NOTES FROM THE PRESIDENT'S DESK
By Betty Barnett
Scotland High School

The joint meeting of the SC²PMT and NC²PMT organizations during the 1997 Carolinas' Conference on October 23 was well attended by members of both groups. Dr. Roger Allen talked about the expected shifts in emphasis on the AP examinations. Changes, which will be gradual in his opinion, will mean the inclusion of some new topics or perspectives and a shift in the scoring weight to possibly 213 for the setup and only 113 for the computation. The BC examination material will now be a continuation of the AB examination material with a degree of difficulty throughout somewhere between former AB and BC examinations. A study of the philosophy, goals, and writing will begin in May.

Readers from both organizations discussed the grading of four problems from the 1997 examination. Steve Davis and Jeff Lucia represented NC² PMT by sharing their insights. As always, the listeners made valuable notes on their copies of the grading standards. The dedication and scholarship of AP teachers is inspiring and contagious.

The two groups then held separate business sessions. NC² PMT thanked retiring board ~ember Margaret Wirth for her willingness to serve the teachers of North Carolina. Judy Buswick of New Hanover High School in Wilmington was welcomed to the Board. The amendment to the Constitution which allows the Constitution to be amended by a majority vote of those voting at the meeting, including those voting by absentee ballots prior to the meeting, was approve. The requirement that members be notified of a proposed amendment to the Constitution at least thirty(30) days prior to the meeting still applies.

The Teachers Teaching with Technology Conference held at the College of Charleston last August was praised, and Judy Broadwin's view of the curriculum changes was shared. Information about a summer institute featuring NC² PMT board member Deborah Britt was distributed.

The network of AP teachers has proven to be vital as we meet current challenges. How else could we hear about local linearity, accumulation functions, slope fields, and other new topics without this new network? The eagerness to use new technology and to gain new perspectives is catching. What an exciting group you are!

CONSTITUTIONAL AMENDMENT APPROVED

The membership of NC2PMT approved an amendment to Article VIII. of the Constitution at the annual meeting on October 23, 1997. Article VIII. was amended

FROM

"The Constitution may be amended at an NCA2PMT meeting by a majority vote of the members present provided that notice has been given at least thirty(30) days prior to the meeting.

TO

"The Constitution may be amended at an NCA2PMT meeting by a majority vote of those voting at the meeting including those members voting by absentee ballots received prior to the meeting provided that notice has been given at least thirty(30) days prior to the meeting."

ADVICE TO NEW AP TEACHERS
By Jeff Lucia
Providence Day School

What should you do to become prepared to teach AP calculus?

• Seek the advice of a mentor. If you are like me, you haven't done much with calculus since college. Talk to your favorite experienced AP calculus teacher to get guidance on important versus unimportant topics from whatever books you use, sequencing topics, places where a theoretical approach is particularly necessary, places where technology is useful and/or essential, etc.
• Know your material. There is really no substitute for this. Learn it or releam it to the point that you are comfortable with the basics and can communicate it effectively to your students. Go back to your old textbooks from when you were studying calculus and to your current school textbook. Do the~roblems as the students would and make conclusions about the questions the way they should. This will give you an idea of where it is going.
• Take an AP Institute andlor College Board Workshop. These are given by people who are "plugged into the network." They can help you to gain insight into the nuances of teaching the AP course and taking AP examinations and lead you to become part of the network. I attended an institute in 1987 at Clemson University to prepare to teach BC calculus for the first time. I had done a decent job teaching AB calculus for the three prior years, but that institute was the beginning for me. I learned more in those two weeks from both classmates and instructors about being a good or better AP calculus teacher than in the three previous years. By 1989 I was an AP reader, and I have continued on from there ever since as a member of the network.
• Go to conferences and workshops to learn about technology and its Impact on the AP calculus syllabus and methods of instruction. The graphing calculator has driven much of the change in both of those areas recently, and it is a very effective teaching tool for certain topics.
• Set a high level of expectation for your students. Give daily assignments that test their thinking skills as much or more than their mechanical ability. Expect them to work independently on their assignments so as to be able to spend a lot of class time discussing answers, things they tried, what worked, what did not work, etc. The students should have much of the floor for much of time spent this way. It's through this discussion that many can finally put some of the difficult concepts into place.
• Use old AP multiple choice and free response questions in daily work and on tests and examinations. Doing this all throughout the year will help students to be aware of the types of questions they can expect on AP examinations. There are many sources of these problems. They are also good practice for the teacher to get into the right mind set to teach AP calculus.
• Expect to be less than completely comfortable for at least the first year. After teaching the entire syllabus and seeing where things went well or did not go well, you can revise your game plan for the next year with a much greater sense of anticipation.

