Student: William Scharen
Professor: R. Moroney
Course: EDU 521-04
Date: June 21, 2010
Grade: 12 Topic: Derivative of a Polynomial (Power Rule)
Content Area: AP Calculus
INSTRUCTIONAL OBJECTIVE
After practicing examples of how the derivative of a polynomial function can be simplified using
the Power Rule, students will be able to find the derivative to any simple polynomial function
with 95% accuracy.
STANDARDS AND INDICATORS
Indicators (per AP Calculus Collegeboard.com Course Goals)
Standards *NY State Standards in Algebra 2-Trigonometry relevant towards this lesson
A2.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric,
graphic, algebraic)
Indicators:
Work with functions represented in a variety of ways: graphical,
numerical, analytical, or verbal. They should understand the connections among
these representations.
Use technology to help solve problems, experiment, interpret results, and
verify conclusions.
A2.CN.1 Understand and make connections among multiple representations of the same
mathematical idea
Indicators:
Work with functions represented in a variety of ways: graphical,
numerical, analytical, or verbal. They should understand the connections among
these representations.
Use technology to help solve problems, experiment, interpret results, and
verify conclusions.
NYS Math, Science and Technology Learning Standard 3: Mathematics
Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in real-world
settings, and by solving
problems through the integrated study of number systems, geometry, algebra, data
analysis, probability, and trigonometry
NETS-S 3. Research and Information Fluency
 a. students apply digital tools to gather, evaluate, and use information. Students:
 b. plan strategies to guide inquiry, locate, organize, analyze, evaluate, synthesize, and
ethically use information from a variety of sources and media.
 c. evaluate and select information sources and digital tools based on the appropriateness
to specific tasks.
 d. process data and report results
NETS-S 4. Critical Thinking, Problem Solving, and Decision Making
 Students use critical thinking skills to plan and conduct research, manage projects, solve
problems, and make informed decisions using appropriate digital tools and resources.
 a. identify and define authentic problems and significant questions for investigation
 c. collect and analyze data to identify solutions and/or make informed decisions
MOTIVATION
To express any given polynomial as a car’s speed as a function of time, students will be
shown on Google Earth the Autobahn highway in Germany, watch a 5 minute video on YouTube
of a person going various speeds. The purpose is to demonstrate his speed and distance as a
function of time. These ties in with the lesson of simplifying how to find the derivative of a
simple polynomial function.
MATERIALS
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Computer
Projector
Smart Board
Smart Notebook
Inspiration Software
Microsoft PowerPoint
Google Earth
You Tube
Virtual TI-84 Software
Textbooks
Notebooks
Pens/Pencils
iTunes - Podcast
STRATEGIES
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Direct Instruction
Open discussion
Problem solving – Inductive Thinking/Learning - PBL (Problem-based learning)
Group work – Listen, Think, Pair and Share
Carolina Teams Improvement
ADAPTATIONS
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The student who has a learning disability in note-taking will have a copy of the day’s
notes
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The student who is an ELL will be provided with pertinent vocabulary words prior to the
lesson
The student who is mathematically gifted will be given the opportunity to use an
overhead calculator (Virtual TI-84 Software) to illustrate a numerical or graphical
concept
DIFFERENTIATION OF INSTRUCTION
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The teacher believes that all students deserve the right to an education
The teacher acknowledges that not every student learns the same way
The teacher confirms that multiple methods will be used for instruction to engage
multiple intelligences
The teacher is aware that while this is a college-based course, students with special needs
will be given a fair opportunity to excel in this class.
DEVELOPMENTAL PROCEDURES
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Activities
The teacher will briefly explain that functions in math can model measuring real life
situations.
The teacher will speak about a function could be a car’s speed/distance over time.
The teacher, after briefly asking if anyone knows what the Autobahn is, quickly
explains/introduces the video, making sure the computer and projector are already turned
on, and then play the YouTube video of the Autobahn.
Key Questions to ask (after the video is played)
What was the fastest speed the Lamborghini went?
At a rate of 189mph, was he consistently going that fast? What about 174mph?
How could we graph these speeds over the timeframe of the video?
The teacher calls upon students to break into groups of 2-4 and do their best to represent
his speed over the time of the video. This should take 5 minutes.
A representative from each group will try to re-create their group graph on the
SmartBoard using the SmartNotebook files.
Teacher may show attached SmartNotebook file as example to better represent the
questions being asked
Teacher will show how a polynomial function can be represented as the speed of the
Lamborghini over a function of time.
The teacher will show the students on the SmartBoard the Power Rule of finding a
derivative; relating it to the instantaneous rate of change of the car’s speed f(x) at any
instantaneous given time (x).
Students will be shown three example of how to obtain the derivative of a polynomial
function via Power Rule.
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Students will be given three to five examples to work in groups for the remaining five
minutes of class and asked to find the derivatives of these functions using the Power
Rule.
ASSESSMENT
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By the end of class, students should be able to identify a polynomial’s derivative using
Power Rule
INDEPENDENT PRACTICE
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Students will be given three to five examples to work in groups for the remaining five
minutes of class and asked to find the derivatives of these functions using the Power
Rule.
Upon the conclusion of the lesson, students will be given five to ten examples to
continue for homework.
FOLLOW-UP: ACADEMIC INTERVENTION AND ACADEMIC ENRICHMENT
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Upon the conclusion of the lesson, students will be given five to ten examples to
continue for homework.
Students will be given the iTunes Podcast to listen/watch for further explanations of the
last two lessons if they would like a challenge; they may construct examples on their own
to find the derivatives and bring in the next day for a bonus three points on their next
quiz. The team of students who get the most extra questions correct will be awarded these
bonus point.
TEACHER REFERENCES
Hughes-Hallett, Deborah, Gleason, Andrew, & McCallum, William, (2009). Calculus: Single
Variable, (5th ed.). New York, NY; Wiley & Sons Publishing.
Driving Lamborghini on Autobahn at 305kph=189mph
We rented a Lamborghini Gallardo & Porsche 911 for incredible fun on Autobahn in Germany.
What a way to end a two week Europe vacation! Footage: airjersti, May 8, 2007
http://www.youtube.com/watch?v=sYPCLtA-ETk
Jerison, David. (Teacher). (2007, Fall). MIT Open Courseware.[MIT 18.01]. Single Variable
Calculus: Lecture01: Derivatives, slope, velocity, rate of change. Podcast retrieved from iTunes