|
| |
MIFFLIN COUNTY SCHOOL DISTRICT
|
MCSD Advanced Placement Courses
|
AP Calculus (AB) Planned Instruction
|
|
Title of Planned Instruction: AP
Calculus (AB) |
| Subject Area:
Mathematics
Grade Level: |
| Prerequisites: Pre-calculus and
teacher recommendation Course Description: This course in
calculus covers and demands work that is comparable to calculus
courses in colleges and universities. Students who take this course
are expected to take the AP exam in the spring. The course covers
functions, graphs, limits, derivatives, integrals and applications
for all the content. Students are expected to produce a superior
effort on a daily basis.
Required Time: 180 days
Course Credit: 1.0
Major Text(s) and Resources: |
|
|
- Calculus: Concepts and Applications
, 2005, Key Curriculum
Press
|
| Names of District Subject Area
Curriculum Writing Committee: |
| |
- Sonya D. Curry Indian Valley High School
- Allen C. Muir Lewistown Area High School
|
Date of Board Approval: December
16, 2004
|
Major Topics
Derivatives Limits
IntegralsCourse Objectives and Performance Indicators |
| |
Strand: 2.2
Standard: Computation and Estimation
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.2.11A |
Develop and use
computation concepts, operations and procedures with real numbers in
problem-solving situations. |
Develop and use
computation concepts, operations and procedures with real numbers in
problem-solving situations. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.2.11B |
Use estimation to solve
problems for which an exact answer is not needed. |
Given a function
specified by a graph, by a table of values, or by an equation,
estimate how fast the y- value is changing. Given the equation or
the graph for a function, estimate on a graph the definite integral of
the function between x = a and x = b by counting squares.
Estimate the value of a definite integral by dividing the region
under the graph into trapezoids.
Use Riemann sums to find approximate values of definite integrals.
Use Simpson’s rule or your grapher’s built-in integrate feature to
approximate a given definite integral. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.2.11C |
Construct and apply
mathematical models, including lines and curves of best fit, to
estimate values of related quantities. |
Construct and apply
mathematical models, including lines and curves of best fit, to
estimate values of related quantities. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.2.11D |
Describe and explain
the amount of error that may exist in a computation using estimates. |
Describe and explain
the amount of error that may exist in a computation using estimates. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.2.11F |
Demonstrate skills for
using scientific and graphing calculators. |
Demonstrate skills for
using scientific and graphing calculators. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
-----
Strand: 2.3
Standard: Measurement and Estimation
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.3.11A |
Select and use
appropriate units required in particular measurement situations. |
Select and use
appropriate units required in particular measurement situations (i.e.
definite integrals, derivative). |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
Return to top
-----
Strand: 2.4
Standard: Mathematical Reasoning and Connections
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.4.11A |
Use direct proofs,
indirect proofs or proof by contradiction to validate conjectures. |
Derive algebraically
the formulas for the derivatives of sin x, cos x, tan x, and the
inverse trigonometric functions. Use direct proofs, indirect proofs
or proof by contradiction to validate conjectures.
Use properties of limits to justify the limit of a function. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.4.11B |
Construct valid
arguments from stated facts. |
Prove that a
(differentiable) function is continuous, and use this property to
prove that certain functions are continuous. Prove that ln has the
properties of logarithms. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.4.11C |
Determine the validity
of an argument. |
Determine the validity
of an argument. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.4.11E |
Demonstrate
mathematical solutions to problems. |
Explain either orally
or in written form how the solution to a problem was reached. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
-----
Strand: 2.5
Standard: Mathematical Problem Solving and Communication
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.5.11A |
Select and use
appropriate mathematical concepts and techniques from different areas
of mathematics and apply them to solving non-routine and multi-step
problems. |
Select and use
appropriate mathematical concepts and techniques from different areas
of mathematics and apply them to solving non-routine and multi-step
problems. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.5.11B |
Use symbols,
mathematical terminology, standard notation, mathematical rules,
graphing and other types of mathematical representations to
communicate observations, predictions, concepts, procedures,
generalizations, ideas and results. |
Use symbols,
mathematical terminology, standard notation, mathematical rules,
graphing and other types of mathematical representations to
communicate observations, predictions, concepts, procedures,
generalizations, ideas and results. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.5.11C |
Present mathematical
procedures and results clearly, systematically, succinctly and
correctly. |
Present mathematical
procedures and results clearly, systematically, succinctly and
correctly. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.5.11D |
Conclude a solution
