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Topic: Solution strategies 

Mathematics

 Standard 1.  Uses a variety of strategies in the problem-solving process
  Level I (Grade K-2)
   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
  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 4.Uses trial and error and the process of elimination to solve problems
    Knowledge/skill statements
     1.Uses trial and error to solve problems
     2.Uses process of elimination to solve problems
   Benchmark 5.Knows the difference between pertinent and irrelevant information when solving problems
    Knowledge/skill statements
     1.Identifies pertinent and irrelevant information in a problem
   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
  Level III (Grade 6-8)
   Benchmark 1.Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem
    Knowledge/skill statements
     1.Breaks problems into smaller parts
   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 3.Understands that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages
    Knowledge/skill statements
     1.Works backward from a solution
     2.Identifies a similar problem type
   Benchmark 4.Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions
    Knowledge/skill statements
     1.Identifies multiple possible solutions
     2. Sets expectations for possible solutions
     3.Determines what information is needed to solve problem
   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 2.Constructs algorithms for multi-step and non-routine problems
    Knowledge/skill statements
     1.Understands the distinction between routine and non-routine problems
     2.Understands the concept of an algorithm
   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 III (Grade 6-8)
   Benchmark 4.Uses number theory concepts (e.g., divisibility and remainders, factors, multiples, prime, relatively prime) to solve problems
    Knowledge/skill statements
     1.Determines whether a problem can be solved by using the concept of divisibility
     2.Determines whether a problem can be solved by using the concept of multiples
     3.Determines whether a problem can be solved by using the concept of factors
     4.Determines whether a problem can be solved by using the concept of prime numbers
     5.Understands the concept of relatively prime
     6.Determines whether a problem can be solved by using the concept of relatively prime
  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 I (Grade K-2)
   Benchmark 2.Solves real-world problems involving addition and subtraction of whole numbers
    Knowledge/skill statements
     1. Determines whether a problem can be solved by subtracting whole numbers
     2. Determines whether a problem can be solved by adding whole numbers
  Level II (Grade 3-5)
   Benchmark 8.Solves word problems (e.g., problems that require students to put things into groups or combine things that come in groups)and real world problems involving number operations, including those that specify units (e.g., dollars and cents, hours and minutes, cubic units )
    Knowledge/skill statements
     1.Solves real world problems involving number operations with money
     2.Solves real world problems involving number operations with time
     3.Solves real world problems involving operations with volume
     4.Determines whether a real-world problem can be solved with computation
     5.Solves real world problems involving number operations.
     6.Solves word problems involving number operations.
     7.Solves word problems that require students to put things into groups
     8.Solves word problems that require students to combine things that come in groups
  Level III (Grade 6-8)
   Benchmark 6.Uses proportional reasoning to solve mathematical and real-world problems (e.g., involving equivalent fractions, equal ratios, constant rate of change, proportions, percents)
    Knowledge/skill statements
     1. Determines whether a problem can be solved by calculating percents
     2. Understands the concept of a constant rate of change
     3. Determines whether a problem can be solved using the concept of a constant rate of change
     4.Determines whether a problem can be solved using the concept of proportions
     5.Determines whether a problem can be solved using the concept of equivalent fractions
     6.Understands the concept of equivalent fractions
     7. Understands the concept of equal ratios
     8. Understands the concept of mathematical proportion
     9.Determines whether a problem can be solved using the concept of equal ratios
     10. Uses proportional reasoning to solve mathematical problems involving percents
     11.Uses proportional reasoning to solve mathematical problems involving equivalent fractions
     12. Uses proportional reasoning to solve mathematical problems involving equal ratios
     13. Uses proportional reasoning to solve mathematical problems involving a constant rate of change
     14. Uses proportional reasoning to solve mathematical problems involving proportions
  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 4.  Understands and applies basic and advanced properties of the concepts of measurement
  Level IV (Grade 9-12)
   Benchmark 1.Solves problems involving rate as a measure (e.g., speed, density)
    Knowledge/skill statements
     1. Understands concept of speed
     2.Determines whether a problem can be solved by speed
     3.Determines whether a problem can be solved by density
     4. Understands the concept of density
     5.Understands rate as the quantity or amount of something measured per units of something else
 Standard 8.  Understands and applies basic and advanced properties of functions and algebra
  Level III (Grade 6-8)
   Benchmark 5.Solves linear equations using concrete, informal, and formal methods (e.g., using properties, graphing ordered pairs, using slope-intercept form)
    Knowledge/skill statements
     1.Understands the concept of a linear equation
     2.Solves linear equations using knowledge of properties
     3.Solves linear equations by graphing ordered pairs
     4.Solves linear equations using slope-intercept form
     5.Solves linear equations using concrete methods
     6.Solves linear equations using informal methods
     7.Solves linear equations using formal methods
   Benchmark 6.Solves simple inequalities and non-linear equations with rational number solutions, using concrete and informal methods
    Knowledge/skill statements
     1. Solves simple inequalities using informal methods
     2. Understands the concept of a non-linear equation
     3. Solves simple non-linear equations using informal methods
     4. Understands the concept of an inequality
     5. Solves inequalities using concrete methods
     6. Solves non-linear equations using concrete methods
  Level IV (Grade 9-12)
   Benchmark 5. Solves polynomial equations (e.g., find the real or complex roots of a quadratic equation) using a variety of methods (e.g., symbolically, graphically, numerically).
    Knowledge/skill statements
     1.Understands the concept of polynomials
     2.Understands the concept of roots
     3.Solves polynomial equations symbolically.
     4.Models real world problems using polynomial equations without the use of technology.
     5.Solves real world problems using polynomial equations with the use of technology.
     6.Solves polynomial equations by finding the real roots of a quadratic equation.
     7.Solves polynomial equations by finding the complex roots of a quadratic equation.
   Benchmark 11.Uses a variety of methods (e.g., with graphs, algebraic methods, and matrices) to solve systems of equations and inequalities
    Knowledge/skill statements
     1.Solves systems of equations using graphs
     2.Solves systems of equations using matrices
     3.Solves systems of equations using algebraic methods
     4. Understands the concept of systems of inequalities
     5.Solves systems of inequalities using algebraic methods
     6.Solves systems of inequalities using graphs
     7.Solves systems of inequalities using matrices
   Benchmark 16.Solves quadratic equations in one variable(e.g. using quadratic formula, by completing the square)
    Knowledge/skill statements
     1.Understands the concept of quadratic equations
     2.Knows the quadratic formula
     3.Knows how completing the square can be used to solve a quadratic equation
  Level V (Grade (College Readiness))
   Benchmark 1.Solves equations and inequalities involving absolute value
    Knowledge/skill statements
     1.Solves equations involving absolute value
     2.Solves inequalities involving absolute value
   Benchmark 3.Factors polynomials using a variety of methods (e.g., difference of squares, perfect square trinomials)
    Knowledge/skill statements
     1.Factors polynomials using the difference of squares
     2.Factors polynomials using perfect square trinomials
   Benchmark 5.Solves problem situations that require linear, quadratic, or exponential modeling
    Knowledge/skill statements
     1.Understands the concept of modeling a situation using equations
     2.Solves problem situations that require linear modeling
     3. Solves problem situations that require quadratic modeling
     4. Solves problem situations that require exponential modeling