UNIVERSITY OF WISCONSIN River Falls

# Linear Algebra Appendix C

ll 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 use the axiomatic structure to prove results about matrices and vector spaces. Students use graphing calculators or a computer algebra system to calculate determinants and solutions to linear systems of equations. *

Facilitating the building of student conceptual and procedural understanding.

Students show understanding of linear algebra from algebraic and geometric perspectives. Students show procedural understanding by performing computations with linear systems and show conceptual understanding by using properties of those procedures to solve problems about linear systems.*

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 with the new matrix and vector space techniques they are learning, and generalize computational procedures to prove general results, gaining confidence in their ability to utilize mathematical knowledge. *

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

Students make and test conjectures in determining whether defined sets are vector spaces, and whether defined functions are linear transformations. *

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 generalize computations and computational procedures to prove generalized results about linear spaces and transformations.

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 write mathematical arguments and construct counterexamples on linear algebra topics including establishing identities, and identifying vector spaces and linear transformations. *

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

Students present their mathematical arguments visually and in writing, using mathematical language and symbolism, Students represent low dimension vector spaces both geometrically and algebraically, and 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 use not only the procedure of computing in linear systems, but also use the concepts of the properties of those computations to solve problems (eg. the row reduction properties are used in arguments about determinants and linear independence). Students demonstrate knowledge of connections between geometry and algebra in their work with low dimensional vector spaces. *

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

Students solve problems that involve translating between geometric and algebraic forms of vector spaces. Students use algebraic, geometric and generalized forms of representing vectors and linear transformations in making mathematical arguments. *

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

Students solve problems that require them to relate concrete examples to a generalized context and axiom structure. Students make formal arguments about linear algebra questions. Students communicate their results mathematically; making use of algebraic, geometric and generalized representations. Students use graphing calculators or a computer algebra system to calculate determinants and solutions to linear systems of equations. *

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 use their numerical and algebraic computation skills in solving linear algebra problems. Students study operations on matrices including inverses of matrices. *

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 perform operations on matrices and vectors, and make mathematical arguments about the results of operations; this contributes to students' higher level understanding of number and operations concepts through investigation of this similar system. This class does not directly address any of the listed sub-topics. *

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 linear algebra techniques to analyze lines, planes and spaces from a vector standpoint, including angle measure relationships and perpendicularity. Students study rotations, reflections and dilations in the context of linear transformations, and translate between the geometric and matrix form of these transformations. Students use vector-defined coordinate systems in two and three dimensions, and more generally in higher dimensions. Students relate some synthetic geometry concepts to the vector coordinate properties. Students use matrix and vector techniques to measure length, area and volume of vector-defined shapes (in particular, lines segments, parallelograms and parallelipipeds). *

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.

Students use coordinate geometry from a vector perspective. Students apply, analyze and describe linear transformations, vectors, coordinates and coordinate systems. Students solve problems and write mathematical arguments about transformations, coordinates and coordinate systems and vectors. Students use vectors and matrices to analyze and describe lines, planes and subspaces in two, three and higher dimensions. Students create and share mathematical arguments, proofs and counterexamples in the context of vector spaces and linear transformations. *

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 geometric properties (sub-spaces which are parallel or perpendicular or projections onto subspaces) and connect these to the corresponding vector calculations using expressions and equations. Students demonstrate knowledge of linear transformations and their attributes as functions on vector spaces. Students translate between standard and vector forms of equations for lines and planes. Students solve and use systems of linear equations using matrix methods. *

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. 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. Students demonstrate knowledge of matrices computationally and in vector space contexts.

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

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