Modeling

Modeling uses mathematics to help us make sense of the real world—to understand quantitative relationships, make predictions, and propose solutions. A model can be very simple, such as a geometric shape to describe a physical object like a coin. Even so simple a model involves making choices. It is up to us whether to model the solid nature of the coin with a three-dimensional cylinder, or whether a two-dimensional disk works well enough for our purposes. For some purposes, we might even choose to adjust the right circular cylinder to model more closely the way the coin deviates from the cylinder. In any given situation, the model we devise depends on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have? The range of models we can create and analyze is constrained as well by the limitations of our mathematical and technical skills. For example, modeling a physical object, a delivery route, a production schedule, or a comparison of loan amortizations each requires different sets of tools. Networks, spreadsheets and algebra are powerful tools for understanding and solving problems drawn from different types of real-world situations. One of the insights provided by mathematical modeling is that essentially the same mathematical structure might model seemingly different situations. The basic modeling cycle is one of (1) identifying the key features of a situation, (2) creating geometric, algebraic or statistical objects that describe key features of the situation, (3) analyzing and performing operations on these objects to draw conclusions and (4) interpreting the results of the mathematics in terms of the original situation. Choices and assumptions are present throughout this cycle. //Connections to Quantity, Equations, Functions, Shape, Coordinates and Statistics.// Modeling makes use of shape, data, graphs, equations and functions to represent real-world quantities and situations. Students understand that: >
 * Core Concepts**
 * 1) Mathematical models involve choices and assumptions that abstract key features from situations to help us solve problems.
 * 1) Even very simple models can be useful.

Students can and do: > > > > > >
 * Core Skills**
 * 1) Model numerical situations.
 * 1) Model physical objects with geometric shapes.
 * 1) Model situations with equations and inequalities.
 * 1) Model situations with common functions.
 * 1) Model situations using probability and statistics.
 * 1) Interpret the results of applying a model and compare models for a particular situation.