Creating an Authentic Precalculus
Curriculum: Why We Study
Mathematics
Dr. Mark A. Jones
Chatham High School
mark.jones@chatham-nj.org
Brief Bio
• High School Teacher (CA, 1 year)
• University Professor (NY, 4 years)
• AT&T Bell Labs Researcher (NJ, 19 years)
• High School Teacher (NJ, 7 years)
Dr. Mark A. Jones -- Chatham High School
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Context is Important
• Everything we learn in life is contextualized.
Context facilitates learning, recall and performance.
• Math should be no exception:
– Historical context.
– Theoretical context.
– Application context.
Dr. Mark A. Jones -- Chatham High School
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Context is Important
• Welcome and initiate “Why” questions. They are a
rich source of context.
• Context arises in higher-order questions:
– What portion of the domain for the trig functions would
you have chosen for the range of the inverse trig
functions? Why?
– Vectors are objects having magnitude and direction. What
representation might convey this information graphically?
Dr. Mark A. Jones -- Chatham High School
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Historical Context
• Convey the Leibniz and Newton calculus controversy (story
telling) .
• Reconstruct the bridges of Konigsberg (kinetic learning).
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Historical Context
• Explore π (on π day!) – its history,
computation, memorization,
Buffon’s Needle, digits of π,
celebration in song, etc. (subjectbased learning).
Dr. Mark A. Jones -- Chatham High School
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Historical Context
• Have students conduct and present their own research on
the history of certain ideas, notation, topics, or influential
people (research).
– The concept of zero was fundamental to the development of placevalue representations and the modern algorithms for basic arithmetic.
– Imaginary numbers arose in the 15th and 16th centuries in the study of
cubic equations.
– Euler was a pioneering 18th century Swiss mathematician and physicist
with numerous important contributions including graph theory. He
introduced much of modern math notation, including f(x). The symbol
e honors Euler.
Dr. Mark A. Jones -- Chatham High School
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Theoretical Context
• Relations vs. functions
– Nondeterminism / Determinism
• Transformations
– Horizontal/vertical shifts (x-h, y-k)
– Horizontal/vertical stretching/compression (ax, ay)
– Horizontal/vertical reflection (-x, -y)
• Symmetry
– Hand graphing
– Even/odd
– Polar symmetries
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The Explanatory Power of Transformations
• Translations [horizontal/vertical shifts from (0, 0) to (h, k)]:
– Slope-intercept (lines)
– Point-slope (lines)
– Vertex form (parabolas, absolute value)
– General form for figures with centers (circles, ellipses, hyperbolas)
• Rates [horizontal compression/stretching]:
– Slope (lines)
– Growth and decay rates (exponentials)
• Positive/Negative Growth [horizontal reflection]
– Negative slope is a horizontal reflection of positive slope (lines)
– Decay is a horizontal reflection of growth (exponentials).
Dr. Mark A. Jones -- Chatham High School
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The Explanatory Power of Transformations
• Transformations in periodic functions
– Amplitude: vertical stretching/compression
– Angular frequency: horizontal stretching/compression
– Phase shift: horizontal shift
– Trig. Identities: sin(x) = cos(x-π/2)
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Application Context
• Applications convince students of the relevance of
mathematics.
– When are we ever going to use that?
– Work with real data when possible.
• Applications allow students to “feel” mathematics.
– Kinesthetic applications:
• Using astrolabes to measure angles in solving trig problems.
• Doing projectile motion experiments to illustrate parametric
equations, derivatives, etc.
– Interactive software:
• Internet applets, Wolfram Demonstration Project demos
• Geometer’s Sketchpad animation
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When are we ever gonna have to use this?
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Application Context
• Applications make math fun and interesting.
– Connect math to music and art.
– Tap into student creativity with exploratory learning (Polar Art
Festival).
• Create a new state of mind in your classroom.
– Skill and drill is conditioning and weight training.
Word problems are playing the game.
– Skill and drill is basic nutrition. Word problems are dessert.
– Lead by example -- emphasize applications in your assessments.
