My Unit Plan
Will Scharen
EDU 521.04
Prof. R. Moroney
Summer 2010
Introduction
• Through the course of this unit, we will be studying a function’s
instantaneous rate of change at any given point.
• Since there are formulas for specific examples that need to be
applied to find the derivative to a function, this can become
confusing to students.
• To assist in the teaching and learning process, we will be using
technology in the forms of Smart Board, Hot List, and Podcasts.
– The Smart Board will engage students who are more visual and,
when called upon to write answers on them, students who are
more kinesthetic learners.
– The Hot List will be used for outlining the unit with helpful internet
links. These links vary from formulaic expressions to You Tube
videos for the visual and audio learners.
– The Podcasts via iTunes show a college classroom setting as a
motivational tool and as an aid for the audio/visual learners.
The Context of My Project
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Participants – High School Students
Grade/Subject – AP Calculus (Senior)
Student Tally – 20
Group Experience with Technology – Moderate
Access to Various Forms of Technology – Internet and
SmartBoard (class), Internet (Home)
• Teachers in Room – 1, Me
• The Two Detailed Lessons – Lessons 1 & 2 of the unit
Aim of the Project
• I want the students to be able to calculate the
derivative to a function using various
methods by the end of this unit.
• They will demonstrate that knowledge by
comprehending which application is used for
certain “special” functions in order to obtain
that derivative.
• I want them to be comfortable using
technology as a means to assist their learning
of the unit’s lessons as well.
Justification
• I am using technology to meet my goals
because I am aware that there is so much
pertinent information to relate math with
“real life”, technology helps merge the
gap between the two.
• Technology supports my curriculum by
engaging students of all intelligence levels
while still making the lessons effective and
relevant towards the AP guidelines.
Materials
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Computer
Projector
Smart Board
Smart Notebook Software
Inspiration Software
Microsoft PowerPoint
Virtual TI-84 Software
Textbooks – Single Variable Calculus 5th Ed.
Notebooks
Pens/Pencils
YouTube website (to view opening video)
Google Earth
iTunes Podcast
Time Frame
• Each lesson of this unit takes
roughly 2-4 days
• With 5 lessons, this unit will be
completed in an estimated 25-35
school days (included are days
scheduled for tests/quizzes)
NYS & ISTE Standards Covered
in this Unit
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Standards *NY State Standards in Algebra 2-Trigonometry relevant towards this lesson
*A2.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally,
numerically, algebraically, graphically)
*A2.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic,
algebraic)
*A2.CM.2 Use mathematical representations to communicate with appropriate accuracy, including
numerical tables, formulas, functions, equations, charts, graphs, and diagrams
*A2.CN.1 Understand and make connections among multiple representations of the same mathematical
idea
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
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Lesson Plan Day 1
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Student: William Scharen
Professor: R. Moroney
Course: EDU 521-04
Date: June 21, 2010
Grade: 12 Topic: Derivative of a Function
Content Area: AP Calculus
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INSTRUCTIONAL OBJECTIVE
After practicing examples of how a function’s derivative can be viewed as a
method of expression graphically, numerically, analytically or verbally,
students will be able to identify a basic function’s derivative with 85% accuracy
choosing any of these four methods.
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STANDARDS AND INDICATORS
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Indicators (per AP Calculus Collegeboard.com Course Goals)
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Standards *NY State Standards in Algebra 2-Trigonometry relevant towards this
lesson
A2.PS.4 Use multiple representations to represent and explain problem
situations (e.g., verbally, numerically, algebraically, graphically)
Indicator:
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.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.
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A2.CM.2 Use mathematical representations to communicate with appropriate accuracy,
including numerical tables, formulas, functions, equations, charts, graphs, and diagrams
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
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MOTIVATION
YouTube video of Michael Jordan’s final shot: Analyzing his shot as a parabolic curve, the
motivation is in getting students to think about the instantaneous rate of change that causes
a basketball to go from one location to another in milliseconds.
http://www.youtube.com/watch?v=PRCTp57LQro&feature=related
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MATERIALS
Computer
Projector
Smart Board
Smart Notebook
Inspiration Software
Microsoft PowerPoint
Virtual TI-84 Software
Textbooks
Notebooks
Pens/Pencils
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STRATEGIES
Direct Instruction
Open discussion
Problem solving – PBL (Problem-based Learning)
Guided Practice (in class and home)
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ADAPTATIONS
The student who has a learning disability in note-taking will have a copy of the day’s notes
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
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DIFFERENTIATION OF INSTRUCTION
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.
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DEVELOPMENTAL PROCEDURES
Activities
The teacher will make sure the computer and projector are already turned on, and then
play the YouTube video of Michael Jordan.
Key Questions to ask (after the video is played)
How is a basketball player’s 3-point shot like a parabolic curve?
What if we were to freeze the frame in mid-shot and predict how far up/down the
basketball is going within a fraction of a second?
The teacher draws a curve on a SmartNotebook file on the SmartBoard and shows a
parabolic curve where there is a maxima. In doing so, the teacher must point out that we
are looking for the instantaneous rate of change from when the ball reaches one height, to
where it goes into the next height.
Key Questions to ask:
Does this resemble anything we’ve studied before? (Slope formula)
If each height is a function of time, how could we differentiate one height at one point in
time from another?
(Pick a student) What is the slope formula in your own words?
Teacher may show attached SmartNotebook file as example to better represent the
questions being asked
Teacher will explain that the derivative to a function is an instantaneous rate of change
represented as the instantaneous change in y, divided by the instantaneous change in x.
Or in this case, the instantaneous change in the height of the basketball over the
instantaneous change in distance/time of the basketball; at any given interval.
