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    Subject:
     
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    SubSubject:
     
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    Course
     
    Title:
     
    Mathematics
     
    for
     
    College
     
    Readiness
     
    Course
     
    Section:
     
    Grades
     
    PreK
     
    to
     
    12
     
    Education
     
    Courses
     
    Abbreviated
     
    Title:
     
    Math
     
    Coll.
     
    Readiness
     
    Number
     
    of
     
    Credits:
     
    1
     
    Course
     
    Length:
     
    Year
     
    Course
     
    Type:
     
    Core
     
    Course
     
    Level:
     
    2
     
    Course
     
    Status:
     
    DRAFT
     ‐
    State
     
    Board
     
    approval
     
    pending
     
    Graduation
     
    Requirements:
     
    Course
     
    Description:
     
    This
     
    course
     
    incorporates
     
    the
     
    Common
     
    Core
     
    Standards
     
    for
     
    Mathematical
     
    Practices
     
    as
     
    well
     
    as
     
    the
     
    following
     
    Common
     
    Core
     
    Standards
     
    for
     
    Mathematical
     
    Content:
     
    an
     
    introduction
     
    to
     
    functions,
     
    linear
     
    equations
     
    and
     
    inequalities,
     
    solving
     
    systems
     
    of
     
    equations,
     
    rational
     
    equations
     
    and
     
    algebraic
     
    fractions,
     
    radicals
     
    and
     
    rational
     
    exponents,
     
    factoring
     
    and
     
    quadratic
     
    equations,
     
    complex
     
    numbers,
     
    and
     
    the
     
    Common
     
    Core
     
    Standards
     
    for
     
    High
     
    School
     
    Modeling.
     
    The
     
    benchmarks
     
    reflect
     
    the
     
    Florida
     
    College
     
    Competencies
     
    necessary
     
    for
     
    entry
    level
     
    college
     
    courses.
     
    RELATED
     
    BENCHMARKS:
     
    Scheme
     
    Descriptor
     
    MACC.K12.MP
     
    Mathematical
     
    Practices
     
    MACC.K12.MP.1
     
    Make
     
    sense
     
    of
     
    problems
     
    and
     
    persevere
     
    in
     
    solving
     
    them
     
    MACC.K12.MP.2
     
    Reason
     
    abstractly
     
    and
     
    quantitatively
     
    MACC.K12.MP.3
     
    Construct
     
    viable
     
    arguments
     
    and
     
    critique
     
    the
     
    reasoning
     
    of
     
    others
     
    MACC.K12.MP.4
     
    Model
     
    with
     
    mathematics
     
    MACC.K12.MP.5
     
    Use
     
    appropriate
     
    tools
     
    strategically
     
    MACC.K12.MP.6
     
    Attend
     
    to
     
    precision
     
    MACC.K12.MP.7
     
    Look
     
    for
     
    and
     
    make
     
    use
     
    of
     
    structure
     
    MACC.K12.MP.8
     
    Look
     
    for
     
    and
     
    express
     
    regularity
     
    in
     
    repeated
     
    reasoning
     
    MACC.7.EE
     
    Expressions
     
    and
     
    Equations
     
    MACC.7.EE.1
     
    Use
     
    properties
     
    of
     
    operations
     
    to
     
    generate
     
    equivalent
     
    expressions.
     
    MACC.7.EE.1.1
     
    Apply
     
    properties
     
    of
     
    operations
     
    as
     
    strategies
     
    to
     
    add,
     
    subtract,
     
    factor,
     
    and
     
    expand
     
    linear
     
    expressions
     
    with
     
    rational
     
    coefficients.
     
    MACC.7.EE.2
     
    Solve
     
    real
    life
     
    and
     
    mathematical
     
    problems
     
    using
     
    numerical
     
    and
     
    algebraic
     
    expressions
     
    and
     
    equations
     
    MACC.7.EE.2.4
     
    Use
     
    variables
     
    to
     
    represent
     
    quantities
     
    in
     
    a
     
    real
    world
     
    or
     
    mathematical
     
    problem,
     
    and
     
    construct
     
    simple
     
    equations
     
    and
     
    inequalities
     
    to
     
    solve
     
    problems
     
    by
     
    reasoning
     
    about
     
    the
     
    quantities.
     
    DRAFT
    ?

     
      
     
     
             
        
      
     
       
      
     
             
               
              
     
                
           
     
             
        
      
     
       
      
     
      
     
          
                 
     
        
            
                    
                 
     
     
       
     
          
     
               
        
      
      
      
           
               
            
     
      
      
     
      
       
     
     
     
            
          
     
     
        
        
     
       
            
        
     
                 
                
       
      
             
     
        
              
     
            
      
     
     
          
                
              
          
        
              
          
     
      
      
     
          
     
            
            
          
     
            
      
      
    MACC.7.EE.2.4a
     
    Solve
     
    word
     
    problems
     
    leading
     
    to
     
    equations
     
    of
     
    the
     
    form
     
    px
     
    +
     
    q
     
    =
     
    r
     
    and
     
    p(x
     
    +
     
    q)
     
    =
     
    r
    ,
     
    where
     
    p,
     
    q,
     
    and
     
    r
     
    are
     
    specific
     
    rational
     
    numbers.
     
    Solve
     
    equations
     
    of
     
    these
     
    forms
     
    fluently.
     
    Compare
     
    an
     
    algebraic
     
    solution
     
    to
     
    an
     
    arithmetic
     
    solution,
     
    identifying
     
    the
     
    sequence
     
    of
     
    the
     
    operations
     
    used
     
    in
     
    each
     
    approach.
     
    For
     
    example,
     
    the
     
    perimeter
     
    of
     
    a
     
    rectangle
     
    is
     
    54
     
    cm.
     
