SCHOLARS ON-LINE

 

 

 

 

Course Syllabus

PRE-CALCULUS

2018-2019


PRE-CALCULUS

 

Instructor

ARTHUR MABBOTT

MATHGUY

SCHOLARS ON-LINE

ART@MABBOTT.ORG

(206) 605-7393

 

 

Course Description 

This course is part of the sequence of courses that make up the SCHOLARS ON-LINE Mathematics Department.

 

General Course Goals

This course will focus on how to do mathematics including functions and trigonometry.  This course will focus on how to do mathematics including functions and trigonometry.  The Fall and Spring Semester Guides at the end are the goal for this course.  Chapter by Chapter Calendars will be distributed just in time and will reflect adjustments to meet the needs of the students each year. 

 

Course Location

WIZIQ

 

Course Dates and Times

 

SYNCHRONIS

SESSIONS

Start

Time

End

Time

MONDAY

11:00AM

12:00PM

WEDNESDAY

11:00AM

12:00PM

FRIDAY

11:00AM

12:00PM

 

 

 

 

 

 

 

 

 


 

UNIT I - FUNCTIONS

A. CHAPTER 1 – FUNCTIONS AND

B.  CHAPTER 7 – PROPERIES OF ELEMENTARY FUNCTIONS

C.  CHAPTER 15 – POLY AND RATIONAL FUNCTIONS

 

UNIT II - TRIGONOMETRY

A. CHAPTER 2 – PERIODIC FUNCTIONS

B.  CHAPTER 3 – CIRC FUNCTIONS

C.  CHAPTER 4 – TRIG IDENTITIES

D.CHAPTER 5 – COMBINED SINUSOID

E. CHAPTER 6 – TRIANGLE TRIG

 

UNIT III – ADVANCED TOPICS

A. CHAPTER 11 – MATRICES

B.  CHAPTER 12 – CONICS & QUAD SURFACES




Chapter 1 – 18 Days

Monday

Tuesday

Wednesday

Thursday

Friday

September

 

Day 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Chapter 7 – 22 Days

Monday

Tuesday

Wednesday

Thursday

Friday

Day 19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

 

 

 

Chapter 15 – 16 Days

Monday

Tuesday

Wednesday

Thursday

Friday

 

 

Day 41

42

42

44

45

46

47

48

49

50

51

52

53

54

55

56

 

 

 


Chapter 2 – 15 Days

Monday

Tuesday

Wednesday

Thursday

Friday

Day 57

58

59

60

61

62

63

64

65

66

67

68

59

70

71

Christmas Break

Monday

Tuesday

Wednesday

Thursday

Friday

December 24

25

26

27

28

31

January 1

2

3

4

 


 

Chapter 3 – 19 Days

Monday

Tuesday

Wednesday

Thursday

Friday

Day 72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

Chapter 4 – 25 Days

Monday

Tuesday

Wednesday

Thursday

Friday

Day 92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

Chapter 5 – 20 Days

Monday

Tuesday

Wednesday

Thursday

Friday

Day 117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

EASTER BREAK

132

133

134

135

136

 


Chapter 6 – 14 Days

Monday

Tuesday

Wednesday

Thursday

Friday

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

 

Chapter 11 – 10 days

Monday

Tuesday

Wednesday

Thursday

Friday

152

153

154

155

156

157

158

159

160

161

Chapter 12 – 15 days

Monday

Tuesday

Wednesday

Thursday

Friday

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176


Course Agenda

Unit I (Functions): Chapter 1 - Functions and Mathematical Models

Suggested Timeframe: 21 Days (approximate unit completion date: October 5)

Unit Rationale: Reviewing functions studied in algebra and generalizing the concepts of transformations and compositions of these functions will allow us to apply them to more complicated function families.

