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 problemâ€“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 |