TOM DICK ADDRESSES AP READERS AT COLORADO STATE
Earl Mitchelle - Asheville School - Asheville, NC

Tom Dick of Oregon State University and Chairman of the Test Development Committee addressed a group of readers during the AP calculus grading at Colorado State University in June. He addressed the group about the work of the Committee.

The Committee meets three or four times each year for three to four days each meeting. The members assess course descriptions and test policies, write and evaluate test questions, and put tests together.

The Committee is concerned about the quality and credibility of the tests so that colleges will continue to accept the results of the examinations, fairness and equity of the questions as related to the technology that is available, and the demands that are placed on AP teachers as the course descriptions change. With no ceiling on technology, equity becomes a very crucial issue that must be watched very closely.

Dick indicated that students should be able to solve problems using multiple representations including numerical, graphical, tabular, and analytical formats. The BC examinations will continue to be extensions of the AB examinations. Topics that are in both the AB and BC course descriptions will be tested at the same level of difficulty on both examinations. Simpson's Rule and Newton's Method are no longer part of the course descriptions, but this should not deter teachers from teaching these topics.

In the future, the Committee will be challenged by rapidly evolving technology, teacher development, and the how the AP scores are used by schools, colleges, and states. As a number of veteran teachers are reaching retirement age, it is imperative that new competent, teachers are found to fill these vacancies to insure the continued success of the advanced placement program.

Dick indicated that the Committee would like to hear from teachers about their concerns as well as about the things they like. Teachers are also invited to submit questions to the Committee for both the multiple choice and free-response sections. The following are the current members of the Committee.

FROM THE PRESIDENT'S DESK
Carolyn Walmsley
Lexington Senior High School - Lexington, NC

I trust that you all are having a restful and rejuvenating summer. Due to busy and conflicting schedules, the annual board meeting of the North Carolina Association of Advanced Placement Mathematics Teachers (NCA²PMT) was postponed from June 26 until August 7 at Lexington Senior High School. Our most important agenda item will be planning the annual general meeting of NCA²PMT. The time has been set for Friday, October 1, 1999, at 10:30-11:50 in Greensboro as part of the North Carolina Council of Teachers of Mathematics (NCCTM) Conference. As usual, the program will include a discussion of the the grading standards for the 1999 AB and BC examinations by experienced readers and a review of new developments from the Educational Testing Service (ETS). If you have any suggestions for the program or organization, please contact me via email at swalms4236@aol.com. I am looking forward to seeing all of you at the Greensboro meeting in October.

Test Development Committee

Thomas P. Dick
email: tpdick@math.orst.edu
Oregon State University, Corvallis, Chairman
108D Kidder Hall
Oregon State University
Corvallis, OR 97331-8507

Stella R. Ashford, Southern University, Baton Rouge, Louisiana
Raymond J. Cannon, Baylor University, Waco, Texas
Mary Ann Conners, United States Military Academy, West Point, New York
Nancy N. Gates, Memphis University School, Memphis, Tennessee
Mark J. Howell, Gonzaga College High School, Washington, District of Columbia
Martha M. Montgomery, Fremont Ross High School, Fremont, Ohio
Lawrence H. Riddle, Agnes Scott College, Decatur, Georgia,
Chief Faculty Consultant
Bernard L. Madison, University of Arkansas, Fayetteville
Immediate Past Chief Faculty Consultant

GENERAL NOTES FROM THE 1999 AP CALCULUS READING
Jeff Lucia - Providence Day School - Charlotte, NC
Earl Mitchelle - Asheville School - Asheville, NC

There were about 160,000 examinations this year, 129,000 of them AB and 31,000 BC.

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There were 535 readers. 53% were from secondary schools and 47% were from colleges or universities. 60% of the readers were men and 40% were women. Readers represented fifty-eight states and countries. The most famous reader was Jaime Escalante of "Stand And Deliver" fame.

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Students who repeatedly misrepresented decimals in a problem, e.g., AB1, lost one point only for the first time the error occurred, unless a second error was made at a different decimal place, e.g., -2.05 for AB1 (b) followed by 3.83 for (c) lost one point, but 2.05 followed by 3.8 lost two points. This did not happen very much.

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A student whose calculator was in degree mode for AB1 lost a point only for the first time the error occurred, provided that the work was consistent in subsequent parts of the problem. Students should be aware that calculators should almost always be set in the radian mode on the AP calculus examinations.

