Mr Hickman's Class 2022-2023
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PreCalculus 2021-2022


Prerequisites


Number Sets
  • Natural Numbers
  • Whole Numbers
  • Set of Integers
  • Rationals 
  • Irrationals
  • Real Numbers
  • Pure Imaginary Numbers
  • Complex Numbers

Notation and Abbreviations
  • Do you understand the shorthand used by advanced mathematics?
  • Can you translate verbal to symbolic?
  • Can you translate symbolic to verbal?
  • Can you describe specific portions of a graph, a function's domain, or a function's range using advanced notation?

Functions
  • Do you understand the notation, vocabulary, and abbreviations used to describe a function?
  • domain of a function
  • range of a function
  • end behavior of a function
  • asymptote of  a function
  • vertical asymptote of a function
  • horizontal asymptote of a function
  • slant asymptote of a function
  • oblique asymptote  of a function
  • reciprocal of a function
  • vertex of a quadratic function
  • vertex of an absolute value function
  • local extremes
  • local minimums
  • local maximums
  • absolute extremes
  • absolute minimum
  • absolute maximum
  • inverse function
  • one to one function
  • even function
  • odd function
  • continuous function
  • constant function
  • piecewise function
  • step functions
  • floor function
  • ceiling function
  • composition of functions
  • index of a radical function
  • radicand of a radical function
  • base of a logarithmic function
  • base of an exponential function
  • Can you perform and interpret the meaning of HLT and the VLT?
    • HLT = horizontal line test
    • VLT = vertical line test
  • Can you state the domain and range of a function using a variety of methods?
    • verbal
    • abbreviations
    • symbolic
  • Can you predict the integer or rational coordinates that lie on functions?
  • Can you perform operations with functions?
  • Can you perform composition with functions?
  • Do you understand the connection between a function and its inverse?
  • Can you write a piecewise function for an absolute value function or a quadratic function?
  • Can you determine a function's inverse when possible and be able to state when whether a singe inverse exists or not?

Exponent and Radical Laws
  • Do you know the exponent laws and definitions?
  • Do you know the radical laws and definitions?
  • Can you rewrite expressions that involve exponents?
  • Can you rewrite expressions that involve radicals? 
  • Can you rewrite an expression with negative exponents?
  • Can you convert an expression with fractional exponents to its related radical expression?
  • Can you convert an expression with a radical to its related  expression with  fractional exponents?
  • Can you convert an expression with fractional exponents to its related radical expression?

Factoring and Rewriting Expressions
  • Can you express a linear function in a variety of notations?
    • slope intercept
    • point slope
    • modified point slope
    • standard form
  • Can you express a quadratic function in a variety of notations?
    • standard form
    • vertex form
    • intercept form
  • ​Can you factor any quadratic expression?
  • Do you know the special patterns related to quadratic expressions?
    • DOTS-difference of two squares
    • PST-perfect square trinomials
    • missing constant quadratics
  • Can you factor specific cubic polynomials?
    • DOC-difference of cubes
    • SOC-sum of cubes
    • Factor by Grouping
    • missing constant cubics
    • Factoring using Rational Root Theorem 
  • Can you factor specific higher degree polynomials?
    • binomials in quadratic form
    • binomials in cubic form
    • ​trinomials in quadratic form
    • four term polynomials using grouping
    • polynomials missing constant
  • Can you use Pascal's Triangle to expand powers of a binomial?

Rational Expressions
  • Can you combine two rationals expressions into a single rational expression?
    • addition
    • subtraction
    • mutiplication
    • division
  • Can you determine the domain restrictions of a rational expression?
  • Can you determine the zeros of a rational expression?
  • Can you rewrite complex fractions

