SCHOLARS ON-LINE
Course Syllabus
PRE-CALCULUS
2018-2019
PRE-CALCULUS
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Instructor |
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ARTHUR MABBOTT MATHGUY SCHOLARS ON-LINE ART@MABBOTT.ORG (206) 605-7393 |
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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
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SYNCHRONIS SESSIONS |
Start Time |
End Time |
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MONDAY |
11:00AM |
12:00PM |
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WEDNESDAY |
11:00AM |
12:00PM |
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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
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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September
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Day
1 |
2 |
3 |
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4 |
5 |
6 |
7 |
8 |
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9 |
10 |
11 |
12 |
13 |
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14 |
15 |
16 |
17 |
18 |
Chapter 7 – 22 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
19 |
20 |
21 |
22 |
23 |
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24 |
25 |
26 |
27 |
28 |
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29 |
30 |
31 |
32 |
33 |
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34 |
35 |
36 |
37 |
38 |
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39 |
40 |
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Chapter 15 – 16 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
41 |
42 |
42 |
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44 |
45 |
46 |
47 |
48 |
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49 |
50 |
51 |
52 |
53 |
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54 |
55 |
56 |
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Chapter 2 – 15 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
57 |
58 |
59 |
60 |
61 |
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62 |
63 |
64 |
65 |
66 |
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67 |
68 |
59 |
70 |
71 |
Christmas Break
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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December
24 |
25 |
26 |
27 |
28 |
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31 |
January
1 |
2 |
3 |
4 |
Chapter 3 – 19 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
72 |
73 |
74 |
75 |
76 |
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77 |
78 |
79 |
80 |
81 |
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82 |
83 |
84 |
85 |
86 |
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87 |
88 |
89 |
90 |
91 |
Chapter 4 – 25 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
92 |
93 |
94 |
95 |
96 |
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97 |
98 |
99 |
100 |
101 |
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102 |
103 |
104 |
105 |
106 |
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107 |
108 |
109 |
110 |
111 |
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112 |
113 |
114 |
115 |
116 |
Chapter 5 – 20 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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Day
117 |
118 |
119 |
120 |
121 |
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122 |
123 |
124 |
125 |
126 |
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127 |
128 |
129 |
130 |
131 |
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EASTER BREAK |
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132 |
133 |
134 |
135 |
136 |
Chapter 6 – 14 Days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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137 |
138 |
139 |
140 |
141 |
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142 |
143 |
144 |
145 |
146 |
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147 |
148 |
149 |
150 |
151 |
Chapter 11 – 10 days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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152 |
153 |
154 |
155 |
156 |
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157 |
158 |
159 |
160 |
161 |
Chapter 12 – 15 days
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Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
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162 |
163 |
164 |
165 |
166 |
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167 |
168 |
169 |
170 |
171 |
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172 |
173 |
174 |
175 |
176 |
Course Agenda
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Unit I
(Functions): Chapter 1 - Functions and Mathematical Models |
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Suggested Timeframe: 21 Days (approximate unit
completion date: October 5) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning, Problem Solving, &
Communication |
Planning |
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Chapter Sequence |
Explorations |
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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) |
á
Domain/range á
Argument of
the function á
Asymptote á
Discontinuity á
Boolean
variable á
Restricted
domain á
Dilation,
reflection, translations, á
Displacement
vs. distance á
Even and odd
functions á
Domain and
range of composite functions á
Invertible,
one-to-one function á
Strictly
increasing and strictly decreasing á
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 |
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PC.F.4 Perform
algebraic operations on functions (polynomial, power, exponential, rational,
absolute value, etc.) and apply transformations. (1.3, 1.6) |
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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) |
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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) |
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Supporting
Standards: |
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As a result of this unit,
students will be able to: á
Use multiple
representations of polynomial, quadratic, linear, direct variation, power,
exponential, inverse variation, and rational functions. á
Use transformations
to identify functions from the parent in a function family. á
Compose
functions. á
Understand and
apply inverse and absolute value functions. |
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Unit I
(Functions): Chapter 7 - Properties of Elementary Functions |
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Suggested Timeframe: 18 Days (approximate unit
completion date: November 1) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning, Problem Solving, &
Communication |
Planning |
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Chapter Sequence |
Explorations |
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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) |
á
Proportionality
constant á
Second and