Dr. Mark A. Jones -- Chatham High School
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Honors Precalculus Units
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Review of Relations and Functions
Exponential and Logarithmic Functions
Trigonometric Functions
Polar Coordinates
Vectors
Conics
Data Analysis
Preview of Calculus (Limits, Derivatives, Integrals)
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Relations and Functions – Big Picture
• Why do we study mathematics?
• Relations vs. functions
– (Non-)Determinism. Your calculator is a function
machine. Computer databases are relational. Why?
• What is an algebra?
– Elementary (numeric) algebra, matrix algebra, vector
algebra, …
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Relations and Functions – Representations
• Importance of representations: mappings, ordered
pairs, equations, set builder notation, graphs
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Relations and Functions – Story-telling
• Story-telling
– What has your altitude been since you got up this morning?
– What does a housing bubble look like? Is it OK to buy one now?
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Relations and Functions -- Transformations
• Transformations
– Transformations underlie most of the formulas students
encounter in algebra.
– Students can transform new functions they have never
seen before: logistic functions, gaussian functions (bell
curves), trig functions.
– Function Mania activity: Contest format. Equation-tograph, graph-to-equation, transformations-to-equation,
transformations-to-graph.
Dr. Mark A. Jones -- Chatham High School
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Exponentials and Logs
• Find ways to dramatize the difference between
linear behavior and exponential/log behavior.
– 42 paper foldings to reach the moon, 94 foldings to reach
the end of the visible universe!!
(http://scienceblogs.com/startswithabang/2009/08/paper_folding_to
_the_moon.php)
– The Million Dollar Mission
You have your choice of two payment options:
(1) One cent on the first day, two cents on the second day,
and double your salary every day thereafter for thirty
days; or
(2) Exactly $1,000,000.
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Exponentials and Logs – Discovery
Experimenting with the equation solver (discovery learning).
When will you be a millionaire? Computing yields.
Present value / future value. Pricing zero coupon bonds.
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Exponentials and Logs – e
For advanced students, you can motivate how e arises:
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Exponentials and Logs – e
Then you can derive the continuous compounding formula from
the discrete compounding formula:
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Exponentials and Logs – Rule of 72
• Rule of 72 in investing.
– Approximate the doubling time by dividing 72 by the interest rate as a
percentage. Why does this work?
– It should be called the rule of 69. Why is 72 used instead?
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Trigonometry
• Trigonometry has two faces.
– As its name implies, it involves measuring triangles.
– But it also is the first exposure to periodic functions.
• Measuring triangles activities.
– Widescreen TV project.
– Distance measuring (with astrolabes) project.
• Periodic functions activities.
– Electromagnetic spectrum project.
– Math and Music assembly.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Widescreen TV Project
Let d be the distance to the TV in feet.
Let w be the screen width in inches.
Let x be the screen diagonal in inches.
Let θ be the viewing angle.
http://www.myhometheater.homestead.com/
1.
Electrohome: Electrohome suggests a viewing distance of three (minimum) to
six (maximum) screen widths for video. This corresponds to the point at which
most people will begin having trouble picking out details and reading the
screen. Probably too far away to be effective for home theater, OK for everyday
TV viewing. Most people are comfortable watching TV between this distance
and half this distance.
2.
SMPTE: The Society of Motion Picture and Television Engineers (SMPTE)
standard EG-18-1994 recommends a minimum viewing angle of 30 degrees for
movie theaters. This seems to be becoming a de facto standard for front
projection home theaters also. Viewing from this distance or closer will result in
a more immersive experience, and also lessen eye strain caused by watching a
smaller image in a dark room.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Widescreen TV Project
Let d be the distance to the TV in feet.
Let w be the screen width in inches.
Let x be the screen diagonal in inches.
Let θ be the viewing angle.
http://www.myhometheater.homestead.com/
3.
THX: THX publishes standards for movie theaters and home systems. THX
certification requires that the back row of seats in a theater have at least a 26
degree viewing angle and recommends a 36 degree viewing angle.
