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A Summer Math Problem

Did you ever have a dream about math?  I do occasionally.  I had this dream the other night that involved math calculations and I have no idea whether I was trying to do the calculations while alseep or in that limbo land.

The dream involved me running a workshop at a school some morning. For some reason, the particpants were only men. A similar workshop was run in another room for women. In both workshops, everyone was required to shake hands.  In the afternoon, both groups got together and all the men shook hands with all the women.

For some reason in the dream, I was interested in whether there were more total handshakes in the morning or in the afternoon. Strange thing to dream about!

I tried to calculate whether there were group sizes when the total morning handshakes and afternoon handshakes were the same.  I couldn't find any, but then, I was sort of asleep.  It wasn't until I got up and 4 hours later, the dream came to me when I biking. I came home and put together a problem about it.

So concerning the number of handshakes, which is correct?  More importantly, can you prove your answer?

(A) There are always more handshakes in the morning.

(B) There are always more handshakes in the afternoon.

(C) There are no group sizes where the number of morning handshakes and afternoon handshakes are the same.

(D) There is a finite number of group sizes where the number of morning handshakes and afternoon handshakes are the same.

(E) There is an infinite number of group sizes where the number of morning handshakes and afternoon handshakes are the same.

Think about it.  If you would like the answer, click on the following link.

Handshake problem

If after reading through my solution, you have further insight into the problem, write me at sschwartz8128@gmail.com.

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Answers to Calculus Manual 12/25/17

We have added answers to the AB and BC Calculus manuals. Teachers can give this to students who want to check homework answers. Purchasers of the AB and BC solution manuals were sent download links and new purchasers of solutions will automatically receive the answers as well. They are also available as a separate purchase.

AB Answers

BC Answers

Available Now! 12/13/17

50 question diagnostic multiple choice test that cover every aspect of the AP Statistics exam. The student version of the exam is free and the solutions include an amazing Excel spreadsheet that will detail every statistics topic and the student understanding of it.  This is a wonderful tool to give a couple of weeks before the AP exam so students can see where their strength and weaknesses lie. Problems are broken down into topics in the 5 major sections of the AP Stat curriculum and further broken down into whether problems are calculation oriented or more based on reading and terminology. This product is quite similar to the wildly successful Test Your Strength: AP Calculus.

More information

R.I.P. - Newton, the Calculus Cat

Newton has made a number of appearances in precalculus, calculus and statistics problems in my materials. Many students fell in love with his antics and in 2006, I decided to make him a part of the website. One of my former students Sam Tsui (now a huge YouTube singing star), drew him into our logo. And when I decided to have cartoons made for my products, Newton played a prominent role. He has been a biker, an archaeologist, and a teacher. He has dived into pools, helped a swami, joined a carnival, and hung from a catenary. He has left this world but he still lives in all these drawings. I won't soon forget him. He was my friend.

Change to Calculus Cache - 10/1/17

Because the AP exam's multiple choice questions contain 4 choices, we have changed Calculus Cache so that all questions now have 4 choices. While the 5-choice version is available upon request, downloading the Calculus Cache student exam questions will get you the 4-choice version.  Purchasers of the solutions in Word format can login and when they download the solutions, they will get the 4-choice version. They will also be contacted via email with the download links.

Available Now - 9/1/17

Getting AP Calculus - Across and Down is a crossword puzzle. However, the clues are calculus problems and the answers are numbers 0 - 9 and negative signs.  It covers all aspects of AB calculus and the problems are more straightforward than the tricky AP calculus type question. This is avalable as a subscription ... MasterMathMentor members will receive a new set of clues every week starting September 14. They will be in order of the AP Calculus manual so you can give it to kids from day one of the course. Or you can give it at the end of the year as a review. It will work terrifically as a review for BC students as well.  If you are a member of MasterMathMentor, you will receive it by email every Thursday starting in September. Or click on the link below to get the student downloads every week.

