Standard 1. Mathematics as Problem Solving

1. use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content
2. apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics
3. recognize and formulate problems from situations within and outside mathematics
4. apply the process of mathematical modeling to real-world problem situations

Standard 2. Mathematics as Communication

1. reflect upon and clarify learner's thinking about mathematical ideas and relationships
2. formulate mathematical definitions and express generalizations discovered through investigations
3. express mathematical ideas orally and in writing
4. read written presentations of mathematics with understanding
5. ask clarifying and extending questions related to mathematics they have read or heard about
6. appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas

Standard 3. Mathematics as Reasoning

1. make and test conjectures
2. formulate counterexamples
3. follow logical arguments
4. judge the validity of arguments
5. construct simple valid arguments
6. construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction

Standard 4. Mathematical Connections

1. recognize equivalent representations of the same concept
2. relate procedures in one representation to procedures in an equivalent representation
3. use and value the connections among mathematical topics
4. use and value the connections between mathematics and other disciplines

Standard 5. Algebra

1. represent situations that involve variable quantities with expressions, equations, inequalities, and matrices
2. use tables and graphs as tools to interpret expressions, equations, and inequalities
3. operate on expressions and matrices, and solve equations and inequalities
4. appreciate the power of mathematical abstraction and symbolism
5. use matrices to solve linear systems
6. demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations

Standard 6. Functions

1. model real-world phenomena with a variety of functions
2. represent and analyze relationships using tables, verbal rules, equations, and graphs
3. translate among tabular, symbolic, and graphical representations of functions
4. recognize that a variety of problem situations can be modeled by the same type of function
5. analyze the effects of parameter changes on the graphs of functions
6. understand operations on, and the general properties and behavior of, classes of functions

Standard 7. Geometry from a Synthetic Perspective

1. interpret and draw three-dimensional objects
2. represent problem situations with geometric models and apply properties of figures
3. classify figures in terms of congruence and similarity and apply these relationships
4. deduce properties of, and relationships between, figures from given assumptions
5. develop an understanding of an axiomatic system through investigating and comparing various geometries

Standard 8. Geometry from an Algebraic Perspective

1. translate between synthetic and coordinate representations
2. deduce properties of figures using transformations and using coordinates
3. identify congruent and similar figures using transformations
4. analyze properties of Euclidean transformations and relate translations to vectors
5. deduce properties of figures using vector
6. apply transformations, coordinates, and vectors in problem solving

Standard 9. Trigonometry

1. apply trigonometry to problem situations involving triangles
2. explore periodic real-world phenomena using the sine and cosine functions
3. understand the connection between trigonometric and circular functions
4. use circular functions to model periodic real-world phenomena
5. apply general graphing techniques to trigonometric functions
6. solve trigonometric equations and verify trigonometric identities
7. understand the connections between trigonometric functions and polar coordinates, complex numbers, and series

Standard 10. Statistics

1. construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations
2. use curve fitting to predict from data
3. understand and apply measures of central tendency, variability, and correlation
4. understand sampling and recognize its role in statistical claims
5. design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes
6. analyze the effects of data transformations on measures of central tendency and variability
7. transform data to aid in data interpretation and prediction
8. test hypotheses using appropriate statistics

Standard 11. Probability

1. use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty
2. use simulations to estimate probabilities
3. understand the concept of a random variable
4. create and interpret discrete probability distributions
5. describe, in general terms, the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed
6. apply the concept of a random variable to generate and interpret probability distributions, including binomial, uniform, normal, and chi square

Standard 12. Discrete Mathematics

1. represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations
2. represent and analyze finite graphs using matrices
3. develop and analyze algorithms
4. solve enumeration and finite probability problems
5. represent and solve problems using linear programming and difference equations
6. investigate problem situations that arise in connection with computer validation and the application of algorithms

Standard 13. Conceptual Underpinnings of Calculus

1. determine maximum and minimum points of a graph and interpret the results in problem situations
2. investigate limiting processes by examining infinite sequences and series and areas under curves
3. understand the conceptual foundations of limit, the area under a curve, the rate of change, and the slope of a tangent line, and learner's applications in other disciplines
4. analyze the graphs of polynomial, rational, radical, and transcendental functions

Standard 14. Mathematical Structure

1. compare and contrast the real number system and its various subsystems with regard to learner's structural characteristics
2. understand the logic of algebraic procedures
3. appreciate that seemingly different mathematical systems may be essentially the same
4. develop the complex number system and demonstrate facility with its operations
5. prove elementary theorems within various mathematical structures, such as groups and fields
6. develop an understanding of the nature and purpose of axiomatic systems