# Mathematics (MTH) CPSO

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Course numbers with the # symbol included (e.g. #400) have not been taught in the last 3 years.

**MTH 801 - Probability and Statistics **

**Credits:** 4

In this course students study topics in data analysis including descriptive and inferential statistics, probability, odds and fair games, probability distributions, normal distributions, and estimation. Among the topics are numerical and graphical summaries for one and two variables, linear regression and correlation, confidence intervals and tests concerning means, sampling and experimentation, basic probability, confidence intervals, hypothesis testing, sampling distributions, two-sample t-tests for means, chi-squared tests, regress and correlation, and possible other topics. A standards statistical software package is used throughout the course to support the course format that includes: hands-on activities; computer-based simulations; creating and implementing student developed investigations; and actual secondary and middle school mathematics classroom activities. Throughout the course students are given opportunities to relate the mathematical concepts studied in this course to the mathematical concepts they will be teaching. Successful completion of PreCalculus required.

**Equivalent(s):** MATH 703G

**Mutual Exclusion:** No credit for students who have taken MATH 823.

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Design investigations, collect data, display data in a variety of ways, and interpret data representations including bivariate data, conditional probability and geometric probability.
- Use appropriate methods to estimate population characteristics, test conjectured relationships among variables, and analyze data.
- Use appropriate statistical methods and technology to analyze data and describe shape, spread, and center.
- [Pivotal Standard] Use both descriptive and inferential statistics to analyze data, make predictions, test hypotheses, and make decisions.
- Draw conclusions involving uncertainty by using hands-on and computer-based simulations.

**MTH 802 - Mathematical Proof for Educators **

**Credits:** 4

This course introduces students to the language and methods used to create and write mathematical proofs and solve problems. Methods of proof will include: direct, contrapositive, contradiction, and induction. Methods of problem solving will be based on Polya's four steps for solving problems. Students will learn about and utilize the many functions of proof including: verification, explanation, communication, discovery, justification, and inquiry. The course will also explore the relationship between problem solving and the process of proving. Students will explore fundamental abstract concepts in mathematics chosen from the following areas: functions and relations, set theory, number theory, and logic, Euclidian and non-Euclidian geometry, algebra, mathematical reasoning, proof, and problem solving. Connections to middle and secondary school mathematics curriculum emphasized. Students enrolled in this course at the 700 level will meet additional academic requirements including an applied project. Pre-calculus required prior to taking this course.

**Equivalent(s):** MATH 700G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- Use problem solving to investigate and understand increasingly complex mathematical content, including, but not limited to the ability to use problem solving to develop one's own mathematical knowledge, reflect upon solutions and the problem solving process, as well as refine strategies as needed. Standard~612.18 NH (2.a.2,3,4); Standard ~ 612.17 NH (2.a.2)
- Use mathematical proof, including, but not limited to, the ability to develop and evaluate mathematical conjectures, to select and use various types of reasoning and methods of proof, and to demonstrate the capacity to articulate an understanding of how reasoning and proof are integral components of mathematics. (Standard ~ 612.18 NH (2.b.1,2,3,4) ; Standard ~ 612.17 NH (2.b.1,2,3,4)

**MTH 803 - Number Systems **

**Credits:** 4

This course examines the structure and properties of mathematics while focusing on the development of mental mathematics strategies and problem solving skills. Topics include sets, functions, applications of rational numbers, integers, fractions, decimals, percentages, and number theory. Appropriate grade level techniques are utilized to investigate algorithms, probability and statistics, counting techniques, scientific notation, complex numbers, exponents, geometry, and measurement. Students will also investigate ratios, proportion, data analysis, patterns, and the connections to algebra and geometry topics in the context of the 5-12 grades mathematics curriculum. Successful completion of PreCalculus required prior to taking this course.

**Equivalent(s):** MATH 701G

**Mutual Exclusion:** No credit for students who have taken MATH 821.

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- Demonstrate a capacity to use models to explore certain relationships, including magnitude, among fractions, decimals, percents, rations, and proportions
- Demonstrate knowledge of the historical development of number and number systems
- Apply, explain, and justify concepts in number and number theory
- Demonstrate computational proficiency and fluency, including the use of a variety of algorithms, estimation strategies, and mental mathematics techniques to judge the reasonableness of answers or approximate solutions
- Demonstrate knowledge of concepts and applications of limits and infinity
- Demonstrate a capacity to apply the concepts of proportional reasoning
- Demonstrate a capacity to make sense of large and small numbers and use scientific notation in mathematical and scientific modeling (Standard ~ 612.18 NH (4.a-g) ; Standard ~ 612.17 NH (4.a-g)

**MTH 804 - Geometric Structures for Teachers **

**Credits:** 4

This course will examine concepts in Euclidean and non-Euclidean geometries. Course topics include area and volume, two- and three-dimensional perspective, congruence and similarity, properties of and relationships among geometric shapes and structures. Students will investigate graphing, vectors, motion and symmetry. Students engage in course concepts through proofs, problem solving, dynamic geometric software, and through activities used in secondary and middle school mathematics. Throughout the course students will be given opportunities to relate the mathematical concepts studied to the mathematical concepts they will be teaching. Successful completion of PreCalculus required prior to taking this course.

