The 2-Minute Rule

I am trying to think of everything and anything that might be helpful for my friend taking over for me during maternity leave (any day now!).  One of the things I included in my list to her was my 2-Minute Rule: "I allow students to pack up two minutes before the bell rings, not a second before. Then they need to stay seated until the bell rings.  The kids are pretty good about it, but some of the kids that transferred at semester are still learning."

My first year of teaching high school, the kids would do two things in particular that really annoyed me:

1. Pack up early.
2. Line up at the door.

I don't know if it's because I was homeschooled or if it's because I taught college classes before switching to high school, but I was totally floored and appalled by this behavior, which I've been told is very normal. Thus, I instituted the 2-minute rule. It's simple and I'm sure lots of teachers have something similar, but it really does work because it gives the students freedom to pack up before the bell rings but it also gives them a guideline as to when it's permissible to do so. Usually, I just have to get on to a class once or twice until they get it.


The other day I noticed a kid (who had transferred to my class just days before) pack up seven minutes before the bell rang. I told her she needed to get her stuff back out and that she could pack up at 1:13, two minutes early. She agreed to the rule. But I was reminded in that moment how important this rule is. In my first year, I felt like the kids just packed up earlier and earlier every day. Not ok.

That's all. Totally simple but totally helps me keep my cool.

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.  

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.

Modifications:

• 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!

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? ;)

Another Review...

I'm always trying to fine-tune review activities.  I used to be really into review games.  I would spend hours creating games that we'd play the day before a test.  They're fine:  I still use several of them.  But my criteria of what constitutes a good review has really simplified to two things:

1. Students do most of the work/explaining (not the teacher)
2. Students can self-correct their errors

These two objectives led me to a very simple review for my PreCalculus classes that I thought went swimmingly.

The kids were given a study guide to review for their Quarter Exam (kind of like the Quarter Quell...just kidding...sort of...).  The next day, they were to come to class with a note card with a question like one from their study guide but with different numbers and multiple choice. I didn't tell them which problem to work; I asked them to pick one that they felt they needed more practice on.  (Because if one student needs more work on an objective, then there will be other students who need help in that area also.) Additionally, they were asked to fill out this Google Form so that I could have a key to their questions without having to work fifty problems:

I took two days to let the kids work through all the problems (the first day they worked through their class's cards and the second day they worked through the other PreCalc class's cards).  I made slips of paper with the numbers 1-49 (each kid was assigned a number) so that they could keep a record of their answers (and I could grade them easily).  I made them go back and correct the ones they missed.

This was absolutely lovely because I really didn't have to do anything these two days.  Normally I walk around and take questions, but I wanted the kids to be answering their own questions.  If someone would try to ask me a question, I would tell them to ask the person who wrote the question.

This is the rubric I used:

Q2 MC Test Question Assignment (10 points)

2 points: Create a question like one from the study guide with at least medium difficulty

4 points:  Four good multiple-choice options:  one correct answer and three good distracters

1 point: Index card formatted correctly:  assigned number on the top left, question with all four answers, name and hour on back

1 point:  Very clean handwriting

2 points:  Correct answer submitted on Google form (tinyurl.com/Q2multiplechoice) by tomorrow’s class

Things I really liked about this:

• I didn't have to write any more problems.
• Students got practice writing good multiple choice problems.
• Students were the ones doing the work; not the teacher.
• Students got lots of practice with the types of questions that they tend to struggle with.

The only thing I didn't like so much:

• Some kids didn't have a correct answer on their card...but kids usually found the mistake on their own.

I really liked how this played out.  Super easy on the teacher's part, and kids got loads of practice.

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.

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.

Adaption of AP Calculus Questions

This is my first year teaching calculus through the AP curriculum, and I love, love, love, LOVE it.  It's such a great mix of pure and applied math; I really feel like it supports teachers as we try to cultivate thinkers in our classrooms...and not just regurgitators.