Essential Questions: How do families of functions relate to the types of transformations that are used and how does each transformation affect an equation, graph, table, or data set? How can a function composition and/or an inverse function be used in applications?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.F.3 Identify parent functions of polynomials, power, exponential, inverse variation and rational from a table, graph, equation or situation. (1.1, 1.2)

            Domain/range

            Argument of the function

             Asymptote

            Discontinuity

            Boolean variable

            Restricted domain

            Dilation, reflection, translations,

            Displacement vs. distance

            Even and odd functions

            Domain and range of composite functions

            Invertible, one-to-one function

            Strictly increasing and strictly decreasing

            f-1(f(x)) = f(f-1(x)) = x

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics

 

PC.8.B Select and apply strategies to solve problems.

 

 

1-1

1-2

1-3

Quiz 1-1 to 1-3

1-4

1-5

Quiz 1-4 to 1-5

1-6

1-7 Optional

1-8 Review

Unit Assessment

 

 

 

1-3a, 1-3c

 

1-4a

1-5a

 

1-6a

PC.F.4 Perform algebraic operations on functions (polynomial, power, exponential, rational, absolute value, etc.) and apply transformations. (1.3, 1.6)

PC.F.5 Write an expression for the composition of one function with another and find the domain, range and graph the composite function. (1.4)

PC.F.6 Determine whether a function (equation or graph) has an inverse. Express the inverse as an equation or graph if it exists. Use function notation for inverses. (1.5)

Supporting Standards:

As a result of this unit, students will be able to:

            Use multiple representations of polynomial, quadratic, linear, direct variation, power, exponential, inverse variation, and rational functions.

            Use transformations to identify functions from the parent in a function family.

            Compose functions.

            Understand and apply inverse and absolute value functions.

 

Unit I (Functions): Chapter 7 - Properties of Elementary Functions

Suggested Timeframe: 18 Days (approximate unit completion date: November 1)

Unit Rationale: In order to apply mathematics, students must understand the differences between different types of functions in order to be able to model real-world situations.

Essential Questions: How can determine different types of functions be determined from data? How can I find a missing exponent?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.F.3 Identify parent functions of polynomials, power, exponential, inverse variation and rational from a table, graph, equation or situation. (7.1, 7.2)

            Proportionality constant

            Second and third difference

            Common logarithm

            Natural logarithm

            Base, exponent

 

 

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

PC.8.B Select and apply strategies to solve problems.

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

7-1

7-2

7-3

Quiz 7-1 to 7-3

7-4

7-5

7-6

Quiz 7-4 to 7-6

*7-7 (Optional)

7-8 (Review)

Unit Assessment

 

7-2a

7-3a, 7-3b

 

7-4a

 

 

 

 

7-8a

PC.F.4 Perform algebraic operations on functions (polynomial, power, exponential, rational, absolute value, etc.) and apply transformations.

PC.F.12 Compare the rates of change of functions in different families (linear, quadratic, exponential, and power).  (7.3)

PC.E.1 Use the inverse relationship between exponential and logarithmic functions to solve equations and problems. (7.4, 7.5)

PC.E.2 Solve exponential and logarithmic equations algebraically and graphically. (7.4, 7.5, 7.6)

PC.E.3 Find an exponential and logarithmic function to model a given data set or situation.  Solve problems involving exponential growth or decay. (7.6, 7.7)

Notes:

1.     Avoid Q-problems from chapters 2 – 6.

2.     Review properties of exponents.

Supporting Standards:

As a result of this unit, students will be able to:

            Recognize various functions from patterns, graphs, equations, and/or tables of values.

            Use the properties of exponents and logarithms to simplify expressions and solve equations.

            Use exponential and logarithmic functions to solve real-world problems involving growth and decay.


 

 

Unit I (Functions): Chapter 15 - Polynomial and Rational Functions

Suggested Timeframe: 15 Days (approximate unit completion date:  November 23)

Unit Rationale: Analyzing polynomial functions will facilitate graphing and will allow us to also work with rational functions that are the quotient of two polynomials. This study will lead to the concept of limits and rates of change at any point on a graph.

