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Unit Name: Rates: It's All Relative
Author: Larry Rodgers
SET-UP
Subject:
Mathematics
Course/Grade: Calculus, AP
School:
Country:
State/Group: AR
UNIT SUMMARY
Students will study an application of the derivative and will hopefully realize that calculus may be used to
solve problems involving real-world situations. Required skills: modeling equations from data, solving
abstract equations, implicit differentiation. The goal is to solve equations involving related rates. Related
rate problems present a situation where one or more related quantities are changing. Calculus is used to
determine both the nature and the value of the change at general and specific instances of time.
UNIT RESOURCES
Printed Materials:
Calculus, 2nd Edition. Gerald L. Bradley & Karl J. Smith. Prentice Hall, 1999.
Calculus with Trigonometry and Analytic Geometry. John Saxon, Frank Wang, with Diana Harvey. Saxon
Publishers, Inc.,1988.
UALR A.P.S.I. 1998, Hot Springs AR. (Notebook containing practice problems, and retired Free Response
questions from the AP Exam since 1995.)
Course Outline Guide with sample multiple Choice and Free Response Questions from AP Testing
Company.
Multiple-Choice and Free Response Questions in Preparation for the AP Calculus (AB) Examination
(Seventh Edition). David Lederman. D&S Marketing Systems, Inc. 1998.
Resources:
Testbuilder software from WK Bradford Company.
Calculus AB Testbank for use with Testbuilder.
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Internet Resource Links:
http://apcentral.collegeboard.com
http://www.seresc.k12.nh.us/www/alvirne.html
http://
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STAGE ONE
GOALS AND STANDARDS
Click above to add content for Understandings
Click above to add content for GOALs
UNDERSTANDINGS
Mathematical equations and graphs can model real-world situations.
Changing one parameter of a situation can lead to a change in other parameters.
Differential calculus is used to describe and solve problems involving rate of change.
ESSENTIAL QUESTIONS
How is the derivative related to the rate of change?
What assumptions must be made before a mathematical model can be used?
How do we determine that the derivative is required to solve a particular problem?
How can implicit differentiation simplify the process of taking a derivative especially when the related
variables are both dependent on time?
KNOWLEDGE AND SKILLS
Make a mathematical model (eg. an equation)Use a model to identify the general and specific situation
and express them as variables or values.Differentiate implicitly and solve for a variable.Solve related rate
equations by substitution.Knowledge of basic differentiation techniques involving polynomial, rational,
trigonometric, exponential, and logarithmic functions.Apply product, quotient, and Chain rules to
differentiate equations.
STAGE TWO
PERFORMANCE TASKS
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This is a performance task designed so that students may demonstrate the knowledge, understanding
and skills garnered from geometry and calculus, especially in the related rates unit.
OTHER EVIDENCE
Process check
Model a set of data with a formula (either known from chemistry, physics, etc) or create an equation
unique to the data. Identify any assumptions required to set up the problem. Use implicit differentiation
to find the related rate equation.
Selected Response/Short-answer test/quiz
Multiple choice questions "Like" the AP Calculus test questions on related rates.
Process check
Write a related rate problem that does not involve volume.
Student self-assessment
Split into two groups, share problems created, and select one to give to the other group.
Student self-assessment
Groups work on traded problems, grade them, and explain solutions.
Selected Response/Short-answer test/quiz
Diagnostics: check on derivatives, solving abstract equations, implicit differentiation.
Product check
Student has completed homework problems and corrected any incorrect work by class discussion, group
work, or teacher answer book.
STAGE THREE
LEARNING ACTIVITIES
Introduce the idea of related rates by discussing two sample problems in the book. A method of
approach, explained by the instructor, of the way to decipher the word problem, to set up the relating
equations, to identify the general and specific values, to differentiate implicitly, and to solve for the
unknown rate, will be emphasized in the lecture and demonstration. Students will watch the first, and
then will participate in the step-by-step procedure of the second one.
Teach students related rates problems using geometry and trigonmetry, such as the "falling ladder"
problem, the "pebble in the pond" problem, the "moving shadow" problem, the "shrinking ice cube"
problem.
Homework assignments will treat individually the various aspects of the related rate problem, such as (1)
writing the relating equation, (2) differentiating implicitly, and (3) evaluating equations.
Teach applications of related rates in other occupations, with emphasis not only on the process, but on
recognizing when and where the calculus process should be applied.
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Homework involving geometry and trigonometry related rates problems.
Homework on practical applications of related rate problems.
Work together several related rate problems gleaned from old AP exams. Students will then try a couple
on their own to be graded and critiqued by the instructor.
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