process with a summary of results and evaluate the degree to which the
results obtained represent an acceptable response to the initial
problem and why the reasoning is valid. |
Conclude a solution
process with a summary of results and evaluate the degree to which the
results obtained represent an acceptable response to the initial
problem and why the reasoning is valid. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
-----
Strand: 2.8
Standard: Algebra and Functions
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.8.11R |
Create and interpret
functional models. |
Create and interpret
functional models. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.8.11S |
Analyze properties and
relationships of functions. |
Define continuity at a
point and use it to answer questions. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.8.11T |
Analyze and categorize
functions by their characteristics. |
Given an equation for a
continuous function f and a value of y between f(a) and f(b), find a
value of x = c between a and b for which f(c) = y. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
Return to top
-----
Strand: 2.11
Standard: Concepts of Calculus
Course: Calculus |
| |
Grade Level
Objectives |
Performance
Indicators |
Assessment Options |
| 2.11.11A |
Determine maximum and
minimum values of a function over a specified interval. |
Determine maximum and
minimum values of a function over a specified interval. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11.11B |
Interpret maximum and
minimum values in problem situations. |
Interpret maximum and
minimum values in problem situations. From information about the
first and second derivatives of a function, tell whether the y-value
is a local maximum or minimum at a critical point, tell whether the
graph has a point of inflection, and use this information to sketch
the graph or find the equation of the function.
Given a plane or solid figure, find the maximum or minimum
perimeter, area, or volume. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11.11E |
Estimate areas under
curves using sequences of areas. |
Estimate areas under
curves (parabolas) using sequences of areas of rectangles or
trapezoids. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. F |
Evaluate and
investigate limits and limit relationships or properties for given
functions. |
Given the graph or the
equation of a function, tell whether or not the function has a limit
as x approaches the given value and tell how your answer relates to
the definition of limit. For a given function (f(x)), make tables of
values that show how close x must be kept to a specified number, c, in
order for f(x) to be within given ranges of the limit.
Given values of the three numbers L, c, and ε, calculate the
corresponding value of δ in the definition of limit and show by the
graph that you understand the significance of these numbers.
State and use the limit properties and explain why they are true.
Find limits of functions where either x goes to infinity, the limit
is infinite, or both.
Given an expression with indeterminate form, find its limit using
l’Hospital’s rule. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. G |
Find and investigate
derivatives of functions. |
Given the equation of a
function, find the value of the derivative at a given point, and learn
the meaning of the derivative as it relates to the graph of the
function. Given the equation of a function and a value of x, use the
definition of derivative to calculate the value of the derivative at
that point, and confirm your answer numerically and graphically.
Given the equation for a function, graph the function and its
(numerical) derivative function on the same set of axes, and make
conjectures about the relationship between the derivative function and
the original function.
Given a power function, f(x) = xⁿ ,
where n stands for a constant, or given a linear combination of power
functions, find an equation expressing f ´(x) in terms of x.
Given an equation for displacement of a moving object, find an
equation for its velocity and an equation for its acceleration, and
use the equations to analyze the motion.
Given an equation for a composite function, write the equation for
its derivative function. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. G |
Find and investigate
derivatives of functions (continued). |
Given the equation for
the derivative of a function, find an equation for the function (that
is, find the antiderivative). Given a function that is a product of
two other functions, find an equation for the derivative function.
Given a function whose equation contains a quotient of two other
functions, find an equation for the derivative function and simplify
the answer.
Given a function whose equation contains any of the six
trigonometric functions, find the equation for the derivative
function.
Differentiate each of the six inverse trigonometric functions.
Given equations for x and y in terms of t, find dx/dt, dy/dt, and
dy/dx.
Given the equation for an implicit relation, find the derivative of
y with respect to x, and show by graph that the answer is reasonable.
Given the equation for the derivative of a function, write the
general equation for the antiderivative function. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11.G |
Find and investigate
derivatives (continued). |
Given a real-world
situation in which the rate of change of y with respect to x is
directly proportional to y, write and solve a differential equation
and use the resulting solution as a mathematical model to make
predictions and interpretations of that real-world situation. Given
the relationship between a function and its rate of change, write a
differential equation, solve it to find an equation for the function,
and use the function as a mathematical model.
