DISCLAIMER: This document does not meet the
current format guidelines of
the Graduate School at The University of Texas at Austin. It has been published for informational use only. Copyright
by
Lesley Kaheana Johnson
2009
Development of Curricular Material for an Exploration Based
Precalculus Workbook
by
Lesley Kaheana Johnson, B.S.
Report
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Master of Education
The University of Texas at Austin
August 2009
Development of Curricular Material for an Exploration Based
Precalculus Workbook
Approved by
Supervising Committee:
Efraim P. Armendariz
Mark L. Daniels
Abstract
Development of Curricular Material for an Exploration Based
Precalculus Workbook
Lesley Kaheana Johnson, MEd
The University of Texas at Austin, 2009
Supervisor: Efraim Armendariz
The body of this report, a workbook titled Creative Discovery Explorations in
Precalculus, is the final outcome of the project of editing and supplementing a
compilation of investigative exercises designed to enhance a Precalculus curriculum. The
addition of an Instructor Support section to each of the original explorations is in
response to research and interviews, and is designed to help Precalculus teachers
incorporate collaborative discovery activities into their classrooms.
iv
Table of Contents
Introduction ..............................................................................................................1
Precalculus Investigation Workbook .......................................................................3
Preface: A Note to Instructors........................................................................4
One: Transformations
Algebra and Geometry Meet ...................................................................5
Instructor Solutions .................................................................................8
Instructor Support .................................................................................10
Two: Exponential and Logarithmic Functions
Logarithms in Carbon Dating ...............................................................11
Instructor Solutions ...............................................................................15
Instructor Support .................................................................................18
Three: Circular Motion
Angular and Linear Speed: Can You Feel the Breeze?.........................20
Instructor Solutions ...............................................................................24
Instructor Support .................................................................................28
Four: Identities in Trigonometry
cos(ϕ − θ ) and Implications .................................................................29
Instructor Solutions ...............................................................................33
Instructor Support .................................................................................38
Five: Polar Graphing
v
Graphing "Cartesian Functions" in Polar Coordinates .........................40
Instructor Solutions ...............................................................................49
Instructor Support .................................................................................56
Six: A Concept from Calculus
Difference Quotients and Rate of Change ............................................58
Instructor Solutions ...............................................................................63
Instructor Support .................................................................................67
Seven: A Special Number
A Number between 2 and 3 ..................................................................69
Instructor Solutions ...............................................................................72
Instructor Support .................................................................................74
Conclusion .............................................................................................................75
References ..............................................................................................................77
Vita .......................................................................................................................78
vi
Creative Discovery Explorations in Precalculus
Introduction
A significant problem found in modern mathematics education is related to the
lack of instructional strategies whereby teachers actively engage students in the learning
process. Boaler (2002), describes a common situation where students who were exposed
to mathematics in a traditional, teacher led classroom, believed that “the mathematics
they encountered in school and the mathematics they met in the real world to be
completely and inherently different” (p.111). In response to such findings, many
educators recognize the importance of shifting the emphasis and paradigm in a
mathematics classroom from teaching students algorithms that will produce solutions to
guiding the students to understand, discover and experience the patterns and properties of
the discipline (Backhouse, Haggarty, Pirie, Stratton, 1992; Boaler, 2002; Glenn, J. et al.,
1999). “If learners discover some mathematics, they are less likely to forget it. This is
partly because of the satisfaction such an achievement will have given them and partly
because they will have formed links in their minds between what they already knew and
what they have discovered” (Backhouse et al., p.82).
The addition of such collaborative investigations and explorations of mathematics
curricula into a course can be a challenging and intimidating task for teachers. As a
secondary teacher with the personal goal of challenging my students to explore and
discover mathematics for themselves, I know how difficult it is to create or obtain
investigations that will successfully enhance the curriculum and engage students. When I
was presented with the opportunity to edit and supplement a compilation of explorations
into Precalculus curricular material, I welcomed the challenge of creating a clear, helpful,
unintimidating guide to aid teachers in accomplishing this shift in the classroom
environment.
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Creative Discovery Explorations in Precalculus
Some of the potential benefits of engaging students in investigations, as outlined
by Backhouse et al. (1992), include: providing the opportunity for instructors to learn
about their individual students and how their minds tackle problems and process
information, creating an opportunity for collaboration, which incorporates learning from,
and explaining results to their peers, and providing students the opportunity to create and
discover mathematics on their own. This process boosts student confidence in
mathematics and instills practicality and purpose to classroom work. As a Calculus
teacher, I am very conscious of the fact that students, who enter my class with experience
related to thinking about, questioning, and discovering mathematical relationships, are
much better prepared for tackling new and unfamiliar concepts in Calculus. Additionally,
many teachers, like me, are interested in building a cooperative, collaborative atmosphere
in their mathematics classrooms; introducing explorations is an excellent way to achieve
that goal. Many of these activities allow for a style of open communication between
students, and with the teacher, which is not always possible in a teacher led setting.
Enabling mathematics instructors to supplement course material with meaningful
investigations allows more students to view mathematics as a useful, dynamic subject
adaptable to many different real world situations, tasks, and careers.
Page 2
Creative Discovery Explorations in Precalculus
Creative Discovery Explorations
in Precalculus
Editors:
Efraim Armendariz
Mark Daniels
George Innis
Lesley Johnson
Contributors:
Efraim Armendariz
Mark Daniels
Brandy Guntel
George Innis
Lesley Johnson
Brian Katz
Emily Landes
Heather Van Ligton
©2009 A Product of the Teaching Strategies Workshop Natural Sciences in
Conjunction with the Mathematics Department and UTeach Natural Sciences at the
University of Texas at Austin
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Creative Discovery Explorations in Precalculus
Preface:
A Note to Instructors
The explorations in this booklet are intended to be used by mathematics
instructors to motivate some of the major topics encountered by students in a typical
Precalculus course. These explorations can be used in many ways. One might have
students work through a relevant exploration at the beginning of a new chapter in an
adopted Precalculus text to provide students with a problem that can only be completely
solved as students master all of the material of the chapter in question. In this regard the
problems of the exploration would be revisited throughout the presentation of the chapter.
The hope is that the explorations of this workbook will be used to create a ‘need to know’
attitude in students’ minds. Alternately, the explorations might be used in a more
traditional way in that a relevant exploration could be assigned as a homework or class
assignment at the end of a chapter as a way of assessing students’ mastery of the chapter.
The authors of this text would also suggest that instructors allow students to work
collaboratively on these explorations in an inquiry or discovery-based classroom
environment. That is, the instructor should allow students ample time to discuss,
collaborate on, and agree upon derived results relating to the problems with the instructor
acting as a facilitator rather than a lecturer throughout the learning process. Consider
also having students or groups present exploration results to the class allowing class
members to critique each other in order to refine arguments. Of course, some good and
timely questions on the instructors’ part will also lead to enhanced student understanding
of exploration results and concepts.
Ultimately, the explorations of this text are intended to be used in tandem with an
adopted Precalculus textbook. Any number of the explorations can be used to enhance a
course as the instructor sees fit; each exploration is designed to stand alone. The
explorations serve as vehicles to get students to think deeply about the major topics in
Precalculus encountered by students.
Lastly, while the individual questions contained within each exploration in this
text are separated for ease of reading, the authors suggest that instructors have students
work the problems on their own paper. Thus, students will not be confined to trying to fit
their work and answers into too small a space provided between consecutive questions.
[Note to secondary instructors only: All topics align with both the National Council of
Teachers of Mathematics and state standards for mathematics education. We encourage
secondary instructors to determine the alignment with specific applicable standards.]
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Creative Discovery Explorations in Precalculus
One:
Algebra and Geometry Meet
Geometric transformations applied to graphs display interesting interactions between
algebra and geometry. In particular, we know that the parabola y = x 2 can be transformed
into any other parabola y = Ax 2 + Bx + C = a( x − h) 2 + k using translations (horizontal and
vertical), reflections, and/or dilations and contractions. For a review of graphing
transformations, see your book or the examples below. Remember that h, k, and a can
represent any real number and the vertex of the parabola y = a( x − h) 2 + k is ( h, k ) .
The graph of y = f ( x ) can be transformed in many ways. The first graph below is
referred to as the “parent function”, while the others represent transformations of this
graph and are labeled with the algebraic notation for these transformations. In particular,
the second graph is a vertical translation (2 units down), the third is a horizontal
translation (1 unit left), the fourth is a dilation using a scale factor of 2, and the fifth is a
reflection through the x-axis.
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Creative Discovery Explorations in Precalculus
Exploration:
1. Suppose that the graph of y = x 2 is transformed by a horizontal translation of h
units. Find the roots of the resulting parabolas. [Note: Be sure to consider both
cases ( h < 0 and h > 0 ) of the translation.]
2. Suppose that the graph of y = x 2 is transformed by a vertical translation of k units.
Find the roots of the resulting parabolas.
3. Suppose that the graph of y = x 2 is transformed by a dilation using a scale factor
a > 1 . Find the roots of the resulting parabolas.
4. Suppose that the graph of y = x 2 is transformed by the composition of a dilation
using a scale factor a > 1 and a vertical translation of k units. Find the roots of
the resulting parabolas.
5. Suppose that the graph of y = x 2 is transformed by a composition of a dilation
using a scale factor a > 1 , a vertical translation of k units, and a horizontal
translation of h units. Find the roots of the resulting parabolas.