For more information, click on the link.

Clue game revision - 7/1/17

The calculus doesn't change but the suspects definitely do. In 2006, we had suspects like OJ Simpson, Paris Hilton, Usher, Kobe Bryant, Martha Stewart, Jack Nicholson, Michael Jordan, John Kerry, Donald Rumsfeld, Kate Winslett, Tony Soprano, Lance Armstrong, Nemo, George Foreman and Emiril Lagasse. What student would recognize these names today? The only remaining names from our original list is Tom Cruise, Bill Gates, and Hillary Clinton. So we revised the clue cards with more contemporary names in the news. Purchasers of the Clue game can email me at team@mastermathmentor.com and I will send you the revised clue cards.

Now Available! - 4/1/17

Test Your Strength is a 50-question multiple choice diagnostic exam that focuses on all aspects of the AP curriculum. The student version is free and the solution version includes an amazing Excel spreadsheet template that quickly grades a student exam and provides a detailed report as to how the student performed in all questions as well as strength and weaknesses in all topics. There is an AB version as well as a BC version that includes both an AB/BC exam, similar to actual AP exam, as well as a BC only exam that focuses as much as possible on strictly BC topics.

AB Test Your Strength

BC Test Your Strength

Available Now!

This free compendium of 32 mathematical curves with exotic names like conchoids, foliums, pearls, conchoids, cissoids, strophoids, and devil curves are subject to students calculus skills in finding tangent lines, areas, volumes, and arc lengths. Algebraic skills are challenged and students are encouraged to use this workbook in conjunction with a computer graphing app.  Read more about it by clicking on the link.

Added 1/1/17:

S.A.T.^® Clue is completely new and represents the first material on MasterMathMentor.com not written by Stu. It was written by Dr. Sheldon Goldberg who has an extensive knowledge of the S.A.T. Mathematics exam including the recent revisions to the exam. The game is like our popular Calculus Clue game but this one has students determining their future college, future profession, and future city residence by solving clues in the form of sample S.A.T.^® math problems. It is fun and instructive and the student version without solutions is available free of charge. Check it out by clicking on this link.

Revised 11/30/16:

We upgraded our AP Statistics material, making downloads easier. Rather than individual files of 2 or 3 pages, they have been combined into larger files. New quizzes, worksheets, and exams have been added and we have also added Powerpoints for the solution versions.  Check it out: Revision.

Added 9/01/16:

Our AB and BC exams have been completely revised.  They are now in-line with our calculus manuals. Every essential topic is tested and in BC, the non-essential topics have been tested as well. New exams and quizzes have been added and for each, a two-day midterm comprehensive exam are included. Check out both versions:

AB Student Version          BC Student Version

Addition:

We have added a new section (chapter 38) to the AB student and solution manual covering integration applications. These deal with real-world problems involving the derivative of the accumulation function as well as the integral of a rate of change. These problems always show up on an AP exam and we are happy to include it as part of the manual.

You can download the student version off the website. People who have purchased the solutions will be sent the links to the new material.

Revised:

With the format of the AB and BC exams being revised for the 2017 exam as well as some extra topics being included,  we have completly revised Demystifying the AP Exam, and given it a spiffy new cartoon. All choices are A,B,C,D as well the new calculus topics covered.

Also,  we are taking the half-exams previously included in Demystifying the AP Exam and expanding them into full sample AP exams, 4 for AB and 4 for BC. You can use these as mock AP exams for student practice. Previous purchasers of both Demystifying products (multiple-choice and free response) will receive this free.  Look for these on our AP Exam Review page.

BC version of Diving In to AP Calculus

Just completed and available is the BC version of Diving In. There are 18 topics and in the same format as our popular AB version (4 multiple-choice questions and 1 free-response question per topic). This version covers the topics new to the 2017 exam (limit comparison test, alternating series errors, and absolute/conditional convergence). Click on this link and navigate to the bottom to get the FREE DOWNLOADS and purchasing information.