    Its
     
    length
     
    is
     
    6
     
    cm.
     
    What
     
    is
     
    its
     
    width?
     
    MACC.7.EE.2.4b
     
    Solve
     
    word
     
    problems
     
    leading
     
    to
     
    inequalities
     
    of
     
    the
     
    form
     
    px
     
    +
     
    q
     
    >
     
    r
     
    or
     
    px
     
    +
     
    q
     
    <
     
    r
    ,
     
    where
     
    p,
     
    q
    ,
     
    and
     
    r
     
    are
     
    specific
     
    rational
     
    numbers.
     
    Graph
     
    the
     
    solution
     
    set
     
    of
     
    the
     
    inequality
     
    and
     
    interpret
     
    it
     
    in
     
    the
     
    context
     
    of
     
    the
     
    problem.
     
    For
     
    example,
     
    As
     
    a
     
    salesperson,
     
    you
     
    are
     
    paid
     
    $50
     
    per
     
    week
     
    plus
     
    $3
     
    per
     
    sale.
     
    This
     
    week
     
    you
     
    want
     
    your
     
    pay
     
    to
     
    be
     
    at
     
    least
     
    $100.
     
    Write
     
    an
     
    inequality
     
    for
     
    the
     
    number
     
    of
     
    sales
     
    you
     
    need
     
    to
     
    make,
     
    and
     
    describe
     
    the
     
    solutions.
     
    MACC.8.EE
     
    Expressions
     
    and
     
    Equations
     
    MACC.8.EE.1
     
    Work
     
    with
     
    radicals
     
    and
     
    integer
     
    exponents
     
    MACC.8.EE.1.2
     
    Use
     
    square
     
    root
     
    and
     
    cube
     
    root
     
    symbols
     
    to
     
    represent
     
    solutions
     
    to
     
    equations
     
    of
     
    the
     
    form
     
    x
    2
     
    =
     
    p
     
    and
     
    x
    3
     
    =
     
    p
    ,
     
    where
     
    p
     
    is
     
    a
     
    positive
     
    rational
     
    number.
     
    Evaluate
     
    square
     
    roots
     
    of
     
    small
     
    perfect
     
    squares
     
    and
     
    cube
     
    roots
     
    of
     
    small
     
    perfect
     
    cubes.
     
    Know
     
    that
     √
    2
     
    is
     
    irrational.
     
    MACC.8.EE.2
     
    Understand
     
    the
     
    connections
     
    between
     
    proportional
     
    relationships,
     
    lines,
     
    and
     
    linear
     
    equations
     
    MACC.8.EE.2.5
     
    Graph
     
    proportional
     
    relationships,
     
    interpreting
     
    the
     
    unit
     
    rate
     
    as
     
    the
     
    slope
     
    of
     
    the
     
    graph.
     
    Compare
     
    two
     
    different
     
    proportional
     
    relationships
     
    represented
     
    in
     
    different
     
    ways.
     
    For
     
    example,
     
    compare
     
    a
     
    distance
    time
     
    graph
     
    to
     
    a
     
    distance
    time
     
    equation
     
    to
     
    determine
     
    which
     
    of
     
    two
     
    moving
     
    objects
     
    has
     
    greater
     
    speed.
     
    MACC.8.EE.2.6
     
    Use
     
    similar
     
    triangles
     
    to
     
    explain
     
    why
     
    the
     
    slope
     
    m
     
    is
     
    the
     
    same
     
    between
     
    any
     
    two
     
    distinct
     
    points
     
    on
     
    a
     
    non
    vertical
     
    line
     
    in
     
    the
     
    coordinate
     
    plane;
     
    derive
     
    the
     
    equation
     
    y
     
    =
     
    mx
     
    for
     
    a
     
    line
     
    through
     
    the
     
    origin
     
    and
     
    the
     
    equation
     
    y
     
    =
     
    mx
     
    +
     
    b
     
    for
     
    a
     
    line
     
    intercepting
     
    the
     
    vertical
     
    axis
     
    at
     
    b
    .
     
    MACC.8.EE.3
     
    Analyze
     
    and
     
    solve
     
    linear
     
    equations
     
    and
     
    pairs
     
    of
     
    simultaneous
     
    linear
     
    equations
     
    MACC.8.EE.3.7
     
    Solve
     
    linear
     
    equations
     
    in
     
    one
     
    variable.
     
    MACC.8.EE.3.7a
     
    Give
     
    examples
     
    of
     
    linear
     
    equations
     
    in
     
    one
     
    variable
     
    with
     
    one
     
    solution,
     
    infinitely
     
    many
     
    solutions,
     
    or
     
    no
     
    solutions.
     
    Show
     
    which
     
    of
     
    these
     
    possibilities
     
    is
     
    the
     
    case
     
    by
     
    successively
     
    transforming
     
    the
     
    given
     
    equation
     
    into
     
    simpler
     
    forms,
     
    until
     
    an
     
    equivalent
     
    equation
     
    of
     
    the
     
    form
     
    x
     
    =
     
    a,
     
    a
     
    =
     
    a,
     
    or
     
    a
     
    =
     
    b
     
    results
     
    (where
     
    a
     
    and
     
    b
     
    are
     
    different
     
    numbers).
     
    MACC.8.EE.3.7b
     
    Solve
     
    linear
     
    equations
     
    with
     
    rational
     
    number
     
    coefficients,
     
    including
     
    equations
     
    whose
     
    solutions
     
    require
     
    expanding
     
    expressions
     
    using
     
    the
     
    distributive
     
    property
     
    and
     
    collecting
     
    like
     
    terms.
     
    MACC.8.EE.3.8
     
    Analyze
     
    and
     
    solve
     
    linear
     
    equations
     
    and
     
    pairs
     
    of
     
    simultaneous
     
    linear
     
    equations.
     
