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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