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Transcript
Objective 1.1A New Vocabulary set elements natural numbers prime number composite number whole numbers integers negative integers positive integers rational numbers irrational numbers real numbers graph of a real number variable additive inverses opposites absolute value New Symbols is an element of () is not an element of ( is less than (<) is greater than (>) absolute value | | Discuss the Concepts 1. Explain the difference between the set of natural numbers and the set of whole numbers. 2. What is a terminating decimal? What is a repeating decimal? Give examples of each. 3. Explain the difference between rational numbers and irrational numbers. 4. What is the additive inverse of a number? 5. What is the absolute value of a number? Concept Check Determine whether the statement is true or false. 1. All integers are rational numbers. True 2. If a number is an integer, then it is a whole number. 3. The absolute value of a number is positive. False False; |0| = 0 4. Given |x| = 3, then x is a positive integer. False 5. If x < 0, then the absolute value of x is -x. True Optional Student Activity Let S be the set of positive integers that have the following properties: a. when divided by 6 leave a remainder of 5 b. when divided by 5 leave a remainder of 4 c. when divided by 4 leave a remainder of 3 d. when divided by 3 leave a remainder of 2 e. when divided by 2 leave a remainder of 1 1. Find three elements of set S. The least common multiple of 2, 3, 4, 5, and 6 is 60; one less than any multiple of 60 will generate an element of S. The form of these elements is 60k - 1, where k is a natural number. Some possible values are 59, 119, 179, and 239. 2. Find the minimum value of S. The minimum value occurs when k = 1. Therefore, the minimum value is 59. Objective 1.1B New Vocabulary roster method infinite set finite set empty set null set set-builder notation New Symbols empty set (or { } ) null set (or { } ) Discuss the Concepts 1. Give an example of a finite set. Give an example of an infinite set. 2. Give an example of the empty set. Concept Check Determine whether the statement is true or false. 1. 7 {2, 3, 5, 7, 9} 2. 4 {-8, -4, 0, 4, 8} True False 3. {0, 1, 2, 4}False 4. {a} {a, b, c, d, e} False 5. 5 {x | x prime numbers } True 6. 19 {x | x integers } 7. 3.9 {x | x real numbers } True True Optional Student Activity 1. List the elements from the set { 30, 22, 22 , 22, 18 14 , , 3} whose value is between -3 and 5. 5 3 14 3 2. Let A = {0, 2, 4, 6} and B = {1, 2, 3, 4, 5}. Use the roster method to list the elements in the set that satisfy {x | x A and x B} . {0, 6} 3. One hundred students were asked if they liked country western music or jazz. The results were that 5 students liked neither, 85 liked country western, and 23 liked jazz. How many students liked both types of music? 13 students Objective 1.1C New Vocabulary union of two sets intersection of two sets interval notation closed interval open interval half-open interval endpoints of an interval New Symbols union ( intersection ( infinity ( negative infinity (- Discuss the Concepts 1. Explain the difference between the union of two sets and the intersection of two sets. 2. Explain the difference between {x|x < 5} and {x|x ≤ 5}. 3. Explain the similarities and differences between open intervals and closed intervals. Concept Check 1. Is the intersection of two infinite sets always an infinite set? Why or why not? 2. Is the union of two infinite sets always an infinite set? Why or why not? 3. Find A B A {x | x whole numbers} B {x | x positive integers} {x | x whole numbers} 4. Find A B A {x | x rational numbers} B {x | x real numbers} {x | x rational numbers} 5. Find A B A {x | x rational numbers} B {x | x irrational numbers} Optional Student Activity Some search engines make use of the operators “AND” and “OR.” For instance, in Windows XP, the instructions for a full-text search state that entering “computer and monitor” returns topics containing both words. Entering a search for “computers or monitors” returns topics containing either word or both words. a. Explain the search described above in the context of the intersection of two sets. b. Explain how entering a search for “inequality and symbol” differs from a search for “inequality or symbol.” Optional Student Activity Given A {0, 2, 4, 6}, B {1, 2, 3, 4}, C {x | x 3 , x integers}, and D {x | x 4 , x integers}, find each union or intersection. 1. A C {4, 6} 2. B C {4} 3. B C {0, 2, 4} 4. B D {1, 2, 3, 4} 5. B D {x| x ≤ 4, x integers} 6. B D {x| x ≥ 1, x integers} Answers to Writing Exercises 108. -3 > x > 5 means the numbers that are less than -3 and greater than 5. There is no number that is both less than -3 and greater than 5. Therefore, this is incorrect. Objective 1.2A New Vocabulary multiplicative inverse reciprocal Discuss the Concepts 1. Explain the meanings of the words minus and negative. 