# 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 |

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A = Assessment items available |