Saturday, December 26, 2015

Three lessons

One of the most important lessons I learned about teaching I learned as a graduate student working in the Math Lab at my university as a tutor.  We had stations where the tutors would sit (and, quite frankly, work on our own homework) until someone approached us with a question.  At this point, we would gladly and enthusiastically put our homework up and help answer whatever question the undergrad student had.  We thought we were so approachable and were awesome tutors.  Or at least I did.

Until we got a scolding from the head of the math department.

Apparently, I was not as awesome as I thought I was.

Our boss (who is one of the kindest human beings on earth, I might add) gently told us that maybe we weren't quite as approachable as we thought we were.  "Math is really intimidating for most of the people taking these classes.  It takes a lot of courage to get up, in front of everyone, and come over to your station to ask a question."

I'm paraphrasing as it's been several years, but that was the gist.

He encouraged us to go to them.  I remember feeling so humbled.  Of course, he was completely correct.  As soon as I started making my "rounds," the amount of questions I got each day skyrocketed.  Furthermore, I started building rapport with several of the students who came consistently.

This experience greatly shaped the way I now teach high school math.  I'm very against sitting at my desk and letting students come to me.  Because that's what I did as a TA in grad school and it clearly does not work.  You know who comes to ask questions?  The kids who are going to figure it out with or without me.  The resourceful ones.  The ones that need me the least, to be honest.

When kids are in the room, I believe they need to be my primary focus--not lesson planning or grading or writing a quiz.  When kids are with me, they must take precedence:  they are reason I'm there, after all.  This philosophy means I've created methods to grade homework as they go (these methods vary with each of my preps) because prioritizing means something's gotta give.  For me, that's homework grading.  I'd rather spend my time with the kids than grading their homework meticulously every day.  The rest--lesson planning, grading quizzes/tests/projects, writing quizzes/tests, writing rec letters, etc.--that all happens when kids are not in the room:  during my plan, after school, or during the weekends.  That's how I've decided to prioritize and manage my time.  Everyone's different, but my main point is:  our kids need us when they're in our rooms.  So whatever you have to cut out to make time to be with your kiddos, I think it's worth it.

This brings me to the next important lesson I've learned as a teacher.

While I'm pretty good about making my rounds and staying away from my desk (on most days...I'm not going to pretend I'm never at my desk during class time), one of the things I've practiced more recently is being able to pull questions out of kids.  During my rounds, I would often ask questions like, "How's it going?" or "Can I help with anything?"

I thought those were perfectly fine questions.

I assure you, they are not.

I've replaced those phrases with "What questions do you have for me?" or "What may I help with?" or "Tell me about your thought process here."

Goodness.  What a difference.  I cannot even begin to describe how many more responses I get when I invite questions in this manner.  It calms the kids when I approach them with an air of "I expect you to have questions for me, and I want to help you reach a deeper level of understanding."

If you're not convinced that these questions are all that different, take this anecdote as an example.

I approached a kid a couple months ago and asked him, "How's it going--can I help with anything?"

"I'm good!"  he responded with a smile.

I was tempted to leave and move on to the next student, but I knew I owed it to him to pry just a little deeper.
"What can I help with?"

"Actually, could we talk about Number 7...?"

As a teacher, the two questions I asked should mean the same thing.  But to students, they clearly elicit different responses. 

The last important method I use on a daily basis is also very simple, but I believe it's really powerful.  When I help students and I know it's going to take a while, I get on my knees right next to them (or, if the seat next to them is open, I might opt for that). I do this even if I'm wearing a skirt.  Even when I'm eight months pregnant.  It's a way for me to physically say, "I'm here to serve you.  I'm not going anywhere."  I believe this small and simple gesture has broken down so many walls.  It's impossible not to be touched by humility.  

Those are my three lessons.  I typically try to stay away from giving advice (I think most people just need us to listen more than talk).  But, these are lessons that I have to intentionally practice every single day.  It's advice for me as much as it is for anyone else. I hope, though, that it helps others, too. Or at least helps others form their own welcoming classroom culture.  

