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List of Benchmarks for Mathematics

Standard 8.Understands and applies basic and advanced properties of functions and algebra
  Level Pre-K (Grade Pre-K)
   1. Understands simple patterns (e.g., boy-girl-boy-girl)
   2. Repeats simple patterns
  Level I (Grade K-2)
   1. Recognizes regularities in a variety of contexts (e.g., events, designs, shapes, sets of numbers)
   2. Extends simple patterns (e.g., of numbers, physical objects, geometric shapes)  A 
  Level II (Grade 3-5)
   1. Recognizes a wide variety of patterns (e.g., basic linear patterns such as [2, 4, 6, 8 . . .] ; simple repeating and growing patterns) and the rules that explain them  A 
   2. Understands that the same pattern can be represented in different ways (e.g., geometrically or numerically; the pattern of numbers [7, 14, 21, 28 . . .] is equivalent to the mathematical relationship 7 × n)  A 
   3. Knows that a variable is a letter or symbol that stands for one or more numbers  A 
   4. Understands the basic concept of an equality relationship (i.e., an equation is a number sentence that shows two quantities that are equal)  A 
   5. Solves simple open sentences involving operations on whole numbers (e.g., ? + 17 = 23)  A 
   6. Knows basic characteristics and features of the rectangular coordinate system (e.g., the horizontal axis is the X axis and the vertical axis is the Y axis)  A 
  Level III (Grade 6-8)
   1. Knows that an expression is a mathematical statement using numbers and symbols to represent relationships and real-world situations (e.g., equations and inequalities with or without variables)  A 
   2. Understands that a variable can be used in many ways (e.g., as a placeholder for a specific unknown, such as x + 8 = 13; as a representative of a range of values, such as 4t + 7)  A 
   3. Understands various representations (e.g., tables, graphs, verbal descriptions, algebraic expressions, Venn diagram) of patterns and functions and the relationships among them  A 
   4. Understands the basic concept of a function (i.e., functions describe how changes in one quantity or variable result in changes in another)
   5. Solves linear equations using concrete, informal, and formal methods (e.g., using properties, graphing ordered pairs, using slope-intercept form)  A 
   6. Solves simple inequalities and non-linear equations with rational number solutions, using concrete and informal methods  A 
   7. Understands special values (e.g., minimum and maximum values, x- and y-intercepts, slope, constant ratio or difference) of patterns, relationships, and functions
   8. Understands basic operations (e.g., combining like terms, expanding, substituting for unknowns) on algebraic expressions  A 
   9. Uses the rectangular coordinate system to model and to solve problems  A 
   10. Solves simple systems of equations graphically  A 
   11. Understands the properties of arithmetic and geometric sequences (i.e., linear and exponential patterns)  A 
  Level IV (Grade 9-12)
   1. Understands appropriate terminology and notation used to define functions and their properties (e.g., domain, range, function composition, inverses)
   2. Uses expressions, equations, inequalities, and matrices to represent situations that involve variable quantities and translates among these representations
   3. Understands characteristics and uses of basic trigonometric functions (e.g., the sine and cosine functions as models of periodic real-world phenomena)
   4. Understands properties of graphs and the relationship between a graph and its corresponding expression (e.g., maximum and minimum points)
   5.  Solves polynomial equations (e.g., find the real or complex roots of a quadratic equation) using a variety of methods (e.g., symbolically, graphically, numerically).
   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)
   7. Uses a variety of models (e.g., written statement, algebraic formula, table of input-output values, graph) to represent functions, patterns, and relationships  A 
   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)
   9. Understands the effects of parameter changes on functions and their graphs  A 
   10. Understands the basic concept of inverse function and the corresponding graph
   11. Uses a variety of methods (e.g., with graphs, algebraic methods, and matrices) to solve systems of equations and inequalities  A 
   12. Understands formal notation (e.g., sigma notation, factorial representation) and various applications (e.g., compound interest) of sequences and series
   13. Adds, subtracts, multiplies, and divides polynomials
   14. Understands the meaning of and writes equivalent forms of expressions, equations, inequalities, and relations, including all of their parts(e.g., terms, factors, coefficients).
   15. Models and solves real world problems using polynomial equations, with and without the use of technology.   A 
   16. Solves quadratic equations in one variable(e.g. using quadratic formula, by completing the square)
  Level V (Grade (College Readiness))
   1. Solves equations and inequalities involving absolute value
   2. Understands graphs of solution sets of quadratic inequalities
   3. Factors polynomials using a variety of methods (e.g., difference of squares, perfect square trinomials)
   4. Knows how to compose functions
   5. Solves problem situations that require linear, quadratic, or exponential modeling
   6. Identifies if a function has an inverse and, if so, determines the inverse function
   7. Understands periodicity in the trigonometric functions and their graphs
   8. Understands unit circle trigonometry
   9. Understands the Law of Sines, Law of Cosines, and basic trigonometric identities (e.g., double angle formulas, sum and difference identities)
   10. Understands the graph of a function of the form f(t)=Asin(Bt+C)+D or g(t)=Acos(Bt+C)+D, in terms of amplitude, frequency, period, and vertical shift, and phase shift

 A  = Assessment items available