Tag Archives: AP Exam Review

Using Practice Exams

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bubble-sheetThe multiple-choice exams from 2003, 2008 and 2012 and all the free-response questions and solutions from past years are available online. The students can easily find them. 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. However, the rules about using the 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)

Practice 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 sometimes 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 to remove the exams when they learn of such a situation. Nevertheless, students have often able to, shall we say, “research” the questions ahead of their practice exams or homework assignments. Teachers are, quite rightly, upset about this and considered the “research” cheating.To deal with this situation I offer …

A Modest Proposal

If you give a practice exam, DON’T GRADE IT or count it as part of the students’ average. Don’t grade their homework if you assign the released questions.

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 they will use to it maintain their skills, identify their weaknesses, and improve on them, and how this will help them on the real exam. By taking the pressure of a grade away, students can focus on improvement.

Make an incentive of this, by not making students concerned about a grade.

This post is a revision of my post of June 6, 2015. There are some good comment and suggestions from readers of the blog. Check them out here

Next posts:

 

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

 

(Confession: When I was teaching I often had nothing to base a fourth quarter grade on. The 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.)

 

 

 

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Resources for Reviewing

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Here are several resources that will help you get started with your review

  • Released free-response questions from the College Board. AB and BC.
  • Released multiple-choice questions.
    • 2012 from the College Board are here for AB and here for BC FREE (.PDF)
    • 2008 AB and BC  College Board store cost $30.00 Paper (Search on-line and you should be able to find a copy, but so can your students.)
    • 2003 AB and BC College Board store cost $42.00 Paper (Search on-line and you should be able to find a copy, but so can your students.)
    • 1998  AB Exam Free (.PDF)
    • The 2013 – 2016 Secure Exams are available at your audit website.
  • Type Analysis 2018 by the 10 type questions that will be discussed in later posts. (by. Lin McMullin)
  • 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.The free-response instruction have changed slightly from previous years. The change is not a policy change, but rather made to emphasize certain things that students should be doing. For more on the changes see NCTM Calculus Panel Notes.
  • Calculator Skills needed on the AP Exams – 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 posts:

Friday February 24: Using Practice Exams

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

Revised 4-17-17

 

 

AP Exam Review

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Don’t panic! It is not time to start reviewing.

I try to keep these posts ahead of the typical AP Calculus timeline so you can have time to think them over and include what you want to use from them (if anything).

Over the next 6 weeks I will post several times each week. The post will be previous posts on reviewing slightly revised and updated. Today’s post is “Ideas for reviewing for the AP Exam” originally posted on February 25, 2013.

Ideas for reviewing for the AP Exam

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 rear. 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 exams 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 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 (there are 10) in turn and give suggestions about what to look for and how to approach the question.

Simulated Exam

Plan to give a simulated (mock) exam. Each year the College Board makes a full exam available. The exams for 1998, 2003, 2008, and 2012 are available at AP Central  and the secure 2013 – 2016 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 day or in the evening. 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 exams. 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 to know is that they can omit or get wrong many questions and still earn a good score. Students may not be used to this (since they skip or get so few questions wrong 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 they 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.

Directions

Print a copy of the directions for both parts of the exam and go over them with your students. Especially, for the free-response questions explain the need to show their work, explain that they do not have to simplify arithmetic or algebraic expressions, and explain the three-decimal place consideration. Be sure they know what is expected of them.The directions are here: 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.

Next Posts:

Thursday February 23, 2017: A list of resources for you and your students in preparation for the exams.

Friday February 24: Using Practice Exams

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

 

 

 

April 2016 – Exam Review

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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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Calculus Camp

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Today I welcome a guest blogger. Robert Vriesman writes about his Calculus Camp. The annual camp is a great review technique. I was honored to be invited this year and had a great time helping the kids. Thank you Robert for the Blog and the weekend with your students

Many high schools around the nation have only eight to fifteen kids taking Calculus in any given school year. So what are the teachers at the Los Angeles Center for Enriched Studies (LACES) doing differently along with generous help from professors, math professionals, and some parents doing to attract upwards of 200 students to take Calculus each year? The answer…Calculus Camp!

Calculus Camp was first organized by me fourteen years ago when I was LACES Department Chair. The camp began with only forty students and just a handful of teachers, but the excitement generated by the opportunity to go to camp to help them prepare for the College Board Advanced Placement Test increased the number of students taking Calculus each year. The past three years LACES has had over 200 students taking Advanced Placement mathematics.

LACES was already a high-achieving school, but this did not mean there were not a lot of challenges. The camp was large scale effort requiring a large-scale commitment on the part of the mathematics department. Our objectives of our Calculus Camp are:
• to create a support structure necessary to make high achievement by all AP mathematics students a reality.
• to enhance all students’ achievement by creating an environment that would cause them to take a new look at higher levels of mathematics.
• to build a mathematics program so strong and inviting that a large percentage of students-perhaps even every student-could be prepared to successfully complete challenging mathematics courses such as calculus before leaving LACES.
• to further increase participation in Advanced Placement Mathematics classes and to improve the pass rate of our students taking Advanced Placement Mathematics classes.
• to provide an opportunity for students to meet and work with people actively involved in a career in mathematics.

