AP CALCULUS AND BLOCK SCHEDULING
Jane R. Barnett

As promised in the last newsletter, we are reporting on concerns about the block schedule and members' experiences. Some are anticipating the change with the expected fears: time, pace, maintenance until the exam.

One teacher who has taught AP Calculus on the block for three years shared her strategies and results generously. Martha Ray of Southeast Guilford High School teaches AP both fall and spring term. She has had an optional second term for fall students assigned to her classroom (during other classes) to continue preparation for the exam until this year. This year she has a second term for the BC students. To prepare the fall students for the exam, she meets with them from 2:30 to 4:30 every other Sunday afternoon during the spring. The spring term students sign a contract to attend sessions on Saturdays from (9:00 until 12:00) eight times and an all day session the week before the exam. They are not required to attend class after the exam date, so the Saturday classes are a trade-off to meet the requirement of 120 contact hours by the state. She did have 21 students pass the exam last year and those who took the exam but didn't participate in the reviews on Sundays didn't pass.

Martha Ray also uses firm expectations and structure to make this work. This includes use of a looseleaf notebook, assignment sheets to enable absent students to keep up, frequent homework checks, tests modeled after the exam, and a calculus manual. The manual is a notebook section which is graded the last grade period of the term. It consists of some thirty entries of the "How to find f'(x) by the definition of the derivative" style, The students state the theorem or definition, state a particular problem that necessitates use of the procedure, and number each step of the symbolic solution completed on one side of the page, while writing a description of the procedure used on the other.

Since this was my first experience with this schedule, Martha gave me moral support, asserting that success is possible. I used a similar pace, I adopted the manual concept to assist the review, and I have scheduled bi-weekly sessions until May. In my experience it is necessary to think globally, tying things together whenever possible and sometimes teaching two topics simultaneously. Thanks to some very agreeable students1 we were able to complete the syllabus by January with some review. The other shoe will fall in May!

Martha Ray has agreed to participate in the Central Regional meeting of NCA'PMT in Greensboro.

AP CALCULUS AB FALL COURSE OUTLINE

I. Preliminaries (1 week)

1.1 The Real Numher System
1.2 Decimals, Denseness, Calculators
1.3 Inequalities
1.4 Absolute Values, Square Roots, Squares
1.5 The Rectangular Coordinate System
1.6 The Straight Line
1.7 Graphs of Equations

II. Functions and Limits (2 weeks)

2.1 Functions and their Graphs
2.2 Operations on Functions
2.3 The Trigonometric Functions
2.4 Introduction to Limits
2.5 Rigorous Study of Limits
2.6 Limit Theorems
2.7 Continuity of Functions

III. The Derivative (34 weeks)

3.1 Two Problems with One Theme
3.2 The Derivative
33 Rules for Finding Derivatives
3.4 Derivatives for Sines and Cosines
3.5 The Chain Rule
3.6 Leibniz Notation
3.7 Higher-Order Derivatives
3.8 Implicit Differentiation
3.9 Related Rates
3.10 Differentials and Approximation

IV. Applications of the Derivatives (2 weeks)

4.1 Maxima and Minima
4.2 Monotonicity and Concavity
4.3 Local Maxima and Minima
4.4 More Max-Min Problems
4.5 Economic Applications
4.6 Limits at Infinity, Infinite Limits
4.7 Sophisticated Graphing
4.8 The Mean-Value Theorem

V. The Integral (2-3 weeks)

5.1 Antiderivatives (Indefinite Integrals)
5.2 Introduction to Differential Equations
5.3 Sums and Sigma Notation
5.4 The Definite Integral
5.5 The Fundamental Theorem of Calculus
5.6 More Properties of the Definite Integral
5.7 More Properties of the Definite Integral
5.8 Aids in Evaluating Definite Integrals

VI. Applications of the Integral (1 week)

6.1 The Area of a Plane Region
6.2 Volumes of Solids: Slabs, Disks, Washers
6.3 Volumes of Solids of Revolution: Shells

VII. Transcendental Functions (2-3 weeks)

7.1 The Natural Logarithm Function
7.2 Inverse Functions and their Derivatives
7.3 The Natural Exponential Function
7.4 General Exponential and Logarithmic Functions
7.5 Exponential Growth and Decay
7.6 The Inverse Trigonometric Functions
7.7 Derivatives of Trigonometric Functions

