1-4 Learn to use number properties to compute mentally. Vocabulary Commutative Property Associative Property Distributive Property Properties and Mental Math Mental math means “doing math in your head.” Shakuntala Devi is extremely good at mental math. When she was asked to multiply 7,686,369,774,870 by 2,465,099,745,779, she took only 28 seconds to multiply the numbers mentally and gave the correct answer of 18,947,668,177,995,426,462,773,730! Many mental math strategies use number properties that you already know to make equivalent expressions that may be easier to simplify. COMMUTATIVE PROPERTY (Ordering) Words Numbers 18 9 9 18 You can add or multiply numbers in any order. The Commutative and Associative Properties do not apply to subtraction or division. EXAMPLE ASSOCIATIVE PROPERTY (Grouping) 1 Words Numbers When you are only adding or only multiplying, you can group any of the numbers together. ( 17 2 ) 9 17 ( 2 9 ) ( 12 2 ) 4 12 ( 2 4 ) Using Properties to Add and Multiply Whole Numbers A 22 15 2 2 15 Simplify 12 4 18 46. 12 4 18 46 Look for sums that are multiples of 10. 12 18 4 46 ( 12 18 ) ( 4 46 ) 50 30 80 Use the Commutative Property. Chapter 1 Whole Numbers and Patterns Use the Associative Property to make groups of compatible numbers. Use mental math to add. Lesson Tutorials Online my.hrw.com B Simplify 5 12 2. 5 12 2 12 5 2 12 ( 5 2 ) 12 10 The expressions in each row of Example 1B are equivalent expressions because they have the same value. Look for products that are multiples of 10. Use the Commutative Property. Use the Associative Property to group compatible numbers. 120 Use mental math to multiply. To multiply a number by a sum, such as 6 (10 4), you can use the order of operations, or you can use the Distributive Property. DISTRIBUTIVE PROPERTY Words Numbers To multiply a number by a sum, multiply by each number in the sum and then add. 6 (10 4) (6 10) (6 4) 60 24 84 When you multiply two numbers, you can “break apart” one of the numbers into a sum and then use the Distributive Property. EXAMPLE 2 Using the Distributive Property to Multiply Use the Distributive Property to find each product. A B 4 23 4 23 4 ( 20 3 ) ( 4 20 ) ( 4 3 ) 80 12 92 8 74 8 74 8 (70 4) (8 70) (8 4) 560 32 592 “Break apart” 23 into 20 3. Use the Distributive Property. Use mental math to multiply. Use mental math to add. “Break apart” 74 into 70 4. Use the Distributive Property. Use mental math to multiply. Use mental math to add. Think and Discuss 1. Give examples of the Commutative Property and the Associative Property. 2. Name some situations in which you might use mental math. Lesson Tutorials Online my.hrw.com 1-4 Properties and Mental Math 23 1-4 Exercises Homework Help Online go.hrw.com, Go MR10 1-4 Exercises 1–30, 31, 39, 41, 45, 51 GUIDED PRACTICE See Example 1 See Example 2 Simplify. 1. 13 ⴙ 9 ⴙ 7 ⴙ 11 2. 19 ⴙ 18 ⴙ 11 ⴙ 32 3. 25 ⴙ 7 ⴙ 13 ⴙ 5 4. 5 ⴛ 14 ⴛ 4 5. 4 ⴛ 16 ⴛ 5 6. 5 ⴛ 17 ⴛ 2 Use the Distributive Property to find each product. 7. 5 ⴛ 24 8. 8 ⴛ 52 9. 4 ⴛ 39 10. 6 ⴛ 14 11. 3 ⴛ 33 12. 2 ⴛ 78 13. 9 ⴛ 12 14. 2 ⴛ 87 INDEPENDENT PRACTICE See Example 1 See Example 2 Simplify. 15. 15 ⴙ 17 ⴙ 3 ⴙ 5 16. 14 ⴙ 7 ⴙ 16 ⴙ 13 17. 6 ⴙ 21 ⴙ 14 ⴙ 9 18. 5 ⴛ 25 ⴛ 2 19. 2 ⴛ 32 ⴛ 10 20. 