Commutative and Associative Properties

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Commutative and Associative Properties
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Commutative and Associative Properties

• Commutative Property means changing the order in
  which you add or subtract numbers does not change
  the sum or product.
• Associative Property means changing the grouping
  of numbers when adding or multiplying does not
  change their sum or product.
• Grouping symbols are typically parentheses (),but
  can include brackets or Braces .

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Commutative Properties
Commutative Property of addition - (Order)
For any numbers a and b , a b b a.
45 5 5 45
50 50
Commutative Property of multiplication - (order)
For any numbers a and b , a ? b b ? a.
6 ? 8 8 ? 6
48 48
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Commute

• What does the word commute mean? It means to go
  from one place to another.
• Pick a partner. When I say commute, you will
  quickly change seats with your partner.
• When I say commute the 2nd time, you will return
  to you seat.
• Ready.commute. Commute! Commute! Commute!
• What does the commutative property mean to you?

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Associative Properties
Associative Property of addition - (grouping
symbols)
For any numbers a, b, and c, (a b) c a
(b c).
(2 4) 5 2 (4 5)
(6) 5 2 (9)
11 11
Associative Property of multiplication -
(grouping symbols)
For any numbers a, b, and c, (ab) c a (bc).
(2 ? 3) ? 5 2 ? (3 ? 5)
(6) ? 5 2 ? (15)
30 30
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Associate

• To associate with someone means that you are with
  them. To demonstrate associating, I want 3 people
  to come to the front of the room.
• You will use your arms as parentheses.
• The person in the middle will associate with the
  person on the right (put your arm around them).
• Now associate with the person on your left?
• Did the number or placement of the people change?
  What does the associative property mean to you?

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Commutative and Associative Properties

• Commutative and Associative properties are very
  helpful to solve problems using mental math
  strategies.

Rewrite the problem by grouping numbers that can
be formed easily. (Associative property) This
process may change the order in which the
original problem was introduced. (Commutative
property)
Evaluate 18 13 16 27 22 24
(18 22) (16 24) (13 27)
(40) (40) (40) 120
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Commutative and Associative Properties

• Commutative and Associative properties are very
  helpful to solve problems using mental math
  strategies.

Rewrite the problem by changing the order in
which the original problem was introduced.
(Commutative property) Group numbers that can be
formed easily. (Associative property)
Evaluate 4 ? 7 ? 25
4 ? 25 ? 7
(4 ? 25) ? 7
(100) ? 7 700
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Additive Identity

• What does identity mean to you? (Your name?)
• What can I add to a number and it will keep its
  name and value?
• What can I multiply to a number and it will keep
  its name and value?
• O is the additive identity property and 1 is the
  mulitiplicative identity property.
• What does the identity property mean to you?

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Identity Property

• 5 0 5 (5 retains its identity)
• 17 x 1 17 (17 retains its identity)

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Multiplicative Property of Zero

• What can you multiply any number by to get
  zero???
• This is the multiplication property of 0!
• 124 x 0 0
• 0 x 1,890 0

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Multiplicative Identity

• What number can you multiply any number by so
  that it keeps its identity?
• 134 x 134?
• The number is the multiplicative
• identity for all numbers!

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Additive Inverse

• What is the inverse (opposite) of adding?
• What can I add to any number to make zero?? 7
  0
• You can add the inverse of opposite to any number
  to get 0.
• 15 -15 0
• -173 173 0

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Multiplicative Inverse

• What can I multiply to a number to make 1?? 1/3
  x 1
• Multiply the reciprocal or the inverse of the
  number to make 1.
• -4 x 1 (You have to multiply -4 by -1/4 to
  get 0.)

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Distributive Property

• Remember to distribute the number on the outside
  of the parenthesis to everything inside of the
  parenthesis!
• 5(x 3) 12
• 5x 15 12 5x 27

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Properties

• If you dont know the properties you cant
  simplify or solve expressions or equations!!
• Which property do you understand well enough to
  explain to someone else?
• Explain it now and provide an example!

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Algebraic Expressions

• What is the difference between an English phrase
  and a sentence?
• What is the difference between a mathematical
  expression and an equation?
• Since the root word of equation is equal, the
  equations have equal signs and the expressions
  dont!

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Algebraic Expressions

• Use the properties to solve these expressions
  Explain the steps
• 3(w 5) 5w 2
• w 4(y 8)
• 7( x y) 3x 5y
• 2(x 19) 3(x 10)