    DRAFT
    ?

     
      
     
     
                
              
          
     
     
             
      
              
     
                    
         
        
     
     
       
          
        
           
                
          
     
     
     
     
         
     
              
     
            
     
             
               
              
     
       
        
           
                 
                    
                  
             
     
     
        
       
     
              
               
              
     
      
                 
                   
                
     
     
          
                
              
              
     
       
     
              
     
        
     
              
              
             
              
      
    MACC.8.EE.3.8a
     
    Understand
     
    that
     
    solutions
     
    to
     
    a
     
    system
     
    of
     
    two
     
    linear
     
    equations
     
    in
     
    two
     
    variables
     
    correspond
     
    to
     
    points
     
    of
     
    intersection
     
    of
     
    their
     
    graphs,
     
    because
     
    points
     
    of
     
    intersection
     
    satisfy
     
    both
     
    equations
     
    simultaneously.
     
    MACC.8.EE.3.8b
     
    Solve
     
    systems
     
    of
     
    two
     
    linear
     
    equations
     
    in
     
    two
     
    variables
     
    algebraically,
     
    and
     
    estimate
     
    solutions
     
    by
     
    graphing
     
    the
     
    equations.
     
    Solve
     
    simple
     
    cases
     
    by
     
    inspection.
     
    For
     
    example,
     
    3x
     
    +
     
    2y
     
    =
     
    5
     
    and
     
    3x
     
    +
     
    2y
     
    =
     
    6
     
    have
     
    no
     
    solution
     
    because
     
    3x
     
    +
     
    2y
     
    cannot
     
    simultaneously
     
    be
     
    5
     
    and
     
    6.
     
    MACC.8.EE.3.8c
     
    Solve
     
    real
    world
     
    and
     
    mathematical
     
    problems
     
    leading
     
    to
     
    two
     
    linear
     
    equations
     
    in
     
    two
     
    variables.
     
    For
     
    example,
     
    given
     
    coordinates
     
    for
     
    two
     
    pairs
     
    of
     
    points,
     
    determine
     
    whether
     
    the
     
    line
     
    through
     
    the
     
    first
     
    pair
     
    of
     
    points
     
    intersects
     
    the
     
    line
     
    through
     
    the
     
    second
     
    pair
    .
     
    MACC.8.F
     
    Functions
     
    MACC.8.F.1
     
    Define,
     
    evaluate,
     
    and
     
    compare
     
    functions
     
    MACC.8.F.1.2
     
    Compare
     
    properties
     
    of
     
    two
     
    functions
     
    each
     
    represented
     
    in
     
    a
     
    different
     
    way
     
    (algebraically,
     
    graphically,
     
    numerically
     
    in
     
    tables,
     
    or
     
    by
     
    verbal
     
    descriptions).
     
    For
     
    example,
     
    given
     
    a
     
    linear
     
    function
     
    represented
     
    by
     
    a
     
    table
     
    of
     
    values
     
    and
     
    a
     
    linear
     
    function
     
    represented
     
    by
     
    an
     
    algebraic
     
    expression,
     
    determine
     
    which
     
    function
     
    has
     
    the
     
    greater
     
    rate
     
    of
     
    change.
     
    MACC.8.F.1.3
     
    Interpret
     
    the
     
    equation
     
    y
     
    =
     
    mx
     
    +
     
    b
     
    as
     
    defining
     
    a
     
    linear
     
    function,
     
    whose
     
    graph
     
    is
     
    a
     
    straight
     
    line;
     
    give
     
    examples
     
    of
     
    functions
     
    that
     
    are
     
    not
     
    linear.
     
    For
     
    example,
     
    the
     
    function
     
    A
     
    =
     
    s
    2
     
    giving
     
    the
     
    area
     
    of
     
    a
     
    square
     
    as
     
    a
     
    function
     
    of
     
    its
     
    side
     
    length
     
    is
     
    not
     
    linear
     
    because
     
    its
     
    graph
     
    contains
     
    the
     
    points
     
    (1,1),
     
    (2,4)
     
    and
     
    (3,9),
     
    which
     
    are
     
    not
     
    on
     
    a
     
    straight
     
    line.
     
    MACC.8.F.2
     
    Use
     
    functions
     
    to
     
    model
     
    relationships
     
    between
     
    quantities
     
    MACC.8.F.2.4
     
    Construct
     
    a
     
    function
     
    to
     
    model
     
    a
     
    linear
     
    relationship
     
    between
     
    two
     
    quantities.
     
    Determine
     
    the
     
    rate
     
    of
     
    change
     
    and
     
    initial
     
    value
     
    of
     
    the
     
    function
     
    from
     
    a
     
    description
     
    of
     
    a
     
    relationship
     
    or
     
    from
     
    two
     
    (x,
     
    y)
     
    values,
     
    including
     
    reading
     
    these
     
    from
     
    a
     
    table
     
    or
     
    from
     
    a
     
    graph.
     
    Interpret
     
    the
     
    rate
     
    of
     
    change
     
    and
     
    initial
     
    value
     
    of
     
    a
     
    linear
     
    function
     
    in
     
    terms
     
    of
     
    the
     
    situation
     
    it
     
    models,
     
    and
     
    in
     
    terms
     
    of
     
    its
     
    graph
     
    or
     
    a
     
    table
     
    of
     
    values.
     
    MACC.8.F.2.5
     
    Describe
     
    qualitatively
     
    the
     
    functional
     
    relationship
     
    between
     
    two
     
    quantities
     
    by
     
    analyzing
     
    a
     
    graph
     
    (e.g.,
     
    where
     
    the
     
    function
     
    is
     
    increasing
     
    or
     
    decreasing,
     
    linear
     
    or
     
    nonlinear).
     