6. Show how this final answer relates to the quadratic formula.
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Creative Discovery Explorations in Precalculus
Historical note:
The combining of algebraic and geometric methods involving the coordinate
plane is commonly attributed to the French mathematician Rene' Descartes (1596-1650).
References:
Burton, D. (1991). History of Mathematics, An Introduction. Dubuque, IA: Wm. C.
Brown Communications.
Eves, H.W. (1989). Introduction to the History of Mathematics 6th edition. New York:
Saunders Publishing.
Page 7
Creative Discovery Explorations in Precalculus
Instructor Solutions:
Algebra and Geometry Meet
Exploration:
1. Suppose that the graph of y = x 2 is transformed by a horizontal translation of h
units. Find the roots of the resulting parabolas. [Note: Be sure to consider both
cases ( h < 0 and h > 0 ) of the translation.]
This transformation translates the parabola to the right h units if h > 0 , and to
the left h units if h < 0 .
The root of the parabola occurs at x = h .
2. Suppose that the graph of y = x 2 is transformed by a vertical translation of k units.
Find the roots of the resulting parabolas.
This transformation translates the parabola up k units if k > 0 , and down k units
k <0.
If k > 0 , the parabola has no real roots. If k < 0 , the roots of the parabola are
x = ± −k .
3. Suppose that the graph of y = x 2 is transformed by a dilation using a scale factor
a > 1 . Find the roots of the resulting parabolas.
Dilation does not affect points on the x-axis, so the root of the parabola is x = 0 .
4. Suppose that the graph of y = x 2 is transformed by the composition of a dilation
using a scale factor a > 1 , and a vertical translation of k units. Find the roots of
the resulting parabolas.
The dilation affects the roots inversely as compared to the translation. If k and a
−k
have opposite signs then the roots of the parabola are x = ±
. If k and a
a
have the same signs then there are no real roots.
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Creative Discovery Explorations in Precalculus
5. Suppose that the graph of y = x 2 is transformed by the composition of a dilation
using a scale factor a > 1 , a vertical translation of k units, and a horizontal
translation of h units. Find the roots of the resulting parabolas.
−k
If k and a have opposite signs then the roots of the parabola are x = h ±
. If
a
k and a have the same signs then there are no real roots.
6. Show how this final answer relates to the quadratic formula.
The most general of these transformations yields the graph of y = a ( x − h) 2 + k . If
we expand this, we get y = ax 2 − 2ahx + (ah 2 + k ) . The general form for the
quadratic polynomial is, y = Ax 2 + Bx + C , where A = a , B = −2ah , and
C = ah 2 + k . Substituting these values into the quadratic formula, yields
x=
− B ± B 2 − 4 AC 2ah ± 4a 2 h 2 − 4a (ah 2 + k )
−k
.
=
= h±
2A
2a
a
The quadratic formula has a cleaner form in this language.
It's also interesting to think about this using an inverse process. If you are given
the polynomial y = Ax 2 + Bx + C , we see that the scaling factor is a = A , the
B2
−B
, and the vertical shift is k = C −
, which can be
4A
2A
verified by completing the square.
horizontal shift is h =
Page 9
Creative Discovery Explorations in Precalculus
Instructor Support
Algebra and Geometry Meet
Prerequisite Skills:
Students should be able to:
− Know what is meant by transformations in a plane
− Find the roots of a quadratic equation using different methods
Goals and Objectives:
− Make a connection between the algebraic interpretation and the graph of a
polynomial function
− Explore the effects of different transformations on the roots of polynomial
functions
− Derive general expressions for the roots of a polynomial function after
translations, dilations, and composite transformations
− Connect the quadratic formula to the derived expressions of the roots of a
polynomial function written in vertex form, y = a( x − h) 2 + k
Teacher tips:
− Students might need the hint to expand the vertex form of the equation of a
parabola in order to get started on exercise 6
Possible student misconceptions:
− Students may have difficulty considering both positive and negative values for
translations and scale factors
Connections to previous concepts:
− Roots of quadratic polynomials
− Transformations in the plane
Materials:
− None
Page 10
Creative Discovery Explorations in Precalculus
Two:
Logarithms in Carbon Dating
This exploration considers the inverse of the exponential function y = f ( x ) = e x . For the
function f, the symbol f −1 will be used to denote the inverse of f. Since we normally
consider f ( x) = y , the inverse function is written as f −1 ( y ) = x . In the case of the
exponential function, we can write f −1 ( y ) = f −1 (e x ) = x .
Background:
1. Explain how you know that the exponential function has an inverse.
2. Knowing the properties of the exponential function can help us decide on some
properties that should be true of its inverse. Write e a = l and eb = m . Consider the
property e a eb = e a + b . Use this to show that f −1 (lm) = f −1 (l ) + f −1 (m) is a property
of the inverse function.
The inverse function we've been exploring is called the natural logarithm, and is written
as ln( y ) . Rather than writing f −1 ( y ) = x , we can write ln( y ) = x , however, with the
understanding that this is an inverse function, it is normal to write y = ln( x ) or
f ( x ) = ln( x ) .
In 1949, Willard Libby and a team of scientists at the University of Chicago discovered
that the age of an organism could be found based on the amount of radioactive Carbon it
contains – a process known as Carbon dating. Every object contains two types of
Carbon; radioactive Carbon, Carbon-14, and non-radioactive Carbon, Carbon-12. In
living objects, the ratio of these two Carbons is fixed, i.e. the amount of each Carbon in
living things remains the same. When a living organism dies, the Carbon-14 is no longer
replenished and starts to decay. The process of carbon dating aids scientists in
determining the amount of time that has passed since an organism was alive. To
calculate how long ago the Carbon-14 stopped being replenished, scientists use
logarithms. Let's investigate how logarithms are used.
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Creative Discovery Explorations in Precalculus
Exploration:
Radioactive elements, such as Carbon-14, have a specific rate of decay. The amount of
time it takes for half of the radioactive component in the element to decay is known as
the element’s half-life, h. If N(t) is the amount of radioactive material as a function of
time, t, and if N 0 is the amount of radioactive material that was originally in the sample,
and t is the amount of time passed since death, then the amount of radioactive material
remaining in the sample after time, t, is represented by N(t), where
N (t ) = N 0 e
−
t ln 2
h
.
3. According to the definition of half life, half of the original amount of the
radioactive element should remain after one half life has passed. Use the equation
provided for N(t) and set the time passed equal to one half life ( t = h ) to illustrate
this relationship.
4. Suppose you know what N, N 0 and the half life h of an element is, but you would
like to know the time that has passed since a specimen was alive. Describe a
procedure to solve for t. Accompany your procedure with the algebraic
manipulation of N(t) to isolate t.
The equation you found in exercise 4 is the general equation scientists utilize to
determine the amount of time a specimen has been dead when they know the current and
original amounts of a decaying element.
5. Use the fact that the half life of Carbon-14 is about 5700 years to rewrite the
general equation you found in exercise 4. This is the specific equation used for
Carbon dating.
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Creative Discovery Explorations in Precalculus
6. Scientists find a wooden spoon and they want to use Carbon dating to figure out
how old it is. If the amount of Carbon left is 85% of its original amount, how old
is this artifact? (Note here that we are actually finding the age of the wood from
which the spoon was made.)
Page 13
Creative Discovery Explorations in Precalculus
Historical note:
Logarithms were introduced by John Napier (1550-1617), who spent twenty years
working on the theory. He published his work in a book entitled Mirifici Logarithmorum
Canonis Descriptio. Logarithms were highly acclaimed at the time of their discovery
because their properties allowed multiplication problems to be turned into addition
problems (see background exercise 2).
Napier also introduced an early form of the calculator that involved lining up
wooden rods or bones, but was it difficult to use. His device was called Napier's rods or
Napier's bones.
References:
Burton, D. (1991). History of Mathematics, An Introduction. Dubuque, IA: Wm. C.
Brown Communications.
Eves, H.W. (1989). Introduction to the History of Mathematics 6th edition. New York:
Saunders Publishing.
Page 14
Creative Discovery Explorations in Precalculus
Instructor Solutions:
Logarithms in Carbon Dating
Background:
1. Explain how you know that the exponential function has an inverse.
The function f ( x ) = e x is one-to-one, and thus has an inverse that is also a
function.
2. Knowing the properties of the exponential function can help us decide on some
properties that should be true of its inverse. Write e a = l and eb = m . Consider the
property e a eb = e a + b . Use this to show that f −1 (lm) = f −1 (l ) + f −1 (m) is a property
of the inverse function.
(
f −1 ( lm ) = f −1 e a eb
−1
)
a +b
= f (e )
= a+b
= f −1 ( l ) + f −1 (m)
Exploration:
3. According to the definition of half life, half of the original amount of the
radioactive element should remain after one half life has passed. Use the equation
provided for N(t) and set the time passed equal to one half life ( t = h ) to illustrate
this relationship.
N ( h) = N 0 e
−
h ln 2
h
= N 0 e − ln 2
ln
= N0e
1
= N0
2
1
2
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Creative Discovery Explorations in Precalculus
4. Suppose you know what N, N 0 and the half life h of an element is, but you would
like to know the time that has passed since a specimen was alive. Describe a
procedure to solve for t. Accompany your procedure with the algebraic
manipulation of N(t) to isolate t.
[Hint: If you are having trouble with the symbols, first try using numbers, e.g.,
N = 1 , N 0 = 4 and h = 1000 .]
