Download PreCalculus - District 303

MATH - Fourth Course
CUSD 303
Year: 2012-2013
Content
Cluster Standard
The Complex
Perform arithmetic
Number System operations with complex
numbers
Represent complex
numbers and their
operations on the complex
plane
Vector
Quantities and
Matrices
Represent and model with
vector quantities
Standard
Skill Statement
4th.NCN3 Find the conjugate of a complex number; use conjugates 4th.NCN3 Find the conjugate of a complex number
to find moduli and quotients of complex numbers
4th.NCN3 Calculate the moduli
4th.NCN3 Apply the conjugate to calculate the quotient of complex
numbers
4th.NCN4 Represent complex numbers on the complex plane in
4th.NCN4 Graph complex numbers on the complex plane in
rectangular and polar form (including real and imaginary numbers), rectangular form (including real and imaginary numbers)
and explain why the rectangular and polar forms of a given complex 4th.NCN4 Graph complex numbers in polar form (including real and
number represent the same number
imaginary numbers)
4th.NCN4 Algebraically convert between rectangular and polar
forms of a given complex number
4th.NCN5 Represent addition, subtraction, multiplication, and
4th.NCN5 Geometrically represent addition, subtraction,
conjugation of complex numbers geometrically on the complex
multiplication and conjugation of complex numbers on the complex
plane; use properties of this representation for computation For
plane
example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and
4th.NCN5 Calculate the addition, subtraction, multiplication and
argument 120
conjugation of complex numbers
4th.NCN6 Calculate the distance between numbers in the complex 4th.NCN6 Calculate the distance between numbers in the complex
plane as the modulus of the difference, and the midpoint of a
plane as the modulus of the difference
segment as the average of the numbers at its endpoints
4th.NCN6 Calculate the midpoint of a segment in the complex plane
as the average of the numbers at its endpoints
4th.NVM1 Recognize vector quantities as having both magnitude
4th.NVM1 Recognize vector quantities as having both magnitude
and direction Represent vector quantities by directed line segments, and direction
and use appropriate symbols for vectors and their magnitudes (e.g., 4th.NVM1 Represent vector quantities by directed line segments,
v, |v|, ||v||, v)
and use appropriate symbols for vectors and their magnitudes (e.g.,
v, |v|, ||v||, v)
4th.NVM2 Find the components of a vector by subtracting the
4th.NVM2 Calculate the components of a vector by subtracting the
coordinates of an initial point from the coordinates of a terminal point coordinates of an initial point from the coordinates of a terminal point
Perform operations on
vectors
4th.NVM3 Solve problems involving velocity and other quantities that
can be represented by vectors
4th.NVM4a Add vectors end-to-end, component-wise, and by the
parallelogram rule Understand that the magnitude of a sum of two
vectors is typically not the sum of the magnitudes
4th.NVM4b Given two vectors in magnitude and direction form,
determine the magnitude and direction of their sum
4th.NVM4c Understand vector subtraction v – w as v + (–w), where
–w is the additive inverse of w, with the same magnitude as w and
pointing in the opposite direction Represent vector subtraction
graphically by connecting the tips in the appropriate order, and
perform vector subtraction component-wise
4th.NVM5a Represent scalar multiplication graphically by scaling
vectors and possibly reversing their direction; perform scalar
multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy)
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4th.NVM3 Solve problems involving velocity and other quantities that
can be represented by vectors
4th.NVM4a Add vectors end-to-end, component-wise, and by the
parallelogram rule
4th.NVM4a Recognize that the magnitude of a sum of two vectors is
typically not the sum of the magnitudes
4th.NVM4b Given two vectors in magnitude and direction form,
determine the magnitude and direction of their sum
4th.NVM4c Interpret vector subtraction v – w as v + (–w), where –w
is the additive inverse of w, with the same magnitude as w and
pointing in the opposite direction
4th.NVM4c Represent vector subtraction graphically by connecting
the tips in the appropriate order
4th.NVM4c Perform vector subtraction component-wise
4th.NVM5a Represent scalar multiplication graphically by scaling
vectors and possibly reversing their direction
4th.NVM5a Perform scalar multiplication component-wise
Resources
Content
Cluster Standard
Represent and Perform operations on
model with
vectors (cont'd)
vector quantities
(cont'd)
Perform operations
on matrices and use
matrices in applications
