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.

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.

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;

and so that, in addition, college-intending students can--

construct proofs for mathematical assertions, including indirect proofs and

proofs by mathematical induction.

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.

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;

and so that, in addition, college-intending students can--

use matrices to solve linear systems;

demonstrate technical facility with algebraic transformations, including

techniques based on the theory of equations.

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;

and so that, in addition, college-intending students can--

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

of functions.

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;

and so that, in addition, college-intending students can--

develop an understanding of an axiomatic system through investigating and

comparing various geometries.

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;

and so that, in addition, college-intending students can-

deduce properties of figures using vectors;

apply transformations, coordinates, and vectors in problem solving.

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;

and so that, in addition, college-intending students can--

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.

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;

and so that, in addition, college-intending students can--

transform data to aid in data interpretation and prediction;

test hypotheses using appropriate statistics.

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;

and so that, in addition, college-intending students can--

apply the concept of a random variable to generate and interpret probability

distributions including binomial, uniform, normal, and chi square.

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;

and so that, in addition, college-intending students can--

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.

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;

and so that, in addition, college-intending students can--

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.

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;

and so that, in addition, college-intending students can--

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.

STANDARD 1. MATHEMATICS AS PROBLEM SOLVING

STANDARD 2. MATHEMATICS AS COMMUNICATION

STANDARD 3. MATHEMATICS AS REASONING

STANDARD 4. MATHEMATICAL CONNECTIONS

STANDARD 5. ALGEBRA

STANDARD 6. FUNCTIONS

STANDARD 7. GEOMETRY FROM A SYNTHETIC PERSPECTIVE

STANDARD 8. GEOMETRY FROM AN ALGEBRAIC PERSPECTIVE

STANDARD 9. TRIGONOMETRY

STANDARD 10. STATISTICS

STANDARD 11. PROBABILITY

STANDARD 12. DISCRETE MATHEMATICS

STANDARD 13. CONCEPTUAL UNDERPINNINGS OF CALCULUS

STANDARD 14. MATHEMATICAL STRUCTURE