1. Mathematical Properties
A property is a “rule” that is true for all situations.

2. Four Properties
Commutative
Associative
Distributive
Identity

3. Commutative and Associative Properties
Commutative Property means changing the order in which you add or multiply 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 and Multiplication
Order just doesn’t matter…
com- with, alongside, mutual
mute- change
commute: to change with, or exchange with
A x B = B x A
A + B = B + A

5. Commutative Properties
Commutative Property of Addition: Addends may be commuted!
For any numbers a and b , a + b = b + a. =
50 = 50
Commutative Property of Multiplication: Factors may be commuted!
For any numbers a and b , a  b = b  a.
6  8 = 8  6
48 = 48

6. Associative Property of Multiplication and Addition
Associative Property  (a · b) · c = a · (b · c)
Example: (6 · 4) · 3 = 6 · (4 · 3)
Associative Property  (a + b) + c = a + (b + c)
Example: (6 + 4) + 3 = 6 + (4 + 3)
socio-, soci-, socia-: group

7. 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

8. 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:
( ) + ( ) + ( )
(40) + (40) + (40) = 120

9. Commutative and Associative Properties
Commutative and Associative properties can help 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

10. Distributive Property
m(n + p) = mn + mp
m(a + b + c + d)= ma + mb + mc + md
4(3 + 5) = 4x3 + 4x5
4(3 + 5) = 4(3) + 4(5)
4(3) + 4(5) = 4(3+ 5)

11. The Distributive Property
The Distributive Property lets you distribute an outside factor by multiplication to each number inside a set of parentheses, and find the sum or difference of those products you get:
To distribute means to break apart and then dispense evenly.
Sometimes it is faster and easier to break apart a multiplication problem and use the distributive property to solve or simplify the problem using mental math strategies.
Later, you will learn that the distributive property can be used in reverse to “factor out” a factor of an expression.
a(b + c) = ab + ac

12. The Distributive Property
For any numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + ca;
a(b - c) = ab - ac and (b - c)a = ba - ca;
When a number or letter is separated by parentheses and there are no other operation symbols – it means to distribute by multiplying the numbers or variables together.
Find the sum (add) or difference (subtract) of the distributed products.
Notice that it doesn’t matter which side of the expression the letter a is written on. If a = b, then b = a.
If a(b + c) = ab + ac, then ab + ac = a(b + c)

13. Also called the distributive property of multiplication over addition . . .
symbolically:
a (b + c) = a × b + a × c
and pictorially (rectangular array area model): b c a
a × b
a × c

14. 6 × 10 6 × 3 Example: 6 x 13 Use your mental math skills . . .
symbolically:
6 × (10 + 3) = 6 × × 3
and pictorially (rectangular array area model): 10 3
6 × 10
6 × 3 6

15. If you add 0 to any number, the number stays the same.
Identity Properties
If you add 0 to any number, the number stays the same.
a + 0 = a or = 5
If you multiply any number times 1, the number stays the same.
a x 1 = a or 5 x 1 = 5

16. Keys to properties
Distributive - distribute – look for a number outside parentheses to distribute to other values inside parentheses!
Commutative – order of terms changes!
Associative – grouping order – look for grouping to change
Identity – look for something to keep its identity.

17. Identify the property at work:
6(4) + 6(10) = 6(4 + 10)
A · 1 = A
26 + B = B + 26
4 + (5 + H) = (4 + 5) + H
6(Q + 7) = 6Q + 6(7)