Category Archives: AP Calculus Exams

Your AP IPR

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The AP Instructional Planning Report, IPR, is available today from your audit website, the same place you found your students’ scores. While we all like to see how or students performed individually on the AP exams, the IPR may be of more use to you. It will help you learn where the strengths and weaknesses of your students and your teaching are. Here are some suggestions on what to look for and how to use the report.

The first page contains graphs and data comparing your classes to everyone who wrote the exam. You can see how your students did overall, on the multiple-choice section, and on the free-response section.

The second page is more detailed and more useful in analyzing the results. Here you will find data by topic from the multiple-choice section, and by question for the free-response section. The numbers in the “group mean” column are your students’ average. The “global mean” column is the average of all the students who took this form of the exam.

At a glance you can compare your students with everyone who wrote the exam. If your results are higher, that’s great. If not, keep in mind that this may not be just a reflection on your teaching. If your school has open enrollment and requires that everyone write the exam, then you have to expect scores lower than average. That is not a bad thing for you or your lower scoring students. Students who write an AP exam and score one or two still do better in college than students who never took an AP course. By better I mean that they require less remediation, have higher GPAs, and more of them graduate from college on time than students who never tried AP.

Now, try this: for each topic on the list, divide your classes’ mean by the global mean. Your results will be greater than one if your students did better than the entire group or less than one if they did not do as well.  Even if the ratios are all under one, look for the topics with higher ratios. These are the topics your students learned well. The topics with low ratios compared to the others are where you need to find a different approach or spend more time next year.  This works even if all the ratios are over one.

I first learned this approach from Dixie Ross. Her take on IPRs which is worth reading can be found in her blog for AP teacher here.

Percentages Don’t Make the Grade

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Well, the AP exams have been written and the dust has settled. Folks are posting their answers on the Community Bulletin Boards. (I never post mine – too many mistakes.) The other thing that always gets discussed at this time of year is whether this year’s exam is more difficult or less difficult than last year’s.

I am sure this year’s was more difficult or less difficult than last year’s because it is impossible to make two exams of the same difficulty.

But it doesn’t matter.

The grades will reflect, as best as possible, that a student knows as much calculus as students with the same score did last year. That’s the important thing.

Because it is impossible for anyone or any group to make two exams of the same difficulty, percentages tell you nothing. The percentage of the number of points that a student earns out of the number possible tells you just that and nothing more. If the tests are not of the exact same difficulty, then percentages are meaningless.

What to do?

The Educational Testing Service (ETS) who writes and administers the AP exams for the College Board carefully pretests each question. Also, there are a number of questions from last year’s exam on this year’s exam. These questions, called equators, allow ETS to judge the difficulty of the other questions on this year’s exam compared to last year’s. It allows them to judge the ability of this year’s student cohort compared to last year’s. Each question is considered individually. Questions that score poorly or questions that identifiable groups of students do far worse compared to the entire group taking the test are not counted in the final score. (For example, in 2008 question AB 19 was not counted; too many missed it.) They compare the results of questions within each exam. With this information they “scale” the exams and decide on the cut points, the high and low raw scores that earn a 5, 4, 3, 2, or 1.

A teacher on a day-to-day basis cannot do so detailed an analysis. Yet we still need to give students grades. We need to scale the exams.  I was quite happy this year using a scheme Dan Kennedy suggested some years ago (see resources tab above). This worked quite well for me in BC Calculus and in 8th grade Algebra 1. Perhaps you have another system.

Percentages just don’t make the grade.

Update September 22, 2014: Matthew Braddock, Mathematics Instructor & Webmaster, at the Dr. Henry A. Wise, Jr., High School in Prince George’s County, Maryland sent me a GeoGebra applet that will calculate the grades using Dan Kennedy’s scheme described in the link above. It runs at a website so you do not need GeoGebra on your computer or iPad to use it. Simply enter the information and it will do the rest. Thank you Matthew. 

Update December 3, 2018. The link above is no longer active. This link is to a similar app by Dan Anderson on Desmos. Thank you Dan. For more on this scaling test see the post: On Scaling.

Updated: September 22,2014, Kennedy link fixed February 9, 2018

Getting Ready for the AP Exams

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Another month and it will be time to start reviewing for the AP exams. The exams this year are on Wednesday morning May 7, 2014.

To help you plan ahead, below are links to previous posts specifically on reviewing for the exam and on the type questions that appear on the free-response sections of the exams. I try to review by topic spending 1-2 days on each so that students can see the things that are asked for each general type. 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. Of course, I will also spend some time on just multiple-choice questions as well.

