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

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Presentation on theme: "Honors Algebra 2 Chapter 1 Real Numbers, Algebra, and Problem Solving."— Presentation transcript:

1

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

3 1.1 Real Numbers and Operations

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

5 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 #

6 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

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

8 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

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

10 Challenge 1.1.1

11 Challenge 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. 1.Write an expression to find how far Carly is from the garage at the end of the day. 2.Determine Carly’s Distance from the garage 3.Write an expression to determine the total amount of miles Carly drove. 4.Determine the total amount of miles Carly drove.

12 Objective: Multiply positive and negative numbers. Multiplication Rules

13 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

14 Objective: Divide positive and negative numbers. Division Theorem

15 Objective: Divide positive and negative numbers. Division Rules

16 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

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

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

19 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

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

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

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

23 Challenge 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?

24 Challenge 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?

25 1.4 The Distributive Property

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

27 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

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

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

30 Challenge 1.4.1

31 Challenge 1.4.2

32 1.5 Solving Equations

33 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

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

35 Challenge 1.5.1

36 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

37 HW #1.1-5 Pg Pg Pg Pg Pg

38 Pg 8 38Pg 13 56Pg 25 78Pg Pg 8 36Pg 13 54Pg 2580Pg HW Quiz #1.1-5 Tuesday, May 05, 2015 Row 1, 3, 5 Row 2, 4, 6

39 1.6 Writing Equations

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

41 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? Objective: Become familiar with and solve simple algebraic problems

42 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.

43 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 Objective: Become familiar with and solve simple algebraic problems

44 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?

45 Challenge 1.6.1

46

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

48 1.7 Exponential Notation

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

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

51 1.8 Properties of Exponents

52 Objective: Multiply or divide with exponents. A m (A n )

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

54 Objective: Use exponential notation in raising powers to powers. Def (a m ) n

55 Objective: Use exponential notation in raising powers to powers. Def. (a m a n ) b

56 Objective: Use exponential notation in raising powers to powers. Division a m /a n

57 Objective: Use the rules for order of operations to simplify expressions. Order of Operations

58 1.9 Scientific Notation

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

60 1.10 Field Axioms

61 Axioms of Real Numbers

62 Props Of Equality

63 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

64 Objective: Use the definition of a field Proving Fields

65 Objective: Use the definition of a field Try This

66 Objective: Write Column Proofs Extended Dist Prop

67 Objective: Write Column Proofs Additive Inverse

68 Objective: Write Column Proofs Products of Additive Inverses

69 HW # Pg Pg Pg Pg

70 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? HW Quiz #1.6 Tuesday, May 05, 2015

71 HW Quiz #1.7-8 Tuesday, May 05, 2015 Yellow: 1, 3, 5 White: 2, 4

72 The End Chapter 1


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