Updated

Several people asked for a review of AB concepts in the revised BC Calculus Manual. So, as of 6/19/16, we have a new review that is included in the BC manual. You can download the student version on the BC Manual page and customers who purchased solutions will receive the updated links via email. In the Non-Essentials section, we also added an extra topic - solving first-order differential equations.  Something to look at after the AP exam.

Updates because of the curriculum changes

Also, Diving In to the AP Calculus Exam AB has undergone a full update. The multiple-choice questions have changed from 5 choices to 4 choices (A,B,C, D) to match the next AP exam format. You can re-download the student version on AP Exam Prep/Diving In section of this website. Purchasers of solutions will find that if they login to the members area, the solutions are available there. We added a problem in L'Hospital's rule because of the curriculum update. It is # 37, the last problem set to open up the possibiliy that questions involving area using L'Hospital's rule can be asked. Also, Diving In to the AP Calculus Exam BC has just been added (see above).

We have made some slight additions to Step-by-Step AB. Because of the addition of L'Hospital's Rule to the AB curriculum, we added a problem that encompasses this concept for 0/0 and infinity/infinity. We also added a problem to BC Step-by-Step that covers the limit comparison test and absolute/conditional convergence.  Go to either page to download the updated version. Customers who purchased solutons can get updates by contacting me at team@mastermathmentor.com.

Finally Our Review pages have been revised as well. L'Hospital's Rule was added to AB and absolute/conditional convergence has been added to BC.  The pages have been cleaned up as well. You can download them by going to our Review page:

Available Now!

A complete revision of the AB Calculus Manual

Available to Order:  Calculus Cache of Hidden Treasures

This catalog of 588 AB questions and 244 BC questions is available now, free of charge, in subscription form. Answers and solutions are available in Word format allowing you to create your own exams or worksheets.  Every topic covered in extreme detail. Click below for more information about this exciting and extremely useful product that will become your own special treasure trove!

Get information about Calculus Cache of Hidden Treasures.

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Calculus on the Wall BC Powerpoints have arrived!

Announcing the new release of Calculus On The Wall BC - 8 animated and narrated powerpoint presentations to bring the classroom to life, similar to the AB on the Wall Powerpoints!  Click the video below to watch our introduction.  Or click the link below to learn more and download sample powerpoint slides.

Click here for info

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 Calculus Word Search Poster 

   Check out Newton the Cat's new
   Word Search Poster!

   Available now!

  Click here.

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 Diving In to the AP Exam 

Check out our FREE subscription. 36 sets of problems, each with 4 multiple-choice and a free response problem, with the difficulty of AP exam questions. And you can use them from day 1 of class.

Interested?  Check it out.

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Calculus On The Wall 

Announcing the new release of Calculus On The Wall - animated and narrated powerpoint presentations to bring the classroom to life!  Click the video below to watch our introduction.  Or click the link below to learn more and download sample powerpoint slides.

Calculus On The Wall information/downloads - Click here

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WHAT DO WE MEAN BY AVERAGE?

Over the years, I taught both AP calculus and AP statistics. While many of the same students took both courses, rarely was there a problem with students confusing ideas from both courses. In fact, rarely was calculus mentioned in statistics class and vice versa. It was almost as if calculus skills and statistics skills came from a different part of the brain.

The one concept that is common to both course is "average" or "mean." In calculus we have the average value of a function and the average rate of change of a function. In statistics, we have the arithmetic mean or average. Are there occasions when these concepts can be confused?

This document is a paper I was moved to write that examines the word "average" and when you complete reading it, you might find that there are times when an average is not what you think it is.  Read it and give it to your students after the AP exam. It will make for a great classroom discussion.

www.mastermathmentor.com/mmm-archive/Average.pdf

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STU's NEW BOOK HAS ARRIVED!

