Tuesday, March 3, 2015

Things I'm doing in AP Calculus

I've been asked about things I do routinely in AP Calculus to help get kids ready for the exam throughout the year.  Last year I wrote about a lot of things my colleague and I did to help kids review at the end of the course (post here).  But this post is more about what I do with the kids from August to March (once we get back from spring break, we start reviewing).

#1:  How I structure tests
The AP Exam looks like this: 45 multiple choice questions worth 1.2 points each (=54 points) and 6 free response questions worth 9 points each (=54 points).  So my six unit tests consist of 15 MC questions worth 1.2 points each and 2 FRQs worth 9 points each, for a total of 36 points.  I also typically throw in an attainable bonus question at the end.  My test questions are taken as much as possible from released AP Exams.  If I run out of questions, I'll occasionally pull something out of a study book.  Full disclosure, the previous AB teacher did a lot of work on this end.  I've simply updated her tests a bit and added more multiple choice.

So, the big question--do I curve on an AP curve[1]?

No, not anymore.  When I did, I felt like the kids had a false sense of confidence.  If they got As on all their unit tests they felt they would easily get a 5 on the AP Exam.  Well, a unit test is a lot different from a cumulative exam...

So, I "curve" by (1) putting a bonus question in (typically worth 5 points BEFORE I calculate the percentage...HUGE) and (2) giving them the FRQs beforehand via what I affectionately call "AP Sets"...

#2: AP Sets
AP Sets are released FRQs that deal with what we're learning in class.  The kids have about one due every week.  On the day the Set is due, I randomly call on kids to do one of the parts on the board until all parts are completed.  They may use their AP Set as they explain to the class.  If it's perfect (and I mean I don't have to correct an iota), then they get 11/10 points.  If I have to make a small correction, they get 10/10, and then it goes down from there.  If they're not at all ready, they receive a zero, but they are welcome to come make up the assignment for full credit any time on their own time (before/after school, lunch).  If a kid doesn't get called on for that unit, then they are simply excused from this grade in the gradebook.

Here are the AP Sets I gave this year.  I print them off by unit, but you could very well give them all at the beginning of the year:

I give three to four AP Sets each unit.  Two of those sets show up on their exam pretty much verbatim.  So, the kids have access to half their test weeks in advance.  Are they memorizing answers?  I prefer to think of it as memorizing how to write their answers.  And I'm totally ok with that because in AP Calculus, students must memorize how to write mathematical justifications correctly.  If we don't give them tons and tons of practice with this and teach them how important writing is in mathematics, then we've really missed what I feel is one of the most valuable aspects of the AP curriculum.

#3:  Multiple Choice Packets
For MC practice, I give a packet of about 20 multiple choice questions for the unit we're currently in.  These are either past AP questions or questions I've found in textbooks or study books.  Since the College Board doesn't release MC questions every year, these can be harder to come by.  I can recommend Rogawski's AP prep (at the end of each chapter) for some very good (although maybe sometimes too difficult) MC questions.

In addition to the packet, I print off a slip of paper with numbers 1-20 on it (or however many questions they have).  I put an asterisk by the questions that are calculator active (I use this notation throughout the course).  Their MC questions are "due" two days before test day, at which point we trade and grade their slips at the beginning of class.  They then have two days to correct the questions they missed for all their points back.  They also have access to the questions up until the test this way.  I usually pull a few of these questions word for word on their test.

One thing I think is really important for practicing multiple choice is to force the kids to go back and find their mistakes.  These questions are just written way too well to have the students practice without fixing their errors.  They need to get used to the common distractors and learn not to fall prey!

#4:  Daily Multiple Choice
Nearly every day I pull up one of the secure practice exams from the College Board and ask the kids to discuss a 2-3 MC problems.  Because these are secure exams, it's important that the questions don't leave the classroom so I just project the PDF on the screen instead of printing them out and then I ask kids to toss any paper they might have used to solve the questions.  As they get better, I have them poll in their answers through something like PollEverywhere.com so that I can  better understand what our strengths and weaknesses are as a class.  But, primarily, this is a really good way to get them talking to each other and debating mathematics.

#5:  End-of-Semester Folders
At the end of each semester, we require the kids to submit a three-pronged folder with the following for half a test grade:

  • All quizzes and tests with corrections
  • AP Calculus AB course description from College Board's website
  • Four pages of formulas that I print off for them of colored paper

My kids joke that taking AP Calc with me is like preparing for some kind of war, and I guess I see where they're coming from.  But I want as many of my kids to pass this exam as possible and gain college credit, so--yes--maybe I do go a bit overboard. :)

What are some things you do throughout the year to review the AP Exam?

[1]  I call an "AP Curve" a curve where students who would have gotten a 5 would get an A; 4s get a B, etc.

Sunday, March 1, 2015

Two more things

Last week, Sam shared two organizational things he does that help keep his classroom running smoothly.  I love reading things like this from real-life teachers as opposed to promotional magazines.  So, I thought I'd share two things I do, too.  They're not life-changing by any means, but they do help me.

#1:  Class Baskets
I give a lot of handouts.  I put all the extra handouts in one of these three baskets:

Baskets labeled "AP Calculus," "Pre-Calculus," and "Algebra"
If students were absent (or if they lost a handout), they know where to look.

#2:  Copy Folders
I, like Sam, try to avoid making too many trips to the copier.  I have a folder labeled "To Copy" that I stick any papers in that need to be copied (revolutionary, I know).  Inside this folder, I also keep a post-it with the total number of students in each course.  After I make copies, I stick them in these folders and then pull them out whenever I need them:

The back folder is my "To Copy" folder;
the rest of the folders hold all my handouts prior to passing them out

Bonus:  Chocolate
I keep a small stash of slightly overpriced chocolate near my desk at all times.  Kids can be awesome.  But they can also be quite terrible.  Sometimes, though, my mind can be cleared with a little bit of sugar.  And, all of a sudden, the kid's words/actions do not seem quite so egregious.  Or the stack of tests to grade doesn't seem too overwhelming.  Or spring break doesn't seem so far away...

