Working in small groups is typical of most work environments, and learning
to work with others to communicate specific ideas is an important skill. Work
with three or four other students to submit a single report based on each of
the following questions.
It is stated in the text that
"Mathematics is alive and constantly changing. As we complete the last decade
of this century, we stand on the threshold of major changes in the mathematics
curriculum in the United States." Report on some of these recent changes.
References
Lynn Steen, Everybody Counts: A Report to the Nation on the Future
of Mathematics Education (Washington, D.C.: National Academy Press, 1989).
See also, Curriculum and Evaluation Standards for School Mathematics from
the National Council of Teachers of Mathematics (Reston, VA: NCTM, 1989).
Do some research on
Pascal's triangle, and see how many properties you can discover. You might
begin by answering these questions:
- What are the successive powers of 11?
- Where are the natural numbers found in Pascal's triangle?
- What are triangular numbers and how are they found in Pascal's
triangle?
- What are the tetrahedral numbers and how are they found
in Pascal's triangle?
- What relationships do the patterns in Figure G.1 have to
Pascal's triangle?
Figure G.1 Patterns in Pascal's triangle
References
James N. Boyd, "Pascal's Triangle," Mathematics Teacher,
November 1983, pp. 559-560.
Dale Seymour, Visual Patterns in Pascal's Triangle, (Palo Alto,
CA: Dale Seymour Publications, 1986)
Karl J. Smith, "Pascal's Triangle," Two-Year College Mathematics
Journal, Volume 4 (Winter 1973).
Invent an original numeration system.
Organize a debate. One side represents the algorists and the other side the
abacists. The year is 1400. Debate the merits of the Roman numeration system
and the Hindu-Arabic numeration system.
Reference:
Barbara E. Reynolds, "The Algorists vs. The Abacists:
An Ancient Controversy on the Use of Calculators," The College Mathematics
Journal, Vol. 24, No. 3, May 1993, pp. 218-223. Includes additional references.
Organize a debate. The issue: "Resolved: Computers can think."
Problem G6 in Algebra.
In a now famous paper, Alan Turing asked, "What would we ask a computer to
do before we would say that it could think?" In the 1950s Turing devised a
test for "thinking" that is now known as the turing test. Dr. Hugh Loebner,
a New York philanthropist, has offered $100,000 for the first machine that
fools a judge into thinking it is a person. In 1991, the Computer Museum in
Boston held a contest in which 10 judges at the museum held conversations
on terminals with eight respondents around the world, including six computers
and two humans. The conversations of about 15 minutes each were limited to
particular subjects, such as wine, fishing, clothing, and Shakespeare, but
in a true turing test, the questions could involve any topic. Work as a group
to decide the questions you would ask. Do you think a computer will ever be
able to pass the test?
References
Betsy Carpenter, "Will Machines Ever Think?" U.S. News & World
Report, October 17, 1988, pp. 64-65.
Stanley Wellborn, "Machines That Think," U.S. News & World Report,
December 5, 1983, pp. 59-62.
Construct an exhibit on ancient computing methods. Some suggestions for your
exhibit are charts of sample computations by ancient methods, pebbles, tally
sticks, tally marks in sand, Roman number computations, abaci, Napier's bones,
and old computing devices. You should consider answering the following questions
as part of your exhibit: How do you multiply with Roman numerals? What is
the scratch system? What is the lattice method of computation? What changes
in our methods of long multiplication and long division have taken place over
the years? How did the old computing machines work? Who invented the slide
rule?
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Four Fours Write the numbers from 1 to 100 (inclusive) using exactly
four fours. See Problem 60, Problem Set 4.3, x-ref ok to help you get started.
Pythagorean Theorem Write out three different proofs of the Pythagorean
theorem.