N = N0e
− tln 2
h
Original equation (explanations may vary)
− tln 2
N
=e h
N0
Isolate the exponential expression by dividing by N 0
− tln 2
⎛ N ⎞
h
=
ln ⎜
lne
⎟
N
⎝ 0⎠
Apply the natural log of both sides of the equation
⎛ N ⎞
tln 2
ln ⎜
⎟=−
h
⎝ N0 ⎠
Simplify the right side of the equation
h ⎛ N ⎞
ln ⎜
⎟
ln2 ⎝ N 0 ⎠
Divide both sides of the equation by –
t=−
ln 2
to isolate t
h
5. Use the fact that the half life of Carbon-14 is about 5700 years to rewrite the
general equation you found in exercise 4. This is the specific equation used for
Carbon dating.
t=−
5700 ⎛ N ⎞
ln ⎜
⎟ years
ln2 ⎝ N 0 ⎠
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Creative Discovery Explorations in Precalculus
6. Scientists find a wooden spoon and they want to use Carbon dating to figure out
how old it is. If the amount of Carbon left is 85% of its original amount, how old
is this artifact? (Note here that we are actually finding the age of the wood from
which the spoon was made.)
[Hint: rewrite N in terms of N 0 and substitute the expression into the equation]
t=−
5700 ⎛ 0.85 N 0 ⎞
ln ⎜
⎟
ln2 ⎝ N 0 ⎠
=−
5700
ln0.85
ln 2
≈ 1336 years
Page 17
Creative Discovery Explorations in Precalculus
Instructor Support
Logarithms in Carbon Dating
Prerequisite Skills:
Students should be able to:
Demonstrate their knowledge of the behavior and properties of the exponential function
f ( x) = ex
− Determine the existence of an inverse function
− Find the inverse of a function
− Perform manipulations of exponential equations and expressions using the
properties of exponents
− Provide valid reasons for performing manipulations of an equation in order to
demonstrate the existence of an identity
− Simplify expressions and equations using basic properties of logarithms
− Solve exponential equations for a variable in the exponent by applying the natural
logarithm function to both sides of the equation
Goals and Objectives:
− Recognize exponential and logarithmic functions as inverses of each other
− Verify and related identities of exponential functions and their inverses
− Challenge students to provide valid justifications for performing manipulations in
equations
− Solve an equation for a term in an exponent
− Demonstrate a common and significant application of logarithms to the field of
science
Teacher tips:
− Make sure to check that students correctly solve for t in exercise 4 before
continuing on to exercises 5 and 6
− When necessary, present students with hints provided in exercises 4 and 6 of the
Instructor Solutions section
Possible student misconceptions:
− Students often think the exponential function and the logarithmic function are the
same
− Students commonly misplace components of a logarithmic equation when
converting to exponential form (or vice versa)
Connections to previous concepts:
− Properties of exponents
− Inverse functions
Page 18
Creative Discovery Explorations in Precalculus
Materials:
− Calculator or CAS
Page 19
Creative Discovery Explorations in Precalculus
Three:
Angular and Linear Speed: Can You Feel the Breeze?
The coordinates of Austin, Texas, are 30°16’2’’ N and 97°45’50’’W. Using 3950 miles
as the radius of Earth, we want to find the linear speed of a person living in Austin, with
respect to the center of Earth, in miles per hour.
In answering the question, we will consider linear speed, vl , to be the distance traveled
along an arc (the arclength), s, per unit of time, t. We will also consider angular speed,
ω , to be the radians traveled that correspond to the arclength, θ , per unit of time, t.
Background:
1. Write an equation for linear speed, vl , in terms of angular speed, ω , using the
fact that the arclength is equal to the radius of the circle, r, times the angle, θ ,
corresponding to the arc ( s = rθ ).
Before we attempt to answer the title question, let’s first consider a slightly simpler
question. We will consider a merry-go-round, a circle, spinning around an axis rather
than the earth, which is a sphere.
Merry-Go-Round
Suppose a merry-go-round has a diameter of 10 feet and revolves once per minute.
2. Find the angular speed of the merry-go-round in radians per minute?
3. If Olivia is standing 3 feet from center of the merry-go-round, what distance does
she travel in one revolution?
Page 20
Creative Discovery Explorations in Precalculus
4. If Libby is standing 4 feet from the center, what distance does she travel in one
revolution?
5. Who is traveling faster? Olivia or Libby? Justify your answer by finding the
linear speeds of Olivia and Libby, with respect to the center of the merry-goround.
6. Is there a place where the linear speed of a person on the merry-go-round is 0 feet
per minute? Where should a person stand so that they have the fastest linear
speed?
Earth
Now let’s consider the question of a person standing at a specified location on Earth.
7. What is the angular speed of Earth, in radians per hour?
8. Suppose Libby lives in Pontianak, Indonesia, right on the equator. What is
Libby’s linear speed, with respect to the center of Earth, in miles per hour?
9. Is there a place on Earth where a person has a linear speed of 0 miles per hour?
Explain your answer.
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Creative Discovery Explorations in Precalculus
10. Suppose Olivia lives in Austin. Do you think Olivia has the same linear speed as
Libby or is she going faster or slower? Explain your answer.
11. Find the radius of the latitudinal circle on which Olivia, in Austin, travels? [Note:
Since latitude is measured in degrees, you should solve this problem using
degrees.]
12. Find Olivia’s linear speed, with respect to the center of Earth, in miles per hour.
Did you guess correctly in exercise 10?
13. You should have found that Olivia and Libby are going very fast – twice as fast as
an airplane. Why don’t we feel a breeze as a result of the earth’s rotation?
Page 22
Creative Discovery Explorations in Precalculus
Historical Note:
The mathematician Archimedes took the question of this exploration one step
farther by studying spirals. We've seen in this exploration that if you stand on a merrygo-round while it's turning, then you travel in a circle. Imagine what happens if, instead
of standing still, you walk from the center out to the edge of the merry-go-round at a
constant speed while it's turning. The path you travel is no longer a circle, but along a
path known as the Spiral of Archimedes.
Archimedes, born in about 287 BCE, was the most famous mathematician and
inventor in ancient Greece. Among his contributions is an invention known as the
Archimedes Screw that is still used today to pump water from the ground. In 218 BCE,
when Archimedes was 69 years old and living in the city of Syracuse, the Second Punic
War began. By 214 BCE the Romans began a siege of city of Syracuse. As the Romans
were capturing the city in 212 BCE, after the two-year-long siege, a soldier killed
Archimedes. He was 75 years old. His last words are said to have been, "Don't disturb
my circles."
References:
Archimedes. (2009). In Encyclopædia Britannica. Retrieved April 01, 2009, from
Encyclopædia Britannica Online:
http://www.britannica.com/EBchecked/topic/32808/Archimedes
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Creative Discovery Explorations in Precalculus
Instructor Solutions:
Angular and Linear Speed: Can You Feel the Breeze?
Background:
1. Write an equation for linear speed, vl , in terms of angular speed, ω , using the
fact that the arclength is equal to the radius of the circle, r, times the angle, θ ,
corresponding to the arc ( s = rθ ).
vl =
s rθ
=
= rω
t
t
vl = rω
Merry-Go-Round
Suppose a merry-go-round has a diameter of 10 feet and revolves once per minute.
2. Find the angular speed of the merry-go-round in radians per minute?
ω=
θ
t
=
2π
= 2π rad/min
1
3. If Olivia is standing 3 feet from center of the merry-go-round, what distance does
she travel in one revolution?
s = rθ = 3 ( 2π ) = 6π ≈ 18.850 ft
4. If Libby is standing 4 feet from the center, what distance does she travel in one
revolution?
s = rθ = 4 ( 2π ) = 8π ≈ 25.133 ft
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Creative Discovery Explorations in Precalculus
5. Who is traveling faster? Olivia or Libby? Justify your answer by finding the
linear speeds of Olivia and Libby, with respect to the center of the merry-goround.
Olivia: vl =
s 18.850
=
= 18.850 ft/min
t
1
Libby: vl =
s 25.133
=
= 25.133 ft/min
t
1
Since Libby travels farther in the same time period as Olivia, Libby is going
faster. The calculations of the two linear speeds support this conclusion.
6. Is there a place where the linear speed of a person on the merry-go-round is 0 feet
per minute? Where should a person stand so that they have the fastest linear
speed?
If a person stands at the center of the merry-go-round, then they turn on a point
and don’t travel in a circle, so their linear speed is 0 ft/min. In order to have the
fastest linear speed a person has to stand on the circle of greatest radius, and
therefore the person should stand on the edge of the merry-go-round.
Earth
Now let’s consider the question of a person standing at a specified location on Earth.
7. What is the angular speed of Earth, in radians per hour?
ω=
s 2π
=
≈ 0.262 rad/hr
t 24
8. Suppose Libby lives in Pontianak, Indonesia, right on the equator. What is
Libby’s linear speed, with respect to the center of Earth, in miles per hour?
vl =
s rθ 3950(2π )
=
=
≈ 1034.108 mph
t
t
24
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Creative Discovery Explorations in Precalculus
9. Is there a place on Earth where a person has a linear speed of 0 miles per hour?
Explain your answer.
Yes, if a person stands on either of the poles, then he or she travels on a circle
with no radius and his or her linear speed will be 0 mph.
10. Suppose Olivia lives in Austin. Do you think Olivia has the same linear speed as
Libby or is she going faster or slower? Explain your answer.
Olivia will have a slower linear speed than Libby because the latitudinal circle
that Olivia travels along has a smaller radius.
11. Find the radius of the latitudinal circle on which Olivia, in Austin, travels? [Note:
Since latitude is measured in degrees, you should solve this problem using
degrees.]