 GOALS FOR REFORMED AP CALCULUS
"Teacher's Guide to Advanced Placement Courses in Mathematics:
Calculus AB and Calculus BC"
by Dan Kennedy

1. Students should be able to work with functions represented in a variety of ways: Graphical, numerical, analytical, or verbal. They should understand the connections among these representations.

2. Students should understand the meaning of the derivative in terms of a rate of change and local linearity and should be able to use derivatives to solve a variety of problems.

3. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of a rate of change and should be able to use integrals to solve a variety of problems.

4. Students should understand the relationships between the derivative and the definite integral as expressed in both parts of The Fundamental Theorem of Calculus.

5. Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

6. Students should be able to model a written description of a physical situation with a function, differential equation, or an integral.

7. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.

8. Students should be able to determine the reasonableness of solutions including sign, size, relative accuracy, and units of measurement.

9. Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

NORTH CAROLINA AP STATISTICS TEACHER ORGANIZATION
TO BE FORMED
By Daren Starnes
Charlotte Country Day School

At the 1997 Carolinas' Conference, the idea of forming an organization to support the teaching of AP statistics in North Carolina was discussed. Much enthusiasm was expressed by those in attendance, and so the effort is moving forward. This group would consist of high school teachers, department heads, and curriculum specialists as well as college and university representatives and other parties interested in the teaching of statistics just as the NCA²PMT is a group of people interested in the teaching of calculus.

Present goals are to spread the word about the formation of NCA²PST, to discuss ways that this group can have a positive impact on the teaching of AP statistics in North Carolina and to enlist sufficient support for the formal creation of NCA²PST at the 1998 NCCTM Conference in Greensboro in October.

Some teachers have formed their own local AP statistics teachers' forums. In Charlotte, Hughiene Lucas of South Mecklenburg High School and Rudene Marlowe of Charlotte Latin School created the Mecklenburg County Teachers Consortium (MCSTATS). Meeting locations rotate through the Charlotte area public and private schools.

The AP statistics course is rapidly evolving into a legitimate college preparatory course. As more colleges and universities become familiar with the course syllabus and the nature of the AP statistics examination, their credit and placement policies will evolve. It is important for those of us who are teaching the course to engage in productive dialogue with college faculty so that we can prepare our students adequately for their college mathematics careers.

AP statistics is an exciting new mathematics course that emphasizes concepts, applications, and problem solving. It provides an alternative to AP calculus for some students. For others, AP statistics could be taken in addition to AP calculus to broaden their mathematics backgrounds. AP calculus and AP statistics do have very different flavors. As a result, it is vital that teachers of these two courses keep in touch with one another.

Whether you are currently teaching AP statistics, in a school where the course is going to be offered, at the college level and have an interest in the AP statistics course, or are just interested in the course, please consider getting involved in NCA²PST. Please contact me, Daren Starnes, one of the following ways.

Telephone: 704-362-7202
e-mail: dstarnes@oven.ccds.charlotte.nc.us
mail: Charlotte Country Day School
1440 Carmel Road
Charlotte, NC 28226

I will be happy to talk with you. In addition, I have agreed to serve as the temporary host of a discussion group, ncapstat, which you are welcome to join. I would like to thank the NCA2PMT for allowing me space in its excellent newsletter. I hope that NCA²PST can produce its own newsletter by this time next year though I don't think that we can compete with the one publised by NCA²PMT. I hope to hear from many of you.

GENERAL DIRECTIONS FOR AP EXAMINATIONS

Students should be familiar with the general directions for the free-response section of the calculus advanced placement examinations- A review of the following instructions for the 1997 examination before going to take the 1998 examination could be very helpful.

GENERAL INSTRUCTIONS

You may wish to look over the problems before starting to work on them since it is not expected that everyone wi!l be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. The problems are printed in the booklet and in the green insert; it may be easier for you to first look over all of the problems in the insert. When you are told to begin, open your booklet, carefully tear out the green insert, and start work.

