1. Chapter 1 Real Numbers, Algebra, and Problem Solving
Honors Algebra 2
Chapter 1
Real Numbers, Algebra, and Problem Solving

2. 1.1 Real Numbers and Operations

3. There is exactly one real number for each point on a number line.
Real Numbers

4. Objective: Show that a number is rational and distinguish between rational and irrational numbers.
If a real number cannot be expressed as a ratio of 2 integers then it is called irrational.
Def Rat/Irrat #

5. Absolute Value of a Number
The absolute value of a number is the distance on a number line the number is from 0.
Distance from 0 is 2
|-2| = 2

6. Objective: Add positive and negative numbers
Objective: Add positive and negative numbers. Objective: Subtract positive and negative numbers.
Adding Rules

7. The additive inverse of a number is the number added to it to get 0.
Objective: Add positive and negative numbers. Objective: Subtract positive and negative numbers.
The additive inverse of a number is the number added to it to get 0.
Subtraction Defined

8. Objective: Add positive and negative numbers
Objective: Add positive and negative numbers. Objective: Subtract positive and negative numbers.
Subtraction Theorem

9. Challenge 1.1.1

10. Carly Sai drives a delivery truck on a road running east and west
Carly Sai drives a delivery truck on a road running east and west. One day, starting at the garage, she drove 4 miles west, 2 miles east, 18 miles west, 12 miles east, 7 more miles east, then 9 miles west. Let east be the positive direction.
Write an expression to find how far Carly is from the garage at the end of the day.
Determine Carly’s Distance from the garage
Write an expression to determine the total amount of miles Carly drove.
Determine the total amount of miles Carly drove.
Challenge 1.1.2

11. Objective: Multiply positive and negative numbers.
Multiplication Rules

12. Division Defined Objective: Divide positive and negative numbers.
The reciprocal/Multiplicative Inverse of a number is the number we multiply it by to get 1.
Division Defined

13. Objective: Divide positive and negative numbers.
Division Theorem

14. Objective: Divide positive and negative numbers.
Division Rules

15. Division By Zero Objective: Recognize division by zero as impossible
Thus we cannot define and must exclude division by 0.
Zero is the only real number that does not have a reciprocal.
Division By Zero

16. Objective: Recognize division by zero as impossible
Try This Div by 0

17. Is it sometimes, always, or never true that, if x is a real number, (- x)( - x) is negative?
Challenge 1.2.1

18. 1.3 Algebraic Expressions and Properties of Real Numbers
Variable: Any symbol that is used to represent various numbers
Constant: Any symbol used to represent a fixed number

19. Def. Equivalent Expressions
Objective: Use number properties to write equivalent expressions.
Equivalent Expressions: Expressions that have the same value for all acceptable replacements.
Def. Equivalent Expressions

20. Real Number Properties
Objective: Use number properties to write equivalent expressions.
Real Number Properties

21. Objective: Use number properties to write equivalent expressions.
Additive Identity: The number 0.
Multiplicative Identity: The number 1
More Properties

22. For Exercises 1-4, use the properties of real numbers to answer each question.
1. If m + n = m, what is the value of n?
2. If m - n = 0, what is the value of n? What is n called with respect to m?
3. If mn = 1, what is the value of n? What is n called with respect to m?
4. If mn = m, what is the value of n?
Challenge 1.3.1

23. Suppose we define a new on the set of real numbers as follows: b = 4a - b. Thus 2 = 4(9) - 2 = 34. commutative? That is, does b = a for all real numbers a and b?
Challenge 1.3.2

24. 1.4 The Distributive Property

25. Thm Distribute over difference
Objective: Use the distributive property to multiply.
Thm Distribute over difference

26. Objective: Use the distributive property to factor expressions.
Factoring: The reverse of multiplying.
To factor an Expression: To find an equivalent expression that is a product.
Definition Factoring

27. Def Like Terms Objective: Collect like terms.
Like Terms: Terms whose variables are the same
Def Like Terms

28. Objective: Write the inverse of a sum.
Theorem 1-4, 1-5

29. Challenge 1.4.1

30. Challenge 1.4.2

31. 1.5 Solving Equations

32. Objective: Solve equations using the addition and multiplication
properties.
A mathematical sentence A = B says that the symbols A and B are equivalent.
Such a sentence is an equation.
The set of all acceptable replacements is the replacement set.
The set of all solutions is the solution set.
Definitions

