GRADES 9-12

**STANDARD 1. MATHEMATICS AS PROBLEM SOLVING**

- use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content

- apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics

- recognize and formulate problems from situations within and outside mathematics

- apply the process of mathematical modeling to real-world problem situations

**STANDARD 2. MATHEMATICS AS COMMUNICATION**

- reflect upon and clarify learner's thinking about mathematical ideas and relationships

- formulate mathematical definitions and express generalizations discovered through investigations

- express mathematical ideas orally and in writing

- read written presentations of mathematics with understanding

- ask clarifying and extending questions related to mathematics they have read or heard about

- appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical ideas

**STANDARD 3. MATHEMATICS AS REASONING**

- make and test conjectures

- formulate counterexamples

- follow logical arguments

- judge the validity of arguments

- construct simple valid arguments

- construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction

**STANDARD 4. MATHEMATICAL CONNECTIONS**

- recognize equivalent representations of the same concept

- relate procedures in one representation to procedures in an equivalent representation

- use and value the connections among mathematical topics

- use and value the connections between mathematics and other disciplines

**STANDARD 5. ALGEBRA**

- represent situations that involve variable quantities with expressions, equations, inequalities, and matrices

- use tables and graphs as tools to interpret expressions, equations, and inequalities

- operate on expressions and matrices, and solve equations and inequalities

- appreciate the power of mathematical abstraction and symbolism

- use matrices to solve linear systems

- demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations

**STANDARD 6. FUNCTIONS **

- model real-world phenomena with a variety of functions

- represent and analyze relationships using tables, verbal rules, equations, and graphs

- translate among tabular, symbolic, and graphical representations of functions

- recognize that a variety of problem situations can be modeled by the same type of function

- analyze the effects of parameter changes on the graphs of functions

- understand operations on, and the general properties and behavior of, classes of functions

**STANDARD 7. GEOMETRY FROM A SYNTHETIC PERSPECTIVE**

- interpret and draw three-dimensional objects

- represent problem situations with geometric models and apply properties of figures

- classify figures in terms of congruence and similarity and apply these relationships

- deduce properties of, and relationships between, figures from given assumptions

- develop an understanding of an axiomatic system through investigating and comparing various geometries

**STANDARD 8. GEOMETRY FROM AN ALGEBRAIC PERSPECTIVE**

- translate between synthetic and coordinate representations

- deduce properties of figures using transformations and using coordinates

- identify congruent and similar figures using transformations

- analyze properties of Euclidean transformations and relate translations to vectors

- deduce properties of figures using vector

- apply transformations, coordinates, and vectors in problem solving

**STANDARD 9. TRIGONOMETRY**

- apply trigonometry to problem situations involving triangles

- explore periodic real-world phenomena using the sine and cosine functions

- understand the connection between trigonometric and circular functions

- use circular functions to model periodic real-world phenomena

- apply general graphing techniques to trigonometric functions

- solve trigonometric equations and verify trigonometric identities

- understand the connections between trigonometric functions and polar coordinates, complex numbers, and series

**STANDARD 10. STATISTICS**

- construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations

- use curve fitting to predict from data

- understand and apply measures of central tendency, variability, and correlation

- understand sampling and recognize its role in statistical claims

- design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes

- analyze the effects of data transformations on measures of central tendency and variability

- transform data to aid in data interpretation and prediction

- test hypotheses using appropriate statistics

**STANDARD 11. PROBABILITY**

- use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty

- use simulations to estimate probabilities

- understand the concept of a random variable

- create and interpret discrete probability distributions

- 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

- apply the concept of a random variable to generate and interpret probability distributions, including binomial, uniform, normal, and chi square

**STANDARD 12. DISCRETE MATHEMATICS **

- represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations

- represent and analyze finite graphs using matrices

- develop and analyze algorithms

- solve enumeration and finite probability problems

- represent and solve problems using linear programming and difference equations

- investigate problem situations that arise in connection with computer validation and the application of algorithms

**STANDARD 13. CONCEPTUAL UNDERPINNINGS OF CALCULUS**

- determine maximum and minimum points of a graph and interpret the results in problem situations

- investigate limiting processes by examining infinite sequences and series and areas under curves

- 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

- analyze the graphs of polynomial, rational, radical, and transcendental functions

**STANDARD 14. MATHEMATICAL STRUCTURE **

- compare and contrast the real number system and its various subsystems with regard to learner's structural characteristics

- understand the logic of algebraic procedures

- appreciate that seemingly different mathematical systems may be essentially the same

- develop the complex number system and demonstrate facility with its operations

- prove elementary theorems within various mathematical structures, such as groups and fields

- develop an understanding of the nature and purpose of axiomatic systems