| Volume 6 |
Summer 1998 |
Issue No. 2 |
ANNUAL MEETING OF
THE BOARD OF DIRECTORS
Earl Mitchelle - Asheville School - Asheville, NC
The Board of Directors of the North
Carolina Association of Advanced Placement Mathematics Teachers held its annual meeting on
June 27, 1998, at Lexington Senior High School in Lexington, North Carolina.
The annual Treasurer's report showed a
gain of $93.90 for the year ending June 27, 1998. At this time, there are 165 active
members and 36 inactive members of the Association. The number of active members is
expected to increase significantly as a result of NCA2PMT's recent Internet
exposure.
NCA2PMT will have a session at
the North Carolina Council of Teachers of Mathematics(NCCTM) meeting in Greensboro, North
Carolina, October 29-30, 1998. The NCA2PMT session is scheduled from 10:30 a.m.
until 11:50 p.m. on Friday, October 30, 1998. After a brief business meeting, grading
standards for some of the free-response questions from the 1998 examination and an overall
view of the examination will be presented.
Tentative plans were made for NCA2PMT's
participation at the regional meetings of the North Carolina Council of Teachers of
Mathematics during the first months of 1998. At the present time, the dates and locations
of these meetings have not been set.
Board members Charles H. Bodine, Bernice
H. Kenan, and Melba R. Tripp are retiring. New member of the Board include regional
representatives Stephen Davis, Davidson, NC; Rona Schriber, Fayetteville, NC; and Betty
Anne Shearin, Creswell, NC. Carolyn, M. Walmsley, Lexington, NC, became President, Jane R.
Barnett, Laurinburg, NC, became Immediate Past President, and Deborah G. Britt, Mars Hill,
NC, was elected President Elect. Frank J. Vrablic, Manteo, NC, was appointed to a second
two-year term as a regional representative.
At the conclusion of the Lexington
meeting, the Board of Directors of NCA2PMT is as follows:
Carolyn M. Walmsley, Lexington, NC -
President
Deborah G. Britt, Mars Hill, NC - President Elect
Jane R. Barnett, Laurinburg, NC - Immediate Past President
Geoffrey A. Lucia, Charlotte, NC - Treasurer
W. Earl Mitchelle, Asheville, NC - Secretary
Judy Busick, Wilmington, NC - Representative
Stephen Davis, Davidson, NC - Representative
Sue W. Sams, Charlotte, NC - Representative
Rona Schriber, Fayetteville, NC - Representative
Betty Anne Shearin, Creswell, NC - Representative
Frank J. Vrablic, Manteo, NC - Representative
The next meeting of the Board is tentatively
scheduled for June of 1999.
AP CALCULUS ONLINE
DISCUSSION GROUP
This is a moderated discussion group (a.
k. a. mailing list) for current and prospective AP Calculus teachers, faculty consultants
to the AP Calculus Reading, and faculty interested in what AP Calculus is all about.
The idea is for you to share with your
colleagues ideas on syllabi, course texts, teaching techniques, use of technology, and to
discuss other AP issues and topics as they arise.
The group is not designed to answer
questions about program policy of exam administration. If you have specific questions, you
can direct them to apexams@ets.org for information about AP policies, exam administration
details, and other aspects of AP or to apreader@ets.org for information about the AP
Reading.
To subscribe to the
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If you are new to online discussion
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about mailing lists at the following address.
http://www.collegeboard.org/ap/listserv
NOTES FROM THE PRESIDENT'S DESK
Jane R. Barnett - Scotland High School - Laurinburg, NC
What are many of you doing about the 4 by
4 block schedule? We had a number of comments about the topic a few years back. Have we
found any good solutions? It would be worthwhile to "open the floor" to that
topic again. If you have some useful suggestions for those on "the block,"
please submit them to the newsletter for publication consideration.
* * *
While my curriculum committee would not
approve of taking two terms of an AP class, we do offer an optional second term for BC
topics now. It bears only Honors weight. I still use the strategy suggested by Martha Ray
at Southeastern Guilford High School in the calculus manual. I am including here my most
recent list of topics, the AB portion which I pass out on the first day of the fall term.
These are to be completed as the topics are studied. I take them up to read, comment on,
and grade before the related test is given, if possible. Students admit that writing about
their procedures assists in learning, and that the notebook is a valuable tool for review
and for use in later courses. Suggestions regarding the topics or the process are welcome.
