Slope Field Activity

I’m getting ready for Twitter Math Camp's calculus working group (ah!).  We’ve been asked to bring some activities to share with the group, so I’ve been frantically searching my blog this afternoon for stuff I can contribute.  I was typing up a list when I realized I don’t have a whole lot for the second semester of Calc AB.  I imagine a lot of that has to do with the fact that we’re in review mode for half the semester, but still.

However, I did realize that there was one pretty good lesson on slope fields that I didn’t blog about.  I probably didn’t write about it because I stole it 100% from my APSI instructor last summer.  Nevertheless, I’m not feeling that trepidation currently.  This magic should be shared.

• On the board, draw or project a blank Cartesian plane along with a differential equation.  There should be at least as many integer coordinates as there are students:

• Give each student a card with a coordinate on it.  {If you're as Type-A as I am, here are cards you can print for up to 35 students.  And if you're REALLY Type-A, you can print them on card stock, laminate them, and cut on the solid lines.  I love laminated cards.  Laminated cards make me feel like I just insured a valuable asset. Moving right along...}

• Each kid figures out the slope of the tangent line at the given point and draws a tiny line segment with that slope at the given point.

• When everyone is finished, they’ve all contributed to a graph that looks something like this:

My class's actual slope field;
not perfect, but whose slope field is?

For this example, we discussed questions like:

• What's the pattern for the slope to be zero? Why?
• What is the slope doing to the left of y=x?  Why?
• What about to the right? Why?

And I mean that was pretty much all they needed in the way of instruction.  Not that it's all that complicated to begin with, but this was a nice, everyone-get-up-and-contribute type of lesson.

Again, I can take zero credit for this.  But I thought it was worth sharing.

Class Consensus

I taught a five-day summer camp last week to prepare our incoming juniors and sophomores for the PSAT/NMSQT.  One of the greatest things about it, for me, was that the other teacher (English) and I have very similar approaches to teaching; that is, make the kids do the work and talk as little as possible.  We split the kids into groups quite a bit (half did math with me and half did English with her and then we'd switch); when we reconvened, we'd often shrug and say, "Well, that was easy."  We got to teach some pretty motivated kids (especially considering it was summer), and they were good at taking ownership for their own instruction.

That said, there were still times when I'd have an internal panic attack that went something like, "WHAT AM I GOING TO DO WITH THESE KIDS FOR THE NEXT HOUR AND TWENTY-FIVE MINUTES?"  Because it wasn't really "normal" school where I have to get through Section 4.1 today, please and thank you.

So, this little idea came from trying to stretch out what was supposed to be a 15-minute activity into a 30-minute activity.  Honesty is the best policy?

Last summer I attended a PD session on literacy.  Apparently, some stuff really stuck, such as this idea which (I think) the instructor called "Class Consensus." I've done this with some reading passages with moderate success.  But how I never thought to use it with math exercises is beyond me.

This is how it went down:  I gave each student a "Mini-PSAT Test," consisting of seven past PSAT questions.  They were given ten minutes to work this test on their own.  After the ten minutes, they compared their answers with their partner and were asked to reach unanimous consent.  Then, the group of two joined another group of two, and the new group of four was asked to also reach unanimity.  Then the groups of four made groups of eight.  At this point, I had written on the board the numbers 1-7:

Class Answers

1
2
3
4
5
6
7 

I asked the students to write down the answer to each question.  However, they'd better make sure the class agrees because, if a question was wrong, I wouldn't tell them which one was wrong.  I would only announce if they were all correct or not all correct.  I was a little worried about this getting hijacked by one or two students, but it really didn't.  Sometimes one person would go up and write the answers to all the questions (s/he had discussed it with the class first), and sometimes kids would go up one by one and write down an answer that they felt they were confident with once they had discussed it with their peers.  I did this with two different tests and with two different groups, and all four times the class got all the questions all correct without asking me anything (well, I refused to answer questions...).  A couple times, someone would say, "I still don't get Number 2," at which point I could say, "Who put up Number 2? Will you explain, please?"

It was kind of magical.  It's not that different from what I do a lot in class ("Do a problem on your own, then check with your partner"), but just tweaking this a bit generated a lot more conversation and forced kids to talk math with people other than just their partner.  Also, since you're getting so many opinions, it's unlikely the answers will be wrong once you've checked with your entire class.

I'm finding that little activities--for lack of a better word--like this are so valuable to have in my teaching arsenal.  While they might not be anything glamorous, they can really get the job done and spark conversation considerably more deep than what I would get through the traditional mode of teaching.