4.
Viewing Distances based on Visual Acuity: These distances are calculated based
on the resolving power of the human eye or visual acuity. The human eye with
20/20 vision can detect or resolve details as small as 1 minute of a degree of
arc. These distances represent the point beyond which some of the detail in the
picture is no longer able to be resolved and "blends" with adjacent detail. At
full resolution, an HDTV picture is 1920 pixels wide by 1080 pixels high.
[Hint: The optimal visual acuity distance is thus the distance at which 1 minute
of arc sweeps out a distance of inches.]
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Trigonometry – Widescreen TV Project
Part 1: General Formulas
For each of the 4 criteria above, determine how big a TV should be. State the TV’s
size in terms of the diagonal, x, rather than width and height. (Note that the
width and height of a TV can both be expressed in terms of x.) Show all of your
work. In each case, write the formula for the diagonal x in terms of the distance
d: x = . . . some function of d . . .
For Electrohome:
...
For SMPTE:
...
For THX:
...
For Visual Acuity:
minimum x = ______________________
maximum x = ______________________
minimum x = ______________________
minimum x =______________________
recommended x = _________________
optimal x =_______________________
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Widescreen TV Project
Part 2: Your Home HDTV
Measure the distance d in your home from the prime viewing location to
the place where your existing or new HDTV will stand (or hang on the wall
if it is a flat screen).
What is d? _________________________________
Use the formulas from part 1 to compute the appropriate size HDTV for
your home. Show all of your work.
Part 3: Your Math Teacher’s HDTV
Suppose that your favorite math teacher has a 46” HDTV. The viewing
distance, d, to his TV is 11 feet. Use the formulas from part 1 to compute
the appropriate size HDTV for his distance of 11 feet.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Measuring Distances
In this assignment, you’ll be using an astrolabe, a device capable of
measuring vertical angles. It is best used by two students, one to do
the sighting (using the straw) while the other student reads the angle
from the side. Therefore, the first task is to select a partner.
To mark off a known distance to use as a measurement in your
calculations, each group must acquire a piece of string of known length
(e.g., 20 feet) or a measuring tape.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Measuring Distances
We will be using the CHS Auditorium as our measurement site.
Objective #1: Position yourself at, near or on the stage. Determine
the height of the bottom of the curtain hanging above the stage.
Objective #2: Position yourself on the stage. Determine how tall the
curtain is.
Objective #3: Position yourself on the stage. Determine how far
above the stage the sound booth is (the bottom of the window at the
back of the auditorium).
Objective #4: Position yourself up near the back of the auditorium at
an elevation even with the bottom of the curtain. Determine the
distance to the curtain.
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Trigonometry – Measuring Distances
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Trigonometry – Measuring Distances
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Trigonometry – Measuring Distances
After obtaining your measurements, complete the assignment by
producing a typed report with your observations. Each team may
submit a single team report. For each objective:
1. (1 pt per objective) Describe and diagram the observation setting,
2. (2 pts per objective) Provide the raw data that you collected and
compute the required values,
3. (2 pts per objective) Discuss the possible errors from your
observation and how they would influence your answers. For
each quantity that you measure, estimate the amount that you
could be off by (the possible minimum and maximum values, as
well as the measured value). For each quantity that you compute,
figure the computed value, but also the range of possible values
given the measurement errors you estimated.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Measuring Distances
Possible Techniques:
#1 Align the eyes of the sighter to be level with the base of the triangle. This may not be
practical if you are measuring something that stands at ground level or the base of the object
is too high for you to reach eye-level. Use right triangle trig to find y, given x and θ.
(more sighting techniques are also given, including ones that utilize Law of Sines, etc.)
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Electomagnetic Spectrum
The EM Project is a short assignment which drills home the ubiquity of the
electromagnetic radiation that surrounds them and the many practical uses
of it. Here is the set up:
Trigonometry naturally models waves of all types. Electromagnetic waves are
produced by the motion of electrically charged particles. These waves are also
called "electromagnetic radiation" because they radiate from the electrically
charged particles. The waves are carried by massless particles called photons.