Essential Questions:  How does knowledge and analysis of the characteristics of polynomial and rational functions help us easily graph and apply them?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.F.13 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (15.2)

            Critical point

            End behavior

            Remainder Theorem

            Discontinuous

            Removable discontinuity

            Indeterminate form

            Infinite form

            Limit

            Synthetic division

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.B Select and apply strategies to solve problems.

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

15-1

15-2

Quiz

*15-3 (optional)

15-4

*15-5 (optional)

15-6 Review

Unit AssessmenT

15-1a

15-2a, 2b

 

 

PC.F.14 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (15.1)

PC.F.15 Graph linear, quadratic, polynomial, rational, exponential, and logarithmic functions expressed symbolically and show key features of the graph, including zeros, intercepts, critical points, and asymptotic and end behavior. (15.1, 15.2)

PC.F.8 Identify and describe discontinuities (removable, step, asymptotes) of a function and how these relate to the graph. (15.4)

Notes:

*Omit concepts typically taught in a calculus course, including limits and derivatives.

Supporting Standards:

PC.F.2 Find the domains and ranges of functions.

As a result of this unit, students will be able to:

            Determine the degree of a polynomial from its graph.

            Find the zeros of a polynomial from graph or equation.

            Use the Remainder Theorem to find the zeros of a polynomial function.

            Determine whether and how many zeros a polynomial function may have.

            Know how complex solutions to a polynomial equation translate to the graphic representation of that equation.

            Determine if a polynomial model fits a set of data.

            Simplify rational expressions, solve rational equations, and analyze the graphs of rational functions.

            Use partial fractions to solve or graph rational equations or functions.

 

 

 

Unit II (Trigonometry):  Chapter 2 - Periodic Functions and Right Triangle Problems

Suggested Timeframe: 15 days (approximate unit completion date: December 16)

Unit Rationale: Understanding the trigonometric ratios and their relationship to right triangles will allow us to solve many real-world applications involving measures of triangles and will lead to an understanding of trigonometric functions.

Essential Questions: What situations can be modeled with periodic equations? Why are there six trigonometric functions? Where do they come from? How do you find an unknown angle in a right triangle?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.T.1 Define (using the unit circle), graph, and use all six trigonometric functions of any angle.  Convert between radians and degrees. (2.2, 2.3, 2.4)

            Periodic, sinusoid,

            Standard position

            Co-terminal

            Reference angle

            Cycle

            Period

            Sine, cosine, tangent, cotangent, secant, and cosecant

            Inverse trigonometric function

            Principal branch

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics

PC.8.B Select and apply strategies to solve problems.

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

2-1

2-2

2-3

2-4

Quiz 2-1 to 2.4

2-5

Quiz 2-5

2-6

Unit Assessment

 

 

2-1a

2-2a

2-3a,b,c

2-4a, b

 

2-5a, b

PC.T.2 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle. (2.4)

PC.T.3 Students can compute (by hand) the values of six trigonometric functions at any standard point on the unit circle. (2.4)

PC.T.4 Know the definitions of the sine and cosine functions (amplitude, period, sinusoidal axis, frequency and phase shift), and explain the relationship between constants in the formula, and transformed graph. (2.3,

PC.T.8 Solve trigonometric equations using basic identities and inverse trigonometric functions. (2.5)

NOTE:  Radian measures will be primarily used in calculus. Students should be fluent with graphs of all six trig functions using radian measures as well as degrees.

PC.T.9 Use trigonometry to determine unknown sides or angles in right triangles. (2.5)

Supporting Standards:

As a result of this unit, students will be able to:

            Use the unit circle values to define trigonometric functions.

            Extend definitions for sine and cosine to tangent, cotangent, secant, and cosecant.

            Use inverse trigonometric functions to solve equations and problems.

            Graph sine and cosine functions.

            Solve for missing parts of right triangles.

