UNIVERSITY OF WISCONSIN River Falls

# Number Theory Appendix C

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.

The structures within the discipline, the historical roots and evolving nature of mathematics, and the interaction between technology and the discipline.

This course deals with properties of integers. Students' understanding of these properties is assessed through homework and exams.

Facilitating the building of student conceptual and procedural understanding.

This course covers the concepts and procedures used the study of integers. Class discussions together with homework and exams are used to assess this.

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' understanding of number theory and its uses are measured by a combination of classroom discussions, homework and exams.

Exploring, conjecturing, examining and testing all aspects of problem solving.

By examining conjectures about integers and producing formal proofs of these conjectures in selected homework problems and exam questions students show their skills at problem solving.

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.

The ability to pose and solve problems in number theory is measured by exam questions and homework problems.

Making convincing mathematical arguments, framing mathematical questions and conjectures, formulating counter-examples, constructing and evaluating arguments, and using intuitive, informal exploration and formal proof.

Student's have the opportunity to show their ability at giving proofs and constructing counter-examples as well as doing informal exploration. This ability is assessed by using homework problems and exam questions.

Expressing ideas orally, in writing, and visually-, using mathematical language, notation, and symbolism; translating mathematical ideas between and among contexts.

Classroom discussion, homework and exams are used to assess the ability of students to convert English statements about integers into formal mathematical statements

Connecting the concepts and procedures of mathematics, drawing connections between mathematical strands, between mathematics and other disciplines, and with daily life.

By exploring the concepts and methods used in number theory in working out homework problems, students demonstrate their understanding of these ideas.

Selecting appropriate representations to facilitate mathematical problem solving and translating between and among representations to explicate problem-solving situations.

The ability to translate English statements into the forms used in number theory in order to solve problems is assessed through selected exam questions and homework problems.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

Formal arguments are used to prove many number-theoretic results. The ability to produce these formal arguments is assessed on exams and in homework.

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.

As a number theory course operations on integers and relationships among the integers is the heart of the course. This is assessed through class discussions, homework and exams.

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).

Not assessed.

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.

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.

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.

Not assessed.

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.

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.

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.

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.

Not assessed.

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.

Not assessed

### Contact Us

Department of Mathematics
Kathy Tomlinson, Department Chair
Email
715-425-3326, Fax: 715-425-32O3
207 North Hall
410 S. 3rd Street, River Falls, WI 54022-5001