They travel through empty space as well as through air and other substances.
The energy E (in Joules) of the photons directly varies as the frequency f (in
cycles/sec) of the wave by the formula
E = h * f,
(sometimes the Greek letter ν (“nu”) is used rather than f for the
frequency)
where h is Planck’s constant, h = 6.626 × 10-34 J·s
The frequency f (in cycles/sec) inversely varies as the wavelength λ (in cm) with the
photons traveling at the speed of light, c:
f=c/λ
where c = 29,979,245,800 cm/sec
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Electomagnetic Spectrum
. . . Fill out the table on the reverse of this sheet with as many different
regions of the spectrum as you can find. (Use additional sheets if
necessary.) Examples of man-made signals include AM radio, TV, FM
radio, CB radio, cordless phones, radio controlled cars, garage door
openers, cell phones, GPS, etc. Natural radiation includes microwaves,
infrared, visible light, ultraviolet, X-rays, gamma rays, etc. Order the table
by increasing frequency ranges for each type of radiation. The ranges may
overlap since different uses may coexist depending upon the power of the
signals and their distribution. (Do not consider each band of light
separately or each radio station separately.)
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Electomagnetic Spectrum
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Trigonometry – Math and Music
• Math and Music is a two hour assembly, a subject-based
presentation that includes perspectives on music from math,
physics, psychology and music theory.
• The relationships to math include:
– The mathematics of scales (reference frequencies, intervals,
etc.). The 12 tone equal temperament scale is a geometric
progression.
– The mathematics of standing waves (wavelength, period,
frequency, harmonic series, overtones, noise cancellation).
– The mathematics of pitch in strings and open pipes.
– The mathematics of timbres (real sampled waveforms of
student voices and instruments).
– The mathematics of digital music (mp3, synthesizers, etc.).
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Math and Music
• The demonstrations include:
– Guitar and synthesizer: Songs and phrases from songs
that illustrate math/music concepts (amplitude, tunings,
scales, harmonics, distortion, human frequency response).
– Decibel readings (log scale for amplitude/volume).
– Software sampling of student voices and instruments.
– Animusic.
– Synthesizer demos.
Dr. Mark A. Jones -- Chatham High School
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Trigonometry – Math and Music
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Trigonometry – Math and Music
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Just vs. Equal Temperament
#
Interval
Note
1
tonic
A
2
minor 2nd
A#
3
major 2nd
4
Just
Freq
12-TET 440
Freq
Diff
440*1/1
440.00
440.00
440.00
0.00
440*16/15
469.33
*2^(1/12)
466.16
3.17
B
440*9/8
495.00
*2^(2/12)
493.88
1.12
minor 3rd
C
440*6/5
528.00
*2^(3/12)
523.25
4.75
5
major 3rd
C#
440*5/4
550.00
*2^(4/12)
554.37
-4.37
6
perf. 4th
D
440*4/3
586.67
*2^(5/12)
587.33
-0.66
7
dim. 5th
D#
440*7/5
616.00
*2^(6/12)
622.25
-6.25
8
perf. 5th
E
440*3/2
660.00
*2^(7/12)
659.26
0.74
9
minor 6th
F
440*8/5
704.00
*2^(8/12)
698.46
5.54
10
major 6th
F#
440*5/3
733.33
*2^(9/12)
739.99
-6.66
11
minor 7th
G
440*16/9
782.22
*2^(10/12)
783.99
-1.77
12
major 7th
G#
440*15/8
825.00
*2^(11/12)
830.61
-5.61
13
octave
440*2/1
880.00
*2^(12/12)
880.00
0.00
A
Dr. Mark A. Jones -- Chatham High School
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The Math of Pitch in Strings
For strings, the velocity depends on such factors as the tension in the string
and the linear density of the string.
The frequency (or pitch) depends on the velocity and length of the string.
