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Preview of Algebra 1 Polynomials 12A Introduction to Polynomials 12-1 Polynomials LAB Model Polynomials 12-2 Simplifying Polynomials 12B Polynomial Operations LAB Model Polynomial Addition Adding Polynomials Model Polynomial Subtraction Subtracting Polynomials Multiplying Polynomials by Monomials Multiply Binomials Multiplying Binomials 12-3 LAB 12-4 12-5 LAB 12-6 KEYWORD: MT8CA Ch12 Polynomials can be used to calculate the height of fireworks. Walt Disney Concert Hall, Los Angeles 586 Chapter 12 Associative Property Vocabulary Choose the best term from the list to complete each sentence. coefficient 1. __?__ have the same variables raised to the same powers. 2. In the expression 4x 2, 4 is the __?__. Distributive Property 3. 5 (4 3) (5 4) 3 by the __?__. like terms 4. 3 2 3 4 3(2 4) by the __?__. Commutative Property Complete these exercises to review skills you will need for this chapter. Integer Operations Add or subtract. 5. 12 4 6. 8 10 7. 14 (4) 8. 9 5 9. 9 (5) 10. 9 (5) 11. 17 8 12. 12 (19) 13. 23 (5) Evaluate Expressions Evaluate the expression for the given value of the variable. 14. (x y ) z for x 8, y 3, z 4 15. 7ab 5 for a 3, b 4 16. 2(n 3)2 for n 1 17. 6(t 4)2 for t 2 Simplify Algebraic Expressions Simplify each algebraic expression. 18. 8 4x x 9 19. 9b a 11 3a 12 20. n 10m 9n 4 Area of Squares, Rectangles, and Triangles Find the area of each figure. 21. 22. 23. 15 cm 6 in. 36 cm 24 m 15 in. 24 m Polynomials 587 The information below “unpacks” the standards. The Academic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard 10.0 Preview of Algebra 1 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques . (Lessons 12-3, 12-4, 12-5, 12-6; Labs 12-3, 12-4, 12-6) Academic Vocabulary multistep more than one step technique a way of doing something Chapter Concept You use your knowledge of exponents to add, subtract, and multiply polynomials, and you use polynomials to solve problems. Example: You simplify expressions such as 3x 2 2x 5x 2 1 and x(3x 4 5x 3 x 2). Standards AF1.2 and AF1.3 are also covered in this chapter. To see these standards unpacked, go to Chapter 1, p. 4 (AF1.2) and Chapter 3, p. 114 (AF1.3). 588 Chapter 12 Study Strategy: Study for a Final Exam A cumulative final exam will cover material you have learned over the course of the year. You must be prepared if you want to be successful. It may help you to make a study timeline like the one below. 2 weeks before the final: • Look at previous exams and homework to determine areas I need to focus on; rework problems that were incorrect or incomplete. • Make a list of all formulas I need to know for the final. • Create a practice exam using problems from the book that are similar to problems from each exam. 1 week before the final: • Take the practice exam and check it. For each problem I miss, find two or three similar problems and work those. • Work with a friend in the class to quiz each other on formulas from my list. 1 day before the final: • Make sure I have pencils and scratch paper. Try This Complete the following to help you prepare for your cumulative test. 1. Create a timeline that you will use to study for your final exam. Polynomials 589 12-1 Why learn this? You can use polynomials to find the height of a firework when it explodes. (See Example 4.) California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.2 EXAMPLE Polynomials Recall that a monomial is a number, variable, or a product of numbers and variables with exponents that are whole numbers. 1 Monomials 2n, x3, 4a4b3, 7 Not monomials 5 p2.4, 2x, x , 2 g Identifying Monomials Determine whether each expression is a monomial. Vocabulary polynomial bionomial trinomial degree of a polynomial 1 x 4y 7 3 10xy 0.3 monomial not a monomial 4 and 7 are whole numbers. 0.3 is not a whole number. A polynomial is one monomial or the sum or difference of monomials. A simplified polynomial can be classified by the number of monomials, or terms, that it contains. A monomial has 1 term, a binomial has 2 terms, and a trinomial has 3 terms. EXAMPLE 2 Classifying Polynomials by the Number of Terms Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 35.55h 19.55g binomial Polynomial with 2 terms 2x 3y monomial Polynomial with 1 term 2 6x 2 4xy x not a polynomial 7mn 4m 5n trinomial 590 Chapter 12 Polynomials 2 x can be written as 2x1, where the exponent is not a whole number. Polynomial with 3 terms The degree of a term is the sum of the exponents of the variables in the term. The degree of a polynomial is the same as the degree of the term with the greatest degree. A polynomial can be classified by its degree. 2 5 4x 123 2x xy 5 123 { { Degree 2 5 Degree 2 0 1 444444Degree 4444 24 444444Degree 4443 Degree 5 EXAMPLE 3 Classifying Polynomials by Their Degrees Find the degree of each polynomial. 6x 2 3x 4 6x 2 Degree 2 3x Degree 1 4 Degree 0 The greatest degree is 2, so the degree of 6x 2 3x 4 is 2. 6 3m2 4m5 6 Degree 0 3m 2 Degree 2 4m 5 Degree 5 The greatest degree is 5, so the degree of 6 3m 2 4m5 is 5. To evaluate a polynomial, substitute the given number for each variable. EXAMPLE 4 Physics Application The height in feet of a firework launched straight up into the air from s feet off the ground at velocity v after t seconds is given by the polynomial 16t 2 vt s. Find the height of a firework launched from a 10 ft platform at 200 ft/s after 5 seconds. A polynomial is an algebraic expression. For help with evaluating algebraic expressions, see Lesson 1-1. 16t 2 vt s 16(5)2 200(5) 10 400 1000 10 610 Write the polynomial expression for height. Substitute 5 for t, 200 for v, and 10 for s. Simplify. The firework is 610 feet high 5 seconds after launching. Think and Discuss 1. Describe two ways you can classify a polynomial. Give a polynomial with three terms, and classify it two ways. 2. Explain why 5x2 3 is a polynomial but 5x 2 3 is not. 12-1 Polynomials 591 12-1 Exercises California Standards Practice Preparation for Algebra 1 10.0; 7AF1.2 KEYWORD: MT8CA 12-1 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Determine whether each expression is a monomial. 1. 2x 2y See Example 2 4. 9 6. 5r 3r 2 6 3 7. 2 2x 8. 2 x Find the degree of each polynomial. 9. 7m5 3m8 See Example 4 3. 3x Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 3 5. 4x y See Example 3 4 2. 3 x 10. x 4 4 11. 52 12. The trinomial 16t 2 24t 72 describes the height in feet of a ball thrown straight up from a 72 ft platform with a velocity of 24 ft/s after t seconds. What is the ball’s height after 2 seconds? INDEPENDENT PRACTICE See Example 1 See Example 2 See Example 3 See Example 4 Determine whether each expression is a monomial. 5y 4 13. 5.2x 3 14. 3x4 15. 6 x 4 16. 7x 4y 2 17. 210 18. 3x Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1 19. 9m2n6 20. 6g 3h2 21. 4x 3 2x 5 3 22. a 3 23. 2x 24. 5v 3s Find the degree of each polynomial. 25. 2x 2 7x 1 26. 3m2 4m3 2 27. 2 3x 4x 4 28. 6p 4 7p2 29. n 2 30. 3y 8 31. The volume of a box with height x, length x 2, and width 3x 5 is given by the trinomial 3x 3 x 2 10x. What is the volume of the box if its height is 2 inches? PRACTICE AND PROBLEM SOLVING Extra Practice See page EP24. 592 32. Transportation The distance in feet required for a car traveling at r2 r mi/h to come to a stop can be approximated by the binomial 2 r. 0 About how many feet will be required for a car to stop if it is traveling at 70 mi/h? Chapter 12 Polynomials Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. If it is a polynomial, give its degree. 33. 4x3 34. 7x0.7 3x 37. 2f 3 5f 5 f 38. 3 x 5 2 3 35. 6x 5x 2 36. 7y 2 6y 39. 6x 4x 40. 6x4 1 43. 2x 2 3x 4 5 41. 3b 2 9b 8b 3 42. 4 5x 44. 5 45. Transportation Gas mileage at speed s in miles per hour can be estimated using the given polynomials. Evaluate the polynomials to complete the table. Gas Mileage (mi/gal) 40 mi/h Compact – 0.025s2 + 2.45s – 30 Midsize – 0.015s2 + 1.45s – 13 Van – 0.03s2 + 2.9s – 53 50 mi/h 60 mi/h 46. Reasoning Without evaluating, tell which of the following binomials has the greatest value when x 10. Explain what method you used. A 3x 5 8 B 3x 8 8 C 3x 2 8 D 3x 6 8 47. What’s the Error? A student says that the degree of the polynomial 4b5 7b9 6b is 5. What is the error? 48. Write About It Give some examples of words that start with mono-, bi-, tri-, and poly-, and relate the meaning of each to polynomials. 49. Challenge The base of a triangle is described by the binomial x 2, and its height is described by the trinomial 2x 2 3x 7. What is the area of the triangle if x 5? NS1.1, NS2.4, AF1.2 50. Multiple Choice The height in feet of a soccer ball kicked straight up into the air from s feet off the ground at velocity v after t seconds is given by the trinomial 16t 2 vt s. What is the height of the soccer ball kicked from 2 feet off the ground at 90 ft/s after 3 seconds? A 3 ft B 15 ft C 90 ft D 128 ft 51. Gridded Response What is the degree of the polynomial 6 7k 4 8k 9? Write each number in scientific notation. (Lesson 4-5) 52. 4,080,000 53. 0.000035 54. 5,910,000,000 Find the two square roots of each number. (Lesson 4-6) 55. 49 56. 9 57. 81 58. 169 12-1 Polynomials 593 Model Polynomials 12-1 Use with Lesson 12-1 KEYWORD: MT8CA Lab12 KEY REMEMBER x2 x 2 0 x x 0 1 1 0 California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomials. To model the polynomial 4x 2 x 3, you need four x 2-tiles, one x-tile, and three 1-tiles. + + + + + 4x 2 − − − x 3 Activity 1 1 Use algebra tiles to model the polynomial 2x 2 4x 6. All signs are positive, so use all yellow tiles. + + 2x 2 594 Chapter 12 Polynomials + + + + 4x + + + + + + 6 2 Use algebra tiles to model the polynomial x 2 6x 4. Modeling x 2 6x 4 is similar to modeling 2x 2 4x 6. Remember to use red tiles for negative values. − − − − − + + + + + + x 2 6x 4 Think and Discuss 1. How do you know when to use red tiles? Try This Use algebra tiles to model each polynomial. 1. 2x 2 3x 5 2. 4x 2 5x 1 3. 5x 2 x 9 Activity 2 1 Write the polynomial modeled by the algebra tiles below. + + 2x 2 + + + + + + + + + + − − − − − 5x 10 The polynomial modeled by the tiles is 2x 2 5x 10. Think and Discuss 1. How do you know the coefficient of the x 2-term in Activity 2? Try This Write a polynomial modeled by each group of algebra tiles. 1. + − − 2. 3. + 12-1 Hands-On Lab 595 12-2 Why learn this? You can simplify polynomials to determine the amount of lumber that can be harvested from a tree. (See Example 4.) California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 Simplifying Polynomials You can simplify a polynomial by adding or subtracting like terms. The variables have the same powers. Like terms Not like terms EXAMPLE 1 The variables have different powers. Identifying Like Terms Identify the like terms in each polynomial. 2a 4a2 3 5a 6a2 2a 4a2 3 5a 6a2 Identify like terms. Like terms: 2a and 5a; 4a2 and 6a2 4x 5y 3 12x 5y 3 4x 3 6x 5y 3 4x 5y 3 12x 5y 3 4x 3 6x 5y 3 5 3 5 3 Identify like terms. 5 3 Like terms: 4x y , 12x y , and 6x y 5m2 3mn 4m 5m2 3mn 4m There are no like terms. Identify like terms. To simplify a polynomial, combine like terms. First arrange the terms from highest degree to lowest degree by using the Commutative Property. EXAMPLE 2 Simplifying Polynomials by Combining Like Terms Simplify. x 2 5x 4 – 6 7x 2 3x 4 – 4x 2 When you rearrange terms, move the operation symbol in front of each term with that term. 596 x 2 5x4 6 7x 2 3x4 4x 2 Identify like terms. 5x 4 3x 4 x 2 7x 2 4x 2 6 8x 4 4x 2 6 Commutative Property Chapter 12 Polynomials Combine coefficients: 5 3 8 and 1 7 4 4. Simplify. 5a2b 12ab 2 4a2b ab 2 3ab 5a2b 12ab 2 4a2b ab 2 3ab Identify like terms. 5a2b 4a2b 12ab 2 ab 2 3ab Commutative Property 9a2b 11ab2 3ab Combine coefficients: 5 4 9 and 12 1 11. You may need to use the Distributive Property to simplify a polynomial. EXAMPLE 3 Simplifying Polynomials by Using the Distributive Property Simplify. 4(3x 2 5x) 4(3x 2 5x) 4(3x 2) 4(5x) 12x 2 20x Distributive Property No like terms 2(4ab 2 5b) 3ab 2 6 2(4ab 2 5b) 3ab 2 6 2(4ab 2) 2(5b) 3ab 2 6 8ab 2 10b 3ab 2 6 11ab 2 10b 6 EXAMPLE 4 Distributive Property Identify like terms. Combine coefficients. Business Application A board foot is equal to the volume of a 1 ft by 1 ft by 1 in. piece of lumber. The amount of lumber that can be harvested from a tree with diameter d in. is approximately 20 0.005(d 3 30d 2 300d 1000) board feet. Use the Distributive Property to write an equivalent expression. 20 0.005(d 3 30d2 300d 1000) 20 0.005d 3 0.15d2 1.5d 5 15 0.005d 3 0.15d 2 1.5d Think and Discuss 1. Tell how you know when you can combine like terms. 2. Give an example of an expression that you could simplify by using the Distributive Property. Then give an expression that you could simplify by combining like terms. 12-2 Simplifying Polynomials 597 12-2 Exercises California Standards Practice Preparation for Algebra 1 10.0; 7AF1.3 KEYWORD: MT8CA 12-2 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Identify the like terms in each polynomial. 1. 3b2 5b 4b2 b 6 2. 7mn 5m2n 2 8m2n 4m2n 2 See Example 2 Simplify. 3. 2x 2 3x 5x 2 7x 5 4. 6 3b 2b4 7b2 9 4b 3b 2 6. 7(2x 2 4x) 7. 5(3a2 5a) 2a2 4a See Example 3 5. 4(3x 8) See Example 4 8. The level of nitric oxide emissions, in parts per million, from a car engine is approximated by the polynomial 40,000 5x(800 x2), where x is the air-fuel ratio. Use the Distributive Property to write an equivalent expression. INDEPENDENT PRACTICE See Example 1 Identify the like terms in each polynomial. 9. t 4t 2 5t 2 5t 2 10. 8rs 3r 2s2 5r 2s 2 2rs 5 See Example 2 Simplify. 11. 2p 3p 2 5p 12p 2 12. 3fg f 2g fg 2 3fg 4f 2g 6fg 2 14. 2(b 3) 5b 3b 2 1 15. 2(6y 3 8) 3y 3 See Example 3 13. 5(x 2 5x) 4x 2 7x See Example 4 16. The concentration of a certain medication in an average person’s bloodstream h hours after injection can be estimated using the expression 6(0.03h 0.002h 2 0.01h 3). Use the Distributive Property to write an equivalent expression. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP24. Simplify. 17. 2s2 3s 10s2 5s 3 18. 5gh 2 4g 2h 2g 2h g 2h 19. 2(x2 5x 4) 3x 7 20. 5(x x 5 x 3) 3x 21. 4(2m 3m2) 7(3m2 4m) 22. 6b4 2b 2 3(b 2 6) 23. 5mn 3m3n2 3(m3n2 4mn) 24. 3(4x y) 2(3x 2y) 25. Life Science The rate of flow in cm/s of blood in an artery at d cm from the center is given by the polynomial 1000(0.04 d 2). Use the Distributive Property to write an equivalent expression. 598 Chapter 12 Polynomials Art Abstract artists often use geometric shapes, such as cubes, prisms, pyramids, and spheres, to create sculptures. 26. Suppose the volume of a sculpture is approximately s 3 0.52s 3 0.18s 3 0.33s 3 cm3 and the surface area is approximately 6s 2 3.14s 2 7.62s 2 3.24s 2 cm2. a. Simplify the polynomial expression for the volume of the sculpture, and find the volume of the sculpture for s 5. b. Simplify the polynomial expression for the surface area of the sculpture, and find the surface area of the sculpture for s 5. Balanced/Unbalanced O by Fletcher Benton 27. A sculpture features a large ring with an outer lateral surface area of about 44xy in2, an inner lateral surface area of about 38xy in2, and 2 bases, each with an area of about 41y in2. Write and simplify a polynomial that expresses the surface area of the ring. 28. Challenge The volume of the ring on the sculpture from Exercise 27 is 49πxy 2 36πxy 2 in3. Simplify the polynomial, and find the volume for x 12 and y 7.5. Give your answer both in terms of π and to the nearest tenth. Pyramid Balancing Cube and Sphere, artist unknown KEYWORD: MT8CA Art NS1.3, AF3.1 29. Multiple Choice Simplify the expression 4x 2 8x 3 9x 2 2x. A 8x 3 5x 4 2x B 8x 3 13x2 2x C 8x 3 5x2 2x D 5x 3 30. Short Response Identify the like terms in the polynomial 3x 4 5x2 x 4 4x 2. Then simplify the polynomial. Find each percent to the nearest tenth. (Lesson 6-3) 31. What percent of 82 is 42? 32. What percent of 195 is 126? Create a table for each quadratic function, and use it to graph the function. (Lesson 7- 4) 33. y x 2 1 34. y x 2 2x 1 12-2 Simplifying Polynomials 599 Quiz for Lessons 12-1 Through 12-2 12-1 Polynomials Determine whether each expression is a monomial. 1 1. 5x2 1 2. 3x2 x 3 3. 7c 2d 8 Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1 4. x x 2 5. a3 2a 17 6. y 2 Find the degree of each polynomial. 7. u 6 7 8. 3c 2 c5 c 1 9. 43 10. The depth, in feet below the ocean surface, of a submerging exploration submarine after y minutes can be approximated by the polynomial 0.001y 4 0.12y 3 3.6y 2. Estimate the depth after 45 minutes. 12-2 Simplifying Polynomials Identify the like terms in each polynomial. 11. 5x 2 y 2 4xy x 2y 2 12. z 2 7z 4z2 z 9 13. t 8 2t 6 14. 8ab 3ac 5bc 4ac 6ab Simplify. 15. 6 3b5 2b3 7 5b3 16. 6y 2 y 7y 2 4y 5 17. 6(x 2 7x) 2x 2 7x 18. y 5 5y 4(5y 2) 19. The area of one face of a cube is given by the expression 3s 2 5s. Write a polynomial to represent the total surface area of the cube. 20. The area of each lateral face of a regular square pyramid is given by the expression 12b 2 2b. Write a polynomial to represent the lateral surface area of the pyramid. 600 Chapter 12 Polynomials California Standards MR2.1 Use estimation to verify the reasonableness of calculated results. Also covered: NS1.7 Look Back • Estimate to check that your answer is reasonable Before you solve a word problem, you can often read through the problem and make an estimate of the correct answer. Make sure your answer is reasonable for the situation in the problem. After you have solved the problem, compare your answer with the original estimate. If your answer is not close to your estimate, check your work again. Each problem below has an incorrect answer given. Explain why the answer is not reasonable, and give your own estimate of the correct answer. 1 The perimeter of rectangle ABCD is 48 cm. What is the value of x? A 2x 9 B x2 C D Answer: x 5 2 A patio layer can use 4x 6y ft of accent edging to divide a patio into three sections measuring x ft long by y ft wide. If each section must be at least 15 ft long and have an area of at least 165 ft2, what is the minimum amount of edging needed for the patio? y ft x ft 3 A baseball is thrown straight up from a height of 3 ft at 30 mi/h. The height of the baseball in feet after t seconds is 16t 2 44t 3. How long will it take the baseball to reach its maximum height? Answer: 5 minutes 4 Jacob deposited $2000 in a savings account that earns 6% simple interest. How much money will he have in the account after 7 years? Answer: $1925 Answer: 52 ft Focus on Problem Solving 601 Model Polynomial Addition 12-3 Use with Lesson 12-3 KEYWORD: MT8CA Lab12 KEY REMEMBER x2 x 2 0 x x 0 1 1 0 California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomial addition. Activity 1 Use algebra tiles to find (2x 2 2x 3) (x 2 x 5). 2x 2x 2 x2 x 3 5 Use tiles to represent all terms from both expressions. Remove any zero pairs. The remaining tiles represent the sum 3x 2 x 2. 3x 2 x 2 Think and Discuss 1. Explain what happens when you add the x-terms in (2x 5) (2x 4). Try This Use algebra tiles to find each sum. 1. (3m2 2m 6) (4m2 m 3) 602 Chapter 12 Polynomials 2. (5b2 4b 1) (b 1) 12-3 Adding Polynomials Why learn this? You can add polynomials to find the amount of material needed to mat and frame a picture. (See Example 3.) California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 EXAMPLE Adding polynomials is similar to simplifying polynomials. One way to add polynomials is to write them horizontally. First write the polynomials as one polynomial, and then use the Commutative Property to combine like terms. 1 Adding Polynomials Horizontally Add. (6x 2 3x 4) (7x 6) 6x 2 3x 4 7x 6 Write as one polynomial. 6x 2 3x 7x 4 6 Commutative Property 6x 2 4x 2 Combine like terms. (4cd 2 3cd 6) (7cd 6cd 2 6) 4cd 2 3cd 6 7cd 6cd 2 6 2 2 Write as one polynomial. 4cd 6cd 3cd 7cd 6 6 Commutative Property 10cd 2 4cd Combine like terms. (ab 2 4a) (3ab 2 4a 3) (a 5) ab 2 4a 3ab 2 4a 3 a 5 2 2 ab 3ab 4a 4a a 3 5 2 4ab 9a 2 Write as one polynomial. Commutative Property Combine like terms. You can also add polynomials in a vertical format. Write the second polynomial below the first one. Be sure to line up the like terms. If the terms are rearranged, remember to keep the correct sign with each term. 12-3 Adding Polynomials 603 EXAMPLE 2 Adding Polynomials Vertically Add. (5a2 4a 2) (4a2 3a 1) 5a 2 4a 2 4a 2 3a 1 Place like terms in columns. 9a 2 7a 3 Combine like terms. (2xy 2 3x 4y) (8xy 2 2x 3) 2xy 2 3x 4y 8xy 2 2x 3 Place like terms in columns. 2 10xy x 4y 3 Combine like terms. (4a2b 2 3a2 6ab) (4ab a 2 5) (3 7ab) 4a 2b 2 3a 2 6ab a 2 4ab 5 7ab 3 Place like terms in columns. Combine like terms. 4a 2b 2 4a 2 3ab 2 EXAMPLE Reasoning 3 Art Application f m 14 in. Mina is putting a mat of width m and a frame of width f around f m 11 in. m f an 11-inch by 14-inch picture. Find an expression for the amount of m framing material she needs. f The amount of material Mina needs equals the perimeter of the outside of the frame. Draw a diagram to help you determine the outer dimensions of the frame. Width 14 m m f f 14 2m 2f Length 11 m m f f 11 2m 2f P 2(11 2m 2f ) 2(14 2m 2f ) P 2w 2 22 4m 4f 28 4m 4f Simplify. 50 8m 8f Combine like terms. She will need 50 8m 8f inches of framing material. Think and Discuss 1. Compare adding (5x 2 2x) (3x 2 2x) vertically with adding it horizontally. 2. Explain how adding polynomials is similar to simplifying polynomials. 604 Chapter 12 Polynomials 12-3 Exercises California Standards Practice Preview of Algebra 1 10.0; 7AF1.3 KEYWORD: MT8CA 12-3 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 Add. 1. (5x 3 6x 1) (3x 7) 2. (22x 6) (14x 3) 3. (r 2s 3rs) (4r 2s 8rs) (6r 2s 14rs) See Example 2 4. (4b 2 5b 10) (6b 2 7b 8) 5. (9ab 2 5ab 6a 2b) (8ab 12a 2b 6) (6ab 2 5a 2b 14) 6. (h4j hj 3 hj 6) (5hj 3 5) (6h4j 7hj) See Example 3 7. Colette is putting a mat of width 3w and a frame of width w around a 16-inch by 48-inch poster. Find an expression for the amount of frame material she needs. 16 in. 3w w 48 in. w 3w INDEPENDENT PRACTICE See Example 1 Add. 8. (5x 2y 4xy 3) (7xy 3x 2y) 9. (5g 9) (7g 2 4g 8) 10. (6bc 2b 2c 2 8bc 2) (6bc 3bc 2) 11. (9h4 5h 4h6) (h6 6h 3h4) 12. (4pq 5p 2q 9pq 2) (6p 2q 11pq 2) (2pq 2 7pq 6p 2q) See Example 2 13. (8t 2 4t 3) (5t 2 8t 9) 14. (5b 3c 2 3b 2c 2bc) (8b 3c 2 3bc 14) (b 2c 5bc 9) 15. (w 2 3w 5) (2w 3w 2 1) (w 2 w 6) See Example 3 16. Each side of an equilateral triangle has length w 3. Each side of a square has length 4w 2. Write an expression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. w3 4w 2 12-3 Adding Polynomials 605 PRACTICE AND PROBLEM SOLVING Extra Practice Add. See page EP25. 17. (3w 2y 3wy 2 4wy) (5wy 2wy 2 7w 2y) (wy 2 5wy 3w 2y) 18. (2p 2t 3pt 5) (p 2t 2pt 2 3pt) (1 5pt 2 p 2t) 19. Geometry Write and simplify an expression for the combined volumes of a sphere with volume 43πr 3, a cube with volume r 3, and a prism with volume r 3 4r 2 5r 2. Use 3.14 for π. Business 20. Business The cost of producing n toys at a factory is given by the polynomial 0.5n 2 3n 12. The cost of packaging is 0.25n2 5n 4. Write and simplify an expression for the total cost of producing and packaging n toys. 21. Reasoning Two airplanes depart from the same airport, traveling in opposite directions. After 2 hours, one airplane is x 2 2x 400 miles from the airport, and the other airplane is 3x 2 50x 100 miles from the airport. How could you determine the distance between the two planes? Explain. According to the Toy Industry Association, $24.6 billion was spent on toys worldwide in 2000. 22. Write two polynomials whose sum is 3m2 4m 6. 23. Choose a Strategy What is the missing term? (6x 2 4x 3) (3x 2 5) 3x 2 6x 8 KEYWORD: MT8CA Toys A 2x B 2x C 10x D 10x 24. Write a Problem A plane leaves an airport heading north at x 3 mi/h. At the same time, another plane leaves the same airport, heading south at x 4 mi/h. Write a problem using the speeds of both planes. 25. Write About It Explain how to add polynomials. 26. Challenge What polynomial would have to be added to 6x 2 4x 5 so that the sum is 3x 2 4x 7? NS1.5, MG1.2 27. Multiple Choice Debbie is putting a deck of width 5w around her 20 foot by 80 foot pool. Which is the expression for the perimeter of the pool and deck combined? A 100 10w B 150 15w C 200 20w D 250 25w 28. Gridded Response What is the sum of (10x 3 4x 4 3x 5 10), (9x 3 8x 4 20x 5 15), and (x 3 4x 4 17x 5 2)? Find the fraction equivalent of each decimal. (Lesson 2-1) 29. 1.1 30. 0.08 31. 0.24 32. 4.05 Using the scale 1 in. 6 ft, find the height or length of each object. (Lesson 5-7) 33. a 14 in. tall model of an office building 606 Chapter 12 Polynomials 34. a 2.5 in. long model of a train Model Polynomial Subtraction 12-4 Use with Lesson 12-4 KEYWORD: MT8CA Lab12 KEY REMEMBER x2 x 2 0 x x 0 1 1 0 California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomial subtraction. Activity 1 Use algebra tiles to find (2x 2 2x 3) (x 2 x 3). 2x 2 2x 3 x 2 x 3 Remember, subtracting is the same as adding the opposite. Use the opposite of each term in x 2 x 3. x2 Remove any zero pairs. 3x 6 The remaining tiles represent the difference x 2 3x 6. Think and Discuss 1. Why do you have to add the opposite when subtracting? Try This Use algebra tiles to find each difference. 1. (6m 2 2m) (4m2) 2. (5b 2 9) (b 9) 12-4 Hands-On Lab 607 12-4 Who uses this? Manufacturers can subtract polynomials to estimate the cost of making a product and the revenue from sales. (See Example 4.) California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 EXAMPLE Subtracting Polynomials Recall that to subtract an integer, you add its opposite. To subtract a polynomial, you first need to find its opposite. 1 Finding the Opposite of a Polynomial Find the opposite of each polynomial. 8x 3y 6z (8x 3y 6z) 8x 3y 6z is a negative term. 12x 2 5x (12x 2 5x) 12x 2 5x Distributive Property 3ab 2 4ab 3 (3ab 2 4ab 3) 3ab 2 4ab 3 Distributive Property The opposite of a positive term To subtract a polynomial, add its opposite. EXAMPLE 2 Subtracting Polynomials Horizontally Subtract. (n3 n 5n2) (7n 4n2 9) (n3 n 5n2) (7n 4n2 9) n3 5n2 4n2 n 7n 9 n3 9n2 8n 9 (2cd 2 cd 4) (7cd 2 2 5cd ) (2cd 2 cd 4) (7cd 2 2 5cd) 2cd 2 7cd 2 cd 5cd 4 2 5cd 2 6cd 2 608 Chapter 12 Polynomials Add the opposite. Commutative Property Combine like terms. Add the opposite. Commutative Property Combine like terms. You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. EXAMPLE 3 Subtracting Polynomials Vertically Subtract. (x 3 4x 1) (6x 3 3x 5) (x 3 4x 1) x 3 4x 1 (6x 3 3x 5) 6x 3 3x 5 3 5x x 4 Add the opposite. (4m2n 3mn 4m) (8m2n 6mn 3) (4m2n 3mn 4m) 4m2n 3mn 4m (8m2n 6mn 3) 8m2n 6mn 3 Add the 12m2n 3mn 4m 3 opposite. (4x 2y 2 xy 6x) (7x 5xy 6) (4x 2y 2 xy 6x) 4x 2y 2 xy 6x (7x 5xy 6) 5xy 7x 6 2 2 4x y 4xy 13x 6 EXAMPLE 4 Rearrange terms as needed. Business Application Suppose the cost in dollars of producing x model kits is given by the polynomial 3x 400,000 and the revenue generated from sales is given by the polynomial 0.00004x 2 20x. Find a polynomial expression for the profit from making and selling x model kits, and evaluate the expression for x 200,000. 0.00004x 2 20x (3x 400,000) 0.00004x 2 20x (3x 400,00) 0.00004x 2 20x 3x 400,000 0.00004x 2 17x 400,000 revenue cost Add the opposite. Commutative Property Combine like terms. The profit is given by the polynomial 0.00004x2 17x 400,000. For x 200,000, 0.00004(200,000)2 17(200,000) 400,000 1,400,000. The profit is $1,400,000, or $1.4 million. Think and Discuss 1. Explain how to find the opposite of a polynomial. 2. Compare subtracting polynomials with adding polynomials. 12-4 Subtracting Polynomials 609 12-4 Exercises California Standards Practice Preview of Algebra 1 10.0; 7AF1.3 KEYWORD: MT8CA 12-4 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Find the opposite of each polynomial. 1. 4x2y 2. 5x 4xy5 3. 3x2 8x 5 4. 5y2 2y 4 5. 8x3 5x 6 6. 6xy2 4y 2 Subtract. 7. (2b 3 5b 2 8) (4b 3 b 12) 8. 7b (4b 2 3b 12) 9. (4m2n 7mn 3mn2) (5mn 4m2n) See Example 3 10. (8x 2 4x 1) (5x 2 2x 3) 11. (2x 2y xy 3x 4) (4xy 7x 4) 12. (5ab 2 4ab 3a 2b) (7 5ab 3ab 2 4a 2b) See Example 4 13. The volume of a rectangular prism, in cubic inches, is given by the expression x 3 3x 2 5x 7. The volume of a smaller rectangular prism is given by the expression 5x 3 6x 2 7x 14. How much greater is the volume of the larger rectangular prism? Evaluate the expression for x 3. INDEPENDENT PRACTICE See Example 1 See Example 2 Find the opposite of each polynomial. 14. 4rn2 15. 3v 5v2 16. 4m2 6m 2 17. 4xy2 2xy 18. 8n6 5n3 n 19. 9b2 2b 9 Subtract. 20. (6w 2 3w 6) (3w 2 4w 5) 21. (14a a2) (8 a 2 9a) 22. (7r 2s 2 5rs 2 6r 2s 7rs) (3rs 2 3r 2s 8rs) See Example 3 23. (4x 2 6x 1) (3x 2 9x 5) 24. (3a2b 2 4ab 2a 4) (4a 2b 2 5a 3b 6) 25. (4pt 2 6p 3 5p 2t 2) (5p 2 6pt 2 7p 2t 2) See Example 4 610 26. The current in an electrical circuit at t seconds is 4 t 3 5t 2 2t 200 amperes. The current in another electrical circuit is 3t 3 2t 2 5t 100 amperes. Write an expression to show the difference in the two currents. Chapter 12 Polynomials PRACTICE AND PROBLEM SOLVING Extra Practice See page EP25. Subtract. 27. (6a 3b 5ab) (6a 5b 7ab) 28. (4pq 2 6p 2q 3pq) (7pq 2 7p 2q 3pq) 29. (9y 2 5x 2y x 2) (3y 2 7x 2y 4x 2) 30. The area of the rectangle is 2a 2 4a 5 cm2. The area of the square is a 2 2a 6 cm2. What is the area of the shaded region? 31. The area of the square is 4x 2 2x 6 in2. The area of the triangle is 2x 2 4x 5 in2. What is the area of the shaded region? 32. Business The price in dollars of one share of stock after y years is modeled by the expression 3y 3 6y 4.25. The price of one share of another stock is modeled by 3y 3 24y 25.5. What expression shows the difference in price of the two stocks after y years? 33. Choose a Strategy Which polynomial has the greatest value when x 6? A x 2 3x 8 C x 3 30x 200 B 2x 4 7x 14 D x 5 100x 4 10 34. Write About It Explain how to subtract the polynomial 5x 3 3x 6 from 4x 3 7x 1. 35. Challenge Find the values of a, b, c, and d that make the equation true. (2t 3 at 2 4bt 6) (ct 3 4t 2 7t 1) 4t 3 5t 2 15t d AF1.2, AF1.3, AF2.2 36. Multiple Choice What is the opposite of the polynomial 4a2b 3ab2 5ab? A 4a2b 3ab2 5ab C 4a2b 3ab2 5ab B 4a2b 3ab2 5ab D 4a2b 3ab2 5ab 37. Extended Response A square has an area of x 2 10x 25. A triangle inside the square has an area of x 2 4. Create an expression for the area of the square minus the area of the triangle. Evaluate the expression for x 8. Multiply. (Lesson 4-4) 38. (3x)(6x) 39. (9m3)(7m2) 40. (8ab4)(5a4) 41. (4r4s)(2r 6s 9) Simplify. (Lesson 12-2) 42. x 3y 2 2x 2y 4x 3y 2 43. 4(zy 3 2zy) 3zy 5zy 3 44. 6(3x 2 6x 1) 12-4 Subtracting Polynomials 611 12-5 Why learn this? You can multiply polynomials and monomials to determine the dimensions of a planter box. (See Example 3.) California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.2, 7AF1.3, 7AF2.2 EXAMPLE Multiplying Polynomials by Monomials Remember that when you multiply two powers with the same bases, you add the exponents. To multiply two monomials, multiply the coefficients and add the exponents of the variables that are the same. (5m2n3)(6m3n6) 5 6 m 2 3n 3 6 30m5n9 1 Multiplying Monomials Multiply. (4r 3s4)(6r 5s 6) 4 6 r 3 5 s4 6 24r 8s 10 Multiply coefficients. Add exponents that (9x 2y)(2x 3yz 6) 9 2 x 2 3 y1 1 z 6 18x 5y 2z 6 Multiply coefficients. Add exponents that have the same base. have the same base. To multiply a polynomial by a monomial, use the Distributive Property. Multiply every term of the polynomial by the monomial. EXAMPLE 2 Multiplying a Polynomial by a Monomial Multiply. 1 x(y z) 4 1 x(y z) 4 1 1 xy xz 4 4 When multiplying a polynomial by a negative monomial, be sure to distribute the negative sign. 612 5a 2b(3a4b 3 6a 2b 3) 5a 2b(3a 4b 3 6a 2b 3) Multiply each term in the parentheses 15a 6b 4 30a 4b 4 by 5a2b. Chapter 12 Polynomials 1 Multiply each term in the parentheses by 4x. Multiply. 5rs 2(r 2s4 3rs 3 4rst) 5rs 2(r 2s 4 3rs 3 4rst) 5r 3s 6 15r 2s 5 20r 2s 3t EXAMPLE 3 Multiply each term in the parentheses by 5rs 2. PROBLEM SOLVING APPLICATION Chrystelle is making a planter box with a square base. She wants the height of the box to be 3 inches more than the side length of the base. If she wants the volume of the box to be 6804 in3, what should the side length of the base be? Reasoning 1 Understand the Problem 1. If the side length of the base is s, then the height is s 3. The volume is s s (s 3) s2(s 3). The answer will be a value of s that makes the volume of the box equal to 6804 in3. 1.2 Make a Plan You can make a table of values for the polynomial to try to find the value of s. Use the Distributive Property to write the expression s 2(s 3) another way. Use substitution to complete the table. 1.3 Solve s2(s 3) s3 3s2 s s 3 3s 2 15 Distributive Property 16 17 18 153 3(15)2 163 3(16)2 173 3(17)2 183 3(18)2 4050 4864 5780 6804 The side length of the base should be 18 inches. 1.4 Look Back If the side length of the base were 18 inches and the height were 3 inches more, or 21 inches, then the volume would be 18 18 21 6804 in3. The answer is reasonable. Think and Discuss 1. Compare multiplying two monomials with multiplying a polynomial by a monomial. 12-5 Multiplying Polynomials by Monomials 613 12-5 Exercises California Standards Practice Preview of Algebra 1 10.0; 7AF1.2, 7AF1.3, 7AF2.2 KEYWORD: MT8CA 12-5 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 Multiply. 1. (5s 2t 2)(3st 3) 2. (x 2y 3)(6x 4y 3) 3. (5h2j 4)(7h 4j 6 ) 4. 6m(4m 5) 5. 7p 3r(5pr 4) 6. 13g 5h3(10g 5h2) 8. 4ab(a 2b ab 2) 7. 2h(3m 4h) 9. 3x(x 2 5x 10) See Example 3 10. 6c 2d(3cd 3 5c 3d 2 4cd ) 11. The formula for the area of a trapezoid is A 12h(b1 b2), where h is the trapezoid’s height and b1 and b2 are the lengths of its bases. Use the Distributive Property to simplify the expression. Then use the expression to find the area of a trapezoid with height 12 in. and base lengths 9 in. and 7 in. INDEPENDENT PRACTICE See Example 1 See Example 2 See Example 3 Multiply. 12. (6x 2y 5)(3xy 4) 13. (gh3)(2g 2h5) 15. (s 4t 3)(2st) 1 16. 12x 9y 7 2x 3y 14. (4a 2b)(2b 3) 17. 2.5j 3(3h 5j 7) 18. (3m3n4)(1 5mn5) 19. 3z(5z 2 4z) 20. 3h2(6h 3h3) 21. 3cd(2c 3d 2 4cd 2) 22. 2b(4b 4 7b 10) 23. 3s 2t 2(4s 2t 5st 2s 2t 2) 24. A rectangle has a base of length 3x 2y and a height of 2x 3 4xy 3. Write and simplify an expression for the area of the rectangle. Then find the area of the rectangle if x 2 and y 1. PRACTICE AND PROBLEM SOLVING Extra Practice See page EP25. Multiply. 25. (3b 2)(8b 4) 26. (4m2n)(2mn4) 27. (2a 2b2)(3ab 4) 28. 7g(g 5) 29. 3m2(m3 5m) 30. 2ab(3a 2b 3ab2) 31. x 4(x x 3y 5) 32. m(x 3) 33. f 2g 2(3 f g 3) 34. x 2(x 2 4x 9) 35. (4m 2p 4)(5m 2p 4 3mp 3 6m 2p) 36. 3wz(5w 4z 2 4wz 2 6w 2z 2) 37. Felix is building a cylindrical-shaped storage container. The height of the container is x 3 y 3. Write and simplify an expression for the volume using the formula V πr 2h. Then find the volume with r 112 feet, x 3, and y 1. 614 Chapter 12 Polynomials 38. Health The table gives some formulas for finding the target heart rate for a person of age a exercising at p percent of his or her maximum heart rate. Target Heart Rate Nonathletic Male Female p(220 a) p(226 a) 1 p(410 2 Fit a) 1 p(422 2 a) a. Use the Distributive Property to simplify each expression. b. Use your answer from part a to write an expression for the difference between the target heart rate for a fit male and for a fit female. Both people are age a and are exercising at p percent of their maximum heart rates. 