Graphs and Transformations
  • ​Can you graph each of the types of function or a relations regardless of form?
    • linear
    • quadratic
    • cubic 
    • absolute value 
    • radical
    • logarithmic
    • exponential
  • Can you transform the parent function?
    • linear
    • quadratic
    • cubic 
    • absolute value 
    • radical
    • logarithmic
    • exponential
  • Do you understand the impact of type and signs related to coefficients, constants, bases, and other related parts on the graph of a function or a relation?
    • linear
    • quadratic
    • cubic 
    • absolute value 
    • radical
    • logarithmic
    • exponential
  • Can you express the domain and range a function?
  • Can you determine the intercepts from a function algebraically or with the assistance of technology? 
  • Can you determine the "special" points from a function algebraically or with the assistance of technology?
    • ​intercepts
    • integer coordinates
    • rational coordinates
    • local extremes 
  • Can you determine the "specific" points from a function algebraically or with the assistance of technology?
    • ​given x find y
    • given y find x
    • tables that increment by a specific amount​

Solving Equations
  • Can you solve an equation algebraically?
    • linear
    • quadratic
    • cubic 
    • absolute value 
    • radical
    • logarithmic
    • exponential
  • Can you connect a graph of a system to its related equation's solutions?
  • Can you use the rational root theorem for polynomials?
  • Can you use tables and the location principle to find an estimate of a solution?
  • Can you use technology to confirm solutions to equations?
  • Can you use technology to find solutions to equations?
  • Do you understand what it means for an equation to not have a solution algebraically?
  • Do you understand what it means for an equation to not have a solution graphically?

Trigonometry


Angles and Measures
  • Can you define a radian and support the definition of radian with a picture?
  • Can you express an angle in a variety of units?
    • degrees
    • radians
    • DMS notation
  • Can you determine the complement of an angle?
  • Can you determine the supplement of an angle?
  • Can you determine the explement of an angle?
  • Can you express an angle in terms of its rotation?
  • Can you express a rotation in terms of its angle?
  • Can you express an angle using positive and negative angles?
  • Can you use reference angles in each quadrant?
  • Do you know the special trigonometric values in quadrant one?
  • Do you know the special trigonometric values in quadrant two?
  • Do you know the special trigonometric values in quadrant three?
  • Do you know the special trigonometric values in quadrant four?
  • Do you know the special trigonometric values on the axes?

Unit Circle
  • Do you know why the unit circle is called the unit circle?
  • Do you know the coordinate of all points on the unit circle in rectangle coordinate form?
  • Do you know the coordinate of all points on the unit circle in polar coordinate form?
  • Do you know the special coordinates in each quadrant and on the axes related to the unit circle?
  • Can you place the sixteen special coordinates on the unit circle?
  • Can you connect the unit circle to the graphs of sine and cosine?
  • Can you connect the unit circle to the graphs of the other trigonometric functions?

Right Triangle Trigonometry
  • Determine angles and sides of right triangle using SOHCAHTOA
  • Determine missing sides of a right triangle using Pythagorean Theorem
  • Classify a triangle based on side lengths
  • Classify a triangle based on angle measurement
  • Use right triangles to find perimeter and area of polygons.
  • Use right triangles to find surface area and volume of solids.
  • Apply trigonometric models in real life context

Trigonometric functions of any angle
  • be able to state all six trigonometric ratios related to any angle
  • connect basic trigonometric functions to their related reciprocal functions
  • Use advanced notation to state all angles that have a specific trigonometric ratio.
  • Connect the points on the graphs of trigonometric functions 
    • sine to cosine
    • sine and cosine to tangent
    • sine to cosecant
    • cosine to secant
    • tangent to cotangent
    • secant and cosecant to cotangent
  • Understand the difference between reciprocal trigonometric function and inverse trigonometric function

Graphs of Trigonometric Functions
  • Graphing a single period of any trigonometric function
  • Graphing multiple periods of any trigonometric function
  • ​Graphing specific domain of any  trigonometric function
  • Determine the special points that lie on any trigonometric function
    • local extremes
    • y intercept
    • x intercept
    • points on the midline
  • ​Determine the cause and location of a domain restriction for reciprocal and quotient trigonometric functions
    • tangent
    • cosecant
    • secant
    • cotangent
  • Determine the cause and location for any vertical asymptotes for trigonometric functions 
    • tangent
    • cosecant
    • secant
    • cotangent