33. Properties Of Equality
Objective: Solve equations using the addition and multiplication
properties.
Properties Of Equality

34. Challenge 1.5.1

35. Def: Identity Proving Identities
Objective: Prove Identities
Identity: An equation that is true for all acceptable replacements.
To Prove an Identity:
Pick one side of the equation and manipulate it using properties of real numbers to show that it can be transformed so that it is exactly the same as the other side.
Def: Identity Proving Identities

36. HW #1.1-5 Pg Pg Pg Pg Pg

37. HW Quiz #1.1-5 Friday, April 14, 2017 Pg 8 38 Pg 13 56 Pg 25 78
Row 1, 3, 5
Row 2, 4, 6

38. 1.6 Writing Equations

39. Problem Solving Strategy
Objective: Become familiar with and solve simple algebraic problems
Problem Solving Strategy

40. Objective: Become familiar with and solve simple algebraic problems
At 6:00 AM the Wong family left for a vacation trip and drove south at an average speed of 40 mph. Their friends, the Heisers, left two hours later and traveled the same route at an average speed of 55 mph. At what time could the Heisers expect to overtake the Wongs?

41. Objective: Become familiar with and solve simple algebraic problems
At the same moment, two trains leave Chicago and New York. They move towards each other with constant speeds. The train from Chicago is moving at speed of 40 miles per hour, and the train from New York is moving at speed of 60 miles per hour. The distance between Chicago and New York is 1000 miles. A bee is flying back and forth between the two trains at a rate of 80 mph, how far did the bee fly if it stops when the trains meet.

42. Objective: Become familiar with and solve simple algebraic problems
It has been found that the world record for the men's 10,000-meter run has been decreasing steadily since 1950.The record is approximately minutes minus 0.05 times the number of years since Assume the record continues to decrease in this way. Predict what it will be in 2010.

43. Objective: Become familiar with and solve simple algebraic problems
An insecticide originally contained ½ ounce of pyrethrins. The
new formula contains oz of pyrethrins. What percent of the
pyrethrins of the original formula does the new formula contain?

44. Challenge 1.6.1

45. Challenge 1.6.1

46. Objective: Become familiar with and solve simple algebraic problems
Try This Word Problems

47. 1.7 Exponential Notation

48. Def. Exponential Notation
Objective: Simplify expressions with integer exponents.
Def. Exponential Notation

49. Thus we can say that bn and b-n are reciprocals
Objective: Simplify expressions with integer exponents.
Thus we can say that bn and b-n are reciprocals
Def. B-n

50. 1.8 Properties of Exponents

51. Objective: Multiply or divide with exponents.
Am(An)

52. Def. Dividing Exponents
Objective: Multiply or divide with exponents.
We do not define 0°. Notice the following.
Def. Dividing Exponents

53. Objective: Use exponential notation in raising powers to powers.
Def (am)n

54. Objective: Use exponential notation in raising powers to powers.
Def. (am an )b

55. Objective: Use exponential notation in raising powers to powers.
Division am/an

56. Objective: Use the rules for order of operations to simplify expressions.

57. 1.9 Scientific Notation

58. Def. Scientific Notation
Objective: Convert between scientific and standard notation.
Def. Scientific Notation

59. 1.10 Field Axioms

60. Axioms of Real Numbers

61. Props Of Equality

62. The set of Real numbers forms a field with Addition and Multiplication
Objective: Use the definition of a field
Field: Any number system with two operations defined in which all of the axioms of real numbers hold
The set of Real numbers forms a field with Addition and Multiplication
Def. Field

63. Objective: Use the definition of a field
Proving Fields

64. Objective: Use the definition of a field
Try This

65. Objective: Write Column Proofs
Extended Dist Prop

66. Objective: Write Column Proofs
Additive Inverse

67. Products of Additive Inverses
Objective: Write Column Proofs
Products of Additive Inverses

68. HW # Pg Pg Pg Pg

69. HW Quiz #1.6 Friday, April 14, 2017
A gallon of one brand of paint covers approximately 2/5 of the total wall area of a room. A gallon of a different brand will cover about 1/3 of the total wall area. After the first gallon is used, what percent of the remaining wall area will the second gallon cover?
Kirra is two years older than Reggie. Reggie is six times as old as Delia. The average of their ages is nine years and four months. How old is each of them?

70. HW Quiz #1.7-8 Friday, April 14, 2017 Yellow: 1, 3, 5 White: 2, 4

71. The End Chapter 1