39. What’s the Question? A square prism has a base area of x 2 and a height of 3x 4. If the answer is 3x 3 4x 2, what is the question? If the answer is 14x 2 16x, what is the question? 40. Write About It If a polynomial is multiplied by a monomial, what can you say about the number of terms in the answer? What can you say about the degree of the answer? 41. Challenge On a multiple-choice test, if the probability of guessing each question correctly is p, then the probability of guessing two or more correctly out of four is 6p2(1 2p p 2) 4p 3(1 p) p 4. Simplify the expression. Then write an expression for the probability of guessing fewer than two out of four correctly. AF1.3, MG2.1 42. Multiple Choice The width of a rectangle is 13 feet less than twice its length. Which of the following shows an expression for the area of the rectangle? A 22 13 B 22 13 C 2 13 D 6 26 43. Short Response A triangle has base 10cd2 and height 3c2d2 4cd2. Write and simplify an expression for the area of the triangle. Then evaluate the expression for c 2 and d 3. Combine like terms. (Lesson 3-2) 44. 8x 3y x 7 45. 4m n 7 2n 5 46. 9a 11 6b 10a 7b Find the surface area of each figure. Use 3.14 for π. (Lesson 10-4) 47. a rectangular prism with base 4 in. by 3 in. and height 2.5 in. 48. a cylinder with radius 10 cm and height 7 cm 12-5 Multiplying Polynomials by Monomials 615 Multiply Binomials 12-6 Use with Lesson 12-6 KEYWORD: MT8CA Lab12 KEY REMEMBER x2 x 2 x x 1 1 The area of a rectangle with base b and height h is given by A bh. California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to find the product of two binomials. Activity 1 x3 1 To model (x 3)(2x 1) with algebra tiles, make a rectangle with base x 3 and height 2x 1. 2x 1 Area (x 3)(2x 1) 2x 2 7x 3 x2 2 Use algebra tiles to find (x 2)(x 1). x 1 Think and Discuss 1. Explain how to determine the signs of each term in the product when you are multiplying (x 3)(x 2). 2. How can you use algebra tiles to find (x 3)(x 3)? 616 Chapter 12 Polynomials Area (x 2)(x 1) x 2 3x 2 Try This Use algebra tiles to find each product. 1. (x 4)(x 4) 2. (x 3)(x 2) 3. (x 5)(x 3) Activity 2 Write two binomials whose product is modeled by the algebra tiles below, and then write the product as a polynomial expression. The base of the rectangle is x 5 and the height is x 2, so the binomial product is (x 5)(x 2). The model shows one x 2-tile, seven x-tiles, and ten 1-tiles, so the polynomial expression is x 2 7x 10. Think and Discuss 1. Write an expression modeled by the algebra tiles below. How many zero pairs are modeled? Describe them. Try This Write two binomials whose product is modeled by each set of algebra tiles below, and then write the product as a polynomial expression. 1. 2. 3. 12-6 Hands-On Lab 617 12-6 Why learn this? You can multiply binomials to determine the area of a walkway around a cactus garden. (See Example 2.) California Standards Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Vocabulary FOIL Multiplying Binomials You can use the Distributive Property to multiply two binomials. 678 (x y)(x z) x(x z) y(x z) x 2 xz yx yz The product can be simplified using the FOIL method: multiply the First terms, the Outer terms, the Inner terms, and the Last terms of the binomials. First Last F O I L Inner Outer EXAMPLE 1 Multiplying Two Binomials Multiply. (p 2)(3 q) (p 2)(3 q) When you multiply two binomials, you will get four products. Then combine like terms. (m n)(p q) FOIL 3p pq 6 2q mp mq np nq (x 2)(x 5) (3m n)(m 2n) (x 2)(x 5) FOIL x 2 5x 2x 10 x 2 7x 10 Combine like terms. 618 (m n)(p q) Chapter 12 Polynomials (3m n)(m 2n) 3m 2 6mn mn 2n 2 3m 2 5mn 2n 2 EXAMPLE 2 Landscaping Application Find the area of a bark walkway of width x ft around a 12 ft by 5 ft cactus garden. Area of Walkway Total Area (5 2x)(12 2x) Area of Flower Bed (5)(12) 60 10x 24x 4x 2 60 34x 4x 2 The walkway area is 4x 2 34x ft2. Binomial products of the form (a b)2, (a b)2, and (a b)(a b) are often called special products. EXAMPLE 3 Special Products of Binomials Multiply. (x 3)2 (a b)2 (x 3)(x 3) (a b)(a b) x 2 3x 3x 32 x 2 6x 9 a 2 ab ab b 2 a2 2ab b2 (n 3)(n 3) (n 3)(n 3) n2 3n 3n 32 n2 9 3n 3n 0 Special Products of Binomials (a b)2 a2 ab ab b2 a2 2ab b2 (a b)2 a2 ab ab b2 a2 2ab b2 (a b)(a b) a2 ab ab b2 a2 b2 Think and Discuss 1. Give an example of a product of two binomials that has 4 terms, one that has 3 terms, and one that has 2 terms. 12-6 Multiplying Binomials 619 12-6 Exercises California Standards Practice Preview of Algebra 1 10.0 KEYWORD: MT8CA 12-6 KEYWORD: MT8CA Parent GUIDED PRACTICE See Example 1 See Example 2 See Example 3 Multiply. 1. (x 5)(y 4) 2. (x 3)(x 7) 3. (3m 5)(4m 9) 4. (h 2)(3h 4) 5. (m 2)(m 7) 6. (b 3c)(4b c) 7. A courtyard is 20 ft by 30 ft. There is a walkway of width x all the way around the courtyard. Find the area of the walkway. Multiply. 8. (x 2)2 9. (b 3)(b 3) 10. (x 4)2 11. (3x 5)2 INDEPENDENT PRACTICE See Example 1 Multiply. 12. (x 4)(x 3) 13. (v 1)(v 5) 14. (w 6)(w 2) 15. (3x 5)(x 6) 16. (4m 1)(3m 2) 17. (3b c)(4b 5c) 18. (3t 1)(t 1) 19. (3r s)(4r 5s) 20. (5n 3b)(n 2b) See Example 2 21. Construction The Gonzalez family is having a pool to swim laps built in their backyard. The pool will be 25 yards long by 5 yards wide. There will be a cement deck of width x yards around the pool. Find the total area of the pool and the deck. See Example 3 Multiply. 22. (x 5)2 23. (b 3)2 24. (x 4)(x 4) 25. (2x 3)(2x 3) 26. (4x 1)2 27. (a 7)2 PRACTICE AND PROBLEM SOLVING Extra Practice See page EP25. Multiply. 28. (m 6)(m 6) 29. (b 5)(b 12) 30. (q 6)(q 5) 31. (t 9)(t 4) 32. (g 3)(g 3) 33. (3b 7)(b 4) 34. (3t 1)(6t 7) 35. (4m n)(m 3n) 36. (3a 6b)2 37. (r 5)(r 5) 38. (5q 2)2 39. (3r 2s)(5r 4s) 40. A metalworker makes a box from a 15 in. by 20 in. piece of tin by cutting a square with side length x out of each corner and folding up the sides. Write and simplify an expression for the area of the base of the box. 620 Chapter 12 Polynomials Life Science A. V. Hill (1886–1977) was a biophysicist and pioneer in the study of how muscles work. He studied muscle contractions in frogs and came up with an equation relating the force generated by a muscle to the speed at which the muscle contracts. Hill expressed this relationship as (P a)(V b) c, where P is the force generated by the muscle, a is the force needed to make the muscle contract, V is the speed at which the muscle contracts, b is the smallest contraction rate of the muscle, and c is a constant. 41. Use the FOIL method to simplify Hill’s equation. 42. Suppose the force a needed to make the muscle contract is approximately 14 the maximum force the muscle can generate. Use Hill’s equation to write an equation for a muscle generating the maximum possible force M. Simplify the equation. 43. Write About It In Hill’s equation, what happens to V as P increases? What happens to P as V increases? (Hint: You can substitute the value of 1 for a, b, and c to help you see the relationship between P and V.) 44. Challenge Solve Hill’s equation for P. Assume that no variables equal 0. The muscles on opposite sides of a bone work as a pair. Muscles in pairs alternately contract and relax to move your skeleton. AF1.3, AF4.1 45. Multiple Choice Which polynomial shows the result of using the FOIL method to find (x 2)(x 6)? A x 2 12 B x 2 6x 2x 12 C 2x 2x 12 D x2 4 46. Gridded Response Multiply 3a 2b and 5a 8b. What is the coefficient of ab? Solve. (Lesson 2-8) 47. 5x 33.6 91.9 48. 14 1.2j 16.4 49. 4.4 0.6y 6.82 50. 3.7 26.2x 9.4 Simplify. (Lesson 12-2) 51. 4(m2 3m 6) 52. 3(a2b 4a 3ab) 2ab 53. x 2y 4(xy 2 3x 2y 4xy) 12-6 Multiplying Binomials 621 Quiz for Lessons 12-3 Through 12-6 12-3 Adding Polynomials Add. 1. (8x 3 6x 3) (2x 6) 3 2 2 2. (30x 7) (12x 5) 3 2 3. (7b c 6b c 3bc) (8b c 5bc 13) (4b 2c 5bc 9) 4. (2w 2 4w 6) (3w 4w 2 5) (w 2 4) 5. Each side of an equilateral triangle has length w 2. Each side of a square has length 3w 4. Write an expression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. 12-4 Subtracting Polynomials w2 3w 4 Find the opposite of each polynomial. 6. 3x 4xy 3 7. 2m2 6m 3 8. 5v 7v 2 Subtract. 9. 10b2 (3b2 6b 8) 10. (13a a2) (9 a 7a) 11. (6x 2 6x) (3x 2 7x) 12. The population of a bacteria colony after h hours is 4h3 5h2 2h 200. The population of another bacteria colony is 3h3 2h2 5h 200. Write an expression to show the difference between the two populations. Evaluate the expression for h 1000. 12-5 Multiplying Polynomials by Monomials Multiply. 13. (4x 3y 3 )(3xy6 ) 14. (3hj 5)(6h4j 5 ) 15. 4s 2t 2(4s2t 3st s 2t 2) 16. A triangle has a base of length 2x 2 y and a height of x 3 xy 2. Write and simplify an expression for the area of the triangle. Then find the area of the triangle if x 2 and y 1. 12-6 Multiplying Binomials Multiply. 17. (x 2)(x 6) 18. (3m 4)(2m 8) 19. (n 5)(n 3) 20. (x 6)2 21. (x 5)(x 5) 22. (3x 2)(3x 2) 23. A rug is placed in a 10 ft 20 ft room so that there is an uncovered strip of width x all the way around the rug. Find the area of the rug. 622 Chapter 12 Polynomials Cooking Up a New Kitchen Javier is a contractor who remodels kitchens. He drew the figure to help calculate the dimensions of a countertop surrounding a sink that is x inches long and y inches wide. 6 in. 1. Write a polynomial that Javier can use to find the perimeter of the outer edge of the countertop. 4 in. y x 6 in. 4 in. 2. Someone orders a countertop for a sink that is 18 inches long and 12 inches wide. Javier puts tape around the outer edge of the countertop to protect it while it is being moved. Use the polynomial to determine how many inches of tape are needed. 3. Write a polynomial that Javier can use to find the area of the countertop for any size sink. 4. The marble for the countertop costs $1.25 per square inch. Write a polynomial that gives the cost of the countertop. 5. Find the cost of the countertop for the 18-inch by 12-inch sink. Explain your answer. Concept Connection 623 Short Cuts You can use properties of algebra to explain many arithmetic shortcuts. For example, to square a two-digit number that ends in 5, multiply the first digit by one more than the first digit, and then place a 25 at the end. To find 352, multiply the first digit, 3, by one more than the first digit, 4. You get 3 4 12. Place a 25 at the end, and you get 1225. So 352 1225. Why does this shortcut work? You can use FOIL to multiply 35 by itself: 352 35 35 (30 5)(30 5) 900 150 150 25 900 300 25 1200 25 1200 30 40 1225 First use the shortcut to find each square. Then use FOIL to multiply the number by itself. 1. 152 2. 452 3. 852 4. 652 5. 252 6. Can you explain why the shortcut works? Use FOIL to multiply each pair of numbers. 7. 11 14 8. 12 16 9. 13 15 10. 14 17 11. 18 19 12. Write a shortcut for multiplying two-digit numbers with a first digit of 1. RRolling olling for for T Tiles iles For this game, you will need a number cube, a set of algebra tiles, and a game board. Roll the number cube, and draw an algebra tile: 1 , 2 , 3 , 4 , 5 , 6 . The goal is to model expressions that can be added, subtracted, multiplied, or divided to equal the polynomials on the game board. A complete set of rules and a game board are available online. 624 Chapter 12 Polynomials KEYWORD: MT8CA Games Materials • 3 sheets of decorative paper • ruler • compass • scissors • glue • markers PROJECT Polynomial Petals A Pick a petal and find a fact about polynomials! Directions 1 Draw a 5-inch square on a sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. Figure A 2 Draw a 312-inch square on another sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. B 3 Draw a 212-inch square on the last sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. 4 Glue the medium square onto the center of the large square so that the squares are at a 45° angle to each other. Figure B 5 Glue the small square onto the center of the medium square in the same way. Taking Note of the Math Write examples of different types of polynomials on the petals. Then use the remaining petals to take notes on the key concepts from the chapter. When you’re done, fold up the petals. 625 Vocabulary binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 590 degree of a polynomial . . . . . . . . . . . . . . . 591 trinomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 FOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Complete the sentences below with vocabulary words from the list above. 1. The expression 4x 3 10x 2 4x 12 is an example of a ___?___ whose ___?___ is 3. 2. Use the ___?___ method to find the product of two ___?___. 3. A polynomial with 2 terms is called a ___?___. A polynomial with 3 terms is called a ___?___. 12-1 Polynomials (pp. 590–593) EXAMPLE Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. ■ ■ ■ Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 4. 5t 2 7t 8 4xy 34 7x 2y 4 6. 12g 7g 3 2 not a polynomial 7. 4a 2b 3c 5 x x 3 2x 1 degree 3 n 3n4 16n2 degree 4 5. r 4 3r 2 5 5 g 8. x 2xy 9. 6st 7s Find the degree of each polynomial. 10. 3x 5 6x 8 5x 11. x 4 4x 2 3x 1 12. 14 8r 2 9r 3 1 1 7 13. 3m 3 6m 5 9m 2 14. 3x 6 5x 5 9x 626 Chapter 12 Polynomials 1A10.0; 7AF1.2 EXERCISES 4x 5 2x 3 7 trinomial Find the degree of each polynomial. ■ Preview of 12-2 Simplifying Polynomials (pp. 596–599) EXAMPLE 7AF1.3 Simplify. 2x 4 5x 3 15. 4t 2 6t 3t 4t 2 7t 2 1 4x 2 16. 4gh 5g 2h 7gh 4g 2h 5x 2 2x 4 5x 3 4x 2 17. 4(5mn 3m) 18. 4(2a 2 4b) 6b 9x 2 7x 1 ■ 1A10.0; EXERCISES Simplify. ■ 5x 2 Preview of 19. 5(4st 2 6t) 16st 2 7t 4(2x 7) 5x 4 8x 28 5x 4 3x 24 12-3 Adding Polynomials (pp. 603–606) EXAMPLE 2x 5x 2 8x 2 x 2 ■ 20. (4x 2 3x 7) (2x 2 5x 12) 21. (5x 4 3x 2 4x 2) (4x2 5x 9) 3x 2 Identify like terms. Combine like terms. 22. (5h 5) (2h2 3) (3h 1) 23. (3xy 2 5x 2y 4xy) (3x 2y 6xy xy 2) (8t 3 4t 6) (4t 2 7t 2) 8t 3 4t 6 Place like terms 4t 2 7t 2 in columns. 8t 3 4t 2 3t 4 Combine like terms. 12-4 Subtracting Polynomials 24. (4n2 6) (3n2 2) (8 6n2) (pp. 608–611) EXAMPLE ■ 7AF1.3 Add. (3x 2 2x) (5x 2 3x 2) 3x 2 1A10.0; EXERCISES Add. ■ Preview of Preview of 7AF1.3 EXERCISES Subtract. Subtract. (6x 2 4x 5) (7x 2 8x 2) 6x 2 4x 5 (7x 2 8x 2) Add the 25. (x 2 4) (4 5x 2) opposite. Associative Property Combine like terms. 6x 2 4x 5 7x 2 8x 2 x 2 4x 3 1A10.0; 26. (w 2 4w 6) (2w 2 8w 8) 27. (3x 2 8x 9) (7x 2 8x 5) 28. (4ab 2 5ab 7a 2b) (3a 2b 6ab) 29. (3p 3q 2 4p 2q 2) (2pq 2 4p 3q 2) Study Guide: Review 627 12-5 Multiplying Polynomials by Monomials EXAMPLE (pp. 612–615) EXERCISES Preview of 1A10.0; 7AF1.2, 7AF1.3, 7AF2.2 Multiply. Multiply. ■ (3x 2y 3)(2xy 2) 30. (4st 3 )(s 3st 8) (3x 2y 3)(2xy 2) 3 2 x2 1 y3 2 6x 3y 5 ■ (2ab 2)(4a 2b 2 Multiply the coefficients and add the exponents. 31. 6a 2b(2a 2b 2 5ab 2 6a 4b) 32. 2m(m 2 8m 1) 33. 5h(3gh 4 2g 3h 2 6h 4g) 1 34. 2j 3k 2(4j 2k 3jk 2 2j 3k 3 ) 3ab 6a 8) 35. 3x 2y 5(5x 4y 7 6x 5y 9 8xy 4xy 2 ) (2ab 2)(4a 2b 2 3ab 6a 8) 8a3b 4 6a 2b 3 12a 2b 2 16ab 2 12-6 Multiplying Binomials (pp. 618–621) EXAMPLE EXERCISES Multiply. Multiply. ■ (r 8)(r 6) 36. (p 6)(p 2) 37. (b 4)(b 6) ■ (r 8)(r 6) FOIL r 2 6r 8r 48 r 2 2r 48 Combine like terms. 38. (3r 1)(r 4) 39. (3a 4b)(a 5b) 40. (m 7)2 41. (3t 6)(3t 6) 42. (3b 7t)(2b 4t) (b 6)2 43. (10 3x)(4 x) 628 (b 6)(b 6) FOIL b 2 6b 6b 36 b 2 12b 36 Combine like terms. Chapter 12 Polynomials 44. ( y 11)2 Preview of 1A10.0 Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1 1. t 2 2t 0.5 4 2. 2a 3b 6 3. 4m 4 5m 8 Find the degree of each polynomial. 4. 6 9b 2m 4 6. 4 y 5. 54 7. The volume of a cube with side length x 2 is given by the polynomial x 3 6x 2 12x 8. What is the volume of the cube if x 3? Simplify. 9. 3(x 2 6x 10) 8. 2a 4b 5b 6a 2b 10. 2x 2y 3xy2 4x 2y 2x 2y 11. 6(4b 2 7b) 3b 2 5b 12. The area of one face of a cube is given by the expression 2s 2 9s. Write a polynomial to represent the total surface area of the cube. Add. 13. (4x 2 2x 1) (2x 5) 2 2 2 14. (12x 5) (9x 5) 2 15. (3bc b c 5bc ) (2bc bc ) 16. (6h 5 3h3 2h6 ) (h6 2h 5h4 ) 17. (b 3c 2 8b2c 5bc) (6b 3c 2 4bc 3) (b2c 3bc 11) 18. Harold is placing a mat of width w 4 around a 16 in. by 20 in. portrait. Write an expression for the perimeter of the outer edge of the mat. Subtract. 19. (4m2n 5mn mn2) (2mn 4m2n) 20. (12a a2) (6 a2 8a) 21. (3a2b 5a2b2 6ab2) (2a2b2 7a2b) 22. ( j 4 7j 2 4j) (5j 3 2j 2 6j 1) 23. A circle whose area is 2x 2 3x 4 is cut from a rectangular piece of plywood with area 4x 2 3x 1 and discarded. Write an expression for the area of the remaining plywood. Evaluate the expression for x 4 in. Multiply. 24. (3x)(5x4) 25. (4x 2y)(5xy 3) 26. (2a2b4)(5a4b5) 27. a(a3 4a 5) 28. 3m3n4(2m3n4 5m2n2) 29. 3a3(ab2 2ab 8a) 30. (x 2)(x 12) 31. (x 2)(x 4) 32. (a 3)(a 7) 33. A student forms a box from a 10 in. by 15 in. piece of cardboard by cutting a square with side length x out of each corner and folding up the sides. Write and simplify an expression for the area of the base of the box. Chapter 12 Test 629 KEYWORD: MT8CA Practice Cumulative Assessment, Chapters 1–12 Multiple Choice 1. Jerome walked on a treadmill for 45 minutes at a speed of 4.2 miles per hour. How far did Jerome walk? A 1.89 miles 3.15 miles C B 2.1 miles 5. If rectangle MNQP is similar to rectangle ABDC, then what is the area of rectangle ABDC? 18 cm M N x3 A 10 cm x 2. Which line has a slope of 3? A 3 C y B 0 -3 3 y 0 x P y x -3 0 D 3 x -5 0 C D A 44 cm2 C 120 cm2 B 66 cm2 D 270 cm2 6. Which statement is modeled by 3f 2 16? y 4 Q x -4 -3 A Two added to 3 times f is at least 16. B Three times the sum of f and 2 is at most 16. C The sum of 2 and 3 times f is more than 16. D The product of 3f and 2 is no more than 16. 3 -3 3. How much water can the cone-shaped cup hold? Use 3.14 for π. 4 cm 7. If the area of a circle is 49π and the circumference of the circle is 14π, what is the diameter of the circle? 10 cm A 41.9 cm3 C 167.47 cm3 B 502.4 cm3 D 1507.2 cm3 4. Twenty-two percent of the sales of a general store are due to snack sales. If the store sold $1350 worth of goods, how much of the total was due to snack sales? 630 B D 5.6 miles A $167 C B $297 D $2970 Chapter 12 Polynomials $1053 A 7 units C 21 units B 14 units D 49 units 8. Nationally, there were 217.8 million people age 18 and over and 53.3 million children ages 5 to 17 as of July 1, 2003, according to the U.S. Census Bureau. How do you write the number of people age 5 and older in scientific notation? A 2.711 102 C 2.711 107 B 2.711 106 D 2.711 108 9. There are 36 flowers in a bouquet. Two-thirds of the flowers are roses. One-fourth of the roses are red. What percent of the bouquet is made up of red roses? A 9% B 163% 2 C 25% D 663% Short Response 14. A quilt is made by connecting squares like the one below. 2x 4 3x 4 2 If a problem involves decimals, you may be able to eliminate answer choices that do not have the correct number of places after the decimal point. x a. Write an expression for the area of the triangle and an expression for the area of the square. b. Write an expression for the area of the gray region. Gridded Response 10. The floor in the entrance way of Kendra’s house is square and measures 6.5 feet on each side. Colored tiles have been set in the center of the floor in a square measuring 4 feet on each side. The remaining floor consists of white tiles. 15. Draw a model for the product of the two binomials (x 3) and (2x 5) with the following tiles. Use the model to determine the product. x2 6.5 ft 4 ft x 1 Extended Response 16. A cake pan is made by cutting four squares from a 18 cm by 24 cm piece of tin and folding the sides as shown. 24 cm How many square feet of the entrance way floor consists of white tiles? 11. Rosalind purchased a sewing machine discounted 20%. The original selling price of the machine was $340. What was the sale price in dollars? 12. What is the length, in centimeters, of the diagonal of a square with side length 8 cm? Round your answer to the nearest hundredth. 13. The length of a rectangle is 2 units greater than the width. The area of the rectangle is 24 square units. What is its width? x x x x 18 cm x x x x a. Write an expression for the length, width, and height of the cake pan in terms of x. b. Multiply the expressions from part a to find a polynomial that gives the volume of the cake pan. c. Evaluate the polynomial for x 1, x 2, x 3, and x 4. Which value of x gives the cake pan with the largest volume? Give the dimensions and the volume of the largest cake pan. Cumulative Assessment, Chapters 1–12 631 Student Handbook Extra Practice EP2 Skills Bank SB2 Place Value to the Billions . . . . . . . . . . . . . . . . . . . . . . . . . . . SB2 Round Whole Numbers and Decimals . . . . . . . . . . . . . . . . . SB2 Compare and Order Whole Numbers. . . . . . . . . . . . . . . . . . SB3 Compare and Order Decimals. . . . . . . . . . . . . . . . . . . . . . . . SB3 Divisibility Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB4 Factors and Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB4 Prime and Composite Numbers . . . . . . . . . . . . . . . . . . . . . . SB5 Prime Factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB5 Greatest Common Divisor (GCD) . . . . . . . . . . . . . . . . . . . . . SB6 Least Common Multiple (LCM). . . . . . . . . . . . . . . . . . . . . . . SB6 Multiply and Divide by Powers of Ten . . . . . . . . . . . . . . . . . SB7 Dividing Whole Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB7 Plot Numbers on a Number Line . . . . . . . . . . . . . . . . . . . . . SB8 Simplest Form of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . SB8 Mixed Numbers and Improper Fractions . . . . . . . . . . . . . . SB9 Finding a Common Denominator . . . . . . . . . . . . . . . . . . . . SB9 Adding Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB10 Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB10 632 Student Handbook Multiplying Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB11 Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB11 Adding Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB12 Subtracting Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB12 Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB13 Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB13 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB14 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB15 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB16 Geometric Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB16 Classify Triangles and Quadrilaterals . . . . . . . . . . . . . . . . SB17 Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB18 Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB19 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB20 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB21 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB22 Bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB22 Compound Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SB23 Inductive and Deductive Reasoning . . . . . . . . . . . . . . . . . SB24 Selected Answers Glossary Index Table of Formulas and Symbols SA1 G1 I2 inside back cover 633 Extra Practice Chapter 1 LESSON 1-1 Evaluate each expression for the given value(s) of the variable(s). 1. 3 x for x 5 2. 6m 2 for m 3 3. 2(p 3) for p 8 4. 4x y for x 1, y 3 5. 2y x for x 3, y 6 6. 5x 1.5y for x 2, y 4 LESSON 1-2 Write an algebraic expression for each word phrase. 7. seven less than a number b 8. eight more than the product of 7 and a 9. a quotient of 8 and a number m 10. five times the sum of c and 18 Write a word phrase for each algebraic expression. x 11. 9 3 12. 19x 14 1 13. 3(x 1) 4 14. x 100 15. Write a word problem that can be evaluated by the algebraic expression x 122, and then evaluate the expression for x 225. LESSON 1-3 16. In a miniature golf game the scores of four brothers relative to par are Jesse 3, Jack 2, James 5, and Jarod 1. Use , , or to compare Jack’s and Jarod’s scores. Then list the brothers in order from the lowest score to the highest. Write the integers in order from least to greatest. 17. 4, 6, 2 18. 1, 16, 9 19. 14, 2, 19 20. ⏐9⏐ ⏐4⏐ 21. ⏐3⏐ ⏐19⏐ 22. ⏐52 12⏐ 23. ⏐5 7⏐⏐8 6⏐ 24. ⏐4⏐ ⏐1 3⏐ 25. ⏐13 9⏐ ⏐15⏐ Simplify each expression. LESSON 1-4 Add. 26. 4 6 27. 3 (8) 28. 6 (2) 29. 7 (11) Evaluate each expression for the given value of the variable. 30. x 9 for x 8 31. x 3 for x 3 33. The middle school registrar is checking her records. Use the information in the table to find the net change in the number of students for this school for the week. EP2 Extra Practice 32. x 5 for x 7 Day Students Students Registering Withdrawing Monday 4 2 Tuesday 6 7 Wednesday 5 5 Thursday 1 4 Friday 4 0 Extra Practice Chapter 1 LESSON 1-5 Subtract. 34. 6 4 35. 8 (3) 36. 6 (3) 37. 5 8 Evaluate each expression for the given value of the variable. 38. 7 x for x 4 39. 8 s for s 6 40. 8 b for b 12 41. ⏐3 x⏐for x 2 42. ⏐7 b⏐4 for b 10 43. ⏐5 m⏐⏐6⏐for m 3 44. An elevator rises 351 feet above ground level and then drops 415 feet to the basement. What is the position of the elevator relative to ground level? LESSON 1-6 Multiply or divide. 63 46. 7 47. 7(3) 52 48. 4 50. 5(9) 11 51. 4(16 4) 52. 3 7(10 14) 53. 4 x 13 54. t 3 8 55. 17 m 11 56. 5 a 7 57. p 5 23 58. 31 y 50 59. 18 k 34 60. g 16 23 45. 8(6) Simplify. 49. 8(4 5) LESSON 1-7 Solve. 61. Richard biked 39 miles on Saturday. This distance is 13 more miles than Trevor biked. How many miles did Trevor bike on Saturday? LESSON 1-8 Solve and check. 62. a 4 2 66. 8b 64 63. 49 7d 67. 144 9y c 8 64. 2 x 68. 2 78 65. 57 3p 69. 19c 152 70. Jessica hiked a total of 36 miles on her vacation. This distance is 4 times as far as she typically hikes. How many miles does Jessica typically hike? LESSON 1-9 Translate each sentence into an equation. 71. 8 less than the product of 5 and a number x is 53. 72. 7 more than the quotient of a number t and 6 is 9. Solve. 73. 3m 5 23 x 74. 9 4 31 75. 1 6y 11 m 76. 16 7 5 Extra Practice EP3 Extra Practice Chapter 2 LESSON 2-1 Write each fraction as a decimal. 1 1. 8 7 2. 1 2 5 9 4. 2 0 3. 3 Write each decimal as a fraction in simplest form. 5. 0.4 6. 0.05 7. 0.125 8. 0.625 Write each repeating decimal as a fraction in simplest form. 10. 0.25 11. 0.45 9. 0.7 12. 0.009 LESSON 2-2 Compare. Write , , or . 6 13. 7 4 5 5 2 17. 18 13 9 11 14. 1 10 5 1 1 18. 28 25 1 15. 3 5 6 12 19. 1 7 4 0.75 1 8 16. 5 7 20. 58 5.9 21. In a sewing class, students were instructed to measure and cut cloth to a width of 3 yards. While checking four students’ work, the teacher found that the four pieces of cloth were cut to the following measurements: 11 3 3.5 yards, 2.75 yards, 21 yards, and 38 yards. List these measurements 6 from least to greatest. LESSON 2-3 22. Hannah and Elizabeth drove to Niagara Falls for vacation. Hannah drove 9834 miles, and Elizabeth drove 106.44 miles. How far did they drive together? Add or subtract. Write each answer in simplest form. 3 7 23. 4 4 19 11 24. 8 8 5 15 25. 4 4 26. 4 4 7 11 9 15 27. 2 2 11 14 28. 2 2 9 22 29. 4 4 30. 3 3 21 16 Evaluate each expression for the given value of the variable. 31. 32.9 x for x 15.8 32. 21.3 a for a 37.6 3 1 33. 5 z for z 35 LESSON 2-4 Multiply. Write each answer in simplest form. 5 7 3 35. 1 5 2 36. 5 1 0 38. 4.7(8) 39. 4.1(8.6) 40. 0.06(5.2) 3 4 9 34. 4 9 3 13 37. 7 1 4 41. 0.003(2.6) 42. Rosie ate 212 bananas on Saturday. On Sunday she ate 12 as many bananas as she ate on Saturday. How many bananas did Rosie eat over the weekend? EP4 Extra Practice Extra Practice Chapter 2 LESSON 2-5 Divide. Write each answer in simplest form. 3 1 1 43. 24 3 47. 5.68 0.2 7 5 44. 55 8 48. 7.65 0.05 3 1 45. 39 4 49. 1.76 0.8 2 46. 38 5 50. 0.744 8 Evaluate each expression for the given value of the variable. 7.4 51. x for x 0.5 11.88 52. x for x 0.08 15.3 53. x for x 1.2 54. Yolanda is making bows that each take 2112 inches of ribbon to make. She has 344 inches of ribbon. How many bows can she make? LESSON 2-6 Add or subtract. Write each answer in simplest form. 8 2 55. 9 7 1 3 2 56. 8 3 1 59. 45 27 2 1 57. 3 7 2 7 60. 33 18 1 5 4 58. 6 9 3 61. 48 15 1 1 62. 87 41 0 Evaluate each expression for the given value of the variable. 1 2 1 63. 82 x for x 49 7 64. n 9 for n 18 1 4 65. 18 y for y 7 66. A container has 1012 gallons of milk. If the children at a preschool drink 734 gallons of milk, how many gallons of milk are left in the container? LESSON 2-7 Solve. 67. x 3.2 5.1 1 5 71. m 3 8 68. 3.1p 15.5 3 1 72. x 7 9 a 7.9 69. 2.3 70. 4.3x 34.4 4 2 73. 5w 3 9 5 74. 1 z 8 0 75. Peter estimates that it will take him 934 hours to paint a room. If he gets two of his friends to help him and they work at the same rate as he does, how long will they take to paint the room? LESSON 2-8 76. A bill from the plumber was $383. The plumber charged $175 for parts and $52 per hour for labor. How long did the plumber work at this job? Solve. a 77. 2 3 8 78. 2.4 0.8x 3.2 79. 0.9m 1.6 5.2 b 80. 5 2 3 x4 81. 3 7 82. 8.6 3.4k 1.8 83. 6c 1.2 4.2 84. 7 4d 8 85. 3 2 y9 Extra Practice EP5 Extra Practice Chapter 3 LESSON 3-1 Name the property that is illustrated in each equation. 1. 5 (6 x) (5 6) x 3. 4 m m 4 2. 8 (1) (1) 8 Simplify each expression. Justify each step. 1 4. 5 9 35 2 5 6. 7 6 7 5. 23 14 7 Write each product using the Distributive Property. Then simplify. 7. 3(94) 8. 8(47) 9. 6(52) LESSON 3-2 Combine like terms. 10. 5x 4x 7x 11. 6x 4x 9 5x 7 12. 2x 3 2x 5 13. 7a 2b 6 4b 5a 14. 4s 9t 9 15. 6m 4n 6m n 17. 3(3b 3) 3b 18. 4(x 2) 3x 8 19. 4a 5 2a 9 28 20. 5 8b 6 2b 61 21. 4x 6 8x 9 21 22. g 9 4g 6 12 23. 2 3f 5 5f 6 24. 4r 8 7 6r 9 4a 7 3 25. 5 5 5 1 2b 7 26. 3 3 3 4z 3 27. 1 1 1 1 1 10c 16 56 29. 4 2 8 9x 45 36x 126 30. 3 9 6 9 Simplify. 16. 6(y 4) y LESSON 3-3 Solve. 2f 24 28. 4 4 1 2 31. A round-trip car ride took 12 hours. The first half of the trip took 7 hours at a rate of 45 miles per hour. What was the average rate of speed on the return trip? LESSON 3-4 Solve. 32. 4z 2 z 1 33. 4a 4 a 11 34. 4p 6 3 4p 35. 6 5c 3c 4 36. 7d 3 2d 5d 8 1 37. 3f 4 5f f 4 f 38. 5k 4 k 3k 6 2k w 5 2w 7 2w 39. 4 8 2 8 4 a 1 5a 9 2a a 40. 3 6 6 6 3 3 2 5 41. 3 9 6 6 1 8 2q 42. A cafeteria charges a fixed price per ounce for the salad bar. A sandwich costs $3.10, and a large drink costs $1.75. If a 7-ounce salad and a drink cost the same as a 4-ounce salad and a sandwich, how much does the salad cost per ounce? EP6 Extra Practice q 5q Extra Practice Chapter 3 LESSON 3-5 Write an inequality for each situation or statement. 43. The cafeteria could hold no more than 50 people. 44. There were fewer than 20 boats in the marina. 45. A number n decreased by 4 is more than 13. 46. A number x divided by 5 is at most 28. Graph each inequality. 47. y 2 48. f 3 49. n 1.5 50. x 4 Write a compound inequality for each statement. 51. A number m is both less than 8.5 and greater than or equal to 0. 52. A number c is either greater than 1.5 or less than or equal to 1. LESSON 3-6 Solve and graph. 1 53. x 3.5 7 54. h 5 13 55. 5 q 32 56. m 1 0.25 2 1 57. n 43 53 58. q 0.5 1 59. 30 h 23 60. 7 x 65 61. 5 2 62. 40 4q 63. 9x 72 64. 2 3 65. 3 6 66. 3 5 67. f 4 68. 5y 55 4 LESSON 3-7 Solve and graph. v p w s 69. Reese is running for student council president. In order for a student to be elected president, at least 13 of the students must vote for him. If there are 432 students in a class, at least how many students must vote for Reese in order for him to be elected class president? LESSON 3-8 Solve and graph. 70. 3a 6 12 71. 5 4x 7 72. 2b 8 16 73. 4z 8 4 x 1 2 74. 6 3 3 1 75. 2k 8 9 d 4 76. 2 5 2 2 7 77. 3 6 6 p 78. Nikko wants to make flyers promoting a library book sale. The printer charges $40 plus $0.03 per flyer. How many flyers can Nikko make without spending more than the library’s $54 budget? Extra Practice EP7 Extra Practice Chapter 4 LESSON 4-1 Write in exponential form. 1. 3 3 3 3 2. 6a 6a 6a 6a 6a 3. (9) (9) 4. b 6. 34 8. (3)5 Simplify. 5. 25 7. (6)2 Evaluate each expression for the given values of the variables. x 9. s 4 y(s 3) for s 1 and y 2 10. 10 x 2 2(y 4) for x 3 and y 2 LESSON 4-2 Simplify the powers of 10. 11. 101 12. 102 13. 103 14. 104 Simplify. 15. (4)2 2 3 18. 4 (9 3)0 3 16. 33 17. (5)4 19. 13 (3) 19(1 2)2 20. 45 32 (3)3 LESSON 4-3 Simplify each expression. Write your answer in exponential form. 21. 24 25 3 x 25. 3 y 9 6 22. w7 w7 4 23. 9 c 24. 2 26. (30)4 27. (32)3 28. (a 3)4 4 c LESSON 4-4 Multiply or divide. Assume that no denominator equals zero. 29. (x5)(4x4) 30. (7a4b2)(3b4) 31. (8m6n3)(5mn7) 10 22m 32. 5 8 7 18x y 33. 8 5 9 45a b 34. 2 5 36. (9p3r)2 37. (4a7b4)3 2m 12x y 9a b Simplify. 35. (3y)5 LESSON 4-5 Write each number in scientific notation. 38. 0.00384 39. 1,450,000,000 40. 0.654 Write each number in standard form. 41. 2.4 103 42. 3.62 105 43. 5.036 104 44. 8.93 102 45. The population of the United States was approximately 2.9 108 people in July 2006. In the same month, the population of Canada was approximately 3.3 107 people. Which country had the greater population in July 2006? EP8 Extra Practice Extra Practice Chapter 4 LESSON 4-6 Find the two square roots of each number. 46. 25 47. 49 48. 289 49. 169 50. The area of a square garden is 1,681 square feet. What are the dimensions of the garden? Simplify each expression. 400x20 51. 52. 49y6 53. m14n12 54. 196a16 b8 LESSON 4-7 Each square root is between two integers. Name the integers. Explain your answer. 55. 30 56. 61 58. 124 57. 93 59. Each tile on Michelle’s patio is 18 square inches. If her patio is square shaped and consists of 81 tiles, about how big is her patio? Approximate each square root to the nearest hundredth. 60. 52 61. 83 62. 166 63. 246 LESSON 4-8 Write all classifications that apply to each number. 64. 5 16 66. 2 65. 61.2 67. 6 State if the number is rational, irrational, or not a real number. 68. 25 4 69. 9 13 71. 0 70. 17 Find a real number between each pair of numbers. 1 2 1 72. 58 and 58 2 5 73. 43 and 43 6 74. 37 and 37 LESSON 4-9 Use the Pythagorean Theorem to find each missing measure. 75. 76. 77. 78. b c y 5 in. 25 ft 20 ft 12 km 8 km 9 cm 10 cm 12 in. x Tell whether the given side lengths form a right triangle. 79. 6, 8, 10 80. 8, 11, 13 81. 7, 10, 12 82. 0.8, 1.5, 1.7 Extra Practice EP9 Extra Practice Chapter 5 LESSON 5-1 Write each ratio in simplest form. 1. 2 cups of milk to 8 eggs 2. 36 inches to 2 feet 3. 4 feet to 20 yards Simplify to tell whether the ratios are equivalent. 5 3 and 1 4. 3 0 8 12 16 5. 2 and 2 1 8 91 52 7. 6 and 112 4 15 10 6. 2 and 1 1 6 LESSON 5-2 8. Nikko jogs 3 miles in 30 minutes. How many miles does she jog per hour? 9. A penny has a mass of 2.5 g and a volume of approximately 0.442 cm3. What is the approximate density of a penny? Estimate the unit rate. 10. 384 mg of calcium for 8 oz of yogurt 11. $57.50 for 5 hours 12. Determine which brand of detergent has the lower unit rate. Detergent Brand LESSON 5-3 Size (oz) Price ($) Pizzazz 128 3.08 Spring Clean 64 1.60 Bubbling 196 4.51 Tell whether the ratios are proportional. 7 3 13. 8 and 4 3 24 14. 4 and 3 2 32 18 15. 4 and 2 8 7 6 12 16. 2 and 1 2 0 17. Mark is making 35 sandwiches for a luncheon. He made the first 15 sandwiches in 45 minutes. If he continues to work at the same rate, how many more minutes will he take to complete the job? 18. An 18-pound weight is positioned 6 inches from a fulcrum. At what distance from the fulcrum must a 24-pound weight be positioned to keep the scale balanced? LESSON 5-4 19. A water fountain dispenses 8 cups of water per minute. Find this rate in pints per minute. 