- The document discusses the associative properties of addition, subtraction, and multiplication. - It shows that addition and multiplication are associative by working through examples with different groupings, but subtraction is not associative as the examples produce different results depending on grouping. - The key idea is that for operations to be associative, the result must be the same regardless of how the numbers are grouped during calculation.

1. Associative
Property

2. • You are tired of studying and need a break. You need
to make a decision as to which group you want to
associate with during your break.
Image from http://
www.ourcommunity.co
m.au/images/photobank/
web/01543.jpg

3. • You are tired of studying and need a break. You need
to make a decision as to which group you want to
associate with during your break.
Image from http://
www.ourcommunity.co
m.au/images/photobank/
web/01543.jpg
• The blue men
Image from http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489

4. • You are tired of studying and need a break. You need
to make a decision as to which group you want to
associate with during your break.
Image from http://
www.ourcommunity.co
m.au/images/photobank/
web/01543.jpg
• The blue men
• or your friends
Image from http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489 Image from http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

5. • Does the total number of people change if you
choose to go with the blue men
http://www.ourcommunity.com.au/images/photobank/web/01543.jpg
http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489
http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

6. • Does the total number of people change if you
choose to go with the blue men
http://www.ourcommunity.com.au/images/photobank/web/01543.jpg
http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489
http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

7. • or your group of friends?
http://www.ourcommunity.com.au/images/photobank/web/01543.jpg
http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489
http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

8. • or your group of friends?
http://www.ourcommunity.com.au/images/photobank/web/01543.jpg
http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489
http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

9. • The number of people didn’t change but the group
you were associated with changed.
http://www.ourcommunity.com.au/images/photobank/web/01543.jpg
http://api.ning.com/files/3V*P52AJjFy8m-
DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489
http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

10. • The Associative Property has the same idea.
http://www.ourcommunity.com.au/ http://api.ning.com/files/3V*P52AJjFy8m-
images/photobank/web/01543.jpg DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489 http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

11. • The Associative Property has the same idea.
• It doesn’t matter which group you add first,
(1 + 3) + 10 = 1 + ( 3 + 10 )
http://www.ourcommunity.com.au/ http://api.ning.com/files/3V*P52AJjFy8m-
images/photobank/web/01543.jpg DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489 http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

12. • The Associative Property has the same idea.
• It doesn’t matter which group you add first,
(1 + 3) + 10 = 1 + ( 3 + 10 )
( 4 ) + 10 = 1 + (13)
http://www.ourcommunity.com.au/ http://api.ning.com/files/3V*P52AJjFy8m-
images/photobank/web/01543.jpg DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489 http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

13. • The Associative Property has the same idea.
• It doesn’t matter which group you add first,
(1 + 3) + 10 = 1 + ( 3 + 10 )
( 4 ) + 10 = 1 + (13)
14 = 14 • the result is the
same.
http://www.ourcommunity.com.au/ http://api.ning.com/files/3V*P52AJjFy8m-
images/photobank/web/01543.jpg DEAbj2KS6dsW*oJeGYBFqrON95zIwu2GCb-fc8ovg-
zuB0Xitc/BlueManGroup.jpg?width=737&height=489 http://farm4.static.flickr.com/3528/3461253839_12244b14f2.jpg

14. • You can add in any order and get the same answer.

15. • You can add in any order and get the same answer.
• How does this look in Algebra?

16. • You can add in any order and get the same answer.
• How does this look in Algebra?
( a + b ) + c = a + (b + c )

17. • You can add in any order and get the same answer.
• How does this look in Algebra?
( a + b ) + c = a + (b + c )
• Try different numbers for a, b, and c to see this is
true.

18. Is Subtraction Associative?

19. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)

20. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)
(5) − 2

21. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)
(5) − 2
3

22. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)
(5) − 2 8 − (1)
3

23. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)
(5) − 2 8 − (1)
3 7

24. Is Subtraction Associative?
• Try these problems to see what you get:
( 8 − 3) − 2 8 − ( 3 − 2)
(5) − 2 8 − (1)
3 7
• Different answers. This means subtraction is not
associative.

25. Is Multiplication Associative?

26. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?

27. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)

28. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)
• Here are the 2 grouping options for solving....
( 200 ⋅12 ) ⋅ 4 200 ⋅ (12 ⋅ 4 )

29. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)
• Here are the 2 grouping options for solving....
( 200 ⋅12 ) ⋅ 4 200 ⋅ (12 ⋅ 4 )
( 2400 ) ⋅ 4

30. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)
• Here are the 2 grouping options for solving....
( 200 ⋅12 ) ⋅ 4 200 ⋅ (12 ⋅ 4 )
( 2400 ) ⋅ 4
9600

31. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)
• Here are the 2 grouping options for solving....
( 200 ⋅12 ) ⋅ 4 200 ⋅ (12 ⋅ 4 )
( 2400 ) ⋅ 4 200 ⋅ ( 48 )
9600

32. Is Multiplication Associative?
• You are buying your first car but need to finance a portion of the
cost. The monthly payment is $200 a month for 4 years. How
much will you have paid at the end of 4 years?
• Solve using (payment)(months in a year)(number of years)
• Here are the 2 grouping options for solving....
( 200 ⋅12 ) ⋅ 4 200 ⋅ (12 ⋅ 4 )
( 2400 ) ⋅ 4 200 ⋅ ( 48 )
9600 9600

33. • Notice you got the same answer. This means
multiplication is associative.

34. • Notice you got the same answer. This means
multiplication is associative.
• You can do multiplication in any order and
get the same result.

35. • Notice you got the same answer. This means
multiplication is associative.
• You can do multiplication in any order and
get the same result.
• Here’s what it looks like in Algebra:
( xy ) z = x ( yz )

36. Is Division Associative?

37. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )

38. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )
(12 ) ÷ 2

39. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )
(12 ) ÷ 2
6

40. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )
(12 ) ÷ 2 48 ÷ ( 2 )
6

41. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )
(12 ) ÷ 2 48 ÷ ( 2 )
6 24

42. Is Division Associative?
• Try these to see what you get...
( 48 ÷ 4 ) ÷ 2 48 ÷ ( 4 ÷ 2 )
(12 ) ÷ 2 48 ÷ ( 2 )
6 24
• Big different, huh? Division is not associative.

43. Key points to remember...

44. Key points to remember...
• Addition and Multiplication are associative.

45. Key points to remember...
• Addition and Multiplication are associative.
• Subtraction and Division are not associative.

46. Key points to remember...
• Addition and Multiplication are associative.
• Subtraction and Division are not associative.
• Parenthesis are necessary when entering in a
calculator so it know which part to evaluate first.