**Equivalent(s):** MATH 702G

**Mutual Exclusion:** No credit for students who have taken MATH 822.

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Solve problems involving Euclidean and non-Euclidean geometry and systems of measurement.
- [Pivotal Standard] Use constructions, models, and dynamic geometric software to explore geometric relationships.
- [Pivotal Standard] Derive and explain formulas found in Euclidean geometry.
- Construct proofs using the axioms of Euclidean and non-Euclidean geometries.
- Analyze and make connections between geometry concepts and the 5 - 12 mathematics curriculum.

**MTH 805 - Calculus I **

**Credits:** 4

The first semester of a calculus sequence dealing with applications and modeling of the differential and integral calculus. Course will focus on functions and their graphs, limits, continuity, differentiation, integration, the derivative and its uses in optimization and mathematical modeling, as well as the Fundamental Theorem. Throughout the course students will be given opportunities to relate the mathematical concepts studied to the mathematical concepts they will be teaching. Graphing calculators are used throughout the course to explore and represent concepts. Students enrolled in this course at the 700 level will meet additional academic requirements including an applied project. Pre-calculus required prior to taking this course.

**Equivalent(s):** MATH 706G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Demonstrate a conceptual understanding of and procedural facility with basic calculus concepts including limits, continuity, differentiation, and integration.
- [Pivotal Standard] Use mathematical modeling and the concepts of calculus to represent and solve problems from real-world contexts.
- [Pivotal Standard] Use technology to explore and represent fundamental concepts of calculus.
- Demonstrate knowledge of the historical development of calculus.
- Describe the connection of calculus to middle and high school mathematics topics.

**MTH 806 - History of Mathematics **

**Credits:** 4

This course addresses the historical development of major themes in mathematics, including calculation, numbers, geometry, algebra, infinity, and formalism in various civilizations ranging from the antiquity of Babylonia and Egypt through classical Greece, the Middle and Far East, and on to modern Europe. The course emphasizes how earlier civilizations influenced or failed to influence later ones and how the concepts evolved in these various civilizations.

**Prerequisite(s):** MTH 805 with a minimum grade of B- or MATH 706G with a minimum grade of B-.

**Equivalent(s):** MATH 708G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- Develop and strengthen their conceptual knowledge of arithmetic, algebra, geometry, trigonometry and calculus through the study of why and how these concepts developed.
- Analyze how the development of mathematical concepts in different cultures influenced the development of those cultures and our present culture.
- Learners will explore the influence of the development of mathematical concepts on other disciplines.
- Follow the development of mathematics from early number systems to the invention of calculus.
- Research historical questions and applications and present conclusions to others.

**MTH 807 - Calculus II **

**Credits:** 4

This course is the second semester of a calculus sequence dealing with applications of differential and multivariable calculus. Topics include the calculus of transcendental functions, applications of integration, some differential equations, sequences and series, differentiation and integration of trigonometric functions multidimensional calculus with applications, and an introduction to multivariable calculus. Throughout the course students are given opportunities to relate the mathematical concepts studies to the mathematical concepts they will be teaching. Graphing calculators are used throughout the course to explore and represent concepts.

**Prerequisite(s):** MTH 805 with a minimum grade of B- or MATH 706G with a minimum grade of B-.

**Equivalent(s):** MATH 707G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Demonstrate procedural facility with basic calculus concepts including integration techniques, sequences and series, parametric and polar curves, and vectors, and vector-valued functions.
- [Pivotal Standard] Use mathematical modeling and the concepts of calculus to represent and solve problems from real-world contexts.
- [Pivotal Standard] Use technology to explore and represent fundamental concepts of calculus.
- Demonstrate knowledge of the historical development of calculus.
- Describe the connection of calculus to middle and high school mathematics topics.
- Demonstrate an understanding of basic concepts of multivariable calculus.

**MTH 808 - Discrete Mathematics **

**Credits:** 4

This course is designed to introduce students to discrete and abstract mathematical topics. Topics include propositional and predicate logic; elementary set theory; introduction to proof techniques including mathematical induction; sets, relations, functions, and relations; recurrence relations, graph theory, as well as the properties of groups, rings, and fields. Students study number systems, mathematical induction, algorithms and complex number systems, matrix manipulation, combinatorics, graph theory, and finite differences. Course activities are based on secondary and middle school mathematics curricula. This course considers the basic objects of mathematics through real-world examples and the methods used to elucidate their properties.