20. Jo’s car uses 1664 quarts of gas per year. Find this rate in gallons per week. 21. Toby walked 352 feet in one minute. What is his rate in miles per hour? 22. A three-toed sloth has a top speed of 0.22 feet per second. A giant tortoise has a top speed of 2.992 inches per second. Convert both speeds to miles per hour, and determine which animal is faster. 23. There are markers every 1000 feet along the side of a road. While cycling, Ben passes marker number 6 at 3:35 P.M. and marker number 24 at 3:47 P.M. Find Ben’s average speed in feet per minute. Use dimensional analysis to check the reasonableness of your answer. EP10 Extra Practice Extra Practice Chapter 5 LESSON 5-5 24. Which triangles are similar? A B A C D G 11 ft 30° 11.2 ft C 60° B 5 ft 10 ft E 60° 45° 10 ft 14.1 ft 45° I 10 ft 22 ft 22.4 ft 30° H 25. Khaled scans a photo that is 5 in. wide by 7 in. long into his computer. If the length of the scanned photo is reduced to 3.5 in., how wide should the scanned photo be for the two photos to be similar? F 26. Mutsuko drew an 8.5-inch-wide by 11-inch-tall picture that will be turned into a 34-inch-wide poster. How tall will the similar poster be? 27. A right triangle has legs that measure 3 cm and 4 cm. The shorter leg of a similar right triangle measures 6 cm. What is the length of the other leg of the similar triangle? LESSON 5-6 28. Brian casts a 9 ft shadow at the same time that Carrie casts an 8 ft shadow. If Brian is 6 ft tall, how tall is Carrie? 29. A telephone pole casts an 80 ft shadow, and a child standing nearby who is 3.5 ft tall casts a 6 ft shadow. How tall is the pole? LESSON 5-7 Use the map to answer each question. 30. On the map, the distance from State College to Belmont is 2 cm. What is the actual distance between the two locations? 31. Henderson City is 83 miles from State College. How many centimeters apart should the two locations be placed on the map? Belmont State College Scale: 1 cm:5 mi 32. What is the scale of a drawing in which a building that is 95 ft tall is drawn 6 in. tall? 33. A model of a skyscraper was made using a scale of 0.5 in:5 ft. If the actual skyscraper is 570 feet tall, what is the height of the model? 34. Julio uses a scale of 81 inch 1 foot when he paints landscapes. In one painting, a giant sequoia tree is 34.375 inches tall. How tall is the actual tree? 35. On a scale drawing of a house plan, the master bathroom is 112 inches wide and 258 inches long. If the scale of the drawing is 136 inches 1 foot, what are the actual dimensions of the bathroom? Extra Practice EP11 Extra Practice Chapter 6 LESSON 6-1 Compare. Write , , or . 3 1. 5 2 2. 3 62% 2 663% 3. 24% 4. 1% 0.25 0.11 Order the numbers from least to greatest. 5. 0.11, 11.5%, 10%, 8 1 1 2 6. 5, 100%, 263%, 0.3 7 7. 6, 115%, 83, 83.3% 8. 67.5%, 3, 160%, 2.2 7 9. A molecule of ammonia is made up of 3 atoms of hydrogen and 1 atom of nitrogen. What percent of an ammonia molecule is made up of hydrogen atoms? LESSON 6-2 Estimate. 10. 51% of 1019 11. 33% of 60 12. 60% of 79 2 13. 663% of 211 14. Approximately 23% of each class walks to school. A student said that in a class of 20 students, approximately 2 students walk to school. Estimate to determine if the student’s number is reasonable. Explain. LESSON 6-3 15. What percent of 364 is 92? 16. What percent of 48 is 5? 17. What percent of 164 is 444? 18. What percent of 50 is 4? 19. Mt. McKinley in Alaska is 20,320 feet tall. The height of Mt. Everest is about 143% of the height of Mt. McKinley. Estimate the height of Mt. Everest. Round to the nearest thousand. 20. A restaurant bill for $64.45 was split among four people. Dona paid 25% of the bill. Sandy paid 15 of the bill. Mara paid $14.25. Greta paid the remainder of the bill. Who paid the most money? LESSON 6-4 21. 38 is 42% of what number? 22. 46 is 74% of what number? 23. 23 is 8% of what number? 24. 93 is 62% of what number? 25. 315 is 92% of what number? 26. 52 is 120% of what number? 27. A certain rock is a compound of several minerals. Tests show that the sample contains 17.3 grams of quartz. If 27.5% of the rock is quartz, find the mass in grams of the entire rock. 28. The Alabama River is 729 miles in length, or about 31% of the length of the Mississippi River. Estimate the length of the Mississippi River. Round to the nearest mile. EP12 Extra Practice Extra Practice Chapter 6 LESSON 6-5 Find each percent of increase or decrease to the nearest percent. 29. from 10 to 17 30. from 38 to 65 31. from 91 to 44 32. from 3 to 25 33. from 86 to 27 34. from 38 to 46 35. from 19 to 60 36. from 88 to 23 37. A stereo that sells for $895 is on sale for 20% off the regular price. What is the discounted price of the stereo? 38. Mr. Schultz’s hardware store marks up merchandise 28% over warehouse cost. What is the selling price for a wrench that costs him $12.45? LESSON 6-6 Find each sales tax to the nearest cent. 39. total sales: $12.89 sales tax rate: 8.25% 40. total sales: $87.95 sales tax rate: 6.5% 41. total sales: $119.99 sales tax rate: 7% Find the total sales. 42. commission: $127.92 commission rate: 8% 43. commission: $32.45 commission rate: 5% 44. An electronics salesperson sold $15,486 worth of computers last month. She makes 3% commission on all sales and earns a monthly salary of $1200. What was her total pay last month? 45. Jon bought a printer for $189 and a set of printer cartridges for $129. Sales tax on these items was 6.5%. How much tax did Jon pay for those items? 46. In her shop, Stephanie earns 16% profit on all of the clothes she sells. If total sales were $3920 this month, what was her profit? LESSON 6-7 Find the simple interest and the total amount to the nearest cent. 47. $3000 at 5.5% per year for 2 years 48. $15,599 at 9% per year for 3 years 49. $32,000 at 3.6% per year for 5 years 50. $124,500 at 8% per year for 20 years 51. Rebekah invested $15,000 in a mutual fund at a yearly rate of 8%. She earned $7200 in simple interest. How long was the money invested? 52. Shu invested $6000 in a savings account for 4 years at a rate of 5%. a. What would be the value of the investment if the account is compounded semiannually? b. What would be the value of the investment if the account is compounded quarterly? Extra Practice EP13 Extra Practice Chapter 7 LESSON 7-1 Identify the quadrant that contains each point. Plot each point on a coordinate plane. 1. M(–1, 1) 2. N(4, 4) y A 2 3. Q(3, –1) B x Give the coordinates of each point. 4. A 2 5. B 6. C 4 C –2 LESSON 7-2 Determine if each relationship represents a function. y 7. 8. 4 O x 4 x 3 1 1 1 0 2 2 0 y 9. y 2 O x 2 LESSON 7-3 Graph each linear function. 10. y 2x 2 11. y x 3 12. y x 2 13. The outside temperature is 52°F and is increasing at a rate of 6°F per hour. Write a linear function that describes the temperature over time. Then make a graph to show the temperature over the first 3 hours. LESSON 7-4 Create a table for each quadratic function, and use the table to graph the function. 14. y x2 2 15. y x2 x 8 16. y x2 x 6 LESSON 7-5 Create a table for each cubic function, and use the table to graph the function. 17. y x3 1 1 18. y 4x3 19. y 4x3 Tell whether each function is linear, quadratic, or cubic. 20. 21. y x EP14 Extra Practice 22. y x y x Extra Practice Chapter 7 LESSON 7-6 Find the slope of each line. 23. 4 24. y 2 ⫺4 ⫺2 x O ⫺2 3 x 2 ⫺4 25. y 2 O 2 1 4 ⫺2 ⫺2 4 ⫺2 y O 1 x 2 4 ⫺3 ⫺4 LESSON 7-7 Find the slope of the line that passes through each pair of points. 26. (3, 4) and (2, 2) 27. (6, 2) and (2, 6) 28. (3, 3) and (1, 4) 29. (2, 4) and (1, 1) 30. The table shows how much money Andy and Margie made while working at the concession stand at a baseball game one weekend. Use the data to make a graph. Find the slope of the line, and explain what the slope means. Time (h) 2 4 6 8 Money Earned ($) 15 30 45 60 LESSON 7-8 31. Abby rode her bike to the park. She had a picnic there with friends and then rode home. Which graph best shows the situation? Graph A Distance from home Distance from home Distance from home Time Graph C Graph B Time Time 32. Greg walked to a café for lunch. Then he walked across the street to a store before returning home. Sketch a graph to show Greg’s distance compared to time. LESSON 7-9 33. Instructions for a chemical-concentrate swimming-pool cleaner state that 2 ounces of concentrate should be added to every 112 gallons of water used. How many ounces of concentrate should be added to 18 gallons of water? 34. The distance d that an object falls varies directly with the square of the time t of the fall. This relationship is expressed by the formula d k t 2. An object falls 90 feet in 3 seconds. How far will the object fall in 15 seconds? Extra Practice EP15 Extra Practice Chapter 8 LESSON 8-1 C Use the diagram to name each figure. N A 1. a line 2. three rays 3. a plane 4. three segments B Use the diagram to name each figure. 5. a right angle 6. two acute angles 7. an obtuse angle 8. a pair of complementary angles B C 55° 35° 90° D E A LESSON 8-2 A Identify two lines that have the given relationship. B D 9. perpendicular lines C 10. skew lines E 11. parallel lines H Identify two planes that appear to have the given relationship. I 12. parallel planes 13. perpendicular planes D E G 14. neither parallel nor perpendicular F LESSON 8-3 Use the diagram to find each angle measure. 15. If m1 107°, find m3. 4 1 3 2 16. If m2 46°, find m4. In the figure, line d⏐⏐line f. Find the measure of each angle. 17. 1 18. 2 19. 3 g 1 120° 2 d f LESSON 8-4 3 Find the missing angle measures in each triangle. 20. 21. 22. 43° 1 23 x° 54° 66° x° and 23. In the figure, B is the midpoint of AC is perpendicular to AC . Find the length BD of AD . EP16 Extra Practice x° y° A 16 m m 16 B 6m D 4x° C Extra Practice Chapter 8 LESSON 8-5 Give all of the names that apply to each figure. A 24. B 2 cm AB || CD D 2 cm 25. 2 cm 2 cm C Find the coordinates of the missing vertex. Then tell which lines are parallel and which lines are perpendicular. 26. rhombus ABCD with A(2, 3), B(3, 0), and D(1, 0) 27. square JKLM with J(1, 1), K(4, 1), and L(4, 2) 28. rectangle ABCD with A(4, 3), B(1, 3), and D(4, 1) 29. trapezoid JKLM with J(2, 1), K(2, 1), and L(1, 1) LESSON 8-6 In the figure, quadrilateral ABCD quadrilateral KLMN. 30. Find x. A 125° 31. Find y. D 32. Find z. M B x2 10 32 C 8y L 95 K z° N LESSON 8-7 Graph each transformation. 33. Rotate PQR 90° counterclockwise about vertex R. 4 P 34. Reflect the figure across the y-axis. y Q 4 2 y E R D 4 y S 2 x R O 35. Translate RST 3 units right and 3 units down. 2 G x F 2 O 2 4 x 2 O T2 4 LESSON 8-8 Create a tessellation with each figure. 36. 37. Extra Practice EP17 Extra Practice Chapter 9 LESSON 9-1 Find the perimeter of each figure. 1. 2. 3. 1 6m 14 m 3 3 in. 11 m 1 3 3 in. 9m Graph and find the area of each figure with the given vertices. 4. (2, 1), (5, 1), (2, 4), (5, 4) 5. (1, 2), (2, 1), (5, 2), (6, 1) 6. Find the perimeter and area of the figure. 10 4 4 6 6 6 6 10 LESSON 9-2 Find the missing measurement for each figure with the given perimeter. 7. perimeter 27 cm 8. perimeter 92 ft 9. perimeter 16 units 3 16 ft 5 cm 10 cm 4 c 16 ft a d 5 32 ft Graph and find the area of each figure with the given vertices. 10. (4, 3), (2, 3), (2, 1) 11. (2, 1), (5, 3), (0, 1), (3, 3) 12. The sail of a toy sailboat forms a right triangle with legs that measure 5 inches each. Find the perimeter and area of the sail. LESSON 9-3 Name the parts of circle I. M L 13. radii 14. diameters J 15. chords I K H Find the central angle measure of the sector of a circle that represents the given percent of a whole. 16. 12% EP18 17. 40.5% Extra Practice 18. 82% 19. 50% Extra Practice Chapter 9 LESSON 9-4 Find the circumference and area of each circle, both in terms of π and to the nearest hundredth. Use 3.14 for π. 20. 21. 4 cm 22. 14 in. 14 ft 23. A wheel has a radius of 14 in. Approximately how far does a point on the wheel travel if it makes 15 complete revolutions? Use 272 for π. LESSON 9-5 Find the shaded area. Round to the nearest tenth, if necessary. 24. 3 ft 25. 9 ft 3 ft 13 m 5m 3 ft 3 ft 26. 27. 8m 3 yd 4 yd 8m 1 yd 7 yd 4m LESSON 9-6 Find the area of each figure. 28. 29. Use composite figures to estimate the shaded area. 30. 31. Extra Practice EP19 Extra Practice Chapter 10 LESSON 10-1 Describe the bases and faces of each figure. Then name the figure. 1. 2. 3. Classify each figure as a polyhedron or not a polyhedron. Then name the figure. 4. 5. 6. LESSON 10-2 Find the volume of each figure to the nearest tenth. Use 3.14 for π. 7. 8. 9. 2 cm 5 ft 7 in. 4 cm 2 ft 8 ft 15 in. 5 in. 10. A can has a diameter of 3 in. and a height of 5 in. Explain whether doubling only the height of the can would have the same effect on the volume as doubling only the diameter. 11. A shoe box is 6.8 in. by 5.9 in. by 16 in. Estimate the volume of the shoe box. 4 12. Find the volume of the composite figure. 2 3 7 5 LESSON 10-3 9 Find the volume of each figure to the nearest tenth. Use 3.14 for π. 10 m 13. 14. 4.2 yd 15. 46 mm 31 mm 15 m 15 m 8 yd 16. A rectangular pyramid has a height of 15 ft and a base that measures 5 ft by 7.5 ft. Find the volume of the pyramid. EP20 Extra Practice 31 mm Extra Practice Chapter 10 LESSON 10-4 Find the surface area of each prism to the nearest tenth. 17. 18. 19. 3m 9 cm 21 ft 9 cm 3m 6m 9 cm 20 ft 40 ft Find the surface area of each cylinder to the nearest tenth. Use 3.14 for π. 20. 21. 6 m 22. 7 ft 20 m 4 cm 9 ft 12 mm 14 mm LESSON 10-5 Find the surface area of each figure to the nearest tenth. Use 3.14 for π. 23. 24. 1m 15 in. 25. 17 yd 1m 10 yd 13 in. 13 in. 10 yd 10 yd Find the surface area of each figure with the given dimensions. Use 3.14 for π. 26. cone: diameter 4 in. slant height 3 in. 27. regular square pyramid: base area 64 ft2 slant height 8 ft LESSON 10-6 Find the volume of each sphere, both in terms of π and to the nearest tenth. Use 3.14 for π. 28. r 5 ft 29. d 40 cm Find the surface area of each sphere, both in terms of π and to the nearest tenth. Use 3.14 for π. 30. r 3.8 mm 31. d 1.5 ft LESSON 10-7 An 8 cm cube and a 5 cm cube are both part of a demonstration kit for architects. Compare the following values of the two cubes. 32. edge length 33. surface area 34. volume Extra Practice EP21 Extra Practice Chapter 11 LESSON 11-1 1. Use a line plot to organize the data showing the number of miles that students cycled over a weekend. What number of miles did students cycle the most? Number of Miles Cycled by Students 12 21 12 8 10 15 15 18 12 11 9 10 9 6 0 5 12 5 14 14 10 8 12 10 9 2. Use the given data to make a back-to-back stem-and-leaf plot. World Series Win/Loss Records of Selected Teams (through 2001) Team Yankees Pirates Giants Tigers Cardinals Dodgers Orioles Wins 26 5 5 4 9 6 3 Losses 12 2 11 5 6 12 4 LESSON 11-2 Find the mean, median, mode, and range of each data set. 3. 13, 8, 40, 19, 5, 8 4. 21, 19, 23, 26, 15, 25, 25 5. The table shows the number of points a player scored in ten games. Find the mean and median of the data. Which measure best describes the typical number of points scored in a game? Justify your answer. Points a Player Scored in Ten Games Game 1 2 3 4 5 6 7 8 9 10 Points 36 34 28 50 30 30 31 47 55 29 LESSON 11-3 Find the lower and upper quartiles for each data set. 6. 27, 31, 26, 24, 33, 31, 24, 28, 31, 24, 22, 27, 31, 28, 26 7. 84, 79, 77, 72, 81, 82, 89, 94, 72, 80, 76, 80, 83, 86, 73 Use the given data to make a box-and-whisker plot. 8. 11, 4, 9, 17, 16, 12, 5, 16, 9, 11, 13 9. 57, 53, 52, 31, 48, 59, 64, 86, 56, 54, 55 LESSON 11-4 10. The table shows the relationship between the number of years of post high school education and salary. Use the given data to make a scatter plot. Then describe the relationship between the data sets. Number of Years of Post High School Education and Salary Years 1 3 4 4 4 5 5 6 6 8 Salary ($1000’s) 18 20.5 28 35 51 43 58 52 64 58 75 73.5 EP22 Extra Practice 1 8 Extra Practice Chapter 11 LESSON 11-5 Refer to the spinner at right. Give the probability for each outcome. 11. not red 12. blue 13. not yellow A game consists of randomly selecting four colored marbles from a jar and counting the number of red marbles in the selection. The table gives the experimental probability of each outcome. Number of Red Marbles Probability 0 1 2 3 4 0.043 0.248 0.418 0.248 0.043 14. What is the probability of selecting 2 or more red marbles? 15. What is the probability of selecting at most 1 red marble? LESSON 11-6 A utensil is drawn from a drawer and replaced. The table shows the results after 100 draws. 16. Estimate the probability of drawing a spoon. 17. Estimate the probability of drawing a fork. A sales assistant tracks the sales of a particular sweater. The table shows the data after 1000 sales. 18. Use the table to compare the probability that the next customer will buy a brown sweater to the probability that the next customer will buy a beige sweater. Outcomes Draws Spoon 33 Knife 36 Fork 31 Outcomes Sales White 361 Beige 207 Brown 189 Black 243 LESSON 11-7 An experiment consists of rolling a fair number cube. There are six possible outcomes: 1, 2, 3, 4, 5, and 6. Find the probability of each event. 19. P(rolling an odd number) 20. P(rolling a 2) 21. P(rolling a number greater than 3) 22. P(rolling a 7) An experiment consists of rolling two fair number cubes. Find each probability. 23. P(rolling a total of 4) 24. P(rolling a total less than 2) 25. P(rolling a total greater than 12) 26. P(rolling a total of 9) LESSON 11-8 27. An experiment consists of rolling a fair number cube three times. For each toss, all outcomes are equally likely. What is the probability of rolling a 2 three times in a row? 28. A jar contains 3 blue marbles and 9 red marbles. What is the probability of drawing 2 red marbles at the same time? Extra Practice EP23 Extra Practice Chapter 12 LESSON 12-1 Determine whether each expression is a monomial. 1 1. 4r 2st 5 2. 6xy 3 3. 2 yx 4 3 3m 4. 5 n Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1 5. 4x 3 x 2 4 9. xw7 7wz 6. 9x 5y 2z4 4 7. 5m4n3 m3 3 z 2 11. 3a 4 a3 5 10. 4 8. h 2h0.5 1 12. 13st 6 Find the degree of each polynomial. 13. 3x 3 4x 5 8 14. b 2 9b3 b 4 8 15. z 2 5z 6 9z 3 16. 4 t 17. The trinomial 16t 2 vt 12 describes the height in feet of a baseball thrown straight up from a 12-foot platform with a velocity of v ft/s after t seconds. What is the height of the ball after 3 seconds if v 55 ft/s? 18. The trinomial 4x 2 270x 2000 gives the net profit in dollars that a custom bicycle manufacturer earns by selling x bikes in a given month. What is the net profit for a month if x 15 bicycles? LESSON 12-2 Identify the like terms in each polynomial. 19. s 5r 2 7r 9r 2 3s 20. 4x 2 9 7x x 5 10 21. 5x 2y 7x 2z 7yz 2 3x 2y xz 2 22. 3mn 3p 2 3p 5p 2 3mn 23. 8s 2 6 7s 6s4 13 24. 25 15xy 10x 2 3xy 5y 2 5 Simplify. 25. 5z 2 2z z 2 11z 9 26. 5 7(a 2 9) 3a 2 7 27. y 2z 8xz 6xy 2 10y 2z xy 2 28. 5c 2 11cd d 2 5(c 2 d 2) 2c(c d) 29. 5(a 2b 2 3ab) 3(ab 2 5ab) 30. 2s 2t 2 st 2 5s 2t 2 7s 2t 3st 2 s 2t 31. A rectangle has a width of 12 cm and a length of (2x 2 6) cm. The area is given by the expression 12(2x 2 6) cm2. Use the Distributive Property to write an equivalent expression. 32. A parallelogram has a base of (3x 2 4) in. and a height of 4 in. The area is given by the expression 4(3x 2 4) in2. Use the Distributive Property to write an equivalent expression. EP24 Extra Practice Extra Practice Chapter 12 LESSON 12-3 Add. 33. (2x 2y 3 7x3y 2 xy) (x 2y 3 2x 3y 2 3xy) 34. (3a 2 4ab 2) (a 2b 2b2) (a2b 7ab2) 35. (m3 3m2n2 4) (6m2n2 9) 36. (10r 3s2 7r 2s 4r) (4r 3s 2 3r) 37. A rectangle has a width of (x 7) cm and a length of (3x 5) cm. An equilateral triangle has sides of length x 2 2x 3. Write an expression for the sum of the perimeters of the rectangle and the triangle. LESSON 12-4 Find the opposite of each polynomial. 38. 6xy 2y 3 39. 4a 3b 7ab 8 40. 2x 6 3x x 3 41. 7g 5 gh4 Subtract. 42. (5x 2 2xy 3y 2) (3x 2 2y 2 8) 43. 14a (5a 3 3a 6) 44. (5r 2s 2 9r 2s rs) (3r 2s 7rs 2r 2) 45. (12y 3 6xy 1) (8xy 2x 1) 46. The area of the larger rectangle is (10x 2 2x 15) cm2. The area of the smaller rectangle is (5x 2 3x) cm2. What is the area of the shaded region? LESSON 12-5 Multiply. 47. (5xy 2)(7x 3y 2) 48. (2a 2bc 2 )(5a 3b 2 ) 49. (6m3n4 )(2mn) 50. 6t(9s 5t ) 51. p (2p 2 pq 5) 52. 3xy 2(2x 3y 2 4x 2 y xy 6 11xy) 53. A rectangle has a width of 3x 2 y ft and a length of (2x 2 4xy 7) ft. Write and simplify an expression for the area of the rectangle. Then find the area of the rectangle if x 2 and y 3. LESSON 12-6 Multiply. 54. (y 5)(y 3) 55. (s 3)(s 5) 56. (3m 2)(4m 3) 57. (y 1)2 58. (d 5)2 59. (a 9)(a 9) 60. (x 11)(x 5) 61. (7b 2)(7b 2) 62. (2m n)(6m 8n) Extra Practice EP25 Skills Bank Review Skills Place Value to the Billions 4NS1.1 A place-value chart can help you read and write numbers. The number 345,012,678,912.5784 (three hundred forty-five billion, twelve million, six hundred seventy-eight thousand, nine hundred twelve and five thousand seven hundred eighty-four ten-thousandths) is shown. Billions Millions Thousands Ones 345, 012, 678, 912 Tenths Hundredths Thousandths Ten-Thousandths . 5 7 8 4 EXAMPLE Name the place value of the digit. A the 7 in the thousands column 7 ten thousands place B the 0 in the millions column 0 hundred millions place C the 5 in the billions column 5 one billion, or billions, place D the 8 to the right of the decimal point 8 thousandths PRACTICE Name the place value of the underlined digit. 1. 123,456,789,123.0594 2. 123,456,789,123.0594 3. 123,456,789,123.0594 4. 123,456,789,123.0594 5. 123,456,789,123.0594 6. 123,456,789,123.0594 Round Whole Numbers and Decimals 4NS1.3, 5NS1.1 To round to a certain place, follow these steps. 1. Locate the digit in that place, and consider the next digit to the right. 2. If the digit to the right is 5 or greater, round up. Otherwise, round down. 3. Change each digit to the right of the rounding place to zero. EXAMPLE A Round 125,439.378 to the nearest thousand. 125,439.378 Locate the digit. The digit to the right, 4, is less than 5, so round down. 125,000.000 125,000 B Round 125,439.378 to the nearest tenth. 125,439.378 Locate the digit. The digit to the right, 7, is greater than 5, so round up. 125,439.400 125,539.4 PRACTICE Round 259,345.278 to the place indicated. 1. hundred thousand SB2 Skills Bank 2. ten thousand 3. thousand 4. hundredth Compare and Order Whole Numbers 4NS1.2 You can use place values to compare and order whole numbers. EXAMPLE Order the numbers from least to greatest: 42,810; 142,997; 42,729; 42,638. Start at the left most place value. There is one number with a digit in the greatest place. It is the greatest of the four numbers. 42,810 Compare the remaining three numbers. All values in the next two places, the ten thousands and thousands, are the same. 142,997 In the hundreds place, the values are different. Use this digit to order the remaining numbers. 42,729 42,638 42,638; 42,729; 42,810; 142,997 PRACTICE Order the numbers in each set from least to greatest. 1. 2,564; 2,546; 2,465; 2,654 2. 6,237; 6,372; 6,273; 6,327 3. 132,957; 232,795; 32,975; 31,999 4. 9,614; 29,461; 129,164; 129,146 Compare and Order Decimals 4NS1.2 You can also use place values to compare and order decimals. EXAMPLE Order the decimals from least to greatest: 14.35, 14.3, 14.05. 14.35 14.30 14.30 14.35 Compare two of the numbers at a time. Write 14.3 as “14.30.” 14.35 14.05 14.05 14.35 Start at the left and compare the digits. Look for the first place the digits are different. 14.30 14.05 14.30 14.05 Graph the numbers on a number line. 14.05 14.30 14.35 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15 The numbers are in order from left to right: 14.05, 14.3, and 14.35. PRACTICE Order the decimals in each set from least to greatest. 1. 9.5, 9.35, 9.65 2. 4.18, 4.1, 4.09 3. 1.56, 1.62, 1.5 4. 6.7, 6.07, 6.23 Skills Bank SB3 Divisibility Rules 4NS4.1 A number is divisible by another number if the division results in a remainder of 0. Some divisibility rules are shown below. A number is divisible by . . . 2 if the last digit is an even number. Divisible Not Divisible 11,994 2175 216 79 1028 621 15,195 10,007 1332 44 25,016 14,100 144 33 2790 9325 3 if the sum of the digits is divisible by 3. 4 if the last two digits form a number divisible by 4. 5 if the last digit is 0 or 5. 6 if the number is even and divisible by 3. 8 if the last three digits form a number divisible by 8. 9 if the sum of the digits is divisible by 9. 10 if the last digit is 0. PRACTICE Determine whether each number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10. 1. 56 2. 200 3. 75 7. 784 8. 501 9. 2345 4. 324 5. 42 10. 555,555 11. 3009 6. 812 12. 2001 Factors and Multiples 4NS4.1 When two numbers are multiplied to form a third, the two numbers are said to be factors of the third number. Multiples of a number can be found by multiplying the number by 1, 2, 3, 4, and so on. EXAMPLE A List all the factors of 48. The possible factors are whole numbers from 1 to 48. 1 48 48, 2 24 48, 3 16 48, 4 12 48, and 6 8 48 The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. B Find the first five multiples of 3. 3 1 3, 3 2 6, 3 3 9, 3 4 12, and 3 5 15 The first five multiples of 3 are 3, 6, 9, 12, and 15. PRACTICE List all the factors of each number. 1. 8 2. 20 3. 9 4. 51 5. 16 6. 27 Find the first five multiples of each number. 7. 9 SB4 8. 10 Skills Bank 9. 20 10. 15 11. 7 12. 18 Prime and Composite Numbers A prime number has exactly two factors, 1 and the number itself. 2 A composite number has more than two factors. Factors: 1 and 2; prime 4 11 Factors: 1 and 11; prime 47 Factors: 1 and 47; prime 4NS4.2 Factors: 1, 2, and 4; composite 12 Factors: 1, 2, 3, 4, 6, and 12; composite 63 Factors: 1, 3, 7, 9, 21, and 63; composite EXAMPLE Determine whether each number is prime or composite. B 16 A 17 Factors 1, 17 prime C 51 Factors 1, 2, 4, 8, 16 Factors 1, 3, 17, 51 composite composite PRACTICE Determine whether each number is prime or composite. 1. 5 2. 14 3. 18 4. 2 7. 13 8. 39 9. 72 10. 49 5. 23 11. 9 6. 27 12. 89 Prime Factorization 5NS1.4 A composite number can be expressed as a product of prime numbers. This is the prime factorization of the number. To find the prime factorization of a number, you can use a factor tree. EXAMPLE Find the prime factorization of 24. 24 2 • 12 2•3•4 2•3•2•2 24 3• 8 3•2•4 3•2•2•2 24 4• 6 2•2•2•3 The prime factorization of 24 is 2 2 2 3, or 23 3. PRACTICE Find the prime factorization of each number. 1. 25 2. 16 3. 56 4. 18 5. 72 6. 40 Skills Bank SB5 Greatest Common Divisor (GCD) 6NS2.4 The greatest common divisor (GCD) of two or more whole numbers is the greatest whole number that divides evenly into each number. EXAMPLE Find the greatest common divisor of 24 and 32. Method 1: List all the factors of both numbers. Then find all the common factors. Method 2: Find the prime factorization. Then find the common prime factors. 24: 1, 2, 3, 4, 6, 8, 12, 24 32: 1, 2, 4, 8, 16, 32 24: 2 2 2 3 32: 2 2 2 2 2 The common factors are 1, 2, 4, and 8, so the GCD of 24 and 32 is 8. The common prime factors are 2, 2, and 2. The GCD is the product of the factors, so the GCD of 24 and 32 is 2 2 2 8. PRACTICE Find the GCD of each pair of numbers by either method. 1. 9, 15 2. 25, 75 3. 18, 30 4. 4, 10 5. 12, 17 6. 30, 96 7. 54, 72 8. 15, 20 9. 40, 60 10. 40, 50 11. 14, 21 12. 14, 28 Least Common Multiple (LCM) 6NS2.4 The least common multiple (LCM) of two or more numbers is the common multiple with the least value. EXAMPLE Find the least common multiple of 8 and 10. Method 1: List multiples of both numbers. Then find the least value that is in both lists. 8: 8, 16, 24, 32, 40, 48, 56 10: 10, 20, 30, 40, 50, 60 The least value that is in both lists is 40, so the LCM of 8 and 10 is 40. Method 2: Find the prime factorization. Then find the most occurrences of each factor. 8: 2 2 2 10: 2 5 The LCM is the product of the factors, so the LCM of 8 and 10 is 2 2 2 5 40. PRACTICE Find the LCM of each pair of numbers by either method. 1. 2, 4 2. 3, 15 3. 10, 25 7. 12, 21 8. 9, 21 9. 24, 30 SB6 Skills Bank 4. 10, 15 10. 9, 18 5. 3, 7 11. 16, 24 6. 18, 27 12. 8, 36 Multiply and Divide by Powers of Ten 4NS3.2, 5NS2.2 When you multiply by powers of ten, move the decimal point one place to the right for each zero in the power of ten. When you divide by powers of ten, move the decimal point one place to the left for each zero in the power of ten. EXAMPLE Find each product or quotient. A 0.37 100 0.37 100 0.37 B 43 1000 43 1000 43.000 37 43,000 C 0.24 10 0.24 10 0.24 0.024 D 1467 100 1467 100 14 6 7. 14.67 PRACTICE Find each product or quotient. 1. 10 8.53 2. 0.55 104 3. 48.6 1000 4. 2.487 1000 6. 103 12.1 7. 3.75 10 8. 8.5 10 9. 6420 103 5. 6.03 103 10. 1.9 10 Dividing Whole Numbers 5NS2.2 Division is used to separate a quantity into equal groups. The number to be divided is the dividend , and the number you are dividing by is the divisor . The answer to a division problem is known as the quotient . EXAMPLE Divide 8208 by 72. 114 728 2 0 8 72 100 72 288 288 0 Write the dividend under the long division symbol. Subtract. Bring down the next digit. Subtract. Bring down the next digit. Subtract. PRACTICE Divide. 1 2 5 1. 1254 2. 1582 0 ,6 9 8 3. 2684 5 5 6 4. 393 4 7 1 5. 994 6 5 3 6. 3213 8 ,8 4 1 7. 1205 0 4 0 8. 1081 0 ,4 7 6 9. 7411 0 7 ,4 4 5 Skills Bank SB7 Plot Numbers on a Number Line 5NS1.5, 6NS1.1 You can order rational numbers by graphing them on a number line. EXAMPLE 3 4 Put the numbers 0.25, 4, 0.1, and 5 on a number line. Then order the numbers from least to greatest. 3 __ 4 __ 0.1 0.25 4 5 0 0.5 1 3 0.75 4 4 0.8 5 The values increase from left to right: 0.1, 0.25, 34, 45. PRACTICE Plot each set of numbers on a number line. Then order the numbers from least to greatest. 2 1 3 4 1. 2.6, 25, 22 2 2. 0.45, 8, 9 3. 0.55, 3, 0.6 4. 5.25, 53, 5.05, 5.5 1 5. 0.4, 5, 4, 0.42 5 4 6 6. 8, 9, 0.6, 7 1 1 8. 0.3, 7, 0.4, 8 2 7 5 4 9. 8, 6, 7, 0.65 7. 0.18, 7, 6, 0.12 3 1 3 Simplest Form of Fractions 6NS2.4 A fraction is in simplest form when the greatest common divisor of its numerator and denominator is 1. EXAMPLE Simplify. 24 A 30 24: 1, 2, 3, 4, 6, 8, 12, 24 Find the greatest common divisor 30: 1, 2, 3, 5, 6, 10, 15, 30 of 24 and 30. 24 6 4 Divide both the numerator and 30 6 5 the denominator by 6. 18 B 28 18: 1, 2, 3, 6, 9, 18 Find the greatest 28: 1, 2, 4, 7, 14, 28 common divisor of 18 and 28. 18 2 9 Divide both the 28 2 14 numerator and the denominator by 2. PRACTICE Simplify. 15 1. 2 0 32 2. 4 0 14 3. 3 5 30 4. 7 5 17 5. 5 1 18 6. 4 2 19 7. 3 8 22 8. 121 10 9. 3 2 39 10. 9 1 SB8 Skills Bank Mixed Numbers and Improper Fractions 6NS1.0 Mixed numbers can be written as fractions greater than 1, and fractions greater than 1 can be written as mixed numbers. EXAMPLE 3 A Write 2 as a mixed number. 5 2 B Write 6 7 as a fraction. 23 Divide the numerator 5 by the denominator. 4 52 3 20 3 3 45 Multiply the denominator by the whole number. Write the remainder as the numerator of a fraction. 2 67 Add the product to the numerator. 7 6 42 42 2 44 44 7 Write the sum over the denominator. PRACTICE Write each mixed number as a fraction. Write each fraction as a mixed number. 22 1. 5 2. 97 1 41 3. 8 4. 59 7 5. 3 6. 41 1 9 47 7. 1 6 8. 38 2 33 11. 5 31 9. 9 10. 83 7 3 1 12. 129 Finding a Common Denominator 6NS2.4 You must often rewrite two or more fractions so that they have the same denominator, or a common denominator . One way to find a common denominator is to multiply the denominators. Or you can use the least common denominator (LCD), which is the LCM of the denominators. EXAMPLE 3 1 Rewrite 4 and 6 so that they have a common denominator. Method 1: Multiply the denominators: 4 6 24 18 3 36 24 4 46 4 1 14 24 6 64 For each fraction, multiply the denominator by a number to get the common denominator. Then multiply the numerator by the same number. Method 2: Find the LCD. The LCM of the denominators, 4 and 6, is 12. So the LCD is 12. 9 3 33 12 4 43 2 1 12 12 6 62 PRACTICE Rewrite each pair of fractions so that they have a common denominator. 5 4 1. 6, 9 1 5 2. 6, 8 3 1 3. 1 , 0 8 1 3 4. 4, 1 4 2 5 5. 9, 1 2 Skills Bank SB9 Adding Fractions 5NS2.3 To add fractions, first make sure they have a common denominator. Then add the numerators and keep the common denominator. EXAMPLE Add. Write your answer in simplest form. 7 3 A 8 8 7 3 73 10 5 8 8 8 8 4 Add the numerators. Keep the denominator. B 5 3 6 5 Step 1 Find the LCD. The LCD is 30. 5 55 25 Step 2 Rewrite the fractions using the LCD: 6 6 5 30 25 18 43 Step 3 Add: 3 30 3 0 0 3 36 18 5 56 30 Add the numerators. Keep the denominator. PRACTICE Add. Write your answer in simplest form. 3 1 6 3 3 1 1. 5 5 2. 7 7 3. 8 4 2 5 4. 3 9 3 2 5. 4 5 Subtracting Fractions 5NS2.3 To subtract fractions, first make sure they have a common denominator. Then subtract the numerators and keep the common denominator. EXAMPLE Subtract. Write your answer in simplest form. 7 3 1 A 1 0 0 7 3 4 73 2 10 10 10 10 5 Subtract the numerators. Keep the denominator. B 3 1 8 6 Step 1 Find the LCD. The LCD is 24. 3 33 9 Step 2 Rewrite the fractions using the LCD: 8 24 8 3 9 4 5 Step 3 Subtract: 2 24 2 4 4 4 1 14 24 6 64 Subtract the numerators. Keep the denominator. PRACTICE Subtract. Write your answer in simplest form. 9 5 8 5 11 2 1. 7 7 2. 9 9 3. 1 3 2 SB10 Skills Bank 8 4 4. 9 5 7 3 5. 1 8 0 Multiplying Fractions 5NS2.4, 5NS2.5 To multiply fractions, you do not need a common denominator. Multiply the numerators, and then multiply the denominators. EXAMPLE 5 2 Multiply 7 5. Write your answer in simplest form. 5 2 52 7 5 75 10 3 5 2 7 Multiply numerators and denominators. Write in simplest form. PRACTICE Multiply. Write your answer in simplest form. 1 4 1. 2 7 3 4 2. 8 5 1 3 3. 5 5 6 7 4. 1 4 12 1 3 5. 3 8 2 14 6. 7 4 3 5 7. 5 7 6 1 8. 9 2 5 7 9. 8 1 0 10 1 10. 1 35 Dividing Fractions 5NS2.4, 5NS2.5 Two numbers are reciprocals if their product is 1. To find the reciprocal of a fraction, switch the numerator and denominator. Dividing by a fraction is the same as multiplying by its reciprocal. So, to divide fractions, multiply the first fraction by the reciprocal of the second fraction. EXAMPLE 2 3 Divide 5 10. Write your answer in simplest form. 3 2 2 10 10 5 5 3 2 10 53 20 1 5 4 3 Change the division to multiplication by the reciprocal. Multiply numerators and denominators. Write in simplest form. PRACTICE Divide. Write your answer in simplest form. 3 5 1. 8 6 5 3 2. 8 4 7 3 3. 1 4 2 2 6 4. 9 7 3 4 5. 5 5 3 3 6. 1 4 2 1 2 7. 6 3 9 3 8. 1 2 1 2 5 5 9. 6 9 7 3 10. 8 4 Skills Bank SB11 Adding Decimals 5NS2.1 When adding decimals, first align the numbers at their decimal points. You may need to add zeros to one or more of the numbers so that they all have the same number of decimal places. Then add the same way you would with whole numbers. EXAMPLE Add 23.7 1.426. 23.700 1.426 25.126 Align the numbers at their decimal points. Place two zeros after 23.7. Add and bring the decimal point straight down. You can estimate the sum to check that your answer is reasonable: 24 1 25 ✔ PRACTICE Add. 1. 4.761 3.41 2. 14.2 16.9 3. 5.785 0.215 4. 