2. Commutative and Associative Properties
• Properties refer to rules that indicate a standard
procedure or method to be followed.
• A proof is a demonstration of the truth of a
statement in mathematics.
• Properties or rules in mathematics are the result
from testing the truth or validity of something by
experiment or trial to establish a proof.
• Therefore, every mathematical problem from the
easiest to the more complex can be solved by
following step by step procedures that are
identified as mathematical properties.
3. 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 {}.
4. Commutative Property
of addition - (Order)
Commutative Property
of multiplication -
(order)
For any numbers a and b , a + b = b + a.
For any numbers a and b , a b = b a.
45 + 5 = 5 + 45
6 8 = 8 6
50 = 50
48 = 48
Commutative Properties
5. Associative Property of
addition - (grouping
symbols)
Associative Property of
multiplication -
(grouping symbols)
For any numbers a, b, and c,
(a + b) + c = a + (b + c).
For any numbers a, b, and c,
(ab) c = a (bc).
(2 + 4) + 5 = 2 + (4 + 5)
(2 3) 5 = 2 (3 5)
(6) + 5 = 2 + (9)
11 = 11
(6) 5 = 2 (15)
30 = 30
Associative Properties
6. Evaluate: 18 + 13 + 16 + 27 + 22 + 24
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)
(18 + 22) + (16 + 24) + (13 + 27)
(40) + (40) + (40) = 120
Commutative and Associative Properties
• Commutative and Associative properties are very helpful
to solve problems using mental math strategies.
7. Evaluate: 4 7 25
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)
4 25 7
(4 25) 7
(100) 7 = 700
Commutative and Associative Properties
• Commutative and Associative properties are very helpful
to solve problems using mental math strategies.
