National Council of Teachers of Mathematics Standards

Number and Operations Standard

1a. Understand numbers, ways of representing numbers, relationships among numbers, and number systems
 use numbertheory arguments to justify relationships involving whole numbers
 understand vectors and matrices as systems that have some of the properties of the realnumber system
 compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions
 develop a deeper understanding of very large and very small numbers and of various representations of them

1b. Understand meanings of operations and how they relate to one another
 develop an understanding of permutations and combinations as counting techniques
 develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices
 judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities

1c. Compute fluently and make reasonable estimates
 judge the reasonableness of numerical computations and their results
 develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paperandpencil calculations for simple cases and technology for morecomplicated cases

Algebra Standard

2a. Understand patterns, relations, and functions
 interpret representations of functions of two variables
 understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions
 understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on morecomplicated symbolic expressions
 analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior
 understand relations and functions and select, convert flexibly among, and use various representations for them
 generalize patterns using explicitly defined and recursively defined functions

2b. Represent and analyze mathematical situations and structures using algebraic symbols
 judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology
 use a variety of symbolic representations, including recursive and parametric equations, for functions and relations
 use symbolic algebra to represent and explain mathematical relationships
 write equivalent forms of equations, inequalities, and systems of equations and solve them with fluencyâmentally or with paper and pencil in simple cases and using technology in all cases
 understand the meaning of equivalent forms of expressions, equations, inequalities, and relations

2c. Use mathematical models to represent and understand quantitative relationships
 draw reasonable conclusions about a situation being modeled
 use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts
 identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships

2d. Analyze change in various contexts
 approximate and interpret rates of change from graphical and numerical data

Geometry Standard

3a. Analyze characteristics and properties of two and threedimensional geometric shapes and develop mathematical arguments about geometric relationships
 use trigonometric relationships to determine lengths and angle measures
 establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others
 explore relationships (including congruence and similarity) among classes of two and threedimensional geometric objects, make and test conjectures about them, and solve problems involving them
 analyze properties and determine attributes of two and threedimensional objects

3b. Specify locations and describe spatial relationships using coordinate geometry and other representational systems
 investigate conjectures and solve problems involving two and threedimensional objects represented with Cartesian coordinates
 use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations

3c. Apply transformations and use symmetry to analyze mathematical situations
 use various representations to help understand the effects of simple transformations and their compositions
 understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices

3d. Use visualization, spatial reasoning, and geometric modeling to solve problems
 use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture
 use geometric models to gain insights into, and answer questions in, other areas of mathematics
 use vertexedge graphs to model and solve problems
 visualize threedimensional objects and spaces from different perspectives and analyze their cross sections
 draw and construct representations of two and threedimensional geometric objects using a variety of tools

Measurement Standard

4a. Understand measurable attributes of objects and the units, systems, and processes of measurement
 make decisions about units and scales that are appropriate for problem situations involving measurement

4b. Apply appropriate techniques, tools, and formulas to determine measurements.
 use unit analysis to check measurement computations
 apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations
 understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders
 analyze precision, accuracy, and approximate error in measurement situations

Data Analysis and Probability Standard

5a. Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
 compute basic statistics and understand the distinction between a statistic and a parameter
 understand histograms, parallel box plots, and scatterplots and use them to display data
 understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable
 know the characteristics of welldesigned studies, including the role of randomization in surveys and experiments
 understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each

5b. Select and use appropriate statistical methods to analyze data
 identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled
 recognize how linear transformations of univariate data affect shape, center, and spread
 display and discuss bivariate data where at least one variable is categorical
 for bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools
 for univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics

5c. Develop and evaluate inferences and predictions that are based on data
 understand how basic statistical techniques are used to monitor process characteristics in the workplace
 evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions
 understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference
 use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions

5d. Understand and apply basic concepts of probability
 understand how to compute the probability of a compound event
 understand the concepts of conditional probability and independent events
 compute and interpret the expected value of random variables in simple cases
 use simulations to construct empirical probability distributions
 understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases

Process Standard

6a. Problem Solving
 build new mathematical knowledge through problem solving
 solve problems that arise in mathematics and in other contexts
 apply and adapt a variety of appropriate strategies to solve problems
 monitor and reflect on the process of mathematical problem solving

6b. Reasoning and Proof
 recognize reasoning and proof as fundamental aspects of mathematics
 make and investigate mathematical conjectures
 develop and evaluate mathematical arguments and proofs
 select and use various types of reasoning and methods of proof

6c. Communication
 organize and consolidate their mathematical thinking through communication
 communicate their mathematical thinking coherently and clearly to peers, teachers, and others
 analyze and evaluate the mathematical thinking and strategies of others
 use the language of mathematics to express mathematical ideas precisely

6d. Connections
 recognize and use connections among mathematical ideas
 understand how mathematical ideas interconnect and build on one another to produce a coherent whole
 recognize and apply mathematics in contexts outside of mathematics

6e. Representation
 create and use representations to organize, record, and communicate mathematical ideas
 select, apply, and translate among mathematical representations to solve problems
 use representations to model and interpret physical, social, and mathematical phenomena