 

Unit II (Trigonometry): Chapter 3 - Applications of Trigonometric and Circular Functions

Suggested Timeframe: 17 Days (approximate unit completion date: January 26)

Unit Rationale: Extending our understanding of periodic functions by using our knowledge of transformations learned (Unit 1 - Chapter 1) will allow us to model real-world sinusoidal relationships.

Essential Questions: How can you write periodic equation from a situation or graph? What is a radian? Why do we need to use radians?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.T.5 Find and use a sinusoidal function to model a given data set or situation. (3.1,3.7)

            Amplitude

            Phase displacement

            Sinusoidal Axis

            Cycle

            Frequency

            Convex/Concave

            Point of inflection

            Upper bound/Lower bound

            Asymptotes

            Radian

            Arc length

            Circular functions

            Arc functions

            Inverse functions

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.B Select and apply strategies to solve problems.

PC.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

3-1

3-2

*3-3 (optional)

3-4

3-5

3-6

3-7

*3-8 (optional)

3-9 Review

Unit Assessment

3-1b

3-2a, b

3-3a

3-4a,b

3-5a

3-6

PC.T.1 Define (using the unit circle), graph, and use all six trigonometric functions of any angle.  Convert between radians and degrees. (3.1, 3.4 - radians, 3.5 – unit circle)

PC.T.4 Graph transformations of the sine and cosine functions (amplitude, period, sinusoidal axis, frequency and phase shift), and explain the relationship between constants in the formula and transformed graph. (3.2)

Supporting Standards:

PC.T.6 Know the definitions of the inverse trigonometric functions, including their domains and ranges; recognize their graphs. (3.6)

NOTE:

As a result of this unit, students will be able to:

            Write a particular equation for a sinusoid that fits any given conditions.

            Graph any trigonometric function from its equation or given amplitude, period or frequency, phase displacement, and sinusoidal axis.

            Find a trigonometric equation from its graph or given amplitude, period, phase displacement, and sinusoidal axis.

            Find the amplitude, period, phase displacement, and sinusoidal axis from an equation or graph.

 

Unit II (Trigonometry): Chapter 4 - Trigonometric Function Properties: Identities

Suggested Timeframe: 20 Days (approximate unit completion date: March 2)

Unit Rationale: Learning the Pythagorean properties will allow us to transform one trigonometric expression into an equivalent form, prove identities, and derive other properties, such as the double- and half-angle properties.

Essential Questions: When we solve a trigonometric equation, how can we know that we have found all possible solutions?

How can we use what we know about proof to show that two trigonometric expressions are equivalent?

 

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.T.7 Use the basic trigonometric identities for sine, cosine and tangent to prove additional trigonometric identities and derive some of the basic ones (e.g. Use cos2x + sin2x = 1 to prove that sec2x = tan2x + 1). (4.1 – 4.3)

            Pythagorean property

            Reciprocal property

            Quotient property

            Identity

            Conjugate

            Closed interval

            Open interval (Notation)

            Principal branch

 

PC.8.B Select and apply strategies to solve problems.

 

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem

4-1

4-2

4-3

Quiz 1

4-4

4-6

Quiz 2

4-7

Test Ch 4/5

 

 

4-2a

4-3a

 

4-4a

4-6a, b

PC.T.8 Solve trigonometric equations using basic identities and inverse trigonometric functions. (4.4)

PC.T.6  Know the definitions of the inverse trigonometric functions, including their domains and ranges; recognize their graphs. (4.6)

Supporting Standards:

PC.5 Write an expression for the composition of one function with another and find the domain, range and graph of the composite function.  Recognize components when a function is composed of two or more elementary functions.

NOTE: Check the recommended explorations for more practice problems.

 

As a result of this unit, students will be able to:

            Transform a trigonometric expression to an equivalent expression.

            Use inverse trigonometric functions to solve equations.

            Find the graphs and characteristics of inverse trigonometric functions.

            Recognize that some triangles have two possible solutions (ambiguous case).