2014 Letters

Last year I wrote every one of my kids a personal note at the end of the year.  It was a valiant attempt, but with 140 kids, I vowed never to do that again.  Even though I tried to personalize all of them, after a while, they all started to sound the same, and they weren't really anything of substance.

I decided to write only two letters this year--one to my seniors and one to my juniors (I don't teach any other grade).  I could try to explain both my love and disdain for these letters, but Sam Shah pretty much summed it up a few days ago here.  

I wish I could put into words all my emotions; I wish I could write exactly what I want these kids to know.  I wish a letter would suffice, but it never will.  Regardless, I attempted a one-page note.

I feel a little vulnerable publishing these for anyone to read.  But when I blog, sometimes I get so wrapped up in the details of lessons, that who I am as a teacher maybe doesn't quite come through.  So this is a small attempt to combat that.

Letter to Seniors 2014

Letter to Juniors 2014

Reviewing for the AP Exam

Throughout the year, the other AB teacher and I introduced kids to past AP questions through different homework assignments and assessments.  I’ll probably blog about that at some point, but this post is about what we did during our review weeks.  We finished covering new material just before spring break.  This gave us about six weeks of review time before test day.

Remind101
If you haven’t registered for Remind101, you need to.  You can send texts to your students from a number other than your own, and they can’t text you back.  It’s magical.  Also, you can schedule texts.  HEAVEN.  So, starting three weeks before the test, I would send out a quick question at 4:00 PM.  Then at 4:30, I’d send out the answer.

Actually, I dropped the ball on this the last week, but I did get two weeks’ worth of texts in!  This is my message history starting on the day I began sending these texts.  You may want to read from bottom to top as the history shows the most recent text first:

Message History for Calculus 2014-15

Free Response by Topic
We started our review by using Lin McMullin’s Topical Review.  Lin has organized past FRQs into various categories (or topics)[1].  It’s a great way to ease the kids into review mode.  We put 3-4 past questions from the same topic on one sheet for the kids.  They would get fifteen minutes to answer the first question.  Then I would show them the rubric.  I made them grade themselves first (with a different color) and then make corrections.  I think it’s really, really important that they grade their own work.  Then, they would work the remaining 2-3 questions for that topic for homework.  When they got back to class the next day, I would spot check for completion.  It’s important to note that they did not get any kind of traditional grade during this time.  It was basically “A for effort.”  These questions are all over the internet, so if we would have taken these for a grade, I’m certain the vast majority of students would just look up the rubric and copy the answers in their notebooks.  Which totally defeats the purpose.  So, I had to be very ok with limited answers in the beginning…because I’d much rather them try totally on their own and then grade themselves than copy something down they haven’t even truly attempted.  But, that said, I did check every single day for some kind of work, and I would say that the kids really did attempt most of the problems.

When we had a little extra time, I also really liked handing out student samples of a particular question (you can find these on AP Central), and having kids grade the student samples.  It creates fabulous discourse.

Free Response by Year
It took us about two weeks to get through Lin’s five AB topics (we sprinkled some multiple choice review in there, too).  After these two weeks, we went back and printed the last two years of FRQs and we assigned homework in a similar fashion: do one in class, grade it, assign two for homework, and then come back and grade those two in class.  Repeat.  Sure, they had already seen some of these problems, but I actually think that’s a good thing.

Multiple Choice Homework
To me, multiple choice is a lot harder to prepare the kiddos for.  The free response topics are at least somewhat predictable (though, of course not completely).  To me, the best practice for multiple choice is to just have the kids work several sets of them throughout the year.  They really do improve.  But it’s pretty brutal at first.  We would give the kids probably about twenty problems or so at a time and then give them a “first due date,” at which point I’d take a completion grade (looking to see if they showed work for the problems that needed work shown).  Then, I’d give them the solutions and they were to mark the correct answers.  They then were given a “second due date,” in which they needed to correct all missed problems on a clean sheet of paper.  I truly believe that not giving kids time to make corrections, or not showing them what they missed may actually be more harmful than not assigning any homework at all.  It this point in the game, if I took a completion grade, I'd follow it up with a "correctness" grade.

