordinate of the point where the graph intersects the -ais. slope m = -intercept b = -. Use the points ( 0, 9 ) and (, 0 ) to find the slope. m = 0-9 - 0 = -9 = - The slope is -. The -intercept is 9, since 9 is the -coordinate of the point where the graph intersects the -ais. slope m = - -intercept b = 9. Use the points (, 7 ) and (, 1 ) to find the slope. m = 1-7 - = = The slope is. The difference between the -values is. The difference between the -values is. Working backward, when = 0, = 1. The -intercept is 1. slope m = -intercept b = 1. Use the points (, 10 ) and ( 10, 100 ) to find the slope. m = 100-10 = 10 - -0 = - The slope is -. The difference between the -values is. The difference between the -values is -0. Working backward, when = 0, =10. The -intercept is 10. slope m = - -intercept b = 10 7. Find the slope b substituting the coordinates of two points on the line into the slope formula. Find the -intercept b identifing the -coordinate of the point where the graph crosses the -ais. Independent Practice 8. Use the points ( 1, 1 ) and (, 17 ) to find the rate of change. rate of change = 17-1 = - 0 1 1 = 0 The rate of change is $0 per room. The difference between the -values is 1. The difference between the -values is 0. Working backward, when = 0, = 7. The initial value is $7, which represents a flat fee no matter how man rooms are cleaned. 9. a. Use the points ( 1, 17 ) and (, 9 ) to find the hourl rate. hourl rate = 9-17 - 1 = 1 1 = 1 The rate to rent a paddleboat is $1 per hour. To find the cost to park for a da, find the initial value. The difference between the -values is 1. The difference between the -values is 1. Working backward, when = 0, =. It costs $ to park for the da. b.. hours $1 per hour + $ = $7 $7 = $.0 Lin will pa $.0.