third difference á
Common
logarithm á
Natural
logarithm á
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 |
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PC.F.4 Perform
algebraic operations on functions (polynomial, power, exponential, rational,
absolute value, etc.) and apply transformations. |
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PC.F.12 Compare
the rates of change of functions in different families (linear, quadratic,
exponential, and power). (7.3) |
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PC.E.1 Use the
inverse relationship between exponential and logarithmic functions to solve
equations and problems. (7.4, 7.5) |
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PC.E.2 Solve
exponential and logarithmic equations algebraically and graphically. (7.4,
7.5, 7.6) |
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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. |
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Supporting
Standards: |
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As a result of this unit,
students will be able to: á
Recognize
various functions from patterns, graphs, equations, and/or tables of values. á
Use the
properties of exponents and logarithms to simplify expressions and solve
equations. á
Use
exponential and logarithmic functions to solve real-world problems involving
growth and decay. |
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Unit I
(Functions): Chapter 15 - Polynomial and Rational Functions |
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Suggested Timeframe: 15 Days (approximate unit
completion date: November 23) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning, Problem Solving, &
Communication |
Planning |
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Chapter Sequence |
Explorations |
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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) |
á
Critical point á
End behavior á
Remainder
Theorem á
Discontinuous á
Removable
discontinuity á
Indeterminate
form á
Infinite form á
Limit á
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 |
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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) |
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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) |
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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. |
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Supporting Standards: PC.F.2 Find the domains and ranges of functions. |
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As a result of
this unit, students will be able to: á
Determine the
degree of a polynomial from its graph. á
Find the zeros
of a polynomial from graph or equation. á
Use the
Remainder Theorem to find the zeros of a polynomial function. á
Determine
whether and how many zeros a polynomial function may have. á
Know how
complex solutions to a polynomial equation translate to the graphic
representation of that equation. á
Determine if a
polynomial model fits a set of data. á
Simplify
rational expressions, solve rational equations, and analyze the graphs of
rational functions. á
Use partial
fractions to solve or graph rational equations or functions. |
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Unit II (Trigonometry): Chapter 2 - Periodic Functions and
Right Triangle Problems |
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Suggested Timeframe: 15 days (approximate unit
completion date: December 16) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning,
Problem Solving, & Communication |
Planning |
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Chapter
Sequence |
Explorations |
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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) |
á
Periodic,
sinusoid, á
Standard
position á
Co-terminal á
Reference
angle á
Cycle á
Period á
Sine, cosine,
tangent, cotangent, secant, and cosecant á
Inverse
trigonometric function á
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 |
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PC.T.2 Students
know the definition of sine and cosine as y- and x-coordinates of points on
the unit circle. (2.4) |
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PC.T.3 Students
can compute (by hand) the values of six trigonometric functions at any
standard point on the unit circle. (2.4) |
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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, |
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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. |
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PC.T.9 Use
trigonometry to determine unknown sides or angles in right triangles. (2.5) |
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Supporting Standards: |
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As a result of this unit,
students will be able to: á
Use the unit
circle values to define trigonometric functions. á
Extend
definitions for sine and cosine to tangent, cotangent, secant, and cosecant. á
Use inverse
trigonometric functions to solve equations and problems. á
Graph sine and
cosine functions. á
Solve for
missing parts of right triangles. |
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|
Unit
II (Trigonometry): Chapter 3 - Applications of Trigonometric and Circular
Functions |
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Suggested Timeframe: 17 Days (approximate unit
completion date: January 26) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning, Problem Solving, &
Communication |
Planning |
|
|
Chapter Sequence |
Explorations |
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Focus Standards: PC.T.5 Find and use a sinusoidal function to model a given data set or
situation. (3.1,3.7) |
á
Amplitude á
Phase
displacement á
Sinusoidal
Axis á
Cycle á
Frequency á
Convex/Concave á
Point of
inflection á
Upper
bound/Lower bound á
Asymptotes á
Radian á
Arc length á
Circular
functions á
Arc functions á
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) |
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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) |
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Supporting
Standards: PC.T.6 Know
the definitions of the inverse trigonometric functions, including their
domains and ranges; recognize their graphs. (3.6) |
NOTE: |
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As a result of
this unit, students will be able to: á
Write a
particular equation for a sinusoid that fits any given conditions. á
Graph any
trigonometric function from its equation or given amplitude, period or frequency, phase displacement,
and sinusoidal axis. á
Find a
trigonometric equation from its graph or given amplitude, period, phase
displacement, and sinusoidal axis. á
Find the
amplitude, period, phase displacement, and sinusoidal axis from an equation
or graph. |
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|