The MC homework sets came from practice books that we have and also from the problems that College Board released in their Course Description.  In addition, I  made this document from the BC Course Description, but I only copied the AB topics:

Selected BC MC

Multiple Choice Secured Exams
The College Board has some secure exams that you can give your kids, but they can’t leave your classroom.  A good chunk of our review time is during block scheduling (because of state tests), so this is the perfect amount of time to do a full multiple-choice test.  At first, these were not at all for a grade—just practice (and endurance-building).  The last one the kids took, we did take a grade on it, but it was basically a grade booster.  We used this scale:

Multiple Choice Score	What past students with this MC score made on the AP Exam	Grade we put in the gradebook (out of 100)	Number of my students with this score
48-54	5	105	0
39-47	4/5	100	0
32-38	3/4/5	95	8
24-31	3/4	90	5
17-23	2/3	85	6
0-16	1/2	75	3

I included my results.  As you can see, they’re not stellar. Not atrocious, either, but not stellar.  But, this did help me narrow down my focus that last week or so.  The kids who scored 0-16…at this point, there’s not really much hope for them, let’s be honest.  But the kids in the 17-23 range are totally capable of passing.  They’re the ones that needed just a couple more questions right.  So, they’re the kids I intentionally watched that last week.  I made sure I answered their questions first.  I made sure I gave them extra encouragement.  I made sure they knew I believed they could pass.

After they completed a secure multiple choice exam, the next day I would give them back their test booklets and answer sheets (they used the answer sheet as scratch paper and they were instructed to clearly label all scratch work as we would be going over the correct answers the next day).  I would project the correct answers, they would mark the ones they needed to go back and look through.  And then they would help each other and discuss how they got to the right answer.  If no one could figure out a problem (which was rare), they made a list on the whiteboard, and I would go through these questions at the end.  I think this worked really well.  I don’t think me standing at the SMART Board all hour long lecturing on the most missed problems would have been too beneficial.  The kids really embraced being each others’ students and teachers.  And I think they got a lot more out of it this way.

Mock Exam
So about two and a half weeks before the actual test, we gave a mock exam on a Saturday morning.  We used the latest secure exam given by the College Board.  Out of my 25 calc kids, I had 24 sign up to take the AP test; of these 24, 20 showed up to take the mock exam, which I think is pretty darn good for 8 AM on a Saturday morning.  If the kids scored a solid 3 or higher, I entered a 100% in for their final exam (they already get to waive the final if they take the AP Exam).  I think about half my kids earned this incentive (and keep in mind, we were still more than two weeks out from the actual test).  I had four kids score a 5.  They were given these “trophies”:

While the incentive was a good thing, I think there were two other reasons the kids came:

1. I told them that May 7 (Exam Day) should NOT be their first experience taking a 3.5-hour calculus test.  They needed to have that experience BEFORE test day.  I reminded them how tired they are after a REGULAR 1-hour calculus test… ;)
2. I told the kids we’d be going over the mock exam the following Monday, so if they didn’t take the test, they would be behind their peers that day.

We spent a block day going over the mock exam.  This time, I selected the groups (typically I just let them work with the peers they feel comfortable working with).  I told the kids that I put them into groups based on their strengths.  So, each group had someone who was strong at, say, the non-calculator MC, the calculator MC, the non-calculator FRQs, and the calculator FRQs.  What really happened was I just made sure each group had at least one strong student based on the mock exam results.  I also thought about those borderline students and made sure they were placed with strong students who could also explain.  The kids rotated through stations.  Each station was dedicated to either 5-8 MC questions or 2-3 FRQs.  I had enough solutions/rubrics printed off so that everyone in a station could have their own.  Again, I think this is way better than me lecturing at them for two hours.

Night Study Sessions
We had a couple night study sessions.  Kids came with questions, and we went over some FRQs from past “Form B” exams.

Book problems
The week before the exam, I started to feel like we were losing momentum.  Also, I was running out of stuff to do.  So, during my plan one day, I frantically made a list of topics (Limits, Differentiability/Continuity, Derivative Rules, Applications of Derivatives, Integrals, Area/Volume, FTC) and selected problems from the text that corresponded to these topics.  I realize this doesn't cover every topic in Calc AB, but here’s the key:  these were not super difficult problems.  Sure, they were “AP-like” in nature, but they were not super hard.  And, this is EXACTLY what the kids needed.  They worked so steadfastly on these problems.  It was just the boost they needed, while still keeping them focused.

Day before test
The day before the test, we did a few problems; not a ton.  Then we went over some test day instructions (where to meet, what to bring, etc.).  Back in January, when were going over the Washer Method, I gave them each a washer to tape into their notes.  The day before the test, I gave them another one, told them to put it in their pockets in the morning, and every time they felt it, to know that I was thinking about them.  And also to remember to use Washer Method.  Then I read them a funny story from Glennon Melton to get their minds off the test.  And that was that.  That’s all I can do.

Oh, I also gave them a heads up and told them that College Board would release the FRQs Friday afternoon, so we’d be chatting about them Monday.  I think it’s a good idea to let your kids know in advance if that’s something you’re planning on doing.  If you take them by surprise, they might freak and shut down on you.