Inverse Trigonometric Functions
  • Determine the domain and range of inverse trigonometric functions 
  • Determine special points that lie on inverse trigonometric functions
  • Determine asymptotes related to inverse trigonometric functions
  • Graph the inverse trigonometric functions

Applications and Models that Involve Trigonometric Functions
  • Arc Length, Linear Speed, and Angular Speed
  • Applications of periodic nature
  • Applications related to a right triangle

Analytic Trigonometry


Fundamental Trigonometric Identities
  • Can you use the trigonometric identities to rewrite expressions?
    • Pythagorean Identities
    • Quotient Identities
    • Reciprocal Identities
    • Even Odd Identities
    • Co-function Identities

Verification of Trigonometric Identities
  • Can you derive trigonometric identities?
    • Pythagorean Identities
    • Quotient Identities
    • Reciprocal Identities
    • Even Odd Identities
    • Co-function Identities

Solving Trigonometric Equations
  • Determine an exact and approximate solution of a trigonometric equation algebraically
  • Determine an exact and approximate solution of a trigonometric equation graphically
  • Determine all exact and approximate solutions of a trigonometric equation algebraically
  • Determine all exact and approximate solutions of a trigonometric equation graphically
  • Determine specific exact and approximate solutions of a trigonometric equation over a given domain algebraically
  • Determine all exact and approximate solution of a trigonometric equation over a given domain graphically

Trigonometric Formulas 
  • Rewrite trigonometric expressions using trigonometric formulas 
    • Sum and Difference Formulas
    • Double Angle Formulas
    • Half Angle Formulas
    • Sum to Product Formulas
    • Product to Sum Formulas

Additional Topics in Trigonometry


Law of Sines
  • Be able to solve an oblique triangle when given three pieces of information related to a triangle
  • Recognition of when to use the Law of Sines to find an angle or a side of an oblique triangle
    • ASA (Angle Included Side Angle)
    • AAS (Angle Angle Non-Included Side)
    • SSA (Side Side Non-Included Side)
  • Be able to determine when and why the ambiguous case occurs
    • SSA -0 Solutions 
    • SSA- 1 Solution
    • SSA- 2 Solutions

Law of Cosines
  • Be able to solve an oblique triangle when given three pieces of information related to a triangle
  • Recognition of when to use the Law of Cosines to find an angle or a side of an oblique triangle
    • SAS (Side Included Angle Side)
    • SSS (Side Side SIde)

Applications of Trigonometry
  • Be able to use trigonometry to find the area of a triangle using a variety of methods
    • Heron's Formula (SSS)
    • SAS Triangle Area Formula
  • Be able to use trigonometry to determine areas and perimeters of regular and irregular polygons
  • Be able to apply the Laws of Sine and Cosine to solve real-life problems 

Vectors in the Plane


basic vector notations​ and vocabulary
  • Be able to use proper notation, abbreviation, and vocabulary as it relates to vector mathematics in the plane
  • Be able to write a direction vector in component form
  • Be able to write a direction vector in unit vector form
  • Be able to write a direction vector using polar coordinates 
  • Be able to distinguish the difference between a unit vector and unit vector form
  • Be able to determine the unit vector related to a direction vector regardless of form
  • Be able to determine if vectors are parallel or perpendicular
  • Be able to determine the angle formed with x axis and a vector
  • Be able to determine the angle formed by two vectors 

vector operations
  • Be able to perform vector operations 
    • addition 
    • subtraction
    • scalar multiplication
    • dot product
  • Be able to write a vector equation related to a vector triangle or vector polygon

plotting vectors
  • Be able to plot a given vector on a coordinate grid
  • Be able to plot a vector that is parallel to a given vector on a coordinate grid
  • Be able to plot a perpendicular (orthogonal) vector to a given vector on a coordinate grid
  • Be able to use the parallelogram law to show vector operations
  • Be able to plot a scalar multiple to a given vector on a coordinate grid
  • Be able to plot a vector equation as a vector polygon on a coordinate grid

applications of vectors 
  • Be able to apply properties and operations with vectors in a real-life setting
    • use vectors to determine speed of an airplane or boat influenced by wind or current.
    • use vectors to model basic physics applications  

