21ST CENTURY LEARNING CURRICULUM FRAMEWORK
PERFORMANCE RUBRICS FOR MATHEMATICS
PRE-CALCULUS
Table of Contents
Functions ............................................................................................................................... 2
Polynomials and Rational Functions ....................................................................................... 3
Exponential Functions ............................................................................................................ 4
Logarithmic Functions ........................................................................................................... 5
Vectors................................................................................................................................... 6
Polar Coordinates ................................................................................................................... 7
Parametric Equations ............................................................................................................. 8
Conic Sections ....................................................................................................................... 9
Systems and Matrices ............................................................................................................. 10
Sequences and Series ............................................................................................................. 11
Statistics................................................................................................................................. 12
Probability ............................................................................................................................. 13
Three Dimensions .................................................................................................................. 14
Analytic Trigonometry ........................................................................................................... 15
1
Functions
Topic: Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving functions, the student:
 determines characteristics of a graph (required: domain, range, inc/dec intervals,
continuity, relative extrema, end behavior, asymptotes)
 decomposes a given function
 sketches graphs of functions using transformations (required: horizontal reflections,
stretches and shrinks)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o identifies basic examples of domain, range, inc/dec intervals, continuity,
relative extrema, end behavior, asymptotes
o identifies examples of a decomposed function
o sketches simple graphs using a basic transformation
 recognizes or recalls specific terminology such as:
o domain, range
o inc/dec intervals, continuity, relative extrema, end behavior, asymptotes
o composition, decomposition
o domain restrictions
o transformation
o relation, function, odd and even functions
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
2
Topic: Polynomials and Rational Functions
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving polynomials and rational
functions, the student:
 analyzes and graphs polynomial functions (required: intermediate value thm)
 analyzes and graphs rational functions (required: end behavior)
 solves a polynomial inequality (required: analytically and graphically)
 solves a rational inequality (required: analytically and graphically)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o domain, range
o relative extrema, end behavior, continuity, asymptotes
o limits at infinity
o increasing/decreasing intervals
o intermediate value theorem, rational root theorem
o synthetic division
o interval and set notation
o test intervals
o domain restrictions
o extraneous solutions
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
3
Topic: Exponential Functions
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving exponential functions, the
student:

solves exponential equations (required: base e, different bases)

analyzes and graphs exponential functions (required: horizontal reflections, stretches
and shrinks)