Since the radius of the earth ( rE ) is given as
3950 miles the radius of the latitudinal circle
of Austin, TX (rA ) can be found using the
relationship:
sin ( 90° − θ ) =
rA
rE
rA = 3950sin ( 90° − θ ) , where
θ = 30°16 ' 2" = 30 +
16
2
+
≈ 30.267°
60 3600
rA ≈ 3411.552 miles
12. Find Olivia’s linear speed, with respect to the center of Earth, in miles per hour.
Did you guess correctly in exercise 10?
⎛ 2π ⎞
vl = rω = 3411.552 ⎜
⎟ ≈ 893.142 mi/hr
⎝ 24 ⎠
Olivia is traveling at a linear speed slower than Libby (answers will vary based
on the conjecture in exercise 10).
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Creative Discovery Explorations in Precalculus
13. You should have found that Olivia and Libby are going very fast – twice as fast as
an airplane. Why don’t we feel a breeze as a result of the earth’s rotation?
Wind occurs when there is a difference between the speed of the movement of the
atmosphere (air) and the speed of a person. If they are both traveling at the same
speed, then there is no wind. Not only are Olivia and Libby traveling with fast
linear speeds, but the atmosphere surrounding them is also traveling at similar
speeds. So, there is little difference between the speed of Olivia and the air
surrounding her, and the speed of Libby and the air surrounding her, therefore
there is no breeze.
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Creative Discovery Explorations in Precalculus
Instructor Support
Angular and Linear Speed: Can you feel the breeze?
Prerequisite Skills:
Students should be able to:
− Represent and read angle measures as degrees, minutes, seconds
− Work with radians as angle measures
− Convert angle measures between degrees and radians
− Find the distance traveled around a circle (the arclength) using s = rθ
Goals and Objectives:
− Work with angles measured in both degrees and radians
− Model and solve a real world problem using angles
− Find the linear speed of an object traveling in a circular motion
Teacher tips:
− Make sure students pay careful attention to the units provided and accurately label
their answers
Possible student misconceptions:
− Students may think that they need more information in order to complete
exercises 2 and 7. If necessary, remind students that θ is the distance of one
revolution in radians (2π ) , and t is the time to complete one revolution.
− In exercise 2, t is given for the merry-go-round, and in exercise 7 it is assumed
that students know the earth revolves once approximately every 24 hours.
− Some students may argue that it takes slightly more than 24 hours for the earth to
complete one revolution of the earth. For this exploration, setting t = 24 hours is
sufficient
− Students often rewrite an angle measure given in degrees, minutes, seconds as a
decimal by simply inserting a decimal point rather than completing the
appropriate conversion. In exercise 11, students may write 30°16'2" as 30.162
rather than 30.267 .
Connections to previous concepts:
− Rates of change
− Speed
− Circular motion vs. linear motion
− Trigonometric functions and their applications
Materials:
− Calculator or CAS
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Creative Discovery Explorations in Precalculus
Four:
Cos(ϕ −θ ) and Implications
At this stage of your study of trigonometric functions you have learned a geometric
definition of each of these functions and properties of their graphs. In applications of
trigonometric functions, expressions such as sin(ϕ + θ ) and cos(ϕ − θ ) are frequently
encountered. Our study of trigonometry continues with the discovery of expressions for
the sine and cosine of sums and differences of angles and for half and double angles.
These types of expressions are part of Analytic Trigonometry and they are used
extensively throughout the sciences, further mathematical studies, and to simplify
complex statements in economics.
Background:
⎛π ⎞
1. If asked to find the exact value of cos ⎜ ⎟ without using a calculator or
⎝ 12 ⎠
trigonometric tables, one might reason since
π
12
=
π
4
−
π
6
, then
⎛π ⎞
⎛π ⎞
⎛π ⎞
cos ⎜ ⎟ = cos ⎜ ⎟ − cos ⎜ ⎟ . Is this statement true? Show work to support
⎝ 12 ⎠
⎝4⎠
⎝6⎠
your answer.
Exploration:
We begin this exploration by finding a formula for cos(ϕ − θ ) with restrictions
0 ≤ θ ≤ ϕ < π . Consider the figures below when completing the exercises.
Figure 1
Figure 2
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Creative Discovery Explorations in Precalculus
Figure 1 is the graph of an angle (ϕ − θ ) in standard position with the unit circle
centered at (0,0). AB is the chord opposite the angle (ϕ − θ ) .
Figure 2 is the graph of the separate angles, ϕ and θ , in standard position with the unit
circle centered at (0,0). CD is the chord opposite the angle (ϕ − θ ) .
2. Are the chords opposite to angles (ϕ − θ ) in the two graphs the same length?
Justify your answer. [Hint: You may want to refer to geometric theorems in
your justification.]
3. Write expressions for the lengths of the chords, in the two diagrams provided
using the coordinates of the end-points of each chord.
4. Use the results of exercise 3 to derive an expression for cos (ϕ − θ ) . [Hint:
Replace the Cartesian coordinates with their equivalents in terms of sines and
cosines. Caution: It looks messy but works out beautifully.]
⎛π ⎞
5. Use the result from exercise 4 to find an algebraic expression for cos ⎜ ⎟ .
⎝ 12 ⎠
Evaluate this expression and compare to a calculated or tabulated value.
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Creative Discovery Explorations in Precalculus
6. Use the result from exercise 4 to find a formula for cos (ϕ + θ ) in terms of
trigonometric functions of ϕ and θ .
7. Find formulas for sin (ϕ + θ ) and sin (ϕ − θ ) . [Hint: Consider the cofunction
identities.]
⎛ϕ ⎞
8. Make a conjecture as to how one could rewrite (2ϕ ) and ⎜ ⎟ in order to derive
⎝2⎠
identities using the sum and difference formulas in terms of sin ϕ and cos ϕ .
⎛ϕ ⎞
Then, find formulas for sine and cosine of angles (2ϕ ) and ⎜ ⎟ in terms of the
⎝2⎠
sine and cosine of ϕ .
⎛π ⎞
9. Use the results above to find an algebraic expression for cos ⎜ ⎟ . Evaluate this
⎝ 24 ⎠
expression and compare your result to a calculated or tabulated value. [Hint: Use
the results from exercises 5 and 8.]
10. Describe how one might use the identities you’ve just derived to aid in creating a
trigonometric table.
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Creative Discovery Explorations in Precalculus
Historical Note:
Augustin-Louis Cauchy, 1789-1857, is credited with the idea behind the proofs
exemplified by this exploration.
[For discussion: The rate at which mathematics and many other fields of human
endeavor advance is rapidly increasing. How elegant is the proof you developed above?
How easy will it be to remember and to communicate to colleagues? Do you think
elegance is important in math? In other fields?]
References:
Burton, D. (1991). History of Mathematics, An Introduction. Dubuque, IA: Wm. C.
Brown Communications.
Eves, H.W. (1989). Introduction to the History of Mathematics 6th edition. New York:
Saunders Publishing.
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Creative Discovery Explorations in Precalculus
Instructor Solutions:
Cos(ϕ −θ ) and Implications
Background:
⎛π ⎞
1. If asked to find the exact value of cos ⎜ ⎟ without using a calculator or
⎝ 12 ⎠
trigonometric tables, one might reason since
π
12
=
π
4
−
π
6
, then
⎛π ⎞
⎛π ⎞
⎛π ⎞
cos ⎜ ⎟ = cos ⎜ ⎟ − cos ⎜ ⎟ . Is this statement true? Show work to support
⎝ 12 ⎠
⎝4⎠
⎝6⎠
your answer.
No, this statement is not true:
2
3
⎛π ⎞
⎛π ⎞
cos ⎜ ⎟ − cos ⎜ ⎟ =
−
2
⎝4⎠
⎝6⎠ 2
⎛π ⎞
≈ −0.159 ≠ cos ⎜ ⎟ ≈ 0.966
⎝ 12 ⎠
Exploration:
2. Are the chords opposite to angles (ϕ − θ ) in the two graphs the same length?
Justify your answer. [Hint: You may want to refer to geometric theorems in
your justification.]
Yes, the chords are the same length because their opposite angles have the same
measure (in the same circle, or in two congruent circles, if two central angles
have the same measure then their chords are congruent). You can also argue
side-angle-side to prove ΔAOB ≅ ΔCOD , and thus all of the corresponding parts
of the triangles must be congruent.
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Creative Discovery Explorations in Precalculus
3. Write expressions for the lengths of the chords, in the two diagrams provided
using the coordinates of the end-points of each chord.
Since AB has endpoints ( x1 , y1 ) and (1, 0 ) ,
AB =
( x1 − 1) + ( y1 )
2
2
Since CD has endpoints ( x2 , y2 ) and ( x3 , y3 ) ,
CD =
( x2 − x3 ) + ( y2 − y3 )
2
2
4. Use the results of exercise 3 to derive an expression for cos (ϕ − θ ) . [Hint:
Replace the Cartesian coordinates with their equivalents in terms of sines and
cosines. Caution: It looks messy but works out beautifully.]