Derivatives of Inverse Functions

This is my fourth year teaching calculus on some level.  Every year (until this one!) my students have really struggled with finding the derivative of inverse functions at a point, especially in the manner these questions are often phrased on AP Exams.

To me, they're some of the most straight-forward multiple choice questions the students encounter on the exam; yet, year after year they miss this question (at least on their unit tests and mock exam).

So, clearly, not as straight-forward as I thought...

This year I formalized a strategy for them in three steps.  Not all three of these steps are necessary every time; but, if my students took the time to follow all three steps, they got these questions correct.

Here are the steps:

If f and g are differentiable functions and g is the inverse of f, then to find g'(a):

  1. List all points given on f as ordered pairs.
  2. List the points you now know are on g (switch x and y).
  3. Follow this formula: g'(a)=1/(f'(g(a)).
Let me show you with a couple examples.  Here's a question I pulled from this website.

Following the steps, we would work this question as follows:

How about one that describes f as an algebraic or numeric function, such as this FRQ from 2007:

Students could certainly start with Step 1 again and work their way down, but I encourage them--once they get comfortable--to feel free to start with Step 2 and fill in the blanks as they see fit.  Here's how I would suggest they work this problem:

That's it!

Tuesday, December 1, 2015

The Peterson Diagram

Here's a short (very low-tech) explanation of a diagram I created a couple years ago to help my students answer questions about how f, f', and f'' are related.  My kids use this especially when--for example--they're given a graph of f' and they have to answer various questions about f, which seems to be a favorite of AP Calculus Exam writers.

Thursday, October 29, 2015

Related Rates Related to You

We did these problems today in AP Calculus.  Part of the problem I have with related rates questions is most of the time they seem so contrived and impractical.  Like, why do I care what rate the radius of the balloon is increasing given the rate the volume is increasing?  So, I had some fun with these types of problems.  The kids loved seeing their names in the stories.

I put them in groups and assigned one problem per group (making sure that if a student was in one of the word problems, s/he was in the group working that same question).  Answers are included so the kids could check themselves easily.  Then I had them make a poster.  These are some from last year:

Afterwards, I asked the students to read the other problems and then use the posters to write the solutions in their notebooks.

Here are the problems I used (I'm sure I stole several of them from other people, so let me know if I owe you credit).  I always change the names for each section.

Download here.

Wednesday, August 19, 2015

What they want is our time

Disclaimer:  I started this post a couple weeks ago and am just now publishing it.


It's August, and I feel like I haven't done nearly enough to get ready for the beginning of school.  Most of my summers have been spent in conferences around the state and/or planning for that new prep I'm about to take on.

This coming year (Year #7) will be my first year teaching exactly what I taught the previous year, and so I've let that be an excuse just to be lazy this summer.  That and I'm pregnant, which I feel is a valid excuse for just about anything these days.  And so now, it's August.  And because I'm prone to guilt, I've been feeling rather convicted about this laziness.

Until a student of mine called me up today.

Her reason?  She wanted to ask my permission to call me her mentor, because that's what she's been calling me all summer as she's counseled young girls at camp.

I don't do particularly well with flattery.  Don't get me wrong, I enjoy it as much as the next gal, but in the moment, I never know how to respond.  I'm trying this new thing where I just sit and listen and let the person tell me what they want to tell me.  Usually all I can muster up is a weak, "Thank you," or "I'm so honored..."  It all sounds so pathetic in light of their gracious words.

As I sit here, trying to remember exactly what this girl told me, all that's really coming to mind is her gratitude to me for taking time for her.  That's all.

I'm attempting to put my thoughts into words mostly for my future self, as a reminder.  Rebecka:  the hours you spend planning a single lesson or activity, it's all well and good (and necessary--the kids need to see this passion).  But the time you spend helping your kids understand how important they are to you...that's what they really remember.