solves applications of exponential functions (required: average rate of change,
instantaneous rate of change)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o base of exponents
o logarithms
o change of base
o properties of exponents
o inverse operations
o definition of a logarithm
o asymptotes
o transformations
o natural base
o average rate change
o instantaneous rate of change
o extraneous solution
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
4
Topic: Logarithmic Functions
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving logarithmic functions, the
student:
 solves logarithmic equations (required: natural logarithms)
 analyzes and graphs logarithmic functions (required: horizontal reflections, stretches
and shrinks)
 solves applications of logarithmic functions
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o logarithms
o change of base
o properties of exponents
o inverse operations
o definition of a logarithm
o exponentiate
o asymptotes
o transformations
o natural base
o average rate change
o instantaneous rate of change
o extraneous solutions
o expanding and condensing logarithms
o domain restrictions
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
5
Topic: Vectors
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving vectors, the student:
 writes vectors in component form
 calculates the angle between two vectors
 projects one vector onto another vector
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic processes:
o add level 2
 recognizes or recalls specific terminology such as:
o linear combination form
o dot product
o orthogonal vectors
o projection
o unit vector
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
6
Topic: Polar Coordinates
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving polar coordinates, the student:
 converts between rectangular and polar coordinates
 analyzes and graphs polar equations
 writes the polar form of a complex number
 solves operations of complex numbers in polar form (required: multiply, divide,
powers, roots)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o rectangular coordinates
o polar coordinates
o complex numbers
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
7
Topic: Parametric Equations
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving parametric equations, the
student:
 eliminates the parameter to write a rectangular equation
 writes a rectangular equation in parametric form (required: conic sections)
 uses parametric equations to simulate the motion of a projectile
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o parameter
o rectangular form
o parametric form
o conic section
o projectile
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
8
Topic: Conic Sections
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving conic sections, the student:
 analyzes and graphs conic sections (required: rotating axes)
 converts between polar equation and rectangular equation of conic sections
 graphs polar equations of conic sections
 analyzes polar equations of conic sections (required: eccentricity, type)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic processes:
o add level 2
 recognizes or recalls specific terminology such as:
o focus
o directrix vertex
o axis
o asymptotes
o degree of rotation
o eccentricity
o standard form, general form
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
9
Topic: Systems and Matrices
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
In addition to Level 3.0 performance, in-depth inferences and applications with partial
Level 3.5
success.
Level 3.0
While engaged in grade appropriate tasks involving systems and matrices, the
student:
 solves operations on matrices
 solves a system (required: Gauss-Jordan elimination, Cramer’s Rule)
 solves a system using an inverse matrix
 decomposes partial fractions
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o matrix
o scalar multiple
o matrix multiplication
o row-reduction
o inverse
o determinant
o Cramer’s Rule
o decomposition of fractions
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
10
Topic: Sequences and Series
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving sequences and series, the
student:
 writes the formula for the nth term of a sequence given critical information (required:
arithmetic, geometric, finite differences)
 evaluates a finite arithmetic and geometric series
 evaluates an infinite geometric series
 writes a series in summation notation
 proves summation formulas using induction
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
Level 1.0
Level 0.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations such:
o add level 2
 recognizes or recalls specific terminology such as:
o sequence
o series
o summation notation
o sigma
o arithmetic
o geometric
o induction
o nth partial sum
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
11
Topic: Statistics
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving statistics, the student:
 calculates and analyzes measures of variation (required: range, and standard
deviation)
 determines and analyzes the intervals in which percentages of data lie (required:
empirical, Chebyshev)
 uses measures of position to compare different data (required: z-score, percentile
rankings, boxplot)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o frequency distribution (categorical, grouped, relative, cumulative,
cumulative relative)
o class, class boundaries, class midpoint, class width, class limit (lower,
upper)
o histogram, ogive, stem-and-leaf, Pareto, pie chart, scatterplot, time-series
graph
o bell-shaped distribution
o uniformity
o skew (left, right, symmetric)
o modality (unimodal, bimodal, multimodal, nonmodal)
o central tendency, mean, median, mode, weighted mean, modal class
o variation, range
o standard deviation,
z-score/standard score
o percentile, decile, quartile, Inter-Quartile range
o outliers,
o Empirical Rule, Chevyshev’s Theorem
o boxplot
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
12
Topic: Probability
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving probability, the student:
 calculates and analyzes the probability of an event (required: addition/multiplication
rule, permutations/combinations)
 uses the binomial theorem to expand a binomial expression
 calculates and analyzes mean and standard deviation for discrete probability
distributions (required: binomial)
 calculates and analyzes probabilities in continuous probability distributions
(required: standard normal, non-standard normal, and Central Limit Thm)
 solves for data values given probabilities in continuous probability distributions and
analyze results
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic processes:
o add level 2
 recognizes or recalls specific terminology such as:
o events (mutually exclusive, independent, dependent, complementary,
equally likely, simple, compound)
o conditional probability
o Law of Large Numbers
o sample space
o classical probability, empirical probability
o factorial
o permutation
o Fundamental Counting Rule
o tree diagram, Venn diagram
o expected value
o binomial experiment, binomial distribution
o discrete probability distribution
o random variable
o normal distribution, standard normal distribution, sampling distribution
of sample means
o z-value
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
13
Topic: Three Dimensions
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving probability, the student:
 finds the equation of a sphere and critical information
 graphs a sphere
 finds the magnitude and direction of a vector
 determines if two vectors are parallel, orthogonal, or neither
 uses cross product to find an orthogonal vector given two vectors
 writes a set of parametric and symmetric equations given a point and parallel to a
given vector
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations:
o add level 2
 recognizes or recalls specific terminology such as:
o sphere
o vector
o magnitude
o direction
o orthogonal
o cross product
o dot product
o parallel
o parametric
o symmetric
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
14
Topic: Analytic Trigonometry
Pre-Calculus
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond what was taught.
Level 3.5
In addition to Level 3.0 performance, in-depth inferences and applications with partial
success.
Level 3.0
While engaged in grade appropriate tasks involving analytic trigonometry, the
student:
 simplifies trigonometric expressions (required: half angle, double angle, sum and
difference, Pythagorean identities, reciprocal identities, quotient identities,
cofunction identities, even and odd identities)
 evaluates trigonometric expressions (required: half angle, double angle, sum and
difference, reciprocal identities, quotient identities, cofunction identities, even and
odd identities)
 verifies trigonometric identities (required: half angle, double angle, sum and
difference, Pythagorean identities, reciprocal identities, quotient identities,
cofunction identities, even and odd identities, 2 column proof format, common
denominators, factoring, the conjugate, multiplying/dividing by an expression)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and processes
as the student:
 performs basic operations such as:
o simplifies trigonometric expressions (required: sum and difference,
Pythagorean identities, reciprocal identities, quotient identities,
cofunction identities, even and odd identities)
o evaluates trigonometric expressions (required: sum and difference,
reciprocal identities, quotient identities, cofunction identities, even and
odd identities
o verifies trigonometric identities (required: Pythagorean identities,
reciprocal identities, quotient identities, cofunction identities, even and
odd identities, 2 column proof format, common denominators, factoring)
 recognizes or recalls specific terminology such as:
o 2-column proof format
o conjugate
o factoring
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but not the
more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
15