Calculus Manual
Tape the table of contents in the cover
of a seventy-page spiral notebook. For each topic the following procedure should be used.
1. On a page numbered the same as the
entry, write the topic as a title.
2. State any relevant definition or
theorem.
3. State the specific problem(s) that
necessitate the use of the theorem or procedure under the general topic.
4. Number each step in the symbolic form
of the solution on one side of the page and then write, in words, the procedure used in
each step on the other side of the page.
Calculus Manual Entries
1. How to solve an absolute value
inequality three ways.
2. How to graph the basic transformations
of a function.
3. How the graphs of three functions for
which the limit as x ® a would not exist could look. Use three different justifications.
4. How to prove that a function is
continuous at a point.
5. How to prove that a function is
continuous on a closed interval.
6. How the graphs of functions which fail
each part of the continuity could look. Use three examples.
7. How to apply the Intermediate Value
Theorem.
8. How to use two definitions which
are limits of difference quotients to find the derivative of a function.
9. How to find an equation of a line
tangent to a graph and a line normal to a graph.
10. How to implement the Mean Value
Theorem for Derivatives. Illustrate the results graphically.
11. How to relate continuity to
differentiability.
12. How to use the First Derivative Test
to graph a function.
13. How to find local extrema using the
Second Derivative Test.
14. How to solve an optimization problem. (Max/Min)
15. How to find absolute extrema on a
closed interval.
16. How to graph a function, f(x), given
the graphs of f'(x) and f"(x).
17. How to find the total distance
traveled by a particle moving on the x-axis. Determine the velocity, speed, and
acceleration at a point where it is moving to the left.
18. How to solve Related Rate problems.
19. How to compute the Riemann sum over
four equal subintervals for y = x2 +4 on the interval [0,8] using left
endpoints, right endpoints and midpoint evaluation points.
20. Define the definite integral as a
limit of a Riemann sum in No.19.
21. How to find an indefinite integral
using the method of substitution.
22. How to find a definite integral using
a change of variable.
23. How to determine the value indicated
by the Mean Value Theorem for Integrals and the average value of the function.
24. How to use the Trapezoidal Rule.
25. How to compute the area between two
curves.
26. How to compute the volume of a solid
of revolution
(a) about the x-axis using the washer method.
(b) about some line y = a for some a.
27 How to compute the volume of a solid of
revolution
(a) about the x-axis using the cylindrical shell method.
(b) about some line y = a for some a.
28. How to find the volume with a given
cross-section.
29. How to find a derivative by
Logarithmic Differentiation.
30. How to solve an exponential
growth/decay problem.
31. How to solve work problems using the
definite integral.
32. How to apply L'Hopital's Rule.
33. How to solve a separable differential
equation with given boundaries.
34. How to interpret differential
equations geometrically using slope fields and their relationship to derivatives of
implicitly defined functions.
35. How to use Euler's method to solve
differential equations numerically.
36. How to apply Newton's Law of Cooling.
37. How to integrate using several inverse
trigonometric functions.
38. How to antidifferentiate a function
using simple integration by parts.
39. How to integrate using simple partial
fractions.
40. How to evaluate improper integrals as
limits of definite integrals.
41. How to solve logistic differential
equations.
42. How to translate between parametric
equations and single x/y equations and graphs them over a range of t-values.
43. How to find the first and second
derivatives of parametric equations.
44. How to compute arclengths of graphs of
f(x), parametric equations, polar equations, and vector form.
45. How to translate between polar points
and rectangular points, and equations.
46. How to differentiate polar equations.
47. How to find areas enclosed by polar
equations.
48. How to analyze planar curves in vector
form including velocity and acceleration vectors.
49. How to use Taylor polynomials
approximations with graphs showing increasing convergence.
50. How to form a series for the cos x
function given the series for sin x.
GENERAL NOTES FROM THE 1998 AP CALCULUS READING
Jeff Lucia - Providence Day School - Charlotte, NC
Earl Mitchelle - Asheville School - Asheville, NC
There will be no changes in the course
descriptions for either the AB or BC examinations for 1998-1999. More functions given in
multiple ways, e.g., AB-3, more separable differential equations, and more applications of
the definite integral, e.g., AB-5 / BC-5, will probably appear on future examinations.