Dr. Mark A. Jones -- Chatham High School
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Frequency – Pitch
• Frequency for musical sounds is usually expressed in terms of
cycles/sec or Hz. If f is the frequency in Hz, then:
ω = f cycles/sec*2 radians/cycle
= 2 f radians/sec
so
y(t) = A sin(ωt - φ) = A sin(2 ft - φ)
• For A 440:
y(t) = A sin((2 440)t – φ) = A sin(880 t – φ)
~ A sin(2764.6t – φ)
Dr. Mark A. Jones -- Chatham High School
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Digital Music
• Compact Discs
– 44.1 kHz, stereo, uncompressed
(sample rate must more than double the frequency)
44,100 samples/sec x 16 bits/sample x 2 [stereo]
= 1,411,200 bits/sec ~ 1,400 Kbps
• MP3
– 32-320 Kbps, stereo, compressed and lossy, uses mathematical and
psychoacoustic compression
• AAC (default Apple iTunes format)
– 96-320 Kbps, stereo, compressed and lossy, uses mathematical and
psychoacoustic compression; better quality at lower bitrate than MP3 but
less widely supported; optionally protected (encrypted)
Dr. Mark A. Jones -- Chatham High School
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Polar Coordinates – Polar Art Festival
• I sponsor an annual Polar Art Festival to motivate students to
explore polar equations, their symmetries, transformations
and shapes.
• Refreshments and low-volume music set the atmosphere.
• I set up a video camera on a copy stand pointing down.
Students first place a paper with their entrant number, name
and the title of their artwork. Then they place their calculator,
display the equations, then display the window settings and
then graph.
• Certificates are awarded for the Most Humorous, Most
Bizarre, and Most Artistic.
Dr. Mark A. Jones -- Chatham High School
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Polar Coordinates – Polar Art Festival
Dr. Mark A. Jones -- Chatham High School
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Polar Coordinates – Polar Art Festival
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Polar Coordinates – Polar Art Festival
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Polar Coordinates – Polar Art Festival
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Vectors – Information Retrieval
One of the fundamental approaches to information theory is the
vector space model. It is adapted extensively in various forms for
similarity matching throughout the Internet.
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Conics
I have designed a number of activities that use the following
examples:
• Parabolas
– Suspension Bridges, Satellite Dishes, Telescopes, Flashlights, St. Louis
Arch, Arranging Students
• Ellipses
– Whispering Rooms, Planets, Arranging Students (such as ‘0’, ‘6’, ‘8’, ‘9’
for the class yearbook picture)
Dr. Mark A. Jones -- Chatham High School
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Data Analysis – Using Excel
• One of the things that I experienced as a researcher in AI and
natural language processing was how the availability of data and its
impact had grown, particularly with the rise of the Internet.
• Data analysis and statistics have transformed every subfield of AI,
linguistics and psychology (fields that I was directly involved in) and
nearly every other academic field as well.
• To prepare our students, they need to be conversant with the
language, tools and techniques of data analysis.
• I devote a full week or more to data analysis. Students learn to use
Excel, including its data analysis facilities. They work with real data.
Dr. Mark A. Jones -- Chatham High School
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Data Analysis – Using Excel
• Part 1 of the data analysis project is scripted. Students learn:
– To obtain data that is reliable, relatively error-free, and appropriate to
the task. They obtain roller coaster data from http://www.rcdb.com/.
– To enter data, convert it to a desired form, and to clean the data of
obvious errors. Skills include copy/paste, find/replace, cell types.
– To obtain summary statistics to gain an initial understanding of the
data (central tendencies, histograms, etc.).
– To learn how to create and interpret scatterplots.
– To learn how to identify and model relationships which exist in the
data using curve fitting, regression and an understanding of functions.
– To understand the concepts of correlation and goodness-of-fit.
Dr. Mark A. Jones -- Chatham High School
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Data Analysis – Using Excel
• Part 2 of the data analysis project is open-ended. Students learn:
– To frame a central question.
– To find data to answer it.
– To analyze that data.
– To write persuasive, quantitative prose with accompanying graphs
and tables to answer the central question.
• Part 2 is far more challenging than part 1 for most students.
