21ST CENTURY LEARNING CURRICULUM FRAMEWORK
PERFORMANCE RUBRICS FOR MATHEMATICS
CALCULUS
Table of Contents
Limits .................................................................................................................................... 2
Derivatives and the Tangent Line ........................................................................................... 3
Implicit Differentiation .......................................................................................................... 4
Curve Sketching ..................................................................................................................... 5
Optimization .......................................................................................................................... 6
Anti-derivatives and the Indefinite Integral ............................................................................ 7
Area Under the Curve ............................................................................................................ 8
Techniques of Integration ....................................................................................................... 9
Exponential and Logarithmic Differentiation .......................................................................... 10
Exponential and Logarithmic Integration................................................................................ 11
Inverse Trigonometric Functions ............................................................................................ 12
Volume of Three Dimensional Solids ..................................................................................... 13
1
Topic: Limits
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 limits, the student:
 uses limits to determine the continuity of a function
 evaluates limits (required: one-sided, infinite)
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 uses limits to determine the continuity of a continuous function
o evaluates limits of functions without removable discontinuities
 recognizes or recalls specific terminology such as:
o epsilon, delta
o direct substitution, factor, conjugate, common denominator, hole
o left-hand limit, right-hand limit, limits that do not exist, two special limits
of sine/cosine
o continuity
o vertical asymptote
o infinite limits, properties of limits
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: Derivatives and the Tangent Line
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 derivative and the tangent line,
the student:
 uses the limit definition of the derivative to find the slope of the tangent line
 applies the appropriate rules/properties to find the derivative of a function (required:
product, quotient and chain rules)
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 uses the limit definition of the derivative to find the slope of the tangent
line of linear and quadratic functions
o applies the appropriate rules/properties to find the derivative of a simple
function (required: product, quotient and chain rules
 recognizes or recalls specific terminology such as:
o tangent line, secant line
o difference quotient
o formal definition of derivative, alternative definition of derivative,
derivative from the right, derivative from the left
o differentiability implies continuity
o power rule, quotient rule, product rule, chain rule
o derivatives of trigonometric functions, the position function, the velocity
function, the acceleration function
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: Implicit Differentiation
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 implicit differentiation, the
student:
 uses implicit differentiation to find the derivative
 solves related rates problems
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 uses implicit differentiation to find the derivative of simple functions
o solves related rates problems given a diagram
 recognizes or recalls specific terminology such as:
o explicit functions and their derivatives
o implicit functions and their derivatives
o related rates
o volume formulas, surface area formulas, area formulas
o angle of depression/elevation
o rate of change with respect to time
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: Curve Sketching
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 curve sketching, the student:
 finds increasing and decreasing intervals using the first derivative
 finds points of inflection of a function using the second derivative and finds
concavity intervals of a function
 sketches a curve using the first and second derivative of the function
 finds the relative and absolute extreme of a function
 finds horizontal asymptotes of a function using infinite limits
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 finds increasing and decreasing intervals using the first derivative of
simple functions
o finds points of inflection of a simple function using the second derivative
and finds concavity intervals of a simple function
o sketches a curve using the first and second derivative of the simple
function
o finds the relative and absolute extreme of a simple function
o compares horizontal asymptotes of a function using infinite limits and
algebra
 recognizes or recalls specific terminology such as:
o absolute minimum, absolute maximum, relative min/max
o critical numbers
o first derivative test, increasing/decreasing intervals, second derivative
test, concavity intervals
o points of inflection, limits at infinity
o horizontal asymptotes
o domain restrictions of the original function relating to the derivative
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: Optimization
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 optimization, the student:
 solves optimization problems (required: business application)
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 solves simple optimization problems (required: business application)
 recognizes or recalls specific terminology such as:
o primary and secondary equations
o marginal/actual profit
o marginal/actual revenue
o marginal/actual cost
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: Anti-derivatives and the Indefinite Integral
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 anti-derivatives and the
indefinite integral, the student:
 finds an anti-derivative of a function
 solves differential equations (required: general solution, particular solution)
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 finds an anti-derivative of a simple function
o solves differential equations (required: general solution)
 recognizes or recalls specific terminology such as:
o constant of integration
o general solution
o particular solution
o indefinite integral
o properties of integration
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: Area Under a Curve
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 area under a curve, the student:
 approximates the area under a curve (required: upper and lower sums)
 finds the exact area under a curve (required: limits of Riemann sums)
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 approximates the area under a curve of functions bounded in quadrant 1
(required: upper and lower sums)
o finds the exact area under a curve of functions bounded in quadrant 1
(required: limits of Riemann sums)
 recognizes or recalls specific terminology such as:
o sigma notation
o summation formulas
o upper sum
o lower sum
o area under a curve
o Riemann sum
o limits of Riemann sums
o definite integrals
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: Techniques of Integration
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 techniques of integration, the
student:
 uses the first fundamental Thm of calculus to find the definite integral of a function
 uses U substitution to find the definite integral of a function
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 uses the first fundamental Thm of calculus to find the definite integral of
a simple function
o uses U substitution to find the indefinite integral of a simple function
 recognizes or recalls specific terminology such as:
o definite integral
o first fundamental Thm of calculus
o upper/lower bounds of integration
o “u” substitution
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: Exponential and Logarithmic Differentiation
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 and logarithmic
differentiation, the student:
 finds the derivative of exponential functions (required: bases other than e)
 finds the derivative of logarithmic functions (required: bases other than e)
 finds the derivative of a function using logarithmic differentiation
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 finds the derivative of simple exponential functions (required: bases
other than e)
o finds the derivative of simple logarithmic functions (required: bases
other than e)
o finds the derivative of a simple function using logarithmic differentiation
 recognizes or recalls specific terminology such as:
o logarithmic differentiation
o properties of logarithms
o properties of exponents
o change of base
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: Exponential and Logarithmic Integration
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 and logarithmic
integration, the student:
 finds the integral of exponential functions (required: bases other than e)
 finds the integral where the general solution is a logarithmic function (required:
bases other than e)
 solves differential equations involving exponents and logarithms
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 finds the integral of simple exponential functions (required: bases other
than e)
o finds the integral where the general solution is a simple logarithmic
function (required: bases other than e)
o solves simple differential equations involving exponents and logarithms
 recognizes or recalls specific terminology such as:
o logarithmic differentiation
o properties of logarithms
o properties of exponents
o change of base
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.
11
Topic: Inverse Trigonometric Functions
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 inverse trigonometric functions,
the student:
 finds the derivative of an inverse trigonometric function
 finds the integral that is an inverse trigonometric function
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 finds the derivative of a simple inverse trigonometric function
o finds the integral that is a simple inverse trigonometric function
 recognizes or recalls specific terminology such as:
o inverse
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: Volume of Three Dimensional Solids
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 volumes of three dimensional
solids, the student:
 finds the area between two curves
 finds the volume of a three dimensional solid using the disc/washer method
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 finds the area between two curves in quadrant 1
o finds the volume of a three dimensional solid using the disc/washer
method given one function in quadrant 1 bounded by the axes
 recognizes or recalls specific terminology such as:
o recognizes or recalls specific terminology such as:
o disc method
o washer method
o axis of revolution
o cross sections
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