32 1.75 Subtracting Decimals 5NS2.1 When subtracting decimals, first align the numbers at their decimal points. You may need to add zeros to one or more of the numbers so that they all have the same number of decimal places. Then subtract the same way you would with whole numbers. EXAMPLE Subtract. A 42.543 6.7 42.543 6.700 35.843 Align the numbers at their decimal points. Place two zeros after 6.7. Subtract and bring the decimal point straight down. Estimate to check that your answer is reasonable: 43 7 36 ✔ B 5.0 0.003 5.000 0.003 4.997 Place two zeros after 5.0. Align the numbers at their decimal points. Subtract and bring the decimal point straight down. PRACTICE Subtract. 1. 4.761 3.41 SB12 Skills Bank 2. 30.79 28.18 3. 24.912 22.98 4. 20.11 8.284 Multiplying Decimals 5NS2.1 When multiplying decimals, multiply as you would with whole numbers. The sum of the number of decimal places in the factors equals the number of decimal places in the product. EXAMPLE Find each product. A 81.2 6.547 6.547 81.2 1.3094 6.5470 523.7600 531.6164 B 0.376 0.12 3 decimal places 1 decimal place 0.376 0.12 752 3760 0.04512 3 decimal places 2 decimal places 5 decimal places 4 decimal places PRACTICE Find each product. 1. 0.97 0.76 2. 0.5 3.761 3. 42 17.654 4. 7.005 32.1 5. 9.76 16.254 6. 296.5 2.4 7. 7.7 6.5 8. 8.92 2.8 9. 3.65 4.2 10. 0.002 8.1 11. 0.03 0.204 12. 98.6 4.9 Dividing Decimals 5NS2.1 When dividing with decimals, set up the division as you would with whole numbers. Pay attention to the decimal places, as shown below. EXAMPLE Find each quotient. A 89.6 16 B 3.4 4 0.85 43 .4 0 32 20 20 0 5.6 168 9 .6 80 96 96 0 Place decimal point. Insert zeros if necessary. PRACTICE Find each quotient. 1. 242.76 68 2. 40.5 18 3. 121.03 98 4. 3.6 4 5. 1.58 5 6. 0.2835 2.7 7. 8.1 0.09 8. 0.42 0.28 10. 36.9 0.003 11. 0.784 0.04 9. 480.48 7.7 12. 15.12 0.063 Skills Bank SB13 Order of Operations 6AF1.3, 6AF1.4 When simplifying expressions, follow the order of operations. 1. Simplify within parentheses. 2. Simplify exponents. 3. Multiply and divide from left to right. 4. Add and subtract from left to right. EXAMPLE Simplify each expression. A 32 (11 4) 32 (11 4) 32 7 Simplify within parentheses. 97 Simplify the exponent. 63 Multiply. 2 22 2 B 53 22 22 53 (22 22) (5 3) The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing. 22 4 53 Simplify the power in the numerator. 26 53 Subtract to simplify the numerator. 26 2 Subtract to simplify the denominator. 13 Divide. PRACTICE Simplify each expression. 2 1. 45 15 3 4 2. 3. 35 (15 8) 0 24 4. 62 5. 24 3 6 12 6. 6 8 22 8. 32 10 2 4 2 9. 27 (3 6) 62 49 4 7. 20 3 4 10. 4 2 8 23 4 2 7 13. 42 53 2 3(6) 2 11. 2 12. 82 4 12 13 5 14. 37 21 7 15. 92 32 8 3 16. 10 2 8 2 2 8 17. 18. 4 12 4 8 2 19. 28 32 27 3 20. 9 (50 16) 2 18 6 21. SB14 Skills Bank 21 2 9 Measurement 6AF2.1 1 min 60 s 1 h 60 min 1 day 24 h The measurements for time are the same worldwide. The customary system of measurement is used in the United States. 1 wk 7 days 1 yr 12 mo 1 yr 365 days Length 12 in. 1 ft 3 ft 1 yd 5280 ft 1 mi The metric system is used elsewhere and in science worldwide. Capacity 8 oz 1 c 2 c 1 pt 2 pt 1 qt Length 1 mm 0.001 m 1 cm 0.01 m 1 km 1000 m 1 leap yr 366 days Weight 16 oz 1 lb 2000 lb 1 ton Capacity 1 mL 0.001 L 1 kL 1000 L Mass 1 mg 0.001 g 1 kg 1000 g Use the table below to convert from metric to customary measurements. Length Capacity Mass/Weight Temperature 1 cm 0.394 in. 1 L 1.057 qt 1 g 0.0353 oz 1 m 3.281 ft 1 L 0.264 gal 1 kg 2.205 lb 1 m 1.094 yd 1 L 4.227 c 1 kg 0.001 ton 1 km 0.621 mi 1 mL 0.338 fl oz 1 metric T 1.102 ton C 32 F 9 5 Use the table below to convert from customary to metric measurements. Length Capacity Weight/Mass Temperature 1 in. 2.540 cm. 1 qt 0.946 L 1 oz 28.350 g 1 ft 0.305 m 1 gal 3.785 L 1 lb 0.454 kg 1 yd 0.914 m 1 c 0.237 L 1 ton 907.185 kg 1 mi 1.609 km 1 fl oz 29.574 mL 1 ton 0.907 metric ton (F 32) C 5 9 EXAMPLE A Write , , or . 35 in. 1 yd 35 in. 3 ft 35 in. 36 in. 35 in. 1 yd B Convert 32 km to mi. C Convert 25°C to °F. 9 F 5 25 32 1 km 0.621 mi 1 yd 3 ft 3 ft 36 in. 32 km 32 0.621 mi F 45 32 F 77°F 32 km 19.872 mi PRACTICE Write , , or . 1. 3 lb 40 oz Convert. 4. 15 mi to km 2. 200 cm 5. 2 weeks to hours 2m 6. 32 fl oz to mL 3. 6 c 2 qt 7. 95°F to °C 8. 14 tons to kg Skills Bank SB15 Polygons 3MG2.1 A polygon is a closed figure with three or more sides. The name of a polygon is determined by its number of sides. If all the sides are the same length, and all the angles have the same measure, the polygon is a regular polygon . Sides and angles with the same measures are marked with the same symbol. Number of Sides Name Number of Sides Name 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n n-gon EXAMPLE Identify each polygon. The mark on each side indicates that the sides are all the same length. The arch inside each angle indicates that the angles all have the same measure. A regular hexagon B pentagon PRACTICE Identify each polygon. 1. 2. 3. Geometric Patterns 3MG2.0 Patterns involving polygons may deal with size, color, position, or shape. EXAMPLE Predict the next term: , , , ... Each term has one more side than the previous term. The next term will have six sides. hexagon PRACTICE 1. Predict the next term. , SB16 , ,... Skills Bank 2. Describe the missing term. , , , ,... Classify Triangles and Quadrilaterals A triangle can be classified according to its angle measurements or according to the number of congruent sides it has. Classifying by Angles 4MG3.7, 4MG3.8 Classifying by Sides Acute Three acute angles Scalene No sides congruent Right One right angle Isosceles At least 2 sides congruent Obtuse One obtuse angle Equilateral All sides congruent EXAMPLE 1 Classify each triangle according to its angles and sides. A B C 12 cm 4 cm 9 cm acute isosceles obtuse scalene acute equilateral Quadrilaterals can also be classified according to their sides and angles. Parallelograms Parallelogram 2 pairs of parallel, congruent sides Rectangle 4 right angles Rhombus 4 congruent sides Other Quadrilaterals Trapezoid exactly 1 pair of parallel sides Isosceles Trapezoid congruent, nonparallel legs Kite 2 pairs of adjacent, congruent sides Square 4 right angles and 4 congruent sides EXAMPLE 2 Tell whether the following statement is always, sometimes, or never true: A square is a rectangle. always A rectangle must have four right angles, and a square always has four right angles. PRACTICE Classify each triangle according to its angles and sides. 1. 2. 35º 35º 110º 3. 4 in. 2 in. 4 in. Tell whether each statement is always, sometimes, or never true. 4. A rectangle is a square. 5. A trapezoid is a parallelogram. Name the quadrilaterals that always meet the given conditions. 6. All sides are congruent. 7. Two pairs of sides are congruent. Skills Bank SB17 Bar Graphs 5SDAP1.2 A bar graph displays data using vertical or horizontal bars that do not touch. Bar graphs are often a good way to display and compare data that can be organized into categories. EXAMPLE 1 Use the bar graph to answer each question. Most Widely Spoken Languages A Which language has the most native speakers? The bar for Mandarin is the longest, so Mandarin has the most native speakers. B About how many more people speak Mandarin than speak Hindi? About 500 million more people speak Mandarin than speak Hindi. English Hindi Mandarin Spanish 0 200 400 600 800 1,000 Number of speakers (millions) You can use a double-bar graph to compare two related sets of data. EXAMPLE 2 The table shows the life expectancies of people in three Central American countries. Make a double-bar graph of the data. Country Male Female El Salvador 67 74 Honduras 63 66 Nicaragua 65 70 Step 1 Choose a scale and interval for the vertical axis. Life Expectancies in Central America Step 2 Draw a pair of bars for each country’s data. Use different colors to show males and females. 80 Step 3 Label the axes and give the graph a title. Age 60 Step 4 Make a key to show what each bar represents. 40 20 0 El Salvador Male SB18 Skills Bank 15 10 5 es ge s an Or ap na s s 0 Gr 3. About how many more pounds of bananas than pounds of oranges were eaten per person? 20 na 2. About how many pounds of apples were eaten per person? 25 Ba 1. Which fruit was eaten the least? 30 Ap ple The bar graph shows the average amount of fresh fruit consumed per person in the United States in 1997. Use the graph for Exercises 1–3. Nicaragua Fresh Fruit Consumption Average per person (lb) PRACTICE Honduras Female Line Graphs 5SDAP1.2 You can use a line graph to show how data changes over a period of time. In a line graph , line segments are used to connect data points on a coordinate grid. The result is a visual record of change. EXAMPLE Make a line graph of the data in the table. Use the graph to determine during which 2-month period the kitten’s weight increased the most. Age (mo) Weight (lb) 0 0.2 2 1.7 4 3.8 6 5.1 8 6.0 10 6.7 12 7.2 Step 1 Determine the scale and interval for each axis. Place units of time on the horizontal axis. Step 2 Plot a point for each pair of values. Connect the points using line segments. Step 3 Label the axes and give the graph a title. Growth Rate of a Kitten Weight (lb) 8 6 4 2 0 0 2 4 6 8 10 12 Age (mo) The graph shows the steepest line segment between 2 and 4 months. This means the kitten’s weight increased most between 2 and 4 months. PRACTICE 1. The table shows average movie theater ticket prices in the United States. Make a line graph of the data. Use the graph to determine during which 5-year period the average ticket price increased the least. Year 1965 1970 1975 1980 1985 1990 1995 2000 2005 Price ($) 1.01 1.55 2.05 2.69 3.55 4.23 4.35 5.39 6.41 2. The table shows the number of teams in the National Basketball Association (NBA). Make a line graph of the data. Use the graph to determine during which 5-year period the number of NBA teams increased the most. Year Teams 1965 1970 1975 1980 1985 1990 1995 2000 2005 9 14 18 22 23 27 27 29 30 Skills Bank SB19 Histograms 5SDAP1.2 A histogram is a bar graph that shows the frequency of data within equal intervals. The bars must be of equal width and should touch, but not overlap. EXAMPLE The table shows survey results about the number of CDs students own. Make a histogram of the data. Number of CDs 1 2 3 4 lll ll llll llll l 5 6 7 8 llll l lll llll lll llll ll 9 10 11 12 llll llll llll llll l llll llll l llll 13 14 15 16 llll llll llll llll llll llll l llll l llll l 17 18 19 20 llll llll llll ll ll llll l Step 1 Make a frequency table of the data. Be sure to use a scale that includes all of the data values and separate the scale into equal intervals. Use these intervals on the horizontal axis of your histogram. Frequency 1–5 22 6–10 34 11–15 52 16–20 35 Step 2 Choose an appropriate scale and interval for the vertical axis. The greatest value on the scale should be at least as great as the greatest frequency. Step 3 Draw a bar for each interval. The height of the bar is the frequency for that interval. Bars must touch but not overlap. Step 4 Label the axes and give the graph a title. CD Survey Results 60 50 Frequency Number of CDs 40 30 20 10 0 PRACTICE 1. The list below shows the ages of musicians in a local orchestra. Make a histogram of the data. 14, 35, 22, 18, 49, 38, 30, 27, 45, 19, 35, 46, 27, 21, 32, 30 2. The list below shows the results of a typing test in words per minute. Make a histogram of the data. 62, 55, 68, 47, 50, 41, 62, 39, 54, 70, 56, 47, 71, 55, 60, 42 SB20 Skills Bank 1– 5 6– 10 0 5 –1 6 – 2 1 11 Number of CDs Circle Graphs 5SDAP1.2 A circle graph shows parts of a whole. The entire circle represents 100% of the data and each sector represents a percent of the total. EXAMPLE Type of Program Number of Students At Mazel Middle School, students were surveyed about their favorite types of TV programs. Use the given data to make a circle graph. Science 25 Cooking 15 Sports 50 Step 1 Find the total number of students surveyed. Sitcoms 150 Movies 60 Cartoons 200 25 15 50 150 60 200 500 Step 2 Find the percent of the total students who like each type of program. Percent 25 500 15 500 50 500 150 500 60 500 200 500 Step 3 Find the angle measure of each sector of the graph. There are 360° in a circle, so multiply each percent by 360°. 5% 0.05 360° 18° 3% 0.03 360° 10.8° 10% 0.1 360° 36° 30% 0.3 360° 108° 12% 0.12 360° 43.2° 40% 0.4 360° 144° Step 4 Use a compass to draw a large circle. Use a straightedge to draw a radius. Step 5 Use a protractor to measure the angle of the first sector. Draw the angle. Step 6 Use the protractor to measure and draw each of the other angles. Step 7 Give the graph a title, and label each sector with its name and percent. Color the sectors. Angle of Sector Favorite TV Programs Cooking 3% Science 5% Sports 10% Cartoons 40% Sitcoms 30% Movies 12% PRACTICE 1. Use the given data to make a circle graph. Favorite Pets Type of Pet Dog Fish Bird Cat Other Number of People 225 150 112 198 65 Skills Bank SB21 Sampling 6SDAP2.2 A population is a group that someone is gathering information about. A sample is part of a population. For example, if 5 students are chosen to represent a class of 20 students, the 5 chosen students are a sample of the population of 20 students. The sample is a random sample if every member of the population had an equal chance of being chosen. EXAMPLE Jamal telephoned people on a list of 100 names in the order in which they appeared. He surveyed the first 20 people who answered their phone. Explain whether the sample is random. Names at the beginning of the list have a greater chance of being selected than those at the end of the list, so the sample is not random. PRACTICE Explain whether each sample is random. 1. Rebecca surveyed every person in a theater who was sitting in a seat along the aisle. 2. Inez assigned 50 people a number from 1 to 50. Then she used a calculator to generate 10 random numbers from 1 to 50 and surveyed those with matching numbers. Bias 6SDAP2.4 Bias is error that favors part of a population and/or does not accurately represent the population. Bias can occur from using sampling methods that are not random or from asking confusing or leading questions. EXAMPLE Jenn went to a movie theater and asked people who exited if they agree that the theater should be torn down to build office space. Explain why the survey is biased. People usually only go to movies if they enjoy them, so those exiting a movie theater would not want it torn down. People who do not use the theater did not have a chance to answer. PRACTICE Explain why each survey is biased. 1. A surveyor asked, “Is it not true that you do not oppose the candidate’s views?” 2. Brendan asked everyone on his track team how they thought the money from the athletic department fund-raiser should be spent. SB22 Skills Bank Compound Events 6SDAP2.2 A compound event consists of two or more single events. EXAMPLE Marilyn randomly draws 1 of 6 cards, numbered from 1 to 6, from a box and then randomly selects 1 of 2 marbles, 1 red (R) and 1 blue (B), from a jar. Find the probability that the card will show an even number and that the marble will be red. 1 2 3 4 5 6 R 1, R 2, R 3, R 4, R 5, R 6, R B 1, B 2, B 3, B 4, B 5, B 6, B Use a table to list all possible outcomes. Circle or highlight the outcomes with an even number and “red.” 3 1 3 ways outcome can occur P(even, red) 1 4 2 12 equally likely outcomes In the Example, a table was used to list the possible outcomes. Another way to list outcomes of a compound event is to use a tree diagram . Card Number Marble R B List all possible outcomes for the six cards: 1, 2, 3, 4, 5, 6. 1 R 2 B R 3 B R 4 5 Then, for each outcome of the six cards, list all possible outcomes for the marbles: red and blue for each card. B R B 6 R B PRACTICE 1. If you spin the spinner twice, what is the probability that it will land on blue on the first spin and on green on the second spin? 2. What is the probability that the spinner will land on either red or yellow on the first spin and blue on the second spin? 3. What is the probability that the spinner will land on the same color twice in a row? Skills Bank SB23 Inductive and Deductive Reasoning 6AF2.0, 7MR2.4 Inductive reasoning involves examining a set of data to determine a pattern and then making a conjecture about the data. In deductive reasoning , you reach a conclusion by using logical reasoning based on given statements, properties, or premises that you assume to be true. EXAMPLE A Use inductive reasoning to determine the 30th number of the sequence. 3, 5, 7, 9, 11, . . . Examine the pattern to determine the relationship between each term in the sequence and its value. Term 1st 2nd 3rd 4th 5th Value 3 5 7 9 11 121213 421819 221415 5 2 1 10 1 11 321617 To obtain each value, multiply the term by 2 and add 1. So the 30th term is 30 2 1 60 1 61. B Use deductive reasoning to make a conclusion from the given premises. Premise: Makayla needs at least an 89 on her exam to get a B for the quarter in math class. Premise: Makayla got a B for the quarter in math class. Conclusion: Makayla got at least an 89 on her exam. PRACTICE Use inductive reasoning to determine the 100th number in each pattern. 1 1 1 1. 2, 1, 12, 2, 22, . . . 2. 1, 4, 9, 16, 25, . . . 3. 4, 6, 8, 10, 12, . . . 4. 0, 3, 6, 9, 12, 15, . . . Use deductive reasoning to make a conclusion from the given premises. 5. Premise: If it is raining, then there must be a cloud in the sky. Premise: It is raining. 6. Premise: A quadrilateral with four congruent sides and four right angles is a square. Premise: Quadrilateral ABCD has four right angles. Premise: Quadrilateral ABCD has four congruent sides. 7. Premise: Darnell is 3 years younger than half his father’s age. Premise: Darnell’s father is 40 years old. SB24 Skills Bank Selected Answers 37. 43 39. 8 41. 30 43. 15 Chapter 1 1-1 Exercises 45a. $1,146,137,000,000 m; m $39 57. C 59. 16 b. $1,763,863,000,000 c. about 61. 12 63. x 9 65. x 26 $618,000,000,000 or $618 billion 1 1. 15 3. 3 5. 18 7. 2 c 9. 22 c 11. 37 13. 57 15. 13 pt 17. 10 pt 49. C 51. 4 53. 12 55. 13 1-5 19. 22 21. 10 23. 20 25. 16 q 4 51. 74 mi 53. $195 $156 Exercises 1-9 Exercises 1. 9x 6 12 3. 2 5. 8 7. 5 9. 24 11. 42 13. 88 15. 18 27. 45 29. 138 31. 20.7 33. 36 1. 14 3. 12 5. 17 7. 18 9. 10 DVDs 17. 4z 18 54 19. 1 35. 105 37. 11 39. 32 41. 21 11. 17 13. 17 15. 10 17. 6 23. 5 25. 2 27. 9 29. 4 43. 19 45. $110 47. B 49. yes (4) 2 19. 3 21. 56 31. 40 33. 55 35. 30 37. 5 h 51. A 53. 15, 21, 61, 71 55. 49, 81 23. 26 25. 20 27. 24 39. Nine less than 4 times a 57. 202 59. 400 61. 200.2 63. 40 29. 13 31. 190 ft above the number is 3; 3 41. 21 43. 3 starting point 33. Great Pyramid x to Cleopatra; about 500 years 1. 3p 5 3. 16 7 5. 18 plus 45. 0 47. 9 3 8; x 99 49. 5 3m 4; m 3 51. 16m 35. Cleopatra takes the throne and 42,000; m 2625 miles per year the product of 43 and s 7. 10 plus Napoleon invades Egypt. the quotient of y and 31 9. 5n; 39. 8 41. 3 43. 12 53. 6m 22; m 132 miles 55. A 1-2 Exercises d 5 $75; $125; $175; $225 11. 1 n 13. 78j 45 15. 14 59q 17. 12 1-6 Exercises more than the product of 16 and g 1. 32 3. 84 5. 49 7. 24 19. 51 less than the quotient of w 9. gains $15 11. 21 13. 84 1 and 182 23. 6y 9 25. 3g m 2 27. 13y 6 29. 23 31. 83 x 5 33. 8 times the sum of m and 5 15. 3 17. 140 19. 60 21. 39 15 6 9 23. $100 25. 3, 5; 3, 2; 3 , 3; 15 , 5 27. 12 29. 4 31. 8 3 1 57. 44 59. 30 60. 6 61. 72 62. 3 63. 147 64. 9 Chapter 1 Study Guide: Review 1. equation 2. opposite 3. absolute value 4. 147 5. 152 6. 278 7. 2(k 4) 8. 4t 5 35. 17 times the quotient of 16 33. 36 35. 11 37. –8 39. –6 9. 10 less than the product of 5 45. D 47. j 18 49. y 22 and b 10. 32 plus the product of 43. 9n; $9450 45. 24 47. 7 51. 9 53. 254 23 and s 11. 12 less than the 1 x7 and w 37. 4(x 7) or 4 41. C quotient of 10 and r 1-3 Exercises 1-7 Exercises 12. 16 more than the quotient of y 1. 4 2 3. 17, 8, 6 5. 7, 1. 12 3. 14 5. 23 7. 39 and 8 13. 1 14. 15 15. 34 0, 3 7. 15 9. 5 11. 21 13. 14 9. 15,635 ft 11. 32 13. 44 16. 21 17. 14 18. 1 19. 2 15. 6, 2, 5 17. 30, 27, 25 15. 74 17. 35 19. 5 21. 24 20. 12 21. 3 22. 1 23. 24 19. 9 21. 11 23. 5 25. 0 27. 23. 3 25. 949 27. 110 29. 17 24. 8 25. 8 26. 16 27. 17 29. 31. 33. 35. 16, 31. 138 33. 1784 28. 3 29. 15 30. 22 31. 4 12, 24 37. 45, 25, 35 39. 50 35. t 600 173; 427°C 37. 40 32. 16 33. 5 34. 35 35. 18 41. 1 43. 72 45. 6 47. 9 49. 38 39. 1 41. 59 47. B 49. 14 36. 52 37. 25 38. 120 39. 2 51. 16 53. Antarctica, Asia, North 51. 13 53. 56 55. 2 40. z 23 41. t 8 42. k 15 43. x 15 44. 1300 lb America, Europe, South America, Africa, Australia 57. 7, 6, 5 59. A 61. 9 62. 64 64. 7t 8 1- 4 Exercises 1-8 Exercises 45. 3300 mi2 46. g 8 47. k 9 1. 7 3. 2 5. 60 7. 6 9. 54 48. p 80 49. w 48 people 11. 15 13. 7 15. 2 50. y 40 51. z 192 17. 4 19. 54 21. 96 23. 161 52. 705 mi 53. 24 months 1. 6 3. 2 5. 7 7. 6 9. 11 25. 252 27. 7 29. 2 31. 56 54. x 6 55. y 96 11. 10 13. 2 15. 5 17. 22 33. 72 35. 17 37. 207 39. 48 56. n 49 57. x 4 19. 16 21. 7 23. 32 25. 23 41. 15 43. 8n 32; n 4 58. n 24 59. y 11 27. 9 29. 3 (4) 7 servings 45. 16 h; $160 47. 7 n 60. 3 hours 31. 12 33. 17 35. 22 15; n 8 49. 12 q (8); Selected Answers SA1 Chapter 2 2-4 5 5 1 1. 3 3. 58 5. 42 miles 7. 0.162 2 7 7 1 13. 2 15. 12 9. 0.42 11. 3 6 4 Exercises 2-1 1. 0.625 3. 0.416 5. 0.1 7. 0.375 3 2 1 9. 1.25 11. 4 13. 5 15. 25 21 9 115 73 19. 21. 23. 17. 3 100 11 333 99 58 25. 9 27. 0.375 29. 1.8 9 3 31. 1.6 33. 4.6 35. 1.3 37. 377 18 1 31. Cullinan III 33. z 3 Exercises 5 2 5 43. 1 45. 39. 2 41. 1 1000 33 5 5 4 3 151 51. 53. yes 47. 1 49. 1 11 333 21 17. 4 19. 27 boys 21. 0.235 0 b. 3 3; 2 3; 2 2; 3 5; 2 2 2 2; 5 5; 2 2 2 c. 0.4, repeating; 0.16, repeating; 0.25, terminating; 0.46, repeating; 47. y 54.2 49. m 1.4 53. 60 carats 55. 3v 64; 1 1 minutes 56. p 15 21 2 2-5 2 1. 7 hours 3. y 11 5. x 5 11 2 1. 3 3. 7 5. 18 7. 15 9. 12.4 0 15. 490 11. 15.3 13. 4.23 1 17. 19.4 19. 45 21. 18 23. 6 9 8 3 serving 25. 13 27. 15 29. 2 5 1 39. 13.6 41. 12 43. 11 45. 370 0.375, terminating 65. GCD 2; in. 47. 11 glasses 49. 34 8 1 13 53. about 3 55. B 57. 11 59. 2 0 723 4 63. 61. 1000 5 1. 3. 5. 7. 5 1 in., 7.5 in., 88 in., 8.25 in. 9. 71 6 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 37a. apricot, sulphur, large orange sulphur, white-angled sulphur, great white b. between the apricot sulphur and the large orange sulphur 2 31 2-6 Exercises 2-2 5 41. ⏐0.62⏐, ⏐3⏐, ⏐0.75⏐, ⏐6⏐ Exercises 43. D 45. 10 47. 47 49. 0.75 meters 55. The company did not 51. 2.5 53. 0.95 find a common denominator 1 when adding 2 and 4. 57. 28 Exercises 2-3 3 5 61. 6 59. 11 1 1 1 4 8 1 21. 3 23. 3 25. 7.9375 3 69 6 1 29. in. 31. 27. 4 200 11 4 5 1 33. 29 35. 1 37. 12 39. 0.186 quadrillion Btu 43. D 11 47. b 1 49. a 5 45. 1 5 51. 53. SA2 Selected Answers 2-7 1. y 82.3 3. m 19.2 5 5. s 97.146 7. x 9 7 9. w 1 11. y 8 13. 8 days 5 15. m 7 17. k 3.6 6 19. c 3.12 21. d 2 5 5 23. x 2 25. r 7 4 1 1 5 45. y 4.4 41. 8 43. 1 2 47. m 25.6 Chapter 2 Study Guide: Review 1. rational number 2. terminating decimal 3 1 21 5 7. 1 inverse 4. 5 5. 4 6. 4 0 1 1 212 10. 1.75 11. 0.26 8. 3 9. 999 12. 0.7 13. 14. 2 1 15. 0.9, 3, 0.25, 2 16. 0.11, 0, 9 6 7 5 17. 1 18. 5 19. 9 0.67, 1 0 3 1 1 12 1 20. 6 21. 1 22. 1 23. 15 1 3 3 8 1 5 1 24. 75 25. 1 26. 4 27. 24 5 1 7 28. 34 29. 13 30. 1 31. 6 8 3 2 5 32. 8 33. 9 34. 16 35. 4 36. 2 3 7 39. 111 40. 42 37. 16 38. 1 8 0 0 17 11 43. 21.8 41. 36 42. 9 0 36 5 95 44. 18 45. 8 46. 2 47. 9 9 48. 2 49. 44.6 50. 6 54. c 16 55. r 3 56. t 16 Exercises 1 72 35. 110,000 37. 25 in. 39. B 51. $108 52. m 10 53. y 8 1. 30.01 3. 24.125 5. 0.56 s 7. 15 9. 5 11. 9 13. 33.67 14 15. 12.902 17. 10.816 s 19. 1 7 3 13. r 64 15. 2.02 17. 17.5 1 1 21. 1 23. 10 25. 64 19. 1 5 8 11 n7 27. 1.3 29. 5 31. 5 13; 3. reciprocal or multiplicative 3 1 7 19 1. 2 3. 41 5. 4 7. 81 0 1 0 2 69 1 11 13. 4 yd 15. 1 9. 2 11. 130 72 29 13 19 1 21. 1 23. 17. 1 19. 126 4 21 3 4 39 11 29. 25. 14 27. 5 833 120 1 1 19 31. 18 miles 33. 1 35. 4 2 2 7 5 145 67 in. 41. 43. 37. 3 39. 4 8 168 72 1 5 45. 43 47. 28 cups 3 1 1 49a. 168 m b. 118 mi c. 272 mi 73 19 meter 53. 19 51. 100 100 1 3 2 7. a 22 9. y 5 11. m 121 0 terminating; 0.625, terminating; 81. 126 4 Exercises 2-8 Exercises 2 5 1 61. 24 57. 2(m 19) 59. 101 2 5 0.5625, terminating; 0.48, 75. 20 77. 21 79. 180 1 43. y 4.2 45. c 2 0 7 5 31. 159 27. 0.368 29. 81 1 53 33. 1.82 35. 0.558 37. 25 4 55 3 1 41. 43a. 1 tsp 38. 192 8 4 1 b. 12 tsp c. 2 tsp 47. A 49. C 5 2 51. 53. 55. 6 57. 9 1 31. 22 33. 83 35. 9.7 37. 40.55 8 21 69. C 71. 9.875 73. 28; 48 34 1 39. d 2 41. v 30.25 23. 4.125 25. 0.496 3 55. no 57. yes 59. yes 61. 38 4 1 1 7 9 12 5 3 63a. 9; 6; 4; 1 ; ; ; ; 5 16 25 8 8 35. j 21.6 37. t 7 4 27. d 12 29. 22581 carats 1 5 57. w 64 58. r 42 59. h 50 60. x 52 61. d 33 62. a 67 63. c 90 64. 11 Chapter 3 3-1 13. k 10 15. w 3 17. y 5 29. The solution is x 4. 33. B 19. h 6 21. m 2 23. x 12 37. 1 39. t 2 8 35. 2 5 8 25. n 2 27. b 13 29. x 17 Exercises 31. y 7 33. $11.80 per hour 1 19 3-7 Exercises 1. Comm. Prop. of Add. 3. 39 35. 31 and 32 37. 212°F 41. C 1. r 18 3. 120 j, or j 120 5. 700 7. 130 9. 168 11. 54 43. n 3 45. x 121 5. 40 a, or a 40 13. 396 15. Comm. Prop. of Mult. 47. 6t 3k 15 7. r 63 9. 104 sandwiches 17. 109 19. 1300 21. 24 23. 133 25. 132 27. 21 29. Distrib. Prop. 3-4 11. 75 x, or x 75 13. 77 p, Exercises or p 77 15. h 12 31. Assoc. Prop. of Add. 1. x 1 3. x 2 5. x 20 17. q 4 19. 6 r, or r 6 33. Comm. Prop. of Add. 35. 40 7. x 1 9. d 5 21. w 3 23. t 95 37. 15 39. $53 43. 8; Distrib. 11. 250 min; $8.75 13. x 1 25. a 120 33. A 37. 27 lb, Prop. 45. y; Assoc. Prop. of Add. 15. all real numbers 17. y 6 135 lb 39. at least $13.34 per 19. x 16.2 21. a 13 23. y 2 43. 83 week 41. 3 3 47. 24 19 26 21 24 26 19 21 (Comm. Prop. of Add.) (24 26) (19 21) (Assoc. Prop. of Add.) 50 40 90 in. 25. n 5 27. x 5 29. 22, 23 5 3-8 31. x 25 10 4x; x 7 1 Exercises 33. 350 units 35a. 17 protons 1. k 2 3. y 8 5. y 7 49. The sentence should read, b. 11 39. C 41. x 3 43. g 12 7. x 3 9. h 1 11. d 1 “You can use the Commutative 45. 47. 13. at least 21 caps 15. x 4 Property of Addition...” 1 1 1 1 51. 12 3 6 4 12 3 12 1 1 12 (Distrib. Prop.) 4 2 6 4 3 (Mult.) (4 2) 3 (Assoc. Prop. of Add.) 6 3 (Add) 9 2 1 1 59. 21 53. C 55. 15 57. 2 5 0 61. 28 3-2 Exercises 3-5 7. 5x 8y 9. 9x y 11. 7g 5h 12 13. r 12 15. 2t 56 17. 10y 17 19. 13y 21. 6a 15 23. 5x 3 25. 7z b 5 22 23. k 3 25. r 3 27. p 3 1 1. p 60 3. m 7 15 5. 7. 29. w 1 31. a 2 33. q 6 35. b 2.7 37. f 2.7 39. 7 5 4 3 2 1 0 1 2 41. at least 31 beads 2 1 0 1 2 3 4 5 43a. $158 b. 17 mo 47. B 9. s 5 or s ≥ 3 11. s 10 49. Comm. Prop. of Mult. y 13. x 11 35 15. 7 10 51. Comm. Prop. of Add. 17. 1. 5x 3. 12f 8 5. 6p 9 17. q 2 19. x 7 21. a 3 Exercises 19. 21. 23. 2 1 0 1 2 3 4 5 2 1 0 1 2 3 4 5 2 1 0 1 2 3 4 5 5 4 3 2 1 0 1 2 53. 7r 60 Chapter 3 Study Guide: Review 1. inequality 2. Comm. Prop. 3. terms 4. Comm. Prop. of Add. 5. Assoc. Prop. of Add. 6. Dist. 35. 3x 48 37. 2(5x x); 12x 1 25. c 2 or c 7 27. 5w 60 2 1 29. m 25 35 31. x 9 33. x 4 35. d 89,000 8. 19m 10 9. 14w 6 39. no; 6r 12m 5m 15 5r 39. B 41. 25, 19, 12 10. 2x 3y 11. 2t2 4t 3t3 7, Distrib. Prop.; 6r 5r 12m 43. 11, 9, 7, 5 45. 1 1 1 47. 13 27. 11x 4y 8 29. 6d 3e 12 31. 4y 10 33. 11x 18 5m 15 7, Comm. Prop.; 6r 5r 12m 5m 15 7; 11r 2 17m 8 41. 7d 1 43. 6r 11r 1 3-6 Exercises Prop. 7. Assoc. Prop. of Add. 12. y 1 13. h 2 14. t 1 15. r 3 16. z 2 17. a 12 1 18. s 7 19. c 24 20. x 6 2 21. y 3 22. no solution 45. k 13 47. 49g 53s 44b 1. x 7 3. f 24 5. k 9.3 23. z 5 24. Let d distance; 49. y2 2(x y2); y2 2x d 1 mi 25. Let c cost; 59. 2; Distrib. Prop. 61. 5; Assoc. 1 1 7. 62 x 16; x 92 9. x 56 1 11. x 15 13. c 33 15. 21.75 x 50; x 28.25 students; s 45 Prop. of Mult. 17. z 0 19. y 1.8 27. 5 1 53. 6x 112 55. 16 57. a 80 3-3 Exercises 1. d 3 3. e 6 5. h 7 7. x 1 9. p 1 11. 6 hours 21. k 1 23. {x : x 20} c 1500 26. Let s number of 3 2 1 0 1 2 3 25. {b : b 3.5} 27a. 98,200 s 201,522; s 103,322 Selected Answers SA3 28. 6 4 2 29. 1 1 –2 –4 0 0 2 4 6 1 4 1 2 3 4 1 1 t 33. 4|y7| 35. a2|b3| 37. 13|p15|q12 43. 712 45. cannot combine 39. 4 41. 2 43. 8 47. y0 1 49. 262, or 676 1 30. r 4 31. n 2 32. x 0.2 33. y 5 34. n 17.75 20; n 2.25 35. m 18 36. n 3 37. t 16 38. p 3 39. b 27 40. a 8 41. z 1 42. h 6 43. a 24 44. x 6 1 45. k 3 46. y 8 Chapter 4 4-1 35. 8 37. 44 39. a7 41. x10 51. 122; 121 53. 4 55. 0 100 57a. 10 200 b. 10 61. B 63. 25 1 1 69. 9 71. 1 65. 22.8 67. 11 1 4-4 Exercises 1. 56y7 3. 6a5b5 5. 4xy 7. 3n3 3 9. 2ab3 11. 6c5 13. 125a3 x 8 x4; x10 |x5|; |x3|; 7 x12 x6 49. C 51. 2 0 9 53. 75 55. 1.97 109 0 57. 3.14 1010 59. 6 103 4-7 Exercises 1. 6 and 7 3. 12 and 13 5. 15 and 16 7. 6.48 9. 12.49 11. 16.58 13. 5.8 15. 13.8 6 1 x 2 |x|; x4 x 2; x6 45. 19. 22z12 21. 2x10 23. 6a2b5 3 1. 121 3. 22b 3 5. 64 7. 8 9. 4096 11. 5 13. 710 15. 56 1 15. 216x6y15 17. 32m15n10 25. 5x2 27. 2q3 29. 2xy5 31. 256y8 33. 256b8 35. 32a15b50 Exercises 9 17. 7 and 8 19. 24 and 25 21. 20 and 21 23. 4.36 25. 11.09 37. 144m4 39. 5a3b5 41. 9 r 43. 3 45. 16m4n5 47. 6x3y2 27. 17.03 29. 9.6 31. 12.2 33. B 49a. The degree of a monomial is , 53, 7.15, 4, 32 43. 25 35. E 37. F 39. 89.6 in. 41. 40 ft 2 17. 32d 2 19. (4)2c 3 21. 256 the sum of the exponents of the 45. 151 km 1 1 25. 27. 173 29. 1 23. 32,768 36 variables in the monomial. b. 8 47. 31. 36 33. 1 35. 325 53. D 55. 20y 18 57. 21 21k 59. g11 61. 71 or 7 37. 1 39. 426 41. 218 262,144 bacteria 43. (3d)2 45. (7x)4 47. 1728 cm3 51. B 53. 625 55. 83 57. 55 4-5 Exercises 1. 15,000 3. 208,000 5. 5.7 105 7. 6.98 106 9. 4170 59. 0.14 61. 0.375 11. 62,000,000 13. no 15. 80,000 4-2 17. 400,000 19. 5,500,000,000 Exercises 1 1 1. 0.01 3. 0.000001 5. 6 7. 2 4 7 1 1 11. 13. 0.1 9. 13 125 8 1 15. 0.00000001 17. 4 1 3 , or 0.0001 21. 1 19. 10,000 4 4 1 3 4 27. 29. 23. 118 25. 1 32 9 4 3 31. 28 33. 13.02 35. 4 1 37. 114 11 11 11 11 1 14,641 1 1 39. 63 666 216 1 meter 45. 38,340 lb 41. 250,000 1 49. C 51. 8x 21 47. 480 53. 7x 24y 55. 81 57. 125 59. 32 4-3 21. 700,300 23. 6.5 106 25. 5.87 106 27. 0.00067 29. 524,000,000 31. 140,000 33. 78 35. 0.000000053 37. 559,000 39. 7,113,000 41. 0.00029 43a. 1.5 104 g b. 1.5 1026 g 45. 367,000 47. 4 49. 340 51. 540,000,000 53. 9.8 108 feet per second 55. 8.58 103 57. 5.9 106 59. 7.6 103 61. 4.2 103 63. 6 1010 65. 7 106 67. 5.85 103, 1.5 102, 2.3 102, 1.2 106, 5.5 106 71. C 73. number of students 35 Exercises 1. 515 3. m4 5. 62 7. 120 1 1 4 1 2 15. r 17. 3 19. y3 21. 54 1 23. t13r 25. 3 27. 60 1 29. 50 1 4 1 31. 4 33. (1)9 or 1 3 9. 320 11. 46 or 6 13. 1017 () SA4 29 Selected Answers 75. 72 77. t3 4-6 Exercises 1. 2 3. 11 5. 16 ft 7. |y3| 9. 7a4 11. 13 13. 19 15. |s3| 17. 6x4 19. 6 21. 15 23. 21 25. 24 29. 104 ft 31. 8x8 49a. about 610 mi/h b. about 7.8 h 51. B 53. 21 55. 18 57. 8 59. 36 4-8 Exercises 1. irrational, real 3. rational, real 5. rational 7. irrational 9. rational 11. rational 17. rational, real 19. integer, rational, real 21. rational 23. irrational 25. irrational 27. not real 31. whole, integer, rational, real 33. irrational, real 35. rational, real 37. rational, real 39. rational, real 41. integer, rational, real 3 43. 0 is undefined so it is not a 0 real number. 3 is 0 so it is a rational number. 55. irrational 57. rational 59. irrational 63. C 65. C 67. Dist. Prop. 69. 32,768 71. 625 4-9 Exercises 1. 20 m 3. 24 cm 5. yes 7. yes 9. 15 ft 11. about 21.2 mi 13. no 15. 5 17. 13 19. 9 21. Yes. 62 82 102 23. 22.6 ft 25a. 110 m b. 155.6 m 27. 208.5 m 31. C 1 33. x 3 35. x 4 notation 3. Pythagorean 2 4 4. real numbers 5. (3) 6. k 9. 625 10. 32 11. 1 12. 256 13. 3 14. 64 1 1 1 27. yes 29. no 31. yes 33. no 5-2 1 16. 17. 15. 125 64 11 1 1 1 19. 1 20. 21. 18. 10,000 36 8 1 22. 1 23. 2 24. 0 25. 47 26. 96 1. 3 mi 3. approximately 40 per serving 7. 38 oz box 35. 4 , or 1 36. y 37. 4 2 cups per batch 13. approximately $4 per lb 15. 16 oz package minute 23. approximately 41. 7x3z11 42. a7b2 43. 20r6s6 $4.50/lb; 3 lb 27. approximately 50. 216x12y3 51. 10,000m8n20 9 52. 8 10 7 53. 7.3 10 54. 9.6 106 55. 5.64 1010 2 $110 per day 31. width: 20 in.; height: 15 in. 33. The bunch of 4 has the lower unit price. 59. 0.000091 60. 3.1 105, 1.7 104, 2.3 105, 4.9 104 61. 4 and 4 62. 30 and 30 63. 26 and 26 64. 7m2 65. 2|a3| 1 1. yes 3. no 5. no 7. 53 in. 9. no 11. no 13. yes 15. 2.25 m 81 27 0.5 1 66. x6 67. 9 and 10 68. 3 and 4 19. 3 , 21. 6, 1 17. 4, 1 2 2 9 13 23. $144 25. 64 minutes 69. 11 and 12 70. 6 and 7 27. 12 computers 71. 16 and 17 72. 13 and 14 29. 14 molecules 31a. about 3:2 73. rational 74. irrational b. about 68 mm Hg 75. rational 76. irrational 33. 154 and 56 35. 16 37. 33 77. rational 78. not a real 39. 6-ounce can number 79. Possible answer: 3.5 80. 10 81. 10 82. no 83. yes 84. no 5-4 Exercises Chapter 5 2 1 5-7 Exercises 1 13. 630 ft 7. 7.5 ft 9. 1 11. 2 0 15. 6.25 ft 17. 18 in. 19. 945 in2; 6.6 ft2 23. D 25. 5 and 6 27. 7 and 8 29. 6 and 7 31. $0.17 per apple Chapter 5 Study Guide: Review rate 3. similar; scale factor 4. 4 27 3 2 5. 4 6. 4 7. 1 8. yes 9. no 0 13. unit prices are the same 14. 8-pack 15. x 15 16. h 6 1 17. w 21 18. y 293 19. 72 min 20. 90,000 m/h 1 21. 4500 ft/min 22. 5833 m/min 23. 0.8 mi/min ( 48 mi/h) 24. 12.5 in. 25. 3.125 in. 26. 18 ft 27. 6.2 ft 28. 43.75 miles 29. 3 in. 30. 46 mi 31. 57.5 mi 32. 153 mi Chapter 6 1. 0.694 km/s 3. 7.5 mi/h 5. 0.075 page/min 7. 1.8 km/h 5-1 1 10. yes 11. no 12. 75 disks Exercises 8 24 1 19. 6 21. 4 23. 12 9 5 37. t 8.4 39. no 41. no 5-3 9. 85 ft 11. 65 ft 15. C 17. 5 1. ratio; proportion 2. rate; unit 1 35. w 13 56. 1620 57. 0.00162 58. 910,000 1. 128 yd 3. 5.6 ft 5. 4 ft 7. 90 ft 12 4 apples per pound 25. $3.75/lb; 7 47. 6m9n3 48. 16t6 49. 49p10q12 Exercises 1. 300 ft 3. 14 in. 5. 25 mi 17. 14 points per game 39. (10)0 1 40. 5m10n10 44. 9p 45. 6t2 46. 2x3y2 23. 70 25. 3328 quarts 1 9. 3.52 g/cm3 11. approximately 21. approximately 50 beats per 1 38. 10 x Exercises 1. triangle A and triangle B 5-6 5. approximately 500 Calories 31. 83 32. 92 33. m5 34. 37 12 5-5 15. 18 in. 17. 15 ft 21. 31.5 cm2 Exercises 19. 16 beats per measure 9 41. $249 per monitor 9. similar 11. yes 13. x 6 ft 27. p4 28. 153 29. 64 30. x10 0 39. $0.175 per oz 3. 16.5 cm 5. 2.98 gal 7. similar 49. 1.1 106 students per bus theorem; legs; hypotenuse 8. 6x 1 45. 0.642 47. 1.2 107 1. irrational number 2. scientific 7. (9) 3 39 Chapter 4 Study Guide: Review 2 3 35. 1 39. C 41. yes 43. 5.44 8 37. 5 and 6 39. 7 and 8 1 2 17. 5 19. 5 5 21. 3 4 23. no 4 12 2 3 2 4 12 9 ; 1; 8 6 25. 4 6; 5 1 0 3 9. 480 cereal boxes 11. 6 fish 13. 5.8 mi 15. 0.88 g 17. 28.75 tons 19. 10.3 m/s Exercises 10 15 4 2 1. 5 3. 7 5. 1 6. 1 1 1 1 7 7 25 12 8. 4 4 11. yes 13. 1 15. 2 1 21. 2.85 gal 23. C 27. B 29. 32 31. 22 33. 1012 35. 27 37. m13 6-1 Exercises 1 1. 4 3. 87.5% 5. 7. 1 3 1 9. 0.3, 333%, 36%, 8 11. 333% 39 15. 125% 17. 19. 13. 100 2 21. 0.04, 5, 42%, 70% 23. 40%, 30%, 20%, 10% 25. 40%, 30%, 25%, 5% 33. B 35. 37. 39. z 5 Selected Answers SA5 Exercises 6-2 13. $81,200 15. Deborah should 1. 50 3. 30 5. 13 7. 16 11. 100 choose the salary option that pays 13. 6 15. 32 17. 9 21. B 23. B $2100 plus 4% of sales. 25. C 27. 150 29. 40 31. 800 17a. $64,208 b. $12,717 33. 30 35. 100 37. 300 cars 39. 475,000 41. 12 hours 43a. no b. yes c. 1 per mi2; 1000 per mi 2 47. B 49. B 51. 9 53. 64 55. 125 57. 8 23 59. 5 61. 0.5 63. 0.525 0 Exercises 6-3 y2 21. x2 23. 2 25. 50% decrease x 6-7 31. Lena: $11.87, Ana: $12.36, Joseph: $12.50, George: $12.71 35. B 37. 1000 g/1 kg 39. 1 mi/5280 ft 41. 16 oz/1 lb 43. 100 45. 40 6-4 Exercises 1. 60 3. 166.7 5. 2.4 oz 7. 135 9. 1333.3 11. 400 cards 13a. 250 b. 125 c. 62.5 15a. 30 b. 20 c. 15 17. 657,000 19. 6.7% 21. 98,000 23. C 25. 5 and 6 27. 7 and 8 29. 