use number-theory arguments to justify relationships involving whole numbers
understand vectors and matrices as systems that have some of the properties of the real-number 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
judge the reasonableness of numerical computations and their results
develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases
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 more-complicated 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
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
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 three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them
analyze properties and determine attributes of two- and three-dimensional objects
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
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
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