Topic: Mathematical reasoning |
Mathematics |
| Standard 1. Uses a variety of strategies in the problem-solving process |
| | Level II (Grade 3-5) |
| | | Benchmark 6. | Understands the basic language of logic in mathematical situations (e.g., "and," "or," "not") |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands logic operator “and” |
| | | | | 2. | Understands logic operator “or” |
| | | | | 3. | Understands logic operator “not” |
| | | Benchmark 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 |
| | | | Knowledge/skill statements |
| | | | | 1. | Determines reasonableness of a problem solution |
| | | | | 2. | Verify results of problem solution |
| | | Benchmark 8. | Understands basic valid and invalid arguments (e.g., counter examples, irrelevant approaches) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands valid counter examples |
| | | | | 2. | Recognizes invalid approaches based on irrelevant information |
| | Level III (Grade 6-8) |
| | | Benchmark 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) |
| | | | Knowledge/skill statements |
| | | | | 1. | Provides supportive arguments for conjectures |
| | | | | 2. | Generalizes from a pattern of observations |
| | | | | 3. | Makes conjectures from generalizations |
| | | Benchmark 9. | Uses a variety of reasoning processes (e.g., reasoning from a counter example, using proportionality) to model and to solve problems |
| | | | Knowledge/skill statements |
| | | | | 1. | Uses counter examples to model problems |
| | | | | 2. | Uses proportionality to model problems |
| | | | | 3. | Uses counter examples to solve problems |
| | | | | 4. | Uses proportionality to solve problems |
| | Level IV (Grade 9-12) |
| | | Benchmark 4. | Constructs logical verifications or counter examples to test conjectures and to justify algorithms and solutions to problems (i.e., uses deductive reasoning) |
| | | | Knowledge/skill statements |
| | | | | 1. | Tests conjectures using counter examples |
| | | | | 2. | Tests conjectures through logical analysis |
| | | Benchmark 5. | Uses formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations |
| | | | Knowledge/skill statements |
| | | | | 1. | Explains relationships among different representations of problems |
| | | | | 2. | Represents word problems using formal notation |
| | Level V (Grade (College Readiness)) |
| | | Benchmark 1. | Translates from specific instances to generalizations and identifies generalizations that follow from specific cases |
| | | | Knowledge/skill statements |
| | | | | 1. | Translates from specific instance to generalizations |
| | | | | 2. | Identifies generalizations that follow from specific cases |
| | | | | 3. | Understands the concept of generalizations |
| Standard 5. Understands and applies basic and advanced properties of the concepts of geometry |
| | Level IV (Grade 9-12) |
| | | Benchmark 10. | Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures (e.g., the relationship among interior angles of parallel lines cut by a transversal) |
| | | | Knowledge/skill statements |
| | | | | 1. | Makes deductions about the relationships among geometric figures |
| | | | | 2. | Makes inductions about the relationships among geometric figures |
| | | | | 3. | Understands the concept of a transversal |
| | | | | 4. | Makes deductions about the properties of figures |
| | | | | 5. | Makes inductions about the properties of figures |
| Standard 6. Understands and applies basic and advanced concepts of statistics and data analysis |
| | Level III (Grade 6-8) |
| | | Benchmark 7. | Understands faulty arguments, common errors, and misleading presentations of data |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands faulty arguments from data |
| | | | | 2. | Understands misleading presentations of data |
| | | | | 3. | Understands common errors with data |
| | Level V (Grade (College Readiness)) |
| | | Benchmark 2. | Evaluates reports in the media in terms of the sources of the data and the design of the study |
| | | | Knowledge/skill statements |
| | | | | 1. | Evaluates reports in the media based on the sources of the data |
| | | | | 2. | Evaluates reports in the media based on the design of the study |