NCTM Curriculum Standards for Grades 9-12

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;

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

• 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;

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.

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;

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

• 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;

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

• 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;

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

• 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;

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.

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;

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

• 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;

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.

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;

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.

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;

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.

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;

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.