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North Carolina Proposed Essential Standards for Mathematics
North Carolina Proposed Essential Standards for Mathematics
Level 1  Level 2  Level 3  Level 4  Level 5  Level 6  Level 7  Level 8  Trajectories / Strands 
ProblemSolving and Mathematical Modeling  
Create, apply and adapt a variety of strategies to solve problems, and create and use representations to organize,record, and express mathematical ideas.  Select, apply, and coordinate among
mathematical representations, indicating what different representations
show and hide, in solving problems and modeling situations. 
Make and investigate mathematical conjectures, generalize and extend mathematical patterns, and develop and evaluate mathematical arguments and proofs.  Problemsolving heuristics 

Use appropriate technology and tools (e.g. computers and graphing calculators ) to measure, model, and simulate phenomena (e.g. probeware, games, spreadsheets), retrieve and analyze data (databases and statistical software) and other dynamic software (for probability, geometry, and functions), and judge the reasonableness of solutions.  Select appropriate technology and
tools (e.g. computers and graphing calculators) to measure, model, and
simulate phenomena (e.g. probeware, games, spreadsheets), retrieve and analyze data (databases and statistical software) and other dynamic software (for probability, geometry, and functions), and judge the reasonableness of solutions. 
Technology and Tools  
Draw connections among mathematical strands.  Draw connections among mathematical strands.  Draw connections among mathematical strands.  Connections  
Given a simple mathematical model for an
indeterminate situation, evaluate what the model explains, how it is constrained, and refine it to improve its usefulness, accuracy, and/or precision. 
Model and solve problems in context using
iterative and recursive forms. 
Model contextual situations using similarity and congruence. 
Model contextual situations with matrices and
graph theory. 
Model contextual situations with power functions.  Modeling  
Model contextual situations with linear functions .  Model contextual situations with systems of
linear functions. 
Model contextual situations with quadratic functions .  Model contextual situations with exponential and logarithmic functions. 
Model contextual situations with polynomial functions.  
Probability and Data Analysis  
Identify research questions that can be addressed
by survey method, design and carry out data collection and sampling methods, and analyze results. 
Identify research questions that can be addressed
by observational method, design and carry out data collection and sampling methods, and analyze results. 
Identify research questions that can be addressed
by experimental method, design and carry out data collection and sampling methods, and analyze results. 
Statistical Investigations 

Associate the sizes of partitioned regions within
regular geometric figures with probabilities. 
Apply and interpret simulations to estimate
probabilities of events for which theoretical values are difficult or impossible to compute, and predict that simulation results vary from one run of the simulation to another and that the results tend to converge as the number of trials increases. 
Apply and interpret various probability
distributions, including discrete vs. continuous, theoretical vs. empirical, and normal distributions. 
Summarize data from simulations using appropriate graphical and numerical summaries, develop an estimate for the probability of an event for which theoretical values are difficult or impossible to compute, and explain the effect of the number of trials on the estimated probability of the event ( Law of Large Numbers). 
Probability  
Name and determine the number of outcomes in
sample spaces for twostage experiment, based on combinations and permutations (advanced counting methods). 
Develop and use the multiplication rule for
probability to calculate probabilities of multistage probability experiments involving independent events. 
Distinguish between independent and dependent compound events and compute their probabilities using representations for such events and using the multiplication rule for probability. 
Understand how to calculate and interpret the
expected value of a random variable having a discrete probability distribution. 

Analyze distributions and detemine the effect of
an outlier on the mean, median, mode, and range of a set of data, including various graphical displays. 
Compare shape, center, and spread of univariate
data using graphical displays, quartiles, percentiles, outliers, means, and standards deviations. 
Univariate and Probability Distributions 

Model trends in bivariate data displayed in
scatterplots, using informal strategies for placement of lines of best fit, and evaluate goodness of fit to linear models. 
Determine, interpret, and compare linear models, including medianfit lines and least squares regression lines. 
Compute and plot residuals from the leastsquares regression line to evaluate the fit of a linear model and determine whether its use is appropriate. 
Model trends in bivariate data displayed in
scatterplots, using informal strategies to evaluate goodness of fit to exponential models. 
Model trends in bivariate data displayed in
scatterplots, using informal strategies to evaluate goodness of fit to quadratic and other models (logistic, higherorder, and periodic) models. 