11 and 12 31. 10 and 11 33. 14 and 15 35. 0.625 37. 0.71 39. 0.75 41. 1.23 43. 3.5 6-5 Exercises 1. 48% increase 3. 100% increase 5. $9773.60 7. 22% increase 9. 8.6% 11. 33% decrease 5. about $2971.89 7. 17 years 9. 5.5% 11. $94.50, $409.50 29. C 31. 25% 33. $7.49; $31.76 35. 50% 37. 311.75 6-6 Exercises 1. $574 3. 11.6% 5. $603.50 7. 3.1% 9. $15.23 11. $38.07 SA6 Selected Answers 2 4 4 21. (4, 4) 23. (5, 4) 25. (5, 6) 27. 13. $446.25, $4696.25 y 6 (3, 6) 4 25. B 27. 1 gal/4 qt 29. 95 (4, 3) 2 (8, 1) money in the savings account? x 6 4 2 O 2 2 Chapter 6 Study Guide: Review 1. percent 2. percent of change 7 1 5. 43.75% 6. 18 7. 112.5% 8. 1 0 9. 0.7 10. 30 11. 62 12. 3.3 31. III 33. (12, 7) 35. a. (68°, 39. B 41. about $22 per hour 43. about 16 students per teacher 1 1 45. 0.14, 7, 15%, 6 13. 18 14. $7.50 15. $16.00 16. 33% 17. 4200 ft 18. 7930 mi 19. 5 lb 7 oz 20. 472,750% 21. 34.4% 22. $21 23. $16,830 24. $3.55 25. $2000 26. $3171.88 7-2 Exercises 5. yes 7. no 13. no 15. no 17. yes 19. a. yes b. input: {0, 20, 40, 60, 80, 100}; 27. $400 28. 7% 29. 0.5 yr 30. $1000 at 3.75% for 3 years; output: {0, 150, 300, 450, 600, 750} 21. a. $0.20 b. any nonnegative $7.50 31. about $574.44 number of hours (x 0) c. 550 hours 25. All real numbers 27. D Chapter 7 7-1 29. triangle; Quadrants I and II 26°) b. (80°, 26°) c. (91°, 32°) 3. commission 4. 0.4375 29. 7 31. 6 33. 100 35. $3 7-3 Exercises Exercises 1. 1. II 3. III 5, 7. y 6 y (2, 5) 4 4 (0, 3) (1, 2) 2 (2, 1) x x 23a. $78 b. $117 c. $39 d. 80% 25. 24,900% 27. decrease; 13% x O 2 (5, 5) 1. $2234.38; $9384.38 3. $1430.24 13. 39% decrease 15. 24% decrease 17. $500 19. 120 21. 50 2 2 21. How long did Alice keep her 23a. 10 b. 20 c. 40 29. 1.0% 4 Exercises 7. 1.6 mi 9. 400% 11. 1% level 17. 24.4 19. 399 21. 500 (1, 1) c. 17.8% d. 19.8% 19. $695 15. $9.26, $626.26 17. 5 years 1 y 4 1. 49.5% 3. 75% 5. 23% 13. 296% 15. 54.0 ft above sea 17, 19. 2 O 2 4 2 (3, ⫺4) 4 9. (6, 3) 11. (4, 0) 13. I 15. IV O 2 2 3. a. y 750x 2 4 b. Amount of water (gal) 5,000 4,000 21. Simon calculated the b. 3 s 21. a. $4860, $4940, $5000, y-coordinates incorrectly. $5040, $5060 b. 7 23. B 23. 32.3 25. 2.3 25. It has no x-intercepts. 27. 7 29. 7.5 ft 31. 12.75 ft 33. 12,444.4 3,000 Exercises 7-4 2,000 1. 1,000 7-5 y 6 Exercises 1. y 4 0 2 x Time (hr) 5. 4 2 1 2 3 4 5 6 7 – 4 –2 O –2 y 4 2 x 4 –4 –2 O –2 –4 2 O 2 4 3. 6 4 3. 4 x 4 O 2 –4 –2 O –4 4 (2, 7) 6 –8 4 5. x O 2 4 2 4 –4 9. $550; $975; $1400; $1825; $2250 y –12 9 8 7 6 5 4 3 2 1 (0, 3) 4 (2, 1) x 4 2 2 y 5. linear 7. 10 6 4 2 –2 –4 2 x 4 –10 –8 –6 –4 –2 O –2 Distance (mi) 4 6 8 10 y –6 –8 –10 2 9. x 30 4 O 2 y 4 4 2 2 10 1 2 x 4 x 0 –4 –2 O –2 2 Time (h) Distance (ft) 2 –4 7. 50 20 y 8 4 13. 4 y 8 2 7. 70 2 y 4 11. 4 –4 x 2 2 9. 15, 6, 15 11. 0, 0, 18 y 13. 37, 7, 31 15. Graph C 2 4 –4 17. Graph B 16 19. a. 8 4 x 0 1 2 Time (s) 15. a. 5 ppm b. about 382 ppm 19. It is not linear. 11. h 12 40 35 30 25 20 15 10 5 0 16 y 12 8 4 x t 0.5 1 1.5 2 2.5 3 –4 –2 O 2 4 –4 Selected Answers SA7 13. quadratic 15. quadratic 17. Exercises 7-6 y 11. y 1 1. constant 3. 3 5. 6 7. variable 4 4 9. 0 11. 4 13. constant 2 15. variable 17. 19 2 x –4 –2 O –2 2 x 19. a. $0.11/yr; $0.32/yr; $0.19/yr 4 ⴚ4 ⴚ2 b. 1998 to 2001 23. D 25. 12. 5 2 4 2 4 ⴚ2 27. 6.9 29. 20 31. 20 ⴚ4 –4 Exercises 7-7 1 1. 1 3. 4 5. 0 7. The slope of the 1 3 6 line is 5. 9. 2 11. 4 13. 7 1 15. 5 17. The slope of the line 4 is 4. 19. y 5x 350 21. The 2z roof is flat. 25. w 27. 3 y 19. 4 2 x –4 –2 O –2 2 4 9 x 2 4 right. (2, 5) linear relationship in which the y-intercept is always 0. 13. y 4x y –4 2 4 (2, 1) (0, 3) 4 y 14. 4 1 15. y 2x 17. y 8x 19. y 13x 2 23. Each watermelon would need 4 y = x3+3 y = x3+1 2 O 6 9. no 11. A direct variation is a 1 –4 –2 O –2 2 2 1 the curve rises or falls from left to x to be exactly the same weight. O 25. 28 27. $348.75; $1123.75 2 4 2 4 x 2 4 y = x3– 2 y = x3– 4 Chapter 7 Study Guide: Review 1. direct variation 2. function 3. linear function 4. J(2, 1), IV b. 1; 3; 2, 4 c. 15 29. B 5. K(2, 3), II 6. L(1, 0), x-axis 31. 38% increase 7. M(4, 2), III 33. 12% increase 8. Possible answer: x 1 0 1 y x 2 4 –4 –6 –8 SA8 x 4 5. y 6x; about 968 kg 7. yes 23. The sign determines whether y 2 1. yes 3. no direct variation –4 –4 –2 O –2 13. Exercises 7-9 –2 25. a. ⴚ2 ⴚ4 15. 9 17. 15 19. 5 21. 31 4 1 23. 11 inches 6 2 2 x ⴚ4 ⴚ2 9. Graph A 10. Graph B 13. B 4 35. 2 1. Graph A 3. Graph B y –4 4 Exercises 7-8 –2 O y 29. 4 miles 31. 20 –4 21. 12. Selected Answers y 11 4 3 9. Possible answer: x 1 0 1 y 2 10. yes 0 15. y x 4 2 O 2 4 2 3 10 17 2 6 3 2 8 18 (2, 8) 8 (2, 4) (0, 6) 16. 22. 31. y 4 4 2 2 y 4 2 x –4 –2 2 –4 –2 O –2 4 –2 –4 2 –4 33. 2 4 y 4 3 27. 4 8. 4 29. 1 2 30. –2 –1 2 x Distance (mi) –4 4 ⫺4 25. constant 26. constant 9 8 7 6 5 4 3 2 1 2 ⫺2 23. quadratic 24. linear 17. x ⫺4 ⫺2 O 4 4 ⫺4 ⫺2 O ⫺2 18 12 ⫺4 6 Time 31. y 5x; $22.50 18. 4 Chapter 8 –4 –2 2 4 8-1 –2 Exercises 3. XY , YZ , ZX 5. AEB or 1. XY –4 DEB 7. AEC 9. AEB and BED, AEC and CED 4 , 11. plane N or plane JKL 13. KJ , LK , MK 15. VWZ, , KM KL 2 YWX 17. VWZ, YWX 19. y 19. false 21. false 23. false , , VX 25. false 27. 30°, 60° 29. YV x ⴚ4 ⴚ2 O 2 4 ⴚ2 33. A 35. m5 37. 711 , YZ XY ⴚ4 39. 2.8 ft 20. 8-2 y 13. none of these 15. skew , and EH 21. plane , FG 17. BC 4 2 x –4 –2 2 4 29. 36m6 31. 27x12y3 33. AEC –4 35. BEC, CED 37. AEC, CED y 4 8-3 2 x –4 –2 O –2 –4 ABF, plane FBC, plane DAE, and plane DCG 23. yes 27. B –2 21. Exercises 2 4 Exercises 1. 105° 3. 62° 5. 118° 7. 126° 9. 70° 11. 110° 13. 4, 5, and 8 17. 129° 19. 180° x° 8-4 Exercises 1. q 77 3. s 120 5. c 56 7. 60°, 30°, 90° 9. r 86 11. t 115 13. m 72 15. 27°, 135°, 18° 17. y 15 19. w 60 21. 64° 23. C 25. x 30, y 100, z 50 27. 105° 29. 8 and 9 CD 31. 17 and 18 33. AB 8-5 Exercises CD 3. Parallelogram, 1. AN rhombus, rectangle, square AB 5. C(2, 1) 7. CD 9. Parallelogram, rhombus, rectangle, square 11. C(4, 1) 13. 3 15. 0 17. 2 21. true 23. false 25. true 27. The slope is also undefined because of CD parallel lines have the same slope. 1 29. D 31. B 37. 3 39. 140 41. 20 8-6 Exercises 1. triangle ABC triangle FED 3. q 5 5. s 7 7. quadrilateral PQRS quadrilateral ZYXW 9. n 7 11. x 19, y 27, z 18.1 13. r 24, s 120, t 48 15. 62° 19. C 21. 3 23. P(4, 1) 29. The measures of the remaining angles are 90°. Selected Answers SA9 Exercises 8-7 17. y 1. rotation 4 y 2 3. 4 D Aʹ A 2 Cʹ Bʹ O ⴚ4 ⴚ2 ⴚ4 2 O(3, 1) 7. y N 4 M′ x O ⴚ2 C′ 4 x D′ A′ 4. LKM 5. LKM and JKM and SV 7. plane SVR ⊥ 6. PQ plane RVT 8. plane PQR plane STV 9. 66° 10. 114° 11. 66° 23. (5, 2) 29. A(0, 4), B(0, 1), 12. 66° 13. 114° 14. m 26 C(5, 1) 31. constant 33. 7 in. 15. 13 cm 16. trapezoid y 17. Exercises 4 1. O ⴚ2 x L N N′ 4 ⴚ2 M ⴚ4 3. 11. K 2 ⴚ4 9. translation y J′ lines 2. rectangle; square; rhombus; square 3. KM 4 2 ⴚ2 1. parallel lines; perpendicular 19. (3, 2) 21. (m, n) 8-8 L M B′ O ⴚ2 x 5. L(2, 3) M(4, 6), N(7, 3), ⴚ4 Chapter 8 Study Guide: Review C B 18. H′ y 4 J F′ ⴚ4 ⴚ2 W H G′ Z x x F O 5. O G ⴚ2 X Y 13. A(1, 3), B(5, 1), C(7, 5), D(3, 7) 8 19. x 19 20. t 2.4 21. q 7 7. y D′ y 22. C′ A′ ⫺4 O 4 B′ B 2 x 8 A ⴚ4 ⫺4 D 15. ⴚ4 y B′ B 4 A′ C′ C 11. yes 15. hexagon 17. D 9 19. 3.4 10 D′ 2 x D A ⴚ2 O 2 B′ 21. 2.8 10 y A′ ⴚ4 2 ⴚ4 x ⫺4 ⫺2 O ⫺2 ⫺4 B′ 4 B 2 4 4 Selected Answers x 4 C′ y 23. 4 23. ⴚ2 SA10 ⴚ2 C A O A′ ⴚ2 9. C ⫺8 B 4 4 2 4 C′ A ⴚ2 O ⴚ2 ⴚ4 C x 2 4 y 24. 4 2 C x A ⴚ4 C′ dimensions are multiplied by 4, Chapter 9 Study Guide: Review the area will be 42 times as great 1. perimeter, area 2. chord and the perimeter will be 4 times 3. about 79 in2, 12 in. 4. 198 m2, 27. 9.1 ft 29. When the B A′ O 2 as great. 31. 153.8 cm2 4 1 11 yd2 7. 9 cm2, 14.2 cm 37. 874.6 ft2; 160.4 ft 39. C 41. 3.7 mi 43. 14 units 25. Possible answer: 8. 16 in2, 26.3 in. 9. 36 ft , FG 11. GI 12. HI , GI , , FI 10. HF 2 13. A 144π 452.2 in2; , JI GJ Exercises 9-3 5 80 m 5. 112 ft; 58 ft2 6. 20 yd; 33. 466.6 ft 35. 49.8 ft B′ 2 , OR , OS , OT 3. RT , RS , 1. OQ ST , TQ 5. CA , CB , CD , CE , CF , DE , FE , AE 9. 10 cm , BF 7. GB C 24π 75.4 in 14. A 17.6π 55.4 cm2; C 8.4π 26.4 cm 15. A 9π 28.3 m2; C 6π 18.8 m 16. A 0.36π 1.1 ft2; 11. 100.8° 13. 252° 15. 133.2° C 1.2π 3.8 ft 17. 57 m2 19. 120° 21. 90° 23. 18 25. 35 18. 40.57 ft2 19. 73.5 cm2 20. 14 units2 21. 15 units2 Exercises 9- 4 2 22. 12 units2 23. 10 units2 1. 6π cm; 18.8 cm 3. 16.8π ft ; 26. Possible answer: 52.8 ft2 5. A 4π units2; Chapter 10 12.6 units2; C 4π units; 12.6 units 7. 18π in.; 56.5 in. 10-1 Exercises 9. 256π cm2; 803.8 cm2 2 2 11. A 16π units ; 50.2 units ; 1. pentagon; triangles; pentagonal C 8π units; 25.1 units pyramid 3. triangles; rectangles; 13. C 10.7 m; A 9.1 m2 triangular prism 5. polyhedron; 2 Chapter 9 9-1 15. C 56.5 in.; A 254.3 in. hexagonal pyramid 7. triangle; 17. 6.4 cm 19. 6 cm 21. 11.7 m triangles; triangular pyramid 2 23. 38.5 m 25. C 30π ft 94.2 ft; A 225π ft2 706.5 ft2 Exercises 27.a. 9.6 in. 1. 28 cm 3. 12.2x ft 5. 18 units2 7. 14 units2 9. 42 cm 11. 26x m 2 b. 28.3 in. pyramid 11. not a polyhedron; 2 c. three regular pancakes 31. C 33. m4 7 35. 8 37. 27 units 2 17. 33 ft; 54 ft 19. 18 ft; 10.5 ft cylinder 13. square prism 15. triangular pyramid 19. rectangular pyramid 13. 24 units2 15. 12 units2 2 9. hexagon; triangles; hexagonal 31 2 2 Exercises 9-5 21. $3375 23. 42,000 mi 1. 48 m 27. B 29. x 4 31. a 37 7. 88.1 ft2 9. 54 cm2 11. 23.4 in2 2 3. 109.5 cm2 5. 59.6 in2 21. cylinder 23. A 25. 4 0 25 27. 13 29. 100 oz for $6.99 6 10-2 Exercises 33. rational 35. rational 37. not 13. 63,800 mi a real number 21. y 5x 23. 10π in.; 31.4 in. 5. 1500 ft3 7. 100 in3 9. 351 m3 25. 8.2π cm; 25.7 cm 11. 60 cm3 13. a. 800 in3 9-2 Exercises 1. 9 units 3. 3 units 5. 21 units 7. 15 units2 9. 5 units 11. 21 units 13. 20 units 19. 33 cm 9-6 2 17. 80 in 19. C Exercises 13. no 17. Approximate the area 2 15. 12 units2 17. 23.2 m2 2 2 2 2 21. 37 m 23. 60 units2 25. 25x units2 1. 463.1 cm3 3. 1256 m3 15. a. 4.62 107 in3 b. about 18.8 ft 19. 180 in3 21. D 23. (5, 9) 25. 4 in.; 12 in2 of the glacier with a trapezoid (b1 2, b2 3, h 1) that has 5 area 2 and a triangle (b 3, 3 h 1) that has area 2. 19. D 21. rational 23. rational 10-3 Exercises 1. 20 cm3 3. 99.7 ft3 5. 9.1 cm3 7. 6,255,000 ft3 9. 35.0 m3 11. 66.2 ft3 13. 5494.5 units3 15. 6 in 17. 11 ft 19. 736 cm3 Selected Answers SA11 21. 600 in3 23. 301,056 ft3 25. 8 27. B 29. x 15 31. x 9 2 33. 56.25π ft ; 176 ft 27. D 29. x 8 31. w 2 2 33. 314 mm 2 5. The medians are equal, but data set B has a much greater range. 35. 50.2 in 7. 45.5; 62.5 2 9. 10-4 Exercises Chapter 10 Study Guide: Review 1. 28 cm2 3. 52 cm2 5. 61.8 m2 1. cylinder 2. surface area 7. 401.9 in2 9. 791.3 cm2 3. cone 4. cylinder 50 54 68 80 85 11. Data set Y has a greater median. 11. 94 cm2 13. 38 cm2 5. rectangular pyramid 6. 364 cm3 3 15. 1160 mm2 17. 791.3 cm2 7. 24 mm 19. 747.3 yd2 21. 1920π 9. 111.9 ft3 10. 60 in3 11. 210 ft3 6028.8 mm2 23. 4 m 25. $34.56 12. 471 cm 27. C 31. 232 33. 4.25 8. 415.4 mm 3 Data set Y has a greater range. 13. 68; 85 15. 35; 57.5 3 17. 3 13. 314 m 2 14. 680 mm 67 15. 944 in2 35. 1.77 37. 110 units2 16. 150.7 in2 17. 180.6 cm2 10-5 Exercises 20. 288π 904.32 in3 1. 105 m2 3. 144 m2 5. 702.5 ft2 21. 7776π 24,416.6 m3 22. 3:1 7. 125.6 mm2 9. no 11. 0.18 km2 23. 9:1 24. 27:1 75 85 93 19. 18. 132 cm2 19. 804.9 in2 0 2 21. 3 34 9 16 c. Khufu; 91,636,272 ft3 19. B 21. 56 23. 17 25. 120 ft3 5 7 10 1. 15 31. 10 and 10 33. 4.8; 5; 5 7. 256π cm2; 803.8 cm2 35. 64.9; 63; 48 and 75 Republicans 3 66 4 8764 5 8411 6 20.7 m3 5. 4π in2; 12.6 in2 2678 1234 34 11- 4 Exercises Key: 冨4冨1 means 41 6冨4冨 means 46 9. The volume of the sphere and 5. 3 7. 4; 31 9. B 11. 61 13. 27 the cube are about equal 23. 1 19. line plot 21. 4 0 0 2 25. 72 units 1 1. 9 775.3 cm3 13. 1.3π in3; 4.2 in3 15. 207.4π m2; 651.2 m2 2 17. 400π cm ; 1256 cm 2 11-2 Exercises 14 12 10 8 6 4 2 0 0 10,000 30,000 50,000 Area (mi2) 1. 20; 20; 5 and 20; 30 3. Mean: 352.5; median: 350 3. positive correlation 21. V 52.41π 164.55 yd3; 5. 83.3; 88; 88; 28 7. 4.3; 4.4; 4.4 5. S 46.24π 145.19 yd2 and 6.2; 4.2 9. mean 6.2; 25. 1767.15 cm3 27. 6.33 in3 median 6.5 11. 151 13. 9; 8; 12 29. 113.04 31. 8 33. 15 21. D 35. 364 m2 23. Stems 4 5 6 10-7 Exercises 1. 4:1 3. 64:1 5. 8.2 cm3 7. 2:1 Leaves 358 17 022 11-3 Exercises 17. 7 cm; 343 cubes 1. 52; 70 19. 1,000,000 cm3 21a. 2508.8 in3 3. 11. positive correlation 13. negative correlation 17. A 19. 78.5 cm2 21. 9 to 1 11-5 Exercises b. about 10.9 gal 23. No; the Selected Answers 5 10 15 20 25 30 35 Price ($1000) 1 cube 15. 9 cm; 729 cubes SA12 35 30 25 20 15 10 5 0 7. positive correlation 9. 73°F 9. 8:1 11. 33,750 in3 13. 1 cm; the volume increases by 8 times. Miles peer gallon 19. 366.17π in3; 1149.76 in3 surface area increases by 4 times; 26 27. B 29. 4 and 4 1. 36π cm3; 113.0 cm3 3. 6.6π m3; ( 268 in3). 11. 246.9π cm3; 17 11-1 Exercises 3. Democrats 10-6 Exercises Tropical storms Chapter 11 Population (millions) b. Menkaure; 191,684 ft2 4.5 5 Hurricanes 13. 279,221,850π mi2 15. a. 481; 277 99 19 26 33 44.5 59 1. 0.6, or 60%; 0.4, or 40% 3. 0.709, or 70.9% 5. 0.49, or 49% 7. 0.885 9. 0.784 11. sample correlation 12. 0.85, or 85%; space: blue, green, red, yellow; 0.15, or 15% 13. 0.15, or 15% outcome shown: yellow 16. 15. 14. 1 5 1296 51 4 1 4 21. 2 23. 0.1 25. 0.0000001 5 27. 5 29. 0 6 13 12- 4 Exercises 13. sample space: 1, 2, 3, 4, 5, 6; outcome shown: 4 15. 0.545 1 11 or 11 31. 2 27. C 29. 1 0 5 0 33. 84 ft 35. 42 ft Chapter 12 1. 4x2y 3. 3x 2 8x 5 5. 8x 3 5x 6 7. 2b 3 5b 2 b 4 12-1 Exercises 9. 8m2n 3mn2 2mn 11-6 Exercises 1. yes 3. no 5. binomial 7. not a 11. 2x2y 5xy 10x 8 1. 0.34, or 34% 3. 0.136; polynomial 9. 8 11. 0 13. yes 13. 4x3 9x2 12x 21 in3; 0.113 5. 0.319, or 31.9% 15. no 17. yes 19. monomial 30 in3 15. 3v 5v2 7. 0.398; 0.239 9. 0.25 21. trinomial 23. not a 17. 4xy2 2xy 19. 9b2 2b 9 11. 0.025 13. 0.35 15. 0.3 polynomial 25. 2 27. 4 29. 1 3 1 19. 51 21. 4.4 23. 8 25. 8 31. 8 in3 33. monomial; 3 21. 5a 8 23. x2 3x 4 25. 6p3 5p2 2p2t2 10pt2 35. binomial; 2 37. trinomial; 5 27. 2b 2ab 39. not a polynomial 29. 6y2 12x2y 5x2 31. 2x2 41. trinomial; 3 43. not a 6x 1 in2 37. 10x 29; 109 Exercises 11-7 1 1 1 1. 2 3. 1 5. 4 7. red 9. 0 2 1 2 1 15. 1 17. 30 19. 8 11. 3 13. 1 8 1 1 1 5 21. 2 23. 8 25. 2 29. 6 polynomial 51. 9 53. 3.5 105 39. 63m5 41. 8r10s10 55. 7 57. 9 43. zy3 5zy 31. yes 33. no 12-2 Exercises 12-5 Exercises 11-8 Exercises 1. 3b2 and 4b2, 5b and b 1. 15s 3t 5 3. 35h6j 10 5. 35p 4r 5 1 10 1. dependent 3. 3 5. 2 253 5 1 7. independent 9. 2 11. 2 4 4 1 13. 2 15. 3 0.03125 21. D 7 2 23. 11; 21 25. 0 Chapter 11 Study Guide: Review 1. median; mode 2. probablilty 3. line of best fit; scatter plot; correlation 4. x x x x x x x x x 2 3. 7x 4x 5 5. 12x 32 7. 17a2 21a 9. t and 5t, 4t 2 and 5t 2 2 11. 9p 7p 21. 6c4d3 12c 2d 3 19. 2x 2 13x 15 23. 12s 4t 3 15s3t3 6s4t4 21. 9m2 20m 23. 17mn 25. 40 1000d 2 27. 82xy 82y in2 29. C 31. 51.2% 35. 20m4p8 12m3p7 24m4p5 43. 15c 3d 4 20c2d 4 x x x x 6. 43; 46; none; 25 7. ⴚ5 O 70 75 12-6 Exercises ⴚ5 1. 5x 3 3x 6 3. 11r 2s 9rs 5. 15ab 2 3ab a2b 8 8. 7. 128 32w in. 9. 7g 2 g 1 50 55 60 65 70 75 80 85 90 11. 3h6 12h 4 h 13. 13t 2 4t 12 9. 15. w 2 4w 2 80 82 84 86 88 90 92 94 10. no correlation 11. positive 45. 4m 3n 2 47. 59 in2 5 12-3 Exercises 65 29. 3m5 15m3 31. x5 x 7y 5 37. V πr 2x3 πr 2y 3; 63π 5. 302; 311.5; 233 and 324; 166 60 25. 24b6 27. 6a3b6 33. 3f 2g2 f 3g2 f 2g 5 y 10 11 12 13 14 15 55 1 19. 15z3 12z2 17. 12s 2 2s 3 5 1 30x 11. A 2b1h 2b2h 13. 2g3h8 15. 2s5t4 17. 7.5h5j10 13. 9x 2 32x 15. 6y 3 4 33. 7. 6hm 8h2 9. 3x3 15x2 17. 7w 2y 2wy 2 4wy 19. 6.19r 3 4r 2 5r 2 1. xy 4x 5y 20 3. 12m2 7m 45 5. m2 9m 14 7. 4x2 100x ft2 9. b2 9 11. 9x2 30x 25 13. v 2 4v 5 15. 3x2 13x 30 17. 12b2 11bc 5c2 19. 12r 2 11rs 5s2 21. 4x 2 60x 125 yd2 23. b2 6b 9 25. 4x2 9 27. a2 14a 49 29. b 2 7b 60 Selected Answers SA13 31. t2 13t 36 33. 3b 2 5b 28 6. not a polynomial 7. monomial 31. 12a4b3 30a3b3 36a3b 35. 4m2 11mn 3n2 37. r 2 25 8. not a polynomial 9. binomial 24a2b2 32. 2m3 16m3 2m 10. 8 11. 4 12. 3 13. 5 14. 6 33. 10g3h3 15gh5 20gh 30h2 2 2 39. 15r 22rs 8s 41. PV bP aV ab c 45. B 47. 25.1 49. 18.7 51. 4m2 12m 24 2 2 53. 11x y 4xy 16xy 2 2 15. 7t 3t 1 16. 11gh 9g h 2 17. 20mn 12m 18. 8a 10b 2 2 19. 36st 23t 20. 6x 2x 5 21. 5x4 x2 x 7 22. 2h2 2 2 8h 7 23. 2xy 2x y 2xy 26. w 12w 14 27. 4x 1. polynomial; degree 2. FOIL; 16x 14 28. 4ab2 11ab 4a2b 24. 13n2 12 25. 6x2 8 3 2 2 2 2 2 binomials 3. binomial; trinomial 29. p q 4p q 2pq 4. trinomial 5. not a polynomial 30. 12s2t4 4s2t3 32st3 SA14 Selected Answers 35. 18x7y14 15x6y12 12x3y7 24x3y6 36. p2 8p 12 37. b2 10b 24 38. 3r2 11r 4 39. 3a2 11ab 20b2 Chapter 12 Study Guide: Review 2 3 34. 2j5k3 2j4k4 j6k5 40. m2 14m 49 41. 9t2 36 42. 6b2 2bt 28t2 43. 3x2 2x 40 44. y2 22y 121 Glossary/Glosario KEYWORD: MT8CA Glossary ENGLISH absolute value The distance of a number from zero on a number line; shown by ⏐⏐. (p. 15) SPANISH valor absoluto Distancia a la que está un número de 0 en una recta numérica. El símbolo del valor absoluto es ⏐⏐. acute angle An angle that ángulo agudo Ángulo que mide measures less than 90°. (p. 379) menos de 90°. acute triangle A triangle with all triángulo acutángulo Triángulo angles measuring less than 90°. (p. 392) en el que todos los ángulos miden menos de 90°. Addition Property of Equality Propiedad de igualdad de la suma Propiedad que establece que The property that states that if you add the same number to both sides of an equation, the new equation will have the same solution. (p. 33) Addition Property of Opposites The property that states that the sum of a number and its opposite equals zero. (p. 18) puedes sumar el mismo número a ambos lados de una ecuación y la nueva ecuación tendrá la misma solución. Propiedad de la suma de los opuestos Propiedad que establece EXAMPLES ⏐5⏐ 5 ⏐5⏐ 5 14 6 8 6 6 14 14 12 (12) 0 que la suma de un número y su opuesto es cero. additive inverse The opposite of inverso aditivo El opuesto de a number. (p. 8) un número. adjacent angles Angles in the same plane that are side by side and have a common vertex and a common side. (p. 388) ángulos adyacentes Ángulos en el mismo plano que comparten un vértice y un lado. algebraic expression An expression that contains at least one variable. (p. 6) expresión algebraica Expresión que contiene al menos una variable. x8 4(m b) algebraic inequality An desigualdad algebraica inequality that contains at least one variable. (p. 136) Desigualdad que contiene al menos una variable. x 3 10 5a b 3 alternate exterior angles ángulos alternos externos Par de A pair of angles on the outer sides of two lines cut by a transversal that are on opposite sides of the transversal. (p. 389) ángulos en los lados externos de dos líneas intersecadas por una transversal, que están en lados opuestos de la transversal. The additive inverse of 5 is 5. b a a c b d a and d are alternate exterior angles. Glossary/Glosario G1 ENGLISH SPANISH alternate interior angles ángulos alternos internos Par de A pair of angles on the inner sides of two lines cut by a transversal that are on opposite sides of the transversal. (p. 389) ángulos en los lados internos de dos líneas intersecadas por una transversal, que están en lados opuestos de la transversal. altitude (of a triangle) A perpendicular segment from a vertex to the line containing the opposite side. (p. 393) altura (de un triángulo) Segmento perpendicular que se extiende desde un vértice hasta la recta que forma el lado opuesto. angle A figure formed by two rays ángulo Figura formada por dos with a common endpoint called the vertex. (p. 379) rayos con un extremo común llamado vértice. angle bisector A line, segment, or ray that divides an angle into two congruent angles. (p. 382) bisectriz de un ángulo Línea, EXAMPLES r t s v r and v are alternate interior angles. N segmento o rayo que divide un ángulo en dos ángulos congruentes. P O M arc An unbroken part of a circle. arco Parte continua de un círculo. , - (p. 446) area The number of non- área El número de unidades overlapping unit squares needed to cover a given surface. (p. 435) cuadradas que se necesitan para cubrir una superficie dada. x Ó The area is 10 square units. Associative Property of Addition The property that states that for all real numbers a, b, and c, the sum is always the same, regardless of their grouping. (p. 116) Associative Property of Multiplication The property that states that for all real numbers a, b, and c, their product is always Propiedad asociativa de la suma Propiedad que establece que para todos los números reales a, b y c, la suma siempre es la misma sin importar cómo se agrupen. Propiedad asociativa de la multiplicación Propiedad que the same, regardless of their grouping. (p. 116) establece que para todos los números reales a, b y c, el producto siempre es el mismo, sin importar cómo se agrupen. average The sum of a set of data promedio La suma de los divided by the number of items in the data set; also called mean. (p. 537) elementos de un conjunto de datos dividida entre el número de elementos del conjunto. También se llama media. G2 Glossary/Glosario a b c (a b) c a (b c) a b c (a b) c a (b c) Data set: 4, 6, 7, 8, 10 Average: 4 6 7 8 10 35 7 5 5 ENGLISH back-to-back stem-and-leaf plot A stem-and-leaf plot that compares two sets of data by displaying one set of data to the left of the stem and the other to the right. (p. 533) SPANISH diagrama doble de tallo y hojas Diagrama de tallo y hojas que compara dos conjuntos de datos presentando uno de ellos a la izquierda del tallo y el otro a la derecha. EXAMPLES Data set A: 9, 12, 14, 16, 23, 27 Data set B: 6, 8, 10, 13, 15, 16, 21 Set A Set B 9 0 68 642 1 0356 73 2 1 Key: 2 1 means 21 3 2 means 23 gráfica de barras Gráfica en la que se usan barras verticales u horizontales para presentar datos. Sunlight’s Travel Time to Planets Time (s) bar graph A graph that uses vertical or horizontal bars to display data. (p. SB17) 4800 5000 4000 2600 3000 2000 760 M ar s Ju pi te r Sa tu rn Ea rt h 1000 500 Planet base (in numeration) When a base Cuando un número es elevado number is raised to a power, the number that is used as a factor is the base. (p. 166) a una potencia, el número que se usa como factor es la base. base (of a polygon) A side of a polygon, or the length of that side. (p. 435) base (de un polígono) Lado de un polígono; cara de una figura tridimensional según la cual se mide o se clasifica la figura. base (of a three-dimensional figure) A face of a three- base (de una figura tridimensional) Cara de una figura dimensional figure by which the figure is measured or classified. (p. 480) tridimensional a partir de la cual se mide o se clasifica la figura. base (of a trapezoid) One of the two parallel sides of a trapezoid. (p. 440) 35 3 3 3 3 3; 3 is the base. L Bases of a cylinder Bases of a prism Base of a cone Base of a pyramid base (de un trapecio) Uno de los dos lados paralelos del trapecio. ÊLÊ£ ÊLÊÓ binomial A polynomial with two terms. (p. 590) binomio Polinomio con dos términos. xy 2a2 3 4m3n2 6mn4 Glossary/Glosario G3 ENGLISH SPANISH bisect To divide into two trazar una bisectriz Dividir en dos congruent parts. (p. 382) partes congruentes. EXAMPLES JK bisects LJM box-and-whisker plot A graph gráfica de mediana y rango that displays the highest and lowest quarters of data as whiskers, the middle two quarters of the data as a box, and the median. (p. 543) Gráfica que muestra los valores máximo y mínimo, los cuartiles superior e inferior, así como la mediana de los datos. capacity The amount a container can hold when filled. (p. 512) capacidad Cantidad que cabe en un recipiente cuando se llena. Celsius A metric scale for Celsius Escala métrica para medir measuring temperature in which 0°C is the freezing point of water and 100°C is the boiling point of water; also called centigrade. (p. SB15) la temperatura, en la que 0 °C es el punto de congelación del agua y 100 °C es el punto de ebullición. También se llama centígrado. center (of a circle) The point inside a circle that is the same distance from all the points on the circle. (p. 446) centro (de un círculo) Punto interior de un círculo que se encuentra a la misma distancia de todos los puntos de la circunferencia. center of rotation The point about which a figure is rotated. (p. 410) centro de una rotación Punto 90˚ Lower quartile Upper quartile Minimum Median Maximum 0 alrededor del cual se hace girar una figura. 2 4 6 8 10 12 14 A large milk container has a capacity of 1 gallon. 90˚ Center 90˚ 90˚ central angle An angle formed ángulo central de un círculo by two radii with its vertex at the center of a circle. (p. 447) Ángulo formado por dos radios cuyo vértice se encuentra en el centro de un círculo. certain (probability) Sure to happen; an event that is certain has a probability of 1. (p. 556) seguro (probabilidad) Que con seguridad sucederá. Representa una probabilidad de 1. chord A segment with its cuerda Segmento de recta cuyos endpoints on a circle. (p. 446) extremos forman parte de un círculo. G4 Glossary/Glosario When rolling a number cube, it is certain that you will roll a number less than 7. À` ENGLISH circle The set of all points in a SPANISH plane that are the same distance from a given point called the center. (p. 446) círculo Conjunto de todos los puntos en un plano que se encuentran a la misma distancia de un punto dado llamado centro. circle graph A graph that uses sectors of a circle to compare parts to the whole and parts to other parts. (p. SB21) gráfica circular Gráfica que usa secciones de un círculo para comparar partes con el todo y con otras partes. EXAMPLES Residents of Mesa, AZ 65+ 13% 45–64 27% Under 18 19% 11% 30% 18–24 25–44 circumference The distance circunferencia Distancia alrededor around a circle. (p. 450) de un círculo. Circumference clockwise A circular movement to the right in the direction shown. (p. 412) coefficient The number that is multiplied by a variable in an algebraic expression. (p. 120) commission A fee paid to a en el sentido de las manecillas del reloj Movimiento circular en la dirección que se indica. coeficiente Número que se multiplica por una variable en una expresión algebraica. 5 is the coefficient in 5b. person for making a sale. (p. 298) comisión Pago que recibe una persona por realizar una venta. commission rate The fee paid to a person who makes a sale expressed as a percent of the selling price. (p. 298) tasa de comisión Pago que recibe una persona por hacer una venta, expresado como un porcentaje del precio de venta. A commission rate of 5% and a sale of $10,000 results in a commission of $500. common denominator A común denominador denominator that is the same in two or more fractions. (p. 70) Denominador que es común a dos o más fracciones. The common denominator of 8 2 and 8 is 8. common factor A number that is factor común Número que es a factor of two or more numbers. (p. SB6) factor de dos o más números. common multiple A number that is a multiple of each of two or more numbers. (p. SB6) común múltiplo Número que es Commutative Property of Addition The property that states Propiedad conmutativa de la suma Propiedad que establece que that two or more numbers can be added in any order without changing the sum. (p. 116) sumar dos o más números en cualquier orden no altera la suma. múltiplo de dos o más números. 5 8 is a common factor of 16 and 40. 15 is a common multiple of 3 and 5. 8 20 20 8; a b b a Glossary/Glosario G5 ENGLISH SPANISH Commutative Property of Multiplication The property that Propiedad conmutativa de la multiplicación Propiedad que states that two or more numbers can be multiplied in any order without changing the product. (p. 116) establece que multiplicar dos o más números en cualquier orden no altera el producto. compatible numbers Numbers números compatibles Números that are close to the given numbers that make estimation or mental calculation easier. (p. 278) que están cerca de los números dados y hacen más fácil la estimación o el cálculo mental. complementary angles Two ángulos complementarios Dos ángulos cuyas medidas suman 90°. angles whose measures add to 90°. (p. 379) EXAMPLES 6 12 12 6; a b b a To estimate 7957 5009, use the compatible numbers 8000 and 5000: 8000 5000 13,000. 37˚ A 53˚ B The complement of a 53° angle is a 37° angle. composite figure A figure made up of simple geometric shapes. (p. 436) figura compuesta Figura formada por figuras geométricas simples. 2 cm 5 cm 4 cm 4 cm 4 cm 5 cm 3 cm composite number A whole number greater than 1 that has more than two positive factors. (p. SB5) número compuesto Número mayor que 1 que tiene más de dos factores que son números cabales. 4, 6, 8, and 9 are composite numbers. compound event An event suceso compuesto Suceso made up of two or more simple events. (p. 569) formado por dos o más sucesos simples. Rolling a 3 on a number cube and spinning a 2 on a spinner is a compound event. compound inequality A combination of more than one inequality. (p. 137) desigualdad compuesta x 2 or x 10; 2 x 10 compound interest Interest earned or paid on principal and previously earned or paid interest. (p. 304) interés compuesto Interés que se gana o se paga sobre el capital y los intereses previamente ganados o pagados. cone A three-dimensional figure with a circular base lying in one plane plus a vertex not lying on that plane. The remaining surface of the cone is formed by joining the vertex to points on the circle by line segments. (p. 481) cono Figura tridimensional con una base circular que está en un plano más un vértice que no está en ese plano. El resto de la superficie del cono se forma uniendo el vértice con puntos del círculo por medio de segmentos de recta. G6 Glossary/Glosario Combinación de dos o más desigualdades. If $100 is put into an account with an interest rate of 5% compounded monthly, then after 2 years, the account will 12 2 $110.49. have 100 1 12 0.05 ENGLISH congruent Having the same size and shape. (p. 406) SPANISH EXAMPLES congruentes Que tienen la misma Q R forma y el mismo tamaño. P S PQ RS congruent angles Angles that have the same measure. (p. 382) ángulos congruentes Ángulos que D tienen la misma medida. C F E A B ABC DEF congruent segments Segments segmentos congruentes that have the same length. (p. 382) Segmentos que tienen la misma longitud. Q R P S PQ S R constant A value that does not constante Valor que no cambia. 3, 0, π constante de variación La constante k en ecuaciones de variación directa e inversa. y 5x change. (p. 120) constant of variation The constant k in direct and inverse variation equations. (p. 357) conversion factor A fraction whose numerator and denominator represent the same quantity but use different units; the fraction is equal to 1 because the numerator and denominator are equal. (p. 237) coordinate One of the numbers of an ordered pair that locate a point on a coordinate graph. (p. 322) factor de conversión Fracción cuyo numerador y denominador representan la misma cantidad pero con unidades distintas; la fracción es igual a 1 porque el numerador y el denominador son iguales. coordenada Uno de los números de un par ordenado que ubica un punto en una gráfica de coordenadas. constant of variation 1 day 24 hours and 24 hours 1 day B 4 y 2 x –4 –2 O 2 4 The coordinates of B are (2, 3) coordinate plane (coordinate grid) A plane formed by the plano cartesiano (cuadrícula de coordenadas) Plano formado por la intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis. (p. 322) intersección de una recta numérica horizontal llamada eje x y otra vertical llamada eje y. y-axis O x-axis Glossary/Glosario G7 ENGLISH correlation The description of the relationship between two data sets. (p. 548) SPANISH correlación Descripción de la relación entre dos conjuntos de datos. EXAMPLES *ÃÌÛiÊVÀÀi>Ì Þ ÊVÀÀi>Ì Þ Ý i}>ÌÛiÊVÀÀi>Ì Þ Ý Ý correspondence The relationship between two or more objects that are matched. (p. 406) correspondencia La relación entre dos o más objetos que coinciden. A and D are corresponding angles. A B C D A B and D E are corresponding sides. E F corresponding angles (for lines) For two lines intersected ángulos correspondientes (en líneas) Dadas dos líneas by a transversal, a pair of angles that lie on the same side of the transversal and on the same sides of the other two lines. (p. 389) cortadas por una transversal, el par de ángulos ubicados en el mismo lado de la transversal y en los mismos lados de las otras dos líneas. corresponding angles (in polygons) Matching angles of ángulos correspondientes (en polígonos) Ángulos que están en la two or more polygons. (p. 244) misma posición relativa en dos o más polígonos. m n o p q s r t m and q are corresponding angles. A and D are corresponding angles. corresponding sides Matching sides of two or more polygons. (p. 244) lados correspondientes Lados que se ubican en la misma posición relativa en dos o más polígonos. AB and DE are corresponding sides. counterclockwise A circular movement to the left in the direction shown. (p. 412) en sentido contrario a las manecillas del reloj Movimiento cross products In the statement productos cruzados En el bc and ad are the cross products. (p. 232) enunciado bc y ad son productos cruzados. a b c , d G8 Glossary/Glosario circular en la dirección que se indica. a b c , d 2 4 3 6 2 4 For the proportion 3 6, the cross products are 2 6 12 and 3 4 12. ENGLISH SPANISH cube (geometric figure) A rectangular prism with six congruent square faces. (p. 477) cubo (figura geométrica) Prisma cube (in numeration) A number cubo (en numeración) Número raised to the third power. (p. 168) elevado a la tercera potencia. cubic function A polynomial function of degree 3. (p. 338) función cúbica Función polinomial customary system of measurement The measurement sistema usual de medidas El sistema de medidas que se usa comúnmente en Estados Unidos. system often used in the United States. (p. SB15) EXAMPLES rectangular con seis caras cuadradas congruentes. 23 2 2 2 8 8 is the cube of 2. y = x3 de grado 3. cylinder A three-dimensional figure with two parallel congruent circular bases. The third surface of the cylinder consists of all parallel circles of the same radius whose centers lie on the segment joining the centers of the bases. (p. 481) cilindro Figura tridimensional con decagon A polygon with ten sides. decágono Polígono de diez lados. inches, feet, miles, ounces, pounds, tons, cups, quarts, gallons dos bases circulares paralelas y congruentes. La tercera superficie del cilindro consiste en todos los círculos paralelos del mismo radio cuyo centro está en el segmento que une los centros de las bases. (p. SB16) deductive reasoning The process of using logic to draw conclusions. (p. SB24) razonamiento deuctivo Proceso degree The unit of measure for angles or temperature. (p. 379) grado Unidad de medida para ángulos y temperaturas. degree of a polynomial The grado de un polinomio La highest power of the variable in a polynomial. (p. 591) potencia más alta de la variable en un polinomio. denominator The bottom denominador Número que está number of a fraction that tells how many equal parts are in the whole. (p. 66) abajo en una fracción y que indica en cuántas partes iguales se divide el entero. Density Property The property Propiedad de densidad Propiedad según la cual entre dos números reales cualesquiera siempre hay otro número real. that states that between any two real numbers, there is always another real number. (p. 199) en el que se utiliza la lógica para sacar conclusions. The polynomial 4x5 6x2 7 has degree 5. 