Standard
4th.NVM5b Compute the magnitude of a scalar multiple cv using
||cv|| = |c|v Compute the direction of cv knowing that when |c|v ≠ 0,
the direction of cv is either along v (for c > 0) or against v (for c < 0)
Skill Statement
4th.NVM5b Compute the magnitude of a scalar multiple cv using
||cv|| = |c|v
4th.NVM5b Compute the direction of cv knowing that when |c|v ≠ 0,
the direction of cv is either along v (for c > 0) or against v (for c < 0)
4th.NVM6 Use matrices to represent and manipulate data, e.g., to
represent payoffs or incidence relationships in a network
4th.NVM6 Create matrices to represent and manipulate data
4th.NVM7 Multiply matrices by scalars to produce new matrices,
e.g., as when all of the payoffs in a game are doubled
4th.NVM7 Multiply matrices by scalars to produce new matrices
4th.NVM8 Add, subtract, and multiply matrices of appropriate
dimensions
4th.NVM9 Understand that, unlike multiplication of numbers, matrix
multiplication for square matrices is not a commutative operation,
but still satisfies the associative and distributive properties
4th.NVM8 Add, subtract, and multiply matrices of appropriate
dimensions
4th.NVM9 Establish that, unlike multiplication of numbers, matrix
multiplication for square matrices is not a commutative operation,
but still satisfies the associative and distributive properties
4th.NVM10 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
4th.NVM10 Establish 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
4th.NVM10 Establish that the determinant of a square matrix is
nonzero if and only if the matrix has a multiplicative inverse
4th.NVM11 Multiply a vector (regarded as a matrix with one column) 4th.NVM11 Multiply a vector (regarded as a matrix with one column)
by a matrix of suitable dimensions to produce another vector Work by a matrix of suitable dimensions to produce another vector
with matrices as transformations of vectors
4th.NVM11 Represent transformations of vectors with matrices
Reasoning with
Equations and
Inequalities
Interpreting
Functions
Building
Functions
4th.NVM12 Work with 2 × 2 matrices as transformations of the
4th.NVM12 Transform figures in the plane by multiplying 2 X 2
plane, and interpret the absolute value of the determinant in terms of matrices
area
4th.NVM12 Interpret the absolute value of the determinant in terms
of area
Solve systems of equations 4th.AREI8 Represent a system of linear equations as a single matrix 4th.AREI8 Represent a system of linear equations as a single matrix
equation in a vector variable
equation in a vector variable
4th.AREI9 Find the inverse of a matrix if it exists and use it to solve 4th.AREI9 Calculate the inverse of a matrix if it exists
systems of linear equations (using technology for matrices of
4th.AREI9 Solve systems of linear equations using inverse matrices
dimension 3 × 3 or greater)
(using technology for matrices of dimension 3 × 3 or greater)
Analyze functions using
different representations
4th.FIF7d Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end
behavior
Build a function that models 4th.FBF1c Compose functions For example, if T(y) is the
a relationship between two temperature in the atmosphere as a function of height, and h(t) is the
quantities
height of a weather balloon as a function of time, then T(h(t)) is the
temperature at the location of the weather balloon as a function of
time
4th.FIF7d Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end
behavior
4th.FBF1c Compose functions
Build new functions from
existing functions
4th.FBF4b Verify by composition that one function is the inverse of
another
4th.FBF4b Verify by composition that one function is the inverse of
another
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Resources
Content
Building
Functions
(cont'd)
Cluster Standard
Build new functions from
existing functions (cont'd)
Trigonometric
Functions
Extend the domain of
trigonometric functions
using the unit circle
Standard
4th.FBF4c Read values of an inverse function from a graph or a
table, given that the function has an inverse
4th.FBF4d Produce an invertible function from a non-invertible
function by restricting the domain
4th.FBF5 Understand the inverse relationship between exponents
and logarithms and use this relationship to solve problems involving
logarithms and exponents
Skill Statement
4th.FBF4c Read values of an inverse function from a graph or a
table, given that the function has an inverse
4th.FBF4d Produce an invertible function from a non-invertible
function by restricting the domain
4th.FBF5 Verify the inverse relationship between exponents and
logarithms
4th.FBF5 Apply inverse relationships to solve problems involving
logarithms and exponents
4th.FTF3 Use special triangles to determine geometrically the