In PreCalculus, we are now starting to discuss actual calculus topics:  limits, formal definition of the derivative, and approximation methods for area under a curve.  I was reading this article from the College Board about vertical alignment, and it got me thinking about how we could be exposing our Algebra 2 and PreCalculus kids to past AP Calculus questions.

And so I created these questions, which are adaptions from the 2012 and 2011B AP Calculus exams (both questions appeared on the AB and BC exams).

AP Area Questions by Rebecka Kermanshahi Peterson

Does anyone else have previous AP questions they've modified to fit earlier classes--not just for calculus but possibly also for statistics?  Or, do you have "vertical alignment" in your pre-AP math courses?

The need to teach creativity in mathematics + FRACTALS!

I really resonated with Sam's most recent post about building time and space into curriculum to let students play with math. Mathematics is incredibly creative and innovative and I know that I (and I'm guessing others?) don't take enough time to let kids tinker. It's only when we sit there and tinker (preferably with something that intrigues us) that we become really good at something. This is a truth I constantly try to convince both kids and adults of: a mathematician wasn't miraculously born a "math person"; she found some kind of math that interested her and played with it for a very long time.  I.e., she had to work for it, but she most likely enjoyed the work.

And I am convinced every person can find some kind of math that he enjoys.

And I'm certainly convinced I can do more to be an advocate of the CREATIVITY needed to be successful in mathematics.

So here's a small first attempt! We're currently in our chapter of sequences and series in Precalc (which also includes the Binomial Theorem and hence Pascal's Triangle !!), so I thought it'd be fun to talk about fractals:

Fractals

(Email me if you would like the SMART Notebook file.)

I was able to find a decent video of the Mandelbrot Set (I muted the audio and played it on 2x speed).  We watched this as we talked about what they noticed/wondered.  They were very quick to point out possible fractals found in nature.

And we also watched one of Vi Hart's fabulous clips to motivate drawing fractals by hand (which is pretty darn addicting, no matter who you are):

After this video, we went over how to draw Sierpinski's Triangle (and how it's related to Pascal's Triangle!) and the Koch Snowflake. Then I let them research other fractals (there's a QR code to a Google doc I made with various good links for instructions on how to draw some fractals). Their assignment was to submit a fractal that they've drawn before spring break (either their own fractal or one that's already been "invented"). They are also to include both a recursive and explicit formula that models their iterations.

The types of things I saw as I walked around and the kinds of questions they were asking and the stuff they were pulling up on their phones made me so very happy. I felt completely justified in taking this time to breathe and to play and to create. I think I've said this before, but when you combine teenagers, art, and mathematics, you're bound to be continually impressed. I need to do this more.  

With that said, hopefully I'll have some good pictures to share next week!

Analyzing Exponential and Logarithmic Graphs

As I was looking ahead in my unit of exponentials and logs in Algebra II, I opened up a lesson plan whose first page read, "Note to self: This lesson sucked. Kids were totally bored."

Must have written that a year ago and forgotten...until now.

This is a topic that I teach in PreCalc also, so I was motivated to change this boring lesson.  But worse than being boring, my lesson honestly did not have kids exploring interesting mathematics.

What I really wanted was for kids to understand the inverse relationship between exponential and logarithmic functions before we talked about solving equations.  I wanted them to start to understand what happens graphically before we explored the analytic implications.

So, I made this matching activity.  I really broke it down for my Algebra II kids, but I think PreCalc students (or advanced Algebra II students) could dive right into it with little to no instruction on the teacher's part.  I limited the transformations of the graphs to shifts only, but, for more advanced students, it could be nice to show reflections also (though I might stay away from stretches/shrinks...).

I had my Algebra II students work ONLY with the exponential graphs first.  They shared a deck of cards with a partner, but each student was to fill in his/her own chart. Once they were done with that side, I had them figure out which log graph was the correct inverse for each exponential graph.  Lastly, I had them analyze the log graphs.

The activity is designed so that students can see the similarities/differences of exponential and log functions, beyond just "x's and y's switch."  Ok, so they switch...what does that mean?  If I have an exponential graph that shifted to the right 2 units, which direction will its inverse graph shift?  Why?