Collect and organize bivariate data and display
them using scatterplots and examine type (positive, negative) and extent (strong, weak, or none) of association. 
Evaluate association of bivariate numerical data
in tables and scatterplots and use the correlation coefficient to measure linear association. 
Bivariate Distributions 

Applications of Numbers  
Apply ratio and rates to solve problems involving derived measures, selecting appropriate units and conducting unit analysis. 
Use matrices to represent crosscategorized data
as multidimensional numbers that both quantify and organize. 
Applications of Numbers 

Apply significant figures , orders of magnitude,
scientific notation, precision, and estimation in using and comparing extreme numbers, including in context. 

Define, graph, and compute with complex numbers .  
Find integer powers of rational numbers, apply
the basic laws of exponents, and apply their meanings to variables. 
Understand and operate with square roots and cube roots, including in context. 
Translate between writing numbers with rational
powers and expressing them using roots. 
Understand, operate with, and solve problems in
context with rational and irrational solutions. 

Algebraic Reasoning  and Functions  
Apply Venn diagrams to illustrate relations among numbers, geometric shapes, and other entities and display sets, subsets, unions, and intersections. 
Complete simple logical truth tables for
conjunction, disjunction, negation, and conditional relations. 
Identifies hypotheses and conclusions, formulates logical statements, and investigates the validity of conditionals, converses, inverses, contrapositives, biconditionals. 
Logic and Argument  
Apply properties of exponents (integral and
rational), to simplify algebraic expressions. 
Add, subtract, multiply and divide (monomial
only) in order to combine algebraic expressions including the application of associativity, communtativity and distributivity. 
Factor simple quadratic expressions (limited to
the removal of monomial terms, perfectsquare trinomials, differences of squares , and quadratics of the form ax^{2}+bx+c that factor over the integers), and multiply the resulting binomials to check result. 
Perform operations on simple (monomial in the denominator) rational expressions (add, subtract, multiply, divide). 
Add, subtract, multiply, simplify and evaluate
rational expressions with linear and quadratic denominators 
Simplifying Algebraic Expressions 

Define and evaluate absolute value and greatest
integer expressions symbolically and in context. 
Simplify algebraic expressions containing roots
or rational powers. 
Convert between exponential and logarithmic
notation for positive integerbase logarithms. 
Apply properties of logarithms (integer and
rational), to simplify algebraic expressions and prove theorems. 

Use literal equations to represent situations
using two or more variables in relation to direct (proportional) and indirect (inverse) variation. 
Solve literal equations including direct
(proportional) and indirect (inverse) variation, by substitution of values for variables. 
Simplify and solve literal equations for any
variable and explain in terms of unit analysis. 
Describe the qualitative effects on literal
equations of changing the values of one variable or constant on the values of the others (increase, decrease). 
Solve literal equations involving joint and
combined variation, by substitution of values for variables, and by solving for any variable and explaining in terms of unit analysis. 
Interpret the meaning of literal equations
involving combinations of families of functions. 
EquationSolving and Inequalities 

Solve linear equations using graphs, tables, and symbols, and in context. 
Solve linear equations with absolute value using
graphs, tables, and symbols, and in context 

Solve pairs of linear equations, two variables , graphically and in context. 
Solve pairs of linear equations, two variables, symbolically and in context. 

Solve pairs of linear inequalities, two
variables, graphically and symbolically, and in context. 
Construct, solve, and interpret solutions for
systems of linear inequalities, including in context. 
Construct, solve, and interpret solutions for
systems of combinations of inequalities (linear, absolute value, and quadratric), including in context. 

Solve quadratic equations using completing
the square, and interpret the (rational) solutions, including in context. 
Solve quadratic equations with real coefficients ,
using the quadratic formula, with and without technology, over the set of complex numbers (real and complex roots). 

Solve simple (monomial in the denominator)
rational equations in one variable. 
Simplify and solve rational equations with linear
and quadratic denominators. 

Solve exponential and logarithmic equations using
tables and graphs. 
Apply properties to solve exponential and
logarithmic equations symbolically. 

Represent sequences (iterative and recursive) and
extend patterns, including analysis on spread sheets, including arithmetic, geometric, Fibonacci, irrational numbers. 
Use appropriate terminology (including domain,
range, and intercepts ) and notation associated with functions, and distinguish functions from relations, including the application of the vertical line test. 
Distinguish and apply the different uses of and
notation for variables, parameters, and constants in equations and functions. 
Families of Functions  
Find solutions for particular values of a
function and interpret their relationship to graphs, tables, and equations, and in context. 
Interpret, using multiple representations, the
sum, difference, product, quotient (monomial divisors), and composition of two given functions, and evaluate for given values of the variable. 
Find the inverse of a function using tables,
graphs, and equations, and demonstrate that the composition of a function and its inverse returns the identity function (f(f 

Distinguish between direct ( proportional
relationships ) and indirect (inverse) variation, in context. 
Determine whether relationship is linear or
nonlinear based on whether it has a constant rate of change, using multiple representations. 
Analyze rates of change of functions (increasing, decreasing, oscillating), using tables and graphs. 