Modular Art Many interesting designs such as those shown here can be
created using patterns based on modular arithmetic. Prepare a report for class
presentation based on the article "Using Mathematical Structures to Generate
Artistic Designs" by Sonia Forseth and Andrea Price Troutman, The Mathematics
Teacher, May 1974, pp. 393-398. Another source is "Mod Art: The Art of Mathematics"
by Susan Morris, Technology Review, March/April 1979.
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Prove that there are infinitely many integers such that the sum of the digits
in their square equals the sum of the digits in their cube.
JOURNAL PROBLEM (From Journal of Recreational Mathematics,
Vol. II, #2) Translate the following message: Wx utgtuz f pbkz tswx wlx xwozm
pbkzr, f exbmwo cxlzm xm ts jzszmfi fsv cxlzm lofwzgzm tswx wlx cxlzmr xe
woz rfnz uzsxntsfwtxs fkxgz woz rzpxsu tr tncxrrtkiz, fsu T ofgz frrbmzuiv
exbsu fs funtmfkiz cmxxe xe wotr kbw woz nfmjts tr wxx sfmmxl wx pxswfts tw.
Ctzmmz Uz Ezmnfw
The entrance of the Aquarium of Americas in New Orleans has a gigantic building-size
curve called a logarithmic spiral. Find out how to construct a logarithmic
spiral, and write a paper about what you learned. Why do you suppose it would
appear on the front of an aquarium?
If we assume that the world population grows exponentially, then it is also
reasonable to assume that the use of some nonrenewable resource (such as petroleum)
will also grow exponentially. In calculus, it is shown that for some constant
k, under these assumptions, the formula for the amount of the resource,
A, consumed from time t = 0 to t = T is given by the
formula
where r is the relative growth rate of annual consumption.
a. Solve this equation for T to find a formula for life expectancy
of a particular resource.
b. According to the Energy Information Administration, the annual world
production (in billions of barrels per day) of petroleum is shown in the
following table:
| Year: |
1975 |
1980 |
1985 |
1990 |
1995 |
2000 |
2003 |
| Quantity: |
52.42 |
62.39 |
52.97 |
60.90 |
61.85 |
66.03 |
67.00 |
Find an exponential equation for these data.
c. If in 1998, the world petroleum reserves are 2.8 trillion barrels,
estimate the life expectancy for petroleum.
Suppose you have just inherited $30,000 and need to decide what to do with the
money. Write a paper discussing your options and the financial implications
of those options. The paper you turn in should offer several alternatives and
then the members of your group should reach a consensus of the best course of
action.
Write a short paper about Fibonacci numbers. You might check The Fibonacci
Quarterly, particularly "A Primer on the Fibonacci Sequence," Parts I
and II, in the February and April 1963 issues. The articles, written by Verner
Hogatt and S. L. Basin, are considered classic articles on the subject. One
member of your group should investigate the relationship of the Fibonacci
numbers to nature, another the algebraic properties of the sequence, and another
the history of the sequence.
Suppose you were hired for a job paying $21,000 per year and were given the
following options:
OPTION A: Annual salary increase of $1,200
OPTION B: Semiannual salary increase of $300
OPTION C: Quarterly salary increase of $75
OPTION D: Monthly salary increase of $10
Each person should write the arithmetic series for the total
amountofmoneyearnedin10 years under a different option.
Your group should reach a consensus as to which is the best
option. Give reasons and show your calculations in the paper that your group
submits.
It is not uncommon for the owner of a home to receive a letter similar
to the one shown below. Write a paper based on this letter.
Different members of your group can work on different parts of the question,
but you should submit one paper from your group.
a. What is the letter about?
A computer printout (above) was included with the letter. Assuming that these
calculations are correct, discuss the advantages or disadvantages of accepting
this offer.
c. The plan as described in the letter costs $375 to sign up.