    Sketch
     
    a
     
    graph
     
    that
     
    exhibits
     
    the
     
    qualitative
     
    features
     
    of
     
    a
     
    function
     
    that
     
    has
     
    been
     
    described
     
    verbally.
     
    MACC.8.NS
     
    The
     
    Number
     
    System
     
    MACC.8.NS.1
     
    Know
     
    that
     
    there
     
    are
     
    numbers
     
    that
     
    are
     
    not
     
    rational,
     
    and
     
    approximate
     
    them
     
    by
     
    rational
     
    numbers
     
    MACC.8.NS.1.1
     
    Know
     
    that
     
    numbers
     
    that
     
    are
     
    not
     
    rational
     
    are
     
    called
     
    irrational.
     
    Understand
     
    informally
     
    that
     
    every
     
    number
     
    has
     
    a
     
    decimal
     
    expansion;
     
    for
     
    rational
     
    numbers
     
    show
     
    that
     
    the
     
    decimal
     
    expansion
     
    repeats
     
    eventually,
     
    and
     
    convert
     
    a
     
    decimal
     
    expansion
     
    which
     
    repeats
     
    eventually
     
    into
     
    a
     
    rational
     
    number.
     
    DRAFT
    ?

     
      
     
       
            
         
         
            
      
                
                   
         
     
     
      
        
     
        
     
     
              
              
     
         
     
       
     
               
      
      
      
      
         
                 
              
      
     
              
          
     
              
       
     
     
      
           
     
                 
             
          
              
     
              
     
     
              
             
          
        
          
       
     
     
               
         
           
         
     
         
     
              
      
                 
               
               
          
    MACC.8.NS.1.2
     
    Use
     
    rational
     
    approximations
     
    of
     
    irrational
     
    numbers
     
    to
     
    compare
     
    the
     
    size
     
    of
     
    irrational
     
    numbers,
     
    locate
     
    them
     
    approximately
     
    on
     
    a
     
    number
     
    line
     
    diagram,
     
    and
     
    estimate
     
    the
     
    value
     
    of
     
    expressions
     
    (e.g.,
     π
    2
    ).
     
    For
     
    example,
     
    by
     
    truncating
     
    the
     
    decimal
     
    expansion
     
    of
     √
    2
     
    (square
     
    root
     
    of
     
    2),
     
    show
     
    that
     
    2
     
    is
     
    between
     
    1
     
    and
     
    2,
     
    then
     
    between
     
    1.4
     
    and
     
    1.5,
     
    and
     
    explain
     
    how
     
    to
     
    continue
     
    on
     
    to
     
    get
     
    better
     
    approximations.
     
    MACC.912.A
    APR
     
    Arithmetic
     
    with
     
    Polynomials
     
    and
     
    Rational
     
    Expressions
     
    MACC.912.A
    APR.1
     
    Perform
     
    arithmetic
     
    operations
     
    on
     
    polynomials
     
    MACC.912.A
    APR.1.1
     
    Understand
     
    that
     
    polynomials
     
    form
     
    a
     
    system
     
    analogous
     
    to
     
    the
     
    integers,
     
    namely,
     
    they
     
    are
     
    closed
     
    under
     
    the
     
    operations
     
    of
     
    addition,
     
    subtraction,
     
    and
     
    multiplication;
     
    add,
     
    subtract,
     
    and
     
    multiply
     
    polynomials.
     
    MACC.912.A
    APR.4
     
    Rewrite
     
    rational
     
    expressions.
     
    MACC.912.A
    APR.4.6
     
    Rewrite
     
    simple
     
    rational
     
    expressions
     
    in
     
    different
     
    forms;
     
    write
     
    a(x)/b(x)
     
    in
     
    the
     
    form
     
    q(x)
     
    +
     
    r(x)/b(x),
     
    where
     
    a(x),
     
    b(x),
     
    q(x),
     
    and
     
    r(x)
     
    are
     
    polynomials
     
    with
     
    the
     
    degree
     
    of
     
    r(x)
     
    less
     
    than
     
    the
     
    degree
     
    of
     
    b(x),
     
    using
     
    inspection,
     
    long
     
    division,
     
    or,
     
    for
     
    the
     
    more
     
    complicated
     
    examples,
     
    a
     
    computer
     
    algebra
     
    system.
     
    MACC.912.A
    APR.4.7
     
    Understand
     
    that
     
    rational
     
    expressions
     
    form
     
    a
     
    system
     
    analogous
     
    to
     
    the
     
    rational
     
    numbers,
     
    closed
     
    under
     
    addition,
     
    subtraction,
     
    multiplication,
     
    and
     
    division
     
    by
     
    a
     
    nonzero
     
    rational
     
    expression;
     
    add,
     
    subtract,
     
    multiply,
     
    and
     
    divide
     
    rational
     
    expressions.
     
    MACC.912.A
    CED
     
    Creating
     
    Equations
     
    MACC.912.A
    CED.1
     
    Create
     
    equations
     
    that
     
    describe
     
    numbers
     
    or
     
    relationships
     
    MACC.912.A
    CED.1.1
     
    Create
     
    equations
     
    and
     
    inequalities
     
    in
     
    one
     
    variable
     
    and
     
    use
     
    them
     
    to
     
    solve
     
    problems.
     