Matrices


basic matrix notations and vocabulary
  • Know and properly use notation and vocabulary related to matrix
    • row of a matrix
    • column of a matrix
    • dimension of a matrix
    • inverse matrix
    • scalar of a matrix
    • inner dimension of a matrix product
    • outer dimension of a matrix product
    • square matrix
    • rectangular matrix
    • identity matrix
    • matrix equation
    • transpose matrix
    • augmented matrix equation

matrix operations
  • Be able to perform operations with matrices and be able to determine when and why an matrix operation is impossible. 
    • matrix addition
    • matrix subtraction
    • scalar multiplication involving a matrix
    • matrix multiplication
  • Be able to find the determinant of square 2 x 2 and 3 x 3 matrices algebraically
  • Be able to find the determinant of any square matrix using technology
  • Be able to find the inverse of a 2 x 2 matrix algebraically
  • Be able to find the inverse of any square matrix
  • Be able to explain why a matrix product is not possible
  • Be able to explain why a matrix would have a determinant of 0
  • Be able to explain why a matrix equation will have no single solution
  • Be able to explain why a matrix equation will have no solution
  • Be able to solve 2 x 2 and 3 x 3 matrix equations algebraically
  • Be able to solve 2 x 2 and 3 x 3 matrix equations using Cramer's Rule and determinants
  • Be able to solve any n x n matrix equation using matrix inverses  

applications of matrices
  • Be able to set up and use matrices in an inventory scenario
  • Be able to use coordinates and determinants to find areas of triangles
  • Be able to use coordinates and matrices to build polynomials from their coordinates

Sequences and Series


sequence and series notation and vocabulary
  • Be able to use a sequence of no less than three values to write general rule for all terms in sequence
  • Be able to use proper notation, abbreviations, and vocabulary associated with sequences and  series
    • common difference
    • common ratio
    • first term in a sequence
    • last term in a finite sequence
    • nth term in a sequence
    • arithmetic sequence
    • geometric sequence
    • recursive sequence
    • sum of a finite sequence 
    • sum of an infinite sequence 
  • Be able to classify a sequence properly

arithmetic sequence and series
  • Be able to determine when a given sequence of no less than three terms is arithmetic
  • Be able to write an arithmetic sequence rule for a given arithmetic sequence
  • Be able to determine the number of terms are in a finite sum of an arithmetic sequence

geometric ​sequence and series
  • Be able to determine when a given sequence of no less than three terms is geometric
  • Be able to write a geometric sequence rule for a given geometric sequence
  • Be able to determine the number of terms are in a finite sum of an geometric sequence
  • Be able to determine the sum of a geometric sequence of infinite number of terms

recursive series and sequence
  • Be able to write a recursive rule for an arithmetic sequence 
  • Be able to write a recursive rule for an geometric sequence 
  • Be able to write a recursive rule for for a given sequence 

factorials, permutations, and combinations
  • Be able to use proper vocabulary and notations associated with factorials, permutations, and combinations
  • Be able to determine the value of factorials 
  • Interpret the meaning of a factorial in relation to arrangement of items
  • Understand and be able to use the algebraic definitions and notations associated with permutation
  • Understand and be able to use the algebraic definitions and notations associated with permutation
  • Be able to distinguish the difference between permutation of n items, permutation of n choose r items, and the combination of n choose r items
  • Connect permutation, combination, and factorial to probability
  • Be able to use technology to calculate permutations, combinations, and factorials 

summation
  • Be able to use summation notation to represent finite and infinite sums
  • Be able to use algebraic methods to determine finite and infinite sums
  • Be able to use technology to find given sums and general rule for a sequence​

Advanced Functions and Relations


Exponential Functions and their Transformations
  • Determine the domain and range of exponential functions
  • Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of an exponential function 
  • Be able to graph the parent exponential function related to a specific base with and without technology
  • Be able to use proper vocabulary related to transformation of exponential functions
  • Be able to distinguish the difference between exponential functions that have a base that is between 0 and 1 and exponential functions that have a base that is greater than 1
  • Connect exponential functions to logarithmic functions
  • Use exponential functions in a real-life scenario 
  • Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a logarithmic function

