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 |