1. Multiplying & Dividing Signed Numbers Dr. Sarah Ledford Mathematics Educator Sarah@SarahLikesMath.com

2. Multiplication Show the following using your counters: 2 x 3 = We can think of this as “2 groups of 3 flowers equal …” Or “2 groups of 3 positives equal …” How many positives do you have altogether? This is not new… Just multiplication facts from 3 rd grade. ;)

3. Multiplication Show the following using your counters: 2 x 3 = 2 groups of 3 positives = 6 blue positives 2 x 3 = 6

4. Multiplication Show the following using your counters: 2 x (−3) = We can think of this as “2 groups of 3 negatives equal …”

5. Multiplication Show the following using your counters: 2 x (−3) = How many negatives do you have altogether? 6 negatives 2 x (−3) = −6

6. Multiplication Show the following using your counters: 5 x (−2) = We can think of this as “5 groups of 2 negatives equal …”

7. Multiplication Show the following using your counters: 5 x (−2) = How many negatives do you have altogether? 10 negatives 5 x (−2) = −10

8. Multiplication Show the following using your counters: 4 x (−1) = We can think of this as “4 groups of 1 negative equal …”

9. Multiplication Show the following using your counters: 4 x (−1) = How many negatives do you have altogether? 4 negatives 4 x (−1) = −4

10. Multiplication 2 x (−3) = −6 5 x (−2) = −10 4 x (−1) = −4 3 x (−3) = …

11. Multiplication 2 x (−3) = −6 5 x (−2) = −10 4 x (−1) = −4 3 x (−3) = −9 4 x (−2) = …

12. Multiplication 2 x (−3) = −6 5 x (−2) = −10 4 x (−1) = −4 3 x (−3) = −9 4 x (−2) = −8 2 x (−2) = …

13. Multiplication 2 x (−3) = −6 5 x (−2) = −10 4 x (−1) = −4 3 x (−3) = −9 4 x (−2) = −8 2 x (−2) = −4 Observations?

14. Multiplication So far, we have (+) x (+) = (+) (+) x (−) = (−)

15. Commutative Property of Multiplication a x b = b x a 2 x (−3) = (−3) x 2 = −6 5 x (−2) = (−2) x 5 = −10 4 x (−1) = (−1) x 4 = −4 3 x (−3) = (−3) x 3 = −9 4 x (−2) = (−2) x 4 = −8 2 x (−2) = (−2) x 2 = −4 Observations?

16. Multiplication So far, we have (+) x (+) = (+) (+) x (−) = (−) (−) x (+) = (−) What other options do we need to consider? (−) x (−) = ?

17. Multiplication Use a calculator to compute the following. (−2) x (−3) = …

18. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = …

19. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = 10 (−4) x (−1) = …

20. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = 10 (−4) x (−1) = 4 (−3) x (−3) = …

21. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = 10 (−4) x (−1) = 4 (−3) x (−3) = 9 (−4) x (−2) = …

22. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = 10 (−4) x (−1) = 4 (−3) x (−3) = 9 (−4) x (−2) = 8 (−2) x (−2) = …

23. Multiplication Use a calculator to compute the following. (−2) x (−3) = 6 (−5) x (−2) = 10 (−4) x (−1) = 4 (−3) x (−3) = 9 (−4) x (−2) = 8 (−2) x (−2) = 4 Observations?

24. Multiplication (+) x (+) = (+) (+) x (−) = (−) (−) x (+) = (−) (−) x (−) = ? In your own words?

25. Multiplication When multiplying two numbers with the same sign, the product is positive. When multiplying two numbers with different signs, the product is negative. Why did I italicize two above?

26. Number Line Models Multiplication: 2 x 3 = ?

27. Number Line Models Multiplication: 2 x 3 = ?

28. Number Line Models Multiplication: 2 x 3 = ?

29. Number Line Models Multiplication: 2 x 3 = ? 2 x 3 = 6

30. Number Line Models Multiplication: 5 x 2 = ?

31. Number Line Models Multiplication: 5 x 2 = ?

32. Number Line Models Multiplication: 5 x 2 = ?

33. Number Line Models Multiplication: 5 x 2 = ?

34. Number Line Models Multiplication: 5 x 2 = ?

35. Number Line Models Multiplication: 5 x 2 = ?

36. Number Line Models Multiplication: 5 x 2 = ? 5 x 2 = 10

37. Number Line Models Multiplication: 2 x (−3) = ?

38. Number Line Models Multiplication: 2 x (−3) = ?

39. Number Line Models Multiplication: 2 x (−3) = ?

40. Number Line Models Multiplication: 2 x (−3) = ? 2 x (−3) = −6

41. Division Show the following using your counters: 6 ÷ 2 = 3 We can think of this as “6 flowers divided into 2 groups equal 3 flowers in each group” Or “6 flowers divided into groups of 2 flowers in each group equal 3 groups”