I think this was a considerably more interesting way to get kids more comfortable with log graphs.  And they were definitely noticing the types of patterns I was hoping they'd notice.  The nice thing is that since a lot of the patterns are obvious, kids can quickly check their own work for errors once you've had a discussion as a class about all the similarities that should occur in their charts.

Chart to record results (and key):

Exp Log Match Table

Deck of cards (6 exponential functions and their corresponding logarithmic inverses)--thanks as always, Desmos!:

Exp Log Match Graphs

A couple of the matches

Just a little review...

In Algebra II, I tried a different kind of review as we were preparing for final.  It worked well, so I thought I'd share:

1. Type up/select some review problems and number them as you go, just like you would a review guide or practice test (all my questions were multiple choice, but free response would work, too).  I'd write a few more problems than there are students so each kid will get 1-2 total.
2. Print off the problems and cut them into strips.
3. Pass out the strips of paper (more advanced students got harder problems).  Also, as students finish before others, you can give the fast workers another problem since you made some extras (mwahahaha).
4. As students finish, have them record their answer(s).  They can use their own paper or something like this for ease in assessing on the teacher's part.  Check students' work for accuracy as they finish.  If the answer is correct, they get a piece of tape.
5. Once everyone is finished, students put their name on the strip(s) of paper they received and are told to place their problem anywhere in the room.  The only two restrictions I gave were (1) each piece of paper had to be put in a place where even a person of my height could see it and (2) don't hang anything from the Smart Board.
6. After the problems are hung, the kids work each problem.  If they have a question on a problem, they are to consult the person whose name is written on that piece of paper.

The students worked all hour and I think I answered like two questions the whole time.  I even had one girl say, "Mrs. Peterson, can you...wait!  Never mind, I'm supposed to ask...[so and so]."

Hoorah!

I printed off the problems on a colored sheet of paper, just to make things more exciting, I guess.  But that turned out to be good because the kids asked if we could do this review again the next day, so I printed off more problems on a different color for the following class period.  My classroom looked like a hot mess for a couple days, but it was definitely worth it.

Do you have any other ways you love to review that put the onus on the students?

My Unit on Rational Functions (Algebra II)

Disclaimer--this unit is fast and very calculator-heavy.  It would need a good deal of reconstruction for an Advanced Algebra II course.  Nevertheless...

Part I:

Review of asymptotes via Asymptote Bingo

Part II:

Introduction to rational functions via this foldable:

Rational Functions Foldable by Rebecka Kermanshahi Peterson

*I think you could use this in an Interactive Notebook if you just deleted Example 3.

Part III:

Exploring rational functions via Desmos

This was my favorite.  Oh, Desmos, how I love thee.  I wrote this literacy/technology activity for my students and then we headed to the Math Lab together to work on the computers:

Rational Functions With Desmos

We have really nice, big screens in the Math Lab so the kids were able to get beautiful and clear pictures of these functions, which (I think) a typical handheld graphing calculator can't quite provide.  Here's what I loved:  The kids would graph the function in question, for example this:

And then they were asked to analyze.  I asked them to graph all their asymptotes and highlight all intercepts.  So, if they accidentally said that the horizontal asymptote was x=0, when they graphed their answer, they (usually) immediately identified their mistake and made the appropriate corrections.  (Or, at the very least, they raised their hands and told me, "This doesn't look right to me...")  If done correctly, their ending picture should have looked something like:

So beautiful and clean!

Part IV:
Solving rational equations through graphing and technology

Including review and assessment, I spent just over a week on this unit (like I said, it was fast).  But I'm pretty happy with it--especially our day in the Math Lab.  I worked out some issues with the activity, so I'm interested to use it again (I want to try it in PreCalculus) and see how it goes the second time around.

Asymptote Bingo

We're starting rational functions next week in Algebra II, but I knew my kids weren't super strong with asymptotes yet.  Truth be told, we still have a hard time recognizing that vertical lines will be an x=____; horizontal will be a y=____.

So, we played some Bingo on Friday, and I'm hoping for a solid start on rational functions tomorrow.

