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Topic: Functions 

Mathematics

 Standard 8.  Understands and applies basic and advanced properties of functions and algebra
  Level III (Grade 6-8)
   Benchmark 3.Understands various representations (e.g., tables, graphs, verbal descriptions, algebraic expressions, Venn diagram) of patterns and functions and the relationships among them
    Knowledge/skill statements
     1. Understands how algebraic expressions can represent functions
     2. Represents functions using algebraic expressions
     3.Understands how verbal descriptions can represent functions
     4.Represents functions using tables
     5.Understands how tables can represent functions
     6.Understands how graphs can represent functions
     7.Represents functions using graphs
     8.Represents functions using verbal descriptions
     9.Understands the concept of a Venn diagram
     10.Represents patterns using Venn diagrams
     11.Understands how Venn diagrams can represent patterns
     12.Understands how algebraic expressions can represent patterns
     13.Represents patterns using algebraic expressions
     14.Understands how verbal descriptions can represent patterns
     15.Represents patterns using tables
     16.Understands how tables can represent patterns
     17.Understands how graphs can represent patterns
     18. Represents patterns using graphs
     19.Represents patterns using verbal descriptions
     20. Understands the relationship between tables and graphs
     21.Understands the relationship between tables and verbal descriptions
     22.Understands the relationship between tables and algebraic expressions
     23.Understands the relationship between tables and Venn diagrams
     24.Understands the relationship between graphs and verbal descriptions
     25.Understands the relationship between graphs and algebraic expressions
     26.Understands the relationship between graphs and Venn diagrams
     27.Understands the relationship between verbal descriptions and algebraic expressions
     28.Understands the relationship between verbal descriptions and Venn diagrams
   Benchmark 4.Understands the basic concept of a function (i.e., functions describe how changes in one quantity or variable result in changes in another)
    Knowledge/skill statements
     1.Understands the concept of a function
   Benchmark 7.Understands special values (e.g., minimum and maximum values, x- and y-intercepts, slope, constant ratio or difference) of patterns, relationships, and functions
    Knowledge/skill statements
     1.Understands x- and y-intercepts
     2.Understands the concept of slope
     3.Understands constant ratio or difference
     4.Understands the concept of maximum
     5.Understands the concept of minimum
  Level IV (Grade 9-12)
   Benchmark 1.Understands appropriate terminology and notation used to define functions and their properties (e.g., domain, range, function composition, inverses)
    Knowledge/skill statements
     1.Understands the concept of range as it relates to functions
     2. Understands the notation used to represent functions
     3.Understands the concept of inverse as it relates to functions
     4.Understands the concept of function composition
     5. Understands the concept of domain as it relates to functions
   Benchmark 6.Understands the concept of a function as the correspondences between the elements of two sets (e.g., in algebra, functions are relationships between variables that represent numbers; in geometry, functions relate sets of points to their images under motions such as flips, slides, and turns; in the "real-world," functions are mathematical representations of many input-output situations)
    Knowledge/skill statements
     1.Understands functions as fixed relationships between elements of two sets
     2. Understands that in the real world, functions relate inputs to outputs
     3.Understands that in geometry, functions relate points to images in motion (e.g., flips, slides, turns)
     4.Understands that in algebra, functions represent relationships between variables
   Benchmark 7.Uses a variety of models (e.g., written statement, algebraic formula, table of input-output values, graph) to represent functions, patterns, and relationships
    Knowledge/skill statements
     1.Represents functions using graphs
     2.Represents functions using table of input/output values
     3.Represents functions using algebraic formula
     4.Represents functions using written statements
     5.Represents functions using models
     6. Represents patterns using graphs
     7.Represents patterns using table of input/output values
     8. Represents patterns using algebraic formula
     9.Represents patterns using written statements
     10. Represents patterns using models
     11. Represents relationships using graphs
     12. Represents relationships using table of input/output values
     13. Represents relationships using algebraic formula
     14. Represents relationships using written statements
     15.Represents relationships using models
   Benchmark 8.Understands the general properties and characteristics of many types of functions (e.g., direct and inverse variation, general polynomial, radical, step, exponential, logarithmic, sinusoidal)
    Knowledge/skill statements
     1. Understands the concept of logarithmic functions
     2. Understands the concept of exponential functions
     3.Understands the concept of polynomial functions
     4. Understands the concept of direct variation functions
     5. Understands the concept of radical functions
     6.Understands the concept of inverse variation functions
     7. Understands the concept of sinusoidal functions
     8. Understands the concept of step functions
     9. Understands the concept of linear functions
     10. Understands the concept of quadratic functions
     11. Understands the concept of families of functions
     12. Understands the concept of absolute value functions
   Benchmark 9.Understands the effects of parameter changes on functions and their graphs
    Knowledge/skill statements
     1. Understands the concept of a parameter
     2. Understands how parameter changes affect functions
     3. Understands how parameter changes affect graphs of functions
   Benchmark 10.Understands the basic concept of inverse function and the corresponding graph
    Knowledge/skill statements
     1. Understands the concept of inverse functions
     2. Graphs inverse functions
  Level V (Grade (College Readiness))
   Benchmark 4.Knows how to compose functions
    Knowledge/skill statements
     1. Knows composite function notation
     2. Knows a function can be substituted into a function
   Benchmark 6.Identifies if a function has an inverse and, if so, determines the inverse function
    Knowledge/skill statements
     1. Identifies if a function has an inverse
     2. Determines the inverse of a function