The students load the buses at noon on a Thursday to travel to Calculus Camp in the San Gabriel Mountains 90 minutes north of Los Angeles. The students are kept quite busy over the course of the weekend with two study sessions on Thursday, and three each on Friday and Saturday. Over the weekend they put in as many as 24 hours doing Calculus. They take a mock test on Sunday morning as a way of gauging their progress over the course of the weekend.

Teachers from LACES, other teachers, professors, and professional mathematicians are invited to come to Calculus Camp to help the students of LACES. Dr. Michael Raugh of Harvey Mudd College (retired), Dr. Kyran Mish formerly of the University of Oklahoma, and Dr. D. Lewis Mingori of UCLA (retired) have come back year after year to help the LACES students. This past year Lin McMullin attended the LACES Calculus Camp for the first time! Former colleagues of mine C. Dean Becker and Ken Bailey have also been a huge help over the years. This interaction with the adult professionals is something that is different from the ordinary in their lives. The benefit to the students is not measurable in a traditional sense, but it is undeniable for all those that see it working during the weekend. The past few years has seen an increase in the number of former LACES students who return to Calculus Camp to help the current LACES students. These former students returning to Calculus Camp is a testament to what the camp has meant to their lives.

The students work in groups of four or five; the teachers and mentors respond when a group needs assistance. It is the other students in the group that are the first resource. Teachers act not as a tutor, but as a mentor ready to help a group of students who are working together on a problem with a direction or a suggestion, not necessarily with a solution. This group of students is sitting in a room with other groups of students working on other problems; a community all working together to the same end. They gain confidence from the group experience to be able later to go it alone.

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In between the study sessions there is lots of interaction among students, teachers, and mentors. Those at the camp see and feel the intangible benefits these students derive from the camp experience from this interaction. It was great to see an actuary named Alejandro Ortega playing volleyball with a group of students and to hear Nick Mitchell (a retired actuary) explain to a group of students at lunch explain just what Actuarial Science is. To see the college professors, interact with high school students, to see the students asking them questions in a comfortable setting in not an everyday occurrence.

What has all this meant to the students of the Los Angeles Center for Enriched Studies? We have had many more students go into fields involving mathematics than in previous years. There are at least three students who went to college intending to study Actuarial Science in the past two years alone. Our pass rate has improved dramatically on the College Board Advanced Placement Calculus Tests. More students are taking Precalculus courses than ever before because they too want to go to Calculus Camp. Out of a department of nine mathematics teachers, five different teachers are teaching a total of seven Advanced Placement Mathematics classes.

On Sunday morning the students take a mock AP test to demonstrate to themselves what they learned. It makes them fully aware of the testing format and the length of various sections of the exam so there are no surprises on the day of the actual AP test.

There is plenty of fun built into the schedule as well. There is a bonfire on Thursday night, and a concert on Friday night and a talent show on Saturday night. This year I invited a friend from mine from college days, Sgt. Major Woodrow English, U.S. Army (Ret.), who was the principal trumpet in the U.S. Army band in Washington, D.C. for 30 years to play a concert on Friday night (and reveille every morning). English came all the way from Virginia to attend the camp this year. The music he provided seemed to “set the tone” for the students. Listening to a world class musician and working with world class mathematicians inspired the students to work hard and to give their best. And “Woody” gained a new appreciation for teachers and their dedication to their students.

If you have questions about starting your own Calculus Camp contact me, Robert Vriesman, at rvriesman@hotmail.com.

April

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Update April 7, 2015: This video may have been more appropriate a week ago, but I had not seen it then.

 

 

Back to work: Review time  for the AP Exams is here. The AP Calculus exams this year are on Tuesday morning May 5, 2015. Most of you will be finishing your new work this month and getting ready to review. So I’m repeating most of my March 1 post here with the links to help you review. But first:

I’d like to invite you to the annual AP Calculus Panel Discussion and Reception at the NCTM Annual Meeting 

Date: Thursday April 16, 2014 from 6:00 PM to 8:00 PM

Location:

     Grand Ballroom Sections D/E 

     Westin Boston Waterfront Hotel,

     425 Summer Street, Boston, MA.  

The speakers will include

–          Stephen Kokoska, Chief Reader for Calculus

–          Vicki Carter – From the exam committee

–          Dennis Donovan – Question leader

–          Benjamin Hedrick – the College Board

–          Lin McMullin – moderator of the AP Calculus Community and host.  

After the panel discussion there will be a question and answer period, and a raffle.

Refreshments and adult beverages will be provided. The reception is free and no advance registration, conference registration, or RSVP is necessary. Just come, meet the panelists, and enjoy the discussion.

The reception is sponsored jointly by D & S Marketing System, Inc., Bedford, Freeman and Worth, and Hewlett-Packard.

As for reviewing: I suggest you review by topic spending 1 – 3 days on each type so that students can see the things that are asked for and the different ways they are asked. Most of the questions include topics taught at different times during the year; students are not used to this. By considering each type separately students will learn how to pull together what they have been studying all year.

Many of the same ideas are tested in smaller “chunks” on the multiple-choice sections, so looking at the type should help with not only free-response questions but many of the multiple-choice questions as well. You may also find multiple-choice questions for each of the types and assign a few of them along with the corresponding free-response type.

Ideas for Reviewing for the AP Calculus Exams

Calculator Use on the AP Exams (AB & BC)

Interpreting Graphs AP Type Questions 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)

Writing on the AP Calculus Exams

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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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