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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
Problem-Solving 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.     Problem-solving
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 two-stage experiment, based on combinations
and permutations (advanced counting methods).
Develop and use the multiplication rule for probability
to calculate probabilities of multi-stage 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 median-fit lines and least -squares regression
lines.
Compute and plot residuals from the least-squares
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, higher-order, 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 cross-categorized 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, perfect-square trinomials,
differences of squares , and quadratics of the form
ax2+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 integer-base 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.
    Equation-Solving 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 non-linear
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, point-slope,
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
real-world phenomena using tables, symbolic forms, or
graphical representations, and solve equations related to
these functions.
         
    Represent and interpret exponential functions based on
mathematical and real-world 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 real-world phenomena using tables,
symbolic forms, or graphical representations, and solve
equations related to these functions.
     
    Represent and interpret quadratic functions based on
mathematical and real-world phenomena using tables,
symbolic forms, or graphical representations, and solve
equations related to these functions.
Represent and interpret power functions (f(x)=axp, p ∈Q), including proportional and inverse-proportional
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 real-world 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 y-intercepts,
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 real-world 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 real-world
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 three-dimensional
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 problem-solving contexts in
mathematics and other disciplines.
Identify and apply conditions that are sufficient to
guarantee similarity of higher-order 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 origin-centered 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 cross-sectional shapes of slices of threedimensional
objects, and identify three-dimensional
objects traced out by rotations of two-dimensional
objects, with and without dynamic geometry software.
 
Discrete Mathematics              
Explore the properties of vertex-edge 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, origin-centered dilations, and
origin-centered 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, points-for-preference,
runoff, pairwise-comparrison, 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 vertex-edge 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
vertex-edge 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 vote-analysis 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 Problem-Solving 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 first-year assessment, in the model of statewide assessments at year 1 and year 3
First-year test: 42 cells

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