1. Adding & Subtracting Signed Numbers Dr. Sarah Ledford Mathematics Educator Sarah@SarahLikesMath.com

2. Addition Show the following using your counters: 2 + 3 = We can think of this as “2 flowers plus 3 flowers equal …” OR “2 positives plus 3 positives equal …” We will let the BLUE counters represent POSITIVES.

3. Addition Show the following using your counters: 2 + 3 =

4. Addition Show the following using your counters: 2 + 3 = 5

5. Negatives Next we will look at (−3) + (−1) = This introduces a problem with negative numbers. We could also write it as −3 + −1 = (-3) + (-1) = -3 + -1 = I think the first is the easiest on the eyes in this ppt.

6. Negatives (−3) + (−1) = The parentheses are around the negative number (for now) so that you get used to seeing negative numbers. We could put parentheses around positive numbers, too, so that +2 looks like (2). Eventually, the parentheses around the negative numbers will be dropped & the other versions of writing this problem will be adopted.

7. Negatives In 6 th grade it is VERY important for students to understand that −3 is the OPPOSITE of 3 and that 4 is the OPPOSITE of −4.

8. Addition Show the following using your counters: (−3) + (−1) = We can think of this as *(−3) means that we have “3 negatives” so “3 negatives plus 1 negative equal …” We will let the WHITE counters represent NEGATIVES.

9. Addition Show the following using your counters: (−3) + (−1) =

10. Addition Show the following using your counters: (−3) + (−1) = 4 negatives = −4

11. Addition Observations? When adding numbers of the same sign, …. Add the numbers and keep the sign.

12. Addition Show the following using your counters: 3 + 0 = This shows the “additive identity” because any number plus zero is itself. We can also think of zero as a “neutral” here because it does not change the value in any way.

13. Addition Show the following using your counters: 3 + 0 =

14. Addition Show the following using your counters: 3 + 0 = When working with signed numbers, we also let a “neutral” be represented by “+/− “ because it has no value and does not change the value of the problem in any way. Show this problem with several “neutrals” being added in.

15. Addition Show the following using your counters: 3 + 0 =

16. Addition Show the following using your counters: 1 + (−1) = 2 + (−2) = 3 + (−3) = 4 + (−4) = 5 + (−5) = x + (−x) =

17. Addition Show the following using your counters: 1 + (−1) = 2 + (−2) = 3 + (−3) = Rearrange the counters to show neutrals or zero pairs.

18. Addition Show the following using your counters: 1 + (−1) = 2 + (−2) = 3 + (−3) =

19. Addition Show the following using your counters: 1 + (−1) = 2 + (−2) = 3 + (−3) = 4 + (−4) = 5 + (−5) = x + (−x) = Observations? Anything plus its opposite is zero (The Additive Inverse)

20. Addition Show the following using your counters: 3 + (−2) = We can think of this as “3 positives plus 2 negatives equal …” Remember We are letting the BLUE counters represent POSITIVES & WHITE counters represent NEGATIVES.

21. Addition Show the following using your counters: 3 + (−2) =

22. Addition Show the following using your counters: 3 + (−2) =

23. Addition Show the following using your counters: 3 + (−2) = This problem has a new “issue.” We need to pair every neutral or zero pair (+/−) that we can.

24. Addition Show the following using your counters: 3 + (−2) = We have 2 “neutrals” that represent zero or nothing & 1 blue chip that represents 1 positive so 3 + (−2) = 1

25. Addition Show the following using your counters: 3 + (−5) =

26. Addition Show the following using your counters: 3 + (−5) =

27. Addition Show the following using your counters: 3 + (−5) = We need to pair every neutral or zero pair (+/−) that we can.

28. Addition Show the following using your counters: 3 + (−5) = We have 3 “neutrals” that represent zero or nothing & 2 white chips that represent 2 negatives so 3 + (−5) = −2

29. Addition You try these: (−3) + 2 = (−3) + 5 = (−4) + 2 = 4 + (−3) = 4 + (−6) = (−1) + 5 =

30. Commutative Property Don’t forget the “commutative property of addition!” a + b = b + a 3 + −2 = −2 + 3 3 + −5 = −5 + 3 −3 + 2 = 2 + −3 −3 + 5 = 5 + −3

31. Addition (+) + (+) = (+) (−) + (−) = (−) (+) + (−) = ? (−) + (+) = ? it depends on the numbers!!

32. Addition In your own words? (+) + (+) = (+) (−) + (−) = (−) If signs are the same, add & keep the sign. (+) + (−) = ? (−) + (+) = ? If signs are different, subtract the “smaller” number from the “larger” number & keep the sign of the “larger” number.

33. Number Line Models Addition: 3 + 2 = ?

34. Number Line Models Addition: 3 + 2 = ?

35. Number Line Models Addition: 3 + 2 = ?

36. Number Line Models Addition: 3 + 2 = ? 3 + 2 = 5

37. Number Line Models Addition: 3 + (−2) = ?

38. Number Line Models Addition: 3 + (−2) = ?

39. Number Line Models Addition: 3 + (−2) = ?

40. Number Line Models Addition: 3 + (−2) = ? 3 + (−2) = 1

41. Number Line Models Addition: 2 + (−3) = ?

42. Number Line Models Addition: 2 + (−3) = ?

43. Number Line Models Addition: 2 + (−3) = ?

44. Number Line Models Addition: 2 + (−3) = ? 2 + (−3) = −1

45. Number Line Models Addition: (−2) + 5 = ?

46. Number Line Models Addition: (−2) + 5 = ?

47. Number Line Models Addition: (−2) + 5 = ?

48. Number Line Models Addition: (−2) + 5 = ? (−2) + 5 = 3

49. Subtraction Show the following using your counters: 3 – 2 = We can think of this as “3 flowers take away 2 flowers equal …” OR “3 positives take away 2 positives equal …” OR “3 blue chips take away 2 blue chips equal …”