    Include
     
    equations
     
    arising
     
    from
     
    linear
     
    and
     
    quadratic
     
    functions,
     
    and
     
    simple
     
    rational
     
    and
     
    exponential
     
    functions.*
     
    MACC.912.A
    CED.1.2
     
    Create
     
    equations
     
    in
     
    two
     
    or
     
    more
     
    variables
     
    to
     
    represent
     
    relationships
     
    between
     
    quantities;
     
    graph
     
    equations
     
    on
     
    coordinate
     
    axes
     
    with
     
    labels
     
    and
     
    scales.*
     
    MACC.912.A
    CED.1.3
     
    Represent
     
    constraints
     
    by
     
    equations
     
    or
     
    inequalities,
     
    and
     
    by
     
    systems
     
    of
     
    equations
     
    and/or
     
    inequalities,
     
    and
     
    interpret
     
    solutions
     
    as
     
    viable
     
    or
     
    non
    viable
     
    options
     
    in
     
    a
     
    modeling
     
    context.
     
    For
     
    example,
     
    represent
     
    inequalities
     
    describing
     
    nutritional
     
    and
     
    cost
     
    constraints
     
    on
     
    combinations
     
    of
     
    different
     
    foods.*
     
    MACC.912.A
    CED.1.4
     
    Rearrange
     
    formulas
     
    to
     
    highlight
     
    a
     
    quantity
     
    of
     
    interest,
     
    using
     
    the
     
    same
     
    reasoning
     
    as
     
    in
     
    solving
     
    equations.
     
    For
     
    example,
     
    rearrange
     
    Ohm’s
     
    law
     
    V
     
    =
     
    IR
     
    to
     
    highlight
     
    resistance
     
    R.*
     
    MACC.912.A
    REI
     
    Reasoning
     
    with
     
    Equations
     
    and
     
    Inequalities
     
    MACC.912.A
    REI.1
     
    Understand
     
    solving
     
    equations
     
    as
     
    a
     
    process
     
    of
     
    reasoning
     
    and
     
    explain
     
    the
     
    reasoning
     
    MACC.912.A
    REI.1.1
     
    Explain
     
    each
     
    step
     
    in
     
    solving
     
    a
     
    simple
     
    equation
     
    as
     
    following
     
    from
     
    the
     
    equality
     
    of
     
    numbers
     
    asserted
     
    at
     
    the
     
    previous
     
    step,
     
    starting
     
    from
     
    the
     
    assumption
     
    that
     
    the
     
    original
     
    equation
     
    has
     
    a
     
    solution.
     
    Construct
     
    a
     
    viable
     
    argument
     
    to
     
    justify
     
    a
     
    solution
     
    method.
     
    DRAFT
    ?

     
      
     
                
           
            
              
         
     
           
                
             
        
         
            
                 
              
               
             
      
      
         
         
                 
                  
           
            
       
              
     
           
     
                  
                
         
        
             
     
      
      
      
            
      
        
         
              
             
        
       
     
                  
                 
                 
       
      
     
        
     
         
                
               
     
      
              
        
              
          
    MACC.912.A
    REI.1.2
     
    Solve
     
    simple
     
    rational
     
    and
     
    radical
     
    equations
     
    in
     
    one
     
    variable,
     
    and
     
    give
     
    examples
     
    showing
     
    how
     
    extraneous
     
    solutions
     
    may
     
    arise.
     
    MACC.912.A
    REI.2
     
    Solve
     
    equations
     
    and
     
    inequalities
     
    in
     
    one
     
    variable
     
    MACC.912.A
    REI.2.3
     
    Solve
     
    linear
     
    equations
     
    and
     
    inequalities
     
    in
     
    one
     
    variable,
     
    including
     
    equations
     
    with
     
    coefficients
     
    represented
     
    by
     
    letters.
     
    MACC.912.A
    REI.2.4
     
    Solve
     
    quadratic
     
    equations
     
    in
     
    one
     
    variable.
     
    MACC.912.A
    REI.2.4a
     
    Use
     
    the
     
    method
     
    of
     
    completing
     
    the
     
    square
     
    to
     
    transform
     
    any
     
    quadratic
     
    equation
     
    in
     
    x
     
    into
     
    an
     
    equation
     
    of
     
    the
     
    form
     
    (x
     
     
    p)
    2
     
    =
     
    q
     
    that
     
    has
     
    the
     
    same
     
    solutions.
     
    Derive
     
    the
     
    quadratic
     
    formula
     
    from
     
    this
     
    form.
     
    MACC.912.A
    REI.2.4b
     
    Solve
     
    quadratic
     
    equations
     
    by
     
    inspection
     
    (e.g.,
     
    for
     
    x
    2
     
    =
     
    49),
     
    taking
     
    square
     
    roots,
     
    completing
     
    the
     
    square,
     
    the
     
    quadratic
     
    formula
     
    and
     
    factoring,
     
    as
     
    appropriate
     
    to
     
    the
     
    initial
     
    form
     
    of
     
    the
     
    equation.
     
    Recognize
     
    when
     
    the
     
    quadratic
     
    formula
     
    gives
     
    complex
     
    solutions
     
    and
     
    write
     
    them
     
    as
     
    a
     
    ±
     
    bi
     
    for
     
    real
     
    numbers
     
    a
     
    and
     
    b
    .
     
    MACC.912.A
    REI.3
     
    Solve
     
    systems
     
    of
     
    equations
     
    MACC.912.A
    REI.3.5
     
    Prove
     
    that,
     
    given
     
    a
     
    system
     
    of
     
    two
     
    equations
     
    in
     
    two
     
    variables,
     
    replacing
     
    one
     
    equation
     
    by
     
    the
     
    sum
     
    of
     
    that
     
    equation
     
    and
     
    a
     
    multiple
     
    of
     
    the
     
    other
     
    produces
     
    a
     
    system
     
    with
     
    the
     
    same
     
    solutions.
     
    MACC.912.A
    REI.3.6
     
    Solve
     
    systems
     
    of
     
    linear
     
    equations
     
    exactly
     
    and
     
    approximately
     
    (e.g.,
     
    with
     
    graphs),
     
    focusing
     
    on
     
    pairs
     
    of
     
    linear
     
    equations
     
    in
     
    two
     
    variables.
     