Feedback from Students
So the test was on a Wednesday.  I didn't have my calc kids that Thursday (due to block), so on Friday we just had a "chill" day.  One thing I did want to do, though, was to get their input on the review time and the class in general.  The problem with this is that I'm really, really little, and so people (usually) have a hard time even thinking about being mean to me.  So, when I first said, "What are some things you liked about this class?  What are some things you'd change?" The response was "EVERYTHING!" and then "NOTHING!"  Lol, kids, come on--puh-lease!

But they did eventually start to open up.  A little.

Here are their thoughts (more for my sake for next year):

Thoughts on the review time:

• Liked having 2-3 FRQs for hw and liked how it was originally split up into topics
• Liked having so much time to review
• Appreciated advice on how to earn "easy points"
• Appreciated the mock exam and the other secure MC exams we did in class

Thoughts on the class as a whole:

• Liked having quizzes not for a grade
• Liked having AP Sets throughout the year but agreed that there needs to be a way to make students more accountable to take them seriously (maybe hw quiz or making them part of their actual tests?)
• Appreciated End of Semester Folders (they had to have all their quizzes/tests corrected)
• Would have liked more Portfolio questions (maybe if I give more, I can make them optional?)
• Would have liked more FRQs in their Portfolios
• Learned a lot from Dixie Ross's Big Picture Review

So that's that.  Let me know if you have any questions about any of this.  And if you have ideas for reviewing for AP tests, please do share!

My 25 UHS Calculus Rockstars
And now all my PreCalc kids think it's really cool to be in calc.
Mission Accomplished.

[1]  Next year I might add two topics to the list: (1) Rate In/Rate Out and (2) “Traditional Calculus Problems” (aka given a function f(x), find the intervals where f is increasing/decreasing and concave up/concave down; state relative extra and POIs, etc.)

Decoding with Matrices//A Scavenger Hunt

I love PreCalculus.  I love that I get to pique kids' interest of calculus (the world's greatest subject).  I love that I get to teach a plethora of topics.  I love that there's no high-stakes test at the end.  I kinda get to do whatever I want to do, within reason. 

Our latest adventure in PreCalculus was a scavenger hunt throughout the third floor of our school.  We're currently in our last unit:  matrices.  The final objective of this unit is to apply matrices in "real-world applications."  One of the most fun applications you can find, in my opinion, is encoding and decoding messages with invertible matrices.  I totally play up this application, telling the kids that I used to want to work for the NSA as a mathematician who encoded and decoded top-secret information for the government (which is true...but I always wanted to teach more).  At this point, they're pretty engaged because I'm a quiet, 5'1", 100-lb teacher who often gets mistaken for a student; so I think they find the thought of me working as a secret agent humorous (and rightfully so).

Once we went over how to encode and decode messages using matrices, I assigned some homework problems from the book to practice.  The next day, I took questions over those problems to make sure the kids were pretty sound on the theory.  And then came the fun part.  I broke each class up into five teams.  Each team was given five matrices (which were printed on different colors of paper) and one string of numbers (which was printed on one of the same colors as their matrices):

Whatever matrix was printed on the matching color was the matrix used originally to ENCODE the message.  Their mission--should they choose to accept it--was to DECODE the mess of numbers and translate it back to the original message, which would take them to a location on our floor where a new string of numbers was hidden.  I sent them to different teachers who would verify that the kids had gotten the correct translation (for example, the clue that sent them to the Advanced Physiology teacher who's known for cat dissection was "CAT MAN").  I also sent them to other well-known locations in my classroom or on the third floor.  At each stop was a new clue that they needed to decode.  All the teams eventually went to all the same locations, but I started them off at different locales, so they weren't really running into each other.

Scavenger Hunt Directions

The kids absolutely loved this activity.  It took a little while (probably two hours) to create, but it was well worth it.  Once they were off on their hunt, I didn't have to do a thing.  Every single class asked if we could do it again (one girl even said, "I want to go back in time and start all over!  That was awesome!"), and several suggested expanding it to the entire building.  But I don't think I'm that brave (there are 3300 students in our building).

The adults who helped (teachers and counselors) said the kids were super respectful (I gave the students a  lecture about even though I wanted them to have lots of fun, to remember that others were in the middle of work and to mind their please's and thank-you's).  One of the adults said, "They were so polite!  Several teams would even give me the code, followed by 'Please?'"  Adorable.  

We had tons of fun.  If you're interested in making something similar, I'd be happy to help you create clues for your specific location.  If you work at a school that is pushing STEM education, like mine is (HOORAY!), this might be a good little activity to add to your arsenal.