VIII. Techniques of Integration (2 days)

8.4 Integration by Parts

IX. Indeterminate Forms and Improper Integrals (2 days)

9.1 Indeterminate Forms of Type 0/0

X. Numerical Methods, Approximations (1 day)

10.3 Numerical Integration (Trapezoidal Rule)
10.4 Solving Equations Numerically (Newton's Method)

Any time remaining will be spent reviewing and preparing for the AP Calculus exam. The AP exam is usually given the first of May. A practice exam will be given at the time determined as convenient for the majority of the students.

Problem: The Incongruity of the 4 X 4
Concentrated Curriculum and AP Calculus
Catherine H. Neagle

If a student takes Calculus first semester, there is a 4 month delay before the AP exam is administered in May. However, if a student takes Calculus second semester, then there is not enough time to cover the curriculum before the AP exam in May.

One Possible Solution: Have the student take AP Calculus AB first semester and AP Calculus BC second semester. Then the teacher and student will jointly decide whether the student is prepared to take the AP Calculus AB exam or the AP Calculus BC exam.

Course Descriptions

Advanced Placement Calculus AB
Course #0211          Grade Levels: 12 One Unit
                                 Prerequisites: Pre-Calculus

AP Calculus AB explores the idea of changing variables. It covers limits, differentiation (formulas and applications), integration (formulas and applications), volumes obtained by disc and cylindrical shell rotation and other concepts dealing with basic polynomial calculus. Students will be expected to take one of the AP Calculus tests.

Advanced Placement Calculus C and Chaos Theory
Course #0212            Grade Levels:   12 One Unit
                                   Prerequisites: Pre-Calculus, AP Calculus AB

AP Calculus C will consist of an independent study of the following topics: vector functions and their derivatives; parametrically defined curves, their derivatives, their graphs, and tangent lines to them, polar equations and areas bounded by them; L'Hopital's Rule for rigorous indeterminate forms; integration by trigonometric substitution, by parts, and by partial fractions; Simpson's Rule; composite functions defined by integrals; work as an integral; sequences and series and their convergence; Maclauren series expansions; Taylor's series. Students will be expected to sit for one of the AP Calculus exams. Chaos theory will introduce the mathematical theory behind chaos, fractals, and dynamic systems.

COLLEGE BOARD ADVANCED PLACEMENT PROGRAM
THE SEMESTER PLAN

A number of secondary schools have inquired about the implications of the semester plan and/or block scheduling on participation in the College Board's AP Program.

Most schools that are adopting a semester plan based on a concentrated curriculum will implement either three or four 90-minute courses each semester. This will replace the traditional six to seven classes of 50 to 56 minutes each. Schools are asking the following questions:

What is the College Board's position on offering AP courses under the semester plan?

The College Board recognizes the many changes occurring in the educational system and is currently studying the impact of the semester plan on the AP Program.

How many schools will use the semester plan in 1994-95?

A recent College Board survey indicates about 350 schools, nationwide, will implement the semester plan next year. Regionally, the semester plan concept is most prevalent in North Carolina and
Virginia. Although there is growing popularity for this method of scheduling, it is still not a major national trend.

Will the College Board offer AP semester exams?

Semester AP exams will not be available for the 1995-96 school year. The College Board is giving serious consideration to the possible development of future semester AP exams if a significant number of schools adopt the semester plan. AP exams for all 29 courses are different each year. The AP test development committees follow a two-year cycle to construct each new AP exam. Thus, to maintain the integrity of the AP exams, we cannot create new ones immediately.

Can the May AP exams be administered in January?

No, because of the obvious loss of exam security.

Will students taking AP exams under the semester plan be evaluated the same and be eligible for the same college credit possibilities?

Yes. All AP exams will be evaluated by the same process no matter what scheduling plan the students studied under. Colleges are interested in the AP exam score results, not the scheduling process for implementing the AP courses.

