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|  | by the AP Calculus Development Committee and Chief Reader
When graphing calculators were first required on the AP Calculus Exams in 1995, the instructions for the free-response section (Section II) of the exams naturally required some significant revisions. With each year's grading experience, new lessons have been learned. The Development Committee and Chief Reader review and revise these instructions annually.
In this report, we include the most recent version of the general instructions for the free-response section, which will be in effect for the 2006 exams. Although the general instructions for the 2006 exams are identical to those for the 2005 exams, we describe some new tips in this document that will help AP students and teachers understand how to interpret these instructions.
It is an excellent idea for AP teachers to go over these free-response instructions with their students well in advance of the exams. By design, the instructions are kept as concise as possible. In a classroom discussion, the teacher has the opportunity to elaborate on and to emphasize key points. The value of this discussion goes beyond the extrinsic motivation of improving students' exam grades. For the most part, the instructions simply stress the importance of communicating written mathematical work clearly. The anonymous graders of the exams play the role of interested readers of the students' work.
Each bulleted instruction is given as it appears on the exams, followed by additional comments that the committee would like to share (in italics). Some of these comments are in the form of answers to frequently asked questions (FAQs). Where appropriate, examples from recent official scoring standards are cited to provide additional illustrations.
GENERAL INSTRUCTIONS FOR SECTION II PART A AND PART B
For each part of Section II, you may wish to look over the problems before starting to work on them, since it is not expected that everyone will 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.
- YOU SHOULD WRITE ALL WORK FOR EACH PART OF EACH PROBLEM WITH A PENCIL OR PEN IN THE SPACE PROVIDED FOR THAT PART IN THE PINK EXAM 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.
Comment: If a student wants the grader to ignore some erroneous work, crossing it out is not only faster, but also a clearer indicator than erasing (some students do not erase as thoroughly as others). Students may use pencil or pen (blue or black ink) for the free-response section.
FAQ: What happens if a student actually provides two solutions for the same question and these solutions do not agree?
Answer: If it is not clear that the student has abandoned one solution attempt for another, graders are usually instructed to grade both solutions and average the two scores, rounding down to the nearest whole number score. For example, if the two solutions are scored 1 and 4 points, respectively, the student is awarded 2 points (the average of 2.5 is rounded down). - Show all your work. Clearly label any functions, graphs, tables, or other objects that you use. You will be graded on the correctness and completeness of your methods as well as your answers. Answers without supporting work may not receive credit.
Comment: In questions where two or more functions are under discussion (for example, a function and one or more of its derivatives), it is very important for students to make unambiguous references in both their labeling and their prose. In a written explanation, ambiguous references to "the function" or "it" are particularly troublesome when there is more than one "thing" to which these could refer. In general, graders do not infer a specific reference when there is more than one possibility.
FAQ: What happens if a student provides a perfectly correct final answer with no supporting work?
Answer: In the language of the grading, a "bald" answer is one unadorned by any supporting reasoning or computations. A correct bald answer may earn no credit, or at best, only minimal partial credit. And, of course, an incorrect bald answer has no chance of earning partial credit.
Comment: The instruction "Show all your work" is important throughout Section II of the exams. In some problems, students may be specifically reminded to include evidence of the reasoning, strategies, or computations they used to arrive at their answers. The phrasing of this reminder can take several forms, as illustrated below. These reminders add emphasis to the "show your work" instruction. The absence of such a reminder is not an invitation to ignore this instruction. The underlying message is the same for all problems: answers need support to be complete.
"Justify your answer."
This requires a mathematical argument to back up the claim or conclusion. This is where the application (often with citation) of a theorem, property, or test is generally needed.
Examples: The existence of a value satisfying a given condition might be justified by showing that the hypotheses are satisfied for the relevant theorem, such as the Intermediate Value Theorem or Mean Value Theorem, or for a test such as the Second Derivative Test. Examples of ways that students can present their work for this type of problem are available in the official scoring standards for 2004 AB3, 2004 AB4/BC4, and 2003 AB4.
FAQ: Are sign charts acceptable in justifying either a local or an absolute extremum of a function?
Answer: A sign chart is an annotated number line that relates the graphical behavior (increasing/decreasing, concave up/down) of one function with the sign behavior (positive/negative) of another. Sign charts can provide a useful tool to investigate and summarize the behavior of a function. We commend their use as an investigative tool. However, sign charts, by themselves, will not be accepted as a sufficient response when a problem asks for a justification for the existence of either a local or an absolute extremum of a function at a particular point in its domain. For more details on this topic, consult "On the Role of Sign Charts in AP Calculus Exams for Justifying Local or Absolute Extrema," which is available as a .pdf file on the Calculus AB and Calculus BC Home Pages at AP Central©.
"Give a reason for your answer."
"Explain your reasoning."
"Explain your answer."
These reminders ask for a basic mathematical connection to back up the answer to a "This or that?" kind of question, especially questions that ask for an interpretation. Another version of "Give a reason for your answer" that might follow a "Yes or no?" question is "Why or why not?" Here are some of the forms of questions that are often followed by such a reminder: Is *blank* increasing or decreasing? concave up or concave down? speeding up or slowing down? OR: Does *blank* happen? How many times does *blank* happen?, etc.
