Benchmark 9B: The Mathematical World - Symbolic Relationships

• In some cases, the more of something there is, the more rapidly it may change (as the number of births is proportional to the size of the population). In other cases, the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting). (1 of 6)
• Standard 13-6, page 116, Grades 5-8
Develop the concepts of rates and other derived and indirect measurements

Standard 8-3, page 98, Grades 5-8
Analyze functional relationships to explain how a change in one quantity results in a change in another

Standard 6-3 page 154, Grades 9-12
Translate among tabular, symbolic, and graphical representations of functions

• Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time. (2 of 6)
• Standard 14-2 page 184, Grades 9-12
Understand the logic of algebraic procedures

Standard 5-3 page 150, Grades 9-12
Operate on expressions and matrices, and solve equations and inequalities

• Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences , or by requiring too much computation. (3 of 6)
• Standard 5-2 page 150, Grades 9-12
Use tables and graphs as tools to interpret expressions, equations, and inequalities

Standard 1-4 page 137, Grades 9-12
Apply the process of mathematical modeling to real-world problem situations

• Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another. (4 of 6)
• Standard 6-3 page 154, Grades 9-12
Translate among tabular, symbolic, and graphical representations of functions

Standard 6-4 page 154, Grades 9-12
Recognize that a variety of problem situations can be modeled by the same type of function

• When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all. (5 of 6)
• Standard 5-3 page 150, Grades 9-12
Operate on expressions and matrices, and solve equations and inequalities
• The reasonableness of a result of a computation can be estimated from what the input and operations are. (6 of 6)
• Standard 7-6 page 94, Grades 5-8
Use estimation to check the reasonableness of results

Standard 12-3 page 176, Grades 9-12
Develop and analyze algorithms