Using Practice Exams

Posted on February 24, 2017 by Lin McMullin

The 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

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.)

Back to School – 2016

Posted on August 10, 2016 by Lin McMullin

BACK TO SCHOOL – The three words we love and we hate.

I’ve been touching up the blog a bit this week. I hope the changes will make it easier to find posts on the topics you are interested in, and the blog more useful in general.

The THRU THE YEAR tab on the navigation bar at the top of the page has been updated. Links to my videos are now included. The monthly listings are organized so that you can stay ahead of your syllabus; to give you time to incorporate a new idea you like. So if your school starts in September, the August posts are where to start. The order of topics is not a day-to-day listing for you to follow or not even the order you must teach them in. So feel free to look around and rearrange.

The latest post appears at the top of the page. Below that are the four FEATURED POSTS. I will keep the current group in place for a while. These four posts concern the new Course and Exam Description (CED) for the AP Calculus program. The AP Calculus Concept Outline and the Mathematical Practices will be a help you understand the AP courses. The Getting Organized post will show you a way to organize your course using a Trello board that is already started for you.

Below the Featured Posts are the latest posts, newest on top. At the end of each post is a comment button so you can add your comments, suggestions, and ideas. Please use it.

You can find topics by using the SEARCH box on the right below the featured posts. Below that is a POST BY TOPIC drop-down list and a chronological ARCHIVES drop-down list.

If you cannot find what you need please let me know and I’ll see what I can do to help. I really could use some suggests for posts. My email is lnmcmullin@aol.com.

My plan for the coming months is to write posts on simple graphing calculator use for AP Calculus – ideas that I hope you can use to help your students understand what’s going on better. Look for them.

I hope the blog will make your year a little easier.

Have a good one!

Summertime

Posted on June 1, 2016 by Lin McMullin

Flowers at Ashford Castle, Ireland.

Well school is over in most of the country, even if the folks here in the Northeast do have a few weeks to go . I am not planning to write much over the summer; I’d like to, but am short on ideas. So if you have any questions, suggestions, or just something from Calculusville you’d like to hear about please let me know, and I’ll see what I can do. My email address is lnmcmullin@aol.com

I have a vacation and then one AP summer institute planned. Other than that, I’m going to perfect my loafing techniques

Since many of you will be planning for next school year and the new Course and Exam Description (CED), I’ll leave the four featured posts below so you can find them easily. The College Board has produced a series of short videos on the new CED. Click here to see these Course Overview Modules.

And take some time off for yourself and your family as well.

Happy Summer.

What is a Solution?

Posted on March 15, 2016 by Lin McMullin

“How does one solve x + ln(x) = c algebraically?” A teacher asked that on the old calculus electronic discussion group (EDG) and it got me to thinking about what a “solution” really is.

We start by teaching students how to solve linear equations; the idea is to do some arithmetic and/or algebraic operations on an equation so that you end up with x = some number. And that’s what students learn: do some operations on an equation so that you end up with x = some number.

Next come quadratic equations. You can perform a series of operations (completing the square) and end up with x = two numbers. And then you learn two short cuts – factoring and the quadratic formula. (Okay, maybe you learn factoring first.) These solutions often involve radicals, and therefore, do not necessarily give a recognizable number.

Next come cubics and higher degree polynomials. Sometimes you can factor, and there is a cubic formula. There is also the rational root theorem and synthetic division that can help you find rational solutions to polynomial equations; but if the solutions are not rational, you may be out of luck.

Equations involving trigonometric functions always have solutions that are the so-called “special angles.” Exponential and logarithmic equations always involve convenient bases. So, everything can be solved that way – or not.

These kinds of answers are usually called “closed form” expressions. That is, they are given in a notation that indicates what arithmetic must be done to get a decimal answer. That is, 5192/13 means to get “the answer” divide 5192 by 13. Since decimals are not always “exact,” the closed-form answers are preferred – they are “exact.”

We rarely consider solving by graphing by-hand, graphing calculators, computer algebra systems (CAS), or searching the internet. Why not? Because by-hand graphing is not very accurate, you only get numerical answers, graphing is “just for checking,” and of course technology is not allowed on the state exams.

So, returning to the EDG here are some of the answers that were offered to the question “How does one solve x + ln(x) = c algebraically?”

- You don’t
- You can’t
- Sure, you can like this:
  - $x+\ln \left( x \right)=c$
  - $\displaystyle {{e}^{x+\ln \left( x \right)}}={{e}^{c}}$
  - $x{{e}^{x}}={{e}^{c}}$
  - $x=\text{Lambert}W\left( {{e}^{c}} \right)$
  - Where Lambert(W) is the Lambert W-function named after Johann Heinrick Lambert (1728 – 1777).

Then folks started complaining.

Someone wrote “’Naming’ a previously unknown function isn’t ‘solving’ a problem.”

To which someone else replied

Before Joe Arcsine invented his function, one could not “solve” sin(x) = 0.123
Before Betty Logarithm invented her function, one could not “solve” ${{e}^{x}}=0.123$
Before Johann Lambert invented his function, one could not “solve” $x{{e}^{x}}=0.123$

And he is correct – well okay, his idea is correct, even if he may not have ascribed the solutions to the correct mathematicians. We actually do it all the time. Functions like the arcsin(x), ln(x), and the like are simply names given to functions for which we may not have a series of arithmetical operations leading to a closed-form solution. (Taylor series, being infinite in length, are not closed form.)