So, my advice to both my current and future self:  Stop stressing so much.  Stop feeling guilty for taking time to relax and recuperate.  And start focusing on how you'll love on this year's group of kids.

Sometimes I think we live under the impression that we can't be too friendly or too nice or too "buddy buddy" with our students:  we're their teachers, not their friends; we need their respect.

Yes, we need their respect.  But, have you ever respected someone you didn't like?  Especially as a kid or as a teenager?

I want to take more time this year to ask about my kids' families, pets, hobbies, and jobs.  I want them all to know that I'm invested in them and that I care about them.  Because that's what they'll remember about math class.

I don't really care whether or not they remember the Fundamental Theorem of Calculus ten years from now.  I care about whether or not they felt safe, loved, and welcomed in Mrs. Peterson's classroom.  That's what I want them to remember:  that they were a priority.


So, as the first day of school approaches tomorrow (AH!), my prayer is that we would bond as a class and as friends faster than ever before.  That mutual respect, love, and admiration would abound.  That we would all recognize that every single person has something to teach us.  That we would be better together than apart.

That I would give my kids my time and my undivided attention.

Tuesday, June 16, 2015

What we did post-AP Exam

I still had a few weeks of class to fill after the AP Exam.  I definitely wanted the kids to still be working, but not necessarily on calculus and not to the point where they had tons of homework.  So, this is what we ended up doing:

1 day to go over released FRQs
1 week to work on Shoe Box Projects
3 days to work on End of Year Folders and fill out surveys
1 week to work on Serve + Create
1 day for goodbyes :(

Here's what we did for the shoe box project.  The kids were asked to create a rate FRQ and then make a shoe box scene that went with their story (like in elementary).  I think they had a lot of fun with it.  We basically just modified an Algebra 2 assignment, so if I owe you credit for this, please let me know!

Click here to download file

Here are some of their finished products:

This one had involved the rate at which Mrs. Peterson adopts cats vs. the rate at which she donates them.
If you can't tell, that's me holding to two cats...

For the End of Year folder, I asked students to correct all quizzes and tests from the semester (I asked this of them last semester, too, so they were expecting it).  Sure, this would have been good to do before the AP Exam, but it just fit better after.

I also had them fill out two surveys:  one that was just a general "What should I keep/change for next year?" (here) and then one that asked them fill in positive adjectives for each of their classmates (here), which I used to create Wordle bookmarks for end-of-the-year gifts (thank you, Pinterest). More on this at the end of this post.


Finally, the week before finals, I gave a totally non-math project, which I called Serve + Create (details here).  I told the kids that in the midst of both joy and sorrow, it's important to learn how to cope with change, and that two of the best ways I know to do that are (1) serve someone else and (2) make something.  So, that's what I asked the kids to do.  I gave them very little guidance.  I wanted it to be organic and come from them.


I wanted to take a day just to say goodbye to my lovely kids, many of whom I've seen every school day for the last two years.  So much is written on things to do the first day of school, but I feel like we are lacking in material on last day of school activities.  Especially if you have seniors, I think it's so important to make a big deal of the last day, because you accomplished a lot together.

So, for the last day, I first passed out my junior and senior letters, which were not too different from last year's letters (here).  One thing I did add was advice from the Class of 2015.  I gave the seniors a whole white board and about a week to add their advice/wisdom for us.  Afterwards, I took a picture and copied it to the back of each of the letters:

{This might be a good time to mention, again, if you've never used the app DocScan, I highly recommend it!  Covert pictures to PDFs easily.}

After passing out the letters, I read Oh, The Places You'll Go! by Dr. Seuss.  

Next, I passed out these bookmarks that had a word cloud with each of their names and the adjectives their peers and I had entered for them in the survey:

I used Word It Out because it was the first word cloud generator I found that wasn't blocked on my school computer, but there are several good ones.  I just had to play with the settings a bit.

I read each one out loud.  The kids seemed to love this.

