UNIVERSITY OF WISCONSIN River Falls

# Probability 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.

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

Origins of probability theory in gambling and insurance are discussed. Calculators are used to generate random variables.

Facilitating the building of student conceptual and procedural understanding.

Students are tested on concepts and methods of probability.

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.

Homework and exam problems require an understanding of prerequisite courses such as calculus and discrete math.

Applications of probability are stressed throughout the course, including problem assignments.

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

Students are encouraged to try different approaches, rather than rely on memorized formulas.

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.

In homework and exam problems, different strategies are used. Students work on probabilistic models involving "real-world" situations such as dice and card games, then generalize this to a more abstract setting.

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 prove basic results using the axioms of probability and such concepts as independence. When a result is not true, students are expected to find counter-examples.

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

In homework and exam problems, students are expected to "bridge the gap" between informal intuitive reasoning and rigorous mathematics.

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

Many problems are based on examples from industry, games and other sciences.

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

Homework and exam problems require geometric visualization, counting techniques and calculus-based methods.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

Solutions (homework and exam) require various problem-solving techniques based on discrete math and calculus.

Calculators and computers are occasionally used for simulations and generation of random variables.

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.

The axiomatic development of probability is set-theoretic and students are tested on this. Throughout the course, numerical calculations are important. Also, students are encouraged to always check whether their solutions "make sense". (e.g. probabilities below 0 or above 1 make no sense.)

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).

Many homework and exam problems are based on sophisticated counting techniques learned in discrete math.

"Real-world" applications of probability (odds, expected payoffs, etc.) are stressed throughout the course.

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.

Geometric models of probability (based on uniform random variables) are used repeatedly. In homework and on exams, students must be able to interpret probabilities as lengths, areas or volumes, depending on the context.

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.

Counter-examples, proofs, and valid mathematical exposition are assessed through homework and tests.

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.

Throughout the course, in exams and homework, students are assessed on their ability to derive probabilistic results and employ probabilistic reasoning in solving "real-world" problems.

Data from random experiments or "real-life" events are frequently examined, including statistics such as mean, median, variance and standard deviation.

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.

Students are expected to understand random variables and know how certain random variables are generated from simulations.

Homework and exams test students' ability to employ methods from discrete math and calculus (for continuous models).

The importance of probability as a necessary tool for statistical decision-making is examined, and "real-life" applications are stressed.

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.

Students need to employ algebraic and calculus-based methods in solving probability problems.

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.

Concepts of calculus reviewed and applied include

i) the fundamental theorem of calculus

ii) chain rule of differentiation

iii) exponential functions

iv) definite vs indefinite integral

v) integration techniques

vi) improper integrals

vii) derivatives

viii) Taylor series

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.

In homework and on exams, students must be able to use counting techniques from discrete math (multiplication principle, combinations, permutations, binomial coefficients, etc.)

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.

Besides basic counting techniques, discrete math concepts such as recursion (for calculating probabilities) and induction (for proving results) are examined.

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