REA's All Access AP Calculus Review book was written by Stu. It covers AB and BC and was written for students to review the course for the AP exam. There are many example problems as well as a 15-question quiz for each chapter, two 22-question mini-tests, both AB and BC full review exams and 100 review flash cards. All problems have explained solutions. Everything is original and there are no duplicates from www.mastermathmentor. So for teachers, this is a wonderful source of both multiple choice and free response problems.

Check it out at http://store.rea.com/0738610844.html. Available from Amazon and other book-sellers. Not available on this website.

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To purchase the solutions - click here

Register Now! click here

Calculus, Ripped From the Headlines give MasterMathMentor members a weekly calculus problem that will engage students and give them a chance to see how calculus can be applied to topics found in the news.  Here are some examples:

. Tiger Woods putting on the Master's treacherous greens
. Shrinking a tumor
. Toyota's sudden acceleration problem
. Marketing Apple's new iWatch
. The change in calls to the police reporting suspicious packages after

   the Boston Marathon bombings
. The speed of the Japanese tsunami as it hit land
. The largest pizza in the world!

. The coming of the cicadas.

There are 40 problems, each based on a story that actually appeared in the news. The problems will be similar to the format of the AP free response questions with several sub-parts. Only calculus topics that are taught in a typical AP calculus course will be needed to solve these problems.

To LEARN MORE or purchase the soltuions - click here

Register Now! click here

BC Manual Changes

There have been some changes to the BC Calculus Student and Solution manuals, notably in various locations through the last 70 pages. Some problems were changed and reworked. Also, the order of certain topics were slighly altered. For instance, I never liked that I never actually named and stressed the nth-term test for convergence, which I believe students should always think of first before trying other tests. For teachers who have purchased the solution version, I would recommend that before the new school year, you download the lastest student version and email me anytime at team@mastermathmentor.com and I'll send you the links for the solutions.

RU Ready for Some Calculus?

After 8 months working on several books (including a calculus one for REA ... more on that later), we have a new product for you!

If you are frustrated that your AP calculus students cannot solve a simple equation, find the domain of a function, simplify a complex fraction, or find tan (p/3) without a calculator, this will certainly help.  This is a 50-page workbook covering 21 vital topics from precalculus with solved problems for students. The student version is completely free  and the solution version has 4 tests that teachers can use. Click here to read about it:

A New AP Calc Review

Check out the new review for the AP Calculus Exam. The paper version including student versions and solutions versions are still available and free. But this new version was created so you can review topic by topic by projecting a general situation on the screen and asking students what actions they would take. Available in both AB and BC versions.  Check it out by clicking below:

Go to the review page

We have a new server!

Our apologies if you have had trouble downloading products over the past several weeks. Our server has been upgraded and now is amazingly fast. What this means to you is that downloads will be much faster than before. I recommend that if you have purchased materials and want the latest versions of solutions, to contact me at sschwartz8128@verizon.net and I will send you links to them.  If you order solutions and the process hits a snag, don't worry. Contact us at the same address and we will get it working for you. Putting in a new server is like transfering one brain to another body and connecting all the "wires."

STEP-by-STEP - the BC Version

Similar to Step-by-Step - the AB version. This version contains 52 free-response problems emphasize the concepts from BC calculus while reinforcing those from AB calculus. People who have purchased AB Step-by-Step or BC Demystifying the AP exam will be contacted via email.

Check out Step-by-Step BC.

STEP-BY-STEP                                         

If you ever gave practice free-response problems to your students, you have probably been frustrated because you can only assign part a) or parts a) and b) because the other parts involve concepts not yet taught.  No more.  Step-by-Step is a set of 75 free-response AB problems that emphasize specific topics rather than the global approach of released AP problems.  As usual, the student version of these will be completely free. People who have purchased solutions to Demystifying the AP Exam will be notified by e-mail!

Check out Step-by-Step.

