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Topic: Representing problems 

Mathematics

 Standard 1.  Uses a variety of strategies in the problem-solving process
  Level I (Grade K-2)
   Benchmark 1.Draws pictures to represent problems
    Knowledge/skill statements
     1.Knows that pictures can represent problems
   Benchmark 2.Uses discussions with teachers and other students to understand problems
    Knowledge/skill statements
     1.Discusses with peers to understand a numerical problem
   Benchmark 3.Explains to others how she or he went about solving a numerical problem
    Knowledge/skill statements
     1. Communicates the steps of a problem–solving process
   Benchmark 4.Makes organized lists or tables of information necessary for solving a problem
    Knowledge/skill statements
     1. Constructs table to solve a numerical problem
     2.Makes organized lists to solve a numerical problem
     3.Makes a chart to solve a numerical problem
   Benchmark 5.Uses whole number models (e.g., pattern blocks, tiles, or other manipulative materials) to represent problems
    Knowledge/skill statements
     1.Understands that manipulatives (e.g., tiles, blocks) can be used to model problems
  Level II (Grade 3-5)
   Benchmark 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)
    Knowledge/skill statements
     1.States problem in own words
     2.Makes diagrams of a problem
     3.Uses physical objects to model a problem
     4.Discusses with peers to understand a problem situation
     5.Uses patterns to solve problems
   Benchmark 2.Represents problems situations in a variety of forms (e.g., translates from a diagram to a number or symbolic expression)
    Knowledge/skill statements
     1.Translates from a diagram to a numeric expression
     2.Translates from a diagram to a symbolic expression
     3.Uses pictures to represent problem situations
     4.Uses manipulatives to model problem situations
   Benchmark 3.Understands that some ways of representing a problem are more helpful than others
    Knowledge/skill statements
     1.Selects best representation of a problem
  Level III (Grade 6-8)
   Benchmark 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)
    Knowledge/skill statements
     1.Restates a problem in a number of ways
   Benchmark 5.Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms
    Knowledge/skill statements
     1.Translates a verbal problem into a graph
     2.Translates a verbal problem into a pictorial representation
     3.Translates a verbal problem into a concrete representation
   Benchmark 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
    Knowledge/skill statements
     1. Understands how written symbols represent mathematical ideas
   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 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
    Knowledge/skill statements
     1.Uses strategies to understand new mathematical content
     2.Uses strategies to develop more efficient solution methods
     3.Uses strategies to extend solution methods to other problems
   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 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")
    Knowledge/skill statements
     1.Knows x-intercept is zero point of function
     2.Understands equivalent representations of a problem situation or mathematical concept
     3.Understands the correspondence of binary multiplication to a series electrical circuit
     4.Understands corresponding procedures for the same problem situation or mathematical concept
   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
 Standard 2.  Understands and applies basic and advanced properties of the concepts of numbers
  Level IV (Grade 9-12)
   Benchmark 3.Uses discrete structures (e.g., finite graphs, matrices, sequences) to represent and to solve problems
    Knowledge/skill statements
     1.Determines whether a problem can be modeled by number sequences
     2.Determines whether a problem can be solved by identifying number sequences
     3.Understands the concept of a number sequence
     4.Understands the concept of finite graphs
     5.Determines whether a problem can be modeled by finite graphs
     6.Understands the concept of discrete structures
     7.Determines whether a problem can be solved by using finite graphs
     8.Determines whether a problem can be modeled using matrices
     9.Determines whether a problem can be solved by using matrices
     10.Determines whether a problem can be modeled using discrete structures
     11.Determines whether a problem can be solved by using discrete structures
 Standard 3.  Uses basic and advanced procedures while performing the processes of computation
  Level IV (Grade 9-12)
   Benchmark 6.Uses recurrence relations (i.e., formulas expressing each term as a function of one or more of the previous terms, such as the Fibonacci sequence or the compound interest equation) to model and to solve real-world problems (e.g., home mortgages, annuities)
    Knowledge/skill statements
     1.Understands Fibonacci sequence is a type of recurrence relationship
     2.Determines whether a problem can be modeled by a recurrence relation
     3.Understands the concept of a recurrence equation
     4.Determines whether a problem can be solved by a recurrence relation
 Standard 8.  Understands and applies basic and advanced properties of functions and algebra
  Level IV (Grade 9-12)
   Benchmark 2.Uses expressions, equations, inequalities, and matrices to represent situations that involve variable quantities and translates among these representations
    Knowledge/skill statements
     1.Understands the relationship between inequalities and equations
     2.Transforms inequalities to equations
     3.Understands the relationship between equations and matrices
     4.Represents equations in matrix form
     5.Understands the relationship between expressions and equations
     6.Understands the relationship between expressions and inequalities
     7.Understands the relationship between expressions and matrices
     8.Understands the relationship between inequalities and matrices
     9.Understands the concept of an algebraic expressions
 Standard 9.  Understands the general nature and uses of mathematics
  Level II (Grade 3-5)
   Benchmark 2.Understands that mathematical ideas and concepts can be represented concretely, graphically, and symbolically
    Knowledge/skill statements
     1.Understands that mathematical ideas can be represented concretely
     2.Understands that mathematical concepts can be represented graphically
     3.Understands that mathematical concepts can be represented symbolically