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Functions_ Statistics and Trigonometry _FST_

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					Functions, Statistics and
Trigonometry (FST)
Course Outcome Summary

Information
Credits                                   1
Organization                              Hartford Union High School:



Mission/Description
Mission Statement: Our mathematics program is designed to enable all students to function successfully
in a number related world, to give students a basis to increase their mathematical knowledge and improve
their quality of life, and to develop an understanding and appreciation of mathematics and technology.


Course Description: This course is designed as a college preparatory course for the student who is not
necessarily majoring in a mathematics or science field. Successful completion of this course should
prepare the student for Pre-Calculus. The topics will include functions, logarithms, trigonometry,
probability, and statistics. The student will be required to have a graphing calculator (TI83 Plus or TI83
recommended. Cost approximately $100.)



Textbooks
Advanced Mathematical Concepts



Units and Timelines
Linear Relations and Functions - 10 days
Exploring Graphs of Functions - 15 days
Solving Polynomial and Rational Functions - 20 days
Exploring Conic Sections - 15 days
Exploring Exponential and Logarithmic Functions - 15 days
Sequences and Series - 15 days
The Trigonometric functions - 15 days
Graphs and Inverses of the Trigonometric functions - 15 days
Trigonometric Identities and Equations - 15 days
Vectors and Parametric Equations - 10 days
Combinatorics and Probability - 15 days
Statistics and Data Analysis - 20 days



Core Abilities
Analyze divergent views to resolve conflicts.
Assess personal goals, values, and motivations.
Assume responsibility for own actions.
Balance a lifestyle that is safe and healthy.
Demonstrate appropriate communication skills.
Devise solutions to complex problems.
Initiate positive leadership skills.
Integrate self awareness and attitudes appropriate in diverse social situations.
Organize data into useable forms.
Utilize appropriate behaviors and attitudes in the work environment.
Utilize technology to improve efficiency.



Competencies, Linked Standards, Objectives and
Performance Standards
1.     Use statistics to describe data sets.
       Properties
       Domain: Cognitive
       Level: Application
       Difficulty: Medium
       Importance: Important
       Linked External Standards
                WI.MA.A.12.1 Use reason and logic
                WI.MA.A.12.2 Communicate logical arguments and clearly show why a result does or
                does not make sense, why the reasoning is or is not valid, and an understanding of the
                difference between examples that support a conjecture and a proof of the conjecture
                WI.MA.A.12.3 Analyze non-routine problems and arrive at solutions by various means
                WI.MA.A.12.4 Develop effective oral and written presentations employing correct
                mathematical terminology, notation, symbols, and conventions for mathematical
                arguments and display of data
                WI.MA.A.12.5 Organize work and present mathematical procedures and results clearly,
                systematically, succinctly, and correctly
                WI.MA.A.12.6 Read and understand mathematical texts and other instructional
                materials, writing about mathematics (e.g., articles in journals), and mathematical ideas
                as they are used in other contexts
                WI.MA.B.12.1 Use complex counting procedures such as union and intersection of sets
                and arrangements (permutations and combinations) to solve problems
              WI.MA.E.12.1 Work with data in the context of real-world situations
              WI.MA.E.12.2 Organize and display data from statistical investigations
              WI.MA.E.12.3 Interpret and analyze information from organized and displayed data
              when given measures of dispersion, including standard deviation and variance; measures
              of reliability; and measures of correlation
              WI.MA.E.12.4 Analyze, evaluate, and critique the methods and conclusions of
              statistical experiments reported in journals, magazines, news media, advertising, etc.
     Performance Standards
     o     learner calculates statistics from a data set and display the results in a graph with 75%
           accuracy.
     o        learner determines the probability that a randomly selected member of a sample lies
              within a specified number of standard deviations from the mean.
     o        learner determines the standard error of the mean to predict the true mean of a
              population with a 1% and 5% level of confidence.
     Learning objectives
     a.    Calculate measures of central tendency and spread for data sets.
     b.       Describe relations between measures of central tendency or measures of spread.
     c.       Use sigma notation to represent a sum, mean, mean deviation, standard deviation or
              variance.
     d.       Interpret data presented in a frequency distribution.
     e.       Read and interpret histograms, stem-and-leaf plots, and box-and-whisker plots.
     f.       Draw a histogram, frequency polygon, stem-and-leaf plot, and box-and-whisker plot, to
              display data.
     g.       Use the normal distribution curve to represent data.
     h.       Use samples to make inferences about populations.


