Author Archives: Lin McMullin

Practice Exams – A Modest Proposal

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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

The Lagrange Highway

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Recently, there was an interesting discussion on the AP Calculus Community discussion boards about the Lagrange error bound. You may link to it by clicking here. The replies by James L. Hartman and Daniel J. Teague were particularly enlightening and included files that you may download with the proof of Taylor’s Theorem (Hartman) and its geometric interpretation (Teague).

There are also two good Kahn Academy videos on Taylor’s theorem and the error bound on YouTube. The first part is here (11:26 minutes) and the second part is here (15:08 minutes).

I wrotean earlier blog post on the topic of error bounds on February 22, 2013, that you can find here.

Taylor’s Theorem says that

If f is a function with derivatives through order n + 1 on an interval I containing a, then, for each x in I , there exists a number c between x and a such that

\displaystyle f\left( x \right)=\sum\limits_{k=1}^{n}{\frac{{{f}^{\left( k \right)}}\left( a \right)}{k!}{{\left( x-a \right)}^{k}}}+\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}}

The number \displaystyle R=\frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n+1}} is called the remainder.

The equation above says that if you can find the correct c the function is exactly equal to Tn(x) + R.

Tn(x) is called the n th  Taylor Approximating Polynomial. (TAP). Notice the form of the remainder is the same as the other terms, except it is evaluated at the mysterious c that we don’t know and usually are not able to find without knowing the value we are trying to approximate.

Lagrange Error Bound. (LEB)

\displaystyle \left| \frac{{{f}^{\left( n+1 \right)}}\left( c \right)}{\left( n+1 \right)!}{{\left( x-a \right)}^{n-1}} \right|\le \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-a \right|}^{n+1}}}{\left( n+1 \right)!}

The number \displaystyle \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right)\frac{{{\left| x-c \right|}^{n+1}}}{\left( n+1 \right)!}\ge \left| R \right| is called the Lagrange Error Bound. The expression \left( \text{max}\left| {{f}^{\left( n+1 \right)}}\left( x \right) \right| \right) means the maximum absolute value of the (n + 1) derivative on the interval between the value of x and c.

The LEB is then a positive number greater than the error in using the TAP to approximate the function f(x). In symbols \left| {{T}_{n}}\left( x \right)-f\left( x \right) \right|<LEB.

Here is a little story that I hope will help your students understand what all this means.

Building A Road

Suppose you were tasked with building a road through the interval of convergence of a Taylor Series that the function could safely travel on. Here is how you could go about it.

Build the road so that the graph of the TAP is its center line. The edges of the road are built LEB units above and below the center line. (The width of the road is about twice the LEB.) Now when the function comes through the interval of convergence it will travel safely on the road. I will not necessarily go down the center but will not go over the edges. It may wander back and forth over the center line but will always stay on the road. Thus, you know where the function is; it is less than LEB units (vertically) from the center line, the TAP.

Building a Wider Road

As shown in the example at the end of my previous post, it is often necessary to use a number larger than the minimum we could get away with for the LEB. This is because the maximum value of the derivative may be difficult to find. This amounts to building a road that is wider than necessary. The function will still remain within LEB units of the center line but will not come as close to the edges of our wider road as it may on the original road.  As long as the width of the wider road is less than the accuracy we need, this will not be a problem: the TAP will give an accurate enough approximation of the function.

Soda Cans

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A typical calculus optimization question asks you to find the dimensions of a cylindrical soda can with a fixed volume that has a minimum surface area (and therefore is cheaper to manufacture).

Let r be the radius of the cylinder and h be its height. The volume, V, is constant and V=\pi {{r}^{2}}h. The surface area including the top and bottom is given by

S=2\pi rh+2\pi {{r}^{2}}

Since \displaystyle h=\frac{V}{\pi {{r}^{2}}}, the surface area, S, can be expressed as

S=2V{{r}^{-1}}+2\pi {{r}^{2}}

To find the value of r that will give the smallest surface area we find the derivative, set it equal to zero and solve for r:

\displaystyle \frac{dS}{dr}=-2V{{r}^{-2}}+4\pi r

This will equal zero when \displaystyle r=\sqrt[3]{\frac{V}{2\pi }} and substituting into the expression above \displaystyle h=\sqrt[3]{\frac{4V}{\pi }}.

Then \displaystyle \frac{h}{r}=\sqrt[3]{\frac{\frac{4V}{\pi }}{\frac{V}{2\pi }}}=2, so h=2r. In the optimum can the height is equal to the diameter.

The thing is that very few cans, especially beverage cans are anywhere near this “square “ shape. The closest I could find in my pantry was a tomato sauce can holding 8 oz. or 277 mL. The inside dimensions are about 65 cm. by 75cm.  Compare this to the 12 oz. soda can holding 355 mL. The usual reason given for this departure from the mathematically best shape is the taller can is easier to hold especially for children.