2. Is subtraction a commutative operation? No 3. When is a ÷ b = b ÷ a? When a = b and a ≠ 0, b ≠ 0 Concept Check Determine whether the statement is always true, sometimes true, or never true. 1. The sum of two numbers with opposite signs is negative. Sometimes true 2. The product of two numbers with the same sign is positive. 3. The sum of a number and its additive inverse is zero. Always true Always true 4. The sum of two integers is greater than either of the two integers. Sometimes true 5. The difference between two integers is smaller than either of the two integers. Sometimes true 6. If two integers are multiplied, the product is greater than either of the two integers. 7. If the value of -5y is a positive integer, then y is a positive integer. 8. If -8x = 0, then x is a negative integer. Never true Optional Student Activity 1. Simplify: 1 – 2 + 3 – 4 + ∙ ∙ ∙ – 98 + 99 50 2. Simplify: – 1 + 2 – 3 + 4 – 5 + ∙ ∙ ∙ + 98 – 99 Objective 1.2B Vocabulary to Review rational number integer [1.1A] [1.1A] -50 Never true Sometimes true New Vocabulary least common multiple (LCM) of the denominators greatest common factor (GCF) Discuss the Concepts 1. Can a number be both a rational and an irrational number? 2. Are there any integers that are not rational numbers? Are there any rational numbers that are not integers? 3. What are the real numbers? 4. Is there a smallest positive rational number? Is there a largest positive rational number? 5. Given any two distinct rational numbers, is it always possible to find a rational number between the two given numbers? If so, explain how to find one. Yes. For example, add the two numbers and then divide by 2. Concept Check 1. When two rational numbers are multiplied, it is possible for the product to be less than either factor, greater than either factor, or a number between the two factors. Give examples of each of these occurrences. 2. Suppose the numerator of a fraction is a fixed number—for instance, 5. How does the value of the fraction change as the denominator increases? The fraction decreases in value. Optional Student Activity 1. If b has a value between 3 and 5 and c has a value between 0.5 and 1, what values is b between? c 3 and 10 2. Express the sum as a single fraction in simplest form. 1 1 1 1 1 2 2 3 3 4 9 10 9 10 3. If x and y are positive real numbers and x + y =10, what is the minimum value of 1 1 x y ? 2 5 Objective 1.2C New Vocabulary exponent base factored form exponential form power Discuss the Concepts For the expression 45, which is the base and which is the exponent? What does the base represent? What does the exponent represent? Concept Check Rewrite each expression as an exponential expression. 1. seven to the fourth power 45 2. x to the third power x3 3. six to the nth power 6n 4. b ∙ b ∙ b ∙ b ∙ b ∙ b ∙ b b7 Optional Student Activity 1. To express 10 as a sum of different powers of 2, we could write 10 = 23 + 21. The sum of the exponents of these powers is 4. When 100 is expressed as a sum of different powers of 2, what is the sum of the exponents of these powers? 26 + 25 + 22; 13 2. Given that n is a positive integer, find the smallest value of n for which 12n is divisible by 29. 5 Objective 1.2D New Vocabulary grouping symbols complex fraction main fraction bar New Procedures Order of Operations Agreement Discuss the Concepts You may want to use Exercises 93 and 94 on page 27 for a class discussion of the Order of Operations Agreement. Concept Check Arrange the expressions in order from the least value to the greatest value. a. 3[(12 ÷ 3) – (-3)] + 5 15 5 3 2 2 9 b. 30 6 13 2 24 1 c. 3 3 1 8 3 d. 0.3 1.3 2.1 4.7 2 23 , d. (4.892), b. (17), and a. (26) 24 c. Optional Student Activity 1. Simplify: 1 1 1 1 2 1 3 1 4 1 10 1 2 3 4 10 45 2. Define a @ b as a ∙ b + b. Use this definition to find the value of x @ (y @ z) when x = 1.7, y = 2.3, and z = -1.8. -16.038 Answers to Writing Exercises 1. a. Students should paraphrase the rule: Add the absolute values of the numbers; then attach the sign of the addends. b. Students should paraphrase the rule: Find the absolute value of each number; subtract the smaller of the two numbers from the larger; then attach the sign of the number with the larger absolute value. 2. To rewrite 8 – (-12) as addition of the opposite, change the subtraction to addition and change -12 to the opposite of -12: 8 – (-12) = 8 + 12. Answers to Writing Exercises 39. a. The least common multiple of two numbers is the smallest number that is a multiple of each of those numbers. b. The greatest common factor of two numbers is the largest integer that divides evenly into both numbers. 