Today and Tomorrow

Today is this blog’s tenth anniversary!

 My first post was on July 15, 2012. At the time I was working with the Arkansas Advanced Initiative for Math and Science. I was thinking of a series of emails with teaching hints for the calculus teachers I was working with. It occurred to me that a blog format would be more useful to them and to others who stumbled across it. So, that’s how all this all got started.

This is my 492nd post in addition to the 98 pages available from the menu bar. As of this morning, the blog has had, 956,803 visitors and 1,628,857 page views – and counting.

Teaching mathematics is more than just proving the theorems and doing the standard examples.  I certainly have not posted about everything there is to know about calculus – which would be difficult, since I don’t know everything. It was never my intent to write an online calculus book or even cover all the topics in the course description. Textbooks do that well enough. I hoped to provide some insight and ideas to help teachers explain things.

But I seem to have little more to add. I have found little new to write about recently. For the past few years, as you’ve probably noticed, many of my post were lists of links to past posts of actual calculus content.

So, I’ve spent some time this month looking at all my past posts and sorting out the ones with real content from those linking to the content posts. I’ve added a new drop-down menu to the navigation bar at the top of the screen called Blog Guide. Here you will find all the content posts organized in a way that I hope you will find useful. (The “link” posts are not there but are still available if you’ve bookmarked any of them.)

Please take a minute to look at the Blog Guide. I hope its organization will help you find your way around. (The “Search, “Posts by Topics,” and the “archives” on the sidebar will also help.)  

From now on, the blog will be on autopilot, so-to-speak. There will be few new posts. If I get an interesting idea, I will share it, but will not be posting regularly.

Some of my best inspiration comes from readers. So, if you have a calculus topic you would like me to discuss or expand on, please email me here and I’ll see what I can do. (The address is also on the navigation bar under “About.”), Also, I would appreciate you letting me know of any typos or broken links.

If you click on the “Follow” link in the sidebar, you will receive an email whenever a new post appears.

I hope to have helped you at least a little and hope to continue to do so. Thanks for reading and supporting

Enjoy your summer and have a good school year.


Let ‘um Try!

Last week, I received an email from a mother whose concern really ticked me off. 

The mother of a student entering eleventh grade this fall wrote because her son wanted to take AP Calculus next fall. He will not be allowed to take the course because he missed the cut-off grade in his precalculus course by less than two percentage points.

That’s wrong!

The College Board, under prerequisites in the current calculus Course and Exam Description, states, “Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students.” Her son has done that. There is nothing about achieving a certain arbitrary average.

A percentage grade does not tell you anything useful. (I’ve discussed this before. See here and the opening paragraphs here.)

How can you pass a student in precalculus and then turn around and tell them they aren’t ready for calculus? If they are not ready, fail them in precalculus.

It is certainly reasonable to council a student with an average or below average grade. You can, and probably should, sit down with them and their parents and explain that they may find the AP course difficult, and to do well they will have to commit spending more time and effort than they may be used to. Offer them extra help – it’s you job! If their grade was a D or D– you can be a little more insistent that they think it over carefully. But to flat out deny them the opportunity is just wrong.

Passing Rate.

Often teachers and administrators are concerned about their passing rate in AP courses. I once attended a session at an NCTM meeting where a teacher explained how he achieved a great passing rate each year in AP Calculus. It is easy to do. He explained that he carefully weeded out those students who were not likely to do really well. Once in the course, if they were struggling, he counseled them not to take the exam. (Why waste your money?) So, if passing rate is your concern, that’s how to do it.

He ended with, “I don’t want those students in my denominator.”

Sorry, it’s not about you!

Let’s say you carefully select the students who take the course and later the exam. You have 10 students who take the exam, and they all get a qualifying score (3, 4, or 5). Well, great you have a 100% pass rate.

Now let’s say you let another 10 students into the course. Twenty kids take the exam and 15 get qualifying scores. Even better. An additional five students have earned qualifying scores. That’s what counts! You’ve done a better job! And the five students who did not qualify will benefit from having taken a college-level course and will be better prepared for math in college.

Your pass rate has dropped to 75%, but you’ve helped more students – and that’s what it’s all about.

When I was working for the National Math and Science Initiative (NMSI), part of my job was to vet schools who wanted to join our program. NMSI insisted that there be no artificial barriers or cut-off points for admittance into all AP courses. Students who had passed the prerequisite courses and wanted to take an AP course had to be admitted. And it worked: the schools had more students qualify every year than the year before.

If all your students are earning 3, 4, or 5, you are being too selective.

I gave the woman what advice I could. I hope it helps. I wish I could have done more for her and her son.