2.   Describe the nature of graphs by examining transformations, symmetry, continuity,
     extrema, end behavior, and asymptotes.
     Properties
     Domain: Cognitive
     Level: Comprehension
     Difficulty: Low
     Importance: Important
     Linked External Standards
              WI.MA.A.12.1 Use reason and logic
              WI.MA.A.12.4 Develop effective oral and written presentations employing correct
              mathematical terminology, notation, symbols, and conventions for mathematical
              arguments and display of data
              WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
              procedures to describe and characterize geometric properties and relationships such as
              slope, intercepts, parallelism, and perpendicularity
              WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
              in a variety of ways
     Performance Standards
     o     learner describes how transformations affect important features of graphs and how those
           effects are represented in the changes in equations with 75% accuracy.
     o        learner analyzes all aspects of a function using appropriate tests with 75% accuracy.
     Learning objectives
     a.    Describe the symmetries of graphs.
     b.       Identify transformations of the constant, identity, polynomial, square root, absolute
              value, greatest integer, and rational functions.
     c.       Graph nonlinear inequalities.
     d.       Solve absolute value inequalities.
     e.       Determine the inverse of a function.
     f.       Graph functions and their inverses.
     g.       Determine whether a function is continuous or discontinuous.
     h.       Identify the end behavior of functions .
     i.       Determine whether a function is increasing or decreasing on an interval.
     j.       Find the critical points of a function.
     k.       Use the first and second derivative tests to determine whether a critical point is a
              maximum, minimum, or inflection point.
     l.       Graph rational functions.
     m.       Determine vertical, horizontal, and slant asymptotes of a function.


3.   Solve exponential and logarithmic equations.
     Properties
     Domain: Cognitive
     Level: Application
     Difficulty: High
     Importance: Important
     Linked External Standards
              WI.MA.A.12.4 Develop effective oral and written presentations employing correct
              mathematical terminology, notation, symbols, and conventions for mathematical
              arguments and display of data
              WI.MA.A.12.5 Organize work and present mathematical procedures and results clearly,
              systematically, succinctly, and correctly
              WI.MA.B.12.3 Perform and explain operations on real numbers
              WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations
              involving the application of different number systems (natural, integers, rational, and
              real)
              WI.MA.B.12.5 Create and critically evaluate numerical arguments presented in a
              variety of classroom and real-world situations (e.g., political, economic, scientific, social
              WI.MA.E.12.1 Work with data in the context of real-world situations
              WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse
              variation) and numerical sequences, and then represent them with algebraic expressions
              and equations
              WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
              in a variety of ways
              WI.MA.F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of
              linear equations and inequalities
              WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
              using algebraic expressions, equations, and inequalities
     Performance Standards
     o     learner applies the properties of logarithms to solve equations with 75% accuracy.
     Learning objectives
     a.    Find the value of logarithmic expressions.
     b.       Describe properties of exponential and logarithmic functions.
     c.       Graph exponential and logarithmic functions.
     d.       Use e and natural logarithms to solve applied problems.
     e.       Use logarithmic and exponential functions to model data.


4.   Use trigonometry to find sides or angles in triangles.
     Properties
     Domain: Cognitive
     Level: Application
     Difficulty: Medium
     Importance: Essential
     Linked External Standards
              WI.MA.A.12.1 Use reason and logic
              WI.MA.A.12.3 Analyze non-routine problems and arrive at solutions by various means
              WI.MA.A.12.4 Develop effective oral and written presentations employing correct
              mathematical terminology, notation, symbols, and conventions for mathematical
              arguments and display of data
              WI.MA.A.12.5 Organize work and present mathematical procedures and results clearly,
              systematically, succinctly, and correctly
              WI.MA.C.12.1 Identify, describe, and analyze properties of figures, relationships
              among figures and relationships among their parts
              WI.MA.C.12.2 Use geometric models to solve mathematical and real-world problems
              WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
              procedures to describe and characterize geometric properties and relationships such as
              slope, intercepts, parallelism, and perpendicularity
              WI.MA.C.12.5 Identify and demonstrate an understanding of the three ratios used in
              right-triangle trigonometry (sine, cosine, tangent)
              WI.MA.D.12.3 Determine measurements indirectly
              WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
              using algebraic expressions, equations, and inequalities
     Performance Standards
     o     learner uses properties of trigonometry to solve for missing dimensions in a triangle with
           75% accuracy.
     Learning objectives
     a.    Convert decimal degree measures to degrees, minutes, and seconds and vice versa.
     b.       Find sines, cosines, and tangents of acute angles using the trigonometric ratios.
     c.       Find missing side and/or angle measures of a triangle using the trigonometric ratios.
     d.       Solve problems involving the Law of Sines and the Law of Cosines.
     e.       Determine the area of a triangle given the measure of two sides and the included angle,
              the measure of two angles and a side, or the measure of the three sides.


