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Properties of Rational Numbers

Properties of Rational Numbers. Algebra and Functions 1.3 Simplify Numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative). Math Objective: Understand and distinguish between the commutative and associative properties.

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Properties of Rational Numbers

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  1. Properties of Rational Numbers Algebra and Functions 1.3 Simplify Numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative)

  2. Math Objective:Understand and distinguish between the commutative and associative properties

  3. Five Properties of Rational Numbers • Commutative • Associative • Identity • Inverse • Distributive

  4. The Commutative Property • Background • The word commutative comes from the verb “to commute.” • Definition on dictionary.com • Commuting means changing, replacing, or exchanging • People who travel back and forth to work are called commuters. • Traffic Reports given during rush hours are also called commuter reports.

  5. Here are two families of commuters. Commuter B Commuter A Commuter A & Commuter Bchangedlanes. Remember… commute means to change. Commuter A Commuter B

  6. Home School Would the distance from Home to School and then from school to home change? Home + School = School + Home H + S = S + H A + B = B + A

  7. 3 groupsof5= 5 groups of 3 3 x 5 = 5 x 3 = = 15 kids 15 kids

  8. The Commutative Property A + B = B + A A x B = B x A

  9. The Commutative Property You can add or multiply numbers in any order. It iscalled the commutative property of addition when we add, and the commutative property of multiplication when we multiply.

  10. Five Properties of Rational Numbers • Commutative • Associative • Identity • Inverse • Distributive

  11. The Associative Property • Background • The word associative comes from the verb “to associate.” • Definition on dictionary.com • Associate means connected, joined, or related • People who work together are called associates. • They are joined together by business, and they do talk to one another.

  12. Let’s look at another hypothetical situation Three people work together. Associate B needs to call Associates A and C to share some news. Does it matter who he calls first?

  13. A C B Here are three associates. B calls A first He calls C last If he called C first, then called A, would it have made a difference? NO!

  14. (The Role of Parentheses) • In math, we use parentheses to show groups. • In the order of operations, the numbers and operations in parentheses are done first. (PEMDAS) So….

  15. A C B A C B The Associative Property The parentheses identify which two associates talked first. (A + B) + C = A + (B + C) THEN THEN

  16. ) ( Notice the first two students are associating with each other in the first situation. In the second situation, the same girl is associating with a different student. Have the students changed? Have the students moved places? ) ( =

  17. The Associative Property When adding or multiplying, you can change the grouping of numbers without changing the sum or product. The order of the terms DOES NOT change. It iscalled the associative property of addition when we add, and the associative property of multiplication when we multiply.

  18. Let’s practice ! Look at the problem. Identify which property it represents.

  19. (4 + 3) + 2 = 4 + (3 + 2) The Associative Property of Addition It has parentheses!

  20. 6 • 11 = 11 • 6 The Commutative Property of Multiplication • Same 2 numbers • Numbers switched places

  21. (1 • 2) •3 = 1 • (2 • 3) The Associative Property of Multiplication • Same 3 numbers in the same order • 2 sets of parentheses

  22. a • b = b • a The Commutative Property of Multiplication

  23. A C B (a • b) •c = a • (b • c) The Associative Property of Multiplication

  24. 4 + 6 = 6 + 4 The Commutative Property of Addition Numbers change places.

  25. A C B (a + b) + c = a + (b + c) The Associative Property of Addition Parentheses!

  26. a + b = b + a The Commutative Property of Addition Moving numbers!

  27. Five Properties of Rational Numbers • Commutative • Associative • Identity • Inverse • Distributive

  28. The Identity Property I am me! You cannot change My identity!

  29. Identity Property of Addition Zero is the only number you can add to something and see no change. This property is also sometimes called the Identity Property of Zero.

  30. Identity Property of Addition A + 0 = A + 0 =

  31. Identity Property of Multiplication One is the only number you can multiply by something and see no change. This property is also sometimes called the Identity Property of One.

  32. Identity Property of Multiplication A • 1 = A • 1 =

  33. Five Properties of Rational Numbers • Commutative • Associative • Identity • Inverse • Distributive

  34. Inverse Property Inverse means “opposite”.

  35. Inverse Property The opposite of addition is… subtraction. So, when I use inverse operations, I can “undo” the original number. Example: 3 + (-3)= 0

  36. Inverse Property The opposite of division is… multiplication. So, when I use inverse operations, I can “undo” the original number. Example:

  37. Let’s practice ! Look at the problem. Identify which property it represents.

  38. a • 1 = a The Identity Property of Multiplication

  39. 12 + 0 = 12 The Identity Property of Addition It is the only addition property that has two addends and one of them is a zero.

  40. 987 • 1 = 987 The Identity Property of Multiplication • Times 1

  41. 7 + (- 7) = 0 The Inverse Property • Undo the operation by using the opposite operation

  42. 9 • 1 = 9 The Identity Property of Multiplication • Times 1

  43. 6 = 1 6 The Inverse Property • Undo the operation by using the inverse operation

  44. 3 + 0 = 3 The Identity Property of Addition See the zero?

  45. a + 0 = a The Identity Property of Addition Zero!

  46. Five Properties of Rational Numbers • Commutative • Associative • Identity • Inverse • Distributive

  47. The Distributive Property • Background • The word distributive comes from the verb “to distribute.” • Definition on dictionary.com • Distributing refers to passing things out or delivering things to people

  48. The Distributive Property a(b + c) = (a • b) + (a • c) A times the sum of b and c = a times b plus a times c Let’s plug in some numbers first. Remember that to distribute means delivering items, or handing them out. Here is how this property works: 5(2 + 3) = (5 • 2) + (5 • 3)

  49. You have sold many items for the BMMS fundraiser! You went to five houses. Every family bought 5 items total, 2 red gifts and three green gifts! How many gifts did you deliver all together? 5(2 + 3) = (5 • 2) + (5 • 3) Think: Five groups of (2+3) or (2+3) + (2+3) + (2+3) + (2+3) + (2+3) How many red gifts were distributed? How many green gifts were distributed?

  50. You will be distributing 5 items to each house.

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