**Prerequisite(s):** MTH 805 with a minimum grade of B- or MATH 706G with a minimum grade of B-.

**Equivalent(s):** MATH 705G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Apply the fundamental ideas of discrete mathematics in a formulation and solution of problems arising from real-world situations.
- Use technology to solve problems involving the use of discrete structures.
- [Pivotal Standard] Demonstrate knowledge of:
- Historical development of discrete mathematics
- Basic elements of discrete mathematics, including but not limited to: -Graph theory -Propositional logic -Mathematical induction -Recurrence relations -Finite differences -Linear programming -Combinatorics

**MTH 809 - Topics in Linear and Abstract Algebra **

**Credits:** 4

This course will examine concepts in algebra including: Patterns and functions, arithmetic sequences, geometric sequences, arithmetic and algebra of the integers, least common multiple and greatest common divisor, inequalities, modular arithmetic and systems of numbers, properties of groups and fields, the field of complex numbers, polynomial arithmetic and algebra, linear equations. Course will develop the mathematical structures, algebraic properties, and applications of matrices, determinants, vectors, vector spaces, systems of linear equations, and linear transformations. Students will engage with these concepts through exploration, analysis, proof, and problem solving based on activities used in secondary and middle school mathematics. Throughout the course students will be given opportunities to relate the mathematical concepts studied to the mathematical concepts they will be teaching. Students enrolled in this course at the 700 level will meet additional academic requirements including an applied project.

**Prerequisite(s):** (MTH 802 with a minimum grade of B- or MATH 700G with a minimum grade of B- and (MTH 807 with a minimum grade of B- or MATH 707G with a minimum grade of B-.

**Equivalent(s):** MATH 704G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- [Pivotal Standard] Demonstrate proficiency in linear and abstract algebra concepts, including vector spaces, matrices, groups, rings, and fields.
- [Pivotal Standard] Connect major concepts of linear and abstract algebra to the complex number systems and other mathematical structures.
- [Pivotal Standard] Identify and apply arithmetic and geometric sequences with connections to linear and exponential functions.
- Use mathematical models to understand, represent, and communicate quantitative relationships, including, but not limited to equality, equations, inequalities, and proportional relationships.
- Using graphing utilities and other technological tools to represent, explain, and explore algebraic ideas including functions, equations, and expressions.
- Generalize patterns and functions using recursive and explicit representations.

**MTH 810 - Algebra Theory for Teachers **

**Credits:** 4

This course will examine concepts in Algebra including patterns, functions, arithmetic sequences, geometric sequences, arithmetic and algebra of the integers, least common multiple and greatest common division, inequalities, basic properties of groups and fields, and polynomial arithmetic and algebra. Throughout the course students will be given opportunities to relate the mathematical concepts studied to the mathematical concepts they will be teaching.

**Prerequisite(s):** (MTH 802 with a minimum grade of B- or MATH 700G with a minimum grade of B- and (MTH 805 with a minimum grade of B- or MATH 706G with a minimum grade of B-.

**Equivalent(s):** MATH 709G

**Grade Mode:** Letter Grading

### View Course Learning Outcomes

- In the subject area of number and operations, the candidate shall have the ability to: -Demonstrate a capacity to use physical materials and models to explore and explain the operations and properties of real and complex numbers with extensions to matrices and vectors. -Represent, use, and apply introductory concepts and properties of complex numbers; -Identify and illustrate the mathematics that underlies the procedures used for operations involving real numbers and their subsets; -Explain the distinctions among real numbers and their subsets with connection to field axioms -Demonstrate a capacity to apply the concepts of exponents, including integer and rational, through modeling and applications
- In the subject area of geometry and measurement, the candidate shall have the ability to: -Connect the ideas of algebra and geometry through the use of coordinate geometry, graphing, vectors, and motion geometry
- In the subject area of functions and algebra, the candidate shall have the ability to: -Model and analyze change and rates of change in various contexts -Use mathematical models to understand, represent, and communicate quantitative relationships, including, but not limited to equality, equations, inequalities, and proportional relationships -Explore, analyze, and generalize a wide variety of patterns and functions using multiple representations including tables, graphs, written word, and symbolic rules; -Represent information and solve problems using matrices -Use graphing utilities and other technological tools to represent, explain, and explore algebraic ideas including functions, equations, and expressions -Demonstrate knowledge of the historical development of algebra -Generalize patterns and functions using recursive and explicit representations -Understand, identify, and apply arithmetic and geometric sequences -Articulate the meaning of functions and their inverse relationships, both formally and informally, with the use of concrete materials and graphing utilities -Understand and compare the properties of classes of functions and their inverses, including exponential, polynomial, rational, step, absolute value, root, logarithmic, and periodic, including trigonometric Represent and analyze group and field properties of real numbers and other mathematical structures