 

 

 


 

Unit II (Trigonometry): Chapter 5 - Properties of Combined Sinusoids

Suggested Timeframe: 6 days (approximate unit completion date: March 12)

Unit Rationale: Learning how to work with trigonometric arguments that are sums, differences, composites, double-, and half-angles allows us to simplify and solve more complicated trigonometric expressions and equations.

Essential Questions:  How can we use identities to simplify and solve more quickly trigonometric expressions and equations involving double and half angles?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.T.7 Use the basic trigonometric identities for sine, cosine and tangent to prove trigonometric identities and derive some of the basic ones (e.g., double-angle formula from sum and difference formula, half-angle formula from double angle formula) (5.2, 5.3, 5.6)

            Cofunctions

            Odd and even functions

            Double and half angle argument

 

PC.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

 

 *5.1 (optional)

5-2

5-3

*5-4 (optional)

*5-5 (optional)

5-6

5-7 Review

Quiz

 

5-2b, c

 

 

 

5-6a

PC.T.8 Solve trigonometric equations using basic identities and inverse trigonometric functions. (5.2, 5.3, 5.6)

Supporting Standards:

 

NOTE: Review problems from 5.7 should include only those sections covered in this unit.

As a result of this unit, students will be able to:

            Derive and use properties for trigonometric functions of sums and differences of angles.

            Derive and use properties of products of arguments, including double- and half-arguments.

            Use algebraic techniques to solve trigonometric equations.

 


 

Unit II (Trigonometry):  Chapter 6 - Triangle Trigonometry

Suggested Timeframe: 14 Days (approximate unit completion date: March 30

Unit Rationale: Learning how to find unknown measurements in oblique triangles (sides, angles, and areas) will allow us to solve any triangle given enough information.

Essential Questions: How can the formulas and techniques that we have learned to solve a right triangle help us solve any oblique triangle?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.T.10 Use law of sines and law of cosines to solve problems. (6-2, 6-4, 6-5, 6-7)

            Oblique triangle

            Law of Sines

            Law of Cosines

            Semi-perimeter

            Heros Formula

            Ambiguous

 

PC.8.B Select and apply strategies to solve problems.

 

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

6-1

6-2

6-3

Quiz 1

6-4

6-5

Quiz 2

*6-6 (optional)

6-7

6-8 Review

Unit Assessment

 

 

6-2a,b

6-3a

 

 

6-5a,b

Supporting Standards:

 

As a result of this unit, students will be able to:

            Solve an oblique triangle, given enough information.

            Apply the Laws of Sines and Cosines to solve triangles.

            Find the area of triangles using trigonometry, including Heros formula.

 


 

Unit III (Advanced Topics): Chapter 11 - Matrices – Operations

Suggested Timeframe: 10 days (approximate unit completion date: April 13)

Unit Rationale: Learning to work with matrices can simplify work with transformations, data display, and system solutions.

Essential Questions: How can we use matrices to transform geometric figures, display data, and/or solve a system?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.M.1 Add, subtract, and multiply matrices of appropriate dimensions. (11-2)

            Matrix

            Scalar

            Determinant

            Adjoint

            Transformation matrix

            Image matrix

            Iterate

 

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

 

11-2

*11-3 (optional)

*11-4 (optional)

*11-5 (optional)

*11-6 (optional)

11-7 Review

Unit Assessment

 

 

PC.M.2 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. (11-2)

PC.M.4 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.  (in skipped chapter 10)

Supporting Standards:

PC.M.3 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

PC.M.5 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

As a result of this unit, students will be able to:

            Perform operations with matrices.

            Use matrices to rotate and dilate figures on the coordinate plane.

 


 

Unit III (Advanced Topics): Chapter 12 - Analytic Geometry of Conic Sections and Quadric Surfaces

Suggested Timeframe: 15 Days (approximate unit completion date: May  11)

Unit Rationale:  Defining and understanding conic sections, algebraically and geometrically, will allow us to apply them to real-world problems, including design, orbital paths, marketing, and three-dimensional rotations of conic sections.

