Students whose number and operation work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of number, operations, and number sense. They demonstrate a deep understanding of the concepts, properties, and operational skills of both the real and complex number systems. These students represent real and complex numbers using coordinate and matrix forms. They apply mental mathematics skills and number facts and relationships in simplifying and evaluating numerical computations, as well as in making reasonable estimates and approximations involving multi-step real number computations. They easily compare and order real numbers in any form, including radicals and powers-integral or rational.
Students whose number and operation work exceeds the Standards determine appropriate use of roots, exponents, and logarithms in representing and computing with real numbers in symbolic and applied settings. These students extend non-routine numeric patterns, including arithmetic and geometric sequences, and produce applicable expressions and formulas to model the sequences, sums of terms, and related patterns.
Students whose number and operation work exceeds the Standards are fluent in their ability to deal with all forms of percentage problems, including exponential growth and decay in both business and scientific applications. They appropriately use graphing calculators and technology to investigate mathematical ideas. These students demonstrate highly developed problem-solving skills and the ability to use mathematical models to model and identify solutions for non-routine problems.
Students whose measurement work exceeds the Standards construct and identify solutions for proportions in a wide variety of non-routine settings. They demonstrate a comprehensive, flexible, and widely applicable command of measurement. They know, apply, and modify formulas in a wide variety of theoretical and applied measurement applications involving perimeter, area, volume, angle, time, temperature, mass, speed, distance, density, and money. These students choose appropriate units and scales, including nonlinear ones, for problem situations involving scale drawings. They use units of measure (dimensional analysis) to set up problems and determine the appropriate unit for the answer. They convert measures within and between the standard and metric systems of measurement.
Students whose measurement work exceeds the Standards determine numerical answers having appropriate degrees of accuracy. They determine the area and perimeter of regular and irregular two-dimensional figures. They find the volume and surface area of regular and irregular three-dimensional figures.
Students whose measurement work exceeds the Standards use ratio and proportion, including trigonometric ratios, to describe the measures of geometric figures. They determine the effect of a change in one measure (for example, side length) on other measures (such as area, volume, angle measure) in the same or related figures in two and three dimensions.
Students whose algebraic work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of algebra. They use appropriate numerical, graphical, and algebraic representations to illustrate their work. They recognize and represent patterns with variables and develop and use expressions to find solutions for non-routine problems. They manipulate a wide range of equations, inequalities, and systems, both linear and nonlinear, in solving problems represented in algebraic form. They recognize, manipulate, simplify, and evaluate algebraic expressions involving both polynomial and rational forms.
Students whose algebraic work exceeds the Standards distinguish between relations and functions and perform appropriate operations on functions, including finding inverses and composition. They productively use tables, graphs, and algebraic expressions to represent functions and their related equations. They interpret the relative rates of change involved in linear, quadratic, and exponential settings.
Students whose algebraic work exceeds the Standards recognize, model, and apply direct (y = kx) and inverse (y = k/x) variation in representing and solving real-world problems. They model such real-world problems using logarithmic, exponential (growth and decay), and trigonometric functions and matrices.
Students whose geometric work exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of geometry. They know and apply the properties and theorems that characterize segments, angles, and lines in polygonal or circular figures in two and three dimensions. These students understand and apply theorems that describe the measures of congruent or similar figures in both two and three dimensions. They construct formal proofs and logical arguments for geometric statements.
Students whose geometric work exceeds the Standards use trigonometric relationships to determine measures in both right and non-right triangles. They apply coordinate geometry in non-routine problems to find distances, prove properties of geometric figures, and describe congruence or similarity in two- or three-dimensional settings. They use transformations to describe and investigate figures and relationships between them.
Probability, Statistics, and Data Analysis
Students whose analysis of data and chance settings exceeds the Standards demonstrate a comprehensive, flexible, and widely applicable command of probability, statistics, and data analysis. They correctly determine the probability or odds of events using counting principles, combinations, and permutations.
Students whose analysis of data and chance settings exceeds the Standards collect, organize, analyze, describe, and make predictions based on raw data. They formulate well-designed questions and describe appropriate data collection methods, gather and analyze data effectively, and communicate their findings concisely and clearly. They understand the role of randomization in surveys and models. They calculate and interpret appropriate measures of central tendency (mean, median, and mode) and variation (range, variance, and standard deviation). They find, graph, and interpret a line of best fit for a given set of data and analyze the relationship between the predicted and observed data. They distinguish between correlation (events that are unrelated) and causation (events that are related).