A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION.

• You should write work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearly and legibly. If you make an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded.
• Show all of your work. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as the accuracy of your final answers. Correct answers without supporting work may not receive credit. Justifications require mathematical(noncalculator) reasons.
• You are permitted to use your calculator to solve an equation, find the derivative at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results.
• Your work must be expressed in standard mathematical notation rather than in calculator syntax. For example, may not be written as fnlnt(X², X, 1, 5).
• Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
• Unless otherwise specified, the domain of a function f is assumed to the set of all real numbers x for which f(x) is a real number.

ALVIRNE HIGH SCHOOL WEB PAGE
http://www.seresc.k12.nh.uslwwwlalvirne.html
By Sandra Ray
Alvirne Nigh School - Hudson, NH

Editor's Note: I learned about the Alvirne High School AP calculus web page last fall. I started going to it on the internet frequently because I found a lot of interesting information at the site. I invited Sandra Ray, faculty sponsor, to write an article about the web page for publication in this newsletter.

The AP calculus class of Alvirne High School in Hudson, New Hampshire, invites you to join them in creating and solving problems as preparation for the AP calculus examinations. Each week a student-created problem from Alvirne High School as well as a problem submitted from someone in another city, state, or country are featured on the page. The problems require knowledge of various calculus concepts to solve. Solutions are then posted at the end of the week, and the problems and their solutions are archived.

There are three years worth of problems currently archived on the site. The page also features multiple-choice questions which are currently being created by students from Greencastle High School in Pennsylvania. Another school will assume this responsibility during the second semester. In addition, multiple choice questions featuring the graphical and tabular requirements for the new 1998 syllabus are provided by D & S Marketing Systems.

This is Alvirne's third year of producing this web page, and we have been quite pleased by the response. We have received countless solutions to our problems and have received responses and problems from virtually every state in the United States, from Canada and Australia, and from countries in South America, Europe, and Asia. The students have been thrilled with the responses and the fact that students from other countries are taking the time to solve problems they created. The students have learned how difficult it can be to clearly state a problem, and often they have to rewrite their problems as their classmates attempt to solve it. They have learned to accept criticism constructively and make appropriate changes in the original problems so that the end product on the web will be clear for others in cyberspace to solve.

Check out the site at http://www.seresc.k12.nh.uslwwwlalvirne.html and solve a problem or submit a problem for others to solve.

SERIES IN THE NEW AP SYLLABUS
Dan Kennedy
The Baylor School - Chattanooga, TN
Ap Workshop Conference
June, 1996

What has changed in the new course description?

The first change is one of omission: sequences are no longer a separate syllabus item. As with many precaiculus topics, a knowledge of sequences is assumed but will not be basically tested unless it is in the context of series (e.g., a sequence of partial sums).

• Concept of series.

The only new wrinkle here is the use of technology to explore convergence and divergence. "Exploration" here does not mean mere number crunching. For example, looking at the partial sums of the harmonic series gives no recogni~able evidence of divergence - until you plot those partial sums against a graph of y - ln x and come to that conclusion based on a previous knowledge of improper integrals.

• Series of constants.

This section has been streamlined in re-packaging, but is essentially the same as in the previous course description. "Motivating examples including decimal expansion" is more of a pedagogical tip than an extension of the syllabus.

The "error approximation" for alternating series is now referred to as an "error bound," which is more appropriate.

The comparison test for convergence is not mentioned by name, but an understanding of the concept Is implied in "Terms of series as areas of rectangles, induding the integral test."

The limit comparison test is rio longer a syllabus Item.

• Taylor series.

The spectacle of Taylor polynomials actually approximating the graphs of functions to which they converge is one of the most fascinating visualizations of calculus that graphing calculators can provide. Lest any student of BC Calculus be deprived of experiencing this vision, it is now a syllabus item.

The "manipulation of series" topic has been re-worded in an attempt to give teachers a better idea of how far students can go with this technique when answering a question on the AP examination. There is a hard way and an easy way to derive the Taylor series for centered at x = 0, for example, and the hard way is not merely a waste of time; it is a waste of a great deal of time. (The easy way: plug x² into the known Maclaurin series for and multiply the result by x.)

The other topics in this section have been in the BC course description for years.

