April 2016 – Exam Review

This is a copy of my March 2016 post. It is review time and here are the links to my post on reviewing for the AP Calculus Exams

This year’s AP Calculus exams are on the morning of Thursday May 5, 2016. This month AP Calculus teachers will continue their review for the exams. The links to past posts on getting ready and reviewing for the exams are below. I’m posting them ahead of time so you will have time to use them in your planning.

Exam questions:

Released free-response questions are available from AP Central. Click here for AB and here for BC. Released multiple-choice questions are available only to teachers at your AP Audit website (click on “Secure Documents” on the lower left of the screen. Remember that these four years’ exams (2012 – 2015) are not allowed to leave your room (literally) and they may not be posted anywhere on-line.

Indices to released exam questions:

Click the links to an index to the multiple-choice (2003, 2008) and free-response (thru 2015) exam questions. These are Excel spreadsheets; click the arrow at the top of any column and narrow your search by checking exactly what you are looking for. These were prepared by Mark Howell and are available thanks to Skylight Publishing. (www.skylit.com)

I have a shorter and much less detailed three-page free-response  (1998 – 2015) index and multiple-choice (2003, 2008, 2012 – 2015 ) index. Click here.The multiple choice question are grouped by the “Type Questions” referred to below. One interesting feature is that you can see at a glance the number of times each type question was asked from year to year; this may help you decide what to emphasize.

Getting Ready for the AP exams (links to past posts)

The AP Calculus Exams

Using AP Questions All Year

Ideas for Reviewing for the AP Calculus Exams

Practice exams – A Modest Proposal All the past exams are available online – what to consider when your students find them.

Writing on the AP Calculus Exam Don’t miss these 7+ FR points.

Interpreting Graphs AP Type Question 1

The Rate / Accumulation Question AP Type Question 2

Area and Volume Questions AP Type Question 3

Motion on a Line AP Type Question 4

The Table Question AP Type Question 5

Differential Equations AP Type Question 6

Implicit Relations and Related Rates AP Type Question 7

Parametric and Vector Equations AP Type Question 8 (BC)

Polar Curves AP Type Question 9 (BC)

Sequences and Series AP Type Question 10 (BC)

Calculator Use on the AP Exams

And some last-minute advice Getting Ready for the Exam

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Practice Exams – A Modest Proposal

Starting in 2012 the College Board provided full actual AP Calculus exams, AB and BC, for teachers who had an audit on file to use with their students as practice exams. These included multiple-choice and free-response questions from the international exam. (The 2012 exam has now been released and is no longer considered secure. All the practice exams since then are considered secure.) The free-response questions from the operational (main USA) exam are released to everyone shortly after the exams are given and their scoring standards are released in the fall. These are not secure and may be shared with your students.

The rules about using the secure practice exams are quite restrictive. I quote:

AP Practice Exams are provided by the College Board for AP Exam preparation. Teachers are permitted to download the materials and make copies to use with their students in a classroom setting only. To maintain the security of the exams, teachers should collect all materials after their administration and keep them in a secure location. Exams may not be posted on school or personal websites, nor electronically redistributed for any reason. Further distribution of these materials outside of the secure College Board site disadvantages teachers who rely on uncirculated questions for classroom testing. Any additional distribution is in violation of the College Board’s copyright policies and may result in the termination of Practice Exam access for your school as well as the removal of access to other online services such as the AP Teacher Community and Online Score Reports. (Emphasis in original)

Bubble SheetPractice exams are a good thing to use to help get your students ready for the real exam. They

  • Help students understand the style and format of the questions and the exam,
  • Give students practice in working under time pressure
  • Help students identify their calculus weaknesses, to pinpoint the concepts and topics they need to brush up on before the real exam.
  • Give students an idea of their score 5, 4, 3, 2, or 1.

Teachers also assign a grade on the exam and count it as part of the students’ averages.