4th.FTF3 Apply special triangle relationships to determine
values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6
circle to express the values of sine, cosine, and tangent for π–x,
π+x, and 2π–x in terms of their values for x, where x is any real
4th.FTF3 Compute the values of sine, cosine, and tangent for x,
number
π–x, π+x, and 2π–x using the unit circle for π/3, π/4 and π/6
Expressing
Geometric
Properties
with Equations
4th.FTF3 Compute the values of sine, cosine, and tangent where x
is any real number
4th.FTF4 Use the unit circle to explain symmetry (odd and even) and 4th.FTF4 Explain symmetry (odd and even) and periodicity of
periodicity of trigonometric functions
trigonometric functions using the unit circle
Model periodic phenomena 4th.FTF6 Understand that restricting a trigonometric function to a
4th.FTF6 Restrict a trigonometric function to a domain on which it is
with
domain on which it is always increasing or always decreasing allows always increasing or always decreasing to construct its inverse
trigonometric functions
its inverse to be constructed
4th.FTF7 Use inverse functions to solve trigonometric equations that 4th.FTF7 Solve trigonometric equations that arise in modeling
arise in modeling contexts; evaluate the solutions using technology, contexts by applying inverse trigonometric operations
and interpret them in terms of the context
4th.FTF7 Evaluate the solutions using technology
4th.FTF7 Interpret the solutions in terms of the context
Prove and apply
4th.FTF9 Prove the addition and subtraction formulas for sine,
4th.FTF 9 Prove the addition and subtraction formulas for sine,
trigonometric identities
cosine, and tangent and use them to solve problems
cosine, and tangent
4th.FTF9 Solve problems by applying addition and subtraction
formulas for sine, cosine, and tangent
Translate between the
4th.GGPE3 Derive the equations of ellipses and hyperbolas given
4th.GGPE3 Derive the equations of ellipses and hyperbolas given
geometric description and
the foci, using the fact that the sum or difference of distances from
the foci, using the fact that the sum or difference of distances from
the equation for a conic
the foci is constant
the foci is constant
section
Geometric
Measurement
and Dimension
Explain volume formulas
and use them to solve
problems
Using
Calculate expected values
Probability to
and use them to solve
Make Decisions problems
4th.GGMD2 Give an informal argument using Cavalieri’s principle
for the formulas for the volume of a sphere and other solid figures
4th.GGMD2 Give an informal argument using Cavalieri’s principle
for the formulas for the volume of a sphere and other solid figures
4th.SMD1 Define a random variable for a quantity of interest by
4th.SMD1 Define a random variable for a quantity of interest by
assigning a numerical value to each event in a sample space; graph assigning a numerical value to each event in a sample space
the corresponding probability distribution using the same graphical
displays as for data distributions
4th.SMD1 Graph the corresponding probability distribution using the
same graphical displays as for data distributions
4th.SMD2 Calculate the expected value of a random variable;
4th.SMD2 Calculate the expected value of a random variable;
interpret it as the mean of the probability distribution
interpret it as the mean of the probability distribution
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Resources
Content
Cluster Standard
Using
Calculate expected values
Probability to
and use them to solve
Make Decisions problems (cont'd)
(cont'd)
Standard
Skill Statement
4th.SMD3 Develop a probability distribution for a random variable
4th.SMD3 Develop a probability distribution for a random variable
defined for a sample space in which theoretical probabilities can be defined for a sample space in which theoretical probabilities can be
calculated; find the expected value For example, find the theoretical calculated
probability distribution for the number of correct answers obtained by
guessing on all five questions of a multiple-choice test where each
question has four choices, and find the expected grade under
various grading schemes
4th.SMD4 Develop a probability distribution for a random variable
4th.SMD4 Develop a probability distribution for a random variable
defined for a sample space in which probabilities are assigned
defined for a sample space in which probabilities are assigned
empirically; find the expected value For example, find a current data empirically; find the expected value
distribution on the number of TV sets per household in the United
States, and calculate the expected number of sets per household
How many TV sets would you expect to find in 100 randomly
selected households?