Finally, I just spoke for about a minute (this is all my heart could handle) about what a great time I had being their teacher, and how excited I am for their futures and for the futures of the people whose lives they will certainly change for the better.  The last half hour kids spent saying goodbye to their friends and to me.  Lots of pictures were taken and some tears wasn't an easy day, but I very much wanted to make the last day of calculus something meaningful, for both the kids and for myself.

These are world changers.  Of this I am certain.

Friday, June 5, 2015

How I did homework this year in PreCalc...

I stole this idea entirely from the teacher who taught APSI last summer.  I was intrigued by it, so I implemented it in my PreCalc classes this past year.  My kiddos are begging me to extend the concept to AP Calculus next year because they loved it so much (you'll see why...).  I'm undecided.  I thought I'd write about it and get your take...

The premise of this homework set-up is that kids get rewarded for doing homework instead of being punished for not doing homework.  This is how I ended up doing it, which is a slight modification from the way the APSI instructor did it.

  1. Once a kid completes a homework assignment, I spot check it for completion and ask if she checked answers in the back of the book (assignments are due the day before quiz/test day).
  2. If the homework assignment looks thorough, I give the kid a hole punch on an index card (I have a star-shaped hole punch, but you could use a stamp, stickers, etc.).
  3. One a kid has ten hole punches, she gets a 100% on a quiz grade (I simply added extra quizzes, called them "Extra Credit Quiz 1," "Extra Credit Quiz 2," etc., and excused everyone from it until/unless she got 10 hole punches).
  4. That's it.

  • I will probably make these Extra Credit Quizzes worth only half a regular quiz grade next year (which should increase their overall percentage by 0.5-1 on average, instead of 1-2 percentage points on average).
  • My APSI instructor only checked assignments on certain days (and hence only checked certain assignments), but I found it easier just to let all assignments count towards the extra credit.  If I didn't have time to do homework checks one day, it wasn't a big deal--I told the kids to just remind me the next day.
What I liked...maybe even loved:

  • I hardly ever had to grade homework!
    • I still had to have at least two grades in a week, so I would take smaller in-class assignments for a grade (typically as a review of that week's warm ups).
    • Occasionally, I would give worksheets that I counted for "an actual grade" in the gradebook and not as a homework check.
  • Kids were super, super honest.
    • When I took homework for a grade, I saw kids half*** their homework ALL THE TIME.  You know what I mean.  Sometimes a kid would just miraculously go from Question 3 to Question 43...and hope I wouldn't notice.  Or, somehow they'd get the answer from the back of the book with no supporting work.  Yet, with this new method, kids would tell me almost daily, "I'm done with 8.3 but I still have two more questions on 8.4." Because there was no punishment for not finishing those two questions (and because they still had more time), they seemed to be much more up-front about how much work they had actually done.  This was good for me, sure, but I think it was also really good for the kids to voice what they still had left to finish.
  • I'm not sure any more or any fewer kids did homework when it was presented in this manner.  You'd think a lot of kids would just stop doing homework, but I honestly don't think it was any more than normal. There are kids who will do the homework no matter what and kids who will not do homework no matter what.  I don't think this changed that.
  • Kids felt less pressure.
  • I try to give kids as much time as possible to work on assignments in class, where they can ask me and their peers questions.  Because of this, they all typically have at least a very good start on their homework.  Is it really the end of the world if they don't finish every single problem?  Especially if they're working hard in class...? I'm asking in sincerity.
  • No one asked at the end of the year if there was any extra credit they could do to raise their grade.  Of course, I warned them at the beginning of the year that this was it. 
My hesitations:

  • Maybe some kids are falling through the cracks?  I'm unsure.
  • If I do implement this in Calculus, then something else has to change, because I'm currently not giving quizzes for a grade either.  Their entire grade cannot depend on tests!  One thing I do want to change is make their quiz corrections a grade (due the next day as opposed to at the end of the semester).  I've also thought about giving short MC assessments every Friday, which could certainly count as a grade.
  • Calculus is a different beast.  Most kids need to wrestle with concepts, and that takes time.  While I prefer that my kids do most of their work in class, they do need to set aside some of their own home time to really understand what's going on.
So...what are your thoughts?  Is this worth extending to calculus?  Or at least trying it?

