Flipping

Posted on June 24, 2015 by Lin McMullin

Today I am happy to welcome a guest blogger. Bobby Barber is a mathematics teacher at Millville Senior High School in Millville, NJ. In this and the next post he shares his experiences with Flipping his AP Calculus class. He asks that you reach out with questions, suggestions, and stories of your own. Use the comment button at the end of the post. Bobby may be contacted directly at robert.barber.jr@millvillenj.gov

What is a “Flipped” Class?

A flipped class is one where the traditional lecture and note taking is done outside of class time; usually by having students watch a video lesson. The students can re-watch all or parts of the video if they don’t understand something without worrying about interrupting the teacher or having the rest of the class know they don’t get something. Students then do practice problems that would previously be done for homework in class with peer and teacher support. Other projects, explorations, and activities are also done in class. Hence, the traditional in-class and at-home routines have been “flipped.”

Why I Flipped:

I started teaching AP Calculus in 2010 after 8 years of teaching regular level math classes. During my second year teaching, our school went to an integrated math curriculum (IMP) for our regular math classes. The classes were discovery-based and very interactive and I loved teaching them. (Why we never used this for our advanced classes, I will never know.) I was one of two teachers that taught primarily IMP classes and we both had a lot of success with our students, both on local and state assessments. When our AP Calculus teacher changed positions, I was asked to teach the class.

I teach at the same high school that I attended, and my AP Calculus teacher was very popular and had a reputation as being a great teacher. I got a five on the AP exam when I was in his class, so I figured I would model my class after what he did: homework questions, lecture, examples, practice, homework, repeat.

Within a couple of weeks, I realized some things. First, I hated lecturing. Second, I wasn’t getting to know the kids at all, being that I was in the front of the room the whole time. My biggest problem, though, was that I was spending about half of each class going over homework problems. Almost every student was doing the homework and asking legitimate questions about it, but most of the questions had simple solutions. They messed up a sign, or a distribution, or some other arithmetic/algebraic concept that caused them to get the calculus question wrong. This gave them a negative attitude towards calculus, which they didn’t deserve. I thought that if they could get help along the way to avoid these types of mistakes, they would enjoy and appreciate calculus more.

I decided to let them watch videos of the concepts at home and do problems in class. During the summer that I thought of this, I found a lot of information on people already doing this and calling it a “flipped” classroom. I decided that I was going to try it during the next school year.

How I Started Flipping

Even though I was hell-bent on trying the flipped class out, I wasn’t sure how to do it. I teach in a Title 1 district where many students don’t have access to computers/internet at home. (This has improved drastically, but wasn’t great when I started). Also, no one had ever tried this at my school and I wasn’t sure what kind of support I would receive from the administration. I had other ideas for improving my classes, especially AP Calculus, so I decided to try them first. I had some success with these changes, but I still wasn’t convinced I was getting the most out of my time with the students, so I decided to give flipping a try.

I started by watching videos in class with the students. I would project the video and have the students take notes, then do a mini lesson afterwards highlighting and adding what I thought was necessary. I continued this for a chapter. The students seemed to like it, so I took it a step further and assigned videos for homework for the next chapter. I would post the videos on my website and e-mail the links to the students for them to watch at home. Students who didn’t have access would watch on my computer at the beginning of class each day. We then did problem sets in class and students helped each other and used me if they couldn’t figure something out themselves.

I was extremely happy with the results (just intuitive, but I knew I was on to something), so I went to my principal and guidance supervisor to request that I run my class like this permanently. Neither of them had heard of flipping, but once I explained what I wanted to do, they were both all for it. From some of the stories I have heard from colleagues in other schools and on social media, administrators can really ruin a school. I am in the exact opposite situation. My principal lets us try pretty much anything we think will help the students. Once I had her blessing, I ran with it and never looked back.

Benefits of flipping

I have seen a marked improvement in AP exam scores since I started flipping. I think there are several reasons for this improvement. Since flipping my class, I am able to cover way more material in class. On top of covering more material, I am able to cover that material at a deeper level. I have time for explorations, discovery, and quality student discussions in class, where I never had much time for that before because I was always rushing to get through the curriculum and answer students’ homework questions.

The one benefit that a flipped classroom gives me that I don’t know how else to get is the interaction with my students. I get to circulate and talk to every one of my students every day while they are doing math. I get to talk to them about their thought process with the math they are doing and about other things going on in their lives. I really get to know my students over the course of a semester and that helps me help them. I don’t know how else you can develop relationships with 50-75 kids seeing them for 85 minutes a day for 90 days (or whatever schedule your high school is on). These relationships are the biggest benefit of a flipped classroom. The students and I get to know and care about each other, which gives us extra motivation to work hard for each other.

Continued in the next post scheduled for Wednesday July 1, 2015

Practice Exams – A Modest Proposal

Posted on June 6, 2015 by Lin McMullin

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)

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 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

Posted on May 20, 2015 by Lin McMullin

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 T_n(x) + R.

T_n(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

Posted on May 13, 2015 by Lin McMullin

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.

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:

Geometry: Why a cylinder? Why not a sphere or a cube?
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.
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?
Geometry: The (spherical) dome is a very strong shape. In what other places are domes used? Why?
Engineering: How does pressurizing the cans make them stronger?
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?
Physics: Look for a discussion of first- and second-class leavers.
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

Posted on May 4, 2015 by Lin McMullin

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

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

Posted on May 1, 2015 by Lin McMullin

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

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

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

Posted on April 1, 2015 by Lin McMullin

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)