Unit II (Trigonometry): Chapter 4 - Trigonometric
Function Properties: Identities |
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Suggested Timeframe: 20 Days (approximate unit
completion date: March 2) |
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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? |
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STANDARDS |
Vocabulary |
Reasoning, Problem Solving, & Communication |
Planning |
|
|
Chapter Sequence |
Explorations |
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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) |
á
Pythagorean
property á
Reciprocal
property á
Quotient
property á
Identity á
Conjugate á
Closed
interval á
Open interval
(Notation) á
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 |
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PC.T.8 Solve trigonometric equations using basic identities and inverse trigonometric functions. (4.4) |
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PC.T.6 Know the definitions of the inverse trigonometric
functions, including their domains and ranges; recognize their graphs. (4.6) |
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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. |
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|
As a result of
this unit, students will be able to: á
Transform a
trigonometric expression to an equivalent expression. á
Use inverse
trigonometric functions to solve equations. á
Find the
graphs and characteristics of inverse trigonometric functions. á
Recognize that
some triangles have two possible solutions (ambiguous case). |
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|
Unit II
(Trigonometry): Chapter 5 - Properties of Combined Sinusoids |
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|
Suggested Timeframe: 6 days (approximate unit
completion date: March 12) |
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|
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? |
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|
STANDARDS |
Vocabulary |
Reasoning, Problem Solving, &
Communication |
Planning |
|
|
Chapter Sequence |
Explorations |
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|
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) |
á
Cofunctions á
Odd and even
functions á
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) |
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|
Supporting
Standards: |
NOTE: Review problems from 5.7 should
include only those sections covered in this unit. |
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|
As a result of
this unit, students will be able to: á
Derive and use
properties for trigonometric functions of sums and differences of angles. á
Derive and use
properties of products of arguments, including double- and half-arguments. á
Use algebraic
techniques to solve trigonometric equations. |
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|
Unit II (Trigonometry): Chapter 6 - Triangle Trigonometry |
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|
Suggested Timeframe: 14 Days (approximate unit
completion date: March 30 |
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|
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? |
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|
STANDARDS |
Vocabulary |
Reasoning,
Problem Solving, & Communication |
Planning |
|
|
Chapter
Sequence |
Explorations |
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|
Focus Standards: PC.T.10 Use law of sines and law of cosines to solve problems. (6-2,
6-4, 6-5, 6-7) |
á
Oblique
triangle á
Law of Sines á
Law of Cosines á
Semi-perimeter á
HeroÕs Formula á
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: |
|
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|
As a result of
this unit, students will be able to: á
Solve an
oblique triangle, given enough information. á
Apply the Laws
of Sines and Cosines to solve triangles. á
Find the area
of triangles using trigonometry, including HeroÕs formula. |
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|
Unit III
(Advanced Topics): Chapter 11 - Matrices – Operations |
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|
Suggested Timeframe: 10 days (approximate unit
completion date: April 13) |
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|
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? |
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|
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) |
á
Matrix á
Scalar á
Determinant á
Adjoint á
Transformation matrix á
Image matrix á
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) |
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|
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) |
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|
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. |
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|
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. |
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|
As a result of
this unit, students will be able to: á
Perform
operations with matrices. á
Use matrices
to rotate and dilate figures on the coordinate plane. |
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|
Unit III
(Advanced Topics): Chapter 12 - Analytic Geometry of Conic Sections and Quadric
Surfaces |
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|
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) |
á
Conic sections á
Ellipse,
hyperbola á
x-radius,
y-radius á
Major axis,
minor axis á
Asymptotes á
Quadric á
Paraboloid á
Transverse axis á
Conjugate axis á
Focus -
foci á
Directrix á
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: á
Graph a conic
section from an equation. á
Given a graph
of a conic section find the equation. á
Recognize
conic sections from equations. á
Translate
between Cartesian and parametric forms of equations for conic sections. á
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) |
á
Sequence,
series á
Common
difference á
Common ratios á
Arithmetic
sequence á
Geometric
sequence á
Partial sum á
Convergent,
divergent á
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 PascalÕs 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: á
Expand a
sequence. á
Find a partial
sum of a series. á
Express a
sequence or series recursively and explicitly. á
Find a
specific term or specific partial sum of a sequence or series. á
Find the sum
of an infinite geometric series, and recognize when an infinite sum is
possible. á
Use sigma
notation to express sums. á
Expand a power
of a binomial as a binomial series, using the Binomial Formula. á
Recognize and
understand what it means to be a converging or diverging series. |
||||
PRE-CALCULUS WITH
TRIGONOMETRY: Concepts and Applications, Paul A. Foerster, 2nd
Edition, 2007
TI Graphing Calculator
– TI- nSpire or TI-84
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.
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.
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.