If you have recently downloaded the free student version of Step-by-Step, please keep your eye on the errata page. When a new product comes out, little errors always show themselves and users are always kind enough to send them in. We update the server immediately so the most current version is always available.

Flash drives replace CD's                                                                   

I had a problem burning some CD's for people who had ordered them. I considered buying a new CD burner when Ted mentioned that CD's are just plain old. So we decided to offer flash drives instead.  Anyone ordering answer keys on CD's will instead receive your material on a cool 256 meg flash drive which is yours to keep. If you'd rather have a CD, just email us and we'll send that out instead.

It's Here! - Demsytifying the Multiple Choice AP Exam          

A companion piece to Demystifying the AP Free Response exam has been written - this one concentrating on multiple choice problems. There will be two versions, 135 sample problems that have been organized as to topic for the AB exam and 80 sample problems that have been organized as to topic for the BC exam. As usual, these will be available FREE OF CHARGE. There will also be an option to get these problems reorganized in AP exam format - 28 non calculator problem and 17 calculator-active problems. There will be 3 such exams for the AB section and 4 such exams for the BC section. No longer serach for multiple choice questions on certain AP Calc topics. Check it out!

The Demystifying Page

AB and BC Clue Games Revised                                                          

Without changing any problems, the AB and BC Clue games have been revised. Graphics have been improved, especially in those problems where students need to use geometric methods to calculate definite integrals. All solutions have been boxed and a few typos fixed. Most important is a 30th problem set that allows teachers to pull out any of the 30 problems to have different possible solutions to the mystery, in case of a former student telling current students the solution. People who have ordered the AB and/or the BC game should have received the new links via email.

Introductory Pages to the AB Manual.                                                 

Many people have asked me why the AB manual starts on page 10. The answer is that I had an introductory section that was meant to give students a feel for calculus before getting down to the nitty-gritty. I never felt comfortable about making it available until I revised it and here it is.  Teachers who ordered the solution version of the AB manual will receive a link to the solutions to this introductory in the next few days. And now the manual starts at page 0!

New and Revised - the AB Exam Review sheet.                                

Many of you have used my AP exam review sheet - the one that asks students what they would do when they see certain words:

Like - Find the equation of the tangent line to f at x = k should be interpreted as taking the derivative of f and using the point-slope formula at x = k.

Many of you have commented on how effective that review sheet is right before the exam. But I have never liked its look and it has been completely revised in order to give students more room to write in the answers.  Best of all, the solutions are included in the document and are COMPLETELY FREE!  It is available now.

There are two versions of the review: The first one (#49) gives the questions in the typical order you taught them - starting with limits, through differential calculus and integral calculus. The second one (#50) gives the question in random order. While the second one is more representative of the AP exam, the first one is better for studying purposes.

Both files are 4.7 megs so they will take some time to download. The BC versions of this review will be out in a couple of weeks.

The reviews are at the bottom of the AB Manual Calculus section of this website or you can download them directly from here:

Review 1                         Review 2

AB and BC Exams Updated                                                                 

There have been updates to the AB and BC exams, with the BC exams completed 2/25/11. The solutions have now been boxed and many explanations are more thorough.

Also, there are now AP free response-type questions that have been added to both sets (7 for AB, 11 for BC). While I would use the new Demystifying the AP exam for review purposes, these new exams can be given throughout the year to show students that the routine nature of typical exams are not representative of what students need to do on the AP exam. For people who have purchased solutions of either set of exams, you are entitled to the reviwed solutions. Feel free to contact me at team@mastermathmentor.com.

New --- Math Madness                                                                       

A Math Tournament reminiscent of the NCAA Basketball Tourney complete with brackets and seedings and upsets is now available under Additional Materials. It isn't calculus but it allows your students to vie for the Kings of the school. 80 questions and answers along with a scoring program and full instructions are all FREE. Just click on the Math Madness link.

