Topic: Proof and empirical verification |
Mathematics |
| Standard 1. Uses a variety of strategies in the problem-solving process |
| | Level II (Grade 3-5) |
| | | 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 7. | Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of a logical argument |
| | | | | 2. | Understands if/then statements |
| | | | | 3. | Justifies solution methods through logical argument |
| | Level IV (Grade 9-12) |
| | | Benchmark 3. | Understands the concept of a mathematical proof |
| | | | Knowledge/skill statements |
| | | | | 1. | Develops proof paragraphs |
| | | | | 2. | Carries out indirect proofs |
| | | | | 3. | Carries out direct proofs |
| | | | | 4. | Carries out proof by example |
| | | | | 5. | Carries out proof by truth tables |
| | | 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 |
| | | Benchmark 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 |
| | | | Knowledge/skill statements |
| | | | | 1. | Knows that a theorem is a statement verified by a mathematical proof or empirical data |
| | | | | 2. | Understands the concept of verification through examples or empirical data |
| | | | | 3. | Understands concept of mathematical proof |
| | | Benchmark 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) |
| | | | Knowledge/skill statements |
| | | | | 1. | Knows problem formulation as a component of mathematical modeling |
| | | | | 2. | Knows problem solving within a model as a component of mathematical modeling |
| | | | | 3. | Interprets a model solution in a real-world situation as a part of mathematical modeling |
| | Level V (Grade (College Readiness)) |
| | | Benchmark 2. | Understands the structure of an axiomatic system (e.g., axioms, definitions, theorems) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands what an axiom is |
| | | | | 2. | Understands what a definition is |
| | | | | 3. | Understands what a theorem is |
| Standard 5. Understands and applies basic and advanced properties of the concepts of geometry |
| | Level IV (Grade 9-12) |
| | | Benchmark 5. | Uses geometric constructions (e.g., the parallel to a line through a given point not on the line, line segment congruent to a given line segment) to complete simple proofs, to model, and to solve mathematical and real-world problems |
| | | | Knowledge/skill statements |
| | | | | 1. | Constructs a line segment congruent to a line segment |
| | | | | 2. | Understands the basic concept of a geometric construction |
| | | | | 3. | Constructs a line through a given point not on the line |
| | | | | 4. | Uses a line segment congruent to given line segment in proofs |
| | | | | 5. | Uses geometric constructions to complete simple proofs |
| | | | | 6. | Determines whether a problem can be modeled by geometric constructions |
| | | | | 7. | Determines whether a problem can be solved by using geometric constructions |
| | | 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 |
| | Level V (Grade (College Readiness)) |
| | | Benchmark 4. | Constructs geometric proofs (e.g., proves the Pythagorean theorem, proves there are 180 degrees in a triangle) |
| | | | Knowledge/skill statements |
| | | | | 1. | Constructs geometric proof of the Pythagorean theorem |
| | | | | 2. | Constructs geometric proof that there are 180 degrees in a triangle |