AB = CD
( x1 − 1) + ( y1 )
2
( x1 − 1) + ( y1 )
2
(x
2
1
)
2
2
( x2 − x3 ) + ( y2 − y3 )
2
=
= ( x2 − x3 ) + ( y2 − y3 )
2
(
2
2
) (
− 2 x1 + 1 + y12 = x22 − 2 x2 x3 + x32 + y22 − 2 y2 y3 + y32
)
Using ( x1 , y1 ) = ( cos (ϕ − θ ) ,sin (ϕ − θ ) ) ; ( x2 , y2 ) = ( cos θ ,sin θ ) ;
( x3 , y3 ) = ( cos ϕ ,sin ϕ ) , and the Pythagorean Theorem in trigonometric form,
the previous equation reduces to:
2 (1 − cos (ϕ − θ ) ) = 2 − 2 ( cos ϕ cos θ + sin ϕ sin θ )
or
cos (ϕ − θ ) = cos ϕ cos θ + sin ϕ sin θ
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Creative Discovery Explorations in Precalculus
⎛π ⎞
5. Use the result from exercise 4 to find an algebraic expression for cos ⎜ ⎟ .
⎝ 12 ⎠
Evaluate this expression and compare to a calculated or tabulated value.
π
12
=
π
4
−
π
6
; so
⎛π ⎞
⎛π π ⎞
cos ⎜ ⎟ = cos ⎜ − ⎟
⎝ 12 ⎠
⎝4 6⎠
⎛π ⎞
⎛π ⎞
⎛π ⎞ ⎛π ⎞
= cos ⎜ ⎟ cos ⎜ ⎟ + sin ⎜ ⎟ sin ⎜ ⎟
⎝4⎠
⎝6⎠
⎝4⎠ ⎝6⎠
⎛ 2 ⎞⎛ 3 ⎞ ⎛ 2 ⎞ ⎛ 1 ⎞
= ⎜⎜
⎟⎜
⎟⎟ + ⎜⎜
⎟⎟ ⎜ ⎟
⎟⎜
⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠
=
6+ 2
4
≈ 0.966
⎛π ⎞
Compare this value to the approximated value of cos ⎜ ⎟ from a calculator or a
⎝ 12 ⎠
trigonometric table.
6. Use the result from exercise 4 to find a formula for cos (ϕ + θ ) in terms of
trigonometric functions of ϕ and θ .
Note that cos (ϕ + θ ) can be written as cos (ϕ − ( −θ ) ) , and, due to the even and
odd characteristics of the functions, cos ( −θ ) = cos θ and sin ( −θ ) = − sin θ .
cos (ϕ + θ ) = cos (ϕ − ( −θ ) )
= cos ϕ cos ( −θ ) + sin ϕ sin ( −θ )
= cos ϕ cos θ − sin ϕ sin θ
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Creative Discovery Explorations in Precalculus
7. Find formulas for sin (ϕ + θ ) and sin (ϕ − θ ) . [Hint: Consider the cofunction
identities.]
π⎞
π⎞
⎛
⎛
Using the cofuction identities, sin α = cos ⎜ α − ⎟ and sin ⎜ α − ⎟ = − cos α :
2⎠
2⎠
⎝
⎝
π⎞
⎛
sin (ϕ + θ ) = cos ⎜ ϕ + θ − ⎟
2⎠
⎝
⎛
π ⎞⎞
⎛
= cos ⎜ ϕ + ⎜ θ − ⎟ ⎟
2 ⎠⎠
⎝
⎝
π⎞
π⎞
⎛
⎛
= cos ϕ cos ⎜ θ − ⎟ − sin ϕ sin ⎜ θ − ⎟
2⎠
2⎠
⎝
⎝
= cos ϕ sin θ + sin ϕ cos θ
A similar argument yields sin (ϕ − θ ) = cos ϕ sin θ − sin ϕ cos θ .
⎛ϕ ⎞
8. Make a conjecture as to how one could rewrite (2ϕ ) and ⎜ ⎟ in order to derive
⎝2⎠
identities using the sum and difference formulas in terms of sin ϕ and cos ϕ .
⎛ϕ ⎞
Then, find formulas for sine and cosine of angles (2ϕ ) and ⎜ ⎟ in terms of the
⎝2⎠
sine and cosine of ϕ .
⎛ϕ ϕ ⎞
Let ( 2ϕ ) = (ϕ + ϕ ) and ϕ = ⎜ + ⎟ ;
⎝2 2⎠
cos ( 2ϕ ) = cos (ϕ + ϕ )
= cos ϕ cos ϕ − sin ϕ sin ϕ
= cos 2ϕ − sin 2ϕ = 2 cos 2ϕ − 1 = 1 − 2 sin 2ϕ
Similarly, it can be found that sin ( 2ϕ ) = 2sin ϕ cos ϕ .
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Creative Discovery Explorations in Precalculus
⎛ϕ ⎞
Applying double angle results to ϕ = 2 ⎜ ⎟ yields, after some algebraic
⎝2⎠
manipulation:
1 + cos ϕ
1 − cos ϕ
⎛ϕ ⎞
⎛ϕ ⎞
cos ⎜ ⎟ = ±
and sin ⎜ ⎟ = ±
2
2
⎝2⎠
⎝2⎠
⎛π ⎞
9. Use the results above to find an algebraic expression for cos ⎜ ⎟ . Evaluate this
⎝ 24 ⎠
expression and compare your result to a calculated or tabulated value. [Hint: Use
the results from exercises 5 and 8.]
⎛π ⎞
π ⎜⎝ 12 ⎟⎠
Using the half-angle formula from exercise 8, and the substitution
=
:
24
2
⎛π ⎞
⎜ ⎟
⎛π ⎞
cos ⎜ ⎟ = cos ⎜ 12 ⎟
⎝ 24 ⎠
⎜ 2 ⎟
⎝ ⎠
=
=
1 + cos
π
12
2
6+ 2
⎛π ⎞
recall from exercise 5, cos ⎜ ⎟ =
4
⎝ 12 ⎠
4+ 6 + 2
8
≈ 0.991
10. Describe how one might use the identities you’ve just derived to aid in creating a
trigonometric table.
Answers will vary, but should include a comment on the ability to find
trigonometric values for all angles using the fact that the angles can be rewritten
as sums, differences, products or quotients of angles with known trigonometric
values.
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Creative Discovery Explorations in Precalculus
Instructor Support
Cos(ϕ −θ ) and Implications
Prerequisite Skills:
Students should be able to:
− Assign, recognize and utilize Cartesian coordinates of points on a unit circle in the
forms ( x1 , y1 ) and ( cos θ ,sin θ )
− Graph (and read the graph of) an angle in standard position on the unit circle
− Recognize proportionality relationships between chords and circles
− Find the length of a chord in a coordinate plane using the distance formula (or
Pythagorean Theorem)
− Describe the relationship between the distance formula and the Pythagorean
Theorem
− Rewrite trigonometric functions of negative angles using the odd and even
identities of the functions
− Rewrite the sine of an angle as the cosine of the angle (and vice versa) using the
cofunction identities
Goals and Objectives:
− Prove a trigonometric difference identity using two graphs of the same angle, the
Pythagorean Theorem, geometric properties, and algebraic manipulations
− Simplify trigonometric expressions using algebra
− Prove trigonometric sum, half angle and double angle identities using the found
difference identity and some basic properties of trigonometric functions
− Find exact values of trigonometric functions using sum, difference, half angle and
double angle identities
Teacher tips:
− Students should not have had any exposure to the sum, difference, half angle and
double angle identities prior to this exploration
− Encourage students to check their work with each other or with the instructor to
catch errors before they affect the outcome of the future exercises
− If students are struggling with exercise 5, refer them back to the Background
π
question or ask them why we might want to rewrite
as a difference of two
12
angles that have an exact value of sine and cosine which we know from the unit
circle
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Creative Discovery Explorations in Precalculus
Possible student misconceptions:
− Students should not assign specific values (in either degrees or radians) to ϕ or θ
when deriving the sum, difference, half angle and double angle identities
− Students need to work carefully in exercise 4 when performing the substitution of
cosines and sines for x’s and y’s, and when manipulating the equation to solve for
cos (ϕ − θ ) . Many of the exercises provide potential for simple errors in
algebraic manipulation.
− Students may have trouble connecting the task in exercise 5 to the result from
exercise 4.
− In exercise 6, students may try to restart the process by drawing new graphs in
order to solve for the cosine of the sum of two angles. Watch out for this and ask
students to rewrite the sum of two angles and the difference of two angles.
Connections to previous concepts:
− The Pythagorean Theorem
− Distance in a plane
− The geometry of circles
− Finding exact values for common trigonometric angles using the unit circle
− Basic properties of trigonometric functions and their graphs
Materials:
− Calculator, CAS, or a trigonometric table
Page 39
Creative Discovery Explorations in Precalculus
Five:
Graphing “Cartesian Functions” in Polar Coordinates
Doppler radar used on television to report weather conditions; radar screens used by air
traffic controllers to monitor aircraft traffic at an airport; flight plans filed by private
aircraft to indicate paths taken as they move from one point to another; sonar positioning
techniques employed in submarines; distances and compass bearings for directions used
by campers in wilderness areas – these are but a few examples of the occurrence of
vectors and polar coordinates in everyday life.
In this exploration we will use vectors to graph familiar equations in Cartesian
coordinates and compare those to the equivalent graph in polar coordinates. We will use
“( , )” for Cartesian coordinates and “ , ” for polar coordinates. In Cartesian
coordinates, a vector will represent the directed line segment from the point ( x, 0 ) to the
point ( x, f ( x) ) while in polar coordinates, a vector will represent the directed line
segment from the pole 0, 0 to the point f (θ ),θ . In the examples and exercises the
domain of the functions will be limited to the set of nonnegative real numbers. We will
explore both linear and quadratic expressions.
Exploration:
Linear Expressions
1. (Example for your consideration) We begin with a constant function y = c, where
c >0.