Pacing for AP Calculus

Some thoughts on pacing and planning your year’s work for AP Calculus AB or BC.  The ideas are my own and are only suggestions for you to consider.

Almost all textbooks provide an AP pacing guide among their ancillary material. You can consult the guide for your book for specific suggestions for the number of days on each topic or section.

Keep a copy of the latest Course and Exam Description handy. Changes in the exam are announced in this book; to keep up to date be sure you always read the following year’s edition which is available at AP Central shortly after the exam is given in May. The book contains the “Topical Outline” for the AB and BC courses. The topics listed here are what may be tested on the exams. What is not listed will not be tested. For example, calculating volumes by the method of Cylindrical Shells is not listed; any volume problem on the exam can be done by other methods. This does not mean you may not or should not teach the topics that are not listed if you believe your students will benefit from them. If you wish to teach them you may still do so. Students may use these methods on the exam; they will not be penalized for correct mathematics. Many teachers teach these topics in the time after the exam.


Get out your school calendar. The AP Calculus exams are usually given during the first week in May; the exact date will be at AP Central.

  • Count back about 2 school weeks from the exam date (don’t count your spring break week). Allow an extra week if you are prone to many snow days. This time will be used for review. (This brings you to a week or so into April.)
  • Count back two more weeks. I’ll discuss what this should time should be used for later. (Mid-march) This is when you should aim to be done the material and ready to begin review. Finishing by the beginning of March is even better.
  • Count the number of weeks between the beginning of school and the week above. (About 26 – 27 weeks if your start just after Labor Day; 28-30 weeks if you start in mid-August). This is the number of week you have to teach the material. Don’t panic: the AB course is taught typically in college in 30 – 35 classes in one semester. You do have time, but by the same token, you still need to stick with the calendar and keep you students on it as well.
  • Take half of this number and find the middle week of the year. This is sometime in early to mid-December. To allow equal time for derivatives and integrals, this is when you should finish derivatives and start integration. Don’t delay starting integration beyond the first class of the New Year.
  • Now plan your work so that you can do it in the time allowed. You all want your students to do well. It is not unknown for teachers to spend a few extra days now and then to give extra work on derivative. But this time adds up. Remember half the exam is integration; you need to cover that too. Don’t get behind.
  • If you are in an area where there are closings due to weather or other reasons, plan for them. You usually get some short warning that snow is coming. Be ready on short notice to post an assignment, a video to watch, or some other useful work on your website. If it looks like several days off, tell the students you will post the assignment daily and make them responsible for finding them and doing them.

Look over past exams. Learn what is tested and how it is tested and plan your time accordingly. Here are some hints as to where you can shave some time.


  • Summer assignments: Personally, I do not see the use in summer assignments. What is their purpose? To keep the material fresh in the kids’ minds, I suppose. But the good students will do it right away and then forget anyway over the summer, and the others, will forget “everything” over the summer and try the assignment at the end of the summer and get nowhere.
  • If you want to keep their minds on mathematics over the summer, assign a good book to read. Maybe they will spread that out over the summer. Reading suggestion: Is God a Mathematician? by Mario Livio.
  • Ideally, limits and continuity should be taught in pre-calculus. Work with your pre-calculus teachers and help them arrange their curriculum so that the things students need to know coming into calculus are taught in pre-calculus. This is one of the things vertical teaming can accomplish. (Incidentally, be sure they do not start learning about derivatives and the slope of tangent lines in pre-calculus as some textbooks do; the time is better spent elsewhere.) Remember the delta-epsilon definition is not tested and is optional.
  • DO NOT begin the year with a week or two (or even a day or two) of review of mathematics up to calculus. It won’t help. Later in the year when you get to one of those topics students “should” know, they will have forgotten it all over again. So instead of a week or two (or more) of review at the beginning of the year, plan 10 – 15 minutes of review when these topics come up during the year. (You’ll have to do this anyway.)
  • If the first chapter of your textbook is review, as most are, skip this chapter. Make your first night’s assignment to read this chapter and ask about anything they don’t remember. This chapter can be used for reference when necessary later in the year.
  • Do begin the year with derivatives (or limits and continuity if students have not studied this before). The very fact that this is new will help get and retain the students’ interest.