Write, interpret, and translate among equivalent
forms of linear functions, including slope intercept, pointslope, and general form, and describe how equivalent forms for a linear relationship reveal less or more information about a given situation. 
Represent and interpret linear, absolute value,
step, and piecewise linear functions based on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, and solve equations related to these functions. 

Represent and interpret exponential functions
based on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, and solve equations related to these functions. 
Represent and interpret logarithmic functions,
including their relationship to exponential functions, based on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, and solve equations related to these functions. 

Represent and interpret quadratic functions based
on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, and solve equations related to these functions. 
Represent and interpret power functions
(f(x)=ax^{p}, p ∈Q), including proportional and inverseproportional functions, root functions based on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, and solve equations related to 
Represent and interpret polynomial functions,
based on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations. 
Perform operations on polynomial functions (add, subtract, multiply, factor polynomials), divide (with monomial divisors only) polynomials. 

Analyze and describe symbolic forms and graphs of polynomial functions by examining their yintercepts, roots (graphically and by substitution), domains and ranges, and local (turning point) and global (end) behavior, distinguishing between odd and even functions. 
Represent and interpret rational functions, based
on mathematical and realworld phenomena using tables, symbolic forms, or graphical representations, including asymptotic behavior and restrictions on domain and range. 
Distinguish among linear, exponential, power, polynomial, rational, and periodic expressions, equations, and functions in graphical and symbolic form. 

Model and graph problem situations involving
repeated motions as a function of time (walking back and forth), varying parameters of starting point, distance, rate, and number of repetitions. 
Represent periodic functions (sine and cosine)
using the unit circle and angles from special triangles. 
Represent and interpret periodic functions (sine
and cosine), based on mathematical and realworld phenomena, by varying parameters (amplitude, phase shift, and period). 

Distinguish horizontal and vertical translation,
dilation, and reflections for step, absolute value, quadratic, and exponential functions 
Distinguish and apply horizontal and vertical
translation, dilation, and reflections for power functions. 
Distinguish and apply horizontal and vertical
translation, dilation, and reflections for logarithmic functions. 
Apply transformations to power and polynomial functions and develop curve sketching capabilities, identifying roots, asymptotes, and extrema. 
Apply transformations to rational and radical
functions and develop curve sketching capabilities, identifying roots, asymptotes, discontinuities, and extrema. 
Apply transformations to periodic functions (sine
and cosine) and link graphically to amplitude, frequency, period, and phase shift. 
Transformations of Functions and Relations 

Identify coordinates on the plane for simple
geometric figures, and calculate slope, distance between points, their midpoint, and the distance from a point to a line 
Determine equations of circles and parabolas,
given particular configurations of points. 
Apply 'completing the square' to equations for
circles in order to identify center and foci. 
Analytic Geometry and Conic Sections 

Geometry  
Identify points, lines, and planes as undefined
terms, and use these terms to define other geometric terms as line segments, angles, and rays. 
Describe the structure and relationships within
an axiomatic system (undefined terms, defined terms, axioms/postulates, methods of reasoning, and theorems). 
Describe and apply inductive and deductive
reasoning to form and then verify or reject (create counterexample or identify inconsistency) conjectures 
Prove directly or indirectly that a valid
mathematical statement is true, by developing short sequences of geometric theorems within a Euclidean axiomatic system 
Properties and Relationships in the Plane 

Analyze basic geometric shapes, identify and
relate their properties, and form logical arguments about necessary and sufficient conditions for defining shapes. 
Perform and investigate geometric constructions,
using compass and straightedge, dynamic geometry software, and others. 
Justify statements about angles formed by
perpendicular lines and transversals of parallel lines. 
Identify and apply conditions that are sufficient
to guarantee congruence of triangles, noting that congruence is a special case of simliarity (SSS, SAS, ASA, AAS, HL). 
Prove theorems involving right triangles, related
to angle bisectors, medians, isosceles triangles, perpendicular bisectors, altitudes, and geometric mean. 
Prove Pythagorean theorem, and its converse in
multiple ways, and apply them to two and threedimensional settings 

Identify and apply conditions that are sufficient
to guarantee similarity of triangles (AA, SAS, SSS). 
Develop and apply properties of special right
triangles, and triangle inequality. 
Use and prove properties of special quadrilaterals.  
Use similarity to calculate the measures of corresponding parts of similar figures, and apply similarity to a variety of problemsolving contexts in mathematics and other disciplines. 
Identify and apply conditions that are sufficient
to guarantee similarity of higherorder polygons. 
Identify and describe relationships among:
central angles, inscribed and circumscribed angles; right triangles in semicircles; radius of circles perpendicular to chords 

Connect rigid transformations (translations,
reflections, rotations) and origincentered dilations with the relations of congruence and similarity. 
Represent geometric transformations algebraically
with matrices. 

Identify and define radius, diameter, chord,
tangent, secant, and circumference. 
Apply formulas and solve problems involving the
areas of circles and triangles. 
Derive and apply area formulas for quadrilaterals
and regular polygons. 
Determine the arc lengths and the areas of
sectors of circles, using proportions. 
Determine the areas of regular polygons and the
sums of the interior and exterior angles. 
Measurement of and Relationships among 2D and 3D objects 

Derive and apply formulas involving the areas of quadrilaterals, using decomposition into rectangles and triangles, including in context. 
Derive and apply area formulas for regular
polygons, including in context. 
Define and apply the trigonometric ratios (sine,
cosine, tangent) to determine side lengths and angle measures in right triangles. 
Apply, individually and in combination, the
Pythagorean theorem, properties of proportionality, trigonometric ratios, and similarity, in solving mathematical and realworld problems. 

Analyze intersections among two and three planes
in space. 
Apply formulas and solve problems involving
volume of right prisms, right circular cylinders, and right pyramids. 
Link surface area of prisms, cylinders, and
pyramids to the sum of the area(s) of their base(s) and lateral surfaces using planar nets to illustrate and sum the relevant measures. 
Apply formulas and solve problems involving
volume and surface area of cones, spheres, and composite figures. 
Identify and apply the 3:2:1 relationship among
volumes of circular cylinders, hemispheres, and cones with same height and circular base and 3:1 relationship between volume of a prism and pyramid with same base area and height. 
Identify crosssectional shapes of slices of
threedimensional objects, and identify threedimensional objects traced out by rotations of twodimensional objects, with and without dynamic geometry software. 

Discrete Mathematics  
Explore the properties of vertexedge graphs and understand the role they play in optimization and avoiding conflict. 
Represent numerical and relational data
characterized with two or more variables using matrices and solves problems involving addition and subtraction of matrices 
Verify the properties of matrix multiplication,
and multiplies matrices solve problems. 
Describe figures on a coordinate plane using
matrix notation, and use matrix operations to model translations, reflections, origincentered dilations, and origincentered rotations (30, 45, 60, 90, …) 
Construct systems of linear equations that model
realworld situations, represent the systems as a matrix equations, and solve with and without technology. 
Explore mathematical strategies for making fair decisions (majority, plurality, pointsforpreference, runoff, pairwisecomparrison, approval, and apportionment). 
Apply mathematical concepts and strategies
related to information processing, particularly on the internet, focusing on access, security, accuracy, and efficiency. 
Discrete Mathematics 

Use vertexedge graphs and algorithmic thinking
to model and solve problems involving paths, networks, and relationships with finite elements. 
Use mathematical models to represent and solve problems finding efficient routes, Euler Circuits, vertex coloring, and avoiding conflict. 
Use minimum spanning trees and Hamilton circuits
to find optimum networks that span all the vertices in vertexedge graphs. 
Use critical path analysis to optimally schedule
large projects that are comprised of many smaller tasks. 
Explore properties of fair division and
fairly divide continuous objects. 
Analyze fair decision strategies in terms
of fairness and Arrow's Thereom. 
Apply voteanalysis strategies to
critically analyze elections in every day life and those reported by the media. 
Notes:
Levels in the chart below indicate increasing
levels of complexity or sophistication of reasoning along a learning trajectory,
NOT grade level in school.
Students should be expected to analzye the extent to which the solution to a
problem is meaningful within the context of that problem.
At the high school level, the verb "model" will be restricted to refer to the
activity of mathematical modeling.
Counts of standards (not counting ProblemSolving and
Mathematical Modeling):
21  21  25  21  25  14  6  5  138 
Important note: in a cell indicates a standard that is proposed for inclusion
in a firstyear assessment, in the model of statewide assessments at year 1 and
year 3
Firstyear test: 42 cells
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