Teacher should focus on the last slide in the SmartNotebook File
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• ASSESSMENT
Students will be given the function to a parabolic curve and find the rate of
change between two x values.
Students will find the instantaneous rate of change between two intervals
of x for a function when the difference in the intervals is 1, .1, .01, and .001.
Students will graph parabolic functions and label the rates of change
between two intervals of x.
• INDEPENDENT PRACTICE
Upon the conclusion of the lesson, students will be given three examples
for the opportunity to graph, find the rates of change between two
intervals, and the instantaneous rate of change between two assigned
intervals with a difference of .1, .01, and .001.
• FOLLOW-UP: ACADEMIC INTERVENTION AND ACADEMIC ENRICHMENT
Upon the conclusion of the class, students will be given an additional three
to six examples for the opportunity to graph, find the rates of change
between two intervals, and the instantaneous rate of change between two
assigned intervals with a difference of .1, .01, and .001
• TEACHER REFERENCES
Hughes-Hallett, Deborah, Gleason, Andrew, & McCallum, William, (2009).
Calculus: Single Variable, (5th ed.). New York, NY; Wiley & Sons Publishing.
Michael Jordan’s Final Shot
Michael Jordan's game winner vs the Utah Jazz in the 1998 NBA Finals.
Footage: NBA Unforgetabulls Video
http://www.youtube.com/watch?v=PRCTp57LQro&feature=related
Worksheets/References
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Worksheet will be supplied as handout
References
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Hughes-Hallett, Deborah, Gleason, Andrew, & McCallum, William, (2009).
Calculus: Single Variable, (5th ed.). New York, NY; Wiley & Sons
Publishing.
Michael Jordan’s Final Shot
Michael Jordan's game winner vs the Utah Jazz in the 1998 NBA Finals.
Footage: NBA Unforgetabulls Video
http://www.youtube.com/watch?v=PRCTp57LQro&feature=related
Lesson Plan Day 2
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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
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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.
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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.
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•
•
•
•
•
•
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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
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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.
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MATERIALS
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STRATEGIES
Computer
Projector
Smart Board
Smart Notebook
Inspiration Software
Microsoft PowerPoint
Google Earth
You Tube
Virtual TI-84 Software
Textbooks
Notebooks
Pens/Pencils
iTunes - Podcast
Direct Instruction
Open discussion
Problem solving – Inductive Thinking/Learning - PBL (Problem-based learning)
Group work – Listen, Think, Pair and Share
Carolina Teams Improvement
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ADAPTATIONS
The student who has a learning disability in note-taking will have a copy of the day’s notes
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
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
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).
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Students will be shown three example of how to obtain the derivative of a polynomial
function via Power Rule.
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.
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ASSESSMENT
By the end of class, students should be able to identify a polynomial’s derivative using
Power Rule
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INDEPENDENT PRACTICE
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.
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FOLLOW-UP: ACADEMIC INTERVENTION AND ACADEMIC ENRICHMENT
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.
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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.
Worksheets/References
• Worksheets will be supplied as handout
• 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
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• Jerison, David. (Teacher). (2007, Fall). MIT Open Courseware.[MIT
18.01]. Single Variable Calculus: Lecture01: Derivatives, slope,
velocity, rate of change.
Math - Problem Solving : Derivatives of Functions
Teacher Name: Mr. Scharen
Student Name: ________________________________________
CATEGORY
4
Strategy/Procedures Typically, uses an
efficient and
effective strategy to
solve the
problem(s).
3
Typically, uses an
effective strategy to
solve the
problem(s).
2
Sometimes uses an
effective strategy to
solve problems, but
does not do it
consistently.
1
Rarely uses an
effective strategy to
solve problems.
Mathematical
Terminology and
Notation
Correct terminology
and notation are
usually used,
making it fairly
easy to understand
what was done.
The work is
presented in a neat
and organized
fashion that is
usually easy to
read.
Almost all (85-89%) of
the steps and
solutions have no
mathematical
errors.
Correct terminology
and notation are
used, but it is
sometimes not easy
to understand what
was done.
The work is
presented in an
organized fashion
but may be hard to
read at times.
There is little use, or
a lot of
inappropriate use,
of terminology and
notation.
Explanation shows
substantial
understanding of
the mathematical
concepts used to
solve the
Explanation shows
some
understanding of
the mathematical
concepts needed to
solve the
Neatness and
Organization
Correct terminology
and notation are
always used,
making it easy to
understand what
was done.
The work is
presented in a neat,
clear, organized
fashion that is easy
to read.
Mathematical Errors 90-100% of the steps
and solutions have
no mathematical
errors.
Mathematical
Concepts
Explanation shows
complete
understanding of
the mathematical
concepts used to
solve the
Date Created: Jun 28, 2010 01:12 pm (UTC)
Most (75-84%) of the
steps and solutions
have no
mathematical
errors.
The work appears
sloppy and
unorganized. It is
hard to know what
information goes
together.
More than 75% of the
steps and solutions
have mathematical
errors.
Explanation shows
very limited
understanding of
the underlying
concepts needed to
solve the problem(s)
Two Different Forms of Technology
• Lesson 1
• Lesson 2
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Teacher’s use of Smart Board to
draw parabolic curves for
students
Using YouTube to demonstrate
a video
Students will be selected to
write on Smart Board their
interpretation of “Instantaneous
Rate of Change”
Students will be given Hot List
website to access for an overall
summary of entire unit
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Teacher’s use of Smart Board to
demonstrate Power Rule
One student from each group
will be selected to draw on the
Smart Board
Students will be assigned for
extra credit a Podcast via iTunes
to access at home relevant to
lessons in this unit
My Weebly site
http://calculusbabycalculus.weebly.com