How NOT to teach this topic

Consistent with the philosophy of the new course, students should NOT see series as a~ "add~n" topic with a new set of rules and formulas. Thus, it is not a good idea to abandon functions for two weeks in order to drill students on the tricks for computing limits of sequences and for testing convergence or divergence of series of constants. Unfortunately, this is exactly how many books approach the topic.

How to TEACH this topic

Explore. Get the students wondering about how ftmctions are represented by these "infinite polynomials" (power series), and let the question of convergence come up eventually in context. Here are some examples of series explorations that have proven to be successful in the classroom:

1. Most BC students wrn have seen infinite geometric series before their calculus course. They will therefore have seen that:

and some might even recall that this is valid if and only if . What happens if we compare the graphs of and and   ?

The "convergence" is clearly good near 0 gets worse as we approach -1 or +1, and then breaks down completely for . Adding more terms to the polynomial makes the agreenment closer for , but not beyond.

2. Using the above series as a stating point, ask students to "invent" series representations for the following functions, and to conjecture for what values of x each representation will converge:

All three would appear to be valid for -1 < x < 1, working from the original series.

3. If f(x) is represented by the power series

then what power series would represent f '(x)?

Students easily see that f '(x) would be represented by the same power series. It is then natural to ask what function could be its own derivative, and students will volunteer f(x) = e^x. They probably will not know why this is the ONLY candidate for f(x), but this is, in fact, a differential

equation with an initial condition so that solution can be reviewed at that point with some new motivation!

4. Have students construct a 4th degree polynomial P such that:

P(0) = 1,
P'(0) = 2;
P''(0) = 3;
P'''(0) = 4;
P⁽⁴⁾(0) = 5.

This is, of course, a Taylor polynomial, but the students do not need to know that in the process of bullding this polynomial, they are discovering for themselves what might otherwise be a bewilde?1ng formula. This sets them up for:

5. Have them construct a 5th degree polynomial P such that P and Its first five derivatives agree exactly with the function sin x and its first five derivatives at x =0. That is,

P(0) = sin(0)
P'(0) = sin'(0)
P'1(0) = sin"(0)
etc.

6. Compare the graph of sin x to the graph of the 5th degree polynomial in (5):

7. Have them extend the polynomial in (5) to a power series for sin x.

8. Compare the graph of sin x with a few more polynomials of higher degree. (The graph shown here is the 11th degree polynomial.)

These are Taylor polynomials, which can now be named after students have met them. Ask students to conjecture for what values of x the power series for sin x will converge. By the graphs, it looks like It COUID be all real numbers! (The need to explore this question is what should lead teachers eventually to introduce the Ratio Test.)

9. Challenge them to find a power series for cos x. (It's amazing how many students repeat the construction rather than simply differentiate the series for sin x)

10. Challenge them to use their power series to show that e^ix = cos x + i sin x.

Exploring convergence

As noted above, looking at geometric series (algebraically and graphically) can go a long way. Also? since BC students will have already seen improper integrals, the Integral Test can be introduced visually at an early stage to give students a "visual" hook for convergence and divergence:

The convergence of alternating series can also be made visual (the standard textbook approach), and an understanding of the picture also carries an irnplidt understanding of the error bound.

The SEQUENCE mode of some graphing calculators can be used for further exploration of convergence and divergence. Here is a look at the first fifty partial sums of the harmonic series:

Hence, the 1000th partial sum of the harmonic series lies between In 1000 and 1 + ln 1000, i.e., bigger than 6.9 but less than 8. To get the partial sum bigger than 100 requires more than e⁹⁹ terms!

So is there anything they just need to memorize?

They should derive as many things as they can, but then they will need to have certain facts and algorithms at their disposal to use on the AP examination. These are Ltemied in the course description, and include:

1) geometric series;
2) harmonic series;
3) alternating series and error bound;
4) p-series and p-test for convergence;
5) integral and Ratio tests;
6) Taylor series and Taylor polynomials centered at x - a;
7) Maclaurin series

8) The Lagrange error bound for Taylor polynomials:

#7 is very important, as students should use these "basic" series whenever possible if they are asked to construct Taylor series on the AP test.

#8 is intimidating, but students should realize that their graphing calculators give them other ways to look at errors on interals. For example, if I wish to find the maximum error that occurs when approximating cos x by its 4th-degree Taylor polynomial on the interval (-2,2], I can graph the error: