1. Chapter 5 Integers

2. Review a is a factor of b if... m is a multiple of n if... p is a divisor of q if...

3. Review A number is divisible by 2 if... A number is divisible by 3 if... A number is divisible by 4 if... A number is divisible by 5 if... A number is divisible by 7 if... A number is divisible by 8 if... A number is divisible by 9 if... A number is divisible by 11 if...

4. How do we “come up with” other divisibility rules?

5. What is the difference in listing all the factors of a number and writing the prime factorization of the number?

6. What is a prime number? What is a composite number?

7. How do you know if a number is prime? What numbers do you check to find out? How do you know when you are finished checking?

8. What does the GCF mean? What have you found when you have it? What does the LCM mean? What have you found when you have it?

9. Homework Questions Chapter 4

10. Lab Questions

11. Counting Numbers = {1, 2, 3,... } Whole Numbers = {0, 1, 2, 3,... } Counting Numbers 1.Closed with respect to addition 2.Closed with respect to multiplication 3.Not closed with respect to subtraction 4.Not closed with respect to division

12. Counting Numbers = {1, 2, 3,... } Whole Numbers = {0, 1, 2, 3,... } Whole Numbers 1.Closed with respect to addition 2.Closed with respect to multiplication 3.Not closed with respect to subtraction 4.Not closed with respect to division

13. {... -2, -1, 0, 1, 2,...} False Numbers Numbers of Integrity Integers

14. Counting Numbers = {1, 2, 3,... } Whole Numbers = {0, 1, 2, 3,... } Integers = {..., -2, -1, 0, 1, 2... }

15. Counting Numbers = {1, 2, 3,... } Whole Numbers = {0, 1, 2, 3,... } Integers = {..., -2, -1, 0, 1, 2... } Integers 1.Closed with respect to addition 2.Closed with respect to multiplication 3.Closed with respect to subtraction 4.Not closed with respect to division

16. Ancient Asian Notation

18. Ancient Asian Notation: +3

19. Indian Notation -5 = 5

20. Chip Model Counters colored black on one side, red on the other. Drop 10 of them.

21. Chip Model Counters colored black on one side, red on the other. Drop 10 of them. Result is -2

22. Hot Air Balloon

25. I Walk the Line Face a positive direction and stand at 0 Addition: –Walk forward for a positive integer, backward for a negative integer Subtraction: –Walk forward for a positive integer, backward for a negative integer –To subtract, do the inverse so turn around

26. +3 + -2 =

27. +3 + -2 = +1 -3 + -2 =

28. +3 + -2 = +1 -3 + -2 = -5 +4 - +6 =

29. +3 + -2 = +1 -3 + -2 = -5 +4 - +6 = -2 -5 - +2 =

30. +3 + -2 = +1 -3 + -2 = -5 +4 - +6 = -2 -5 - +2 = -7 +2 - -3 =

31. +3 + -2 = +1 -3 + -2 = -5 +4 - +6 = -2 -5 - +2 = -7 +2 - -3 = +5 -4 - -7 =

32. +3 + -2 = +1 -3 + -2 = -5 +4 - +6 = -2 -5 - +2 = -7 +2 - -3 = +5 -4 - -7 = +3

33. Absolute Value of an Integer Page 290 The absolute value of an integer is the distance that integer is from 0 on the number line. |-11| =|13| = |0| =|-9| =

34. Absolute Value of an Integer Page 290 The absolute value of an integer is the distance that integer is from 0 on the number line. |-11| = 11|13| = 13 |0| =0|-9| = 9 |x| = x if x ≥ 0 |x| = -x if x < 0

35. | 5 + (-7)| = “The absolute value of 5 + -7.” | 5 + (-7)| = | -2 | = 2 | 5 | + | -7 | = “The absolute value of 5 plus the absolute value of -7.” | 5 | + | -7 | = 5 + 7 = 12

36. Mail-Time Model At mail time you are delivered a check for $20. What happens to your net worth. At mail time you are delivered a bill for $35. What happens to you net worth? At mail time you receive a check for $10 and a bill for $10. What happens to your net worth?

37. Example 5.10 Page 297

38. Example 5.19 Page 306

39. Adding Integers Subtracting Integers

40. Multiplication by repeated addition (3)(-4) = (-4) + (-4) + (-4) = -12

41. Multiplication by patterns: (4)(-3) (4)(3) = 12

42. Multiplication by patterns: (4)(-3) (4)(3) = 12 (4)(2) = 8 (4)(1) = 4 (4)(0) = 0

43. Multiplication by patterns: (4)(-3) (4)(3) = 12 (4)(2) = 8 (4)(1) = 4 (4)(0) = 0 (4)(-1) = (4)(-2) = (4)(-3) =

44. Multiplication by patterns: (4)(3) = 12 (4)(2) = 8 (4)(1) = 4 (4)(0) = 0 (4)(-1) = -4 (4)(-2) = -8 (4)(-3) = -12

45. Multiplication by patterns: (-3)(-2) (3)(-2) = -6

46. Multiplication by patterns: (-3)(-2) (3)(-2) = -6 (2)(-2) = -4 (1)(-2) = -2 (0)(-2) = 0

47. Multiplication by patterns: (-3)(-2) (3)(-2) = -6 (2)(-2) = -4 (1)(-2) = -2 (0)(-2) = 0 (-1)(-2) = (-2)(-2) = (-3)(-2) =

48. Multiplication by patterns: (3)(-2) = -6 (2)(-2) = -4 (1)(-2) = -2 (0)(-2) = 0 (-1)(-2) = +2 (-2)(-2) = +4 (-3)(-2) = +6

49. Multiplication of Integers: (+)(+) = + (+)(-) = - (-)(+) = - (-)(-) = +

50. Division: Family of Facts (3)(-4) = -12 (-4)(3) = -12 (-12) ÷ 3 = -4 (-12) ÷ (-4) = 3

51. Multiplication of Integers: (+)(+) = + (+)(-) = - (-)(+) = - (-)(-) = + Division of Integers: (+) ÷ (+) = + (-) ÷ (-) = + (-) ÷ (+) = - (+) ÷ (-) = -

52. More Mail-Time Example 5.23, Page 317

53. Properties Page 296 Closure Commutative Associative Identity Element Existence of Negative – For every integer n, there exists –n called “the additive inverse of n” or “the opposite of n” such that n + -n = 0 (the identity element for addition)