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3. Subunit: 1.Positive and Negative Numbers 2.Addition and subtraction of Integers 3.Multiplication and division of integers 14.Integers

4. Number line When points on a line represent numbers, then that line is called number line. 0123456-2-3-4-5-6

5. Definition Positive number – a greater than zero. 0123456

6. Definition Negative number – a less than zero. 0123456-2-3-4-5-6

7. Definition Opposite Numbers – numbers that are the same distance from zero in the opposite direction 0123456-2-3-4-5-6

8. Definition Integers – are all the whole numbers and all of their opposites on the negative number line including zero. 7 opposite -7

9. Negative Numbers Are Used to Measure Temperature

10. Negative Numbers Are Used to Measure Under Sea Level 0 10 20 30 -10 -20 -30

11. Hint If you don’t see a negative or positive sign in front of a number it is positive. 9 +

12. Subunit: Addition and subtraction of Integers 14.Integers-2

13. Addition Rule 1) When the signs are the same, ADD and keep the sign. (-2) + (-4) = -6 2) When the signs are different, SUBTRACT and use the sign of the larger number. (-2) + 4 = 2 2 + (-4) = -2

14. -1 + 3 = ? 1.-4 2.-2 3.2 4.4 Answer Now

15. -6 + (-3) = ? 1.-9 2.-3 3.3 4.9 Answer Now

16. Solve the Problems -3 + -5 = 4 + 7 = (+3) + (+4) = -6 + -7 = 5 + 9 = -9 + -9 = -8 -18 14 -13 7 11

17. Solve These Problems 3 + -5 = -4 + 7 = (+3) + (-4) = -6 + 7 = 5 + -9 = -9 + 9 = -2 5 – 3 = 2 0 -4 1 3 9 – 9 = 0 9 – 5 = 4 7 – 6 = 1 4 – 3 = 1 7 – 4 = 3

18. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 When the number is positive count to the right. When the number is negative count to the left. +-

19. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + - +3 + -5 =-2

20. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + - +6 + -4 =+2

21. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + - +3 + -7 =-4

22. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - + -3 + +7 =+4

23. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - + -5 + +3 =+4

24. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - + -2 + +8 =+6

25. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - + -5 + +2 =-3

26. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - - -4 + -2 =-6

27. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - - -2 + -3 =-5

28. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - - -1 + -4 =-5

29. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 - - -4 + -1 =-5

30. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + + 4 + 1 =5

31. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + + 3 + 2 =5

32. One Way to Add Integers Is With a Number Line 0123456-2-3-4-5-6 + + 1 + 5 =6

33. The additive inverses (or opposites) of two numbers add to equal zero. -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems. Example: The additive inverse of 3 is

34. What’s the difference between 7 - 3 and 7 + (-3) ? 7 - 3 = 4 and 7 + (-3) = 4 The only difference is that 7 - 3 is a subtraction problem and 7 + (- 3) is an addition problem. “SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.”

35. When subtracting, change the subtraction to adding the opposite and then follow your addition rule. Example #1: - 4 - (-7) - 4 + (+7) Diff. Signs --> Subtract and use larger sign. 3 Example #2: - 3 - 7 - 3 + (-7) Same Signs --> Add and keep the sign. -10

36. 11b + (+2b) Same Signs --> Add and keep the sign. 13b Okay, here’s one with a variable! Example #3: 11b - (-2b)

37. Which is equivalent to -12 – (-3)? Answer Now 1.12 + 3 2.-12 + 3 3.-12 - 3 4.12 - 3

38. 7 – (-2) = ? Answer Now 1.-9 2.-5 3.5 4.9

39. Operations with Integers

40. What is an Integer? A whole number that is either greater than 0 (positive) or less than 0 (negative) Can be visualized on a number line:

41. What is a Number Line? A line with arrows on both ends that show the integers with slash marks Arrows show the line goes to infinity in both directions ( + and -) Uses a negative sign (-) with negative numbers but no positive sign (+) with positive numbers Zero is the origin and is neither negative nor positive