Examples: The mathematical (calculus) connection given as a reason might make some reference to the sign or value or behavior of a function or derivative or integral. Examples of ways that students can present their work for this type of problem are available in the official scoring standards for 2004 AB1/BC1, 2003 AB6, and 2003 BC6.
"Show the analysis (or work) that leads to your conclusion."
This reminder prompts the student to indicate the methods used.
Examples: In finding an absolute extremum of a function on a closed interval, the student should show that all critical points as well as the endpoints of the interval were considered as candidates. In finding a slope value, the student should show how it was obtained. See the official scoring standard for 2003 AB2 and 2003 AB4 for examples.
"Show the computations that lead to your answer."
This reminder emphasizes that it is particularly important for the student to show how a final numerical result was obtained.
Examples: In finding an approximation to a value using a technique such as Euler's method or a Riemann sum or a difference quotient, the student should show the basis of the computation, not just the final numerical result. See the official scoring standard for 2003 AB3 for examples.
FAQ: Are there any computations that a student can perform with the calculator without need for showing intermediate computational steps?
Answer: Yes. On Part A (the first three free-response problems), students are assumed to have a graphing calculator that can 1) graph a function, 2) numerically solve an equation, 3) numerically compute the value of a derivative at a point, and 4) numerically calculate the value of a definite integral. A student can freely use a calculator for any of these purposes without showing any intermediate work, as long as the student clearly indicates using mathematical language (not calculator syntax) what the calculator is used for (i.e., what we refer to in the directions as the "setup"). With respect to the four capabilities just mentioned, this means: 1) labeling the function, the axes, and the scaling for a graph sketched from the calculator, 2) stating what equation the calculator solver is used for, 3) stating the function and the point at which the numerical derivative was calculated, and 4) stating the definite integral that has been calculated. Note, however, that while a student can sketch a graph from one obtained from a calculator, that graph may not be sufficient as the basis of a mathematical argument. - Justifications require that you give mathematical (noncalculator) reasons.
Comment: When instructed to "justify" an answer, students are expected to provide an explanation of the mathematical basis for their results or conclusions. For example, to justify the location of a relative extremum of a function, a student could invoke the First or Second Derivative Test accompanied by evidence that the hypotheses are satisfied. In other cases, a student could show that the hypotheses are satisfied for the relevant theorem, such as the Intermediate Value Theorem or the Mean Value Theorem. Statements of the form "from my calculator I can see that..." will not suffice. - Your work must be expressed in standard mathematical notation rather than calculator syntax. For example,
may not be written as fnInt(X2, X, 1, 5).
Comment: The terminology and notation of calculus provide the common language that students and teachers can use to communicate and share ideas. We might draw an analogy with standard language and regional slang. Calculator syntax (that varies with make and model) might make sense to those who use the same type of calculator, but could be nonsensical to someone using another type. On the other hand, it is reasonable to expect all students of calculus to recognize and use the standard mathematical notation for objects studied in calculus. - Unless otherwise specified, answers (numeric or algebraic) need not be simplified.
Comment: Yes, unless specifically instructed otherwise, it really is OK on the AP Exams to write 2+3 instead of 5, or x+5+2x-3 instead of 3x+2 and other such unsimplified expressions. Graders are looking to assess a student's knowledge of calculus, and not for these numerical or algebraic simplification skills. Nevertheless, if a computational result must be interpreted or used in a later part of the problem, a student may find it very useful to write that result in a simpler form. Thus, teachers need not feel apologetic for demanding a more stringent simplification standard in their own AP classes.
FAQ: Does a definite integral count as an unsimplified final numerical answer?
Answer: No. Unless the problem specified that the answer be reported as an integral expression without further evaluation, a student would be expected to compute the value of a definite integral. - If you use decimal approximations in calculations, you will be graded on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point.
Comment: With the use of calculators on the exam, some line had to be drawn in evaluating the accuracy of numerical answers reported in decimal form. The three decimal place standard has been used every year since 1995. Note that the standard can be overridden in a specific problem. For example, in an application problem the student could be asked for an answer rounded to the nearest whole number. A recurring difficulty encountered in the grading is the inappropriate application of this standard to all calculations. If a student rounds intermediate computations to only three places on the road to the final answer, this premature rounding may result in a failure to achieve the desired accuracy in that final answer. A useful calculator strategy for dealing with these intermediate results as accurately as possible is to store them. For example, the limits of integration on a definite integral might need to be computed using the calculator solver to find the intersection points of two graphs. Rather than rounding these values, a student should simply store the computed values as A and B (provided these particular letters are not used in some other way in the problem). This would eliminate any need to retype several digits repeatedly while maintaining as much precision as possible. In writing the integral, the student should show these limits given to at least three decimal places, but in computing the integral, the student could use the stored values. Alternatively a student could write out the more precise numerical values of A and B, and then simply use the labels A and B as appropriate. Examples of ways that students can present their work for this type of problem are available in the official scoring standards for 2003 AB1/BC1.
FAQ: MUST answers reported in decimal form be rounded to only three decimal places?
Answer: Definitely not -- reporting more accuracy would not be penalized! The standard refers to the minimal accuracy expected in a final decimal answer. It should not be read as a requirement to round or truncate decimal answers. - Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
Comment: Nothing new here. This domain convention, common to most first-year courses in calculus, has been in effect for many years.
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