Updates/Errata now available                                                           

With so much material available on the MMM site, it is inevitable that errors creep in.  Many users of this website have kindly informed me of errors and I check out them all and if necessary, make changes to the documents and upload them to our server. But how do you know about the errors?  Some of you have told me of errors that have long been changed.

The Update/Errata page lists the documents that have been changed in the past three months, whether it is to student versions or solution versions. Checking this page periodically ensures you will have the most current version of all documents.  Student versions can be re-downloaded from the website, and if you purchased a solution, you can get the replacement file as well.  Update/Errata can be obtained using the yellow option on the left menu or by clicking on this link:

Click here to go to the Update/Errata page.

Demystifying the AB Calculus Exam - now available                     

Demystifying the BC Calculus Exam - (NEW!)  now available      

The story is always the same: A teacher has covered the entire AP Calculus AB or BC curriculum and feels good that he or she has addressed the entire gamut of problem types.  Yet, when the students look at the actual free-response questions in the exam, they wonder "when was that taught?' and "did we ever do a problem like that?' Frequently, these students are befuddled, especially when they haven't spent a lot of time reviewing past AP released exams.

With that concept in mind, these guides covers a good many topics that research has shown to have been part of most Advanced Placement Calculus AB and BC free-response questions in the past 20 years. I first look at them as separate entities, and then quickly see how they are incorporated into an actual AP test type problem. Finally, several of these topics are put together to mimic what students will see in the actual AP Calculus Exam.  Best of all, they are FREE!

There are two manuals available - one for AB and one for BC;

  AB - a 74-page worksbook with 101 examples.

  BC - a 56-page workbook with 79 examples.

Most of the problems have subparts, just like actual AP free response questions.

     Click here to access the new AP Exam Prep Materials!

Multivariable Calculus                                                                       

More and more schools are looking to put in a course beyond BC Calculus. I was in that position 3 years ago and developed such a course as part of a dual credit program with a local community college. I have taken the extensive amount of material I developed and made it available on this website.  If you are in this situation, not having seen this material in many a year (let alone teach it), this is something you will want to check out. Either go to the Calculus section on this website or simply click the link below:

   Multivariable Calculus Course - Click Here!

 Calculus JEOPARDY                                                                         

Did you ever wish that you had more AP-type multiple choice problems for review? The released exams are great but they are few and far between. Well, get ready for Calculus JEOPARDY. This game, played like the Alex Trebek version will give you 5 problems in increasing order of difficulty in 24 different categories. There will be a JEOPARDY, Double JEOPARDY, Triple JEOPARDY and a BC Triple JEOPARDY plus two final JEOPARDY problems. Add that to the 15 sample questions that will be available on the web page and that is 137 questions that cover the entire curriculum.  

These questions are challenging as well - not the usual "take the derivative of 2x".  Included in the game will be a spreadsheet scoring device that will allow you to keep track of how much money students have won.  This will be a game that you can use to review for the AP exam or use daily for warmup problems.

     CLICK HERE FOR CALCULUS JEOPARDY!

New AP Statistics Game                                                                        

Download PERDI now! A Bingo-like game that will help students prepare for the AP Statistics exam. And it is FREE!  Click here.

Free Calculus and Precalculus Lessons on Video                                 

We recently became aware of a fantastic resource online.  Brightstorm (www.Brightstorm.com) has high quality videos from very good teachers, explaining most of the topics in Pre-Calculus and Calculus. The videos are FREE with registration on the site. With each topic, the teacher explains the concept, and usually walks through a few sample problems. Here are a few examples of specific Pre-Calculus and Calculus topics at Brightstorm:
Product Rule    Chain Rule    Graphs of the Sine and Cosine Functions

We have more extensive list of Brightstorm's Calculus and Pre-Calculus videos on our resources page.  Click here.

 A.P. Statistics On-Line                                                                        

As of 9/2009, A.P. Statistics has come on line with packets, worksheets, exams, projects and Fathom assignments. There are over 80 items you can download, all for free.