We play Bingo a good deal in Algebra II, so the kids are pretty familiar with my set-up by now.  Here's how we do it:

1. On a personal whiteboard, draw four horizontal lines and four vertical lines, creating a 5x5 game board.  Mark the middle box as your FREE SPACE!
2. Fill in  your board with these equations.  You may fill them in however you'd like, but you must use all 24 boxes--so keep track as you go!

Now we're ready to play!

I showed a graph of a function and then told the kids to cross out the correct asymptote.  The first several functions were strictly exponential and log functions--which we've already studied.  Anytime someone got a Bingo, s/he got a piece of candy (thank you, dear parents and guardians).  We played all hour, which gave nearly every kid a chance to get at least one piece of candy.  I kept track of the equations we had used and had the students call off their equations so I could check their answers.

After awhile the students started asking, "When are we going to get to the ones with two or three equations?  I need one for a Bingo!"

Mwahahahahaaaaa!  They were asking to learn about rational functions, and they didn't even know.

Before they knew it, they were analyzing the asymptotes of rational functions without any real instruction from me.  Just good progressions from what they already knew to what they needed to learn.

And now I feel at least a little better about their knowledge of asymptotes.

If you want the Notebook Bingo file, click here.

Solving Exponential and Log Equations Flow Chart

I guess I've been kinda into flow charts this year; I created another one for solving exponential and log equations.  The idea is that kids start with the top box, if they can't complete that task, then they go on to the next box.  We put examples in each box.  I used this for both Algebra II and Pre-Calc this year.

It's not flawless, because mathematics requires more creativity than a flow chart can provide.  But it gets the basic ideas across.

Solving Exponential and Log Equations Flow Chart

Fail Friday...on Saturday

I had been meaning to get help on this activity a while back, so Fail Friday seems to be the perfect opportunity for this.

Anyway...somehow I managed to totally suck at explaining intercepts this year in Algebra II.  Even after an entire semester with these kids, when I say, "What's the y-coordinate of an x-intercept?" all I get is *chirp, chirp, chirp.*

I wrote up this literacy strategy, that I was quite proud of.  I felt like they finally understood the algebraic definition of an x-intercept, and not just the geometric definition (i.e., I want more out of them than just "An x-intercept is where the graph crosses the x-axis.").  But...the next week I felt like we were back to square one.

Help!  How can I help them understand and generalize the concept of an intercept?  I especially want them to understand how factors and zeros are related.  What have you tried that you have had success with?  Class composition is juniors and seniors.

X-Intercepts and Factors by Rebecka Kermanshahi Peterson

Conic Sections Flow Chart

I just finished conic sections with my Algebra II kids.  I used the Conic Cards, created by the wonderful Cindy Johnson (@Johnsonmath), with whom I get to teach in the same building!  Both Kristen Fouss and Amy Gruen have used the Conic Cards and written about the awesomeness of them here and here.  They are truly amazing.  I was dreading teaching conic sections, but after two weeks of card matching (and the two weeks were right after Christmas break, I might add), the kids were able to knock the socks off their first test of the semester.  Which leaves us all happy.

In order to emphasize the similarities and differences of the equations for conic sections, I created this flow chart that the students filled out and used once they had learned all four conics.  Before the flow chart, every time a kid would say, "Mrs. Peterson, is this a hyperbola?!" I would go through the same questions with her:

"Are both variables squared?"

"No."

"Good.  So you know it's not a...?"

"Parabola."

"Awesome.  Now are the squared terms being SUBTRACTED?"

"Yes.  Oh!  Yeah, it IS a hyperbola."

I got tired of going through these questions over and over and over.  Also, I don't think the kids realized the order in which I was asking the questions, which didn't do much for them except answer the immediate question.

So, now, instead of answering their question with a string of my own questions, all I have to say is, "Do you have your flow chart out?"  Much less work for me.  A little more work for them.  And they're reading.