2 In the fraction 5, 5 is the denominator. Glossary/Glosario G9 ENGLISH SPANISH dependent events Events for sucesos dependientes Dos which the outcome of one event affects the probability of the other. (p. 569) sucesos son dependientes si el resultado de uno afecta la probabilidad del otro. diagonal A line segment that diagonal Segmento de recta que connects two non-adjacent vertices of a polygon. (p. 403) une dos vértices no adyacentes de un polígono. EXAMPLES A bag contains 3 red marbles and 2 blue marbles. Drawing a red marble and then drawing a blue marble without replacing the first marble is an example of dependent events. B A C E Diagonal D diameter A line segment that passes through the center of a circle and has endpoints on the circle, or the length of that segment. (p. 446) diámetro Segmento de recta que pasa por el centro de un círculo y tiene sus extremos en la circunferencia, o bien la longitud de ese segmento. difference The result when one number is subtracted from another. (p. 10) diferencia El resultado de restar un número de otro. dimensions (geometry) The dimensiones (geometría) length, width, or height of a figure. (p. 480) Longitud, ancho o altura de una figura. direct variation A relationship variación directa Relación entre between two variables in which the data increase or decrease together at a constant rate. (p. 357) dos variables en la que los datos aumentan o disminuyen juntos a una tasa constante. In 16 5 11, 11 is the difference. 4 y 2 x ⫺4 ⫺2 2 4 ⫺4 y 2x discount The amount by which descuento Cantidad que se resta the original price is reduced. (p. 295) del precio original de un artículo. disjoint events Two events are disjoint if they cannot occur in the same trial of an experiment. (p. 566) sucesos desunidos Dos sucesos Distributive Property The Propiedad distributiva Propiedad property that states if you multiply a sum by a number, you will get the same result if you multiply each addend by that number and then add the products. (p. 117) que establece que, si multiplicas una suma por un número, obtienes el mismo resultado que si multiplicas cada sumando por ese número y luego sumas los productos. dividend The number to be dividendo Número que se divide en un problema de división. In 8 4 2, 8 is the dividend. divisible Can be divided by a divisible Que se puede dividir entre 18 is divisible by 3. number without leaving a remainder. (p. SB4) un número sin dejar residuo. divided in a division problem. (p. SB7) G10 Glossary/Glosario son desunidos si no pueden ocurrir en la misma prueba de un experimento. When rolling a number cube, rolling a 3 and rolling an even number are disjoint events. 5 21 5(20 1) (5 20) (5 1) ENGLISH Division Property of Equality The property that states that if you divide both sides of an equation by the same nonzero number, the new equation will have the same solution. (p. 37) SPANISH Propiedad de igualdad de la división Propiedad que establece que puedes dividir ambos lados de una ecuación entre el mismo número distinto de cero, y la nueva ecuación tendrá la misma solución. EXAMPLES 4x 12 4x 12 4 4 x3 divisor The number you are dividing by in a division problem. (p. SB7) divisor El número entre el que se divide en un problema de división. In 8 4 2, 4 is the divisor. domain The set of all possible input values of a function. (p. 326) dominio Conjunto de todos los The domain of the function y x2 1 is all real numbers. double-bar graph A bar graph gráfica de doble barra Gráfica de barras que compara dos conjuntos de datos relacionados. Students at Hill Middle School 100 Number of students that compares two related sets of data. (p. SB17) posibles valores de entrada de una función. 80 60 40 20 0 Grade 6 Grade 7 Grade 8 Boys edge The line segment along arista Segmento de recta donde se which two faces of a polyhedron intersect. (p. 480) intersecan dos caras de un poliedro. endpoint A point at the end of a line segment or ray. (p. 378) extremo Un punto ubicado al final de un segmento de recta o rayo. Girls Edge equally likely Outcomes that have the same probability. (p. 564) igualmente probables Resultados que tienen la misma probabilidad de ocurrir. When tossing a coin, the outcomes “heads” and “tails” are equally likely. equation A mathematical sentence that shows that two expressions are equivalent. (p. 32) ecuación Enunciado matemático que indica que dos expresiones son equivalentes. x47 6 1 10 3 equilateral triangle A triangle triángulo equilátero Triángulo con with three congruent sides. (p. 393) tres lados congruentes. equivalent expressions expresión equivalente Las Equivalent expressions have the same value for all values of the variables. (p. 120) expresiones equivalentes tienen el mismo valor para todos los valores de las variables. 4x 5x and 9x are equivalent expressions. Glossary/Glosario G11 ENGLISH SPANISH EXAMPLES razones equivalentes Razones que representan la misma comparación. 1 2 and are equivalent ratios. 2 4 estimate (n) An answer that is estimación (s) Una solución close to the exact answer and is found by rounding or other methods. (v) To find such an answer. (p. 278) aproximada a la respuesta exacta que se halla mediante el redondeo u otros métodos. estimar (v) Hallar una solución aproximada a la respuesta exacta. 500 is an estimate for the sum 98 287 104. evaluate To find the value of a evaluar Hallar el valor de una numerical or algebraic expression. (p. 6) expresión numérica o algebraica. even number A whole number that is divisible by two. (p. SB4) número par Número cabal divisible event An outcome or set of outcomes of an experiment or situation. (p. 556) suceso Un resultado o una serie de resultados de un experimento o una situación. When rolling a number cube, the event “an odd number” consists of the outcomes 1, 3, and 5. experiment (probability) In probability, any activity based on chance (such as tossing a coin). (p. 556) experimento (probabilidad) En Tossing a coin 10 times and noting the number of “heads”. experimental probability The probabilidad experimental Razón ratio of the number of times an event occurs to the total number of trials, or times that the activity is performed. (p. 560) del número de veces que ocurre un suceso al número total de pruebas o al número de veces que se realiza el experimento. exponent The number that exponente Número que indica indicates how many times the base is used as a factor. (p. 166) cuántas veces se usa la base como factor. exponential form A number is forma exponencial Se dice que un in exponential form when it is written with a base and an exponent. (p. 166) número está en forma exponencial cuando se escribe con una base y un exponente. expression A mathematical phrase that contains operations, numbers, and/or variables. (p. 6) expresión Enunciado matemático face A flat surface of a polyhedron. (p. 480) cara Superficie plana de un poliedro. equivalent ratios Ratios that name the same comparison. (p. 224) G12 Glossary/Glosario Evaluate 2x 7 for x 3. 2x 7 2(3) 7 67 13 2, 4, 6 entre 2. probabilidad, cualquier actividad basada en la posibilidad, como lanzar una moneda. Kendra attempted 27 free throws and made 16 of them. Her experimental probability of making a free throw is 16 number made 0.59. number attempted 27 23 2 2 2 8; 3 is the exponent. 42 is the exponential form for 4 4. 6x 1 que contiene operaciones, números y/o variables. Face ENGLISH SPANISH factor A number that is factor Número que se multiplica multiplied by another number to get a product. (p. SB4) por otro para hallar un producto. Fahrenheit A temperature scale in which 32°F is the freezing point of water and 212°F is the boiling point of water. (p. SB15) Fahrenheit Escala de temperatura en la que 32° F es el punto de congelación del agua y 212° F es el punto de ebullición. fair When all outcomes of an justo Se dice de un experimento donde todos los resultados posibles son igualmente probables. experiment are equally likely, the experiment is said to be fair. (p. 564) first quartile The median of the lower half of a set of data; also called lower quartile. (p. 542) When tossing a coin, heads and tails are equally likely, so it is a fair experiment. Lower quartile Upper quartile Minimum Median Maximum mitad inferior de un conjunto de datos. También se llama cuartil FOIL Sigla en inglés de los términos que se usan al multiplicar dos binomios: los primeros, los externos, los internos y los últimos (First, Outer, Inner, Last). formula A rule showing fórmula Regla que muestra relationships among quantities. (p. 434) relaciones entre cantidades. fraction A number in the form ab, where b 0. (p. 66) fracción Número escrito en la forma ab, donde b 0. frequency table A table that lists tabla de frecuencia Una tabla en la que se organizan los datos de acuerdo con el número de veces que aparece cada valor (o la frecuencia). items together according to the number of times, or frequency, that the items occur. (p. SB19) 7 is a factor of 21 since 7 3 21. primer cuartil La mediana de la inferior. FOIL An acronym for the terms used when multiplying two binomials: the First, Inner, Outer, and Last terms. (p. 618) EXAMPLES function An input-output función Relación de entrada-salida relationship that has exactly one output for each input. (p. 326) en la que a cada valor de entrada corresponde exactamente un valor de salida. 0 2 4 F 6 8 10 12 14 L (x 2)(x 3) x 2 3x 2x 6 x2 x 6 I O A w is the formula for the area of a rectangle. 2 3 Data set: 1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 6 Frequency table: Data Frequency 1 2 2 2 3 1 4 1 5 3 6 2 6 5 2 1 Glossary/Glosario ⫺4 ⫺1 0 G13 ENGLISH SPANISH graph of an equation A graph gráfica de una ecuación Gráfica of the set of ordered pairs that are solutions of the equation. (p. 123) del conjunto de pares ordenados que son soluciones de la ecuación. EXAMPLES yx 1 4 y 2 x ⫺4 ⫺2 O 2 4 ⫺4 great circle A circle on a sphere círculo máximo Círculo de una such that the plane containing the circle passes through the center of the sphere. (p. 508) esfera tal que el plano que contiene el círculo pasa por el centro de la esfera. greatest common divisor (GCD) máximo común divisor (MCD) El The largest whole number that divides evenly into two or more given numbers. (p. SB6) mayor de los factores comunes compartidos por dos o más números dados. height In a triangle or quadrilateral, the perpendicular distance from the base to the opposite vertex or side. (p. 435) altura En un triángulo o cuadrilátero, la distancia perpendicular desde la base de la figura al vértice o lado opuesto. In a trapezoid, the perpendicular distance between the bases. (p. 440) En un trapecio, la distancia perpendicular entre las bases. In a prism or cylinder, the perpendicular distance between the bases. (p. 413) En un prisma o cilindro, la distancia perpendicular entre las bases. In a pyramid or cone, the perpendicular distance from the base to the opposite vertex. (p. 420) En una pirámide o cono, la distancia perpendicular desde la base al vértice opuesto. hemisphere A half of a sphere. hemisferio La mitad de una esfera. (p. 508) heptagon A seven-sided heptágono Polígono de siete lados. polygon. (p. SB16) hexagon A six-sided polygon. (p. SB16) G14 Glossary/Glosario hexágono Polígono de seis lados. Great circle The GCD of 27 and 45 is 9. b2 h b1 h Height histogram A bar graph that shows the frequency of data within equal intervals. (p. SB19) SPANISH EXAMPLES histograma Gráfica de barras que Starting Salaries muestra la frecuencia de los datos en intervalos iguales. Frequency ENGLISH 40 30 20 10 0 9 9 –3 –2 20 30 9 –4 40 9 –5 50 Salary range (thousand $) hypotenuse In a right triangle, hipotenusa En un triángulo the side opposite the right angle. (p. 203) rectángulo, el lado opuesto al ángulo recto. Identity Property of Multiplication The property that Propiedad de identidad del uno states that the product of 1 and any number is that number. (p. 37) Propiedad que establece que el producto de 1 y cualquier número es ese número. Identity Property of Addition Propiedad de identidad del cero The property that states the sum of zero and any number is that number. (p. 32) Propiedad que establece que la suma de cero y cualquier número es ese número. image A figure resulting from a imagen Figura que resulta de una transformation. (p. 410) transformación. hypotenuse 414 3 1 3 404 3 0 3 B⬘ B C A C⬘ A⬘ impossible (probability) Can imposible (en probabilidad) Que never happen; an event that is impossible has a probability of 0. (p. 556) no puede ocurrir. Suceso cuya probabilidad de ocurrir es 0. improper fraction A fraction in which the numerator is greater than or equal to the denominator. (p. SB9) fracción impropia Fracción cuyo numerador es mayor que o igual al denominador. 17 3 , 5 3 independent events Events for sucesos independientes Dos which the outcome of one event does not affect the probability of the other. (p. 569) sucesos son independientes si el resultado de uno no afecta la probabilidad del otro. A bag contains 3 red marbles and 2 blue marbles. Drawing a red marble, replacing it, and then drawing a blue marble is an example of independent events. indirect measurement The technique of using similar figures and proportions to find a measure. (p. 248) medición indirecta La técnica de When rolling a standard number cube, rolling a 7 is an impossible event. usar figuras semejantes y proporciones para hallar una medida. Glossary/Glosario G15 ENGLISH SPANISH EXAMPLES inductive reasoning The process of conjecturing that a general rule or statement is true because specific cases are true. (p. SB24) razonamiento inductivo Proceso de razonamiento por el que se determina que una regal o enunciado son verdaderos porque ciertos casos específicos son verdaderos. inequality A mathematical statement that compares two expressions by using one of the following symbols: , , , , or . (p. 136) desigualdad Enunciado matemático que compara dos expresiones por medio de uno de los siguientes símbolos: , , , , ó . 58 5x 2 12 input The value substituted into valor de entrada Valor que se usa an expression or function. (p. 326) para sustituir una variable en una expresión o función. For the function y 6x, the input 4 produces an output of 24. integers The set of whole enteros Conjunto de todos los numbers and their opposites. (p. 14) números cabales y sus opuestos. interest The amount of money charged for borrowing or using money. (p. 303) interés Cantidad de dinero que se interior angles Angles on the ángulos internos Ángulos en los inner sides of two lines cut by a transversal. (p. 389) lados internos de dos líneas intersecadas por una transversal. . . . 3, 2, 1, 0, 1, 2, 3, . . . cobra por el préstamo o uso del dinero. r t s v r, s, t, and v are interior angles. intersecting lines Lines that líneas secantes Líneas que se cross at exactly one point. (p. 388) cruzan en un solo punto. m n interval The space between marked values on a number line or the scale of a graph. (p. SB18) inverse operations Operations intervalo El espacio entre los valores marcados en una recta numérica o en la escala de una gráfica. operaciones inversas Operaciones que se cancelan mutuamente: suma y resta, o multiplicación y división. Adding 3 and subtracting 3 are inverse operations: 5 3 8; 8 3 5 Multiplying by 3 and dividing by 3 are inverse operations: 2 3 6; 6 3 2 irrational number A real number that cannot be expressed as a ratio of two integers. (p. 198) número irracional Número real que no se puede expresar como una razón de dos enteros. 2, π isolate the variable To get a despejar la variable Dejar sola la variable en un lado de una ecuación o desigualdad para resolverla. x 7 22 7 7 x 15 that undo each other: addition and subtraction, or multiplication and division. (p. 32) variable alone on one side of an equation or inequality in order to solve the equation or inequality. (p. 32) G16 Glossary/Glosario 12 3x 3 3 4x ENGLISH isosceles triangle A triangle with at least two congruent sides. (p. 393) lateral area The sum of the areas of the lateral faces of a prism or pyramid, or the area of the lateral surface of a cylinder or cone. (p. 498) SPANISH EXAMPLES triángulo isósceles Triángulo que tiene al menos dos lados congruentes. área lateral Suma de las áreas de las caras laterales de un prisma o pirámide, o área de la superficie lateral de un cilindro o cono. £ÓÊV ÈÊV nÊV Lateral area 2(8)(12) 2(6)(12) 336 cm2 lateral face In a prism or a pyramid, a face that is not a base. (p. 498) cara lateral En un prisma o pirámide, una cara que no es la base. Bases Lateral face Right prism lateral surface In a cylinder, the superficie lateral En un cilindro, curved surface connecting the circular bases; in a cone, the curved surface that is not a base. (p. 499) superficie curva que une las bases circulares; en un cono, la superficie curva que no es la base. Lateral surface Right cylinder least common denominator (LCD) The least common mínimo común denominador (mcd) El mínimo común múltiplo multiple of two or more denominators. (p. 70) de dos o más denominadores. least common multiple (LCM) mínimo común múltiplo (mcm) El menor de los números cabales, distinto de cero, que es múltiplo de dos o más números dados. The least number, other than zero, that is a multiple of two or more given numbers. (p. SB6) legs In a right triangle, the sides that include the right angle; in an isosceles triangle, the pair of congruent sides. (p. 203) catetos En un triángulo rectángulo, los lados adyacentes al ángulo recto. En un triángulo isósceles, el par de lados congruentes. 3 5 The LCD of 4 and 6 is 12. The LCM of 6 and 10 is 30. leg leg like terms Two or more terms that have the same variable raised to the same power. (p. 120) line A straight path that extends without end in opposite directions. (p. 378) términos semejantes Dos o más términos que contienen la misma variable elevada a la misma potencia. In the expression 3a 5b 12a, 3a and 12a are like terms. línea Trayectoria recta que se extiende de manera indefinida en direcciones opuestas. Glossary/Glosario G17 SPANISH line graph A graph that uses line gráfica lineal Gráfica que muestra segments to show how data changes. (p. SB18) cómo cambian los datos mediante segmentos de recta. EXAMPLES Marlon’s Video Game Scores Score ENGLISH 1200 800 400 0 1 2 3 4 5 6 Game number line of best fit A straight line línea de mejor ajuste La línea recta that comes closest to the points on a scatter plot. (p. 549) que más se aproxima a los puntos de un diagrama de dispersión. 160 120 80 40 0 line of reflection A line that a ínea de reflexión Línea sobre la figure is flipped across to create a mirror image of the original figure. (p. 410) cual se invierte una figura para crear una imagen reflejada de la figura original. R Q line plot A number line with marks or dots that show frequency. (p. 532) diagrama de acumulación Recta numérica con marcas o puntos que indican la frecuencia. 40 80 120160 S S′ T T′ X X X X X X X X 0 2 1 R′ Q′ X 3 4 Number of pets line segment A part of a line between two endpoints. (p. 378) segmento de recta Parte de una línea con dos extremos. linear equation An equation ecuación lineal Ecuación cuyas whose solutions form a straight line on a coordinate plane. (p. 330) soluciones forman una línea recta en un plano cartesiano. linear function A function whose graph is a straight line. (p. 330) función lineal Función cuya gráfica GH H G y 2x 1 es una línea recta. yx 1 4 y 2 x ⫺4 ⫺2 O 2 4 ⫺4 lower quartile The median of cuartil inferior La mediana de la the lower half of a set of data; also called first quartile. (p. 542) mitad inferior de un conjunto de datos; también se llama primer cuartil. G18 Glossary/Glosario Lower quartile Upper quartile Minimum Median Maximum 0 2 4 6 8 10 12 14 ENGLISH SPANISH EXAMPLES markup The amount by which a margen de beneficio Cantidad wholesale cost is increased. (p. 295) que se agrega a un costo mayorista. maximum The greatest value in a máximo El valor mayor de un conjunto de datos. Data set: 4, 6, 7, 8, 10 Maximum: 10 mean The sum of a set of data divided by the number of items in the data set; also called average. (p. 537) media La suma de todos los Data set: 4, 6, 7, 8, 10 Mean: measure of central tendency medida de tendencia dominante A measure used to describe the middle of a data set. (p. 538) Medida que describe la parte media de un conjunto de datos. median The middle number, or the mean (average) of the two middle numbers, in an ordered set of data. (p. 537) mediana El número intermedio metric system of measurement sistema métrico de medición A decimal system of weights and measures that is used universally in science and commonly throughout the world. (p. SB15) Sistema decimal de pesos y medidas empleado universalmente en las ciencias y de uso común en todo el mundo. midpoint The point that divides a line segment into two congruent line segments. (p. 393) punto medio El punto que divide minimum The least value in a data set. (p. 543) mínimo El valor meno de un conjunto datos. Data set: 4, 6, 7, 8, 10 Minimum: 4 mixed number A number made número mixto Número compuesto up of a whole number that is not zero and a fraction. (p. SB9) por un número cabal distinto de cero y una fracción. 48 mode The number or numbers that occur most frequently in a set of data; when all numbers occur with the same frequency, we say there is no mode. (p. 537) moda Número o números más frecuentes en un conjunto de datos; si todos los números aparecen con la misma frecuencia, no hay moda. Data set: 3, 5, 8, 8, 10 Mode: 8 monomial A number or a monomio Un número o un 3x 2y 4 product of numbers and variables with exponents that are whole numbers. (p. 178) producto de números y variables con exponentes que son números cabales. data set. (p. 543) elementos de un conjunto de datos dividida entre el número de elementos del conjunto. También se llama promedio. o la media (el promedio) de los dos números intermedios en un conjunto ordenado de datos. 4 6 7 8 10 35 7 5 5 mean or median Data set: 4, 6, 7, 8, 10 Median: 7 centimeters, meters, kilometers, gram, kilograms, milliliters, liters un segmento de recta en dos segmentos de recta congruentes. C. B is the midpoint of A 1 Glossary/Glosario G19 ENGLISH SPANISH EXAMPLES multiple The product of any múltiplo El producto de cualquier number and a non-zero whole number is a multiple of that number. (p. SB4) número y un número cabal distinto de cero es un múltiplo de ese número. Multiplication Property of Equality The property that states Propiedad de igualdad de la multiplicación Propiedad que that if you multiply both sides of an equation by the same number, the new equation will have the same solution. (p. 38) establece que puedes multiplicar ambos lados de una ecuación por el mismo número y la nueva ecuación tendrá la misma solución. multiplicative inverse One of inverso multiplicativo Uno de dos two numbers whose product is 1; also called reciprocal. (p. 82) números cuyo producto es 1; también llamado recíproco. 3 4 mutually exclusive Two events mutuamente excluyentes Dos are mutually exclusive if they cannot occur in the same trial of an experiment. (p. 566) sucesos son mutuamente excluyentes cuando no pueden ocurrir en la misma prueba de un experimento. When rolling a number cube, rolling a 3 and rolling an even number are mutually exclusive events. negative correlation Two data correlación negativa Dos conjuntos sets have a negative correlation if one set of data values increases while the other decreases. (p. 549) de datos tienen correlación negativa si los valores de un conjunto aumentan a medida que los valores del otro conjunto disminuyen. negative integer An integer less entero negativo Entero menor que than zero. (p. 14) cero. net An arrangement of twodimensional figures that can be folded to form a polyhedron. (p. 496) 30, 40, and 90 are all multiplies of 10. 1 x 7 3 1 (3) 3x (3)(7) x 21 The multiplicative inverse of is 43. y x 2 is a negative integer. 4 3 2 1 plantilla Arreglo de figuras bidimensionales que se doblan para formar un poliedro. 0 1 2 3 10 m 10 m 6m 6m no correlation Two data sets sin correlación Caso en que los have no correlation when there is no relationship between their data values. (p. 549) valores de dos conjuntos no muestran ninguna relación. y x numerator The top number of a fraction that tells how many parts of a whole are being considered. (p. 66) numerador El número de arriba de numerical expression An expresión numérica Expresión que incluye sólo números y operaciones. expression that contains only numbers and operations. (p. 6) G20 Glossary/Glosario una fracción; indica cuántas partes de un entero se consideran. 4 5 numerator (2 3) 1 4 ENGLISH SPANISH obtuse angle An angle whose measure is greater than 90° but less than 180°. (p. 379) ánlgulo obtuso Ángulo que mide más de 90° y menos de 180°. obtuse triangle A triangle triángulo obtusángulo Triángulo que tiene un ángulo obtuso. containing one obtuse angle. (p. 392) octagon An eight-sided polygon. EXAMPLES octágono Polígono de ocho lados. (p. SB16) odd number A whole number that is not divisible by two. (p. SB4) opposites Two numbers that are an equal distance from zero on a number line; also called additive inverse. (p. 14) número impar Número cabal que no es divisible entre 2. 1, 3, 5 opuestos Dos números que están a la misma distancia de cero en una recta numérica. También se llaman 5 and 5 are opposites. inversos aditivos. order of operations A rule for orden de las operaciones Regla evaluating expressions: First perform the operations in parentheses, then compute powers and roots, then perform all multiplication and division from left to right, and then perform all addition and subtraction from left to right. (p. SB14) para evaluar expresiones: primero se hacen las operaciones entre paréntesis, luego se hallan las potencias y raíces, después todas las multiplicaciones y divisiones de izquierda a derecha, y por último, todas las sumas y restas de izquierda a derecha. ordered pair A pair of numbers par ordenado Par de números que sirven para ubicar un punto en un plano cartesiano. that can be used to locate a point on a coordinate plane. (p. 322) 5 units 5 units ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 42 8 2 16 8 2 16 4 20 0 1 2 3 4 5 6 Simplify the power. Divide. Add. B 4 y 2 x –4 –2 O 2 4 The coordinates of B are (2, 3). origin The point where the x-axis and y-axis intersect on the origen Punto de intersección entre el eje x y el eje y en un plano coordinate plane; (0, 0). (p. 322) cartesiano: (0, 0). origin O outcome (probability) A possible result of a probability experiment. (p. 556) resultado (en probabilidad) Posible resultado de un experimento de probabilidad. When rolling a number cube, the possible outcomes are 1, 2, 3, 4, 5, and 6. Glossary/Glosario G21 ENGLISH outlier A value much greater or SPANISH much less than the others in a data set. (p. 538) valor extremo Un valor mucho mayor o menor que los demás valores de un conjunto de datos. output The value that results valor de salida Valor que resulta from the substitution of a given input into an expression or function. (p. 326) después de sustituir una variable por un valor de entrada determinado en una expresión o función. parabola The graph of a quadratic function. (p. 334) parábola Gráfica de una función cuadrática. parallel lines Lines in a plane líneas paralelas Líneas que se encuentran en el mismo plano pero que nunca se intersecan. that do not intersect. (p. 384) parallel planes Planes that do planos paralelos Planos que no se not intersect. (p. 385) cruzan. EXAMPLES Most of data Mean Outlier For the function y 6x, the input 4 produces an output of 24. r s Plane AEF and plane CGH are parallel planes. parallelogram A quadrilateral paralelogramo Cuadrilátero con with two pairs of parallel sides. (p. 399) dos pares de lados paralelos. pentagon A five-sided polygon. pentágono Polígono de cinco lados. (p. SB16) percent A ratio comparing a porcentaje Razón que compara un number to 100. (p. 274) número con el número 100. percent of change The amount stated as a percent that a number increases or decreases. (p. 294) porcentaje de cambio Cantidad en que un número aumenta o disminuye, expresada como un porcentaje. percent of decrease A percent change describing a decrease in a quantity. (p. 294) porcentaje de disminución G22 Glossary/Glosario Porcentaje de cambio en que una cantidad disminuye. 45 45% 100 An item that costs $8 is marked down to $6. The amount of the decrease is $2, and the percent 2 of decrease is 8 0.25 25%. ENGLISH SPANISH EXAMPLES percent of increase A percent change describing an increase in a quantity. (p. 294) porcentaje de incremento Porcentaje de cambio en que una cantidad aumenta. The price of an item increases from $8 to $12. The amount of the increase is $4 and the percent 4 of increase is 8 0.5 50% perfect square A square of a cuadrado perfecto El cuadrado de 52 25, so 25 is a perfect square. whole number. (p. 190) un número cabal. perimeter The sum of the lengths of the sides of a polygon. (p. 434) perímetro La suma de las 18 ft longitudes de los lados de un polígono. 6ft perimeter 18 6 18 6 48 ft perpendicular bisector A line mediatriz Línea que cruza un that intersects a segment at its midpoint and is perpendicular to the segment. (p. 382) segmento en su punto medio y es perpendicular al segmento. perpendicular lines Lines that intersect to form right angles. (p. 384) líneas perpendiculares Líneas que perpendicular planes Planes that intersect at 90° angles. (p. 385) planos perpendiculares Planos pi (π) The ratio of the pi (π ) Razón de la circunferencia de un círculo a la longitud de su diámetro; π 3.14 ó 272. circumference of a circle to the length of its diameter; π 3.14 or 272. (p. 450) Ű n al intersecarse forman ángulos rectos. m que se cruzan en ángulos de 90°. plane A flat surface that extends plano Superficie plana que se forever. (p. 378) extiende de manera indefinida en todas direcciones. point An exact location in space. punto Ubicación exacta en el (p. 378) espacio. polygon A closed plane figure polígono Figura plana cerrada, formed by three or more line segments that intersect only at their endpoints (vertices). (p. 399) formada por tres o más segmentos de recta que se intersecan sólo en sus extremos (vértices). polyhedron A three-dimensional figure in which all the surfaces or faces are polygons. (p. 480) poliedro Figura tridimensional polynomial One monomial or polinomio Un monomio o la suma the sum or difference of monomials. (p. 590) o la diferencia de monomios. population The entire group of objects or individuals considered for a survey. (p. SB22) población Grupo completo de objetos o individuos que se desea estudiar. A C B R P cuyas superficies o caras tiene forma de polígonos. 2x2 3xy 7y2 In a survey about study habits of middle school students, the population is all middle school students. Glossary/Glosario G23 ENGLISH SPANISH positive correlation Two data correlación positiva Dos conjuntos sets have a positive correlation when their data values increase or decrease together. (p. 549) de datos tienen una correlación positiva cuando los valores de ambos conjuntos aumentan o disminuyen al mismo tiempo. EXAMPLES y x positive integer An integer greater than zero. (p. 14) entero positivo Entero mayor que power A number produced by potencia Número que resulta al raising a base to an exponent. (p. 166) elevar una base a un exponente. prime factorization A number factorización prima Un número written as the product of its prime factors. (p. SB5) escrito como el producto de sus factores primos. prime number A whole number greater than 1 that has exactly two factors, itself and 1. (p. SB5) número primo Número cabal mayor que 1 que sólo es divisible entre 1 y él mismo. principal The initial amount of money borrowed or saved. (p. 302) capital Cantidad inicial de dinero depositada o recibida en préstamo. principal square root The raíz cuadrada principal Raíz 25 5; the principal square nonnegative square root of a number. (p. 190) cuadrada no negativa de un número. root of 25 is 5. prism A three-dimensional figure with two congruent parallel polygonal bases. The remaining edges join corresponding vertices of the bases so that the remaining faces are rectangles. (p. 480) prisma Figura tridimensional con dos bases poligonales congruentes y paralelas. El resto de las aristas se unen a los vértices correspondientes de las bases de manera que el resto de las caras sean rectángulos. probability A number from 0 to 1 (or 0% to 100%) that describes how likely an event is to occur. (p. 556) probabilidad Un número entre 0 y product The result when two or more numbers are multiplied. (p. 26) producto Resultado de multiplicar dos o más números. The product of 4 and 8 is 32. profit The difference between ganancia Diferencia entre el total total income and total expenses. (p. 299) de ingresos y de gastos. If total income is $2,400, and total expenses are $2,100, the profit is $2,400 $2,100 $300. proportion An equation that proporción Ecuación que establece states that two ratios are equivalent. (p. 232) que dos razones son equivalentes. 2 4 3 6 protractor A tool for measuring angles. (p. 478) G24 Glossary/Glosario cero. 1 (ó 0% y 100%) que describe qué tan probable es un suceso. transportador Instrumento para medir ángulos. { Î Ó £ ä £ Ó Î { 2 is a positive integer. 23 8, so 2 to the 3rd power is 8. 10 2 5, 24 23 3 5 is prime because its only factors are 5 and 1. A bag contains 3 red marbles and 4 blue marbles. The probability of randomly choosing a red marble 3 is 7. ENGLISH SPANISH pyramid A three-dimensional figure with a polygonal base lying in one plane plus one additional vertex not lying on that plane. The remaining edges of the pyramid join the additional vertex to the vertices of the base. (p. 480) pirámide Figura tridimensional con una base poligonal en un plano más un vértice adicional que no está en ese plano. El resto de las aristas de la pirámide unen el vértice adicional con los vértices de la base. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. (p. 203) Teorema de Pitágoras En un quadrant The x- and y-axes cuadrante El eje x y el eje y dividen divide the coordinate plane into four regions. Each region is called a quadrant. (p. 322) el plano cartesiano en cuatro regiones. Cada región recibe el nombre de cuadrante. EXAMPLES 13 cm triángulo rectángulo, la suma de los cuadrados de los catetos es igual al cuadrado de la hipotenusa. 