50. Subtraction Show the following using your counters: 3 – 2 =

51. Subtraction Show the following using your counters: 3 – 2 = 1

52. Subtraction Show the following using your counters: (−4) – (−2) = We can think of this as “4 negatives take away 2 negatives equal …” OR “4 white chips take away 2 white chips equal …”

53. Subtraction Show the following using your counters: (−4) – (−2) =

54. Subtraction Show the following using your counters: (−4) – (−2) = 2 white chips = 2 negatives = −2

55. Subtraction Show the following using your counters: 3 – 5 = We can think of this as “3 positives take away 5 positives equal …” However, we do not have enough positives to take away 5 of them. We can add in neutrals that do not change the value of the problem. Then we can take away 5 positives once we have enough.

56. Subtraction Show the following using your counters: 3 – 5 = We must start with 3 positives. Then add in neutrals or zero pairs.

57. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Still +3

58. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Still +3

59. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Still +3

60. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Still +3

61. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Still +3

62. Subtraction Show the following using your counters: 3 – 5 = What is the value now? Get the idea?

63. Subtraction Show the following using your counters: 3 – 5 = We can take away 5 positives now.

64. Subtraction Show the following using your counters: 3 – 5 = Let’s clear away the neutrals or zero pairs.

65. Subtraction Show the following using your counters: 3 – 5 =

66. Subtraction Show the following using your counters: 3 – 5 = We are left with 2 white chips that represent 2 negatives so 3 – 5 = –2

67. Subtraction Show the following using your counters: 3 – (−1) = We can think of this as “3 positives take away 1 negative equal …” However, we do not have any negatives to take away. We can add in neutrals that do not change the value of the problem. Then we can take away 1 negative once we have it.

68. Subtraction Show the following using your counters: 3 – (−1) = We must start with 3 positives. Then add in neutrals or zero pairs.

69. Subtraction Show the following using your counters: 3 – (−1) = What is the value now? Still +3

70. Subtraction Show the following using your counters: 3 – (−1) = What is the value now? Still +3

71. Subtraction Show the following using your counters: 3 – (−1) = What is the value now? Still +3

72. Subtraction Show the following using your counters: 3 – (−1) = What is the value now? Still +3

73. Subtraction Show the following using your counters: 3 – (−1) = What is the value now? Still +3 Get the idea?

74. Subtraction Show the following using your counters: 3 – (−1) = We can take away 1 negative now.

75. Subtraction Show the following using your counters: 3 – (−1) = Let’s clear away the neutrals or zero pairs.

76. Subtraction Show the following using your counters: 3 – (−1) = We are left with 4 blue chips that represent 4 positives so 3 – (−1) = 4

77. Subtraction Show the following using your counters: (−2) – (−4) = We can think of this as “2 negatives take away 4 negatives equal …” We do not have enough negatives to take away. We can add in neutrals that do not change the value of the problem. Then we can take away 4 negatives once we have enough.

78. Subtraction Show the following using your counters: (−2) – (−4) = We must start with 2 negatives. Then add in neutrals or zero pairs.

79. Subtraction Show the following using your counters: (−2) – (−4) = What is the value now? Still −2

80. Subtraction Show the following using your counters: (−2) – (−4) = What is the value now? Still −2

81. Subtraction Show the following using your counters: (−2) – (−4) = What is the value now? Still −2

82. Subtraction Show the following using your counters: (−2) – (−4) = What is the value now? Still −2 Get the idea?

83. Subtraction Show the following using your counters: (−2) – (−4) = We can take away 4 negatives now.

84. Subtraction Show the following using your counters: (−2) – (−4) = Let’s clear away the neutrals or zero pairs.

85. Subtraction Show the following using your counters: (−2) – (−4) = We are left with 2 blue chips that represent 2 positives so (−2) – (−4) = 2

86. Subtraction Show the following using your counters: (−5) – 2 = We can think of this as “5 negatives take away 2 positives equal …” We do not have any positives to take away. We can add in neutrals that do not change the value of the problem. Then we can take away 2 positives once we have them.

87. Subtraction Show the following using your counters: (−5) – 2 = We must start with 5 negatives. Then add in neutrals or zero pairs. How many neutrals/zero pairs do you think we need to add in?

88. Subtraction Show the following using your counters: (−5) – 2 = What is the value now? Still −5

89. Subtraction Show the following using your counters: (−5) – 2 = What is the value now? Still −5

90. Subtraction Show the following using your counters: (−5) – 2 = We can take away 2 positives now.

91. Subtraction Show the following using your counters: (−5) – 2 = Because we only added in the number of neutrals or zero pairs that we needed, we don’t have to clear any. If you have neutrals or zero pairs, clear them.

92. Subtraction Show the following using your counters: (−5) – 2 = We are left with 7 white chips that represent 7 negatives so (−5) – 2 = −7

93. Subtraction Practice a. (−6) – (−3) = b. (−4) – (−3) = c. 2 – 6 = d. 4 – 8 = e. 7 – (−3) = f. 5 – (−7) = g. (−1) – (−5) = h. (−3) – (−8) = i. (−4) – 1 = j. (−2) – 5 = Show the following using your counters:

94. Practice Compute each pair. What do you notice? a. 3 – 2 = & 3 + (−2) = b. (−4) – (−2) = & (−4) + 2 = c. 3 – 5 = & 3 + (−5) = d. 3 – (−1) = & 3 + 1 = e. 6 – 3 = & 6 + (−3) = f. 2 – 6 = & 2 + (−6) = g. (−3) – 5 = & (−3) + (−5) =

95. Relationship between Subtraction & Addition? Subtracting is the same as adding the opposite