NUMBER AND OPERATIONS

Understand numbers

• Develop a deeper understanding of very large and very small numbers and various representations of them.

(This would be evident if a student created a model using very large or very small numbers—such as models involving orbits of the planets, or drug elimination time constants.)

Compute fluently and make reasonable estimates

• Develop fluency in operations with real numbers, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
  
  
• Judge the reasonableness of numerical computations and their results.

(Both of these standards are addressed every time a student must create reference graphs or analyze the tabular output of various components of the models—to determine if the output seems reasonable.)

ALGEBRA

Understand patterns, relations, and functions

• Generalize patterns using explicitly defined and recursively defined functions; understand relations and functions and select, convert flexibly among, and use various representations for them;

(Students are required to have algebra as a pre-requisite for the modeling course, so they can recognize patterns observed in graphs displayed in news articles. Also, there are explicit lessons on exponential growth, exponential decay, convergent, logistic, and sinusoidal patterns.)

• Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;

(Both the math and modeling courses practice recognizing and developing rates of change that meet certain criteria. Intercepts are evident in various lessons, especially the drug models and spread of epidemic lesson. Asymptotes, local and global behavior are demonstrated, although not explicitly using the math terms, in almost all of the lessons.)

• Understand and perform transformations such as arithmetically combining, and composing, commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;

(Exponential and sinusoidal functions appear in multiple lessons and students are expected to explain the behavior.)

Represent and analyze mathematical situations and structures using algebraic symbols

• Understand the meaning of equivalent forms of expressions;

(This is especially evident in the transferability of structure lesson and the carrying capacity lesson, but is part of other lessons as well.)

• Write equivalent forms of systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

(The mathematical engine behind the software used to build System Dynamics (SD) models is numerical approximation techniques that solve systems of differential equations. Students building models are solving systems when they execute their models. They are performing mental solutions to equations when they construct reference or behavior over time graphs before construction actual simulations. The software can be used to set up and solve simple linear [or linear/exponential, or linear/quadratic, etc.] systems so students could work with both equation and SD software representations.)

• Use symbolic algebra to represent and explain mathematical relationships;

(The STELLA software is used as a symbolic representation of systems involving accumulations and rates of change. Students then explain the mathematical relationships when they explain the shape of the graphs or explaining the feedback controlling the behavior of the system.)

• Use a variety of symbolic representations, including recursive and parametric equations, for functions;

(Creating a model in STELLA is designing a recursively defined system—as that is how the software calculates each variable value. When more than one dependent variable is defined, we have parametric representations, since the independent variable is time.)

• Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology;

(Students explain what is going on in the model design and model (output) behavior in every lesson throughout the modeling course.)

Use mathematical models to represent and understand quantitative relationships

• Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;

(This is the essence of the model-building process.)

• Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;

(This occurs throughout the lessons.)

• Draw reasonable conclusions about a situation being modeled.

(Again, this occurs throughout the lessons.)

Analyze change in various contexts

• Approximate and interpret rates of change from graphical and numerical data.

(This type of modeling focuses on rates of change and students explain the change by viewing the graphs and tables produced by the models and explaining the shape.)

MEASUREMENT

Understand measurable attributes of objects and the units, systems, and processes of measurement

• Make decisions about units and scales that are appropriate for problem situations involving measurement.

(Unit consistency and the understanding of appropriate units is required in all the lessons.)

Apply appropriate techniques, tools, and formulas to determine measurements

• Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;

(As students manipulate parameters to try to replicate pre-defined reference or behavior over time graphs they are using successive approximation. Testing the robustness of a model requires an understanding of reasonable upper and lower bounds of behavior. Limit behavior is viewed as the simulation time is extended, again to determine if reasonable behavior continues as the simulation progresses.)

• Use unit analysis to check measurement computations.

(Built in to all lessons.)

DATA ANALYSIS AND PROBABILITY

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

• Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable;

(Students must have a pre-conception about a variable, as they use variables throughout the lessons. Students must analyze measurement data when building model sketches from news articles and when analyzing data for their modeling project.)

• Develop and evaluate inferences and predictions that are based on data.

(Students are building simulations and experimenting with them in the modeling course. They can alter parameters and analyze how the shape of the resultant behavior is changed. Doing sensitivity analysis [changing parameters automatically among several values] produces a family of curves from which they can analyze variability.)

PROBLEM SOLVING

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve problems;
• Monitor and reflect on the process of mathematical problem solving.

(All of these requirements occur as part of the model building process.)

REASONING AND PROOF

• Make and investigate mathematical conjectures;
• Develop and evaluate mathematical arguments;
• Select and use various types of reasoning.

(All of these requirements occur as part of the model building process.)

COMMUNICATION

• Organize and consolidate their mathematical thinking through communication;

(Students are expected to explain model [output] behavior.)

• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

(Students are expected to predict and explain the behavior of their models in stages, in the modeling class. There is a significant amount of communication expected from the students in the modeling course, both in writing and verbally.)

• Analyze and evaluate the mathematical thinking and strategies of others;

(Students work in teams of two on all lessons and on their modeling projects.)

• Use the language of mathematics to express mathematical ideas precisely;

(This is an expectation for the written communication in the course.)

CONNECTIONS

• Organize and consolidate their mathematical thinking through communication;
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• Recognize and apply mathematics in contexts outside of mathematics.

(All of these requirements occur as part of the model building process.)

REPRESENTATION

• Create and use representations to organize, record, and communicate mathematical ideas;
• Select, apply, and translate among mathematical representations to solve problems;
• Use representations to model and interpret physical, social, and mathematical phenomena.

(All of these requirements occur as part of the model building process.)

¹ The sub-list of standards are referenced from the full list at the National Council of Teachers of Mathematics (NCTM) web site.
² The focus on rate of change and accumulation provides a conceptual introduction to the foundation concepts of calculus.