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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