40. To divide two fractions, change the division sign to a multiplication sign, write the reciprocal of the second fraction, and then multiply the two fractions. Answers to Writing Exercises 93. We need an Order of Operations Agreement to ensure that there is only one way in which an expression can be correctly simplified. 94. Students should describe the steps in the Order of Operations Agreement: Step 1: Perform operations inside grouping symbols. Step 2: Simplify exponential expressions. Step 3: Do multiplication and division as they occur from left to right. Step 4: Do addition and subtraction as they occur from left to right. Objective 1.3A New Properties Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Addition Property of Zero Multiplication Property of Zero Multiplication Property of One Inverse Property of Addition Inverse Property of Multiplication Distributive Property Vocabulary to Review additive inverse [1.1A] multiplicative inverse [1.2A] reciprocal [1.2A] Concept Check Classify each statement below as illustrating the Commutative Property of Addition, the Associative Property of Addition, the Commutative Property of Multiplication, or the Associative Property of Multiplication. a. 13 + 8 = 8 + 13 Comm. Prop. of Add. b. (6 + 2) + 4 = 6 + (2 + 4) Assoc. Prop. of Add. c. 10(5) = 5(10) Comm. Prop. of Mult. d. (3 ∙ 8)9 = 3(8 ∙ 9) e. x + 27 = 27 + x Assoc. Prop. of Mult. Comm. Prop. of Add. f. y + (1 + 6) = (y + 1) + 6 g. pq = qp Assoc. Prop. of Add. Comm. Prop. of Mult. h. x(yz) = (xy)z Assoc. Prop. of Mult. Optional Student Activity A, B, C, and D are four distinct real numbers such that A+B=A B∙A=B C+D=B C(B + A) = A C–D=A Find the values of a. A, b. B, c. C, and d. D. a. A = 2, b. B = 0, c. C = 1, and d. D = -1. Optional Student Activity In Objective 1.2D, complex fractions were simplified by rewriting the numerator and denominator of the complex fraction as single fractions and then dividing the numerator by the denominator. However, a different approach is to multiply the numerator and denominator by the LCM of the denominators. Now that the Distributive Property has been presented, you might have students simplify a complex fraction by this alternative method. Here is the simplification of the complex fraction on page 23. 3 1 3 1 3 1 60( ) 60( ) 60( ) 4 3 4 3 4 3 1 1 1 2 60( 2) 60( ) 60(2) 5 5 5 45 20 25 25 12 120 108 108 Objective 1.3B New Vocabulary variable expression terms variable terms constant term numerical coefficient variable part of a variable term evaluating a variable expression Discuss the Concepts Can the value of a variable ever equal 0? If not, explain why not. If so, give an example of an actual situation in which it makes sense for the value of a variable to be 0. Concept Check Find the smallest possible value for the expression x 4 1 (when x = 2) 2 5 when x is a positive integer. x Optional Student Activity Consider the following variable expressions: a b 2 , a2 b2 , a b , a3 b3 , a b , a4 b4 . 3 4 1. By trying different values of a and b, is a b 2 a 2 b 2 always true? 2. By trying different values of a and b, is a b 3 a 3 b3 always true? 3. By trying different values of a and b, is a b 4 a 4 b 4 always true? 4. On the basis of your answers to 1–3, is a b n No No No a n b n always true when n is a natural number? No Objective 1.3C New Vocabulary like terms combining like terms Concept Check Name the property that justifies each lettered step used in simplifying the expression. 1. 3 x y 2 x a. 3x 3 y 2x b. 3 y 3x 2 x c. 3 y 3x 2 x Distributive Prop. Comm. Prop. of Add. Ass. Prop. of Add. d. 3 y 3 2 x 3 y 5x Distributive Prop. 2. y 3 y a. y y 3 Comm. Prop. of Add. b. y y 3 Ass. Prop. of Add. c. 1y 1y 3 d. 1 1 y 3 2y 3 Mult. Prop. of One Distributive Prop. 3. 5 3a 1 a. 5 3a 5 1 Distributive Prop. b. (5 3)a 5(1) 15a 5 1 c. 15a 5 Ass. Prop. of Mult. Mult. Prop. of One Optional Student Activity Simplify: x a x b x c . . . x y x z 0 Objective 1.4A New Vocabulary See the list of verbal phrases that translate into mathematical operations. Concept Check 1. Write five phrases that would translate into the expression y + 6. For example: the sum of y and 6; the total of y and 6; 6 more than y; 6 added to y; y increased by 6 2. Write four phrases that would translate into the expression c – 18. For example: 18 less than c; the difference between c and 18; c minus 18; c decreased by 18 3. Write three phrases that would translate into the expression 5b. For example: 5 times b; the product of 5 and b; 5 multiplied by b 4. Write three phrases that would translate into the expression x . 