Figure 1a
Page 40
Creative Discovery Explorations in Precalculus
In this example, the vectors in Cartesian coordinates easily translate to vectors of
fixed length bound at the origin with the tip of the vector lying on a circle of
radius c.
Figure 1b
2. One should have little difficulty in graphing y = x , and can use this to interpret
what should take place with the graph of r = θ for θ ≥ 0 . Use the ‘vector
approach’ of the previous example as a guide to do this.
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Creative Discovery Explorations in Precalculus
Quadratic Expressions:
We next consider polar quadratic functions of the form: r = (θ − a )(θ − b ) , where
0 < a < b ; r = (θ − a ) , where a > 0 ; and r = θ 2 + aθ + b , where r (θ ) ≠ 0 for all θ .
2
3. Use the same ‘vector approach’ to graph y = x 2 − 3x + 2 = ( x − 1)( x − 2 ) and the
corresponding polar graph r = (θ − 1)(θ − 2 ) on the grids provided below (you
may have to adjust your scale on each axis). The vertex of the parabolic graph is
3
⎛3 1⎞
at ⎜ , − ⎟ with axis of symmetry at x = . For the polar graph, consider three
2
⎝2 4⎠
3
rays corresponding to the values of θ = 1 rad, θ = rad, and θ = 2 rad.
2
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Creative Discovery Explorations in Precalculus
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Creative Discovery Explorations in Precalculus
4. Consider a quadratic that has only one positive real root, y = x 2 − 4 x + 4 = ( x − 2 )
2
and the corresponding polar curve, r = θ 2 − 4θ + 4 = (θ − 2 ) . When constructing
2
the polar graph there is one important ray to consider, the ray θ = 2 rad. Use the
same ‘vector approach’ to explore the connection between these graphs in the two
systems.
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Creative Discovery Explorations in Precalculus
5. Next consider the quadratic, y = x 2 − 4 x + 8 which has no real roots and is
positive for all values of x. Perform the same systems exploration using the
Cartesian and polar grids below.
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Creative Discovery Explorations in Precalculus
Extension:
6. Use what you learned previously about Rectangular-Polar graphing connections
and the Cartesian graph provided in Figure 2 to create a ‘polar version’ of the
graph for 0 ≤ θ ≤ 6 .
Figure 2
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Creative Discovery Explorations in Precalculus
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Creative Discovery Explorations in Precalculus
Historical Note:
While ancient Greek mathematicians such as Archimedes made references to
functions of chord length that depended upon angles measured, it was a Persian
geographer, Abu Rayhan Biruni (circa 1000) who is credited with developing an early
foundation for a polar coordinate system. The polar coordinate system and known and
used today, however, is credited as having been developed by Issac Newton circa 1671,
and further refined and used by Jacob Bernoulli circa 1691.
References:
Eves, H.W. (1989). Introduction to the History of Mathematics 6th edition. New York:
Saunders Publishing.
Boyer (1949). Newton as the Originator of Polar Coordinates. The American
Mathematical Monthly, 56(2), 73-78.
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Creative Discovery Explorations in Precalculus
Instructor Solutions:
Graphing “Cartesian Functions” in Polar Coordinates
Exploration:
Linear Expressions
1. (Example for your consideration) We begin with a constant function y = c, where
c >0.
Figure 1a
In this example, the vectors in Cartesian coordinates easily translate to vectors of
fixed length bound at the origin with the tip of the vector lying on a circle of
radius c.
Page 49
Creative Discovery Explorations in Precalculus
Figure 1b
2. One should have little difficulty in graphing y = x , and can use this to interpret
what should take place with the graph of r = θ for θ ≥ 0 . Use the ‘vector
approach’ of the previous example as a guide to do this.
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Creative Discovery Explorations in Precalculus
Quadratic Expressions:
3. Use the same ‘vector approach’ to graph y = x 2 − 3x + 2 = ( x − 1)( x − 2 ) and the
corresponding polar graph r = (θ − 1)(θ − 2 ) on the grids provided below (you
may have to adjust your scale on each axis). The vertex of the parabolic graph is
3
⎛3 1⎞
at ⎜ , − ⎟ with axis of symmetry at x = . For the polar graph, consider three
2
⎝2 4⎠
3
rays corresponding to the values of θ = 1 rad, θ = rad, and θ = 2 rad.
2
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Creative Discovery Explorations in Precalculus
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Creative Discovery Explorations in Precalculus
4. Consider a quadratic that has only one positive real root, y = x 2 − 4 x + 4 = ( x − 2 )
2
and the corresponding polar curve, r = θ 2 − 4θ + 4 = (θ − 2 ) . When constructing
2
the polar graph there is one important ray to consider, the ray θ = 2 rad. Use the
same ‘vector approach’ to explore the connection between these graphs in the two
systems.
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Creative Discovery Explorations in Precalculus
5. Next consider the quadratic, y = x 2 − 4 x + 8 which has no real roots and is
positive for all values of x. Perform the same systems exploration using the
Cartesian and polar grids below.
r (θ ) = θ 2 − 4θ + 8
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Creative Discovery Explorations in Precalculus
Extension:
6. Use what you learned previously about Rectangular-Polar graphing connections
and the Cartesian graph provided in Figure 2 to create a ‘polar version’ of the
graph for 0 ≤ θ ≤ 6 .
Figure 2
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Creative Discovery Explorations in Precalculus
Instructor Support
Graphing “Cartesian Functions” in Polar Coordinates
Prerequisite Skills:
Students should be able to:
− Recognize an appropriate linear or quadratic graph from the equation
− Convert angle measures from radians to degrees
− Draw a vector that passes through two specific coordinate points
Goals and Objectives:
− Introduce vectors and polar coordinates through graphing familiar linear and
quadratic functions
− Reinforce the use of radian measures of angles
− Reinforce the use of planar vectors
− Compare the graphs of linear and quadratic expressions in Cartesian and polar
coordinates
− Provide students the opportunity to observe patterns in the polar graphs of linear
and quadratic expressions so they may formulate conjectures about the similarities
between the two systems
− Enable students to visualize the domain-range relationship of polar graphs in a
way that is connected to their prior understanding of function relationships in the
Cartesian plane
− Provide students with a solid foundation of understanding of polar coordinates
prior to the introduction of trigonometric functions, so they will be able to make
the transition into the classic polar relations and graphs that involve trigonometric
functions
Teacher tips:
− Depending on students’ prior exposure to polar graphing, it is recommending that
instructors spend time during the introduction of this exploration explaining the
convention used to plot points in the polar system
− The explorations of linear expressions can be expanded to include general
expressions in the form of y = ax + b
− All explorations can be extended to include negative numbers into the domainthis addition can lead to questions regarding symmetry and points of intersection,
for example
Possible student misconceptions:
− Students often struggle with the notation of polar coordinates; function
expressions usually written in the form of y = f ( x) are now written in the form
r = f (θ )
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Creative Discovery Explorations in Precalculus
− The roles of domain and range are reversed when points in the plane are assignedCartesian coordinates use ( x, y ) where y = f ( x) , while polar coordinates use
(r ,θ ) where r = f (θ )
− Students may try to sketch polar graphs as if the domain appears on a straight line.
The fact that the domain is still the set (or a subset of) the real numbers, yet no
longer appears along a straight line often causes confusion and insecurity in
drawing graphs
− Students may try to graph specific vectors corresponding to θ measured in
radians by arbitrarily guessing the rotation of 1 or 2 radians
− Students should convert the radians to degrees in order to produce an accurate
graph.
Connections to previous concepts:
− Basic functions, domain and range
− Polynomials and their roots
− Utilizing vectors to complete graphs of functions
Materials:
− Blank Cartesian and polar grids
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Creative Discovery Explorations in Precalculus
Six:
Difference Quotients and Rate of Change
In Exploration 3, you investigated some properties of exponential and logarithmic
functions. In this exploration we will use a situation modeled by an exponential function
to investigate the rate of change of that function. There will be two different types of rate
of change examined, the average rate of change of the function and the instantaneous
rate of change of the function. The average rate of change of a function gives
information about how the dependent variable of the function changes with respect to the
independent variable between two independent values of the function. This can be
expressed as the average rate of change of f, which is the quantity
f ( x ) − f (c )
,
x−c
where x is an independent value of choice and c is a particular value of interest in the
domain of f.
Background:
1. In what context have you encountered this equation in past mathematics classes?
The algebraic quantity,
f ( x ) − f (c )
x−c
is known as a difference quotient. We will eventually use difference quotients to help us
define and find the instantaneous rate of change of the function. That is, the rate of
change of a function f at one specific domain value of the function, namely x = c.
2. Is it possible to construct a difference quotient when x = c ? Explain your answer.
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Creative Discovery Explorations in Precalculus
Exploration:
We will examine a tire on a car being punctured by a sharp object. As the air escapes
from the tire the distance d, in inches, between the rim of the tire and the street is a
function of time t in seconds, where t ≥ 0 . From data collected, it was determined that
this situation can be modeled by the exponential equation
d (t ) = 6(1.490− t ) .
Notice that this equation suggests that, at t = 0, the rim was 6 inches above the ground.
Our goal is to estimate the rate of change of d(t) at precisely the time t = 3 seconds. This
will be accomplished by making use of the average rate of change and difference quotient
concepts that were previously defined.
3. On graph paper, make a table of values and plot the graph of d(t) for t ≥ 0 .
[Hint: You may want to use a graphing calculator or a software program to
compute the values.]