Logarithmic Functions and their Transformations
  • Determine the domain and range of logarithmic functions 
  • Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a logarithmic function
  • Be able to graph the parent logarithmic function related to a specific base with and without technology
  • Be able to use proper vocabulary related to transformation of logarithmic functions
  • Be able to distinguish the difference between logarithmic functions that have a base that is between 0 and 1 and logarithmic functions that have a base that is greater than 1
  • Connect exponential functions to logarithmic functions
  • Use logarithmic functions in a real-life scenario 

Polynomial ​Functions and their Transformations
  • Determine the domain and range of polynomial functions 
  • Be able to determine intercepts, integer coordinates, and rational coordinates that lie on the graph of a polynomial function
  • Be able to graph the polynomial function with and without technology
  • Be able to use proper vocabulary related to transformation of polynomial functions
  • Use polynomial functions in a real-life scenario 

Rational ​Functions and their Transformations
  • Determine the domain and range of rational functions 
  • Be able to determine intercepts, asymptotes, integer coordinates, and rational coordinates that lie on the graph of a rational function
  • Be able to graph the rational function with and without technology
  • Be able to use proper vocabulary related to transformation of rational functions
  • Use rational functions in a real-life scenario 

Piecewise Functions
  • Be able to describe the domain and range of a piecewise function
  • Be able to graph a piecewise function with and with out technology
  • Be able to classify a piecewise function as continuous or not continuous
  •  Be able to write a piecewise function from its graph
  • Be able to write a piecewise function for absolute value functions.
  • Be able to write a piecewise function for quadratic functions.

Conic Sections 
  • Be able to use proper notation and vocabulary related to a conic section
    • Center
    • points on graph
    • vertices
    • co-vertices
    • focus and foci
    • minor axis
    • major axis
    • conjugate axis
    • transverse axis 
  • Be able to write a standard form equation of a conic section based on its parts
  • Graph a conic section from its standard form
    • Parabola
    • Circle
    • Ellipse
    • Hyperbola
  • Graph a section from its general form
    • Parabola
    • Circle
    • Ellipse
    • Hyperbola
  • Apply conic sections in a real-lef scenario
  • Parabola
  • Circle
  • Ellipse
  • Hyperbola
  • Parabola
  • Circle
  • Ellipse
  • Hyperbola

  • Polar Graphs
    • Be able to plot points on a polar grid
    • Be able to use proper notation and vocabulary related to polar graphs
      • circles with center at (0,0)
      • circles tangent to the x axis
      • circles tangent to the y axis
      • diagonal lines 
      • vertical lines
      • Limacons
      • Cardiods
      • Rose Curves
      • Lemniscates
    • Be able to plot a variety of polar graphs with and without technology

    ​Probability


    Probability
    • Be able to use proper notation and vocabulary related to single and multiple event probabilities​
      • ​single event probability
      • multiple event probability
      • conditional probability 
      • simple events
      • compound events
      • independent events
      • dependent events
      • mutually exclusive events 
      • complement of an event
      • sample space
      • Venn diagram
    • Be able to determine probabilities algebraically
    • Be able to determine probabilities with the assistance of technology
    • Be able to use regression to model data
    • Connect probability to permutations and combinations
    • Be able to use the Fundamental Counting Principle.
    • Be able to read and create Venn Diagrams to aid in the comprehension and determination of probabilities 

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    Prerequisites

    • Number Sets
    • Notation and Abbreviations
    • ​Functions
    • ​Exponent and Radical Laws
    • ​Factoring and Rewriting Expressions
    • ​​Rational Expressions
    • ​Graphs and Transformations
    • Solving Equations​​

    Trigonometry

    • Angles and Measures
    • ​​Unit Circle
    • ​Right Triangle Trigonometry
    • ​​Trigonometric functions of any angle
    • ​Graphs of Trigonometric Functions
    • ​Inverse Trigonometric Functions
    • ​Applications and Models that Involve Trigonometric Functions

