2. ADDITION AND SUBTRACTION of INTEGERS The University of Texas at Dallas

3. There are three ways to think of adding and subtracting with integers: Use Algebra Tiles Use a Number Line Use Mathematical Rules We will go over all three methods. You may use any method – all will give the correct answer.

4. Algebra Tiles

5. ADDING INTEGERS We can model integer addition with tiles. Represent -2 with the fewest number of tiles Represent +5 with the fewest number of tiles. The University of Texas at Dallas

6. ADDING INTEGERS What number is represented by combining the 2 groups of tiles? Write the number sentence that is illustrated. -2 + + 5 = + 3 The University of Texas at Dallas +3+3

7. ADDING INTEGERS Use your red and yellow tiles to find each sum. -2 + -3 = ? The University of Texas at Dallas ANSWER + = -5 -2 + -3 = - 5

8. ADDING INTEGERS The University of Texas at Dallas - 6 + + 2 = ? + = - 4 + = +1+1 - 3 + + 4 = ? ANSWER - 6 + + 2 = - 4 - 3 + + 4 = + 1

9. INTRODUCTION TO INTEGERS The diagrams below show 2 ways to represent -3. Represent -3 in 2 more ways. The University of Texas at Dallas -3 NOTE TO TUTOR

10. SUBTRACTING INTEGERS The University of Texas at Dallas We can’t take away 5 yellow tiles from this diagram. There is not enough tiles to take away!! This diagram also represents +3, and we can take away +5. When we take 5 yellow tiles away, we have 2 red tiles left. We often think of subtraction as a “take away” operation. Which diagram could be used to compute + 3 - + 5 = ?

11. SUBTRACTING INTEGERS Use your red and yellow tiles to model each subtraction problem. The University of Texas at Dallas - 2 - - 4 = ? ANSWER

12. SUBTRACTING INTEGERS The University of Texas at Dallas This representation of - 2 doesn’t have enough tiles to take away - 4. Now if you add 2 more reds tiles and 2 more yellow tiles (adding zero) you would have a total of 4 red tiles and the tiles still represent - 2. Now you can take away 4 red tiles. - 2 - - 4 = + 2 2 yellow tiles are left, so the answer is…

13. SUBTRACTING INTEGERS The University of Texas at Dallas Work this problem. + 3 - - 5 = ? ANSWER

14. SUBTRACTING INTEGERS The University of Texas at Dallas Add enough red and yellow pairs so you can take away 5 red tiles. Take away 5 red tiles, you have 8 yellow tiles left. + 3 - - 5 = + 8

15. SUBTRACTING INTEGERS The University of Texas at Dallas Work this problem. - 3 - + 2 = ? ANSWER

16. SUBTRACTING INTEGERS The University of Texas at Dallas Add two pairs of red and yellow tiles so you can take away 2 yellow tiles. Take away 2 yellow tiles, you have 5 red tiles left. - 3 - + 2 = - 5

17. Number Line

18. Addition and Subtraction of Integers In this tutorial, we will learn how to add and subtract signed numbers with the help of a toy car. The line the car sitting on is called the number line, where the positive numbers are on the right and the negative numbers are on the left.

19. Addition and Subtraction of Integers The car will stick to the following rules: 1. It always starts at 0 (its home) facing right. 2. If it sees a positive number, it moves forward. 3. If it sees a negative number, it backs up. 4. If it sees an addition sign, it continues to read the next number. 5. If it sees a subtraction sign, it turns around and then continues to read the next number. Click to see the first example.

20. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right (click to see animation)

21. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right

22. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right

23. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right.

24. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3. Next the car will move 4 more units forward because it sees the number 4. (click to see next animation.)

25. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3. Next the car will move 4 more units forward because it sees the number 4. (click to see next animation.)

26. Example 1: 2 + 4 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3. Next the car will move 4 more units forward because it sees the number 4. Since the car now stops at 6, the answer to 2 + 4 is 6. (click to see the next example)

27. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign. (Click to see animation)

28. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign.

29. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign.

30. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign. 3.Next it will move forward by 5 units. (click to see animation)

31. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign. 3.Next it will move forward by 5 units.

32. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign. 3.Next it will move forward by 5 units.

