From the website of the National Council of Teachers of Mathematics (http://www.nctm.org).

**Standard 1: Mathematics as Problem Solving.**-
In grades 9-12, the mathematics curriculum should include the
refinement and extension of methods of mathematical problem solving so
that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
continued development of language and symbolism to communicate
mathematical ideas so that all students can--
- reflect upon and clarify their 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.**-
In grades 9-12, the mathematics curriculum should include numerous and
varied experiences that reinforce and extend logical
reasoning skills so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include
investigation of the connections and interplay among various
mathematical
topics and their applications so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the continued
study of algebraic concepts and methods so that all students
can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
continued study of functions so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
continued study of the geometry of two and three dimensions so that
all
students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
study of the geometry of two and three dimensions from an algebraic
point of view so that all students can--
- 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 vectors;
- apply transformations, coordinates, and vectors in problem solving.

**Standard 9: Trigonometry.**-
In grades 9-12, the mathematics curriculum should include the study of
trigonometry so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
continued study of data analysis and statistics so that all students
can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the
continued study of probability so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include topics from
discrete mathematics so that all students can--
- 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.**-
In grades 9-12, the mathematics curriculum should include the informal
exploration of calculus concepts from both a graphical and a
numerical perspective so that all students can--
- 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 their applications in other disciplines;
- analyze the graphs of polynomial, rational, radical, and transcendental functions.

**Standard 14: Mathematical Structure.**-
In grades 9-12, the mathematics curriculum should include the study of
mathematical structure so that all students can--
- compare and contrast the real number system and its various subsystems with regard to their 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.

Wed Sep 16 09:09:16 EDT 1998