1. Commutative, Associative, Distributive, and Identity Properties
𝐴∗𝐵=𝐵∗𝐴
𝐴+𝐵=𝐵+𝐴
𝐴+ 𝐵+𝐶 = 𝐴+𝐵 +𝐶
Math Properties
𝐴∗1=𝐴
𝐴∗0=0
Commutative, Associative, Distributive, and Identity Properties
𝐴 𝐵+𝐶 =𝐴𝐵+𝐴𝐶
𝐴+0=𝐴

2. What are properties?
In life, there are rules and laws that tell us what we can and can not do. . .
In math, properties are like those laws or rules they tell us what we can and can not do with sets of numbers

3. Commute To commute means to travel from one place to another.
For example, you commute to school in the morning.

4. Commutative Property
Just like you commute from home to school, a number may commute from one spot to another.
a + b = b + a (The numbers change places.)
This is called the commutative property of addition.
Ex) = 3 + 2
Both and equal 5.

5. Commutative Property
The commutative property may be used with addition as seen previously and also with multiplication.
a · b = b · a
Ex) 3 · 5 = 5 · 3
Both 3 · 5 and 5 · 3 equal 15.
This is called the commutative property of multiplication.

6. Associate An associate is a friend or someone you work with.
For example, the head cheerleader is an associate of the school mascot.

7. Suddenly the cheerleader associates with someone else.
Now imagine the football team played a late game and the cheerleader and mascot forgot to study for the math test.
Suddenly the cheerleader associates with someone else.

8. Associative Property A + (B + C) = (A + B) + C or
The associative property is when a number associates with a different number.
A + (B + C) = (A + B) + C or
2 + (6 + 5) = (2 + 6) + 5

9. Associative Property
says that when we ADD or MULTIPLY sets of numbers, how we (GROUP) THEM DOES NOT MATTER because our answer will be the same
Grouping” means putting numbers inside (parentheses)
(A + B) + C = A + (B + C) is called the associative property of addition.
Ex) (2 + 3) + 4 = 2 + (3 + 4)

10. Associative Property
The associative property may be used with addition as seen previously and also with multiplication.
A · (B · C) = (A · B) · C is called the associative property of multiplication.

11. A few notes about the associative and commutative properties. . .
They DO NOT APPLY to subtraction and division because grouping and ordering numbers using those operations DOES CHANGE THE ANSWER
Look at 1, 2, and 3.
Solve: 1 – 2 – 3 =
Then solve: 2 – 1 – 3 =
Are the answers the same?
Who can name the main difference between the two properties?
Answer: The presence of parentheses is always with the associative property!

12. So let’s try a few. . . Name that property!
1) (34 · 24) · 55 = 34 · (24 · 55)
2) (92 + 3) = (3 + 46) + 92
3) 76 · 23 · 1 = 1 · 23 · 76
4) What is the difference between number 2 and 3?

13. Identity Your identity is who you are.
Changing your clothes or getting a new haircut does not change your identity.
Your identity remains the same.

14. Property of Addition A number also has an identity
The identity of a number is the value of the number
The additive identity is the number that when added to another number does not change the identity of the original number
3 + __ = 3 (What goes in the blank?)

15. Zero The additive identity is zero.
We can add zero to any number and the answer is the original number.

16. Identity Property of Multiplication
We also have a multiplicative identity
3 · __ = 3 (What goes in this blank?)
We can multiply any number by one and the answer will be the original number. 1

17. Identity Properties A + 0 = A A · 1 = A Identity Property of Addition
Identity Property of Multiplication
A + 0 = A
A · 1 = A

18. Distribute Distribute means to deliver or pass out
If we distribute food to three boxes, we put food in each of the three boxes

19. Distributive Property
The A is the food and the boxes are B and C.
We pass out A to each of B and C.
In this case that means that we multiply A by both B and C separately and then add the resulting products.

20. = 4 · · 3
Ex) 4(2 + 3)
4 · 5 20 =
= 20

21. Now you try these examples.
5(6 + 3) =
7(2 + 4) =
2(6 -3) =
5 · · 3
7 · · 4
2 · 6 – 2 · 3

22. So let’s try a few. . . Fill in the blank. 5) 4(2+10) = ______________
5) 4(2+10) = ______________
6) 9(12+15) = _____________
7) __________ = 8 · · 17
8) __________ = 12 · · 7

23. Now you try. . . (in class work!)
Identify the property.
9. (12 · 4) · 9 = (4 · 9) · 12 =
11. 3( ) = 3 · · 10
· 88 · 16 = 88 · 45 · 16

24. Now you try. . . (in class work!)
Fill in the blanks then identify the property.
13. ____________ = (87 · 23) · 19
(3 + 5) = ______________
15. _________ = 6 · · 8
= _____________
17. What two operations do the associative and commutative property NOT apply to? Explain why.