    MACC.912.A
    REI.4
     
    Represent
     
    and
     
    solve
     
    equations
     
    and
     
    inequalities
     
    graphically
     
    MACC.912.A
    REI.4.10
     
    Understand
     
    that
     
    the
     
    graph
     
    of
     
    an
     
    equation
     
    in
     
    two
     
    variables
     
    is
     
    the
     
    set
     
    of
     
    all
     
    its
     
    solutions
     
    plotted
     
    in
     
    the
     
    coordinate
     
    plane,
     
    often
     
    forming
     
    a
     
    curve
     
    (which
     
    could
     
    be
     
    a
     
    line).
     
    MACC.912.A
    REI.4.11
     
    Explain
     
    why
     
    the
     
    x
    coordinates
     
    of
     
    the
     
    points
     
    where
     
    the
     
    graphs
     
    of
     
    the
     
    equations
     
    y
     
    =
     
    f(x)
     
    and
     
    y
     
    =
     
    g(x)
     
    intersect
     
    are
     
    the
     
    solutions
     
    of
     
    the
     
    equation
     
    f(x)
     
    =
     
    g(x);
     
    find
     
    the
     
    solutions
     
    approximately,
     
    e.g.,
     
    using
     
    technology
     
    to
     
    graph
     
    the
     
    functions,
     
    make
     
    tables
     
    of
     
    values,
     
    or
     
    find
     
    successive
     
    approximations.
     
    Include
     
    cases
     
    where
     
    f(x)
     
    and/or
     
    g(x)
     
    are
     
    linear,
     
    polynomial,
     
    rational,
     
    absolute
     
    value,
     
    exponential,
     
    and
     
    logarithmic
     
    functions.*
     
    MACC.912.A
    REI.4.12
     
    Graph
     
    the
     
    solutions
     
    to
     
    a
     
    linear
     
    inequality
     
    in
     
    two
     
    variables
     
    as
     
    a
     
    half
    plane
     
    (excluding
     
    the
     
    boundary
     
    in
     
    the
     
    case
     
    of
     
    a
     
    strict
     
    inequality),
     
    and
     
    graph
     
    the
     
    solution
     
    set
     
    to
     
    a
     
    system
     
    of
     
    linear
     
    inequalities
     
    in
     
    two
     
    variables
     
    as
     
    the
     
    intersection
     
    of
     
    the
     
    corresponding
     
    half
    planes.
     
    MACC.912.A
    SSE
     
    Seeing
     
    Structure
     
    in
     
    Expressions
     
    MACC.912.A
    SSE.1
     
    Interpret
     
    the
     
    structure
     
    of
     
    expressions
     
    MACC.912.A
    SSE.1.1
     
    Interpret
     
    expressions
     
    that
     
    represent
     
    a
     
    quantity
     
    in
     
    terms
     
    of
     
    its
     
    context.*
     
    MACC.912.A
    SSE.1.1a
     
    Interpret
     
    parts
     
    of
     
    an
     
    expression,
     
    such
     
    as
     
    terms,
     
    factors,
     
    and
     
    coefficients.*
     
    MACC.912.A
    SSE.1.1b
     
    Interpret
     
    complicated
     
    expressions
     
    by
     
    viewing
     
    one
     
    or
     
    more
     
    of
     
    their
     
    parts
     
    as
     
    a
     
    single
     
    entity.
     
    For
     
    example,
     
    interpret
     
    P(1+r)
    n
     
    as
     
    the
     
    product
     
    of
     
    P
     
    and
     
    a
     
    factor
     
    not
     
    depending
     
    on
     
    P.*
     
    DRAFT
    ?

     
      
     
                  
      
      
      
      
            
          
         
             
                 
            
     
                 
     
                
            
              
      
            
     
        
       
           
     
     
      
     
              
     
                
                 
        
     
                
          
     
          
      
     
          
      
     
               
               
     
     
         
          
                  
         
         
         
            
                
     
     
     
               
               
           
               
               
      
               
     
           
      
           
                 
               
            
    MACC.912.A
    SSE.1.2
     
    Use
     
    the
     
    structure
     
    of
     
    an
     
    expression
     
    to
     
    identify
     
    ways
     
    to
     
    rewrite
     
    it.
     
    For
     
    example,
     
    see
     
    x
    4
     
     
    y
    4
     
    as
     
    (x
    2
    )
    2
     
     
    (y
    2
    )
    2
    ,
     
    thus
     
    recognizing
     
    it
     
    as
     
    a
     
    difference
     
    of
     
    squares
     
    that
     
    can
     
    be
     
    factored
     
    as
     
    (x
    2
     
     
    y
    2
    )(x
    2
     
    +
     
    y
    2
    ).
     
    MACC.912.A
    SSE.2
     
    Write
     
    expressions
     
    in
     
    equivalent
     
    forms
     
    to
     
    solve
     
    problems
     
    MACC.912.A
    SSE.2.3
     
    Choose
     
    and
     
    produce
     
    an
     
    equivalent
     
    form
     
    of
     
    an
     
    expression
     
    to
     
    reveal
     
    and
     
    explain
     
    properties
     
    of
     
    the
     
    quantity
     
    represented
     
    by
     
    the
     
    expression.*
     
    MACC.912.A
    SSE.2.3a
     
    Factor
     
    a
     
    quadratic
     
    expression
     
    to
     
    reveal
     
    the
     
    zeros
     
    of
     
    the
     
    function
     
    it
     
    defines.*
     
    MACC.912.A
    SSE.2.3b
     
    Complete
     
    the
     
    square
     
    in
     
    a
     
    quadratic
     
    expression
     
    to
     
    reveal
     
    the
     
    maximum
     
    or
     
    minimum
     
    value
     
    of
     
    the
     
    function
     
    it
     
    defines.*
     
    MACC.912.A
    SSE.2.3c
     
    Use
     
    the
     
    properties
     
    of
     
    exponents
     
    to
     
    transform
     
    expressions
     
    for
     
    exponential
     
    functions.
     