I called the company and asked what their plan would do that I could not do
myself by simply making 13 payments a year to my mortgage holder. The answer
I received was that the plan would do nothing more, but the reason people
do sign up is because they do not have the self-discipline to make the midmonthly
payments to themselves. Why is a biweekly payment equivalent to 13 annual
payments instead of equivalent to a monthly payment?
d. The representative of the company told me that more than
250,000 people have signed up. How much income has the company received from
this offer?
e.You calculated the income the company has received from this
offer in part d, but that is not all it receives. It acts as a bonded and
secure "holding company" for your funds (because the mortgage company does
not accept "two-week" payments). This means that the
company receives the use (interest value) on your money for
two weeks out of every month. This is equivalent to half the year. Let's assume
that the average monthly payment is $1,000 and that the company has 250,000
payments that they hold for half the year. If the interest rate is 5% (a secure
guaranteed rate), how much potential interest can be received by this company?
Investigate the topic of conic sections. Build models and/or find three-dimensional
models for the conic sections. What did the Greeks know of the conic sections?
Prepare a list of women mathematicians from the history of mathematics. Answer
the question, "Why were so few mathematicians female?"
Reference:
Teri Perl, Math Equals: Biographies of Women Mathematicians
plus Related Activities. (Reading, MA: Addison-Wesley Publishing Co.,
1978).
Loretta Kelley, "Why Were So Few Mathematicians Female?" The Mathematics
Teacher, October 1996. cBarbara Sicherman and Carol H. Green, eds.
Notable American Women: The Modern Period. A Biographical Dictionary. (Cambridge,
MA: Belknap Press, Harvard University Press, 1980).
Outstanding Women in Mathematics and Science (National Women's History
Project, Windsor, CA 95492, 1991).
Prepare a list of black mathematicians from the history of mathematics.
Reference:
Virginia Newell et al., eds. Black Mathematicians
and Their Works (Ardmore, PA: Dorrance & Company, 1980).
Prepare a list of mathematicians with the first name of Karl.
Write a news article about a historical mathematician as if you were a contemporary
of the person you are writing about. Put it in newspaper style and include
other newsworthy items from the period.
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What exactly are fractals?
To get you started on your paper, we ask the following question
that relates the ideas of series and fractals using the snowflake curve. Cut
an equilateral triangle of side a out of paper, as shown in Figure G.a. Next,
three equilateral triangles, each of side a/3, are cut out and placed in the
middle of each side of the first triangle, as shown in Figure G.b. Then 12
equilateral triangles, each of side a/9, are placed halfway along each of
the sides of this figure, as shown in Figure G.c. Figure G.d shows the result
of adding 48 equilateral triangles, each of side a/27, to the previous figure.
As part of the work on this paper, find the perimeter and the area of the
snowflake curve formed if you continue this process indefinitely.
Figure G.9 Construction of a snowflake curve
References:
Anthony Barcellos, "The Fractal Geometry of Mandelbrot," The College
Mathematics Journal, March 1984, pp. 98-114.
"Interview, Benoit B. Mandelbrot," OMNI, February 1984, pp. 65-66.
Benoit Mandelbrot, Fractals: Form, Chance, and Dimension (San
Francisco: W. H. Freeman, 1977).
Benoit Mandelbrot, The Fractal Geometry of Nature (San Francisco:
W. H. Freeman, 1982).
Research the 2000 U. S. Presidential election, but do not focus on the "hanging
chads." Rather, investigate the effect that third party candidate Ralph Nader
had on the outcome of the election.
Write a history of apportionment in the United States House of Representatives.
Pay particular attention to the paradoxes of apportionment.
Your group should investigate some item of interest to your group.
It might be to predict the outcome of an upcoming election, your favorite song
or movie. Your group should make up a list of 5 or 6 choices; for example, you
might be researching what is the best of the Star Wars movies. Make up a written
ballot and ask at least 50 people to rank the items on your list. Summarize
the outcome of your poll. Was there a majority winner; how about a plurality
winner. Who wins the Borda count or the Hare methods? What about the pairwise
comparison method. Present a summary of your results.
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