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Lesson 1.6

Lesson 1.6. Properties of Real Numbers. California Standards. 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.

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Lesson 1.6

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  1. Lesson 1.6 Properties of Real Numbers

  2. California Standards 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. Also covered: 25.1

  3. Words to Know Associate- to group or join together. Commute- to change locations. Distribute- to pass out or to give out shares

  4. Commutative Property… of Addition of Multiplication

  5. Let’s Try It…Commutative Property 7 + 3 7 + 3 10 10 = = 7 3 7 3 . . 21 21 = =

  6. Associative Property… of Addition of Multiplication

  7. Let’s Try It…Associative Property ( ) ( ) 7 + 3 + 2 7 + 3 + 2 7 + 5 12 10 + 2 12 = = ( ) ( ) 7 3 2 7 3 2 . . . . 42 7 6 = . 21 2 42 . =

  8. 7 – 3 = 4 NOT EQUAL 3 – 7 = - 4 (9 – 2) – 3 = 4 NOT EQUAL 9 – (2 – 3) = 10 8 ÷ 4 = 2 NOT EQUAL 4 ÷ 8 = 0.5 8 ÷ (4 ÷ 2) = 4 NOT EQUAL (8 ÷ 4) ÷ 2 = 1 Does It Work With Subtraction or Division?

  9. Name the Property . The grouping is different. A. 7(mn) = (7m)n Associative Property of Multiplication The grouping is different. B. (a + 3) + b = a + (3 + b) Associative Property of Addition The order is different. C. x + (y + z) = x + (z + y) Commutative Property of Addition

  10. Name the Property The order is different. a. n + (–7) = –7 + n Commutative Property of Addition b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3 The grouping is different. Associative Property of Addition The order is different. c. (xy)z = (yx)z Commutative Property of Multiplication

  11. Distributive Property

  12. Let’s try it … ( ) a b + c = + a c a b

  13. Write each product using the Distributive Property. Then simplify. A. 5(71) 5(71) = 5(70 + 1) Rewrite 71 as 70 + 1. = 5(70)+5(1) Use the Distributive Property. = 350 + 5 Multiply (mentally). = 355 Add (mentally). B. 4(38) Rewrite 38 as 40 – 2. 4(38) = 4(40 – 2) Use the Distributive Property. = 4(40) – 4(2) Multiply (mentally). = 160 – 8 = 152 Subtract (mentally).

  14. Now You Try 1. 2(3x – 5) = 2(3x) – 2(5)= =6x – 10 2. -4(y + 9) = -4(y) + -4(9) = -4y - 36

  15. Now You Try (8 – x)3 = 3(8) – 3(x) = = 24 – 3x -4(12 + x - y) = (-4)(12) + (-4)(x) – (-4)(y) = -48 - 4x + 4y

  16. Use Mental Math to Evaluate the Following (9 + 14) + 1 = 10 + 14 = = 24 191 + 12 + 9 = 200 + 12 = = 212

  17. Compare and Contrast Commutative Property Associative Property

  18. A set of numbers is said to be closed, or to have closure, under an operation if the result of the operation on any two numbers in the set is also in the set. Ex: {-1, 0, 1}; the set is closed under multiplication (-1)(0) = 0 (-1)(1) = -1 (0)(1) = 0 yes

  19. Finding Counterexamples to Statements About Closure Find a counterexample to show that each statement is false. A. The prime numbers are closed under addition. Find two prime numbers, a and b, such that their sum is not a prime number. Try a = 3 and b = 5. a + b = 3 + 5 = 8 Since 8 is not a prime number, this is a counterexample. The statement is false.

  20. Lesson Quiz: Part I Name the property that is illustrated in each equation. 1. 6(rs) = (6r)s Associative Property of Multiplication 2. (3 + n) + p = (n +3) + p Commutative Property of Addition 3. (3 + n) + p = 3 + (n + p) Associative Property of Addition 4. Find a counterexample to disprove the statement “The Commutative Property is true for division.” Possible answer: 3 ÷ 6 ≠ 6 ÷ 3

  21. Lesson Quiz: Part II Write each product using the Distributive Property. Then simplify. 5. 8(21) 8(20) + 8(1) = 168 6. 5(97) 5(100) – 5(3) = 485 Find a counterexample to show that each statement is false. 7. The natural numbers are closed under subtraction. Possible answer: 6 and 8 are natural, but 6 – 8 = –2, which is not natural. 8. The set of even numbers is closed under division. Possible answer: 12 and 4 are even, but 12 ÷ 4 = 3, which is not even.

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