I'm fairly certain that I'm keeping this method in PreCalc next year, but what about for Calculus?  Help!

Monday, June 1, 2015

Serve + Create

I've posted a lot about our AP Calculus AB final project (Serve + Create) on the One Good Thing Blog; it was a two-part project in which students were asked to (1) serve someone else and (2) create something they enjoyed making.  I haven't explicitly written what I gave the kids or where this idea came from; hence, some details about the project:

I gave the students four class days (there was also a three-day weekend in between) to work on whatever they wanted.  I told them that some of them would want to use this time for their final project, but most of them would probably want to work mostly out of class.  So, I didn't feel too bad giving a few "free days."  Several did work on their art project and several went to serve a teacher during these four days.  Others relaxed and played math games and worked entirely on the project on their own time.

This idea stemmed from an interview I listened to with Glennon Doyle Melton (here).  In the interview, she was asked how she deals with all the hardships she sees and reads about on a daily basis.  How do we get ourselves out of our own misery?  She said she always comes back to two things:  art and service.

I couldn't agree more with Melton's advice.  I had never realized that art and service were two ways I dealt with change and loss, too.  Thus, this interview was a catalyst to a project I've been mulling over for quite some time...


UHS AP Calculus AB Final Project

You’re almost done!  I recognize this can be a stressful and bittersweet time for both juniors and seniors alike.  While I hope you’ve learned lots of calculus throughout the year, there are things that are more important than mathematics:  one of those being—How do you cope with change?  There are two ingredients I want you to practice before you leave our class—service and art.  When life gets chaotic, one of the best things you can do is remember that there are others in this world who are suffering, and do something to help them.  Furthermore, creating something through an art medium is a great way to express yourself, especially in the moments when you’re feeling overwhelmed.  As a way to encourage you to practice both the art of service and creativity, I've made this our final project for the year (worth 25 points in the test category).

SERVE:  Think of something you can do that would positively impact someone else’s life in a significant way.  Spend no more than $20/person.  You may work in groups or as individuals. 

CREATE:  Create a piece of art that you’re proud of.  This is not limited to drawing or painting:  think outside the box!  Express yourself in a way that is unique to you.  Please work individually on this.

On Thursday, May 28, you will be asked to share both your projects with the class.  I do not want to stifle you in any way, so grading will be based on completion:  do the project and you’ll get full credit.

I'm so proud of all your work this year.  It’s been an honor to be your teacher!


I've never done anything like this, but the results were absolutely wonderful.  As I posted on the One Good Thing blog, here are some of the things the kids shared on Thursday.  They...

  • Helped teachers in elementary and middle schools
  • Played with and loved on elementary children
  • Made cookies, brownies, spring rolls (not even kidding) for the class
  • Sang a song for our class
  • Painted a picture only using spray paint
  • Made calligraphy signs
  • Passed out cards to fellow students with hand-written positive sayings and Hershey Kisses on them
  • Passed out burgers to homeless
  • Made origami
  • Wrote poems
  • Drew pictures
  • Mowed lawns
  • Visited nursing homes
  • Bought flowers for their moms
  • Threw a party for their families
  • Helped renovate a run-down church
  • Made a “Countdown to the 2016 AP Calculus Exam” for their calculus teacher (<3)
  • Made canvas art for their dorm rooms
  • Made canvas art for their teacher
  • Donated money to a student from El Salvador looking to pursue higher education
  • Made French food
  • Made a wreath for their room
  • Made food and cards for their teachers
  • Made shirts for their teammates
  • Made jewelry charms for their friends
  • Volunteered time to help coach the pom squad
  • Worked on a car for free
  • Made a “College Survival Kit” for friends
  • Made a poster to celebrate twelve years of friendship
  • Posted slips around school with motivation sayings for students to take
  • Made a Father’s Day platter
  • Bought reading glasses to accompany a family member going on a missions trip
  • Played the recorder for the class
  • Made paintings for a newly-married family member
  • Gave flowers to teachers
  • Helped at a cultural fair
  • Taught the class a hip-hop dance (this was a dream come true for me…so fun!!)