    ​Analytic Trigonometry

    • Fundamental Trigonometric Identities
    • ​Verification of Trigonometric Identities
    • ​Solving Trigonometric Equations
    • ​​Trigonometric Formulas 

    ​Additional Topics in Trigonometry

    • Law of Sines
    • ​Law of Cosines
    • ​​Applications of Trigonometry

    Vectors in the Plane

    • basic vector notations​ and vocabulary
    • ​vector operations
    • ​plotting vectors
    • ​​applications of vectors 

    Matrices

    • basic matrix notations and vocabulary
    • ​​matrix operations
    • ​applications of matrices

    ​Sequences and Series

    • sequence and series notation and vocabulary
    • ​​arithmetic sequence and series
    • ​geometric ​sequence and series
    • ​recursive series and sequence
    • ​​factorials, permutations, and combinations
    • ​summation

    Advanced Functions and Relations

    • Exponential Functions and their Transformations
    • ​Logarithmic Functions and their Transformations
    • ​Polynomial ​Functions and their Transformations
    • ​Rational ​Functions and their Transformations
    • ​​Piecewise Functions
    • Conic Sections 
    • ​Polar Graphs

    Polar Graphs

    • Polar Graphing

    Probability

    • Probability

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    LARSON PreCalculus 10th Edition
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    Algebra, Algebra 2, PreCalculus
    • GeoGebra Algebra 1 TB Book
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    • GeoGebra Functions Linear Quadratic and Exponential TB Book​
    • GeoGebra Exponential and Logarithmic Functions TB Book​
    • GeoGebra Vectors & Advanced Geometry Book
    • GeoGebra PreCalculus BS Book
    • GeoGebra PreCalculus GR Book
    • GeoGebra Alg 1 Geometry Alg 2 IM Book
    • GeoGebra HS Math Book​

    Geometry
    • ​GeoGebra Geometry AE Book
    • GeoGebra Geometry TB Book
    • GeoGebra Geometry CCCS-G SO Book
    • GeoGebra Geometry CCCS G-GPE MVTB Book
    • GeoGebra Right Triangle Trigonometry TB Book
    • GeoGebra Similarity & Right Triangle Trigonometry TB Book
    • GeoGebra Geometry CCCS G-CO MVTB Book
    • GeoGebra Geometry CCCS G-SRT ADJWTB Book
    • GeoGebra Geometry CCCS G-CO RKTB Book​
    • GeoGebra Coordinate and Analytic Geometry TB Book​
    • GeoGebra Circles TB Book​
    • GeoGebra Congruence 1 TB Book
    • GeoGebra Congruence 2 TB Book​
    • GeoGebra Geometric Proof Challenge TB Book​
    • GeoGebra Alg 1 Geometry Alg 2 IM Book
    • GeoGebra HS Math Book​

    Trigonometry
    • GeoGebra Trigonometry CCCS F-TF RDTB Book
    • GeoGebra Trigonometry CCCS F-TF DETB Book
    • GeoGebra Right Triangle Trigonometry TB Book
    • GeoGebra Trigonometry ABC Book
    • GeoGebra Trigonometry RD Book
    • GeoGebra Similarity & Right Triangle Trigonometry TB Book
    • GeoGebra Trigonometry M TB Book
    • GeoGebra Trigonometry Bearings TB Book
    • GeoGebra Trigonometry GE Book
    • GeoGebra Trigonometry TB Book
    • GeoGebra Trigonometric Functions TB Book
    • GeoGebra Trigonometric Identities TB Book
    • GeoGebra Vector Operations TB Book​
    • GeoGebra Coordinate and Analytic Geometry TB Book​
    • GeoGebra Vectors & Advanced Geometry Book
    • GeoGebra HS Math Book​


    ​
    ​

    Common Core Mathematics Standards


    Illinois Mathematics Standards


    Mr Hickman is available for assistance after school practically anytime.  Talk to me to be certain that I am aware you are coming and 99% of the time I will be able to help you