5 cm 12 cm 2 2 5 12 132 25 144 169 Quadrant II Quadrant I O Quadrant III Quadrant IV quadratic function A function of the form y ax 2 bx c, where a 0. (p. 334) función cuadrática Función del tipo y ax 2 bx c, donde a 0. quadrilateral A four-sided polygon. (p. 399) cuadrilátero Polígono de cuatro quarterly Four times a year. trimestral Cuatro veces al año. y x 2 6x 8 lados. (p. 305) quartile Three values, one of which is the median, that divide a data set into fourths. (p. 542) cuartil Cada uno de tres valores, uno de los cuales es la mediana, que dividen en cuartos un conjunto de datos. Lower quartile Upper quartile Minimum Median Maximum 0 2 4 6 8 10 12 14 quotient The result when one number is divided by another. (p. SB7) cociente Resultado de dividir un número entre otro. In 8 4 2, 2 is the quotient. radical symbol The symbol símbolo de radical El símbolo 36 6 used to represent the nonnegative square root of a number. (p. 192) con que se representa la raíz cuadrada no negativa de un número. Glossary/Glosario G25 ENGLISH SPANISH EXAMPLES radius A line segment with one endpoint at the center of the circle and the other endpoint on the circle, or the length of that segment. (p. 446) radio Segmento de recta con un random sample A sample in muestra aleatoria Muestra en la which each individual or object in the entire population has an equal chance of being selected. (p. SB21) que cada individuo u objeto de la población tiene la misma posibilidad de ser elegido. range (in statistics) The rango (en estadística) Diferencia entre los valores máximo y mínimo de un conjunto de datos. Data set: 3, 5, 7, 7, 12 Range: 12 3 9 range (of a function) The set of all possible output values of a function. (p. 326) rango (en una función) El The range of y ⏐x⏐ is y 0. rate A ratio that compares two quantities measured in different units. (p. 228) tasa Una razón que compara dos cantidades medidas en diferentes unidades. The speed limit is 55 miles per hour or 55 mi/h. rate of change A ratio that tasa de cambio Razón que compara la cantidad de cambio de la variable dependiente con la cantidad de cambio de la variable independiente. The cost of mailing a letter increased from 22 cents in 1985 to 25 cents in 1988. During this period, the rate of change was change in cost 25 21 3 change in years 1988 1985 3 difference between the greatest and least values in a data set. (p. 537) compares the amount of change in a dependent variable to the amount of change in an independent variable. (p. 344) Radius extremo en el centro de un círculo y el otro en la circunferencia, o bien la longitud de ese segmento. Mr. Henson chose a random sample of the class by writing each student’s name on a slip of paper, mixing up the slips, and drawing five slips without looking. conjunto de todos los valores de salida posibles de una función. 1 cent per year. rate of interest The percent charged or earned on an amount of money; see simple interest. (p. 302) tasa de interés Porcentaje que se cobra por una cantidad de dinero prestada o que se gana por una cantidad de dinero ahorrada; ver interés simple. ratio A comparison of two razón Comparación de dos numbers or quantities. (p. 224) números o cantidades. rational number A number that can be written in the form ab, número racional Número que se puede expresar como ab, donde a y b son números enteros y b 0. ray A part of a line that starts at one endpoint and extends forever. (p. 378) rayo Parte de una línea que where a and b are integers and b 0. (p. 66) real number A rational or irrational number. (p. 198) 12 12 to 25, 12:25, 25 6 6 can be expressed as 1. 1 0.5 can be expressed as 2. D comienza en un extremo y se extiende de manera indefinida. número real Número racional o irracional. Real Numbers Rational Numbers () 27 ___ 4 − 0.3 Integers () -3 Natural Numbers () 2 G26 Glossary/Glosario - √ 11 √2 0 3 1 4.5 √ 17 -2 Whole Numbers () -1 Irrational Numbers 10 -___ 11 e __5 9 π ENGLISH SPANISH reciprocal One of two numbers recíproco Uno de dos números cuyo whose product is 1; also called multiplicative inverse. (p. 82) producto es igual a 1. También se llama inverso multiplicativo. rectangle A parallelogram with four right angles. (p. 399) rectángulo Paralelogramo con cuatro ángulos rectos. rectangular prism A three- prisma rectangular Figura tridimensional que tiene tres pares de caras opuestas, paralelas y congruentes que son rectángulos. dimensional figure that has three pairs of opposite parallel congruent faces that are rectangles. (p. 480) reflection A transformation of a figure that flips the figure across a line. (p. 410) EXAMPLES 2 reflexión Transformación que ocurre cuando se invierte una figura sobre una línea. B A regular polygon A polygon in which all angles are congruent and all sides are congruent. (p. SB16) polígono regular Polígono en el que todos los ángulos y todos los lados son congruentes. regular pyramid A pyramid whose base is a regular polygon and whose lateral faces are all congruent. (p. 504) pirámide regular Pirámide que regular tessellation A repeating pattern of congruent regular polygons that completely covers a plane with no gaps or overlaps. (p. 416) teselado regular Patrón que se repite formado por polígonos regulares congruentes que cubren completamente un plano sin dejar espacios y sin superponerse. repeating decimal A rational decimal periódico Número racional number in decimal form in which a group of one or more digits (where all digits are not zero) repeat infinitely. (pp. 66, 191) en forma decimal en el que un grupo de uno o más dígitos (donde todos los dígitos son distintos de cero) se repiten infinitamente. rhombus A parallelogram with all sides congruent. (p. 399) rombo Paralelogramo en el que todos los lados son congruentes. right angle An angle that ángulo recto Ángulo que mide exactamente 90°. measures 90°. (p. 379) 3 The reciprocal of 3 is 2. B⬘ C C⬘ A⬘ tiene un polígono regular como base y caras laterales congruentes. 0.757575. . . 0.7 5 Glossary/Glosario G27 ENGLISH right cone A cone in which a perpendicular line drawn from the base to the tip (vertex) passes through the center of the base. (p. 504) SPANISH EXAMPLES cono recto Cono en el que una línea perpendicular trazada de la base a la punta (vértice) pasa por el centro de la base. Axis Right cone right triangle A triangle containing a right angle. (p. 392) triángulo rectángulo Triángulo que rise The vertical change when the slope of a line is expressed as rise the ratio run , or “rise over run.” (p. 345) distancia vertical El cambio rotation A transformation in rotación Transformación que ocurre cuando una figura gira alrededor de un punto. which a figure is turned around a point. (p. 410) tiene un ángulo recto. vertical cuando la pendiente de una línea se expresa como la razón distancia vertical distancia horizontal , o "distancia vertical sobre distancia horizontal". For the points (3, 1) and (6, 5), the rise is 5 (1) 6. F′ E′ G′ D′ G D F E run The horizontal change when distancia horizontal El cambio the slope of a line is expressed as ris e the ratio run , or “rise over run.” (p. 345) horizontal cuando la pendiente de una línea se expresa como la razón distancia vertical , o "distancia vertical distancia horizontal sobre distancia horizontal". sales tax A percent of the cost of an item, which is charged by governments to raise money. (p. 298) impuesto sobre la venta sample A part of the population. muestra Una parte de la población. In a survey about the study habits of middle school students, a sample is a group of 100 randomly chosen middle school students. sample space All possible espacio muestral Conjunto de outcomes of an experiment. (p. 556) todos los resultados posibles de un experimento. When rolling a number cube, the sample space is 1, 2, 3, 4, 5, 6. scale The ratio between two sets escala La razón entre dos conjuntos of measurements. (p. 252) de medidas. Porcentaje del costo de un artículo que los gobiernos cobran para recaudar fondos. (p. SB21) G28 Glossary/Glosario For the points (3, 1) and (6, 5), the run is 6 3 3. 1 cm : 5 mi ENGLISH SPANISH scale drawing A drawing that uses a scale to make an object smaller than (a reduction) or larger than (an enlargement) the real object. (p. 252) dibujo a escala Dibujo en el que se usa una escala para que un objeto se vea menor (reducción) o mayor (agrandamiento) que el objeto real al que representa. EXAMPLES A E F D B C H G A blueprint is an example of a scale drawing. scale factor The ratio used to enlarge or reduce similar figures. (p. 253) factor de escala Razón empleada para agrandar o reducir figuras semejantes. scale model A proportional model of a three-dimensional object. (p. 252) modelo a escala Modelo proporcional de un objeto tridimensional. scalene triangle A triangle with triángulo escaleno Triángulo que no congruent sides. (p. 393) no tiene lados congruentes. scatter plot A graph with points plotted to show a possible relationship between two sets of data. (p. 548) diagrama de dispersión Gráfica de puntos que muestra una posible relación entre dos conjuntos de datos. y 8 6 4 2 x 0 scientific notation A method of writing very large or very small numbers by using powers of 10. (p. 182) notación científica Método que se second quartile The median of segundo cuartil Mediana de un a set of data. (p. 476) conjunto de datos. sector A region enclosed by two sector Región encerrada por dos radii and the arc joining their endpoints. (p. 447) radios y el arco que une sus extremos. usa para escribir números muy grandes o muy pequeños mediante potencias de 10. 2 4 6 8 12,560,000,000,000 1.256 1013 Data set: 4, 6, 7, 8, 10 Second quartile: 7 segment A part of a line between segmento Parte de una línea entre two endpoints. (p. 378) dos extremos. side One of the segments that lado Uno de los segmentos que form a polygon. (p. 399) forman un polígono. GH H G -`i similar Figures with the same shape but not necessarily the same size are similar. (p. 244) semejantes Figuras que tienen la misma forma, pero no necesariamente el mismo tamaño. Glossary/Glosario G29 ENGLISH SPANISH EXAMPLES simple interest A fixed percent interés simple Un porcentaje fijo of the principal. It is found using the formula I Prt, where P represents the principal, r the rate of interest, and t the time. (p. 302) del capital. Se calcula con la fórmula I Cit, donde C representa el capital, i, la tasa de interés y t, el tiempo. simplest form A fraction is mínima expresión Una fracción 8 Fraction: 12 in simplest form when the numerator and denominator have no common factors other than 1. (p. SB8) está en su mínima expresión cuando el numerador y el denominador no tienen más factor común que 1. Simplest form: 3 simplify To write a fraction or expression in simplest form. (p. SB8) simplificar Escribir una fracción o expresión numérica en su mínima expresión. skew lines Lines that lie in different planes that are neither parallel nor intersecting. (p. 384) líneas oblicuas Líneas que se encuentran en planos distintos, pore so no se intersecan ni son paralelas. $100 is put into an account with a simple interest rate of 5%. After 2 years, the account will have earned I 100 0.05 2 $10. 2 A B E F C D G H and CD are skew lines. AE slant height The distance from altura inclinada Distancia de la the base of a cone to its vertex, measured along the lateral surface. (p. 504) base de un cono a su vértice, medida a lo largo de la superficie lateral. slope A measure of the steepness pendiente Medida de la inclinación de una línea en una gráfica. Razón de la distancia vertical a la distancia horizontal. of a line on a graph; the rise divided by the run. (p. 345) Slant height rise 3 Slope run 4 y 4 2 (2, 2) (⫺2, ⫺1) ⫺4 O 2 x 4 ⫺2 ⫺4 solution of an equation A value solución de una ecuación Valor o or values that make an equation true. (p. 32) valores que hacen verdadera una ecuación. solution of an inequality A solución de una desigualdad value or values that make an inequality true. (p. 140) Valor o valores que hacen verdadera una desigualdad. solution set The set of values that make a statement true. (p. 136) conjunto solución Conjunto de valores que hacen verdadero un enunciado. Equation: x 2 6 Solution: x 4 Inequality: x 3 10 Solution: x 7 Inequality: x 3 5 Solution set: x 2 ⫺4 ⫺3 ⫺2 ⫺1 solve To find an answer or a resolver Hallar una respuesta o solution. (p. 32) solución. G30 Glossary/Glosario 0 1 2 3 4 5 6 ENGLISH sphere A three-dimensional figure with all points the same distance from the center. (p. 508) SPANISH EXAMPLES esfera Figura tridimensional en la que todos los puntos están a la misma distancia del centro. square A rectangle with four cuadrado Rectángulo con cuatro congruent sides. (p. 399) lados congruentes. square (numeration) A number cuadrado (en numeración) raised to the second power. (p. 192) Número elevado a la segunda potencia. square root One of the two equal factors of a number. (p. 190) raíz cuadrada Uno de los dos factores iguales de un número. stem-and-leaf plot A graph used to organize and display data so that the frequencies can be compared. (p. 533) diagrama de tallo y hojas Gráfica que muestra y ordena los datos, y que sirve para comparar las frecuencias. In 52, the number 5 is squared. 16 4 4, or 16 4 4, so 4 and 4 are square roots of 16. Stem 3 4 5 Leaves 234479 0157778 1223 Key: 3 2 means 3.2 straight angle An angle that measures 180°. (p. 379) ángulo llano Ángulo que mide exactamente 180°. substitute To replace a variable sustituir Reemplazar una variable with a number or another expression in an algebraic expression. (p. 6) por un número u otra expresión en una expresión algebraica. Subtraction Property of Equality The property that states Propiedad de igualdad de la resta that if you subtract the same number from both sides of an equation, the new equation will have the same solution. (p. 33) Propiedad que establece que puedes restar el mismo número de ambos lados de una ecuación y la nueva ecuación tendrá la misma solución. sum The result when two or suma Resultado de sumar dos o more numbers are added. (p. 18) más números. supplementary angles Two angles whose measures have a sum of 180°. (p. 379) ángulos suplementarios Dos surface area The sum of the areas of the faces, or surfaces, of a three-dimensional figure. (p. 498) área total Suma de las áreas de las ángulos cuyas medidas suman 180°. Substituting 3 for m in the expression 5m 2 gives 5(3) 2 15 2 13. x6 8 6 6 x 2 The sum of 6 7 1 is 14. 30° 150° caras, o superficies, de una figura tridimensional. 12 cm 6 cm 8 cm Surface area 2(8)(12) 2(8)(6) 2(12)(6) 432 cm2 Glossary/Glosario G31 ENGLISH SPANISH EXAMPLES sistema de ecuaciones Conjunto de dos o más ecuaciones que contienen dos o más variables. x y 3 term (in an expression) The término (en una expresión) Las 3x2 parts of an expression that are added or subtracted. (p. 120) partes de una expresión que se suman o se restan. terminating decimal A decimal number that ends or terminates. (p. 66) decimal finito Decimal con un número determinado de posiciones decimales. tessellation A repeating pattern of plane figures that completely cover a plane with no gaps or overlaps. (p. 416) teselado Patrón repetido de figuras planas que cubren totalmente un plano sin superponerse ni dejar huecos. theoretical probability The probabilidad teórica Razón del ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. (p. 564) número de resultados igualmente probables en un suceso al número total de resultados posibles. third quartile The median of the tercer cuartil La mediana de la upper half of a set of data; also called upper quartile. (p. 542) mitad superior de un conjunto de datos. También se llama cuartil system of equations A set of two or more equations that contain two or more variables. (p. 131) superior. transformation A change in the size or position of a figure. (p. 410) x y 1 Term Term Lower quartile Upper quartile Minimum Median Maximum 0 2 4 6 figura a lo largo de una línea recta. trapecio Cuadrilátero con un par de exactly one pair of parallel sides. (p. 399) lados paralelos. Glossary/Glosario 12 14 C J A′ A′B′C′ C′ J′ M′ K′ L′ M L transversal Línea que cruza dos o más líneas. trapezoid A quadrilateral with G32 10 B′ K transversal A line that intersects two or more lines. (p. 389) 8 B ABC of a figure along a straight line. (p. 410) Term When rolling a number cube, the theoretical probability of 1 rolling a 4 is 6. A traslación Desplazamiento de una 8 6.75 transformación Cambio en el tamaño o la posición de una figura. translation A movement (slide) 6x 1 2 5 6 3 4 7 8 B A Transversal C D ENGLISH SPANISH tree diagram A branching diagram that shows all possible combinations or outcomes of an event. (p. SB23) diagrama de árbol Diagrama ramificado que muestra todas las posibles combinaciones o resultados de un suceso. trial In probability, a single repetition or observation of an experiment. (p. 556) prueba En probabilidad, una sola triangle A three-sided polygon triángulo Polígono de tres lados. EXAMPLES H T 1 2 3 4 5 6 1 2 3 4 5 6 When rolling a number cube, each roll is one trial. repetición u observación de un experimento. (p. 392) Triangle Sum Theorem The theorem that states that the measures of the angles in a triangle add up to 180°. (p. 392) Teorema de la suma del triángulo triangular prism A threedimensional figure with two congruent parallel triangular bases and whose other faces are rectangles. (p. 485) prisma triangular Figura trinomial A polynomial with three terms. (p. 590) trinomio Polinomio con tres unit analysis The process of changing one unit of measure to another. (p. 237) conversión de unidades Proceso que consiste en cambiar una unidad de medida por otra. unit conversion factor A factor de conversión de unidades fraction used in unit conversion in which the numerator and denominator represent the same amount but are in different units. (p. 237) Fracción que se usa para la conversión de unidades, donde el numerador y el denominador representan la misma cantidad pero están en unidades distintas. unit price A unit rate used to precio unitario Tasa unitaria que compare prices. (p. 229) sirve para comparar precios. unit rate A rate in which the second quantity in the comparison is one unit. (p. 228) tasa unitaria Una tasa en la que la segunda cantidad de la comparación es la unidad. upper quartile The median of the upper half of a set of data; also called third quartile. (p. 542) cuartil superior La mediana de la Teorema que establece que las medidas de los ángulos de un triángulo suman 180°. tridimensional con dos bases triangulares congruentes y paralelas cuyas otras caras son rectángulos. 4x2 3xy 5y2 términos. 1h 60 min or 60 min 1h Cereal costs $0.23 per ounce. 10 cm per minute Lower quartile Upper quartile Minimum Median Maximum mitad superior de un conjunto de datos; también se llama tercer cuartil. 0 2 4 6 8 Glossary/Glosario 10 12 G33 14 ENGLISH SPANISH variable A symbol used to variable Símbolo que representa represent a quantity that can change. (p. 6) una cantidad que puede cambiar. vertex (of an angle) The vértice de un ángulo Extremo común de los lados del ángulo. common endpoint of the sides of the angle. (p. 379) EXAMPLES In the expression 2x 3, x is the variable. A is the vertex of CAB. vertex (of a polygon) The intersection of two sides of the polygon. (p. 399) vértice (de un polígono) La intersección de dos lados del polígono. 6iÀÌiÝ A, B, C, D, and E are vertices of the polygon. vertex (of a polyhedron) A point at which three or more edges of a polyhedron intersect. (p. 480) vértice (de un poliedro) Un punto en el que se intersecan tres o más aristas de un poliedro. vertical angles A pair of ángulos opuestos por el vértice opposite congruent angles formed by intersecting lines. (p. 388) Par de ángulos opuestos congruentes formados por líneas secantes. 6iÀÌiÝ 2 1 3 4 1 and 3 are vertical angles. vertical line test A test used to prueba de linea vertical Prueba determine whether a relation is a function. If any vertical line crosses the graph of a relation more than once, the relation is not a function. (p. 327) utilizada para determinar si una relación es una función. Si una línea vertical corta la gráfica de una relación más de una vez, la relación no es una función. volume The number of cubic volumen Número de unidades cúbicas que se necesitan para llenar un espacio. units needed to fill a given space. (p. 485) 5 y -5 5 -5 4 ft 3 ft 12 ft Volume 3 4 12 144 ft3 x-axis The horizontal axis on a eje x El eje horizontal del plano coordinate plane. (p. 322) cartesiano. x-axis O G34 Glossary/Glosario x ENGLISH SPANISH EXAMPLES x-coordinate The first number in an ordered pair; it tells the distance to move right or left from the origin (0, 0). (p. 322) coordenada x El primer número de 5 is the x-coordinate in (5, 3). y-axis The vertical axis on a eje y El eje vertical del plano coordinate plane. (p. 322) cartesiano. un par ordenado; indica la distancia que debes moverte hacia la izquierda o la derecha desde el origen, (0, 0). y-axis O y-coordinate The second coordenada y El segundo número number in an ordered pair; it tells the distance to move up or down from the origin (0, 0). (p. 322) de un par ordenado; indica la distancia que debes avanzar hacia arriba o hacia abajo desde el origen, (0, 0). zero pair A number and its par nulo Un número y su opuesto, cuya suma es 0. opposite, which add to 0. (p. 42) 3 is the y-coordinate in (5, 3). 18 and 18 Glossary/Glosario G35 Index Absolute value, 15 Abstract art, 599 Academic Vocabulary, 4, 62, 114, 166, 222, 272, 320, 376, 432, 476, 528, 588 Acute angles, 379 Acute triangles, 392, SB17 Addition Associative Property of, 116–117 of decimals, 74, SB12 of fractions, 75, SB10 with unlike denominators, 87–89, SB10 of integers, 18–19 of mixed numbers, 87–89 of polynomials, 603–604 modeling, 602 of rational numbers, 74–75 properties, 23, 33, 116 solving equations by, 32–34 solving inequalities by, 140–141 Addition Property of Equality, 33 Additive inverse, 18 Adjacent angles, 388 Alaska, 282 Algebra The development of algebra skills and concepts is a central focus of this course and is found throughout this book. absolute value, 15 combining like terms, 120–121, 124, 596–597 equations, 32 addition, 32–33 checking solutions of, 32, 33, 37–38 decimal, solving, 94, 98 division, 37–38 linear, 330, 331 multi-step, see Multi-step equations multiplication, 37–38 solutions of, 32 subtraction, 32–33 systems of, 131 two-step, see Two-step equations expressions, 6–7, 10–11, 120–121 algebraic, 6–7 numerical, 6–7 variables and, 6–7 functions, 326 cubic, 338–339 graphing, 326–327, 334–335, 338–339 linear, see Linear functions quadratic, 334–335 tables and, 326 inequalities, see Inequalities I2 Index linear functions, 330–331 multi-step equations, 124–125 properties Addition, of Equality, 33 Associative, 116 of circles, 446–447 Commutative, 116 Distributive, 117 Division, of Equality, 37 Identity, 32, 37 Multiplication, of Equality, 38 Subtraction, of Equality, 33 proportions, and indirect measurement, 248–249 in scale drawings, models, and maps, 252–253, 257–258 solving, 232–234 solving equations by adding or subtracting, 32–34 modeling, 41–42 by multiplying or dividing, 37, 38 two-step, 43–45, 98–99 with variables on both sides, 129–131 solving inequalities by adding or subtracting, 140–141 by multiplying or dividing, 144–145 two-step, 148–149 tiles, 41–42, 128, 594–595, 602, 607, 616–617 translating between words and math, 63 translating words into math, 10–11 variables, 6–7 on both sides, solving equations with, 128–131 dependent, 344 independent, 344 isolating, 32 solving for, 32 Algebra tiles, 41–42, 128, 594–595, 602, 607, 616–617 Algebraic expressions, 6, 10–11 evaluating, 6–7 simplifying, 120–121 writing, 10–11 Algebraic inequalities, 136 Alternate exterior angles, 389 Alternate interior angles, 389 American Samoa, 396 Analysis dimensional, 237–239 unit, 237 Anamorphic images, 498 Angles, 379 acute, 379 adjacent, 388 alternate exterior, 389 alternate interior, 389 bisector, 382 central, of a circle, 447 classifying, 379 complementary, 379 congruent, 406 corresponding, 389 in polygons, 403 obtuse, 379 of quadrilaterals, 403 of triangles, 392–393 points and lines and planes and, 378–380 relationships of, 388–389 right, 379 straight, 379 supplementary, 379 vertical, 388 Animals, 81 Answer choices, eliminating, 56–57 Answering context–based test items, 582–583 Ant lions, 505 Applications Animals, 81 Architecture, 23, 195, 254, 255, 352, 493 Art, 194, 247, 500, 514, 604 Astronomy, 35, 139, 178, 187, 335 Banking, 28 Business, 28, 39, 123, 131, 132, 139, 143, 178, 226, 281, 337, 513, 597, 606, 609, 611 Chemistry, 17 Computer, 193 Construction, 352, 620 Consumer Economics, 80, 300 Consumer Math, 45, 46, 100, 118, 142, 559 Cooking, 226 Economics, 21, 151 Energy, 77 Entertainment, 9, 150, 226, 231, 453 Environment, 225, 333 Finance, 9, 126, 281 Food, 240, 453 Games, 195, 573 Geography, 248, 254, 277, 285 Geometry, 122, 127, 169, 182, 183, 606 Health, 19, 81, 540, 615 History, 187, 208, 209 Hobbies, 46, 122, 183, 195, 227, 336 Home Economics, 329 Language Arts, 143, 287 Life Science, 73, 101, 171, 174, 175, 187, 240, 241, 253, 284, 289, 290, 295, 361, 489, 505, 511, 568, 598 Literature, 297 Manufacturing, 280, 352 Measurement, 69, 90 Meteorology, 73 Money, 307 Multi-Step, 39, 80, 90, 91, 97, 199, 209, 231, 277, 287, 297, 442, 502 Music, 40, 449, 487 Nutrition, 39, 96, 141 Patterns, 174, 286 Recreation, 40, 79, 88, 126, 196, 284, 515 Safety, 352, 561 School, 72, 80, 149, 573 Science, 7, 17, 28, 29, 36, 91, 96, 127, 133, 187, 199, 226, 228, 281, 288, 297, 336, 507, 545 Social Studies, 25, 35, 71, 86, 146, 188, 282, 286, 396, 438, 489, 491, 507, 545 Sports, 14, 27, 38, 68, 72, 74, 76, 123, 127, 151, 194, 240, 281, 328, 336, 453, 502, 541 Transportation, 240, 241, 593 Travel, 125, 126 Weather, 119, 325 Approximating square roots, 197 Arc, of a circle, 446 Architecture, 23, 195, 254, 255, 352, 493 Are You Ready?, 3, 61, 113, 165, 221, 271, 375, 431, 475, 527, 587 Area, 435 of circles, 451 of irregular figures, 458–459 lateral, 498 of parallelograms, 435–436 of rectangles, 435–436 of squares, 192 surface, 498 of cones, 504–505 of cylinders, 496–499 of prisms, 496–499 of pyramids, 504–505 of spheres, 509 of trapezoids, 440–441 of triangles, 440–441 Arguments, writing convincing, 223 Art, 194, 247, 500, 514, 604 Ashurbanipal, King, 446 Aspect ratio, 231 Assessment Chapter Test, 55, 109, 159, 217, 265, 315, 369, 427, 469, 523, 581, 629 Cumulative Assessment, 58–59, 110–111, 162–163, 218–219, 268–269, 316–317, 372–373, 428–429, 472–473, 524–525, 584–585, 630–631 Mastering the Standards, 58–59, 110–111, 162–163, 218–219, 268–269, 316–317, 372–373, 428–429, 472–473, 524–525, 584–585, 630–631 Ready to Go On?, 30, 48, 92, 102, 134, 152, 190, 210, 242, 258, 292, 308, 342, 404, 420, 444, 462, 494, 516, 554, 574, 600, 622 Strategies for Success Any Type: Using a Graphic, 470–471 Extended Response: Write Extended Responses, 370–371 Gridded Response: Write Gridded Responses, 160–161 Multiple Choice Answering Context-Based Test Items, 582–583 Eliminate Answer Choices, 56–57 Short Response: Write Short Responses, 266–267 Study Guide: Review, 52–54, 106–108, 156–158, 214–216, 262–264, 312–314, 366–368, 424–426, 466–468, 520–522, 578–580, 626–628 Associative Property of Addition, 116–117 of Multiplication, 116–117 Astronomy, 35, 139, 178, 187, 335 Axes, 322 Back-to-back stem-and-leaf plot, 533–534 Bacteria, 171, 211 Balance of trade, 21 Balance scale, 32 Banking, 28 Bar graphs, 531, SB18 histograms, SB20 Base, 480–481 of polyhedrons, 480 Bases, 168 Benchmarks, 278 Berkeley, 119 Best fit, lines of, 549 Biased samples, SB22 Binary fission, 171 Binomials, 590 multiplication of, 616–617, 618–619 special products of, 619 Bisecting figures, 382 Bisectors constructing, 382–383 perpendicular, 382 Blood pressure, 236 Board foot, 597 Boeing 747, 443 Book, using your, for success, 5 Box-and-whisker plots, 542–544, creating, 543, 547 British thermal units (Btu), 77 Buffon, Comte de, 576 Business, 28, 39, 123, 131, 132, 139, 143, 178, 226, 281, 337, 513, 597, 606, 609, 611 Calculator approximating square roots on a, 197 graphing, see Graphing calculator California locations Berkeley, 119 Carlsbad, 515 Joe Matos Cheese Factory, 240 Legoland, 515 Mammoth Lakes, 374 Menlo Park, 164 Monterey Bay Aquarium, 240 Rincon Park, 220 San Diego, 270 San Francisco, 220 Santa Monica International Chess Park, 195 Santa Rosa, 240 Stanford Linear Accelerator Center, 164 Yosemite National Park, 252 California Link, 9, 77, 143, 195, 240, 515 Capacity, 512 customary units of, SB15 metric units of, SB15 Carlsbad, CA, 515 Caution! 10, 26, 169, 200, 436 Celsius temperature scale, 103, 331 Center of a circle, 446 of rotation, 410 Central angles, 447 Central tendency, measures of, 538 Certain event, 556 Challenge Challenge exercises are found in every lesson. Some examples: 9, 13, 17, 21, 25 Change percents of, 294 rates of, 344–345, see also Rates of change Chapter Project Online, 2, 60, 112, 164, 220, 270, 318, 374, 474, 526, 586 Chapter Test, 55, 109, 159, 217, 265, 315, 369, 427, 469, 523, 581, 629, see also Assessment Chemistry, 17 Chess, 195 Choose a Strategy, 9, 29, 86, 133, 171, 241, 287, 297, 337, 396, 502, 515, 606, 611 Chord, of a circle, 446 Circle(s), 446–447, 450–451 arcs of, 446 area of, 451 central angles, 447 chords of, 446 circumference, 450 diameter, 446, 450 graphs, 447, SB21 great, 508 properties of, 446–447 radius, 446, 450 Circle graphs, 447, SB21 interpreting, 447 making, SB21 reading, 447 Circumference, 450–451 Classifying angles, 379 polygons, 399, SB16–SB17 polynomials, 590–591 quadrilaterals, 399 real numbers, 200–201 three-dimensional figures, 481 triangles, 392–393, SB17 Closed circle, 137 Coefficients, 120 Index I3 Combining like terms, 120–121 Combining transformations, 415 Commission, 298 Commission rate, 298 Common denominator, 70, 87 Common multiple, SB6 least (LCM), SB6 Communicating Math apply, 487 choose, 229 compare, 19, 27, 137, 149, 169, 289, 335, 436, 505, 509, 544, 561, 604, 609, 613 compare and contrast, 481 decide, 193 demonstrate, 285 describe, 19, 23, 34, 71, 99, 117, 121, 141, 149, 186, 193, 225, 234, 253, 280, 331, 335, 359, 412, 417, 441, 487, 491, 513, 539, 550, 566, 591 determine, 79, 186, 197, 280 discuss, 197 draw, 447 explain, 7, 15, 19, 23, 34 , 67, 71, 75, 84, 89, 95, 99, 117, 131, 137, 141, 145, 169, 177, 181, 186, 193, 201, 207, 225, 234, 239, 249, 253, 275, 289, 323, 331, 351, 389, 394, 400, 407, 417, 447, 481, 500, 505, 509, 513, 539, 544, 591, 604, 609 express, 11, 173, 436 give, 11, 67, 131, 145, 225, 229, 239, 561 give an example, 7, 75, 79, 89, 225, 275, 351, 539, 566, 571, 597, 619 identify, 327 list, 27, 125, 177 model, 487, 491 name, 285, 323 show, 169, 275, 285 suppose, 27, 323 tell, 7, 84, 89, 121, 125, 173, 181, 201, 234, 389, 407, 571, 597 Think and Discuss Think and Discuss is found in every lesson. Some examples: 7, 11, 15, 19, 23 use, 201 Write About It Write About It exercises are found in every lesson. Some examples: 9, 13, 17, 21, 25 Commutative Property of Addition, 116–117 of Multiplication, 116–117 Comparing customary units and metric units, SB15 data sets, 544 numbers in scientific notation, 186 rational numbers, 70–71 ratios, 225, 232 volumes and surface areas, 509 Comparing and ordering whole numbers, SB3 I4 Index Compass, 382–383 Compatible numbers, 117, 278 Complementary angles, 379 Composite figures area of, 436 perimeter of, 436 volume of, 487 Composite numbers, SB5 Compound events, 569–571 Compound inequalities, 137 Compound interest, 302, 304 computing, 304–305 exploring, 302 Computer graphics, 398 Computer spreadsheets, 530–531 Computer, 193 Computing compound interest, 302, 304–305 Concept Connection, 49, 103, 153, 211, 259, 309, 363, 421, 463, 517, 575, 623 Concept maps, 433 Conclusions, see Reasoning Concorde, 443 Cones, 481, 490 nets of, 503 right, 504 surface area of, 504–505 volume of, 490–491 Congruence, 406–407 Congruent angles, constructing, 382–383 Congruent figures, 382 Congruent triangles, 406 Constant of variation, 357 Constant rate of change, 344–345 Constants, 120 Constructing bisectors and congruent angles, 382–383 graphs, using spreadsheets for, 530–531 nets, 496–497, 503 scale drawings and scale models, 256–257 Construction, 352, 620 Consumer application, 559 Consumer Economics, 80, 300 Consumer Math, 45, 46, 100, 118, 142, 559 Context-based test items, answering, 582–583 Converse of the Pythagorean Theorem, 206–207 Conversion factors, 237, SB15 units of measure, 237–239 Conversions, metric, SB15 Converting, 224 customary units to metric units, SB15 metric units to customary units, SB15 Convincing arguments, writing, 223 Cooking, 226 Coordinate geometry, 398–400 Coordinate plane, 322–323 graphing on a, 330, 334, 338 Coordinates, 322–323 Cornell system of note taking, 167 Correlation, 549–550 Correlation types, 549 Correspondence, 406 Corresponding angles, 389 Corresponding sides, 406–407 Countdown to Mastery, CA4–CA27 Creating box-and-whisker plots, 547 graphs, using spreadsheets for, 530–531 histograms, SB20 scatter plots, 553 tessellations, 416–417 Cross products, 232 Cubic functions, 338–339 Cubic units, 485 Cullinan diamond, 96 Cumulative Assessment, 58–59, 110–111, 162–163, 218–219, 268–269, 316–317, 372–373, 428–429, 472–473, 524–525, 584–585, 630–631, see also Assessment Cupid’s Span, 220 Customary system of measurement, 237–239, SB15 converting between metric and, SB15 Cylinders, 481, 485 nets of 497 surface area of, 498–500 exploring, 496–497 volume of, 485–487 da Vinci, Leonardo, 506 Data, see also Displaying and organizing data bar graphs, 531, SB18 box-and-whisker plots, 543–544 circle graphs, 477, SB21 collecting, 531 displaying, 447, 530–534, 543–544, SB18–SB21 double-bar graphs, SB18 histograms, SB20 line graphs, SB19 line plots, 532 organizing, 532–534 stem-and-leaf plots, 533 back-to-back, 533–534 Decagon, SB16 Decimals addition of, 74, 75, 94 comparing, 71 converting between percents and fractions and, 274–275 division of, 83 equivalent fractions and, 67 fractions and, 66 multiplication of, 79, 94 by powers of ten, SB7 ordering, 71 repeating, 64–65, 66–67 rounding, SB2 solving equations with, 94 subtraction of, 74, 75, 94 terminating, 64–65, 66–67 writing as percents, 274–275 Decrease, percent of, 294–295 Deductive reasoning, SB24 Degrees of polynomials, 591 Denominator(s), 66 like, adding and subtracting with, 75 unlike, adding and subtracting with, 87–88 Density, 228 Density Property of real numbers, 201 Dependent events, 569–571 finding probability of, 570–571 Dependent variable, 344 Devils Postpile National Monument, 374 Diagonals, 169 Diagram tree, SB23 Diameter, 446, 450 Diastolic blood pressure, 236 Dimensional analysis, 237–239 Dimensions changing, exploring effects of, 486, 505, 512–513 Direct variation, 357–359 Discounts, 295 Disjoint events, 566 Displaying and organizing data, 447, 530–534, 543–544, SB18–SB21 bar graphs, 531, SB18 box-and-whisker plots, 543–544 circle graphs, 447, SB21 double-bar graphs, SB18 histograms, SB20 line graphs, SB19 line plot, 532 stem-and-leaf plots, 533 back-to-back, 533–534 Distracter, 56 Distributive Property, 117, 121, 597 Divisibility rules, SB4 Division of decimals, 83, SB13 and mixed numbers, 82 by powers of ten, SB7 of fractions, 82–84 of integers, 26–27 long, SB7 of monomials, 180–181 of numbers in scientific notation, 189 of powers, 176–177 by powers of ten, SB7 of rational numbers, 82–84, SB11, SB13 solving equations by, 37–39 solving inequalities by, 144–145 of whole numbers, SB7 Division Property of Equality, 37 DNA model, 253 Double-bar graphs, SB18 Draw three-dimensional figures, 477 Drawings, scale, 252–253 Duckweed plants, 187 Earth, 186, 187, 508 Earthquakes, 563 Economics, 21, 151, 301 Edge, of a three-dimensional figure, 480 Edison, Thomas, 329 Effective notes, taking, 167 Eggs, 511 Eliminating answer choices, 56–57 Energy, 77 Enlargement, 253 Entertainment, 9, 150, 226, 231, 453 Environment, 225, 333 Equality