    For
     
    example
     
    the
     
    expression
     
    1.15
    t
     
    can
     
    be
     
    rewritten
     
    as
     
    [1.15
    1/12
    ]
    12t
     ≈ 
    1.012
    12t
     
    to
     
    reveal
     
    the
     
    approximate
     
    equivalent
     
    monthly
     
    interest
     
    rate
     
    if
     
    the
     
    annual
     
    rate
     
    is
     
    15%.*
     
    MACC.912.F
    IF
     
    Interpreting
     
    Functions
     
    MACC.912.F
    IF.1
     
    Understand
     
    the
     
    concept
     
    of
     
    a
     
    function
     
    and
     
    use
     
    function
     
    notation
     
    MACC.912.F
    IF.1.1
     
    Understand
     
    that
     
    a
     
    function
     
    from
     
    one
     
    set
     
    (called
     
    the
     
    domain)
     
    to
     
    another
     
    set
     
    (called
     
    the
     
    range)
     
    assigns
     
    to
     
    each
     
    element
     
    of
     
    the
     
    domain
     
    exactly
     
    one
     
    element
     
    of
     
    the
     
    range.
     
    If
     
    f
     
    is
     
    a
     
    function
     
    and
     
    x
     
    is
     
    an
     
    element
     
    of
     
    its
     
    domain,
     
    then
     
    f(x)
     
    denotes
     
    the
     
    output
     
    of
     
    f
     
    corresponding
     
    to
     
    the
     
    input
     
    x
    .
     
    The
     
    graph
     
    of
     
    f
     
    is
     
    the
     
    graph
     
    of
     
    the
     
    equation
     
    y
     
    =
     
    f(x).
     
    MACC.912.F
    IF.1.2
     
    Use
     
    function
     
    notation,
     
    evaluate
     
    functions
     
    for
     
    inputs
     
    in
     
    their
     
    domains,
     
    and
     
    interpret
     
    statements
     
    that
     
    use
     
    function
     
    notation
     
    in
     
    terms
     
    of
     
    a
     
    context.
     
    MACC.912.F
    IF.2
     
    Interpret
     
    functions
     
    that
     
    arise
     
    in
     
    applications
     
    in
     
    terms
     
    of
     
    the
     
    context
     
    MACC.912.F
    IF.2.5
     
    Relate
     
    the
     
    domain
     
    of
     
    a
     
    function
     
    to
     
    its
     
    graph
     
    and,
     
    where
     
    applicable,
     
    to
     
    the
     
    quantitative
     
    relationship
     
    it
     
    describes.
     
    For
     
    example,
     
    if
     
    the
     
    function
     
    h(n)
     
    gives
     
    the
     
    number
     
    of
     
    person
    hours
     
    it
     
    takes
     
    to
     
    assemble
     
    n
     
    engines
     
    in
     
    a
     
    factory,
     
    then
     
    the
     
    positive
     
    integers
     
    would
     
    be
     
    an
     
    appropriate
     
    domain
     
    for
     
    the
     
    function.
    *
     
    MACC.912.F
    IF.2.6
     
    Calculate
     
    and
     
    interpret
     
    the
     
    average
     
    rate
     
    of
     
    change
     
    of
     
    a
     
    function
     
    (presented
     
    symbolically
     
    or
     
    as
     
    a
     
    table)
     
    over
     
    a
     
    specified
     
    interval.
     
    Estimate
     
    the
     
    rate
     
    of
     
    change
     
    from
     
    a
     
    graph.*
     
    MACC.912.F
    IF.3.7
     
    Graph
     
    functions
     
    expressed
     
    symbolically
     
    and
     
    show
     
    key
     
    features
     
    of
     
    the
     
    graph,
     
    by
     
    hand
     
    in
     
    simple
     
    cases
     
    and
     
    using
     
    technology
     
    for
     
    more
     
    complicated
     
    cases.*
     
    MACC.912.F
    IF.3.7a
     
    Graph
     
    linear
     
    and
     
    quadratic
     
    functions
     
    and
     
    show
     
    intercepts,
     
    maxima,
     
    and
     
    minima.*
     
    MACC.912.F
    IF.3.7b
     
    Graph
     
    square
     
    root,
     
    cube
     
    root,
     
    and
     
    piecewise
    defined
     
    functions,
     
    including
     
    step
     
    functions
     
    and
     
    absolute
     
    value
     
    functions.*
     
    MACC.912.F
    IF.3.8a
     
    Use
     
    the
     
    process
     
    of
     
    factoring
     
    and
     
    completing
     
    the
     
    square
     
    in
     
    a
     
    quadratic
     
    function
     
    to
     
    show
     
    zeros,
     
    extreme
     
    values,
     
    and
     
    symmetry
     
    of
     
    the
     
    graph,
     
    and
     
    interpret
     
    these
     
    in
     
    terms
     
    of
     
    a
     
    context.
     
    DRAFT
    ?

     
      
     
     
              
     
            
      
              
               
     
        
     
         
     
          
           
     
      
     
        
        
      
          
        
     
        
       
             
                 
         
             
              
     
         
             
                
              
               
     
                   
     
      
            
              
       
            
                  
                 
                
     
     
       
      
     
       
     
                
             
     
     
     
               
          
      
     
          
              
            
     
    MACC.912.F
    IF.3.9
     
    Compare
     
    properties
     
    of
     
    two
     
    functions
     
    each
     
    represented
     
    in
     
    a
     
    different
     
    way
     
    (algebraically,
     
    graphically,
     
    numerically
     
    in
     
    tables,
     
    or
     
    by
     
    verbal
     
    descriptions).
     