Does the semester plan provide an advantage for any specific AP courses?

This is hard to determine since there is no significant research available at this time, but there could be some advantage for AP science courses where a 90-minute block of time could be helpful for conducting laboratory experiments.

Are there any possible disadvantages of offering AP courses and exams under the semester plan?

The key issue is the number of direct teacher contact hours the student would have under the semester plan following the most popular choice of 90 minutes per period each day. Offering an AP course in the second semester would result in fewer contact hours than under the traditional method of class scheduling. Offering an AP course first semester would result in students having a three-month break between the end of the course and the current May AP exam administration. Several North Carolina schools report that there would be a loss of over 20 hours of instructional time even if the AP course is offered first semester. There is also a concern, valid or not, that students could have difficulty in understanding and retaining advanced academic studies in a concentrated curriculum structure.

Yes. Teachers and students will face a major time management problem to prevent concentrated curriculum from becoming "watered down." Several schools report that they are making local in-service plans to help support AP teachers with training for time management and curriculum adjustments. This is probably the key issue for quality and success.

Several schools have indicated that they will do some of the following:

• Create pre-AP classes that will begin initial preparation for the AP courses and eliminate time problem.
• Create AP review sessions from February to May for AP courses taught first semester.
• One school system will implement a 3-1 schedule with three semester courses of 90 minutes each and one full-year AP course of 52 minutes.
• Some schools are experimenting with block scheduling and/or modular scheduling without adopting a full semester plan.

The College Board needs your help in assisting us with ideas, successes and problems involving the implementation of the Advanced Placement Program under the semester system. Your help will enable us to share resources and help provide possible future assistance. We recognize that the "C" word presents new concerns and exciting possibilities as we strive to serve students.

POSSIBLE CHANGES TO THE AP CALCULUS
EXAM (BEGINNING 1995) AND THE AP
CALCULUS SYLLABUS (BEGINNING 1997)
Jeff Lucia
Providence Day School

The following is a compilation of information presented by Dan Kennedy of the Baylor School and Tom Tucker of Colgate University. Dan and Tom have worked for many years on the AP Calculus Test Development Committee, including the time during which the transition to the use of graphing calculators has been made.

AP Calculus syllabus and examination revisions have become necessary now that graphing calculators are part of the course, both in teaching and testing. It is clear that some traditional types of calculus questions will receive less emphasis, while some "new" types of calculus questions will receive more emphasis.

There are three new Issues to be considered when writing problems for future calculus examinations:

1. Is the intent of the problem compromised by technology? That Is, does it test what we want it to test, regardless of how the student uses the calculator when solving it?

2. Is the problem vulnerable to differences in calculator capabilities? That is, might students with a particular type of sanctioned calculator have an advantage over students with another?

3. Does the problem test something worthwhile? That is. does the problem reflect something in the curriculum that we ought to be emphasizing in these reformed times?

The conjectures listed on the following pages represent the collective wisdom of Dan and Tom and do not necessarily reflect a consensus of the Test Development Committee nor official AP policy.

PROBLEMS WE ARE LIKELY TO SEE LESS OF

1. Problems that specifically test techniques of differentiation and integration (quotient rule, chain rule, trig substitution, etc.)

2. Problems intended merely to locate maxima, minima, and points of Inflection of curves for given functions.

3. Problems that use algebraically derived results to produce sketches of graphs of functions or other results now obtainable with graphing calculators (e.g., volumes of solids of revolution, particle motion, polar graphs).

4. Problems that require fancy algebra to find limits (e.g., L'Hopital's Rule problems).

5. Problems that lead to equations that are to be solved exactly, using algebra and pre-calculus techniques.

6. Numerical integration for Its own sake.

7. Algebraic problems requiring algebraic solutions.

8. Derivation of infinite series using Taylor coefficients

In general, there will likely be a reduction in problems requiring algebraic solutions, and the major area of added emphasis. may be differential equations, especially in the BC syllabus.