Dr. Mark A. Jones -- Chatham High School
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Data Analysis – Using Excel
Suicide Rate (# suicides / 100,000 people) v. Population Density (people / sq mile)
For year 2005
Suicide Rate (# suicides / 100,000 people)
25
20
15
10
y = -2.077ln(x) + 21.676
R² = 0.721
5
0
0
200
400
600
800
1000
1200
Population Density (people/ square mile)
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Data Analysis – Using Excel
Total Cigarettes (billions)
700.0
y -.0033984727x^3+19.82997364x^2-38556.34462x+24981008.05
R² = 0.98955
600.0
Cigarettes (billions)
500.0
400.0
300.0
200.0
100.0
0.0
1880
1900
1920
1940
1960
1980
2000
2020
Time (years)
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus / Parametric Equations
• The Projectile Motion Project is a group project that brings many
topics together and has the following features:
– It is a good example of kinesthetic learning. Students directly interpret
aspects of physical motion with a mathematical model.
– It illustrates the value of parametric equations and gives them practice
in using them in their calculator.
– It features group dynamics and cooperation.
– It relates concepts from algebra (vertex of a parabola) to concepts of
calculus (maxima).
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus – Projectile Motion
This is a team project which will give you some direct experience with
projectile motion equations. Create a group with either four or five team
members that will play the following roles. (Some roles may overlap in a four
person team.) Fill in your names.
The thrower will throw the ball:
The receiver will catch the ball:
The timer will time the throw:
The recorder will record the data
The spotter will measure arm height, distances, etc.
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus – Projectile Motion
Planning
On the day of this activity, each participant must bring the appropriate
equipment:
– The thrower is responsible for bringing a glove and ball.
– The receiver is responsible for bringing a glove.
– The timer is responsible for bringing a stopwatch or a watch with a
second hand.
– The recorder is responsible for bringing paper and a pencil to record
the data.
– The spotter is responsible for bringing a measuring tape.
– The timer, recorder and/or spotter must bring a calculator so that
results can be checked.
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus – Projectile Motion
Project Description
The group will submit a single report by completing and handing in this
project description. Each member will separately complete and submit the
final critique page.
You may use the equation solver in your calculator to check your answers,
but you must show the algebra for how your computed values were
obtained. Be sure to include the units in each of your answers. Each group
will carry out the objectives below. You may need to conduct several trials for
each objective before you get trustworthy data.
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus – Projectile Motion
1.
Vertical Toss: The thrower will throw the ball as close to straight up as
possible and allow it to hit the ground. The timer will start timing when
the ball is released and stop when the ball hits the ground. The recorder
will record the data. The spotter will observe the vertical release point
(thrower's hand height). The team should calculate the other parameters
and analyze the situation.
2.
Distance Toss: The thrower will throw the ball at least 100 feet to the
receiver. The timer will start timing when the ball is released and stop
when the ball is caught. The recorder will record the data. The spotter
will observe the initial height (thrower's hand height), the final height
(receiver's glove height) and the distance. The team should calculate the
other parameters and analyze the situation.
Dr. Mark A. Jones -- Chatham High School
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Preview of Calculus – Projectile Motion
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Preview of Calculus – Projectile Motion
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Preview of Calculus – Projectile Motion
For each scenario:
• Students sketch the experimental situation.
• Students work through a series of computations which lead them to a
computation for the:
–
–
–
–
–
–
initial horizontal/vertical/combined speed of the throw
angle of the throw
time to reach maximum height
height of the throw
instantaneous speed at the maximum height
instantaneous speed at the receiver
• Students complete a questionnaire on participation in their project, and
how the project contributed to their understanding.
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Summary
• Skill and drill is important, but for math to become ingrained, intuitive,
memorable, and useful . . . context is everything!!
• Math begins to seep into their everyday thought processes.
– One student reported that the laser patterns at a rock concert
reminded him of hyperbolas.
• Context, particularly application context, makes math come alive.
– A student commented after Math and Music, “Now I understand why
we study mathematics.” Indeed.
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