The problem is that some of the exams in whole or part have found their way onto the internet. (Imagine.) The College Board does act when they learn of such a situation. Nevertheless, students have often be able to, shall we say, “research” the questions ahead of their practice exams. Teachers are, quite rightly, upset about this and considered the “research” cheating.

To deal with this situation I offer …

A Modest Proposal

Don’t grade the practice exam or count it as part of the students’ averages.

Athletes are not graded on their practices, only the game counts. Athletes practice to maintain their skills and improve on their weakness. Make it that way with your practice tests.

Calculus students are intelligent. Explain to them why you are asking them to take a practice exam; how it will help them find their weaknesses so they can eliminate them, how they will use the exam to maintain their skills and improve on their weakness, and how this will help them on the real exam.  By taking the pressure of a grade away, students can focus on improvement.

Make it an incentive not to be concerned about a grade.

______________________________

(Confession: When I was teaching, I often had nothing to base a fourth quarter grade on. School started after Labor Day and the fourth quarter began about two weeks before the AP exam (and ran another 6 or 7 week after it). Students were required to take a final exam given the week after the AP exam and then they were done. The fourth quarter grade was usually the average of the first three quarters.)

Update June 7, 2015: There are some good ideas in the replies below. Check them out.

Update 2 April 7, 2018. Several updates to the first paragraph.

Update 3: March 13, 2019


Writing on the AP Calculus Exams

The goals of the AP Calculus program state that, “Students should be able to communicate mathematics and explain solutions to problems both verbally and in well written sentences.” For obvious reasons the verbal part cannot be tested on the exams; it is expected that you will do that in your class. The exams do require written answers to a number of questions. The number of points riding on written explanations on recent exams is summarized in the table below.

 Year AB BC
2007 9 9
2008 7 8
2009 7 3
2010 7 7
2011 7 6
2012 9 7
2013 9 7
2014 6 3

The average is between 6 and 8 points each year with some years having 9. That’s the equivalent of a full question. So this is something that should not be overlooked in teaching the course and in preparing for the exams. Start long before calculus; make writing part of the school’s math program.

That a written answer is expected is indicated by phrases such as:

  • Justify you answer
  • Explain your reasoning
  • Why?
  • Why not?
  • Give a reason for your answer
  • Explain the meaning of a definite integral in the context of the problem.
  • Explain the meaning of a derivative in the context of the problem.
  • Explain why an approximation overestimates or underestimates the actual value

How do you answer such a question? The short answer is to determine which theorem or definition applies and then show that the given situation specifically meets (or fails to meet) the hypotheses of the theorem or definition.

Explanations should be based on what is given in the problem or what the student has computed or derived from the given, and be based on a theorem or definition. Some more specific suggestions:

  • To show that a function is continuous show that the limit (or perhaps two one-sided limits) equals the value at the point. (See 2007 AB 6)
  • Increasing, decreasing, local extreme values, and concavity are all justified by reference to the function’s derivative. The table below shows what is required for the justifications. The items in the second column must be given (perhaps on a graph of the derivative) or must have been established by the student’s work.
Conclusion Establish that
y is increasing y’ > 0  (above the x-axis)
y is decreasing y’ < 0   (below the x-axis)
y has a local minimum y’ changes  – to + (crosses x-axis below to above) or {y}'=0\text{ and }{{y}'}'>0
y has a local maximum y’ changes + to –  (crosses x-axis above to below) or {y}'=0\text{ and }{{y}'}'<0
y is concave up y’ increasing  (going up to the right) or {{y}'}'>0
y is concave down y’ decreasing  (going down to the right) or {{y}'}'<0
y has point of inflection y’ extreme value  (high or low points) or {{y}'}' changes sign.
  •  Local extreme values may be justified by the First Derivative Test, the Second Derivative Test, or the Candidates’ Test. In each case the hypotheses must be shown to be true either in the given or by the student’s work.
  • Absolute Extreme Values may be justified by the same three tests (often the Candidates’ Test is the easiest), but here the student must consider the entire domain. This may be done (for a continuous function) by saying specifically that this is the only place where the derivative changes sign in the proper direction. (See the “quiz” below.)
  • Speed is increasing on intervals where the velocity and acceleration have the same sign; decreasing where they have different signs. (2013 AB 2 d)
  • To use the Mean Value Theorem state that the function is continuous and differentiable on the interval and show the computation of the slope between the endpoints of the interval. (2007 AB 3 b, 2103 AB3/BC3)
  • To use the Intermediate Value Theorem state that the function is continuous and show that the values at the endpoints bracket the value in question (2007 AB 3 a)
  • For L’Hôpital’s Rule state that the limit of the numerator and denominator are either both zero or both infinite. (2013 BC 5 a)
  • The meaning of a derivative should include the value and (1) what it is (the rate of change of …, velocity of …, slope of …), (2) the time it obtains this value, and (3) the units. (2012 AB1/BC1)
  • The meaning of a definite integral should include the value and (1) what the integral gives (amount, average value, change of position), (2) the units, and (3) what the limits of integration mean. One way of determining this is to remember the Fundamental Theorem of Calculus \displaystyle \int_{a}^{b}{{f}'\left( x \right)dx}=f\left( b \right)-f\left( a \right). The integral is the difference between whatever f represents at b and what it represents at a. (2009 AB 2 c, AB 3c, 2013 AB3/BC3 c)
  • To show that a theorem applies state and show that all its hypotheses are met. To show that a theorem does not apply show that at least one of the hypotheses is not true (be specific as to which one).
  • Overestimates or underestimates usually depend on the concavity between the two points used in the estimates.

A few other things to keep on mind:

  • Avoid pronouns. Pronouns need antecedents. “It’s increasing because it is positive on the interval” is not going to earn any points.
  • Avoid ambiguous references. Phrases such as “the graph”, “the derivative” , or “the slope” are unclear. When they see “the graph” readers are taught to ask “the graph of what?” Do not make them guess. Instead say “the graph of the derivative”, “the derivative of f”, or “the slope of the derivative.”
  • Answer the question. If the question is a yes or no question then say “yes” or “no.” Every year students write great explanations but never say whether they are justifying a “yes” or a “no.”
  • Don’t write too much. Usually a sentence or two is enough. If something extra is in the explanation and it is wrong, then the credit is not earned even though the rest of the explanation is great.

As always, look at the scoring standards from past exam and see how the justifications and explanations are worded. These make good templates for common justifications. Keep in mind that there are other correct ways to write the justifications.

QUIZ

Here is a quiz that can help your students learn how to write good explanations.

Let f\left( x \right)={{e}^{x}}\left( x-3 \right) for 0\le x\le 5. Find the location of the minimum value of f(x). Justify your answer three different ways (without reference to each other).

The minimum value occurs at x = 2. The three ways to justify this are the First Derivative Test, the Second Derivative Test and the Candidates’ Test. (Don’t tell your students what they are – they should know that.) Then compare and contrast the students’ answers. Let them discuss and criticize each other’s answers.


 

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Getting Ready for the Exam

I think the idea of writing this blog came to me about this time last year when folks were looking for last-minute advice to give their students before the AP calculus exams. I had some ideas of my own and collected some from others. Here is a list in no particular order.

The review time

  • Concentrate your reviewing on the things you don’t know (yet). Try to pick up those details you are not too sure of.
  • Work as many actual AP problems as you can, but concentrate on the form and ideas. None of these questions will be on the test, but many very much like these will be.
  • With, or without your class, find one (or more) of the released exams and take it in one sitting with the time allowed for each section.  This is to get you used to the real timing and the fact that you may not finish one or more sections.

The day before the test

  • Take a good look at the various formulas you will need; be sure you have them memorized correctly.
  • Put fresh batteries in you calculator and be sure it is in radian mode.
  • Take the afternoon and evening off. Relax. Do something fun.
  • Get to bed early and get a good night’s sleep.
  • Have a good breakfast.
  • Bring a snack for the short break between the two sections of the test.
  • Get Psyched!

