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Combinatorics and Graph Theory 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. |
Homework assignments on Euler circuits, Hamilton paths and the 4-color Theorem pertain to the historical roots of the subject. Assignments on generating functions deal with the interaction between technology and the subject. |

Facilitating the building of student conceptual and procedural understanding. |
In class discussions combined with homework assignments are used to assess the conceptual and procedural understanding of the student. |

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. |
Most topics covered in the course deal with applications of mathematics. Each assignment is used to assess the students' knowledge of these applications. |

Exploring, conjecturing, examining and testing all aspects of problem solving. |
Various problem solving strategies are applied in the classroom discussions of solving counting problems. |

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. |
Counting problems are first carefully posed, then solved with direct methods and indirect methods such as inclusion-exclusion, recurrence relations and generating functions. The students' ability is then assessed on an exam. |

Making convincing mathematical arguments, framing mathematical questions and conjectures, formulating counter-examples, constructing and evaluating arguments, and using intuitive, informal exploration and formal proof. |
Students are expected to be able to produce conjectures concerning various general counting problems. In the exercises, the students are to give proofs or counter-examples as appropriate. |

Expressing ideas orally, in writing, and visually-, using mathematical language, notation, and symbolism; translating mathematical ideas between and among contexts. |
Students are expected to participate in classroom discussions on the topics and then write solutions to the exercises using correct mathematical language. |

Connecting the concepts and procedures of mathematics, drawing connections between mathematical strands, between mathematics and other disciplines, and with daily life. |
The interaction between graph theory, counting topics and tasks from daily life are stressed in discussions and on exams. |

Selecting appropriate representations to facilitate mathematical problem solving and translating between and among representations to explicate problem-solving situations. |
Students are expected to show various ways that graphs can be represented and chose the best representation for the given application |

Mathematical processes including: . Problem solving. . Communication. . Reasoning and formal and informal argument. . Mathematical connections. . Representations. . Technology. |
Each exam and homework assignment is used to assess these skills in the context of the various topics of the course. |

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. |
Not assessed in this course. |

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). |
Students are assigned problems dealing with advanced counting procedures including the use of generating functions for sequences and two variable recurrence relations for the Stirling Numbers. These techniques are also assessed on the exams. |

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. |
Not assessed in this course. |

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. |
Not assessed in this course |

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. |
The techniques of solving counting problems are applied to elementary probability questions in the assignments and the exams. |

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. |
Not assessed in this course. |

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. |
Not assessed in this course. |

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. |
Not assessed in this course. |

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. |
The students' understanding of all of these topics is assessed throughout the course on the assignments, in classroom discussions, and on the exams. |

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. |
The students' understanding of mathematical induction, finite graphs, matrices as a method for representing graphs, and analyzing algorithms is assessed on the assignments, in classroom discussions, and on the exams. |