4 For example: x divided by 4; the quotient of x and 4; the ratio of x to 4 Optional Student Activity Complete each statement with the word even or odd. 1. If k is an odd natural number, then k + 1 is an ______ natural number. 2. If n is a natural number, then twice n is an _____ natural number. Even Even 3. If m and n are even natural numbers, then the product of m and n is an ______ natural number. Even 4. If m and n are odd natural numbers, then the sum of m and n is an _____ natural number. Even 5. If m and n are odd natural numbers, then the product of m and n is an _______ natural number. Odd 6. If m is an even natural number and n is an odd natural number, then the sum of m and n is an _____ natural number. Odd Objective 1.4B Concept Check 1. The sum of two numbers is 14. Express both numbers in terms of the same variable. Then show that the sum of the two expressions is 14. 2. A wire for a guitar is 12 ft long and is cut into two pieces. Express the lengths of the two pieces in terms of the same variable. Then show that the sum of the two lengths is 12. Optional Student Activity Translate each of the following formulas into a variable expression, replacing the word is with an equals sign. Use the letter P for perimeter and the letter A for area. The symbol for pi is . 1. The perimeter of a rectangle is the sum of twice the length and twice the width. P = 2L + 2W 2. The area of a rectangle is the product of the length and the width. 3. The area of a triangle is one-half of the base times the height. 4. The circumference of a circle A = LW 1 A bh 2 is the product of pi and the diameter. 5. The area of a circle is the product of pi and the square of the radius. C = d A = r2 Answers to Focus on Problem Solving: Polya’s Four-Step Process 1. (Exercise 1 is on page 44.) Understand the problem. We must determine the number of ounces of water the cup will hold. To do so, we need the volume of the cup. The dimensions of the cup are given in inches, so the volume will be in cubic inches. We will need to convert cubic inches to fluid ounces. Devise a plan. Consult a reference book to find the formula for the volume of a cone (V 1 r 2 h) . The 3 conversion rate for cubic inches to fluid ounces is given on page 44 of the text: 1 in3 0.55 fl oz. The plan is to find the volume of the cup in cubic inches and then convert the volume to fluid ounces. Carry out the plan. Find the volume of the cone. r 1.5 , h 4 V 1 r 2h 3 1 (1.5)2 (4) 3 9.424778 in3 3 Convert 9.424778 in to fluid ounces. V 9.424778 0.55 5.183628 fl oz The cup will hold about 5.18 fl oz. Review the solution. It seems reasonable that a conical cup 4 in. tall would hold about 5 fl oz. 2. Understand the problem. We are asked to determine the dimensions of a 12-ounce soft drink can. We need to approximate the length of a hand. We also need to know the formula for the volume of a right circular cylinder, and we need to convert 12 fl oz to cubic inches. Devise a plan. 2 From a reference book, find the formula for the volume of a right circular cylinder V r h . We will approximate the length of a hand to be 8 in. We can use this approximation and the formula C 2 r to find the radius of the can. The conversion rate for cubic inches to fluid ounces is 0.55 fl oz 1 in . After finding the radius of the can and converting 12 fl oz to cubic inches, we will find the height of the can. 3 Carry out the plan. The length of the hand is 75% of the circumference. 0.75C 8 C 10.67 in. Use the formula C 2 r to find the radius. 2 r C 2 r 10.67 r 1.70 in. Convert 12 fl oz to cubic inches. 12 0.55 21.82 in 3 Use the formula for the volume of a right circular cylinder to find the height of the can. r2h V (1.70)2 h 21.82 h 2.40 in. The radius of the can is approximately 1.70 in. The height is approximately 2.40 in. Review the solution. The radius of a conventional soda can is smaller than 1.70 in. (only about 1.25 in.). The height is greater than 2.40 in. (almost 5 in.). But we can see that the dimensions are “in the right ball park.” Answers to Projects and Group Activities: Water Displacement 1. The volume of the cylinder is V r 2 h , where r 2 and h 10 . V r2h V (2) 2 (10) V 40 The volume of water displaced is V = LWH, where L 30 , W 20 , and H x . V LWH 40 (30)(20)x 0.21 x The water will rise approximately 0.21 cm. 2. The volume of 2 2 4 of the sphere is V r 3 where r 6 . 3 3 3 2 4 V r3 3 3 8 V (6)3 192 9 The volume of the water displaced is V LWH V LWH 192 (20)(16)x 1.88 x , where L 20 , W 16 , and H x . The water will rise approximately 1.88 in. 3. Find the volume of the statue by finding the volume of the water displaced by the statue. V LWH , where L 12 , W 12 , and H 0.42 . V 1212 0.42 60.48 3 The volume of the statue is 60.48 in . Density weight ÷ volume Density 15 60.48 0.25 3 The density of the statue is approximately 0.25 lb/ in .