Also, my colleagues know I keep chocolate behind my desk and if they've had a bad day, all they have to say is, "Do you have some chocolate...?"

That's my bonus advice for you.  Deep, huh? ;)

Tuesday, February 10, 2015

More Volumes in Calculus {Student Edition}

A couple summers ago, I made some really beautiful (I think) models to represent the kinds of figures we find the volumes of in calculus (post here).  The models worked well last year; I think it made the "formulas" make sense to the kids.  But, I thought it'd be even better if the kids actually got to create some of these in class.  I just couldn't quite figure out how I wanted to do it, without making it a project and without taking up too much class time.  And then, a year late, an idea finally came to me.

First, I bought a package of forty 5.5x8.5" foam sheets that were self-adhesive on one side ($5 at Wal-Mart).  I stacked three sheets together (so I have thirteen "boards") to produce the base of the desired solid.

Then, I graphed two functions (y=2cos(x/2) and y=e^(x/4)) on Desmos, printed them off, and used them as stencils.  So, each foam board has a graph on both sides:

I don't really know why I chose these two graphs other than the fact that I wanted one increasing and one decreasing function, both only in the first quadrant.

Next, I made and printed different kinds of cross sections for the kids to use on cardstock (see file below: squares, rectangles, semi-circles, equilateral triangles, and isosceles right triangles.

And after that, the students did the rest of the work.  They worked in groups of 2-3 to create a solid with either base f(x) or g(x) (I assigned).  Then, they calculated the volume of their solid and put their answer in this table:

Here are some examples of their finished products:

Finally, here is everything you'll need if you want to do this with your calc kids, too!

The first two pages are the two graphs I used.  The next five pages are the cross sections that I printed off on cardstock.  The graphs/cross sections are sized to fit together.  All you'll need is some foam sheets, pins, and Sharpies. :)

Thursday, January 22, 2015


When we work u-substitution problems in calc, the kids sometimes drop things like powers or a base of e while they're re-writing their integral.  Also, sometimes they don't quite see which parts of the original integral they've taken care of, and which they still need to work on.

So, I had an idea about five minutes before I was to teach u-substitution this year.  I call it the highlight-out method.  I think it's easier just to show a slide with two examples rather than try to explain in words:

I had the kids "highlight out" the du portion so they could focus on what's left.  Alternatively, you could have them highlight u in one color and du in another.

It may help some; it won't help others, but I think it's a step in the right direction for me.


Another thing I get asked a lot is, "What happened to the du?"  This is a way I explain indefinite integrals that I've found helpful:
  • The indefinite integral symbol and the differential dx (or du or d-whatever) TOGETHER are a command that mean "Find the family of antiderivatives."
  • Once you have found an antiderivative, the two symbols disappear because you have completed the command.
  • You cannot have an integral symbol without a differential[1]; they're akin to a capital letter and period.
That has seems to help a little.  Nothing ground-breaking here, but just some thoughts on u-substitution.  Would love to hear other ideas!

Here's a slide that seemed to clear things up a little bit more:

One kid told me the last example actually shed a lot of light.  Hooray!

[1]  Yes, I know, technically you can; I've taken Calculus on Manifolds, but these are Calc AB kids, ok?

Sunday, January 11, 2015

Unit Circle Trig

I have an odd love for the unit circle.  I bet most math teachers do.  I had a professor in graduate school who said, "There's nothing left to be discovered in the area of trigonometry.  Just draw the damn unit circle and you're done."  I think that's why I love it so much.  There's so much information you can gather from such a simple representation.

This year in PreCalc, my team and I actually started the year with trigonometry.  So, the kids were introduced to the unit circle on the second or third day of school, I believe (I know...this post is like five months late).  Since we use the unit circle so much, I really wanted to give the kids a visual understanding of where all the ordered pairs come from.  So, in addition to giving them blank unit circles to fill out, I also gave them three triangles that fit onto their circles:

This is, obviously, completely blank, but the kids' triangles' sides
were all labeled (both on the front and back).

Here are the unit circles (I stole this off the Internet sometime ago...let me know if they're yours!).

And here are the three triangles that I created; each hypotenuse should be the same length as the radius of the circles in the previous document:

Using what they remembered from geometry and given that each hypotenuse has a length of 1, the students labeled the remaining sides of the triangles (on both sides of the paper).  Then, they placed the triangle that fit on each coordinate and the x- and y- coordinates were (hopefully) clear to see.

I had the students tape both their completed circle and their three triangles to the very front of their composition notebooks.


Another activity that we did, which I adapted from an article in Mathematics Teacher, was I created a huge "human unit circle."  I bought a cheap plastic tablecloth and drew a circle on it, but there are lots of ways to make one.  Then, I made cards for each coordinate on the circle.  I printed this twice on two separate colors: one for x-values and one for y-values:

I had half the kids pick up a green card (x-value) and half the kids pick up a yellow card (y-value).  Then, I asked them to find a student who had the other part of their ordered pair (we discussed how for most of the cards there were two options).  Once they found their partner, I asked them to place their ordered pair on the correct location on the unit circle:

Once they finished placing their cards, I picked up random ordered pairs and had them give me the corresponding angle, in both degree and radian measure.  

I thought the kids did really well with the unit circle this year.  Now the trick is to keep practicing it with them even though we're done with our trig units... :)