Here are some places you may shave a few days off while teaching derivatives:

  • Computing derivatives is important. Product rule, Quotient rule, Chain rule are all tested on the exam. But look at some past exams: the questions are not that complicated. It is rare to find “monster” problems involving all three rules together along with radicals and trig functions. Sure, give one or two of those, but the basics are what are tested. Furthermore, you can and should include these all thru the year, so students stay in practice.
  • Optimization problems: Building a cheaper box or fencing in the largest field with a given amount of fence are great problems. They do not appear on the AP exams (at least not since 1982). They do not appear because the hard part is writing the model (the equation); if a student misses this they cannot earn anymore points in the problem. If these problems were on the exam, missing the equation means the student could not go on and cost the student all 9 points on a free-response question. Finding maximums and minimum, which require the same calculus thinking and techniques, are tested in other ways. On the multiple-choice section, optimization questions, if any, are of the easiest sort. The model may even be given, and there will  be no more than one such question. Spend only a day or two on the modeling.
  • Related Rate problems: These questions do appear on the exams. A multiple-choice question on related rates may appear. As with any multiple-choice question it cannot be too difficult. Every few years a related rate question shows as part of a free-response question. You cannot cut this out completely, but you can shave some time off here if you are short of time.
  • Practice the differentiation skills, and later the antidifferentiation skills, and the concepts associated with derivatives by including them on all your tests. Make all tests cumulative from the beginning of the year; just a random question or two will keep them on their toes.
  • Look for and assign differentiation problems based on graphs and tables of values in addition to the usual analytic (equation) questions. Use your textbook; however, some textbooks are rather thin on questions with tables and graphs in the stem. Use released exams or a review book for sources.


  • As with derivatives, the finding of antiderivatives is important, but the antiderivatives, definite and indefinite integrals are not very difficult. There are no trig substitution integrals, and nothing too monstrous. Integration by Parts is only on the BC exam.  Give students lots of practice spread over the second half of the year.
  • Trapezoidal Rule is not really tested on the exams. Students do not need to know the formula or the error bound formula for the Trapezoidal Rule. Questions do ask for a “trapezoidal approximation.” Like the left-, right; and midpoint-Riemann sums approximations, these questions can be answered by actually drawing a small number of trapezoids and computing their areas. This should be done from equations, graphs and tables. This tests the concept and often the graphical interpretation, not the mindless use of a formula. Error analysis is tested based on whether the approximating rectangles or trapezoids lie above or below the graph. Simpson’s Rule is not tested.
  • Look for and assign integration problems based on graphs and tables of values in addition to the usual analytic (equation) questions. Use your textbook, released exams or a review book for sources.


The free-response and the multiple-choice sections of the exam contain some questions very similar to questions that are in textbooks and in contiguous sections of the textbook. These include:

The free-response and the multiple-choice sections contain some questions that are very different from questions that are in textbooks. This is because these questions are on topics from different parts of the year (limit, differentiation and integration concepts in the same question), and these questions are just not asked in the same way in textbooks. These include:

  • Rate/accumulation questions
  • Graph Analysis Differentiation and integration questions about a function given the graph of its derivative and functions defined by integrals
  • Motion on a line (AB), or motion in a plane (BC – parametric and vector equations)
  • Polar Equations (BC only)
  • Questions, both differentiation and integration, given a table of values.
  • Overlapping topics in the same question such as a particle motion question based on a graph or table stem, or a question about an important theorem based on value in a table.

The topics in this latter list pull the entire year’s work together. At first students find this disconcerting since they have rarely seen questions like these; so be sure they do see them before the test. Use these two weeks to pull these topics together and get your students thinking more broadly. This will lead naturally into the full-scale review; in fact, some of this work may profitably spill over into the review time.  Spend 2 – 3 days on each type using actual AP questions for each so the students can see the different variations on the same idea, and the different ways the same idea can be tested. (This is preferable starting the review with one complete free-response exam with 6 different type questions to do. However, later in the review you should do this.)

Another way to approach these problems is to include parts of them throughout the year as the students learn the topics tested in each part. Released multiple-choice problems can be used for this purpose as well.


Once the students are familiar with the style of questions, give them a mock exam. For the multiple-choice questions use one of the released exams or one of the genuine-fake exams in a good review book. Give the free-response questions from a recent year. If possible, give the mock exam under the same conditions and timing as the exam. This can be done on a Saturday. If you cannot get 3.25 hours in a row, then give the parts with their proper timing during class periods. Grade the exam according to the standards which are available at AP Central.  Teach them some good test taking strategies.

Spend a fair amount of time doing multiple-choice questions. The released exams from 1998, 2003 and 2008, 2012 and 2013 (and soon 2014) are available. You can also use questions from a good review book (AB or BC). Pay attention to the style and wording, as well as the concepts tested.

Make your calendar up in advance and stick to it. You won’t help the students by getting behind; in college they will have to go a lot faster than in high school. Help them get used to it.

I hope this helps you get started and keep a proper pace through the year.

Revised and updated June 6, 2021