  • Tuesday, April 21, 2015

    More on Reviewing for the AP Calculus Exam

    My kids just took a mock AP Exam this Saturday (I use one of the secured exams that College Board posts in the course audit page).  My scores this year were much, much better than last year's mock.  Last year, nine out twenty-four kids passed (37.5%); this year, forty-one out forty-eight passed (85.4%).  I revamped the structure of the course a bit this year, and I think it must have made a difference;  I don't think these numbers are a fluke (I guess we'll find out for sure this summer).

    A few weeks ago, I wrote about some of the changes I made in the structure of the course here.

    However, I did not include my plan for review (which starts immediately after spring break, giving us about six weeks to review...remember, these are AB kids; not BC...most of them need this time, in my opinion).

    This year, I structured our review time quite differently also.  Each week--up until the mock--the kids were responsible for one FRQ topic and one or two MC topics.  The weekly format looked like this:

    Weekly Format (until Mock Exam)

    Monday (FRQ Day): 4-5 problems due by Friday

    Tuesday (FRQ Day): 2 problems in class; grade own; grade sample responses (if time allows)

    Wednesday (MC Day): Work on MC problems, due Thurs

    Thursday (MC Day): Bonus Problem due; go over MC; corrections due Friday
    Friday (Quiz Day): 35-min quiz {10 MC, 1 FRQ}; 10-min discussion (if time allows)

    For homework, I graded their FRQs by picking one to grade for accuracy (out of the typical 9 points). For their MC, I allowed them to gain all their points back if they did corrections. I also encouraged them to look up the scoring guidelines for their FRQs (I made a TinyURL, if you're interested: I also posted worked-out solutions for the MC each Thursday afternoon. So, there was no reason not to make a 100% on each weekly homework assignment: all answers were given.

    Then, came Friday: Quiz Day. For this, I would pick a free response question (typically from a study book), that was worth 9 points. The multiple-choice, however, came from the kids. They were each allowed to submit one multiple choice question (based off the questions in their weekly packet) for an extra point on their quiz. They also had to have four good distractors (and show the work on why they were good distractors). If their question ended up being one I used for that week's quiz, they got an extra two points (total of three bonus points). The MC questions were worth 1.2 points each (like on the actual exam), so the quiz was out of 21 points.

    This did mean that I only had Thursday evening to write Friday's quiz; but, I pretty much just had to choose which ten MC questions I wanted to use and then type them up.

    If time permitted, we would go over the quiz after they were all turned in. But, honestly, this only happened the first week. Typically, we had to wait until the following week to go over the quiz.

    I think these quizzes were a huge help to the kids because it narrowed down the focus each week, and the students were (for the most part) really able to master the given topics each week.

    Here is the review schedule for this year. It also includes what we'll do the next two weeks:

    Here is an example of what their Monday problems (FRQs) looked like:

    For MC, I just grabbed questions from wherever I could find them.  I used Baron's study book a lot because it has MC questions separated by topic.  Typically, each MC packet had about thirty questions in it.  I'd be willing to email you any/everything I used for their homework assignments each week.

    I hope some of this is helpful to you.  What kinds of things do you do to review that you find are beneficial to your students?

    "Trophies"/commemorative tumblers for the eleven kids who made a 5 on the Mock!

    Thursday, April 16, 2015

    Calculus Flip Book!

    Shireen, at Math Teacher Mambo, posted this fabulous flip book that she had her AP Calculus AB kids make last year.  I was in love the moment I saw it.  In the comments, she also posted a link of a video that gives directions on how to make said flip book.

    I made my own this week (I changed a few things, but the general format is the same):

    I plan to have my kids make this in class next week.  Since I did change a few pieces, and I didn't really want my kids on their phones looking at the video the whole class period, I made written directions:

    The kids will write everything (and I do mean everything) that I have in the directions; they will find their own examples for the highlighted portions.