Use probability to evaluate
outcomes of decisions
Literacy of Math Craft and Structure
4th.SMD5a Find the expected payoff for a game of chance For
example, find the expected winnings from a state lottery ticket or a
game at a fast-food restaurant
4th.SMD5b Evaluate and compare strategies on the basis of
expected values For example, compare a high-deductible versus a
low-deductible automobile insurance policy using various, but
reasonable, chances of having a minor or a major accident
4th.SMD5a Find the expected payoff for a game of chance
4th.SMD5b Evaluate and compare strategies on the basis of
expected values
RST4 Interpret words and phrases as they are used in a text,
11/12RST4 Determine the meaning of symbols, key terms, and
including determining technical, connotative, and figurative
other domain-specific words and phrases as they are used in a
meanings, and analyze how specific word choices shape meaning or specific scientific or technical context
tone
Integration of Knowledge
and Ideas
RST7 Integrate and evaluate content presented in diverse media
11/12RST7 Integrate and evaluate multiple sources of information
and formats, including visually and quantitatively, as well as in words presented in diverse formats and media (e.g., quantitative data,
video, multimedia) in order to address a question or solve a problem
Text Type and Purposes
WHST2 Write informative/explanatory texts to examine and convey
complex ideas and information clearly and accurately through the
effective selection, organization, and analysis of content
11/12WHST2 Write informative/explanatory texts, including the
narration of historical events, scientific procedures/ experiments, or
technical processes
11/12WHST2a Introduce a topic and organize complex ideas,
concepts, and information so that each new element builds on that
which precedes it to create a unified whole; include formatting (e.g.,
headings), graphics (e.g., figures, tables), and multimedia when
useful to aiding comprehension
11/12WHST2b Develop the topic thoroughly by selecting the most
significant and relevant facts, extended definitions, concrete details,
quotations, or other information and examples appropriate to the
audience’s knowledge of the topic
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Resources
Content
Cluster Standard
Literacy of Math Text Type and Purposes
(cont'd)
(cont'd)
Standard
Skill Statement
11/12WHST2c Use varied transitions and sentence structures to link
the major sections of the text, create cohesion, and clarify the
relationships among complex ideas and concepts
11/12WHST2d Use precise language, domain-specific vocabulary
and techniques such as metaphor, simile, and analogy to manage
the complexity of the topic; convey a knowledgeable stance in a
style that responds to the discipline and context as well as to the
expertise of likely readers
11/12WHST2e Provide a concluding statement or section
that follows from and supports the information or explanation
provided (e.g., articulating implications or the significance of the
topic)
Mathematical
Practices
MP1 Make sense of problems and persevere in solving them
MP2 Reason abstractly and quantitatively
MP3 Construct viable arguments and critique the reasoning of
others
MP4 Model with mathematics
MP5 Use appropriate tools strategically
MP6 Attend to precision
MP7 Look for and make use of structure
MP8 Look for and express regularity in repeated reasoning
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Resources