42. What are Opposites? Two integers the same distance from the origin, but on different sides of zero Every positive integer has a negative integer an equal distance from the origin Example: The opposite of 6 is -6 Example: The opposite of -2 is 2

43. What is Absolute Value? Distance a number is from zero on a number line (always a positive number) Indicated by two vertical lines | | Every number has an absolute value Opposites have the same absolute values since they are the same distance from zero Example: |-8| = 8 and |8| = 8 Example: |50| = 50 and |-50| = 50

44. What Can We Do to Integers? Integers are numbers, so we can add, subtract, multiply, and divide them Each operation has different rules to follow

45. Adding Rules – Same Signs If the integers have the SAME signs: ADD the numbers & keep the same sign! Positive + Positive = Positive Answer Negative + Negative = Negative Answer Examples: -3 + (-10) = ? ? = -13 6 + (8) = ? ? = 14

46. Adding (Same Signs) - Examples #1. -3 + (-10) Step 1: 13 Add the #s Step 2: -13 Keep same sign (Both #s are negative – Answer is negative!) #2. 6 + (8) Step 1: 14 Add the #s Step 2: 14 Keep same sign (Both #s are positive – Answer is positive!)

47. Adding Rules – Different Signs If the integers have the DIFFERENT signs: SUBTRACT the numbers & use sign of the BIGGER number! Bigger # is Positive = Positive Answer Bigger # is Negative = Negative Answer Examples: -13 + (7) = ? ? = -6 23 + (-8) = ? ? = 15

48. #1. -13 + (7) Step 1: 6 Subtract the #s Step 2: -6 Use sign of bigger # (Bigger # is negative - Answer is negative!) #2. 23 + (-8) Step 1: 15 Subtract the #s Step 2: 15 Use sign of bigger # (Bigger # is positive - Answer is positive!)

49. Subtracting Rules Put ( ) around second number & its sign Change SUBTRACTION sign to an ADDITION sign Change sign of 2 nd number to its opposite Follow the rules for ADDITION: -SAME signs: Add & keep the same sign -DIFFERENT signs: Subtract & use sign of bigger # Examples: -5 – -10 = ? ? = 5 9 - 23 = ? ? = -14

50. Subtracting - Examples #1. -5 – -10 #2.9 - 23 Step 1: -5 – (-10) Insert ( ) 9 – (23) Step 2: -5 + (-10) Change – to + 9 + (23) Step 3: -5 + (10) Change 2 nd sign 9 + (-23) Step 4: 5 Follow adding rules -14 d

51. Multiplying Rules Multiply the numbers like usual If the integers have the SAME signs: ANSWER will be POSITIVE If the integers have DIFFERENT signs: ANSWER will be NEGATIVE Examples: -3 · (-5) = ? ? = 15 -9 · (-10) = ? ? = 90 -7 · 7 = ? ? = -49 6 · -6 = ?? = -36

52. Multiplying - Examples #1. -3 · (-5) #2. -9 · (-10) 15 Multiply the numbers 90 15 Same signs = Positive Answer 90 #3. -7 · 7 #4. 6 · -6 49 Multiply the numbers 36 -49 Different signs = Negative Answer -36

53. Dividing Rules Divide the numbers like usual If the integers have the SAME signs: ANSWER will be POSITIVE If the integers have DIFFERENT signs: ANSWER will be NEGATIVE Examples: -33 ÷ (-3) = ? ? = 11 -90 ÷ (-10) = ? ? = 9 -20 ÷ 2 = ? ? = -10 6 ÷ -6 = ?? = -1

54. Dividing - Examples #1. -33 ÷ (-3) #2. -90 ÷ (-10) 11 Divide the numbers 9 11 Same signs = Positive Answer 9 #3. -20 ÷ 2 #4. 6 ÷ -6 10 Divide the numbers 1 -10 Different signs = Negative Answer -1

55. Mixed Practice Solve the following problems: -9 + - 9 -18 7 · -4 -28 -10 - (-19) 9 -35 ÷ -7 5 15 + -25 -10 -23 - 9 -32

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