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Many students do not understand that inflection points occur when a continuous function has a second derivative which is zero OR undefined and is accompanied by a change in concavity as demonstrated with some type of derivative analysis rather than a reference to a picture, at the point in question. See 1999 AB5/BC5.

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After two consecutive years of being asked to do a midpoint Riemann sum to approximate a definite integral, many students still do no know how to do this. See AB3/BC3 on the 1999 examination and AB3 on the 1998 examination. The problem may be that students do not recognize the term "Riemann." Teachers might want to consider reviewing the use of numerical integration techniques such as the trapezoidal rule and Riemann sums just before the examination in May. Emphasize that Riemann sums use left, right, and midpoint rectangles.

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Many students do not understand the concept of a function defined as an integral (Second Fundamental Theorem) and its representation of the area under a curve. They had extreme difficulty earning any points on AB5/BC5 this year. Similar questions include AB5/BC5 on the 1997 examinations, and AB6 and BC6 on the 1995 examinations.

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Students should try to be careful with notation such as in including dx or dy with an integral expression and in using parentheses accurately when appropriate. The readers try to be generous about such matters, but often the identification of the independent variable with a dx or dy is an important aspect of a problem, and missing or incorrect expressions can cost the student points. The nature of the error with missing parentheses can also lead to points being deducted even though it is reasonably clear that the student understands the problem. In general, the student has to tell the reader what the correct answer is rather than conveying some correct ideas laden with errors or incorrect notation and/or poor arithmetic or algebra.

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LABEL ANY GRAPH OR SIGN TEST!!!!! Justification points will not be awarded if it is not clear to the reader which function a graph or sign test represents. A verbal description must specifically refer to a function by name. A sentence with references to the "graph," the "function," the "slope," the "derivative," etc., without reference to which function, is considered a verbal version of an unlabeled graph. See 1999 AB5/BC5.

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 staff 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~hoice questions, and the total raw scores are converted by the Chief Faculty Consultant to the AP Program's 5-point scale of grades.

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.

NO-CALCULUS CALCULUS
Earl Mitchelle - Asheville School - Asheville, NC

All of the following questions can be answered without actually finding derivatives or anti derivatives. Results can be obtained using formulas from geometry, formulas and concepts from precalculus1 and concepts from calculus.

Part I.
Use the graph of f(x) to find each of the following.  (answers)

1.	f(3.6) correct to three(3) decimal places.
2.	f(5.8) correct to three(3) decimal places.
3.	Estimate the value(s) of x for which f(x) = 1.5 for 2<x<6.
4.	The area of the region bounded by f(x), for 2<x<4, x=2, x=4, and the x-axis.
5.	f(x)dx.
6.	The mean(average) value of f(x) on [1,7).
7.	Estimate the value(s) of x for which f(x) equals the mean value of f(x) on [1,7]
8.	[f(x)+2]dx.
9.	The mean(average) value of f(x)+2 on [1,7].
10.	Estimate the value(s) of x for which f(x)+2 equals the mean value of f(x)+2 on [1,7].
11.	Estimate the value(s) of x on [1,7] for which the instantaneous rate of change of f(x) equals the average rate of change of f(x).
12.	The volume of the solid of revolution formed by revolving the area bounded by f(x),for 4<x<5, x=4, x=5, and the x-axis about the line y=2.
13.	The volume of the solid of revolution formed by revolving the area bounded by f(x),for 5<x<6, and the line y=1 about the line y= 1.
14.	The volume of the solid of revolution formed by revolving the area bounded by f(x)1 for 6<x<8, and the x-axis about the x-axis.
15.	The volume of the solid of revolution formed by revolving the area bounded by the y-axis, the x-axis, f(x), and x=2 about the x-axis.
16.	The volume of the solid of revolution formed by revolving the area bounded by the y=1, x=8, and f(x) about the line y=2.

Part II.

On the given graph, replace x with t, measured in seconds, and f(x) with x(t), measured in feet. Let x(t) define the position of a body executing rectilinear motion on the x-axis. Assume that the particle is in an environment that permits the unorthodox motion defined by x(t). Find each of the following for 0 < t < 8, unless otherwise indicated.  (answers)

Part III.

On the given graph, replace x with t, measured in seconds, and f(x) with v( t), measured in feet per second. Let v(t) define the velocity of a body executing rectilinear motion on the x-axis. Assume that the particle is in an environment that permits the unorthodox motion defined by v(t). Find each of the following for  o < t < 8, unless otherwise indicated. (answers)

Answers for Part I

Answers for Part II

Answers for Part III