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[1] Math in the Life of a 99-Year-Young Quilt MakerJackie Giles
Texas
Southern University
Flora Jackson, a 99-year-young lady, quilts while watching NFL and NBA sports
into the wee hours of the night. Her quilting and interest in sports has
helped make her sharp mind, sharper. This sewer of patterns has lived a rich
life during her nearly 100 year journey in America. Born October 15, 1909 in
Pitts View, Alabama (now living in Cleveland, Ohio), she raised eight children
of her own and, as a midwife, assisted in the delivery of fifty
babies.
Flora Jackson has sown more than 100 original quilts, and is still going
strong. Although never formally educated in mathematics beyond arithmetic, her
understanding of geometrical concepts is clearly evident in her quilt patterns. Several were filled with reflections, rotations, and
scaling. Never having studied the theory of rotational matrices, she saw
beauty in transformations and designed her quilts to give a beautiful visual
impression. One quilt was made with triangles and squares. The triangles were
inscribed in a square, four in each square, and each one was rotated 90
degrees to yield the position of the next one. In the world of matrix algebra,
this transformation can be represented by
I know what you are saying, "Mrs. Jackson does not know what you are talking
about." I would answer, "Mrs. Jackson does not need to know my language. Her
language is expressed in the design of each and every one of her
quilts."
She told me that it takes her five weeks to make a quilt, working about eight
hours per weekday designing, sewing and finishing her mathematical artwork.
Flora charges only $250 for one of her fine quilts. This means that she is
making $50 per week for each quilt. An interesting computation is to figure
out her hourly wage, assuming a forty hour work week, and compare it to the
official minimum wage. How much should she charge for one of her quilts if she
were to receive the minimum wage for her work? This beautiful, creative, and wise woman hums her favorite hymn "Amazing Grace" and when asked for her favorite scripture replies Psalm 119:105: "Thy word is a lamp unto my feet, and a light unto my path." Her words of advice to young people: "Tend to your business, and let everybody else's alone." [2] Interpreting Slope
The concept of rate of change is central to any study of change. Thus the
concept of slope, as a rate of change, deserves a full class discussion,
possibly seeded with several "Explain" type questions. The following list of
statements and questions are offered as stimulants to an in-depth discussion
of slope. \Slope is presented graphically as rise over run, in function language as change in output over change in input, and algebraically as an average rate of change. How can a quotient represent a rate of change? \When $100 is invested at 4% annually, the plot of interest versus time (years) is a straight line. How is the slope of the line interpreted in terms of the investment? \Hook's Law says that (for small displacements) the restorative force of a spring is proportional to the spring's displacement (i.e., the length that the spring has been stretched or compressed). Thus the force is a linear function of the spring's displacement and the corresponding graph is a straight line. How is the slope of the line interpreted in terms of the spring? \Explain why a linear function is said to be increasing (decreasing) if its slope is positive (negative). \A distance (ft) versus time (sec) of a car that speeds up from a stop position is probably not a straight line. Suppose (2,20) and (10, 800) are two points on the plot. The average rate is (800-10)/(10-2) is approximately 75 mph. This average rate is the slope of what line? \What is the difference between average rate of change and instantaneous rate of change? What does the speedometer of your car register?
\Slope,
average rate of change, is associated with a straight line. Would it make
sense to speak of the instantaneous rate of change of a curve at a point as
the slope of the line tangent to the curve at that point?
Explain. [3] Polynomials
The purpose of this exercise is to let students discover the relationships
between the roots and coefficients of a quadratic polynomial equation. It is
assumed that students understand the terms factor and root
and their relation to one another. Begin by computing the following: a. Determine the quadratic equation whose roots are 2 and 3.
b. Determine the quadratic equation whose roots are
c. Determine the quadratic equation whose roots are
d. Determine the quadratic equation whose roots are 2 and
Based on the results for parts a, b, c, and d, conjecture the relationship
between the coefficients in the quadratic polynomial equation
Follow-on exercise: Experiment, conjecture, and verify the relationships between the roots and the coefficients of a cubic polynomial equation. [4] Planning a Reunion DinnerCandy Hodges
Bethune-Cookman University
A committee has been formed to plan for a Refocused College Algebra Reunion
dinner for the Joint Mathematics Meetings scheduled for January 2010. The
committee is working to determine a ticket price which will cover the banquet
room rental cost of $750 and the $20 (each) dinner charge. The committee also
plans to price the tickets so that the three guests of honor - Don Small,
Archie Wilmer, and Tony Johnson - will not pay to attend. The committee began by completing a table, realizing that if 30 people attended the reunion dinner, only 27 would be charged for tickets. Complete the following table after being certain that you understand the computations for the row the committee completed.
Write a function for the cost per paying attendee,
Sketch the graph of the cost function on a 200 by 200 grid, being sure to
label the axes. (The grid is omitted.) Note To Instructor. Once the tasks above are completed, a discussion about the equation of the asymptote and the lowest possible value for the cost per paying attendee can be discussed.
Candy Hodges: "I have a lot of fun in class with the reunion dinner problem.
We talk about keeping the price down and "play" the --I'll bet if 200 people
come, THAT will get the price under $20 --Maybe if 250 come, THAT will get the
price under $20. etc. It is eventually fun to talk about the equations of the
asymptotes, and, for example, though the one is 3, how within the context of
the problem, actually FOUR people must come (3 guests of honor PLUS ONE) and
that fourth person will not be pleased to have to pay for his and the three
guests of honors tickets. It's just a lot of fun!"
[5] Queries
1. (Volume versus Area) Can the surface area of a cube ever be the same as the
volume of the cube? If so, give the dimensions of such a cube. If not, explain
why not.
2. (Volume versus Area) Can the surface area of a sphere ever be the same as
the volume of the sphere? If so, give the radius of such a sphere. If not,
explain why not.
3. (Area versus Circumference) Can the area of a circle ever be the same as
the length of the circumference of that circle? If so, give the radius of such
a circle. If not, explain why not.
4. (Volume versus Label Area) Can the volume of a cylindrical can be the same
as the label area of the can? (Label area is the surface area of the can not
including the top or bottom of the can.) [6] Wolfram\Alpha Website
The website for computations, Wolfram\Alpha, is (should be) raising questions
in the minds of college algebra instructors, as well as others, as to how this
website will effect their teaching. Not only does this site provide answers to
the computational questions found in traditional college algebra textbooks, it
also provides an explanation of the process for determining the answer. Check
out the site by entering http://WolframApha.com and clicking on Wolfram\Alpha.
Experiment by posing questions (e.g., type Solve
(
The site is much more than a computational site for mathematics. The mission
statement states "Wolfram\Alpha's long-term goal is to make all systematic
knowledge immediately computable and accessible to everyone. We aim to collect
and curate all objective data; implement every known model, method, and
algorithm; and make it possible to compute whatever can be computed about
anything. Our goal is to build on the achievements of science and other
systematizations of knowledge to provide a single source that can be relied on
by everyone for definitive answers to factual queries.
Wolfram\Alpha aims to bring expert-level knowledge and capabilities to the
broadest possible range of people---spanning all professions and education
levels. Our goal is to accept completely free-form input, and to serve as a
knowledge engine that generates powerful results and presents them with
maximum clarity.
Wolfram\Alpha is an ambitious, long-term intellectual endeavor that we intend
will deliver increasing capabilities over the years and decades to come. With
a world-class team and participation from top outside experts in countless
fields, our goal is to create something that will stand as a major milestone
of 21st century intellectual achievement." [7] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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