Essential Questions: How are the equations and graphs of different conic sections related to each other and how are they applied?

STANDARDS

Vocabulary

Reasoning, Problem Solving, & Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.C.1 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. (12-4)

            Conic sections

            Ellipse, hyperbola

            x-radius, y-radius

            Major axis, minor axis

            Asymptotes

            Quadric

            Paraboloid

            Transverse axis

            Conjugate axis

            Focus  - foci

            Directrix

            Eccentricity

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.B Select and apply strategies to solve problems.

PC.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

 

12-1

12-2

Quiz

*12-3 (optional)

12-4

Review

Unit Assessment

 

 

12-2a,b

 

 

12-4a, b, c

PC.C.2 Given a Cartesian or parametric equation of a conic section, sketch or plot the graph, and given the graph, find an equation. (12-1, 12-2)

Supporting Standards:

 

Set aside an Exploration day for 12-4, using the explorations listed above.

As a result of this unit, students will be able to:

            Graph a conic section from an equation.

            Given a graph of a conic section find the equation.

            Recognize conic sections from equations.

            Translate between Cartesian and parametric forms of equations for conic sections.

            Analyze conic sections to find foci, directrix, and eccentricity.

 


 

Unit III (Advanced Topics): Chapter 14 - Sequences and Series

Suggested Timeframe: 14 days (approximate unit completion date: June 1)

Unit Rationale: Geometric and arithmetic sequences and series are logical mathematical models for functions, including those involving compound interest, general growth and decay, and movement.

Essential Questions: How can sequences of numbers and their sums be applied to real-life problems?

STANDARDS

Vocabulary

Reasoning, Problem Solving,

& Communication

Planning

Chapter Sequence

Explorations

Focus Standards:

PC.S.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (14-1)

            Sequence, series

            Common difference

            Common ratios

            Arithmetic sequence

            Geometric sequence

            Partial sum

            Convergent, divergent

            Binomial series

PC.8.A Analyze a problem situation and represent it mathematically.

PC.8.B Select and apply strategies to solve problems.

PC.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

 

14-1

14-2

14-3

14-4 Review

Unit Assessment

 

14-1a

14-2a

14-3b,c

 

 

PC.S.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (14-2)

PC.S.3 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (14-2)

PC.S. 4 Understand, explain and use formulas for the sums of finite arithmetic and geometric sequences. (14-3)

PC.S.5 Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascals Triangle. (14-3)

PC. S. 6 Compute the sums of infinite geometric series; understand and apply the convergence criteria for geometric series. (14-3)

Supporting Standards:

 

 

 

As a result of this unit, students will be able to:

            Expand a sequence.

            Find a partial sum of a series.

            Express a sequence or series recursively and explicitly.

            Find a specific term or specific partial sum of a sequence or series.

            Find the sum of an infinite geometric series, and recognize when an infinite sum is possible.

            Use sigma notation to express sums.

            Expand a power of a binomial as a binomial series, using the Binomial Formula.

            Recognize and understand what it means to be a converging or diverging series.


Learning Resources Required and/or Suggested Texts

Text

PRE-CALCULUS WITH TRIGONOMETRY: Concepts and Applications, Paul A. Foerster, 2nd Edition, 2007

 

Other resources

TI Graphing Calculator – TI- nSpire or TI-84

 

Course Requirements

Mastery

Students are expected to reach mastery level of all concepts explored in this class.  Our goal is to work at a speed so that the students meet that goal.  We get to where we get in preparation for Calculus.

 

Attendance

Participants must attend all course sessions.  If a participant must miss any portion of a session, the instructor should be notified prior to the absence in order to make appropriate arrangements. All sessions are archived on WIZIQ and students can re-watch any sessions – use the same url as the class session.

 

Grades

This course is for letter grade only: A or Inc.  Successful completion includes mastering (100%) all quizzes & exams.  Students will have ample opportunity to retake every exam t reach mastery level.