5.   Solve problems involving circular functions and their graphs.
     Properties
     Domain: Cognitive
     Level: Application
     Difficulty: Medium
     Importance: Important
     Linked External Standards
              WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
              procedures to describe and characterize geometric properties and relationships such as
              slope, intercepts, parallelism, and perpendicularity
              WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
              using algebraic expressions, equations, and inequalities
     Performance Standards
     o     learner solves equations involving circular functions with 75% accuracy.
     o        learner graphs the 6 circular functions and be able to graph their image under a linear
              transformation with 75% accuracy.
     o        learner uses sinusoidal functions to solve problems with 75% accuracy.
     Learning objectives
     a.    Identify the amplitude, period, frequency, vertical translation, and phase shift of circular
           functions.
     b.       Evaluate inverse and reciprocal circular functions.
     c.       State properties of inverse and reciprocal circular functions.
     d.       Graph or identify graphs of circular functions.
     e.       Graph or identify graphs of inverse functions.
     f.       Model real-world data using sine and cosine functions.


6.   Use the unit circle to find the values of the trigonometric functions for any angle.
     Linked External Standards
              WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations
              involving the application of different number systems (natural, integers, rational, and
              real)
              WI.MA.C.12.2 Use geometric models to solve mathematical and real-world problems
              WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
              procedures to describe and characterize geometric properties and relationships such as
              slope, intercepts, parallelism, and perpendicularity
              WI.MA.C.12.5 Identify and demonstrate an understanding of the three ratios used in
              right-triangle trigonometry (sine, cosine, tangent)
     Performance Standards
     o     learner labels the unit circle with 100% accuracy.
     o        learner uses properties of the unit circle to find the value of the six trigonometric ratios
              with 75% accuracy.
     Learning objectives
     a.    Identify angles that are conterminal.
     b.    Identify the key points on the unit circle.
     c.       Convert between degrees and radians.
     d.       Find the values of the six trigonometric functions of an angle in standard position given
              a point on its terminal side.
     e.       Identify the number of degrees or radians in a given number of rotations.


7.   Solve problems using combinatorics and probability.
     Properties
     Domain: Cognitive
     Level: Application
     Difficulty: High
     Importance: Essential
     Linked External Standards
              WI.MA.A.12.1 Use reason and logic
              WI.MA.B.12.1 Use complex counting procedures such as union and intersection of sets
              and arrangements (permutations and combinations) to solve problems
              WI.MA.E.12.1 Work with data in the context of real-world situations
              WI.MA.E.12.5 Determine the likelihood of occurrence of complex events
     Performance Standards
     o     learner applies the properties and formulas for counting to solve probability problems
           with 75% accuracy.
     Learning objectives
     a.    Solve problems related to the Basic Counting Principle.
     b.       Distinguish between dependent and independent events.
     c.       Use combinations to compute the number of ways of selecting objects.
     d.       Solve problems involving permutations and/or combinations.
     e.       Solve problems involving permutations with repetitions.
     f.       Solve problems involving circular permutations.
     g.       Find the probability of an event.
     h.       Find the odds for an event.
     i.       Find the probability of independent and dependent events.
     j.       Identify mutually exclusive events.
     k.       Find the probability of mutually exclusive and inclusive events.
     l.       Find the probability of an event given the occurrence of another event.