Addition Property of, 32 Division Property of, 37 Multiplication Property of, 38 Subtraction Property of, 33 Equally likely outcomes, 564 Equations, 32 graphs of, 330–331, 334–335, 338–339 linear, see Linear equations multi-step, see Multi-step equations with no solutions, 130 one-step, see One-step equations solutions of, 32 solving, see Solving equations systems of, see Systems of equations and tables and graphs, 326, 330, 331, 334, 335, 338, 339 two-step, see Two-step equations with variables on both sides, modeling, 128 Equilateral triangles, 393, SB17 Equivalent expressions, 120 Equivalent fractions decimals and, 274 Equivalent ratios, 224–225, 232 Escher, M. C., 419 Estimate, 278 Estimating square roots, 196–197 with percents, 278–280 Estimation, 24, 86, 90, 171, 179, 194, 231, 281, 493, 541 Evaluating algebraic expressions, 6–7 expressions using the order of operations, 169 expressions with rational numbers, 75, 83, 89 negative exponents, 172–173 Events, 556 disjoint, 566 independent, see Independent events mutually exclusive, 566 Exam, preparing for your final, 589 Experiment, 556 Experimental probability, 560–561 Exploring compound interest, 302 effects of changing dimensions, 486, 505, 512–513 Pythagorean Theorem, 204 rational numbers, 64–65 three-dimensional figures, 478–479 volume of prisms, 484 Exponential form, 168 Exponents, 168–169 integer, 172–173 negative, 172–173 properties of, 176–177 Expressions algebraic, 6–7, 10–11 simplifying, 120–121 writing, 10–11 equivalent, 120 evaluating, with rational numbers, 75, 83, 89 numerical, 6–7 variables and, 6–7 Extended Response, 91, 175, 231, 297, 370–371, 391, 515, 536, 546, 611 Write Extended Responses, 370–371 Extra Practice, EP2–EP25 Face lateral, 498 of a three-dimensional figure, 480 Factor(s), SB4 common, SB6 conversion, 237 scale, 253 Factorization, prime, SB5 Fahrenheit scale, 331 Fahrenheit temperature converting between Celsius and, 103, 331 scale, 103 Fair objects, 564 Ferris wheel, 430, 453 Figures bisecting, 382 built of cubes, 498 composite, 436, 487 congruent, 382 geometric, 374–525 irregular, area of, 458–459 similar, see Similar figures three-dimensional, Index I5 see Three-dimensional figures two-dimensional, see Two-dimensional figures Final Exam, study for a, 589 Finance, 9, 126, 281 Finding numbers when percents are known, 288–289 percents, 283–285 probability of dependent events, 570–571 probability of independent events, 569–570 surface area of prisms and cylinders, 498–500 surface area of pyramids, 504 surface area of similar solids, 513 unit prices to compare costs, 229 volume of prisms and cylinders, 485–487 volume of pyramids and cones, 490–491 volume of similar solids, 513 Fireworks, 591 Flip, see Reflection Fluid ounce, SB15 Focus on Problem Solving Look Back, 93, 601 Make a Plan, 135, 293, 495, 555 Solve, 31, 191, 243, 343 Understand the Problem, 405 FOIL mnemonic, 618 Food, 240, 453 Foot, SB15 Formulas, learning and using, 477, see also inside back cover Fraction(s) addition of, 75 with unlike denominators, 87–89 decimals and percents and, 274–275 division of and mixed numbers, 82–84 equivalent, see Equivalent fractions improper, SB9 multiplication of, and mixed numbers, 78–79 relating, to decimals and percents, 274–275 simplest form, SB8 solving equations with, 94–95, 124–125 subtraction of, 75 with unlike denominators, 87–89 unit, 104 writing as mixed numbers, SB9 writing as terminating and repeating decimals, 66 Fraction form, dividing rational numbers in, 82 Frequency tables, SB20 Fulcrum, 232 Functions, 326–339 cubic, 338–339 linear, 330–331 I6 Index quadratic, 334–335 tables and, 326, 330, 331, 334, 335, 338, 339 graphs and, 330, 331, 334, 335, 338, 339 Gallon, SB15 Game Time 24 Points, 154 Buffon’s Needle, 576 Circles and Squares, 464 Coloring Tessellations, 422 Copy-Cat, 260 Crazy Cubes, 50 Egg Fractions, 104 Egyptian Fractions, 104 Equation Bingo, 212 Magic Cubes, 518 Magic Squares, 212 Math Magic, 50 Percent Puzzlers, 310 Percent Tiles, 310 Planes in Space, 518 Polygon Rummy, 422 Rolling for Tiles, 624 Shape Up, 464 Short Cuts, 624 Squared Away, 364 Tic-Frac-Toe, 260 Trans-Plants, 154 What’s Your Function?, 364 Game Time Extra, 104, 154, 212, 260, 310, 364, 422, 464, 518, 576, 624 Games, 195, 573 Gas mileage, 593 GCD (greatest common divisor), SB6 Geography, 248, 254, 277, 285 Geometric patterns, SB16 Geometry angles, 379–380, 388–389, 392–394, 406–407 area, 435–436, 440–441, 451 building blocks of, 378–380 circles, 446–447, 450–451 area of, 451 circumference of, 450 circumference, 450 cones, 481 volume of, 490–492 surface area of, 504–505 constructing bisectors and congruent angles, 382–383 congruent figures, 406–407 coordinate, 398–400 cylinders, 481 volume of, 485–487 surface area of, 496–497, 498–499 lines, 378–380 parallel, 384–385, 388–389 perpendicular, 384–385, 388 of reflection, 410 skew, 384 transversal, 388–389 nets, 496–497, 503 parallel line relationships, 389 parallelograms, 399 area of, 435 perimeter of, 434 planes, 378 polygons, 399 finding angle measures in, 403 regular, SB16 polyhedrons, 480 prisms, 484, 485–487 volumes of, 484, 485–487 surface area of, 496–497, 498–499 pyramids, 480, 490–491 volume of, 490–491 surface area of, 504–505 quadrilaterals, 399, 403, SB16, SB17 rays, 378 rectangles, 434–436 area of, 435–436 perimeter of, 434–436 rhombuses, 399, SB17 rhombuses, 399, SB17 scale drawings and scale models, 252–253, 256–257, 259 similar figures, 244–245 surface area and volume, 512–513 sphere, 508–509 squares, 399, SB17 surface area, 498–500, 503, 504–505, 509 tessellations, 416–417 three-dimensional figures, 474–525 introduction to, 480–481 modeling, 256–257 trapezoids area of, 440–441 perimeter of, 439 triangles, 392–394, SB16–SB17 area of, 440–441 perimeter of, 439 two-dimensional figures, 430–473 volume, 484–487, 490–491 Giant Ocean Tank, 489 Giant shark, 288, 289 Gigabyte, 179 Global temperature, 73 go.hrw.com, see Online Resources Googol, 179 Gram, SB15 Graph(s) bar, 531, SB18 circle, 447, SB21, see also Circle graphs constructing, using spreadsheets for, 530–531 creating, using technology for, 530 double-bar, SB18 of equations, 330–331, 334–335, 338–339 interpreting, 353–354 line, 330–331, SB19 tables and, 326, 330, 334, 338 Graphing on a coordinate plane, 322–323 inequalities, 137, 140–141, 144, 148–149 integers on a number line, 14 linear equations, 330 linear functions, 330–331 points, 14, 322–323 Technology Labs, see Technology Lab transformations, 410–412, 415 Graphing calculator creating box-and-whisker plots, 547 creating scatter plots, 553 multiplying and dividing numbers in scientific notation, 189 Great circle, 508 Great Lakes, 31 Great Pyramid of Giza, 491 Greatest common divisor (GCD), SB6 Gridded Response, 9, 21, 25, 36, 40, 69, 77, 86, 101, 123, 127, 147, 160–161, 171, 209, 236, 241, 247, 287, 291, 301, 337, 352, 361, 381, 402, 409, 443, 449, 453, 471, 502, 507, 511, 559, 568, 573, 593, 606, 621 Write Gridded Responses, 160–161 Hands-On Lab Angles in Polygons, 403 Combine Transformations, 415 Construct Bisectors and Congruent Angles, 382–383 Construct Scale Drawings and Scale Models, 256–257 Explore Compound Interest, 302 Explore Rational Numbers, 64–65 Explore the Pythagorean Theorem, 204 Explore Three-Dimensional Figures, 478–479 Explore Volume of Prisms, 484 Find Surface Areas of Prisms and Cylinders, 496–497 Find Volumes of Prisms and Cylinders, 484 Identify and Construct Altitudes, 397 Make a Scale Model, 257–258 Model Equations with Variables on Both Sides, 128 Model Polynomial Addition, 602 Model Polynomial Subtraction, 607 Model Polynomials, 594–595 Model Two-Step Equations, 41–42 Multiply Binomials, 616–617 Nets of Cones, 503 Nets of Prisms and Cylinders, 496 Hawaiian alphabet, 287 Health, 19, 81, 540, 615 Heart rate maximum, 615 target, 615 Height of parallelograms, 435–436 slant, 504 using indirect measurement to find, 248–249 using scales and scale drawings to find, 512–513 Helpful Hint, 11, 18, 32, 34, 38, 44, 67, 117, 120, 129, 130, 137, 140, 180, 193, 205, 207, 232, 285, 406, 410, 435, 480, 508, 618 Hemispheres, 508 Heptagons, SB16 Hexagons, 403, SB16 Hill, A. V., 621 Histograms, SB20 History, 187, 208, 209, 483 Hobbies, 46, 122, 183, 195, 227, 336 Home Economics, 329 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 8, 12, 16, 20, 24 Hot Tip!, 57, 59, 111, 161, 163, 219, 267, 269, 317, 371, 471, 525, 583, 585, 631 Hurricanes, 325, 545 Hypotenuse, 205 Ice Hotel, 17 Identifying irrational numbers, 200–201 proportions, 232–233 right triangles, 206–207 Identity Property of Addition, 32 Identity Property of Multiplication, 37 Image(s), 410 anamorphic, 498 Impossible event, 556 Improper fractions, SB9 Inch, SB15 Increase, percent of, 294 Independent events, 569–571 finding probability of, 569–570 Independent variable, 344 Indirect measurement, 248–249 Inductive reasoning, SB24 Industrial supplies, 21 Inequalities, 136–137 algebraic, 136 compound, 137 graphing, 137, 140–141, 144 solving, by adding or subtracting, 140–141 by multiplying or dividing, 144–145 translating word phrases into, 136 two-step, solving, 148–149 writing, 136, 141 compound, 137 Input, 326 Integer exponents, 172–173 Integers, 14 absolute value, 15 addition of, 18–19 comparing, 14 division of, 26–27 multiplication of, 26–27 and order of operations, 27 ordering, 14–15 subtraction of, 22–23 Interest, 303 compound, see Compound interest rate of, 303 simple, 303–304 Interpreting circle graphs, 447 graphics, 529 graphs, 353–354 Interstate highway system, 47 Inverse operations, 32 Inverse Property of Multiplication, 82 Inverse additive, 18 multiplicative, 82 Investment time, 303 Ions, 36 Irrational numbers, 200 Irregular figures, area of, 458–459 Isolating the variable, 32 Isosceles triangles, 393, SB17 It’s in the Bag! A Worthwhile Wallet, 261 Canister Carry-All, 105 Graphing Tri-Fold, 365 It’s a Wrap, 213 Note-Taking Taking Shape, 51 Origami Percents, 311 Perfectly Packaged Perimeters, 465 Picture Envelopes, 155 Polynomial Petals, 625 Probability Post-Up, 577 Project CD Geometry, 423 The Tube Journal, 519 Journal, math, keeping a, 321 Jupiter, 186 Kasparov, Garry, 195 Kente cloth, 421 Kilobyte, 179 Kilogram, SB15 Kiloliter, SB15 Kilometer, SB15 Kilowatt-hour, 39 Kite(s), SB17 Krill, 175 Kwan, Michelle, 410 Index I7 Lab Resources Online, 41, 64, 128, 189, 204, 382, 478, 484, 496, 503, 530, 547, 553, 594, 602, 607, 616 Landscaping, 455 Language Arts, 143, 287 Lateral area, 498 Lateral faces, 498 Lateral surface, 499 LCD (least common denominator), 70 LCM (least common multiple), 70 Leaf, 533 Least common denominator (LCD), 70, 87–88 Least common multiple (LCM), 70, SB6 Lee, Harper, 297 Leg, of a triangle, 205 LEGOLAND, 515 Length customary units of, SB15 metric units of, SB15 Lessons, reading, for understanding, 115 Life Science, 73, 101, 171, 174, 175, 187, 240, 241, 253, 289, 290, 295, 361, 489, 505, 511, 568, 598, 621 Lift, 443 Light bulb filament, 329 Light sticks, 281 Like denominators, adding and subtracting with, 74–75 Like terms, 120, 596 solving equations with, 124–125 Lincoln, Abraham, 88 Line graphs, SB19 Line plots, 532 Line segments, 378 Line(s), 378 of best fit, 549 graphing, 330–331 parallel, 384–385 perpendicular, 384–385 points and planes and angles and, 378–380 of reflection, 410 segments, 378 skew, 384–385 slope of a, 349–351 transversals to, 389 Linear equations, 330 graphing, 330–331 Linear functions, 330 graphing, 330–331 Link Animals, 81 Architecture, 255, 493 Art, 247, 419, 599 Business, 606 Economics, 21, 301 Energy, 77 Entertainment, 9 I8 Index Environment, 333 Games, 195 Health, 236 History, 209, 483 Home Economics, 329 Language Arts, 143 Life Science, 101, 171, 175, 187, 289, 361, 489, 505, 511, 568, 621 Literature, 297 Meteorology, 73 Money, 307 Recreation, 88, 515 Science, 17, 29, 91, 133, 175, 281 Social Studies, 25, 47, 291, 414, 591 Sports, 127, 541 Weather, 325 Liquid mirror, 335 Liter, SB15 Literature, 297 Long division, 83, SB7 Louvre Pyramid, 245, 493 Lower quartile, 542–543 Luxor Hotel, 352 Magic squares, 212 Making circle graphs, SB21 conjectures, SB24 scale models, 257–258 Mammoth Lakes, CA, 374 Manufacturing, 280, 352 Map, concept, 433 Mars, 187 Mass, metric units of, SB16 Mastering the Standards, 58–59, 110–111, 162–163, 218–219, 268–269, 316–317, 372–373, 428–429, 472–473, 524–525, 584–585, 630–631, see also Assessment Math translating between words and, 63 translating words into, 10–11 Math expressions translating, into word phrases, 11 translating word phrases into, 10 Math journals, keeping, 321 Matrushka dolls, 86 Maximum, 543 Mean, 537–538 Measurement, 69, 90 conversion tables, SB15 customary system of, 237–239 indirect, 248–249 metric system of, 237–239, SB15 Measures of central tendency and range, 537–538 Median, 537–538 Menkaure Pyramid, 209 Merced River, 112 Mercury (planet), 35, 186 Metamorphoses, 419 Meteorites, 535 Meteorology, 73 Meter, SB15 Methane, 333 Metric conversions, 237, SB15 Metric system of measurement, SB15 converting between customary and, SB15 Midpoint, 393 Mile, SB15 Milligram, SB15 Milliliter, SB15 Millimeter, SB15 Minimum, 543 Mirror, liquid, 335 Mixed numbers, addition of, 87–88 dividing fractions and, 82–84 equivalent fractions and, SB9 multiplying fractions and, 78–79 subtraction of, 87 Mode, 537 Modeling equations with variables on both sides, 128 polynomial addition, 602 polynomial subtraction, 607 polynomials, 594–595 two-step equations, 41–42 Models scale, 252–253, 259, see also Scale models Mona Lisa, 247 Money, 38, 307 Monomials, 180 division of, 180–181 multiplication of, 180 multiplication of polynomials by, 612–613 raising, to powers, 181 square roots of, 193 Monterey Bay Aquarium, 240 Mountain bikes, 541 Multiple, least common (LCM), 70, SB6 Multiple Choice Multiple Choice test items are found in every lesson. Some examples: 9, 13, 17, 21, 25 Answering Context-Based Test Items, 582–583 Eliminate Answer Choices, 56–57 Multiplication Associative Property of, 116–117 of binomials, 616–617, 618–619 of decimals, 79, 94 of fractions and mixed numbers, 78–79 of integers, 26–27 of monomials, 180 of numbers in scientific notation, 189 of polynomials, by monomials, 612–613 of powers, 176 by powers of ten, 184, SB7 properties, 38 of rational numbers, 78–79 solving equations by, 37–39 solving inequalities by, 144–145 Multiplication Property of Equality, 38 Multiplicative inverse, 82 Multi-Step, 39, 80, 90, 91, 97, 199, 209, 231, 287, 296, 438, 440, 502 Multi-Step Application, 439, 619 Multi-step equations solving, 124–125, 128–131 Muscle contractions, 621 Music, 40, 449, 487 Mutually exclusive events, 566 Negative correlation, 549 Negative exponents, 172–173 Negative slope, 349 Neptune, 187 Nets of cones, 503 of prisms and cylinders, 496–497 Newborns, 274 Newtons (N), 34 n-gons, SB16 Niagara Falls, 91 Nickels, 275 Nielsen Television Ratings, 283 No correlation, 549 Nonagons, SB16 Notation scientific, see Scientific notation set-builder, 142 Note-Taking Strategies, see Reading and Writing Math Notes, taking effective, 167 Number line, 18 Numbers compatible, 278 composite, SB5 division of, in scientific notation, 189 finding, when percents are known, 288–289 irrational, 200 mixed, SB9, see also Mixed numbers multiplication of, in scientific notation, 189 percents of, 283–284 prime, SB5 rational, 60–111, 200–201, see also Rational numbers real, 200–201 Numerator, 66 Numerical expressions, 6–7 Nutrition, 39, 96, 141, 356 Obtuse angles, 379 Obtuse triangles, 392, SB17 Ocean trenches, 29 Octagons, SB16 Of Mice and Men (Steinbeck), 143 One-step equations solving, with decimals, 94 solving, with fractions, 94–95 solving, with integers, 32–34, 37–38 solving, with rational numbers, 94–95 Online Resources Chapter Project Online, 2, 60, 112, 164, 220, 270, 318, 374, 474, 526, 586 Game Time Extra, 104, 154, 212, 260, 310, 364, 422, 464, 518, 576, 624 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 8, 12, 16, 20, 24 Lab Resources Online, 41, 64, 128, 189, 204, 382, 478, 484, 496, 530, 547, 553, 594, 602, 607, 616 Parent Resources Online Parent Resources Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 8, 12, 16, 20, 24 Standards Practice Online, 58, 110, 162, 218, 268, 316, 372, 428, 472, 524, 584, 630 Web Extra!, 25, 101, 195, 209, 236, 255, 291, 419, 443, 483, 563, 599, 606 Open circle, 137 Operations inverse, 32 order of, 6, 23–24, 27 Opposites, 14 Order of operations, 6, SB14 using, 169, 173 Ordered pairs, 322 Ordering measurements, customary and metric, 237 rational numbers, 71 Organizing data, 532–534, SB18–21 Origami, 311 Origin, 322 Ounce, fluid, SB15 Outcomes, 556 equally likely, 564 Outlier, 538–539 Output, 326 Pacific Wheel, 430, 453 Pairs, ordered, 322 Pandas, 6 Parabola, 334 Parallel lines, 388–389 properties of transversals to, 389 and skew lines, 384–385 Parallelograms, 399, SB17 area of, 435–436 perimeter of, 434–436 Parent Resources Online Parent Resources Online are available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 8, 12, 16, 20, 24 Parentheses, 6 Parthenon, 483 Patterns, 174, 286 in integer exponents, looking for, 172–173 Pediment, 445 Pei, I. M., 493 Peirsol, Aaron, 74 PEMDAS mnemonic, 6 Pentagons, 403, SB16 Percent problems, solving, 283–285, 288–289 Percent(s) applications of, 298–299 of change, 294 of decrease, 294–295 defined, 274 estimating with, 278–280 finding, 283–285 using an equation, 283, 284 using a proportion, 283, 285 fractions and decimals and, 274–275 of increase, 294–295 known, finding numbers for, 288–289 of numbers, 283–284 Perfect squares, 192 Perimeter, 196, 434 of parallelograms, 434–436 of rectangles, 434–436 of trapezoids, 439–441 of triangles, 439–441 Periscopes, 391 Perpendicular bisectors, 382 Perpendicular lines, 384–385, 388–389 Person-day, 39 Person-hour, 38 Physics, 591 Pi (), 203, 450 Pint, SB15 Pixels, 193 Place value, SB2 Planes, 378 points and lines and angles and, 378–380 Pluto, 187 Points, 378 on the coordinate plane, 322–323 graphing, 14, 322–323 Index I9 lines and planes and angles and, 378–380 plotting, 323 Polygons, 399 angles in, 392–393, 403 classifying, 399, SB16–17 regular, SB16 Polyhedrons, 480–481 Polynomials classifying, 590–591 addition of, 603–604 modeling, 602 defined, 590–591 degrees of, 591 modeling, 594–595 multiplication of, by monomials, 612–613 simplifying, 596–597 subtraction of, 608–609 modeling, 607 Population(s), SB22 samples and, SB22 Positive correlation, 549 Positive slope, 349 Pound, SB15 Powers, 168 division of, 176–177 multiplication of, 176 of products, 181 raising monomials to, 181 raising powers to, 177 and roots, 192 simplifying, 168–169 of ten, 184 zero, 173 Preparing for your final exam, 589 Price, unit, 229 Prime factorization, SB5 Prime numbers, SB5 Principal, 303 Principal square root, 192 Prisms, 480, 485 lateral faces, 498 naming, 480 nets of, 496 rectangular, 485, see also Rectangular prism surface area of, 498–500 triangular, 485 volume of, 485–487 exploring, 484 Probability defined, 556–558 of dependent events, finding, 570–571 of disjoint events, 566 experimental, 560–561 of independent events, finding, 569–570 theoretical, 564–566 Problem Solving Problem solving is a central focus of this course and is found throughout this book. Problem Solving Application, 34, 84, 98–99, 145, 206, 238, 249, 613 I10 Index Problems, reading, for understanding, 273 Production costs, 586 Profit, 298–299 Properties Addition, of Equality, 33 Associative, 603 of Addition, 116–117 of Multiplication, 116–117 Commutative of Addition, 116–117 of Multiplication, 116–117 Density, of real numbers, 201 Distributive, 117 Division, of Equality, 37 Identity of Addition, 32 of Multiplication, 37 Inverse of Multiplication, 82 Multiplication, of Equality, 38 of circles, 446–447, 450–451 of exponents, 176–177 Subtraction, of Equality, 33 Proportions, 232–234 identifying, 232 and indirect measurement, 248 and percent, 283 ratios and, 232 similar figures and, 244 solving, 233–234 using, to find scales, 252 writing, 233–234 Protractors, 382–383 Punnett squares, 568 Pyramid of Khafre, 209 Pyramids, 480, 490 naming, 480 regular, 504 stone, 483 surface area of, 504–505 volume of, 490–491 Pythagorean Theorem, 205–207 and area, 439 converse of the, 206–207 exploring the, 204 Quadrants, 322 Quadratic functions, 334–335 Quadrilaterals angles of, 403, SB16 classifying, 399 Quart, SB15 Quartiles, 542–543 Radical symbol, 192 Radius, 446, 450 Raising monomials to powers, 181 powers to powers, 177 Random samples, SB22 Range, 537 Rate of change, 344–345 slope and, 344–345 Rate of interest, 303 Rates, 228–229 commission, 298 of interest, 303 unit, 228–229, see also Unit rates Rational numbers, 60–111, 200–201 addition of, 74–75 comparing, 70–71 decimal expansion of, 64–65 defined, 66 division of, 82–84 exploring, 64–65 multiplication of, 78–79 ordering, 71 solving equations with, 94–95, 98–99 subtraction of, 74–75 Ratios, 224–225 comparing, 225, 232 equivalent, 224–225, 232 and proportions, 232 writing, in simplest form, 224 Rays, 378 Reading circle graphs, 77 graphics, 529 lessons for understanding, 115 numbers in scientific notation, 185 problems for understanding, 273 Reading and Writing Math, 5, 63, 115, 167, 223, 273, 321, 377, 433, 477, 529, 589 Reading Math, 82, 137, 168, 177, 252, 274, 349, 379, 411, 440, 446, 486, 513 Reading Strategies, see also Reading and Writing Math Interpret Graphics, 529 Read a Lesson for Understanding, 115 Read Problems for Understanding, 273 Use Your Book for Success, 5 Ready to Go On?, 30, 48, 92, 102, 134, 152, 190, 210, 242, 258, 292, 308, 342, 362, 404, 420, 444, 462, 494, 516, 554, 574, 600, 622, see also Assessment Real numbers, 200–201 Density Property of, 201 Reasoning Reasoning is a central focus of this course and is found throughout this book. Some examples: 13, 16, 17, 29, 39, 40, 45, 69, 71, 72, 76, 83, 90, 95, 98, 119, 122, 127, 142, 143, 150, 171, 174, 183, 195, 199, 208, 226, SB24 deductive, SB24 inductive, SB24 proportional, 220–269 Reciprocals, 82 Recreation, 40, 79, 88, 126, 196, 284, 515 Rectangles, 399, SB17 area of, 435–436 perimeter of, 434–436 Rectangular prism, 485 volume of, 485–487 exploring, 484 Rectangular pyramid, 490 Recycling, 225 Reduction, 253 Reflection(s), 410 line of, 410 translations and rotations and, 410–412 Regular polygons, SB16 Regular pyramids, 504 Regular tessellations, 416 Relating decimals, fractions, and percents, 274–275 Relationships, angle, 388–389 Remember!, 6, 14, 67, 70, 75, 94, 121, 125, 144, 148, 149, 181, 275, 345, 450, 480 Repeating decimals defined, 66 exploring, 64–65 period of, 65 writing, as fractions, 67 Representations of data, see also Displaying and organizing data multiple, using, 377 Reptiles, 361 Reptiles, (M.C. Escher), 419 Reticulated python, 289 Rhombuses, SB17 Right angles, 379 Right cones, 504 Right triangles, 392, SB17 finding angles in, 392 finding lengths of sides in, 205–206 identifying, 206–207 Rincon Park, 220 Rise, 345–346 Roots square, 192–193 square, estimating, 196–197 Rotation(s), 410 center of, 410, 412 translations and reflections and, 410–412 Rounding decimals, SB2 whole numbers, SB2 Rules, divisibility, SB4 Run, 345–346 Safety, 352, 561 Sales tax, 298–299 Sample(s), SB22 biased, SB22 populations and, SB22 random, SB22 surveys and, SB22 Sample spaces, 556 San Diego, 270 San Francisco, CA, 220 Santa Rosa, CA, 240 Scale, 252 Scale drawings, 252 constructing, 256 Scale factors, 253 Scale models, 252–253, 259 constructing, 256–257 Scalene triangles, 393, SB17 Scaling three-dimensional figures, 512–513 Scatter plots, 548–550 creating, 553 School, 72, 80, 149, 234 Science, 7, 17, 28, 29, 36, 91, 96, 127, 133, 187, 199, 226, 228, 281, 289, 297, 336, 507 Scientific notation, 184–186 comparing numbers in, 186 division of numbers in, 189 multiplication of numbers in, 189 reading numbers in, 185 writing numbers in, 184–185 Sections, 463 Sector, 447 Segments, line, 378 Selected Answers, SA1–SA14 Semiregular tessellations, 418 Set-builder notation, 142 Short Response, 13, 29, 97, 151, 179, 188, 195, 227, 251, 255, 266–267, 333, 356, 396, 414, 419, 438, 470, 471, 563, 599, 615 Write Short Responses, 266–267 Sides, corresponding, 406–407 Silos, 488 Similar figures, 244 finding missing measures in, 245 and indirect measurement, 248–249 and scale drawings and models, 252–253 proportions and, 244 three-dimensional surface area of, 513 volume of, 513 using, 248–249 Similar polygons, 244–245 Simple interest, 303–304 Simplest form, 67 writing ratios in, 224 Simplifying algebraic expressions, 120–121 negative exponents, 172–173 polynomials, 596–597 powers, 168–169, 172–173 Skew lines, 384 Skills Bank, SB2–SB24 Slant height, 504 Slide, see Translations Slope, 345–346, 398 of a line, 349–351 rates of change and, 344–345 Snakes, 101 Snowboard half-pipe, 502 Social Studies, 25, 35, 47, 71, 86, 146, 188, 282, 286, 291, 396, 438, 489, 491, 507 Solid figures, see Three-dimensional figures Solution set, 136 Solutions, 32 of equations, 32 Solving equations, see Solving equations inequalities, see Solving inequalities multi-step equations, 124–125, 129–131 percent problems, 283–285, 288–289 proportions, 232–234 two-step equations, 43–45, 98–99 Solving equations by addition, 32–34 with decimals, 94, 98–99 by division, 37–38 with fractions, 94–95, 99 linear, 330–331 by multiplication, 37–38 multi-step, 124–125, 129–131 with rational numbers, 94–95, 98–99 by subtraction, 32–34 two-step, 43–45, 98–99 using addition and subtraction properties, 32–34 using multiplication and division properties, 37–38 with variables on both sides, 129–131 Solving inequalities, by adding or subtracting, 140–141 by multiplying or dividing, 144–145 two-step, 148–149 Special products, 619 Speed, 228–229, 234, 238, 239 Spheres, 508–509 surface area of, 509 volume of, 508 Spiral Standards Review Spiral Standards Review questions are found in every lesson. Some examples: 9, 13, 17, 21, 25 Sports, 14, 27, 38, 68, 72, 74, 76, 123, 127, 151, 194, 240, 281, 328, 336, 453, 541 Spreadsheets using, to construct graphs, 530–531 Square roots approximating, to the nearest hundredth, 197 using a calculator, 197 estimating, 196–197 of monomials, 193 principal, 192 squares and, 192–193 Square units, 485 Square(s), 192, 399, SB17 area of, 435 Index I11 magic, 212 perfect, 192 square roots and, 192–193 Standard form, 185 Standards Practice Online, 58, 110, 162, 218, 268, 316, 372, 428, 472, 524, 584, 630 Stanford Linear Accelerator Center, 164 Statisticians, 270 Steinbeck, John, 143 Stem, 533 Stem-and-leaf plots, 533 back-to-back, 533–534 Step Pyramid of King Zoser, 507 Straight angles, 379 Strategies for Success, see also Assessment Any Type: Using a Graphic, 470–471 Extended Response: Write Extended Responses, 370–371 Gridded Response: Write Gridded Responses, 160–161 Multiple Choice Answering Context-Based Test Items, 582–583 Eliminate Answer Choices, 56–57 Short Responses, 266–267 Study for a Final Exam, 589 Study Guide: Review, 52–54, 106–108, 156–158, 214–216, 262–264, 312–314, 366–368, 424–426, 466–468, 520–522, 578–580, 626–628, see also Assessment Study Strategies, see also Reading and Writing Math Concept Map, 433 Study for a Final Exam, 589 Take Effective Notes, 167 Subscripts, 440 Substitute, 6 Subtraction of decimals, 74 of fractions, 75 with unlike denominators, 87–89 of integers, 22–23 of mixed numbers, 87–89 of polynomials, 608–609 modeling, 607 of rational numbers, 74–75 solving equations by, 32–34 solving inequalities by, 140–141 Subtraction Property of Equality, 33 Supplementary angles, 379 Surface area, 498 of cones, 504–505 exploring, 503 of cylinders, 498–500 exploring, 496–497 of figures built of cubes, 498 of prisms, 498–500 exploring, 496–497 of pyramids, 504–505 I12 Index of similar three-dimensional figures, 512–513 of spheres, 509 Surface, lateral, 499 Surveys, samples and, SB22 Systems of equations, 131 solving, 131 writing, 131 Systolic blood pressure, 236 Tables frequency, SB20 functions and, 326, 327, 330, 331, 334, 335, 338, 339 graphs and, 330, 331, 334, 335, 338, 339 Taiwan, 188 Taking effective notes, 167 Target heart rate, 615 Tax, sales, 298 Tax brackets, 301 Technology Lab Make a Box-and-Whisker Plot, 547 Make a Scatter Plot, 553 Multiply and Divide Numbers in Scientific Notation, 189 Use a Spreadsheet to Make Graphs, 530–531 Television Ratings, Nielsen, 283 Temperature conversions, 103 global, 73 scales, 103, 331 Terabyte, 179 Terminating decimals defined, 66 exploring, 64–65 writing, as fractions, 67 Terms, 120 like, 120 not like, 596 Tessellations, 416–417 regular, 416 semiregular, 418 Test, cumulative, studying for a, 589 Test items, context-based, answering, 582–583 The Grapes of Wrath (Steinbeck), 143 Theorem, Pythagorean, see Pythagorean Theorem Theoretical probability, 564–566 Think and Discuss Think and Discuss is found in every lesson. Some examples: 7, 11, 15, 19, 23 Three-dimensional figures classifying, 481 drawing, 477 exploring, 478–479 introduction to, 480–481 scaling, 512–513 surface area of, 496, 509 volume of, 484–487, 508 Tides, 28 Tiles, algebra, 41–42, 128, 594–595, 602, 607, 616–617 Time, investment, 303 Timeline, 25 Tips, 279 To Kill a Mockingbird (Lee), 297 Ton, SB15 Torus, 518 Toys, 606 Transamerica Pyramid, 492 Transformations, 410 combining, 415 graphing, 410–412 Translating math expressions into word phrases, 11 sentences into two-step equations, 43 word phrases into inequalities, 136 word phrases into math expressions, 10 between words and math, 63 Translations, 410 rotations and reflections and, 410–412 Transportation, 240, 241, 593 Transversals, 389 to parallel lines, properties of, 389 Trapezoids, 399, SB17 area of, 440–441 isosceles, SB17 perimeter of, 439 Travel, 125, 126 Tree diagrams, SB23 Trenches, ocean, 29 Trial, 556 Triangle Sum Theorem, 392 Triangles, 392–394, SB16–SB17 acute, 392, SB17 angles in, 392–394 area of, 440–441 classifying, 392–393 congruent, 406 equilateral, 393, SB17 isosceles, 393, SB17 obtuse, 392, SB17 perimeter of, 439 right, 392, SB17, see also Right triangles scalene, 393, SB17 similar, 244–245 Triangular prism, 485 Triangular pyramid, 490 Trinomials, 590 Trump Tower, 257 Turns, see Rotation(s) Twins, 406 Two-dimensional figures, 430–473 Two-step equations modeling, 41–42 solving, 43–45 solving, with rational numbers, 98–99 Two-step inequalities, solving, 148–149 Umbra, 507 Undefined slope, 350 Understanding reading lessons for, 115 reading problems for, 273 Unisphere, 483 Unit(s) conversion factors, 237 customary, SB15 metric, SB15 Unit analysis, 237 Unit conversion factor, 237 Unit fractions, 104 Unit price, 229 Unit rates, 228–229 estimating, 229 rates and, 228–229 United States census, 291 Unlike denominators addition of fractions with, 87–89 subtraction of fractions with, 87–89 Unpacking the Standards, 4, 62, 114, 166, 222, 272, 320, 376, 432, 476, 528, 588 Upper quartile, 542–543 Use your book for success, 5 Use your own words, 377 Using formulas, 477 graphics, 470–471 nets to build prisms and cylinders, 496–497 similar figures, 248–249 slopes, 350 spreadsheets to construct graphs, 530–531 technology to make graphs, 530–531 your book for success, 5 your own words, 377 Value, absolute, 15 Variable(s) on both sides modeling equations with, 128 solving equations with, 129–131 expressions and, 6–7 Variable rate of change, 344 Variation, direct, 357–359 Venn diagrams, 200 Venus, 187 Vertex of an angle, 379 of a cone, 481 of a polygon, 399–400 of a polyhedron, 480 of a three-dimensional figure, 480 Vertical angles, 388 Vertical line test, 327 Volume, 485 of cones, 490–491 of cylinders, 485–487 of prisms, 485–487 exploring, 484 of pyramids, 490–491 of similar three-dimensional figures, 512–513 of spheres, 508 Weather, 119, 325 Web Extra!, 25, 101, 195, 209, 236, 255, 291, 419, 443, 483, 493, 563, 599, 606, Weight, customary units of, SB15 Whales, 175 What’s the Error?, 13, 21, 36, 69, 73, 81, 97, 119, 123, 127, 139, 147, 179, 183, 195, 203, 227, 231, 241, 277, 325, 352, 391, 453, 489, 493, 546, 559, 593 What’s the Question?, 307, 329, 333, 402, 541, 615 Wheel, 446 Whole numbers comparing and ordering, SB3 dividing, SB7 long division, SB7 rounding, SB2 Word phrases translating, into inequalities, 136 translating, into math expressions, 10 translating math expressions into, 11 Words into math, translating, 136 and math, translating between, 63 use your own, 377 Write a Problem, 40, 77, 143, 151, 188, 208, 247, 251, 282, 361, 409, 507, 573, 606 Write About It Write About It exercises are found in every lesson. Some examples: 9, 13, 17, 21, 25 Writing convincing arguments, 223 extended responses, 370–371 gridded responses, 160–161 inequalities, 136, 141 compound, 137 numbers in scientific notation, 184–185 numbers in standard form, 185 proportions, 232 short responses, 266–267 Writing Math, 66, 192, 388 Writing Strategies, see also Reading and Writing Math Draw Three-Dimensional Figures, 477 Keep a Math Journal, 321 Translate Between Words and Math, 63 Use Your Own Words, 377 Write a Convincing Argument, 223 x-axis, 322 x-coordinate, 322 Yard, SB15 y-axis, 322 y-coordinate, 322 Yosemite National Park, 88, 112, 252 Zero power, 173 Zero slope, 349 Index I13 Credits Staff Credits Bruce Albrecht, Margaret Chalmers, Tica Chitrarachis, Lorraine Cooper, Marc Cooper, Jennifer Craycraft, Martize Cross, Nina Degollado, Julie Dervin, Michelle Dike, Lydia Doty, Sam Dudgeon, Kelli R. Flanagan, Stephanie Friedman, Jeff Galvez, Pam Garner, Diannia Green, Jennifer Gribble, Liz Huckestein, Jevara Jackson, Simon Key, Jane A. Kirschman, Kadonna Knape, Cathy Kuhles, Jill M. Lawson, Liann Lech, Virginia Messler, Susan Mussey, Kim Nguyen, Nathan O’Neal, Manda Reid, Michael Rinella, Annette Saunders, Kay Selke, Robyn Setzen, Patricia Sinnott, Victoria Smith, Dawn Marie Spinozza, Jeannie Taylor, Karen Vigil, Kira J. Watkins, Sherri Whitmarsh, David W. Wynn Photo Credits Frontmatter: viii © Gary Crabbe/Enlightened Images; ix © Ed Young/CORBIS; x © Dr. Eric Chalker/Index Stock Imagery, Inc.; xi © Peter Ginter/Bilderberg; xii Jenny Thomas/HRW; xiii © Ron Vesely/MLB Photos via Getty Images; xiv © John Kelly/Getty Images; xv © VEER/Christopher Talbot Frank/Getty Images; xvi © Richard Cummins/CORBIS; xvii © Robert Landau/CORBIS; xviii © Court Mast/Marling Mast/Getty Images; xix © Armando Arorizo/epa/Corbis Chapter One: 2-3 (bkgd), © Gary Crabbe/Enlightened Images; 9 (l), The Granger Collection, New York; 9 (r), The Kobal Collection; 10 Robert Landau/CORBIS; 14 Don Couch/HRW; 17 © Layne Kennedy/CORBIS; 18 Victoria Smith/HRW; 21 Peter Van Steen; 25 (tc), Araldo de Luca/CORBIS; 25 (tl), Steve Vidler/SuperStock; 25 (br), The Art Archive/Napoleonic Museum Rome/Dagli Orti; 25 (tr), Bettmann/ CORBIS; 26 Dennis MacDonald/PhotoEdit Inc.; 29 (l), Peter David/Getty Images; 34 Sam Dudgeon/HRW; 49 (t), iStockphoto; 49 (b), Sam Dudgeon/HRW; 50 Randall Hyman/HRW; 51 Sam Dudgeon/HRW Chapter Two: 60-61 (bkgd), © Ed Young/CORBIS; 70 Getty Images; 73 NASA; 77 © Lester Lefkowitz/CORBIS; 78 Sam Dudgeon/HRW; 81 John Giustina/Bruce Coleman, Inc.; 86 © Mark Tomalty/Masterfile; 88 Library of Congress; 91 (t), © Lester Lefkowitz/CORBIS; 91 (b), Sam Dudgeon/HRW; 93 (b), © Dean Conger/CORBIS; 98 Eric Gaillard/Reuters/CORBIS; 101 (t), © Torsten Blackwood/AFP/Getty Images; 101 (b), Karl H. 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