42. Division Show the following using your counters: 6 ÷ 2 = 3 We can think of this as “6 positives divided into 2 groups equal 3 positives in each group” or “6 positives divided into groups of 2 positives in each group equal 3 groups”

43. Division Show the following using your counters: 6 ÷ 2 =

44. Division Show the following using your counters: 6 ÷ 2 = 6 positives divided into 2 groups equal 3 positives in each group

45. Division Show the following using your counters: 6 ÷ 2 = 6 positives divided into groups of 2 positives in each group equal 3 groups

46. Division Show the following using your counters: (−6) ÷ 2 = … We can think of this as “6 negatives divided into 2 groups equal … negatives in each group”

47. Division Show the following using your counters: (−6) ÷ 2 = … (−6) ÷ 2 = (−3)

48. Division Show the following using your counters: (−6) ÷ (−2) = … We can think of this as “6 negatives divided into groups of 2 negatives in each group equal … groups”

49. Division Show the following using your counters: (−6) ÷ (−2) = … (−6) ÷ (−2) = 3

50. Division Unfortunately, we can’t use this logic for 6 ÷ (−2) = … “6 positives divided into (−2) groups equal … in each group” “6 positives divided into groups of 2 negatives in each group equal … groups” Neither of these make sense.

51. Division Another way to think about division… Division is the inverse of multiplication. We can discover some rules for division using the manipulatives like we have been doing. Or we can use this idea of inverses. 6 ÷ 2 = …  … x 2 = 6 6 ÷ 2 = …  3 x 2 = 6 6 ÷ 2 = 3  3 x 2 = 6

52. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = …

53. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = …  … x (−2) = (−6)

54. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = …  3 x (−2) = (−6)

55. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = …

56. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = …  … x (−4) = (−8)

57. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = …  2 x (−4) = (−8)

58. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = …

59. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = …  … x (−3) = (−6)

60. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = …  2 x (−3) = (−6)

61. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = …

62. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = …  … x (−2) = (−8)

63. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = …  4 x (−2) = (−8)

64. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = 4  4 x (−2) = (−8) (−4) ÷ (−2) = …

65. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = 4  4 x (−2) = (−8) (−4) ÷ (−2) = …  … x (−2) = (−4)

66. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = 4  4 x (−2) = (−8) (−4) ÷ (−2) = …  2 x (−2) = (−4)

67. Division 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ (−2) = 3  3 x (−2) = (−6) (−8) ÷ (−4) = 2  2 x (−4) = (−8) (−6) ÷ (−3) = 2  2 x (−3) = (−6) (−8) ÷ (−2) = 4  4 x (−2) = (−8) (−4) ÷ (−2) = 2  2 x (−2) = (−4) Observations?

68. Division (+) ÷ (+) = (+) (−) ÷ (−) = (+) When dividing two signed numbers, if the signs are the same, then the quotient is positive. Same rule as for multiplication!! Make a conjecture: When dividing two signed numbers, if the signs are different, then...

69. Division We can develop the other rules of division using inverses as we just did. 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ 2 = …

70. Division We can develop the other rules of division using inverses as we just did. 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ 2 = …  … x 2 = (−6)

71. Division We can develop the other rules of division using inverses as we just did. 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ 2 = …  (−3) x 2 = (−6)

72. Division We can develop the other rules of division using inverses as we just did. 6 ÷ 2 = 3  3 x 2 = 6 (−6) ÷ 2 = (−3)  (−3) x 2 = (−6)

73. Division We can develop the other rules of division using inverses as we just did. OR we can do a lot of examples on a calculator really fast!!

74. Division 10 ÷ (−2) = … 7 ÷ (−1) = … 4 ÷ (−2) = … 50 ÷ (−2) = … 9 ÷ (−3) = … 15 ÷ (−5) = … 100 ÷ (−10) = … (−45) ÷ 9 = … (−18) ÷ 6 = … (−6) ÷ 3 = … (−25) ÷ 5 = … (−21) ÷ 7 = … (−24) ÷ 12 = … (−48) ÷ 8 = …

75. Division 10 ÷ (−2) = (−5) 7 ÷ (−1) = (−7) 4 ÷ (−2) = (−2) 50 ÷ (−2) = (−25) 9 ÷ (−3) = (−3) 15 ÷ (−5) = (−3) 100 ÷ (−10) = (−10) (−45) ÷ 9 = (−5) (−18) ÷ 6 = (−3) (−6) ÷ 3 = (−2) (−25) ÷ 5 = (−5) (−21) ÷ 7 = (−3) (−24) ÷ 12 = (−2) (−48) ÷ 8 = (−6)

76. Division (+) ÷ (+) = (+) (−) ÷ (−) = (+) When dividing two signed numbers, if the signs are the same, then the quotient is positive. (+) ÷ (−) = (−) (−) ÷ (+) = (−) When dividing two signed numbers, if the signs are different, then the quotient is negative.