February 25, 2013: Ideas for Reviewing for the AP Calculus Exams

February 25, 2013: The AP Calculus Exams

February 27, 2013: Interpreting Graphs AP Type Questions 1

March 2, 2013: The Rate/Accumulation Question AP Type Question 2 

March 4, 2013: Area and Volume Questions AP Type Question 3

March 6, 2013: Motion on a Line AP Type Question 4

March 8, 2013: The Table Question AP Type Question 5 

March 10, 2013: Differential Equations AP Type Question 6 

March 15, 2013: Implicit Relations and Related Rates AP Type Question 7 

March 15, 2013: Parametric and Vector Equations AP Type Question 8 (BC)

March 18, 2013: Polar Curves AP Type Question 9 (BC)

March 20, 2013: Sequences and Series AP Type Question 10 (BC)

March 22, 2013: Calculator Use on the AP Exams (AB & BC)

At Just the Right Time

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This is about a little problem that appeared at just the right time. My class had just learned about derivatives (limit definition) and the fact that the derivative is the slope of the tangent line. But none of that was really firm yet. I had assigned this problem for homework:1

Find f (3) and f ‘ (3), assuming that the tangent line to y = f (x) at a = 3 has equation y = 5x + 2

To solve the problem, you need to realize that the tangent line and the function intersect at the point where x = 3. So, f (3) was the same as the point on the line where x = 3. Therefore, f (3) = 5(3) + 2 = 17.

Then you have to realize that the derivative is the slope of the tangent line, and we know the tangent line’s equation and we can read the slope. So f ‘ (3) = 5

In my previous retired years, I wrote a number of questions for several editions of a popular AP Calculus exam review book.2 I found it easy to write difficult questions. But what I was after was good easy questions; they are more difficult to write. One type of good easy question is one that links two concepts in a way that is not immediately obvious such as the question above. I am always amazed at the good easy questions on the AP calculus exams. Of course, they do not look easy, but that’s what makes them good.

Now a month from now this question will not be a difficult at all – in fact it did not stump all of my students this week. Nevertheless, appearing at just the right time, I think it did help those it did stump, and that’s why I like it.

______________________

1From Calculus for AP(Early Transcendentals) by Jon Rogawski and Ray Cannon. © 2012, W. H. Freeman and Company, New York  Website p. 126 #20

2 These review books are published by D&S Marketing Systems, Inc. Website

Today’s the Day

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For all the AP Calculus teachers out there, my best wishes for your students today as they take the big test. I hope they all do well and all your hard work and effort pays off.

Getting Ready for the Exam

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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!

What’s a Mean Old Average Anyway?

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Students often confuse the several concepts that have the word “average” or “mean” in their title. This may be partly because not just the names, but the formulas associated with each are very similar, but I think the main reason may be that they are keying in on the word “average” rather than the full name.

Here are the three items. We will assume that the function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b):

1.  The average rate of change of a function over the interval is simply the slope of the line from one endpoint of the graph to the other.

 \displaystyle \frac{f\left( b \right)-f\left( a \right)}{b-a}

2. The mean (or average) value theorem say that somewhere in the open interval (a, b) there is a number c such that the derivative (slope) at x = c is equal to the average rate of change over the interval.

\displaystyle {f}'\left( c \right)=\frac{f\left( b \right)-f\left( a \right)}{b-a}

3. The average value of a function is literally the average of all the y-coordinates on the interval. It is the vertical side of a rectangle whose base extends on the x-axis from x = a to x =b and whose area is the same as the area between the graph and the x-axis and the function over the same interval.

\displaystyle \frac{\int_{a}^{b}{f\left( x \right)dx}}{b-a}

Notice that when you evaluate the integral, the result looks very much like the ones above. This formula is also called the mean value theorem for integrals or the integral form of the mean value theorem. No wonder people get confused.

The three are closely related. Consider a position-velocity-acceleration situation. The average rate of change of position (#1 above) is the average value of the velocity (#3) and somewhere the velocity must equal this number (#2). Similarly, the average rate of change of velocity (#1) is the average acceleration (#3) and somewhere in the interval the acceleration (derivative of velocity) must equal this number (#2).

These ideas are tested on the AP calculus exams sometimes in the same question. See for example 2004 AB 1 parts c and d.

So, help your students concentrate on the entire name of the concepts, not just the “average” part.