Simplifying

Then there are the answers themselves. Simplifying makes things easier.

x = 3, x = 0.125, $x=3.7\overline{53}$ are fine, but x = 4.567… not so much.
x = ½ and x = $\displaystyle \frac{5192}{13}$ are great, but x = $\displaystyle \frac{5192}{33}$ is not.
We like x = $\sqrt{6}$ , we like $\sqrt{7}$ , we’re not too sure about x = $\sqrt{8}$ , x = $\sqrt{9}$ is just plain wrong, but x = $\sqrt{10}$ is okay.
x = $\displaystyle \frac{2}{\sqrt{3}-1}$ cannot be correct because we have to rationalize the denominator; try that with x = $\displaystyle \frac{1}{\pi }$
Likewise, x = $\displaystyle \arcsin \left( \tfrac{7}{8} \right)$ is okay, but x = $\displaystyle \arcsin \left( \tfrac{1}{2} \right)$ is not.
Which of x = $\displaystyle \frac{\ln \left( 2 \right)}{\ln \left( 7 \right)-\ln \left( 3 \right)}$ and the equivalent x = $\displaystyle \ln \left( {{2}^{\frac{1}{\ln \left( \frac{7}{3} \right)}}} \right)$ is simplified?
When we differentiate tan(x) using the quotient rule we get $\displaystyle \frac{1}{{{\cos }^{2}}\left( x \right)}$ , but that has to be changed to ${{\sec }^{2}}\left( x \right)$ even though there is no secant button on a calculator.
And then there is LambertW(e⁴)

So, what have we learned?

Each kind of equation has a different process for finding its solution.
We are allowed to make up new functions to solve equations.
To the outsider (read: student) this looks like a hodgepodge – and for good reason.
Most of our “solutions” are really directions for finding the solution.

What’s Your Favorite?

Posted on January 20, 2016 by Lin McMullin

A very short post today. Audrey Weeks, author of Calculus in Motion, sent me this link to a BBC article by Melissa Hogenboom discussing the 10 most beautiful equations. Each equation can be clicked for more details and some of the links even have videos discussing the equation. Click here for You decide: What is the most beautiful equation? After reading about them, you can vote for your favorite and see the results so far.

When are we ever …?

Posted on December 10, 2015 by Lin McMullin

The following answer to a question we’ve all been asked was posted yesterday on a private Facebook page for AP Calculus readers. The author, Allen Wolmer is a teacher and AP Calculus reader. He teaches at Yeshiva Atlanta High School, in Atlanta, Georgia. I reprint it here with his kind permission.

Thank you, Allen.

A colleague of mine teaching high school math asked me the following:

“So some of my kids are struggling with the math concepts and ideas (why do I need to learn this? When will I ever use factoring/polynomials?). I would say most of my students have not had a particularly good relationship with math in the past.

I have answered these questions, with responses like financial jobs, accounting, econ, but they have no interest in these types of jobs/real world applications.

I have also talked about logic, problem solving, and puzzles.
Do you have any advice for me, and or ideas to share with them?”

Here is my reply:

“Glad to help.
First of all, recognize that, for the most part, the kids aren’t really interested in your answer. They are just being lazy and looking for a reason, any reason, to not do work, any work.
Now, how do you answer? Well, you can try the face value approach, that is describing engineering, actuary, etc., but that will be meaningful to just a handful of students, and they’re not the ones asking the question anyway.

So, I turn it around. I ask the boys how many go the gym or health club or weight room to work out. Hands quickly shoot up (after all, they want to be cool in front of their buds and the chicks). I then ask them how many use the pec machine (it’s the one where you raise your arms to shoulder level and bring the elbows together). I demonstrate with the motion. Many of them admit they use the machine. I then ask them when they would ever use that motion in real life. The answer, of course, is NEVER! So, why do you do it, I ask them. I answer for them “because it strengthens your muscles!”

And that’s why we study mathematics: to strengthen our mental muscles. We’re not really studying math, I tell them. We’re studying how to analyze and understand problems when we read them. What do we know? What do we not know? What are we trying to find?

We’re learning how to pay attention to detail. We’re learning how to be precise in our thinking and our work. We’re learning how to be neat in our work. We’re learning to look for patterns. We’re learning to look for similarities to other problems we have solved. We’re learning how to look at our results and determine if they make sense or not.
In other words, we’re learning how to think critically and solve problems. We’re doing it in the context of learning mathematics, but that’s only because that’s the best environment to do it in.

Then I tell them never to ask that #$%^%$# question again and to shut the #$%^&*&^% up and get back to work!

Sincerely,
Allen Wolmer”

December 2015

Posted on December 1, 2015 by Lin McMullin

Here are the past posts for December. The topics are those leading to integration and the Fundamental Theorem of Calculus. The four featured post below are the most popular December posts from past years.

I hope you all have a happy holiday season as you get ready for the New Year.

November 18, 2012 Antidifferentiation

November 26, 2012 Integration Itinerary

November 30, 2012 The Old Pump

December 3, 2012 Flying In to Integration Land

December 5, 2012 Jobs, Jobs, Jobs

December 7, 2012 Under is a Long Way Down

December 8, 2012 Inequalities

December 10, 2012 Working Towards Riemann Sums

December 12, 2012 Riemann Sums

December 14, 2012 The Definition of the Definite Integral

December 17, 2012 The Fundamental Theorem of Calculus