6 ⴛ 12 ⴛ 5 Use the Distributive Property to find each product. 21. 3 ⴛ 36 22. 4 ⴛ 42 23. 6 ⴛ 71 24. 2 ⴛ 94 25. 6 ⴛ 23 26. 5 ⴛ 25 27. 6 ⴛ 62 28. 7 ⴛ 21 29. 8 ⴛ 41 30. 2 ⴛ 94 PRACTICE AND PROBLEM SOLVING Extra Practice See page EP2. Use mental math to find each sum or product. 31. 8 ⴙ 13 ⴙ 7 ⴙ 12 32. 2 ⴛ 25 ⴛ 4 33. 4 ⴙ 22 ⴙ 16 ⴙ 18 34. 5 ⴛ 8 ⴛ 12 35. 5 ⴙ 98 ⴙ 95 36. 6 ⴛ 5 ⴛ 14 37. 11 ⴙ 75 ⴙ 25 38. 8 ⴛ 11 ⴛ 5 39. 19 ⴙ 1 ⴙ 11 ⴙ 39 40. Paul is writing a story for the school newspaper about the landscaping done by his class. The students planted 15 vines, 12 hedges, 8 trees, and 35 flowering plants. How many plants were used in the project? 41. Earth Science The temperature on Sunday was 58 °F. The temperature is predicted to rise 4 °F on Monday, then rise 2 °F more on Tuesday, and then rise another 6 °F by Saturday. What is the predicted temperature on Saturday? 42. Multi-Step Janice wants to order disks for her computer. She needs to find the total cost, including shipping and handling. If Janice orders 7 disks, what will her total cost be? 24 Chapter 1 Whole Numbers and Patterns Description Computer Disk Number 7 Unit Cost with Tax Price $24.00 Shipping & Handling Total $7.00 Multiply using the Distributive Property. 44. 4 27 43. 9 17 Life Science 45. 11 18 46. 7 51 Determine if each pair of expressions is equivalent. If so, state the property shown. 47. 8 35 and (8 30)(8 5) 48. 14 7 6 and 6 14 7 49. (9 7) 6 and 9 (7 6) 50. 24 6 2 and 6 24 2 51. Life Science Poison-dart frogs can breed underwater, and the females lay from 4 to 30 eggs. What would be the total number of eggs if four female poison-dart frogs each laid 27 eggs? Poison-dart frogs are members of the family Dendrobatidae, which includes about 170 species. Many are brightly colored. Use the table for Exercises 52 and 53. 52. Rickie wants to buy 3 garden hoses. How much will they cost? Home Center Clearance Sale 53. The boys in Josh’s family are saving money to buy 4 ceiling fans. How much will they need to save? Table lamp $15 Garden hose $16 Ceiling fan $52 54. Critical Thinking Give a problem that you could simplify using the Commutative and Associative Properties. Then, show the steps to solve the problem and label the Commutative and Associative Properties. 55. What’s The Error? A student wrote 5 24 25 6 5 25 24 6 by the Associative Property. What error did the student make? 56. Write About It Why can you simplify 5( 50 3 ) using the Distributive Property? Why can’t you simplify 5( 50 ) 3 using the Distributive Property? 57. Challenge Explain how you could find the product of 52 112 using the Distributive Property. Simplify the expression. Test Prep and Spiral Review 58. Multiple Choice Which expression does NOT have the same value as 7 ( 4 23 )? A 7 27 B ( 7 4 ) ( 7 23 ) C 7 4 23 D 28 ( 7 23 ) 59. Gridded Response Michelle flew 1,240 miles from Los Angeles to Dallas, and another 718 miles from Dallas to Atlanta. From Atlanta, she flew 760 miles to New York City. How many miles did Michelle fly in all? Estimate each sum or difference by rounding to the nearest thousands place. (Lesson 1-1) 60. 5,237 1,586 61. 915,178 451,836 62. 39,187 24,999 Simplify each expression. (Lesson 1-3) 63. 4 14 12 2 64. 16 42 15 2 65. 62 14 (5 4) 1-4 Properties and Mental Math 25