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

1.1 Real Numbers and Operations

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

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 #

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

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

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

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

Challenge 1.1.1

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

Objective: Multiply positive and negative numbers. Multiplication Rules

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

Objective: Divide positive and negative numbers. Division Theorem

Objective: Divide positive and negative numbers. Division Rules

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

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

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

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

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

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

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

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

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

1.4 The Distributive Property

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

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

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

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

Challenge 1.4.1

Challenge 1.4.2

1.5 Solving Equations

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

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

Challenge 1.5.1

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

HW #1.1-5 Pg 8 35-38 Pg 13 53-57 Pg 19 52-53 Pg 25 77-80 Pg 29 54-56

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

1.6 Writing Equations

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

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?

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.

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 28.87 minutes minus 0.05 times the number of years since 1950. Assume the record continues to decrease in this way. Predict what it will be in 2010.

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?

Challenge 1.6.1

Challenge 1.6.1

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

1.7 Exponential Notation

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

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

1.8 Properties of Exponents

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

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

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

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

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

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

1.9 Scientific Notation

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

1.10 Field Axioms

Axioms of Real Numbers

Props Of Equality

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

Objective: Use the definition of a field Proving Fields

Objective: Use the definition of a field Try This

Objective: Write Column Proofs Extended Dist Prop

Objective: Write Column Proofs Additive Inverse

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

HW #1.6-10 Pg 34 23-24 Pg 37 37-41 Pg 43 58-65 Pg 52-53 12-25

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?

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

The End Chapter 1