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Mathematical Modeling Appendix C

All professional education content courses leading to certification shall include teaching and assessment of the Wisconsin Content Standards in the content area.

In this column, list the Wisconsin Content Standards that are included in this course. The Standards for each content area are found in the Wisconsin Content Standards document. |
In this column, indicate the nature of the performance assessments used in this course to evaluate student proficiency in each standard. |
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The structures within the discipline, the historical roots and evolving nature of mathematics, and the interaction between technology and the discipline. |
Students will use a computer algebra system, dynamic graphing software, spreadsheets, applets and graphing calculators in their solutions to real-world problems. |

Facilitating the building of student conceptual and procedural understanding. |
Conceptual and procedural understanding will be developed during classroom discussions while students engage in the modeling process: create a mathematical model based on given assumptions about a real-world situation, solve the mathematical model and interpret the given result(s) within the context of the original problem. Student understanding of mathematical modeling concepts and procedures will be assessed on exams and homework. |

Helping all students build understanding of the discipline including: . Confidence in their abilities to utilize mathematical knowledge. . Awareness of the usefulness of mathematics. . The economic implications of fine mathematical preparation. |
Students engage in the modeling process by making assumptions about the situation, creating an appropriate mathematical model, solving their mathematical model, and reporting their results. Their ability to do so will be analyzed based on classroom discussion, quiz items, test items and a final mathematical modeling project. |

Exploring, conjecturing, examining and testing all aspects of problem solving. |
Homework exercises, mathematical modeling project with a written report. |

Formulating and posing worthwhile mathematical tasks, solving problems using several strategies, evaluating results, generalizing solutions, using problem solving approaches effectively, and applying mathematical modeling to real-world situations. |
Homework exercises, computer activities, classroom discussion, mathematical modeling project with a written report. |

Making convincing mathematical arguments, framing mathematical questions and conjectures, formulating counter-examples, constructing and evaluating arguments, and using intuitive, informal exploration and formal proof. |
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Expressing ideas orally, in writing, and visually-, using mathematical language, notation, and symbolism; translating mathematical ideas between and among contexts. |
Classroom discussion, homework exercises, mathematical modeling project with a written report. |

Connecting the concepts and procedures of mathematics, drawing connections between mathematical strands, between mathematics and other disciplines, and with daily life. |
Homework exercises, mathematical modeling project with a written report. |

Selecting appropriate representations to facilitate mathematical problem solving and translating between and among representations to explicate problem-solving situations. |
Students will develop mathematical models to solve real-world problems. They will explain their solutions and methods symbolically, verbally and graphically. Their ability to do so will be assessed with exam items, computer activities, mathematical modeling project with a written report. |

Mathematical processes including: . Problem solving. . Communication. . Reasoning and formal and informal argument. . Mathematical connections. . Representations. . Technology. |
Students will engage in using mathematics to solve real-world problems. Their solutions will require clear communication of methods and results; connections between mathematics and real-world situations; symbolic, verbal and graphical representations of their methods and results; and the use of a variety of technologies, such as computer algebra systems and spreadsheets. Their work will be assessed through classroom discussions, computer activities, homework exercises, mathematical modeling project with a written report. |

Number operations and relationships from both abstract and concrete perspectives identifying real world applications, and representing and connecting mathematical concepts and procedures including: . Number sense. . Set theory. . Number and operation. . Composition and decomposition of numbers, including place value, primes, factors, multiples, inverses, and the extension of these concepts throughout mathematics. . Number systems through the real numbers, their properties and relations. . Computational procedures. . Proportional reasoning. . Number theory. |
Exercises on descriptive statistics, sampling and confidence intervals. |

Mathematical concepts and procedures, and the connections among them for teaching upper level number operations and relationships including: . Advanced counting procedures, including union and intersection of sets, and parenthetical operations. . Algebraic and transcendental numbers. . The complex number system, including polar coordinates. . Approximation techniques as a basis for numerical integration, fractals, and numerical-based proofs. . Situations in which numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social) can be created and critically evaluated. . Opportunities in which acceptable limits of error can be assessed (e.g., evaluating strategies, testing the reasonableness of results, and using technology to carry out computations). |
Mathematical modeling project includes students doing background research on a real-world problem, applying technology to address the problem and testing the validity of their model. Homework exercises and exam items will require students to use numerical reasoning in real-world situations and to test reasonableness of results. |