Work problems tend to be a bit too specifics for the examination but will not be
eliminated completely. Look for applications of infinite series on the BC examination
soon.
* * *
AB and BC common topics are to be taught
and tested at the same level of rigor. The difference in difficulty between AB-1 and BC-1
seems to challenge this statement.
* * *
There were 116,500 AB, 26,700 BC, and
1,100 alternate examinations given this year.
* * *
A college comparability study was done
this year to measure the continuity of the calculus courses from the high school level to
the college level.
* * *
Throughout the examination there
was renewed emphasis on NOT reading an answer from a graph. Symbolic or analytical work is
required for full credit. The exception to this is problem AB-3, part (a), where the graph
is the primary source of information.
* * *
There continues to be a grading emphasis
on labeling any kind of graph that students produce as part of a solution.
* * *
The Texas Instrument TI-89 and the Casio
CFX-9970 calculators have been added to the approved list of calculators for the 1998-1999
examinations. These two calculators have computer algebra systems(CAS) which allow the
operator to perform symbolic manipulations. The physical appearances of these two
calculators are very similar to other calculators which are on the approved list. This
would make it difficult for test proctors who are not mathematics teachers to identify
these calculators. The Test Development Committee has already modified the 1999
examinations to eliminate any advantages for test takers who use these two calculators.
* * *
Beginning with the 2000
examinations, each student will be required to bring to the examinations a graphing
calculator with the following built-in functions:
1. Produce the graph of a function
within an arbitrary viewing window.
2. Find the zeros of a function.
3. Compute the derivative of a function numerically.
4. Compute definite integrals numerically.
The calculators on the approved list for
the 1998-1999 examinations which have these four functions built in are Casio FX-9700,
FX-97501 CFX-9800, CFX-9850, CFX-9950, and CFX-9970 series calculators, Hewlett-Packard
HP-28, HP-38G, and HP-48 series calculators, Sharp EL-9200, EL-9300, and EL-9600 series
calculators, and Texas Instruments TI-82, TI-83, TI-85, TI-86, and TI-89 calculators.
* * *
Members of the Test Development
Committee who teach in secondary schools do not stop teaching the AP courses in their
schools while they are working on examinations for future years. Because the questions
undergo many changes and members of the committee are not permitted to keep copies of the
questions as the questions are being developed, edited, revised, etc., the students of
these teachers do not gain any advantage.
* * *
The complete 1997 AB and BC
examinations will be released in October of 1998.
*
* *
The complete 1998 AB and BC
examinations will be released in October of 1999.
* * *
The introduction of
computer algebra system (CAS) calculators should not stop the teaching of analytical
methods because these concepts are still tested in the multiple choice questions of the AP
examinations, and college courses still require students to use these rules. Some topics
such as Newton's Method, volume by shells, L'Hopital's Rule, and integration by parts are
still good topics for students to know. Students should be prepared for college calculus,
and many colleges courses are not taught in a CAS environment.
* * *
With calculators containing
computer algebra systems(CAS), the standard rules of calculus will still be tested on the
non-calculator parts. The Test Development Committee has discussed the possibility of
making some portion of the free-response sections non-calculator. An alternative to this
would be to find a a calculator-neutral way to ask certain questions, e.g., the series
questions on the BC examination the past two years where the coefficients of the first few
terms of a Taylor polynomial were given and not the particular function f(x). A question
such as AB-2/BC-2, part (a), could be rephrased to say, "Discuss the end behavior of
f(x)," instead of asking for the limits as x approaches + „. This will become more
difficult for the Committee as the capabilities of calculators increase.
* * *
Free-response questions now
generally have more parts than in previous years, and more points are being allotted for
setups and justifications with, at most, one point being given for the answer in any given
part.
* * *
The Test Development Committee
is constantly reviewing the technology which is available as they develop questions for
future examinations so that students with certain calculators will not be at any advantage
or disadvantage.
* * *
Students are EXPECTED to
evaluate all definite integrals on the free-response section of the examinations with a
calculator and not to use analytical methods.
* * *
The use of conceptual questions
and justifications will continue to be an important part of each examination. It should be
noted that justifications must be analytical in nature, i.e., non-calculator in nature.
