familiar descriptive model—for example, graphs of global temperature High School—Modeling, CONTINUED and atmospheric CO2 over time. One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical structure can Analytic modeling seeks to explain data on the basis of deeper sometimes model seemingly different situations. Models can also theoretical ideas, albeit with parameters that are empirically based; shed light on the mathematical structures themselves, for example, as for example, exponential growth of bacterial colonies (until cut-off when a model of bacterial growth makes more vivid the explosive mechanisms such as pollution or starvation intervene) follows from a growth of the exponential function. constant reproduction rate. Functions are an important tool for analyzing such problems. The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent Graphing utilities, spreadsheets, computer algebra systems, and essential features, (2) formulating a model by creating and selecting dynamic geometry software are powerful tools that can be used to geometric, graphical, tabular, algebraic, or statistical representations model purely mathematical phenomena, e.g., the behavior of that describe relationships between the variables, (3) analyzing and polynomials as well as physical phenomena. performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original MODELING STANDARDS situation, (5) validating the conclusions by comparing them with the Modeling is best interpreted not as a collection of isolated topics but situation, and then either improving the model or, if it is acceptable, rather in relation to other standards. Making mathematical models is a (6) reporting on the conclusions and the reasoning behind them. Standard for Mathematical Practice, and specific modeling standards Choices, assumptions, and approximations are present throughout appear throughout the high school standards indicated by a star this cycle. symbol (★). PROBLEM FORMULATE VALIDATE REPORT COMPUTE INTERPRET In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a
High School—Modeling, CONTINUED One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical structure can sometimes model seemingly different situations. Models can also shed light on the mathematical structures themselves, for example, as when a model of bacterial growth makes more vivid the explosive growth of the exponential function. The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle. familiar descriptive model—for example, graphs of global temperature and atmospheric CO2 over time. Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are empirically based; for example, exponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from a constant reproduction rate. Functions are an important tool for analyzing such problems. Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that can be used to model purely mathematical phenomena, e.g., the behavior of polynomials as well as physical phenomena. MODELING STANDARDS Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). PROBLEM FORMULATE VALIDATE REPORT COMPUTE INTERPRET In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a
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