Honors Algebra 2 Chapter 1 Real Numbers, Algebra, and Problem Solving.
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Transcript of Honors Algebra 2 Chapter 1 Real Numbers, Algebra, and Problem Solving.
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Honors Algebra 2
Chapter 1
Real Numbers, Algebra, and Problem Solving
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1.1 Real Numbers and Operations
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There is exactly one real number for each point on a number line.
Real Numbers
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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 #
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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
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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
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Objective: Add positive and negative numbers.Objective: Subtract positive and negative numbers.
Subtraction Theorem
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Challenge 1.1.1
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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
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Objective: Divide positive and negative numbers.
Division Theorem
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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
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Challenge 1.2.1
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?
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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
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Objective: Use number properties to write equivalent expressions.
Def. Equivalent Expressions
Equivalent Expressions: Expressions that have the same value for all acceptable replacements.
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Objective: Use number properties to write equivalent expressions.
Real Number Properties
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Objective: Use number properties to write equivalent expressions.
Additive Identity: The number 0.
Multiplicative Identity: The number 1
More Properties
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Challenge 1.3.1
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?
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Challenge 1.3.2
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?
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1.4 The Distributive Property
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Objective: Use the distributive property to multiply.
Thm Distribute over difference
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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
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Objective: Collect like terms.
Like Terms: Terms whose variables are the same
Def Like Terms
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Objective: Write the inverse of a sum.
Theorem 1-4, 1-5
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Challenge 1.4.1
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Challenge 1.4.2
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1.5 Solving Equations
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Objective: Solve equations using the addition and multiplicationproperties.
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
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Objective: Solve equations using the addition and multiplicationproperties.
Properties Of Equality
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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
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HW #1.1-5Pg 8 35-38
Pg 13 53-57Pg 19 52-53Pg 25 77-80Pg 29 54-56
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Pg 8 38 Pg 13 56 Pg 25 78 Pg 29 56
Pg 8 36 Pg 13 54 Pg 2580 Pg 29 54
HW Quiz #1.1-5Tuesday, April 18, 2023
Row 1, 3, 5
Row 2, 4, 6
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1.6 Writing Equations
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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
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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
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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?
5
8
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Challenge 1.6.1
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1.7 Exponential Notation
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Objective: Simplify expressions with integer exponents.
Def. Exponential Notation
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Objective: Simplify expressions with integer exponents.
Thus we can say that bn and b-n are reciprocals
Def. B-n
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1.8 Properties of Exponents
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Objective: Multiply or divide with exponents.
Am(An)
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Objective: Multiply or divide with exponents.
We do not define 0°. Notice the following.
Undefined
0 1 10 0 1 10 0 1
1
0 00 0
Def. Dividing Exponents
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Objective: Use exponential notation in raising powers to powers.
Def (am)n
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Objective: Use exponential notation in raising powers to powers.
Def. (am an )b
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Objective: Use the rules for order of operations to simplify expressions.
Order of Operations
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1.10 Field Axioms
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Props Of Equality
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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
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Objective: Use the definition of a field
Proving Fields
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Objective: Use the definition of a field
Try This
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Objective: Write Column Proofs
Extended Dist Prop
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Objective: Write Column Proofs
Additive Inverse
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Objective: Write Column Proofs
Products of Additive Inverses
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HW #1.6-10Pg 34 23-24Pg 37 37-41Pg 43 58-65
Pg 52-53 12-25
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The End Chapter 1