* * *
Teachers are encouraged to
review the instructions for the examinations with students BEFORE the examination.
Students are continuing to make errors which seem to indicate that many of them are not
reading the instructions completely or do not understand the instructions. Every student
should be familiar with the calculator that he or she will use on the examination.
Teachers should stress the directions for calculator use as they pertain to rounding,
using specific methods to arrive at an answer, and how much work must be shown for
justification, etc. There directions are sent to every school well before the AP
examinations are administered.
* * *
The time will probably come soon
that a calculator with a CAS will be the MINIMUM requirement for the AP examinations. The
questions will then likely be written so that students must decide how, why, and when to
use a calculator, i.e., more of this tool (point to your head) and less of this tool
(point to your calculator).
* * *
Different calculators can use
different algorithms to produce answers with slight variations, and the grading standards
thus allow a range of answers to compensate for these variations.
* * *
Many students were unable to
answer the questions about Riemann sums, slope fields, and Euler's method on the 1998
examinations. These students appeared to be totally unfamiliar with these topics.
* * *
Several readers stated that they
hoped that the abstract/pure mathematics type problem would not disappear from the
examination.
CALCULUS AB SUBSCORE GRADE FOR CALCULUS BC EXAMINATION
Beginning with the May 1998
Calculus BC examination, a Calculus AB subscore grade will be reported based on
performance on the portion of the examination devoted to Calculus AB topics (approximately
60% of the examination). The Calculus AB subscore grade is designed to give colleges and
universities more information about the student Although each college and university sets
its own policy for awarding credit and/or placement for AP Exam grades, it is hoped that
institutions will apply the same policy to the Calculus AB subscore grade that they apply
to the Calculus AB grade. It is also hoped that more students who take a Calculus BC
course will take the Calculus BC examination as a result of knowing that they will receive
this Calculus AB subscore grade to assist with college credit and/or placement
The AP Calculus Development Committee
[the group of college faculty and high school AP teachers responsible for developing the
AP Calculus course description and examinations] believes that reporting this subscore
grade is consistent with the philosophical changes of the new course description, since
common topics are tested at the same conceptual level in both Calculus AB and Calculus BC.
An increase in the number of multiple~hoice questions in Section I of the examination
[from 40 in 1997 to 45 in 1998] was necessary in order to report a subscore grade that
meets the statistical standards of the AP Program.
THE GRADING SETTING PROCESS FOR AP
CALCULUS
The Chief Faculty Consultant for AP Calculus (Bernard Madison,
University of Arkansas-Fayetteville) works in conjunction with ETS statistical analysis
and mathematics stats to establish the AP grades. Direct comparisons are made between the
performance of the current year's candidates and that of former candidates on a set of
identical multiple~hoice questions. ETS statistical analysis staff can then determine the
relative difficulty of the free-response questions and calibrate them with a high degree
of accuracy, thereby minimizing the effects on candidates' grades on different questions
from one year to the next
The Chief Faculty Consultant compares the general distributions of
scores to those of the past several years and considers other pertinent data, including
college validity studies and reports of table leaders from the AP Calculus Reading, to
arrive at decisions on grades. (Calculus is conducting a college validity study during the
1997-98 academic year.) The Chief Faculty Consultant's judgments on the free-response
questions are combined with the results of scoring the multiple choice questions, and the
total raw scores are converted by the Chief Faculty Consultant to the AP Program's 5-point
scale of grades.
The AP grade ranges (i.e., the range of scores that determine a
particular AP grade) for the Calculus AB exam will be set first This process will take
into account the performance of the students who took the AB exam. It will not consider
the performance of those students who took the BC exam.
The grade ranges for the Calculus BC exam will then be set This process
will take into account the performance of the students who took the BC exam. It will not
consider the performance of those students who took the AB exam.
The procedures for setting the AB and BC grade ranges will be the same
as for the last 15 years.
Lastly, the grade ranges for the AB
subscore grade will be set The process will take into account both the performance of the
BC students on the AB-level material of the BC exam and the performance of the students
who took the AB exam. Scores on the AB portion of the BC exam will be scaled to scores on
the AB exam using a statistical procedure called equating. The equating will determine the
grade ranges for the AB subscore grade that best represent performance equivalent to the
grade ranges of the 5 AP grades for the AB exam.
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