Use slope fields to solve problems of population growth in an
environment where one population relies on another population for its
food supply. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. H |
Find and investigate
integrals. |
Given the equation for
a function f and a fixed point on its graph, find an equation for the
linear function that best fits the given function. Use the equation to
find approximate values for f(x) and values for the differentials dx
and dy. Become familiar with the symbol used for an indefinite
integral or an antiderivative by using this symbol to evaluate
indefinite integrals.
Calculate Riemann sums for given sets of sample points and reach
some conclusions about how the sample points were chosen.
Learn what the fundamental theorem of calculus says. Show that it
produces reasonable answers by comparing them with approximations
obtained by using the familiar techniques of Riemann summing and the
trapezoidal rule.
Be able to evaluate quickly a definite integral, in an acceptable
format, using the fundamental theorem of calculus.
Given a problem in which a quantity y varies with x, learn a
systematic way to write a definite integral for the product of y and
x, and evaluate the integral by using the fundamental theorem. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11.H |
Find and investigate
integrals (continued). |
Learn the Mean Value
Theorem and Rolle ’s Theorem and learn how to find the point in an
interval at which the instantaneous rate of change of the function
equals the average rate of change. Find the derivative of a function
that contains ln.
Differentiate algebraically a function whose equation has a
variable exponent.
Use the properties of ln to differentiate products, powers, and
quotients.
Find out algebraically what number is the base of the ln function.
Differentiate algebraically a logarithm function with any
permissible number as its base.
Given a function whose equation involves a variable power of e,
find equations for its derivative function. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. H |
Find and investigate
integrals (continued). |
Find the antiderivative
of the reciprocal function. Find the integral for the natural log
function.
Given a function whose equation involves a variable power of e,
find equations for its integral functions.
Given a plane region bounded by the graphs of one or more
relations, write a definite integral representing the area of the
region, and evaluate the integral numerically or algebraically (using
the fundamental theorem of calculus) to find the area.
Given a solid whose cross-sectional area varies along its length,
find its volume by slicing into slabs and performing the appropriate
calculus, and show that your answer is reasonable.
Find the surface of revolution by slicing the surface into frustums
of cones.
Given the equation of a polar function, find the area of a region
bounded by the graph and the length of the graph. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
| 2.11. I |
Find averages, extremes
and vectors using the calculus of motion. |
Given an equation for
the velocity of a moving object, find the distance traveled and the
displacement from the starting point for a specified time interval.
Given velocity or acceleration as a function of time for an object in
linear motion, find the displacement at a given time and distance
traveled in a given time interval.
Calculate the average value for a function given its equation.
Given a situation in which several quantities vary, predict the
rate at which one of them is changing when you know other related
rates.
Given a situation in which something goes from one place to another
through two different media at different rates, find the path that
minimizes a total time or cost.
Given a situation in the real or mathematical world where a
function is to be maximized or minimized, write an equation for the
function and find the maximum or minimum values. |
Observation Evaluate
written work
Performance assessments
Tests, quizzes
Problem solving journal/activity
Evaluate oral response |
Return to top
Suggested Paul A. Forster’s
Calculus Concepts and Applications section omissions:
- Chapter 2- No formal definition of limit is necessary
No need to teach finding epsilon and delta problems until after
the AP exam.
- Chapter 3- Use definitions to prove, but no need to memorize
them.
- Chapter 4- Omit section 4-7 until after the AP exam.
- Chapter 5- Omit section 5-11 (Simpson’s Rule) until after the
AP exam.
- Chapter 6- Omit sections 6-8 and 6-9 until after the AP exam.
- Chapter 7- Omit section 7-5 until after the AP exam.
- Chapter 8- Omit sections 8-6 and 8-7 until after the AP exam.
- Chapter 10- Omit section 10-7 until after the AP exam.
Topics to supplement: Volume slicing and related rate problems.
NOTE: The Topic Outline for Calculus AB from
http://apcentral.collegeboard.com is attached.
|
|
District Recommended Instructional Approaches
For the Course
To Drive Teacher’s Instructional Activities |
- Whole group instruction
- Small group instruction
- Projects
- Class discussion
- Peer evaluation
- Teacher and peer conferencing
- Oral presentations
- Individual instruction
- Research
- Dramatization
- Role playing
- Independent reading
- Read aloud
- Directed reading-thinking activities
- Modeling process
- Games
- Self-reflection
- Self-evaluation
- Independent study
- Guest speaker
- Guest reading
- Writing activities
- Thematic units
- Notebooks
- Study guides
- Computer technology
|
|
Return to top
|