During the test

  • Don’t panic! There is no extra credit for 100%. You may miss quite a few points and still get a 5; and quite a few more and get a 3.
  • Concentrate on the things you know. If you don’t know a how to do a problem, go onto the next one.
  • Keep your eye on the clock. Just before the multiple-choice sections are over, bubble in anything you left blank – there is no penalty for guessing.
  • On the free-response section, do not do arithmetic or algebraic simplification – it is not required and simplifying a correct answer incorrectly will lose a point. And it wastes time.
  • Don’t get bogged down in a problem – if you are not getting anywhere, stop and go to the next part or next question.

Good Luck!

Ideas for Reviewing for the AP Exams

Part of the purpose of reviewing for the AP calculus exams is to refresh your students’ memory on all the great things you’ve taught them during the year. The other purpose is to inform them about the format of the exam, the style of the questions, the way they should present their answer and how the exam is graded and scored.

Using AP questions all year is a good way to accomplish some of this. Look through the released multiple-choice exam and pick questions related to whatever you are doing at the moment. Free-response questions are a little trickier since the parts of the questions come from different units. These may be adapted or used in part.

At the end of the year, I suggest you review the free-response questions by type – table questions, differential equations, area/volume, rate/accumulation, graph, etc. That is, plan to spend a few days doing a selection of questions of one type so that student can see how the way that type question can be used to test a variety of topics. Then go onto the next type. Many teachers keep a collection of past free-response questions filed by type rather than year. This makes it easy to study them by type.

In the next few posts I will discuss each type in turn and give suggestions about what to look for and how to approach the question.

Simulated Exam

Plan to give a simulated exam. Each year the College Board makes a full exam available. The exams for 1998, 2003, 2008 are available at AP Central and the 2012 and the 2013 exams are available through your audit website. If possible find a time when your students can take the exam in 3.25 hours. Teachers often do this on a weekend. This will give your students a feel for what it is like to work calculus problems under test conditions. If you cannot get 3.25 hours to do this give the sections in class using the prescribed time. Some teachers schedule several simulated exam. Of course you need to correct them and go over the most common mistakes.

Explain the scoring

There are 108 points available on the exam; each half is worth the same – 54 points. The number of points required for each score is set after the exams are graded.

For the AB exam the points required for each score out of 108 point are, very approximately:

  • for a 5 – 69 points,
  • for a 4 – 52 points,
  • for a 3 – 40 points,
  • for a 2 – 28 points.

The numbers are similar for the BC exams are again very approximately:

  • for a 5 – 68 points,
  • for a 4 – 58 points,
  • for a 3 – 42 points,
  • for a 2 – 34 points.

The actual numbers are not what is important. What is important is that students can omit or get wrong a large number of questions and still get a good score. Students may not be used to this (since they skip or get wrong so few questions on your tests). They should not panic or feel they are doing poorly if they miss a number of questions. If they understand and accept this in advance they will calm down and do better on the exams. Help them understand they should gather as many points as they can, and not be too concerned if the cannot get them all. Doing only the first 2 parts of a free-response question will probably put them at the mean for that question. Remind them not to spend time on something that’s not working out, or that they don’t feel they know how to do.

Resources

Here are several resources that will help you get started:

  • “The AP Calculus Exam: How, not only to Survive, but to Prevail…” – Advice for students on the format of the exam and do’s and don’ts for the exam. Print this and share it with your students.
  • The  AB Directions and BC Directions. Yes, this is boiler plate stuff, but take a few minutes to go over it with your students. They should not have to see the directions for the first time on the day of the exam. I have highlighted some of the more important directions
  • Calculator Skills – share this information with your students, if you have not already done so. There are only about 12 -15 points on the entire exam which require a calculator. A calculator alone will not get anyone a 5 (or even a 2). Nevertheless, the points are there and usually pretty easy to earn. The real reason calculators and other technology are so important is that when used throughout the year, they help students better understand the calculus.

The next post: The Graph Stem Question