Conic Sections Flow Chart by

The wording isn't perfect.  I don't know how to succinctly differentiate between the ellipse and the circle in standard form.  This is the best I came up with.  Of course, then kids think as soon as an equation has fractions in it, it can't be a circle.  That's not really what the wording says, but I totally understand the confusion.  I combated the confusion the lazy way:  all our circles' centers were (m,n) when m and n were both integers.

Another good thing:  this flow chart can be easily changed to classifying conic sections in general form.  I just had the students take a few extra notes on the side (such as changing different denominators to different coefficients), and they were good to go.

All in all, conic sections went very smoothly.  And now onto exponential and logarithmic functions!

The Importance of Change

I was at a car parts store the other day, and I saw a sun shade for your windshield that advertised this:

I had to double-take.

A change of 44F is the same as a change of 7C?  Now, I'm not as savvy at converting from Celsius to Fahrenheit as Kate Nowak, but even I knew this was some faulty converting by a marketing department.

No.  I take that back.  I admit, they did the conversion fairly well.  44F is indeed (approximately) equal to 7C.  But a change of 44F and a change of 7C?  That's a little different.

Couldn't they just have tried an example?  One example.  That's all I ask.  Like, take two numbers whose change is 7, say 0C and 7C.  Covert those to Fahrenheit (32F and 45F, respectively), and you'd see right away--that's not a 44 degree change.

Check for the reasonableness of your answer, as I tell my students.

So, what is a change of 7C equal to in Fahrenheit?  Let's do a little algebra!

We know to get from Celsius to Fahrenheit we can use:  

So, let's take two temperatures, in Celsius and call them C1 and C2.  Then their change, in Fahrenheit, can be calculated through the following:

In other words, the change in Fahrenheit, is equal to 9/5 times the change in Celsius.[1]  Which means that a change of 7C is only equal to a change of 12.6F, a far cry from 44F, in my opinion.  And this verifies the consistency of our example that we tried earlier.  Which is good since this is a linear function we're talking about.  No change in slope here.

So, which is it?  Does this sun shade keep our vehicles up to 7C (12.6F) cooler, or 44F cooler?  Maybe we just get to choose.


[1]  This can be changed from an algebra problem to a calculus problem rather nicely.  I think this would be a lovely introduction to using differentials as approximations for actual change (or in this case, since the function is linear, the differential will be equal to the true  change).

Step 1:  Give the kids the function for F with respect C.
Step 2:  Have them calculate dF when dC=7. [dF=F'*dC=12.6]
Step 3:  Show them the  picture at the top.
Step 4:  Watch them pee their pants (they're math nerds too, right?).

Function Composition

I love composing functions.  I always have.  It's one of the topics in algebra I get most excited about.  I don't really know why.  It's cool that functions have this operation that the real numbers don't have.  It's cool that this operation shows up repeatedly in calculus.  But, ultimately, I just think it's really fun.

Every time I go to teach function compositions, I think I finally have it nailed.  I use all the right colors to differentiate between the inside function and outside function; I start out with an application problem to motivate the use; and I'm pretty sure my enthusiasm during this lesson is off the charts.

Still, something was missing.

In the past, I've started with composing at a general expression.  We would start by finding f(g(x)), for example.  I guess I did it that way because that's how the textbooks always did it.

This year, I started with composing at a number.  We started by finding f(g(3)), for example.  I'm not sure why I never thought of this before.  I feel like a total idiot that it took me this long, but it is what it is.  I did 2-3 examples of evaluating at a real number, and the kids took off from there.  As I walked around the room, checking their work, it was clear they nailed it (at least for the day, right?).

Now onto the general case.  But first, a bit of a digression.  I showed them this slide:

We worked these together.  The magic bullet?  f(a+1).  In the past, I would skip from f(a) to f(x+1).  This is where I lost the kids.  x+1 was too much too fast because the original function is in terms of x.  But adding that one little line made all the difference in the world.  From here, we could find f(g(x)) where g(x)=x+1.  No problemo.  Here is my notebook file.

Now if they can just remember that (x+1)^2 isn't x^2+1 I will be a very happy lady.