    For
     
    example,
     
    given
     
    a
     
    graph
     
    of
     
    one
     
    quadratic
     
    function
     
    and
     
    an
     
    algebraic
     
    expression
     
    for
     
    another,
     
    say
     
    which
     
    has
     
    the
     
    larger
     
    maximum
     
    MACC.912.F
    IF
    3
     
    Analyze
     
    functions
     
    using
     
    different
     
    representations
     
    MACC.912.N
    CN
     
    The
     
    Complex
     
    Number
     
    System
     
    MACC.912.N
    CN.1
     
    Perform
     
    arithmetic
     
    operations
     
    with
     
    complex
     
    numbers
     
    MACC.912.N
    CN.1.1
     
    Know
     
    there
     
    is
     
    a
     
    complex
     
    number
     
    i
     
    such
     
    that
     
    i
    2
     
    =
     −
    1,
     
    and
     
    every
     
    complex
     
    number
     
    has
     
    the
     
    form
     
    a
     
    +
     
    bi
     
    with
     
    a
     
    and
     
    b
     
    real.
     
    MACC.912.N
    CN.1.2
     
    Use
     
    the
     
    relation
     
    i
    2
     
    =
     
    –1
     
    and
     
    the
     
    commutative,
     
    associative,
     
    and
     
    distributive
     
    properties
     
    to
     
    add,
     
    subtract,
     
    and
     
    multiply
     
    complex
     
    numbers.
     
    MACC.912.N
    CN.1.3
     
    Find
     
    the
     
    conjugate
     
    of
     
    a
     
    complex
     
    number;
     
    use
     
    conjugates
     
    to
     
    find
     
    moduli
     
    and
     
    quotients
     
    of
     
    complex
     
    numbers.
     
    MACC.912.N
    CN.3
     
    Use
     
    complex
     
    numbers
     
    in
     
    polynomial
     
    identities
     
    and
     
    equations
     
    MACC.912.N
    CN.3.7
     
    Solve
     
    quadratic
     
    equations
     
    with
     
    real
     
    coefficients
     
    that
     
    have
     
    complex
     
    solutions.
     
    MACC.912.N
    RN
     
    The
     
    Real
     
    Number
     
    System
     
    MACC.912.N
    RN.1
     
    Extend
     
    the
     
    properties
     
    of
     
    exponents
     
    to
     
    rational
     
    exponents
     
    MACC.912.N
    RN.1.1
     
    Explain
     
    how
     
    the
     
    definition
     
    of
     
    the
     
    meaning
     
    of
     
    rational
     
    exponents
     
    follows
     
    from
     
    extending
     
    the
     
    properties
     
    of
     
    integer
     
    exponents
     
    to
     
    those
     
    values,
     
    allowing
     
    for
     
    a
     
    notation
     
    for
     
    radicals
     
    in
     
    terms
     
    of
     
    rational
     
    exponents.
     
    For
     
    example,
     
    we
     
    define
     
    5
    1/3
     
    to
     
    be
     
    the
     
    cube
     
    root
     
    of
     
    5
     
    because
     
    we
     
    want
     
    [5
    1/3
    ]
    3
     
    =
     
    5
    [(1/3)
     
    x
     
    3]
     
    to
     
    hold,
     
    so
     
    [5
    1/3
    ]
    3
     
    must
     
    equal
     
    5.
     
    MACC.912.N
    RN.1.2
     
    Rewrite
     
    expressions
     
    involving
     
    radicals
     
    and
     
    rational
     
    exponents
     
    using
     
    the
     
    properties
     
    of
     
    exponents.
     
    MACC.912.N
    RN.2
     
    Use
     
    Properties
     
    of
     
    rational
     
    and
     
    irrational
     
    numbers
     
    MACC.912.N
    RN.2.3
     
    Explain
     
    why
     
    the
     
    sum
     
    or
     
    product
     
    of
     
    rational
     
    numbers
     
    is
     
    rational;
     
    that
     
    the
     
    sum
     
    of
     
    a
     
    rational
     
    number
     
    and
     
    an
     
    irrational
     
    number
     
    is
     
    irrational;
     
    and
     
    that
     
    the
     
    product
     
    of
     
    a
     
    nonzero
     
    rational
     
    number
     
    and
     
    an
     
    irrational
     
    number
     
    is
     
    irrational.
     
    MACC.912.S
    ID
     
    Interpreting
     
    Categorical
     
    and
     
    Quantitative
     
    Data
     
    MACC.912.S
    ID.3
     
    Interpret
     
    linear
     
    models
     
    MACC.912.S
    ID.3.7
     
    Interpret
     
    the
     
    slope
     
    (rate
     
    of
     
    change)
     
    and
     
    the
     
    intercept
     
    (constant
     
    term)
     
    of
     
    a
     
    linear
     
    model
     
    in
     
    the
     
    context
     
    of
     
    the
     
    data.*
     
    Modeling
     
    standards
     
    Modeling
     
    is
     
    best
     
    interpreted
     
    not
     
    as
     
    a
     
    collection
     
    of
     
    isolated
     
    topics
     
    but
     
    rather
     
    in
     
    relation
     
    to
     
    other
     
    standards.
     
    Making
     
    mathematical
     
    models
     
    is
     
    a
     
    Standard
     
    for
     
    Mathematical
     
    Practice,
     
    and
     
    specific
     
    modeling
     
    standards
     
    appear
     
    throughout
     
    the
     
    high
     
    school
     
    standards
     
    indicated
     
    by
     
    a
     
    star
     
    symbol
     
    (*).
     
    DRAFT
    ?

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