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What got me interested in this was the video below. While there is no overt calculus mentioned, there is a lot of math. There are also STEM considerations, specifically engineering. As you watch look for the math and engineering ideas that are mentioned and discuss them with your class. Here are a few:

  1. Geometry: Why a cylinder? Why not a sphere or a cube?
  2. Engineering: When cutting circles out of rectangular sheets of aluminum there is a lot of unused metal. Why is all this waste not a problem? This goes to materials engineering; steel is more difficult to recycle than aluminum.
  3. Math: Efficient packing is also a consideration. Check the calculations in the video as to the most efficient way (least empty space) to pack containers. Why do they not use the most efficient?
  4. Geometry: The (spherical) dome is a very strong shape. In what other places are domes used? Why?
  5. Engineering: How does pressurizing the cans make them stronger?
  6. Geometry and Engineering: The elongated ridges on the sides of non-pressurized steel cans strengthen the sides. How are these ridges similar to the dome or circular arch?
  7. Physics: Look for a discussion of first- and second-class leavers.
  8. Engineering: What other advantages are there to using the very thin aluminum can.

At the end of the video 6 other videos are mentioned. These are also interesting and show the same process in cartoon form and in video of the machines making cans. The links to these are here:

Rexam: http://www.youtube.com/watch?v=7dK1VV…
How It’s Made: http://www.youtube.com/watch?v=V7Y0zA…
Anim1: https://www.youtube.com/watch?v=WU_iS…
Anim2:https://www.youtube.com/watch?v=hcsDx…
Drawing: https://www.youtube.com/watch?v=DF4v-…
Redrawing: http://www.youtube.com/watch?v=iUAijp…

Teaching AP Calculus – The Book

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I am happy to announce that the third edition of my book Teaching AP Calculus is now available.

Teaching AP Calculus - Third Edition

Teaching AP Calculus is a summer institute in book form. The third edition is one-third longer than the previous edition and contains more insights, thoughts, hints, and ideas that you will not find in textbooks. There are references to actual AP Calculus exam questions to help you understand how the concepts are actually tested. New teachers will find a place to begin, and experienced AP teachers will find a wealth of new ideas. Whether this is your first year or your twenty-fifth, there is something here for you.

The book has 295 pages of information with 23 chapters in three sections, plus 4 appendices and an index.

Section I The first section of Teaching AP Calculus is about what you should know to get started teaching an AP calculus course. It will tell you where to find resources. The Philosophy and Goals are explained. There is a chapter on finding and recruiting students, pacing and planning the year. A chapter is devoted to technology, especially the use of graphing calculators; this is an important part of the course. The last chapter in the section talks about the prerequisites and things students should know before they start AP calculus.

Section 2 The middle section of Teaching AP Calculus is the longest. In it all of the topics that should be included in the AB and BC courses are discussed: limits, derivatives and their applications, definite integrals and their applications, differential equations, and the additional topics of parametric and polar equations, and power series that are tested on only the BC exam.

These chapters present ideas about how to present the topics. The chapters include some classroom activities. The last chapter is concerned with the writing that students must do on the exams: how to justify and explain their answers.

Margin references lead the reader to actual AP Calculus exam questions on all the important concepts.

Section 3 The last section of Teaching AP Calculus is about the AP exams. Here you will learn how the exams are made up and graded. You will learn how to read the scoring standards. The “type” questions on the exams are each discussed in detail along with what your students should know about them. The final chapter is for you and especially your students. It has lots of information and hints on how to do well on the AP calculus exams.

Teaching AP Calculus may be ordered online at http://www.dsmarketing.com/teapca.html. The website includes sample sections from the book and downloads of calculator programs mentioned in the book.

I hope both new and experienced teachers will find Teaching AP Calculus useful  and informative.

AP Summer Institute leaders: To obtain complimentary examination copy of Teaching AP Calculus, third edition, to show your participants email info@dsmarketing.com. Please include your full name, complete shipping address with zip code, and the location and date of your APSI. 

May

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Only a few days until the AP Calculus Exams!

Time to get psyched-up!

Here is some final advice to your students about How, not only to Survive the AP Calculus exam, but to prevail …

And a previous post on Getting Ready for the Exam with last-minute advice.

Good Luck to all your students – but you’ve done a good job so luck won’t really be necessary.

 

Looking forward to the summer, I am leading two BC Calculus Advanced Placement Summer Institutes. Here is the information:

AP Summer Institute at TCU in Fort Worth, Texas

TCU pix

  • For experienced BC teachers
  • Monday June 15 to Thursday June 18, 2015 from 8:00 AM to 4:30 PM
  • Information and registration: ap.tcu.edu.
  • TCU’s Office of Extended Education
    Telephone: 817.257.7132
    Fax: 817.257.7134

 

 

AP Summer Institute at Metropolitan State University in Denver, Colorado

Metro in Denver

  • For new and experienced BC teachers
  • Tuesday July 14 to Friday July 17, 2015 from 8:00 am to 4:30 pm
  • At the Metropolitan State University, 890 Auraria Parkway, Denver, 80204
  • Information and registration:  http://www.coloradoedinitiative.org/2015-apsi/
  • The Colorado Education Initiative
    1660 Lincoln Street, Suite 2000
    Denver, CO 80264
    (303) 736-6477 | (866) 611-7509 (f)
    info@coloradoedinitiative.org

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)

微积分

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 微积分 is Chinese for calculus.