    I haven't vetted this yet, so there very well may be typos, but I thought some of you might want this sooner rather than later, since the AP Exam is in nineteen days...but who's counting...? ;)

    Wednesday, March 25, 2015

    In which I attempt to defend public educators...

    I got a new student today.  Today.  Eleven weeks before school lets out for summer.  At the end of class, the kid told me that was the first time she understood what was going on in math class for as long as she could remember.  Her presence, along with some other things I've been thinking about, prompted me to pull out my computer and write what's been on my mind...

    I teach in one of the lowest paid states for public education teachers. I work with unbelievably dedicated people who make daily sacrifices to do what they do for a living: to serve other people's kids.

    Sadly, I often hear comments like, "I'm just going to go work at Starbucks. I'll make more money there."

    Now, let me be clear: there is no question in my mind that we are grossly underpaid.


    I understand that we all have frustrating days, where anything sounds better than our current position. But, I also know that--at least for me--the good days far, far outweigh the bad. And if they don't for you, you might want to consider leaving. You're right: we don't get paid enough to be miserable all the time.

    But, I imagine most of you became teachers knowing full well the monetary sacrifice you would make. As I recently heard it put so elegantly, you didn't show up for the income, you showed up for the outcomes.

    You enlisted because even though we're underpaid and unappreciated by adults, we have kids who adore us and would do just about anything for us.

    You kept coming because you knew your kids needed a safe, encouraging, hopeful place. And you are sure as hell going to give that to them. Even if for just one hour a day, a hundred eighty days a year.

    You kept reporting for duty because you believe in public education. Who else is going to take the kids who change schools every few months? Who else would take a kid eleven weeks before summer and say, "Hey, we don't know each other yet, but we will. And I will do everything in my power to get you caught up and to be successful in my class."

    You continued to show up because you're committed to leveling the playing field. Because you believe every kid should have an opportunity at a quality education--regardless of zip code. Because you're driven by the idea that you want every kid to feel better about themselves leaving your doors than they did entering them. Because you see kids not for who they are, but for who they can be. Because you believe they can make this world a better place. Because you know that they are the hope of the future.  Because you understand that education is key to overcoming poverty, prejudice, and ignorance.  And because you feel this indescribably fulfilling joy every time you're reminded of why you do what you do because of something kind a kid said or did.

    Yeah, we don't make a lot financially. But what we do make is not quantifiable.

    And so, to all those who recognize this (and that is the vast majority of my fellow teachers), thank you. Thank you for believing in this profession. Thank you for persevering through the bad days, through the media bashing, through the uninformed comments from both loved ones and strangers.

    Thank you for choosing to see the positive in public education. For knowing that it's not perfect, but for striving for perfection regardless. Please, keep doing what you're doing. We need you. We need each other.

    And your kids need you, more than they'll ever be able to express.

    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 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?

    Update {3/24}

    I got an email asking if I keep kids from Googling the FRQs.  I thought I would post my response here also, in case anyone else wondered:

    I don't keep the students from Googling the answers; I actually encourage them to do so (attempt first, then Google). If you give past FRQs as homework, I don't think there's any way to keep them from looking up the answers, unless you say it's not for a grade and we're going to check answers together as a class tomorrow (so please don't look these up). I have done that before, but it wasn't super successful. Kids didn't take it seriously enough. 

    The truth is, no one is going to get perfect wording on FRQs in the beginning because kids are not used to this type of writing. If students are encouraged to look up the answers, I think it pushes the responsibility back on them--instead of you--and they can also spend time learning the part(s) they need help with. It individualizes the learning process a bit more. 

    They do get the problems in advance, so they can always come ask me questions if they can't find a satisfying enough answer on AP Central (or elsewhere). 

    [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:

    Here's how I told the kids to use the pins:

    Finally, visit my Teachers Pay Teachers site HERE for 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... :)