# List of Benchmarks for Mathematics |

**Standard 1.** | Uses a variety of strategies in the problem-solving process |

| ** Level Pre-K (Grade Pre-K)** |

| 1. | Not appropriate at this level |

| ** Level I (Grade K-2)** |

| 1. | Draws pictures to represent problems |

| 2. | Uses discussions with teachers and other students to understand problems |

| 3. | Explains to others how she or he went about solving a numerical problem |

| 4. | Makes organized lists or tables of information necessary for solving a problem |

| 5. | Uses whole number models (e.g., pattern blocks, tiles, or other manipulative materials) to represent problems |

| ** Level II (Grade 3-5)** |

| 1. | Uses a variety of strategies to understand problem situations (e.g., discussing with peers, stating problems in own words, modeling problem with diagrams or physical objects, identifying a pattern) A |

| 2. | Represents problems situations in a variety of forms (e.g., translates from a diagram to a number or symbolic expression) |

| 3. | Understands that some ways of representing a problem are more helpful than others |

| 4. | Uses trial and error and the process of elimination to solve problems |

| 5. | Knows the difference between pertinent and irrelevant information when solving problems A |

| 6. | Understands the basic language of logic in mathematical situations (e.g., "and," "or," "not") A |

| 7. | Uses explanations of the methods and reasoning behind the problem solution to determine reasonableness of and to verify results with respect to the original problem A |

| 8. | Understands basic valid and invalid arguments (e.g., counter examples, irrelevant approaches) |

| ** Level III (Grade 6-8)** |

| 1. | Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem |

| 2. | Uses a variety of strategies to understand problem-solving situations and processes (e.g., considers different strategies and approaches to a problem, restates problem from various perspectives) A |

| 3. | Understands that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages |

| 4. | Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions A |

| 5. | Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms A |

| 6. | Generalizes from a pattern of observations made in particular cases, makes conjectures, and provides supporting arguments for these conjectures (i.e., uses inductive reasoning) A |

| 7. | Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods) A |

| 8. | Understands the role of written symbols in representing mathematical ideas and the use of precise language in conjunction with the special symbols of mathematics |

| 9. | Uses a variety of reasoning processes (e.g., reasoning from a counter example, using proportionality) to model and to solve problems A |

| ** Level IV (Grade 9-12)** |

| 1. | Uses a variety of strategies (e.g., identify a pattern, use equivalent representations) to understand new mathematical content and to develop more efficient solution methods or problem extensions |

| 2. | Constructs algorithms for multi-step and non-routine problems |

| 3. | Understands the concept of a mathematical proof |

| 4. | Constructs logical verifications or counter examples to test conjectures and to justify algorithms and solutions to problems (i.e., uses deductive reasoning) |

| 5. | Uses formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations |

| 6. | Understands the difference between a statement that is verified by mathematical proof (i.e., a theorem) and one that is verified empirically using examples or data |

| 7. | Understands connections between equivalent representations and corresponding procedures of the same problem situation or mathematical concept (e.g., a zero of a function corresponds to an x-intercept of the graph of the function, the correspondence of binary multiplication to a series electrical circuit and the logical operation "and") |

| 8. | Understands the components of mathematical modeling (i.e., problem formulation, mathematical model, solution within the model, interpretation of solution within the model, validation in original real-world problem situation) |

| ** Level V (Grade (College Readiness))** |

| 1. | Translates from specific instances to generalizations and identifies generalizations that follow from specific cases |

| 2. | Understands the structure of an axiomatic system (e.g., axioms, definitions, theorems) |

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A = Assessment items available |