4. Find the average rate of change of d(t) between the values (a) t = 1 sec. and t = 3
sec. Also find the average rate of change of d(t) between the values (b) t = 2 sec.
and t = 3 sec. and between (c) t = 2.5 sec. and t = 3 sec.
5. What do the average rates of change found in exercise 4 represent with respect to
the rim of the tire and its proximity to the street as described in the problem set-up
of the exploration paragraph above?
6. Make a conjecture as to how one might use difference quotients to find out how
fast the rim of the tire is approaching the street at the instant t = 3 seconds. That
is, can we use difference quotients to determine the rate of change of d(t) at the
moment of t = 3 seconds?
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Creative Discovery Explorations in Precalculus
7. Find the average rates of change of d(t) on the t intervals (2.8, 3), (2.9, 3),
(2.95, 3), and (2.99, 3) seconds.
8. Notice that your answers for each interval supplied above seem to be approaching
what is known as a limit or limiting value. Can you estimate this value? Also, for
this application, what does this limiting value represent in terms of the motion of
the rim of the tire as air escapes from the tire?
9. What is the significance of the numerical sign of the limiting value in relation to
the motion of the rim of the tire?
Extension:
10. Plot the d(t) function on graph paper. On this graph, plot the d(1) and the d(3).
Draw a line through these points. Now do the same with the d(2) and the d(3).
Lastly, do the same with the d(2.5) and the d(3). These lines that you have drawn
are known a secant lines. What is the slope of each of these lines? Have you seen
these values previously in this exploration?
11. Find the equation of the secant line through d(2.99) and d(3).
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Creative Discovery Explorations in Precalculus
12. Explain what would happen to the proximity of these secant lines to the d(t) curve
if you plot points on d(t) closer and closer to the d(3) value and draw secant lines
from each of these points through the d(3).
13. Construct a tangent through (3, d (3)) using the limiting value as the slope of the
tangent line. Compare your result to your answer in exercise 11.
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Creative Discovery Explorations in Precalculus
Historical Note:
The concept of the limiting value has been present in mathematics for over two
thousand years. Archimedes (circa 250BC), mentioned in Exploration Four, used a
limiting process to estimate the value of π . Both Isaac Newton (circa 1700) and
Gottfried Leibnitz (circa 1700 - Leibnitz is also mentioned in an earlier Exploration 1)
made use of the limit concept in each of their individual developments of the Calculus.
However, it was not until the mid-eighteen hundreds that Augustin-Louis Cauchy applied
rigorous mathematical meanings to the concept of limit. Cauchy’s ε − δ definition of
limit is still in standard use today.
The concept of instantaneous rate of change of a function was addressed and
formalize chiefly by Gottfried Leibnitz and Issac Newton in their works relating to the
development of the Calculus. Other mathematicians of the time that contributed to this
topic were Pierre de Fermat, Rene Descartes (mentioned in Exploration 1), Christian
Huygens, and Isaac Barrow. The development of this concept was born out of
investigations involving continuous functions by these mathematicians having to do with
the motion of objects, the tangent line problem, maximum-minimum problems, and area
problems associated with non-standard shapes.
References:
Burton, D. (1991). History of Mathematics, An Introduction. Dubuque, IA: Wm. C.
Brown Communications.
Cajori, F. (1931). A History of Mathematics 2nd edition. New York, NY: Chelsea.
Larson, Hoestetler, Edwards (1994). Calculus With Analytic Geometry. Lexington, MA:
D.C. Heath and Co.
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Creative Discovery Explorations in Precalculus
Instructor’s Solutions:
Difference Quotients and Rate of Change
Background:
1. In what context have you encountered this equation in past mathematics classes?
This equation is used to represent the slope of a line on a coordinate plane
passing through the two points (c, f (c )) and ( x, f ( x )) .
2. Is it possible to construct a difference quotient when x = c ? Explain your answer.
No, the denominator will have a value of zero so the quotient will be in an
indeterminate form.
Exploration:
3. On graph paper, make a table of values and plot the graph of d(t) for t ≥ 0 .
[Hint: You may want to use a graphing calculator or a software program to
compute the values.]
Sample values may include:
d
-0.2000
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.4000
1.6000
1.8000
2.0000
2.2000
2.4000
2.6000
2.8000
3.0000
3.2000
d(t)
6.4981
6.0000
5.5401
5.1154
4.7232
4.3612
4.0268
3.7182
3.4331
3.1700
2.9270
2.7026
2.4954
2.3041
2.1275
1.9644
1.8138
1.6748
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Creative Discovery Explorations in Precalculus
4. Find the average rate of change of d(t) between the values (a) t = 1 sec. and t = 3
sec. Also find the average rate of change of d(t) between the values (b) t = 2 sec.
and t = 3 sec. and between (c) t = 2.5 sec. and t = 3 sec.
(a) av. rate of change between t = 1 sec. and t = 3 sec is approx.
4.0268 − 1.8138
= −1.107 inches/sec
1− 3
(b) av. rate of change between t = 2 sec. and t = 3 sec is approx.
2.7026 − 1.8138
= −0.889 inches/sec
2−3
(c) av. rate of change between t = 2.5 sec. and t = 3 sec is approx.
2.2140 − 1.8138
= −0.800 inches/sec
2.5 − 3
5. What do the average rates of change found in exercise 4 represent with respect to
the rim of the tire and its proximity to the street as described in the problem set-up
of the exploration paragraph above?
These values represent approximately how fast the rim of the tire is approaching
the ground.
6. Make a conjecture as to how one might use difference quotients to find out how
fast the rim of the tire is approaching the street at the instant t = 3 seconds. That
is, can we use difference quotients to approximate the instantaneous rate of
change of d(t) at t = 3 seconds?
Use difference quotients consisting of function values associated with t values
closer and closer to t = 3, along with t = 3.
7. Find the average rates of change of d(t) on the t intervals (2.8, 3), (2.9, 3),
(2.95, 3), and (2.99, 3) seconds.
(a) av. rate of change between t = 2.8 sec. and t = 3 sec is approx.
- 0.753 inches/sec.
(b) av. rate of change between t = 2.9 sec. and t = 3 sec is approx.
- 0.738 inches/sec.
(c) av. rate of change between t = 2.95 sec. and t = 3 sec is approx.
- 0.731 inches/sec.
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Creative Discovery Explorations in Precalculus
(d) av. rate of change between t = 2.99 sec. and t = 3 sec is approx.
- 0.730 inches/sec.
8. Notice that your answers for each interval supplied above seem to be approaching
what is known as a limit or limiting value. Can you estimate this value? Also, for
this application, what does this limiting value represent in terms of the motion of
the rim of the tire as air escapes from the tire?
This limiting value is approximately - 0.73 inches/sec. This value is the
instantaneous rate of change of the rim height at t = 3. This is the velocity at
which the rim is moving after 3 seconds have passed.
9. What is the significance of the numerical sign of the limiting value in relation to
the motion of the rim of the tire?
The negative value indicates that the rim is moving closer to the ground at t = 3
seconds.
For Further Investigation
10. Plot the d(t) function on graph paper. On this graph, plot (a) d(1) and d(3).
Draw a line through these points. Now do the same with (b) d(2) and d(3). Lastly,
do the same with (c) d(2.5) and d(3). These lines that you have drawn are known
a secant lines. What is the slope of each of these lines? Have you seen these
values previously in this exploration?
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Creative Discovery Explorations in Precalculus
(a) The slope of the line from d(1) to d(3) is
approx.
4.0268 − 1.8138
≈ −1.107
1− 3
(b) The slope of the line from d(2) to d(3) is
approx.
2.7026 − 1.8138
≈ −0.889
2−3
(c) The slope of the line from d(2.5) to d(3) is
approx.
2.2140 − 1.8138
≈ −8.00
2.5 − 3
These values are the same as the average rate of change over each time interval
found in exercise 4.
11. Find the equation of the secant line through d(2.99) and d(3).
The slope of the line through d(2.99) and d(3) is approx.
1.8211 − 1.8138
≈ −0.73
2.99 − 3
The equation of the secant line, in slope-intercept, form can be found by:
y − 1.8138 = −0.73 ( x − 3)
y = −0.73 x + 4.0083
12. Explain what would happen to the proximity of these secant lines to the d(t) curve
if you plot points on d(t) closer and closer to the d(3) value and draw secant lines
from each of these points through the d(3).
The secant lines will approach the line tangent to d(t) at the point (3, d(3)).
13. Construct a tangent through (3, d (3)) using the limiting value as the slope of the
tangent line. Compare your result to your answer in exercise 11.
The line constructed in exercise 11 is approximately the same as the tangent line.
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Creative Discovery Explorations in Precalculus
Instructor Support
Difference Quotients and Rate of Change
Prerequisite Skills:
Students should be able to:
− Compute values, using technology, of an exponential function
− Sketch a graph of an exponential function from a table of values
− Substitute values from a graph into a provided formula
− Formulate conjectures and draw conclusions regarding a provided real-world
situation
− Find the equation of a line from two points on the line
− Work with exponential growth and decay functions
Goals and Objectives:
− Expose Precalculus students to a topic of Calculus using their prior mathematical
knowledge
− Provide students the opportunity to make a connection between the slope of a line
and the average rate of change of a function
− Introduce the concept of a limit or limiting value
− Demonstrate the difference and correlation between average rate of change versus
instantaneous rate of change, and secant lines versus tangent lines
Teacher tips:
− Students may be tempted to use their graphing calculators or computer software to
draw a graph of the exponential equation. Please discourage them from using
these capabilities and simply ask them to use the technology to compute values of
the function.