I spent the last week in China and Taiwan doing two workshops for AP Calculus teachers for the College Board. It was an interesting and fun trip for me.

To get to China from Dallas you fly due north almost to the North Pole and then turn left and fly south over Siberia, Mongolia, and into China.

Siberia from 37,000 feet looking east
Siberia from 37,000 feet looking west

The flight took 15 hours. In Shanghai I met the others in our group. Shanghai is a modern city of over 14 million people. There are many tall office buildings and apartment houses. Driving in from the airport in the evening I was struck by how dark the city appeared. The office workers had gone home and all the office lights were out. Neither the offices nor the apartments had the outside lighting so everything looked dark. There were many cars and of course the resulting traffic jams. Electric motor scooters were popular.We spent the next day sightseeing and shopping and resting.

Our first workshop was in Suzhou a city of 4.3 million a two-hour drive west from Shanghai. Actually, the two cities sort of run together with little open land between them. We toured the old part of the city – an area connected by narrow canals and bridges.

Then we continued another hour to the modern part of the city and got down to work setting up our classrooms and getting organized.

The two-day workshops started the next morning. The sessions were held at the Suzhou Foreign Language School, a K – 12 boarding school. The participants were teachers from all over China who, for the most part, were planning to teach AP next year. We had over 270 participants in AP Calculus, Economics, English Language and Composition, Geography, Psychology, Statistics, and pre-AP English, and an additional session for administrators. Most of the participants were Chinese teachers, the others were ex-pats from several countries teaching in China all with a good command of English.

IMG_0342

My calculus group at the workshop

My session was one of two in AP Calculus. The other was led by Tim Zitur, an American living in Singapore, who is an experienced table leader and workshop leader.  The sessions were conducted in English which all of the participant understood. The questions and discussions were very much the same as any workshop in the USA.

Students in China take AP courses so that they can apply to schools in the USA. In China, students take one test before they apply to college. Talk about high-stakes testing: a high score on the exam allows them to apply to the best colleges in China; a lower score prohibits them from applying to the best colleges. A difference of one point can move over 100,000 students from one category to the other. To avoid this, students whose families can afford it send their child to a school in America and use AP credit help then get accepted here.

We were always made to feel welcome. The school took the presenters to a very nice restaurant on the fourth floor of a local mall (you’d recognize a lot of the stores) where we had a Chinese dinner of about two-dozen courses! We all sat at a round table with a huge lazy-Susan as the dishes went around.

After the second day we were driven back to Shanghai; everyone fell asleep on the two-hour trip.

The next morning we were up early for a flight to Taiwan for our next meeting in Taichung. Taichung is a two-hour ride from the airport in Taipei. It is the third largest city in Taiwan with a population of 2.6 million.

The night we arrived we visited the Taichung Lantern Festival. The Chinese New Year’s season was in full swing. We entered what looked like a typical American fair – lots of small booths each serving a different kind of food. Then we headed to the display area. There were acres and acres of large colorful figures made of cloth stretched over heavy wire frames and lit from the inside with colored lights. Beautiful and difficult to describe. Notice the size of the people silhouetted in the pictures below.

Food at the Taichung Lantern Festival
It is the Year of the Ram
Taichung Lantern Festival
Taichung Lantern Festival

There were thousands of people of all ages in attendance, yet I saw no one smoking and there was not a bit of trash or litter on the ground.

Our meetings the next day were at the National Taichung Girls’ Senior High School. The purpose of this meeting was to introduce Advanced Placement to the teachers and administrators. The plenary sessions were in Chinese. While they were going on Tai Hus-Chang, the principal of the school, showed us around. The classes had over 40 girls each. The girls stay in the rooms and the teachers move from class to class. The pupils were very eager to speak to us and spoke very good English.

IMG_0407

Students from the National Taichung Girls’ High School with Tai Hsu-Chang, the principal, me, and Marty Sternstein, the AP Statistics presenter.

I had a look at an eleventh grade math textbook. The book was in Chinese of course. It included solid geometry, the three-dimensional equations of lines, solving systems of three equations by determinants (in the context of the intersection of planes), matrices (including translations and rotation matrices), and a complete chapter on the conic sections. The book had very little text, no sidebars, and very little in the way of pictures not related to the problems.  The mathematics was written in standard notation with English letters for the variables.

I lead two breakout sessions. The first for teachers was a quick introduction to AP Calculus. Students joined us for the second breakout session. I taught a demonstration lesson using the Rule of Four and technology to present the average value of a function. Since the “calculus” came only at the end the students seemed to understand what was going on well enough. My translator (for the teachers) was Dr. Bo Wang, vice-president of the College Board and the leader of the trip.

We were treated to another nice dinner by the president of the Parent Teachers Organization: ten delicious courses.

Shrimp
Tuna
Dessert soup
Braised pork

Then the next day was the long trip home. It was quite a trip and interesting to see mathematics, calculus, teachers, and students in another part of the world.