8.   Calculate or approximate zeros of polynomial functions.
     Properties
     Domain: Cognitive
     Level: Application
     Difficulty: Medium
     Importance: Essential
     Linked External Standards
              WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
              procedures to describe and characterize geometric properties and relationships such as
              slope, intercepts, parallelism, and perpendicularity
              WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
              in a variety of ways
              WI.MA.F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of
              linear equations and inequalities
     Performance Standards
     o     learner uses the techniques of solving polynomials to approximate all the zeroes of a
           function with 75 % accuracy.
     Learning objectives
     a.    Determine roots of polynomial equations.
     b.       Apply the Fundamental Theorem of Algebra.
     c.       Solve quadratic equations by factoring and using the Quadratic Formula.
     d.       Describe the roots of a quadratic function using the discriminant.
     e.       Find factors using the Remainder and Factor Theorems.
     f.       Apply the Rational Root Theorem to identify all possible rational roots.
     g.       Determine the number of positive and negative real roots of a function.
     h.       Approximate the real zeros of a polynomial function.
     i.       Solve rational equations and inequalities.
     j.       Decompose a fraction into partial fractions.
     k.       Solve radical equations and inequalities.
     l.       Model real-world data using polynomial functions.
     m.       Interpret real-world data using polynomial functions.
9.    Examine the conic sections.
      Properties
      Domain: Cognitive
      Level: Analysis
      Difficulty: High
      Importance: Important
      Linked External Standards
               WI.MA.A.12.1 Use reason and logic
               WI.MA.C.12.1 Identify, describe, and analyze properties of figures, relationships
               among figures and relationships among their parts
               WI.MA.C.12.2 Use geometric models to solve mathematical and real-world problems
               WI.MA.C.12.4 Use the two-dimensional rectangular coordinate system and algebraic
               procedures to describe and characterize geometric properties and relationships such as
               slope, intercepts, parallelism, and perpendicularity
               WI.MA.F.12.2 Use mathematical functions (e.g., linear, exponential, quadratic, power)
               in a variety of ways
               WI.MA.F.12.3 Solve linear and quadratic equations, linear inequalities, and systems of
               linear equations and inequalities
      Performance Standards
      o     learner identifies a conic section from its graph or equation with 75% accuracy.
      o     learner writes an equation to represent a graph of a conic section with 75% accuracy.
      o        learner solves a system of second degree equations with 75% accuracy.
      Learning objectives
      a.    State the properties of each of the conic sections.
      b.    Use properties of conic sections (circles, ellipses, hyperbolas, and parabolas) to write
            equations describing them.
      c.    Rewrite the general equations of conic sections in standard form.
      d.       Identify conic sections by their equations in standard form.
      e.       Identify conic sections by their equations in general form.
      f.       Graph the conic sections.
      g.       Graph systems of second degree equations and inequalities.
      h.       Solve systems of second degree equations and inequalities.


10.   Study the polar coordinate system.
      Properties
      Domain: Cognitive
      Level: Analysis
      Difficulty: Low
      Importance: Essential
      Performance Standards
      o     learner converts between the rectangular coordinate system and the polar coordinate
            system by using the conversion formulas with 75% accuracy.
      o        learner graphs in the polar coordinate system with 75% accuracy.
      Learning objectives
      a.    Graph points in polar coordinates.
      b.       Determine the distance between two points with polar coordinates.
      c.       Graph polar equations.
      d.       Convert between polar and rectangular coordinates.
      e.       Write the polar from of a linear equation.
      f.       Graph the polar form of a linear equation.


11.   Calculate powers and roots of complex numbers.
      Properties
      Domain: Cognitive
      Level: Application
      Difficulty: Medium
      Importance: Useful
      Performance Standards
      o     learner finds powers and roots of complex numbers with 75% accuracy.
      Learning objectives
      a.    Add, subtract, multiply and divide complex numbers.
      b.       Graph complex numbers.
      c.       Represent complex numbers in different forms.
      d.       Explore geometric relationships in the complex plane.
      e.       Find the product and quotient of complex numbers in polar form.
      f.       Use De Moivre's Theorem to find powers and roots of complex numbers in polar form.


12.   Perform operations with and solve problems using vectors.
      Linked External Standards
               WI.MA.D.12.2 Select and use tools with appropriate degree of precision to determine
               measurements directly within specified degrees of accuracy and error (tolerance)
               WI.MA.D.12.3 Determine measurements indirectly
      Performance Standards
      o     learner performs operations and solves problems using vectors with 75% accuracy.
      Learning objectives
      a.    Find equal, opposite, and parallel vectors.
      b.       Add and subtract vectors geometrically.
      c.       Find ordered pairs that represent vectors.
      d.       Add, subtract, multiply, and find the magnitude of vectors algebraically.
e.   Add, subtract,and find the magnitude of the vectors in three-dimensional space.
f.   Find the inner and cross products of two vectors.
g.   Determine whether two vectors are perpendicular.
h.   Solve problems using vectors and trigonometry.

				
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