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MIFFLIN COUNTY SCHOOL DISTRICT
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MCSD Mathematics Curriculum
Calculus (Honors)
Planned Instruction
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Title of Planned Instruction:
Calculus (Honors) |
| Subject Area: Mathematics Grade Level:
Grades 12 |
| Prerequisites: Precalculus or
Trigonometry/Algebra III Course Description: The focus of
this course is limits, derivatives, and integrals. Daily
experience with higher order thinking skills will provide an
opportunity to make connections between previously learned skills
and concepts culminating in a rewarding senior course.
Required Time: 180 days
Major Text(s) and Resources: |
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Calculus: Concepts and Applications,
Copyright 1998 by Key Curriculum Press |
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Names of District Subject Area Curriculum Writing Committee: |
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- Sonya D. Curry -- Indian Valley High School
- R. Joseph Lauver -- Lewistown High School
- Allen C. Muir -- Lewistown High School
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Date of Board Approval: July 24, 2003 |
| Major Topics |
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- Derivatives
- Limits
- Integrals
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| Course Objectives and Performance Indicators |
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Strand: 2.2
Standard: Computation and Estimation
Course: Calculus |
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Course
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
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| 2.2.11B |
Use estimation to
solve problems for which an exact answer is not needed. |
Estimate the
instantaneous rate of change of the dependent variable with respect
to the independent variable at a given point when given an equation
for a function. 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
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| 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
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| 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
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| 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
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Strand: 2.3
Standard: Measurement and Estimation
Course: Calculus
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Course
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. |
- Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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Strand: 2.4
Standard: Mathematical Reasoning and Connections
Course: Calculus |
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Course 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 inversed trigonometric
functions. Use direct proofs, indirect proofs or proof by contradiction to
validate conjectures. |
- Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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| 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
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| 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
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| 2.4.11E |
Demonstrate mathematical solutions
to problems. |
Demonstrate mathematical solutions
to problems. |
- Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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Strand: 2.5
Standard: Mathematical Problem Solving and Communication
Course: Calculus
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Course 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
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| 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
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| 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
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| 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
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Strand: 2.8
Standard: Algebra and Functions
Course: Calculus
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Course 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
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| 2.8.11S |
Analyze properties and relationships of
functions. |
Define continuity and use it to answer
questions. |
- Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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| 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
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Strand: 2.11
Standard: Concepts of Calculus
Course: Calculus |
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Course 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
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| 2.11.11B |
Interpret maximum and minimum
values in problem situations. |
Interpret maximum and minimum
values in problem situations. |
- Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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| 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
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| 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 f(c).
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, use, and explain why the limit properties 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
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| 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) = x
, 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
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| 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, in one step, 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 in one step and
simplify the answer.
Given a function whose equation contains any of the six trigonometric
functions, find the equation for the derivative function in one step.
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
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| 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
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| 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
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| 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. |
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Observation
- Evaluate written work
- Performance assessments
- Tests, quizzes
- Problem solving journal/activity
- Evaluate oral response
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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
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