UNIVERSITY OF WISCONSIN River Falls

# Calculus I 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.

Students calculate integrals in both the Riemannian and Newton-Leibnitz methods. Students calculate derivatives using both limit and derived formula methods.* Students use calculators to investigate limits, and to analyze graphs showing extreme values.

Facilitating the building of student conceptual and procedural understanding.

Students show understanding of calculus concepts from algebraic, numerical, and graphical perspectives. Students apply calculus techniques to analyze functions.*

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 integrate their knowledge from previous courses in Algebra and Trigonometry with the new calculus they are learning, gaining confidence in their ability to utilize mathematical knowledge. Students investigate applications of calculus to science and economics: most notably in extreme values applications, but also in related rates, volume and work applications. *

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

Students make and test conjectures when determining substitutions for integrals. *

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.

Students calculate derivatives using both limit and derived formula methods, and calculate integrals in both the Riemannian and Newton-Leibnitz methods. Students use calculator based methods to evaluate their answers for finding extreme values using calculus techniques. Students use problem solving approaches in the context of applications of calculus to science and economics: most notably in extreme values applications, but also in related rates, volume and work applications.*

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 make mathematical arguments by using theorems to show the continuity of functions, and verify that the hypotheses of theorems such as the Mean Value Theorem and the Fundamental Theorem of Calculus are satisfied.*

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

Students express their understanding of calculus concepts algebraically, numerically and graphically, and answer questions that require them to translate between these contexts.*

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

Students connect the different definitions and techniques for derivatives and integrals by using and comparing them. Students study integrals as anti-derivatives, thus connecting differential and integral calculus. Students investigate applications to science and economics. Students apply techniques of algebra to calculus (in many calculations) and techniques of calculus to geometry (area and volume).*

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

Students solve problems using algebraic, graphical and numeric approaches choosing one approach to understand the question, another to solve the problem and perhaps, a third to demonstrate or check the result. This is done with problems such as evaluating limits, derivatives and definite integrals and finding extreme values. *

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

Students solve problems both in an isolated context, and problems with connections between math topics and applications outside of mathematics. Students communicate their results mathematically; making use of algebraic, graphical and numerical representations. Students use graphing calculators to study functions.*

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.

Students approximate definite integrals and limits numerically. Students use their numerical and computational skills in many calculations. Students use composition, decomposition and factorization of polynomials in many computations of integrals and derivatives. Students use proportional reasoning in some application problems.*

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 do beginning work in approximating definite integrals. Students compare results of finding extreme values using technology and using calculus techniques. Students compute relative errors for approximations using differentials.*

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.

Students use calculus to measure areas and volumes by rotation. Students use geometric properties including measurement formulas to solve related rates problems. Students use coordinate geometry to describe and analyze functions. *

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.

Not assessed in this course.

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.

Students use functions to represent situations that involve variable quantities with expressions, and equations and that include algebraic and geometric relationships in applications of extreme values techniques and related rates. Students represent functions algebraically, graphically, and numerically; and convert from one representation to another. Students perform calculations and algebraic manipulations on algebraically described polynomial, rational, algebraic and trigonometric functions. Students work with limits, derivatives and integrals of polynomial, rational, trigonometric and algebraic functions. Students demonstrate knowledge of the underlying concepts of calculus, including, rate of change, limits, and approximations for irregular areas.*

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.

Students demonstrate knowledge of the concepts and techniques of calculus, including limits (from an intuitive viewpoint), tangents, derivatives and integrals. Students use modeling to solve related rates and extreme values problems. Students use calculus techniques to find limits, derivatives, and integrals. Students apply calculus to the problems of optimization, velocity and acceleration, area and volume. Students use Newton's method of approximation and linearization.*

 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 in this course. 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 in this course.

*Students demonstrate their proficiency in these areas on in-class work, homework, quizzes or exam problems.