33. Example 2: (-2 ) + 5 1.Our car starts from 0 facing right. 2.It then backs up 2 units (to the left) because it sees the - sign. 3.Next it will move forward by 5 units. Now it stops at +3, therefore the answer to (-2) + 5 is 3. Please go to the next tutorial for subtractions.

34. Subtraction 1.There is a big difference between addition and subtraction. 2.In addition, our car is always facing right, because that is the positive direction, 3.but in subtraction, the car has to turn around (180 deg) first. Click when you are ready.

35. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. (Click to start animation)

36. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units.

37. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units.

38. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units.

39. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –. Click to see animation.

40. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –.

41. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –.

42. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –. 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –. 4.Finally it will move forward (to the left) by 3 units. click to see animation.

43. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –. 4.Finally it will move forward (to the left) by 3 units.

44. Example 3: 5 – 3 1.Our car still starts at 0 facing right. 2.It then moves forward 5 units. 3.Next it will turn around because it sees the subtraction symbol –. 4.Finally it will move forward (to the left) by 3 units. Since the car stops at 2, the answer to 5 – 3 must be 2. Click to see the next example.

45. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4. (Click to see animation.)

46. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4.

47. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4.

48. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4.

49. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4. 3.Now it has turn around because of the subtraction symbol –. (click to go on)

50. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4. 3.Now it has turn around because of the subtraction symbol –.

51. Example 4: (-4) – 2 1.Our car still starts at 0 facing right. 2.It then backs up 4 units because it sees the negative symbol - in front of 4. 3.Now it has turn around because of the subtraction symbol –.

52. Example 4: (-4) – 2 4.Finally it moves forward by 2 units. (click to go on)

53. Example 4: (-4) – 2 4.Finally it moves forward by 2 units.

54. Example 4: (-4) – 2 4.Finally it moves forward by 2 units.

55. Example 4: (-4) – 2 4.Finally it moves forward by 2 units.

56. Example 4: (-4) – 2 4.Finally it moves forward by 2 units.

57. Example 4: (-4) – 2 4.Finally it moves forward by 2 units. The car now stops at -6, therefore the answer to (-4) – 2 is -6. Click to see the next example.

58. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. (Click to see animation)

59. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right.

60. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right.

61. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right.

62. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right.

63. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign. (Click to see the next animation)

64. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign.

65. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign.

66. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign. 4.It then backs up 3 steps because it sees the negative sign. (click to see animation)

67. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign. 4.It then backs up 3 steps because it sees the negative sign.

68. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign. 4.It then backs up 3 steps because it sees the negative sign.

69. Example 5: 2 – (-3) 1.Our car starts from 0 facing right. 2.It then moves 2 units to the right. 3.Now the car has to turn around because it sees the subtraction sign. 4.It then backs up 3 steps because it sees the negative sign. Since the car stops at 5, the answer to 2 – (-3) is 5. Click to see the next example.

70. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. (click to see animation)

71. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3.

72. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3.

73. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3.

74. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3.

75. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. (click to see animation)

76. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign.

77. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign.

78. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. 4.Finally it has to back up 5 units because the next number is -5. (click to see animation)

79. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. 4.Finally it has to back up 5 units because the next number is -5.

80. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. 4.Finally it has to back up 5 units because the next number is -5.

81. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. 4.Finally it has to back up 5 units because the next number is -5.

82. Example 6: (-3) – (-5) 1.Our car starts from 0 facing right. 2.It backs up 3 units in the beginning because it sees a - sign in front of the 3. 3.Next it has to turn around because it sees the – sign. 4.Finally it has to back up 5 units because the next number is -5. The answer is therefore +2.

83. Mathematical Rules

84. Adding Integers Two positive numbers: Add the numbers, sign stays positive Two negative numbers: Add the absolute value of the numbers, sign stays negative A positive and a negative number: Subtract the absolute values of the numbers – sign is the same as the number with the largest absolute value

85. Examples 3 + 4 = 7 –Add the numbers, sign stays positive -3 + -4 = -7 –Add the numbers, sign stays negative -3 + 4 = 1 –Subtract 4-3 to get 1, sign is positive, since |4|>|-3| 3 + -4 = -1 –Subtract 4-3 to get 1, sign is negative, since |-4|>|3|

86. Subtracting Mathematically Change the minus to a plus Change the sign of the number that follows Follow the rules for addition 4 - (-5) + (+5) -3 - (-6) + (+6)