Geometry and measurement from both abstract and concrete perspectives and to identify real world applications, and mathematical concepts, procedures and connections among them including: . Formal and informal argument. . Names, properties, and relationships of two- and three-dimensional shapes. . Spatial sense. . Spatial reasoning and the use of geometric models to represent, visualize, and solve problems. . Transformations and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships. . Coordinate geometry systems including relations between coordinate and synthetic geometry, and generalizing geometric principles from a two-dimensional system to a three-dimensional system. . Concepts of measurement, including measurable attributes, standard and non-standard units, precision and accuracy, and use of appropriate tools. . The structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems. Measurement including length, area, volume, size of angles, weight and mass, time, temperature, and money. . Measuring, estimating, and using measurement to describe and compare geometric phenomena. . Indirect measurement and its uses, including developing formulas and procedures for determining measure to solve problems. |
Exam items and homework exercises dealing with measurements and units in real-world situations. |

Mathematical concepts, procedures, and the connections among them for teaching upper level geometry and measurement including: . Systems of geometry, including Euclidean, non-Euclidean, coordinate, transformational, and projective geometry. . Transformations, coordinates, and vectors and their use in problem solving. Three-dimensional geometry and its generalization to other dimensions. Topology, including topological properties and transformations. . Opportunities to present convincing arguments by means of demonstration, informal proof, counter-examples, or other logical means to show the truth of statements and/or generalizations. |
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Statistics and probability from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections between them including: . Use of data to explore real-world issues. . The process of investigation including formulation of a problem, designing a data collection plan, and collecting, recording, and organizing data. . Data representation through graphs, tables, and summary statistics to describe data distributions, central tendency, and variance. . Analysis and interpretation of data. . Randomness, sampling, and inference. . Probability as a way to describe chances or risk in simple and compound events. . Outcome prediction based on experimentation or theoretical probabilities. |
Classroom discussion, homework exercises, computer activities and exam items on descriptive statistics, sampling, confidence intervals, hypothesis testing and their applications in real-world situations. |

Mathematical concepts, procedures, and the connections among them for teaching upper level statistics and probability including: . Use of the random variable in the generation and interpretation of probability distributions. . Descriptive and inferential statistics, measures of disbursement, including validity and reliability, and correlation. . Probability theory and its link to inferential statistics. . Discrete and continuous probability distributions as bases for inference. . Situations in which students can analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc. |
Classroom discussion, homework exercises, computer activities and exam items are used to assess students ability to use methods from discrete math and calculus to solve stochastic mathematical models. |

Functions, algebra, and basic concepts underlying calculus from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including: . Patterns. . Functions as used to describe relations and to model real world situations. . Representations of situations that involve variable quantities with expressions, equations and inequalities and that include algebraic and geometric relationships. . Multiple representations of relations, the strengths and limitations of each representation, and conversion from one representation to another. . Attributes of polynomial, rational, trigonometric, algebraic, and exponential functions. . Operations on expressions and solution of equations, systems of equations and inequalities using concrete, informal, and formal methods. . Underlying concepts of calculus, including rate of change, limits, and approximations for irregular areas. |
Computer activities fitting data to function models. |

Mathematical concepts, procedures, and the connections among them for teaching upper level functions, algebra, and concepts of calculus including: . Concepts of calculus, including limits (epsilon-delta) and tangents, derivatives, integrals, and sequences and series. . Modeling to solve problems. . Calculus techniques including finding limits, derivatives, integrals, and using special rules. . Calculus applications including modeling, optimization, velocity and acceleration, area, volume, and center of mass. . Numerical and approximation techniques including Simpson's rule, trapezoidal rule, Newton's Approximation, and linearization. . Multivariate calculus. . Differential equations. |
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Discrete processes from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including: . Counting techniques. . Representation and analysis of discrete mathematics problems using sequences, graph theory, arrays, and networks. . Iteration and recursion. |
Homework exercises and exam items focusing on discrete dynamical systems. |

Mathematical concepts, procedures, and the connections among them for teaching upper level discrete mathematics including: . Topics, including symbolic logic, induction, linear programming, and finite graphs. . Matrices as a mathematical system, and matrices and matrix operations as tools for recording information and for solving problems. . Developing and analyzing algorithms. |
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