− Ask students to round (or use significant figures) with enough accuracy to
illustrate the limiting value without having it be reached until extremely close to
d(3).
Possible student misconceptions:
− Students may think that simple substitution will suffice in finding the limiting
f (c ) − f ( c ) 0
value, and claim
= =1
c−c
0
Connections to previous concepts:
− The slope of a line
dist. ⎞
⎛
− Average rate of change ⎜ rate =
⎟
time ⎠
⎝
− Non-polynomial functions
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Creative Discovery Explorations in Precalculus
Materials:
− Calculator or CAS
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Creative Discovery Explorations in Precalculus
Seven:
A Number Between 2 and 3
In this activity we will investigate an accumulation function related to the algebraic
measure of the area under the curve of the function
1
f (t ) = .
t
We will refine our activities to the interval t ∈ [1, 3] .
Exploration:
1. On graph paper create coordinate axes with a scale 0.1 unit on each axis. The
values on the f (t ) axis should range from 0 to 1.1, and the values on the t axis
should range from 0 to 3.1. Plot the graph of f (t ) on the domain interval [1, 3] by
plotting at least 8 evenly-spaced values on the interval. Carefully draw a
continuous curve through the values.
2. The function that represents the accumulated area under f (t ) on the interval [1, x]
where x ∈ [1,3] will be called L(x). What is the value of L(1)?
3. With a scale of 0.1 on each of your axes, what is the value represented by the area
of each square of the grid on your graph paper? Use this fact to approximate the
value of L(2).
4. Approximate the value of L(3).
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Creative Discovery Explorations in Precalculus
5. Use what you have discovered so far to estimate the value for x ∈ [1,3] such that
L(x) = 1. Do you know the special name given to this value?
Research Extension:
6. Since L(x) is a continuous function, there is a theorem from calculus that
guarantees that there is a unique value x ∈ [1,3] such that L(x) = 1. Research this
theorem and explain how it applies to this problem.
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Creative Discovery Explorations in Precalculus
Historical Note:
The mathematician John Napier (circa 1600) is credited as being the first to
introduce the number e. Napier alluded to e as a “special number” associated with his
development of the theory of logarithms. It is Leonard Euler (circa 1720), however, that
defined and used the symbol e to refer to Napier’s “special number”. Euler also
discovered many of this number’s special properties. It is likely that Euler chose the
symbol e for this natural number as a reference to the “exponential”.
The irrational number e is used extensively in the mathematics associated with
finance and economics. In fact the number can be derived from exploring a special case
of the Amortization Formula
nt
⎛ r⎞
A = P ⎜1 + ⎟ ,
⎝ n⎠
Where A is the total value of an investment after t years, P is the initial or principal
investment, r is the rate of interest, and n is the number of times that the interest is
compounded per year. You are encouraged to research the connection between e and the
mathematics of finance.
References:
Burton, D. (1991). History of Mathematics, An Introduction. Dubuque, IA: Wm. C.
Brown Communications.
Maor, E. (1994), “e”: The Story of a Number. Princeton, NY: Princeton Publishing.
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Creative Discovery Explorations in Precalculus
Instructor Solutions:
A Number between 2 and 3
Exploration:
1. On graph paper create coordinate axes with a scale 0.1 unit on each axis. The
values on the f (t ) axis should range from 0 to 1.1, and the values on the t axis
should range from 0 to 3.1. Plot the graph of f (t ) on the domain interval [1, 3] by
plotting at least 8 evenly-spaced values on the interval. Carefully draw a
continuous curve through the values.
2. The function that represents the accumulated area under f (t ) on the interval [1, x]
where x ∈ [1,3] will be called L(x). What is the value of L(1)?
L (1) = 0
3. With a scale of 0.1 on each of your axes, what is the value represented by the area
of each square of the grid on your graph paper? Use this fact to approximate the
value of L(2).
Area = 0.01 units 2
L (2) ≈ 0.67
(Answers will vary but should be close to this approximation)
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Creative Discovery Explorations in Precalculus
4. Approximate the value of L(3).
L (3) ≈ 1.08
(Answers will vary but should be close to this approximation)
5. Use what you have discovered so far to estimate the value for x ∈ [1,3] such that
L(x) = 1. Do you know the special name given to this value?
x ≈ 2.7 ≈ e
The exact value of x for which L(x) = 1 is the number e (Euler’s constant).
Research Extension:
6. Since L(x) is a continuous function, there is a theorem from calculus that
guarantees that there is a unique value x ∈ [1,3] such that L(x) = 1. Research this
theorem and explain how it applies to this problem.
Intermediate Value Theorem:
If f is continuous on [ a, b] and k is a value between f ( a ) and f (b) then
there must be a number, c, in
[ a, b] such that
f (c ) = k .
In this case, L(x) is a continuous function on [1,3] , L (1) = 0 , and L (3) ≈ 1.08
(this value may vary depending on student approximations from exercise 4).
Since the k-value in this exercise, 1, is between L(1) and L(3), the IVT guarantees
that there exists a unique value x ∈ [1,3] , such that L ( x ) = 1 .
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Creative Discovery Explorations in Precalculus
Instructor Support
A Number Between 2 and 3
Prerequisite Skills:
Students should be able to:
− Draw and label an accurate graph of a function by plotting specific points
Goals and Objectives:
− Expose Precalculus students to a topic of Calculus using their prior mathematical
knowledge
− Introduce the problem (to be solved through Calculus) of finding the exact area
under a curve
− Find an approximation for the area under a curve using the sum of the areas of
rectangles
Teacher tips:
− Students might need some guidance in recognizing the Intermediate Value
Theorem should be the topic of research in exercise 6.
Possible student misconceptions:
− Students may approximate the area under the curve between 2 and 3 (rather than
the accumulation of the area between 1 and 3) when approximating a value of
L(3).
Connections to previous concepts:
− Basic graphing techniques
− Calculating the area of simple geometric figures
Materials:
− Graph paper
− Calculus resource
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Creative Discovery Explorations in Precalculus
Conclusion
Backhouse et al. (1992) share the sentiments of many mathematics educators
when they state their belief that, “. . . investigations in mathematics should be part of
every learner’s experience, otherwise they will have missed the opportunity of meeting
one of the central characteristics of the subject” (p.139). The primary goal of this project
was to edit and enhance a product with helpful tools that will allow Precalculus teachers
to incorporate such exploratory investigations into their mathematics classrooms.
The explorations were edited for accuracy, consistency in voice, and
consideration for the level of student ability. For each exploration, the added section
called Instructor Support outlines:
•
the goals and objectives
•
the prerequisite skills required for successful completion
•
the connections to previous material in secondary or post secondary
curricula
•
a list of necessary materials
•
potential errors to look for in student work
•
additional tips to the teacher for smooth implementation
In order to compile the Instructor Support material for each exploration, I
consulted with two of the original contributors to help determine the specific desired
goals and objectives. I also interviewed several current Precalculus teachers who shared
their experiences of common student misconceptions, and I referred to several
Precalculus text books for connections to previous concepts and for compiling the list of
prerequisite skills. Witnessing and experiencing the amount of time and effort required
to compile and refine the workbook proves not only the need for the accessibility of such
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Creative Discovery Explorations in Precalculus
teacher support, but also the daunting task ahead to change mathematics education to
better prepare our students to use mathematics, inquiry and problem solving in their
futures.
In order to accomplish this project, I approached each exploration from the
perspective of a student with little or no prior knowledge of the material, and also as an
instructor looking to supplement a Precalculus course with investigative work. Thinking
as a student inspired clarification of background material and the creation of questions
designed to gently lead students into the process of discovery for themselves. With the
goal of creating a finished product useful to teachers with all levels of experience, I had
to consider the needs of a new teacher, potentially intimidated to stray from a textbook,
as well as a veteran teacher, perhaps reluctant to change the dynamic of a traditional math
classroom into a less structured environment conducive to inquiry, discovery, and
reasoning. The practices of viewing each classroom activity through the eyes of my
students, along with working on the creative side of curricular materials, were both very
valuable experiences to me as a professional.
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Creative Discovery Explorations in Precalculus
References
Backhouse, J., Haggarty, L., Pirie, S., & Stratton, J. (1992). Improving the Learning of
Mathematics. London: Cassell.
Boaler, J. (2002). Experiencing School Mathematics: Traditional and Reform
Approaches to Teaching and Their Impact on Student Learning, Revised and
Expanded Edition. Mahwah, NJ: Lawrence Erlbaum Associates.
Glenn, J. et al. (2000). Before It’s Too Late: A Report to the Nation from The National
Commission of Mathematics and Science Teaching for the 21st Century. Jessup,
MD: Educational Publications Center.
Retrieved July 19, 2009, from:
http://www.ed.gov/inits/Math/glenn/report.pdf
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Creative Discovery Explorations in Precalculus
Vita
Lesley Kaheana Johnson was born in Honolulu, Hawai`i on September 22nd,
1980, the daughter of Wendy Brandt Johnson and William Paul Johnson.
After
completing her work at Punahou School, Honolulu, Hawai`i, in 1998, she entered The
College of William & Mary in Williamsburg, Virginia. She received the degree of
Bachelor of Science in Applied Mathematics with a minor in Biology from The College
of William & Mary in 2002. Since that time she has been employed as a secondary
mathematics teacher by Le Jardin Academy. In the summer of 2007 she entered the
Graduate School at the University of Texas at Austin.
Permanent address:
1179 Lunahaneli Pl.
Kailua, HI 96734
This report was typed by Lesley Kaheana Johnson.
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