PROBLEMS WE ARE LIKELY TO SEE MORE OF

1. Problems asking for properties of f, given graphs of f, f ', f ", etc.

2. The derivative as a rate of growth and the integral as an accumulation.

3. Applications and modeling problems.

4. Problems utilizing approximate roots found by calculators.

5. Problems calling for interpretations of graphs (range, domain, continuity, differentiability, modeled behavior).

6. Problems requiring justification of calculated results.

7. Problems involving generic functions.

8. Problems Involving parametrically defined functions, both in AB and BC.

9. Problems requiring theoretical understanding (AB 6 types).

10. Geometric problems requiring algebraic solutions.

11. Algebraic problems requiring geometric solutions.

12. Problems constructed around numerical data.

13. Problems requiring an understanding of certain basic differential equations (y' = kx, y' = k/x, y' = ky, y'' = ky, and y' = ky(1-y), to name a few), both in AB and BC.

SAMPLE MULTIPLE CHOICE PROBLEMS

1.  The slope of the line tangent to the graph of f(x) = tan x / ln x at x • 3pi/4 is nearest

A. 2.911
B. 3.034
C. 3.187
D. 3.249
E. 4.000

2.  If x = a is a vertical line which divides in half the area
enclosed between the x-axis and the curve y = x^{2 + 1} on
the interval x < [0, 1], then a is nearest

A. 0.5
B. 0.585
C. 0.596
D. 0.606
E. 0.617

3. The function f(x) = e^{x ln} x has a point of inflection nearest x

A. 0.497
B. 0.592
C. 0.606
D. 0.638
E. 0.670

4. The length of the curve f(x) = tan x from x = 0 to x = pi/4 is nearest x =

A. 1
B. 1.182
C. 1.359
D. 1.278
E. 2.119

5. How many zeroes does the function f(x) • x sin(1/x) have on the interval x { [.05, .20]?

A. 1
B. 2
C. 3
D. 4
E. 5

6. The area of the region enclosed by the graphs of x = 0. x = 1, y = 0, and y = e^-x2 is nearest

A. 0.632
B. 0.747
C. 0.856
D. 0.903
E. 1.718

Answers:   1. A    2. C    3. B    4. D    5. E    6. B

NORTH CAROLINA ASSOCIATION
ADVANCED PLACEMENT MATHEMATICS TEACHERS
Carolinas Mathematics Conference, October 7, 1994

The joint meeting of the North Carolina and South Carolina Associations of Advanced Placement Mathematics Teachers was very informative. The handouts gave our members excellent information to take back to the classroom Attendance was great; approximately 100 attended the combined session with S.C. and 30 attended the N.C. business session.

The AP Standards and the grading of the two common problems on the AB and BC tests in 1994 were presented. Jeff Lucia (NC) presented AB2/BC1 and Brenda Morrow (SC) presented AB5/BC2. Debbie Crawford (SC) and Sam Gough (NC) presented TIMES, They are "a quickly-changing"! Debbie and Sam provided insight into the format of new calculator active AP questions.

Two topics at our NC breakout session were discussed.. First, in approximately 1997, ETS will be giving an AP Statistics Test. The first "Acorn" book for statistics will be published in 1995. Should we encompass statistics and computer science into our organization? Second, how will the 4 x 4 Concentrated Curriculum affect our AP Calculus program? Several teachers already teaching with the 4 x 4 concentrated curriculum shared their experiences and many other teachers expressed their concerns as their district makes this transition.

Melba Tripp thanked Charles Bodine for being our Founding Father. His efforts certainly gave our organization a great jump-start. We presented Charlie with a TI82 graphics calculator and a NCTM text on fractals on behalf of NCA²PMT.

A NOTE FROM THE PRESIDENT
Melba Tripp

The Carolinas Mathematics Conference in October, 1994, gave our members an opportunity to share ideas and to learn new methods. After LINKING with others, we were energized and our batteries were charged!

The big question still is: What are the Calculator Active AP questions in 1995?  The teachers fortuate enough to attend the Carolinas conference left thinking that they could now make an intelligent guess.

I would like to once again thank Charlie Bodine for his excellent leadership throughout 1992-1994. Jeff Lucia and Earl Mitchelle worked by his side. I thank the three of you on behalf of our entire membership.

