Understanding Properties of Addition and Multiplication in Mathematics

1. Multiplication Properties Lesson 2-1

2. Do you remember these Properties of Addition? • Commutative Property of Addition • The numbers move around • a + b = b + a • Associative Property of Addition • Grouping with parentheses • (a + b) + c = a + (b + c) • Identity Property of Addition • The identity of the problem does not change • a + 0 = a

3. In multiplication, you will see these same properties, plus 2 more…

4. Five Properties of Multiplication • These are the basically the same as addition • Commutative • Associative • Identity • These belong to multiplication only • Zero • Distributive

5. Let’s review the addition properties— from the multiplicative perspective… Multiplicative (Do you see most of the word “multiply” in this word?

6. Property #1 The Commutative Property of Multiplication

7. The Commutative Property • Background • The word commutative comes from the verb “to commute.” • Definition on dictionary.com • Commuting means changing, replacing, exchanging, switching places, trading places • People who travel back and forth to work are called commuters.

8. Here are two families of commuters. Hi! Remember us? Commuter B Commuter A Commuter A & Commuter Bchangedlanes. Remember… commute means to switch places. Commuter A Commuter B

9. The Commutative Property A • B = B • A

10. Here is another example…

11. 3 groups of 5 = 5 groups of 3 3 x 5 = 5 x 3 = = 15 kids 15 kids

12. Remember… in Lesson 1-11 we saidthat the word “of” means multiply What commutative means to multiplication… 3 groups of 5 = 5 groups of 3 3 • 5 = 5 • 3 a • b = b • a

13. Property #2 The Associative Property of Multiplication

14. The Associative Property • Background • The word associative comes from the verb “to associate.” • Definition on dictionary.com • Associate means connected, joined, or related • People who work together are called associates. • They are joined together by business, and they have to talk to one another.

15. Let’s look at another hypothetical situation Three people work together. Associate B needs to call Associates A and C to share some news. Does it matter who he calls first?

16. A C B NO! Here are three associates. B calls A first He calls C last If he called C first, then called A, would it have made a difference?

17. (The Role of Parentheses) • In math, we use parentheses to show groups. • In the order of operations, the numbers and operations in parentheses are done first. (PEMDAS) So….

18. A C B A C B The Associative Property The parentheses identify which two associates talked first. (A  B) C = A (B  C) THEN THEN

19. Property #3 The Identity Property of Multiplication

20. The Identity Property I am me! You cannot change My identity!

21. One is the only number you can multiply something by and see no change.

22. Identity Property of Multiplication a x 1 = a x 1 =

23. Identity Property of Multiplication a x 1 = a x 1 = x 1 = x 1 =

24. These are 3 of the Properties of Multiplication • Commutative Property of Multiplication • The numbers move around • a •b = b • a • Associative Property of Multiplication • Grouping with parentheses • (a • b) • c = a • (b • c) • Identity Property of Multiplication • The identity of the problem does not change • a •1 = a

25. There are two more properties which are unique to multiplication • The Zero Property • The Distributive Property

26. Property #4 The Zero Property of Multiplication

27. The Zero Property of Multiplication • This looks like a mixture of the identity property of addition and the identity property of multiplication… • Be careful not to mix them up!

28. The Zero Property • Any time you multiply a number by zero, your answer is zero! If I have 2 pockets with NO money in them, then I have NO money! 2 • 0 = 0 The End

29. Property #5 The Distributive Property of Multiplication

30. The Distributive Property • Background • The word distributive comes from the verb “to distribute.” • Definition on dictionary.com • Distributing refers to passing things out or delivering things to people

31. The Distributive Property a(b + c) = (a • b) + (a • c) A times the sum of b and c = a times b plus a times c Let’s plug in some numbers first. Remember that to distribute means delivering items, or handing them out. Here is how this property works: 5(2 + 3) = (5 • 2) + (5 • 3)

32. You have sold many items for the RCMS fundraiser! You went to two houses on one street and three houses on a different street. Every family bought 5 items! 5(2 + 3) = (5 • 2) + (5 • 3) You went to two houses on one street and three houses on a different street. Every family bought 5 items!

33. You will be distributing 5 items to each house. 2 3 5 4 1

34. 5(2 + 3) = (5 • 2) + (5 • 3) You distributed (delivered) these all in one trip. There are (2+3) five houses all together. You need to deliver 5 gifts to each house. You need to put 25 items on your wagon at one time. 5 items x 5 houses = 25 items all together

35. 5(2 + 3) = (5 • 2) + (5 • 3) and 10 You distributed your items in two trips (+). On the first trip you distributed 5 items to each of 2 houses (5 x 2 = 10). On the second trip you distributed 5 items to each of 3 houses (5 x 3 = 15). That means you distributed (delivered) 10 items plus 15 items. That makes 25 items altogether. + 15 25

36. The Distributive Property Make 1 trip. You have 5 houses. You need to bring 5 items to each house. You need 25 items on your wagon. DISTRIBUTION CENTER 5(2 + 3)

37. The Distributive Property Make 2 trips. You have 2 houses for your first trip and you need to bring 5 items to each house. You have 3 houses on your second trip and need to bring 5 items to each house. When your second trip is over, you will have distributed 25 items. DISTRIBUTION CENTER (5 • 2) + (5 •3)

38. How do I tell the properties apart? • Commutative • Numbers switch places • Associative • Parentheses on both sides • Only multiplication on each side • Identity • Multiply by 1 • Zero Property • Multiply by zero • Distributive • Parentheses on each side • One side has a multiplication sign AND a plus sign

39. Let’s practice ! Look at the problem. Identify which property it represents.

40. 4(5 + 6) = (4 • 5) + (4 • 6) The Distributive Property of Multiplication • 3 numbers on one side—4 on the other • Multiplication AND addition • 3 sets of parentheses

41. 987 • 1 = 987 The Identity Property of Multiplication • Times 1

42. 3 • 0 = 0 Zero Property of Multiplication • Times zero

43. (1 • 2) •3 = 1 • (2 • 3) The Associative Property of Multiplication • Same 3 numbers • Multiplication only • 2 sets of parentheses

44. 6 • 11 = 11 • 6 The Commutative Property of Multiplication • Same 2 numbers • Numbers switched places

45. 9 • 7 = 7 • 9 The Commutative Property of Multiplication • Same 2 numbers • Numbers switched places

46. 12 • 0 = 0 Zero Property of Multiplication • Times zero

47. (9 • 8) •7 = 9 • (8 • 7) The Associative Property of Multiplication • Same 3 numbers • Multiplication only • 2 sets of parentheses

48. 9(8 + 7) = (9 • 8) + (9 • 7) The Distributive Property of Multiplication • 3 numbers on one side—4 on the other • Multiplication AND addition • 3 sets of parentheses

49. 9 • 1 = 9 The Identity Property of Multiplication • Times 1

50. a • 1 = a The Identity Property of Multiplication