As we continue to wonder where technology will lead us, let us continue to attend our conferences, read our literature, study, and train. Submit your comments, questions, and ideas to me or to Earl Mitchelle (for publication).

GRADING CALCULATOR - ACTIVE PROBLEMS
Earl Mitchelle

Early last fall it became apparent to me that some adjustments in the grading process would have to be made in evaluating calculator-active problems. My students were using the graphing utility on their calculators but were recording only the answer. If the answer was incorrect, it was not possible to analyze what had been done and to determine whether any partial credit could be given for an incorrect answer.

At times, a student will make a mistake entering the function to be graphed into the calculator. I ask my students to write the code they entered into the calculator and to make a sketch of the graph on their papers. If the answer they give is incorrect, it is possible to take the incorrect function they used and work through the problem to see if they had done the analysis correctly even though the function entered into the calculator was incorrect.

1.  Let f(x) = x + 5e^{-x + 2 sin x.} For what value of x does the graph of f(x) have a point of inflection on the interval 0< x < 2?

(A)0.0
(B) 0.667
(C)1.055
(D)1.479
(E)2.0

2. =

(A)  0.107
(B)  0.169
(C)  0.214
(D)  0.227
(E)  0.453

3. Let f(x) be the sum of the first three nonzero terms of the Taylor series for in x, centered at x What is the maximum value of (f(x) - ln x) on the interval 0.5<x <2.0?

(A)  0.0
(B)  0.026
(C)  0.085
(D)  0.140
(E)  0.162

4. A particle moves along the x - axis with velocity

 v(t) = e^{-2t + cos t -t + 2.}

Find the total distance traveled by the particle from  t = 0  to      t = 3.

(A)  1.35
(B)  2.14
(C)  3.66
(D)  4.74
(E)  5.52

5. An object moves along the x - axis, its position at each time    t > 0 given by the function x(t) = 1/4*e^{-2t + cos t + t/2.} Find the least value of t for which the object moves to the right.

(A)  0.0
(B)  0.619
(C)  1.523
(D)  2.142
(E)  2.621

6. Let f(x) be the sum of the first three nonzero terms of the Taylor series for g(x) = x ln x, centered at x = 1. Which of the following could be the graph of g(x) - f(x) for 0 < x < 2?

(A)	(C)
(B)	(D)

7. Let f(x) be given by the function
               
 The value of x for which f(x) = 0 lies in the interval

(A) (1.0,1.5)
(B) (1.5, 2.0)
(C) (2.0, 2.5)
(D) (2.5,3.0)
(E) f(x) > 0 for all x > 1

8. What is the largest value of x such that
              
 on the interval 1 < x < 3?

(A) 1.82
(B) 2.05
(C) 2.24
(D) 2.43
(E) 2.64

9. Find all values of c that satisfy the Mean Value Theorem for     f(x) = x⁴ - 5x² + 2x - 3 on the interval 0 < x < 2.

(A)  0.17
(B)  0.203
(C)  1.47
(D)  0.203 and 1.47
(E)  0.17 and 1.47

10. Find the value of s that satisfies the following condition:
if |x-(p/4)|< s, then | sin x  - 0.7071 | < 0.05

i.) 0.06
ii.) 0.08
iii.) 0.10

(A)  i. only
(B)  i. and ii.
(C)  ii. only
(D)  i, ii, and iii.
(E)  iii. only

Answers: 1. C,   2. A,    3. D,   4. D,   5. E,   6. B,   7. C,   
                8. C,   9.  D,   10. A

11.A rectangle of length 2K is inscribed in the region between the x-axis and the graph of y = cos x, as shown. For what value of k does the rectangle have maximum area, and what is the maximum area? (Answer: x= 0.860 and A = 1.122)

12. A segment tangent to the graph of y = cos x forms the hypotenuse of a right triangle with legs along the coordinate axes, as shown. What is the x-coordinate of the point of tangency yielding the triangle of minimum area, and what is the minimum area? (Answer: x = 0.860 and A = 1.122)