Kate Nowak wrote this in one of her posts that I've always remembered:

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get [it]...and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.

Dear Mom, I know something you don't know

This is something I did in the very beginning of the year that I meant to blog about and just never did.

At our school, students are introduced to the graphing calculator in Algebra II, which means I've gotten to show kids lots of awesome techniques to problem-solving that they haven't been exposed to yet.

One of our first tasks of the year was solving systems of linear equations by graphing, specifically using the graphing calculator (TI-84).  I asked the kids if they knew when the graphing calculator was invented and when it became standard use in high school mathematics courses.  I asked if their parents would know how to use such a device.

The students assured me their parents would have no idea how to use a graphing calculator, so I asked them to write their parents a letter describing how to solve a system using the TI-84.  I was somewhat surprised--every student (I have about 100 young algebraists) sat quietly and wrote.  And wrote.  And wrote.

This was my favorite:

Yo momma, it’s yo boy ________ and I’m telling you how to solve for y.  Add x on both sides then you will end up with y=x+0.  On the other problem you are going to subtract 8x on both sides and then divide by 4 on both sides.  You should end up with y=-2x-6.  Then you want to put this in your graphing calculator and press graph.  BOOM!  There’s the graph.  Then will want to press 2^nd then calc, then press enter three times and you should have your coordinates.

Quadratic RAFTs

I felt like my students were not yet ready to test on quadratics again (all methods of solving), so on Friday we took a day to recap what they've learned so far by writing RAFTs.

When I first heard about RAFTs, I was pretty excited, but I wasn't sure how juniors and seniors would respond.  To be honest, we do some pretty cheese-ball stuff in my classes, and I think this qualifies as such.  But, cheese-ball can be hilarious.  Here's evidence of the hilarity:

Role: Discriminant

Audience: America

Format: Campaign Ad[1]

Topic: The usefulness of the discriminant

Dear people of America,

They drew this on their paper,
but I'm too lazy to scan it

Are you tired of solving quadratic equations and wondering what the answer should be?  You waste minute upon upon trying to figure out when to stop solving.  Not with the discriminant.  With the discriminant you can instantly know what to look for while solving.  The problem under the square rot becomes hardly a problem at all if you vote to keep discriminant around.  Do yourself a favor, and check 'Yes" for "Vision D"!

Sincerely,

Board of Discriminants  

*****

Role:  i

Audience: Negativity

Format: Letter

Topic: How i and the negative numbers work together

Dear Negativity,

Your square root is always bringing us together.  At first we had a problem because you were always being fake, but then I came around and made being a real a possibility.  I know that sometimes your square root makes you feel imaginary but I'm always there to rescue you when he does that.  Many, many years ago you were a problem to everyone and no one knew how to fix you.  When I came along things changed and your negativity no longer was a problem.  I love you and your square root.

Sincerely,

i

*****

Role:  i, The Illusionist

Audience: Potential magic show-goers

Format: Ad

Topic: The coolness of i

Hi, my name is "i."  Some call me "Imaginary," and some call me "Illusion."  If you come out to this amazing show you won't regret it!  There are many fascinating things about me that I would like to show you!  Depending on when you catch me at the show, depends on my reality.  Let's just say there is a certain pattern to me.  Sometimes I am just imaginary when I feel like being myself, but I can also be in the form of -1, -i, and 1.  Do you think you can figure me out?  Come to the show and you will see!  Or will you...?

*****

I let my students work in "groups" of ones, twos, or threes.  They brainstormed on whiteboards, and then wrote their final product on a clean sheet of paper.  The activity took about thirty minutes.  A few of my students had written RAFTs before, but most of them had not.  So there was a lot of "I don't get what you want us to do."  And there were a few kids who just sat there for the first few minutes, which I'm pretty ok with.

I definitely had to encourage some students more than I did others.  But, reading the final drafts was both fun and enlightening for me.  It's clear that there are some topics that the students really understand, and some that we really need to discuss further.  This is what I love about writing in math class:  it's incredibly revealing, is it not?

[1] Sounds more like an infomercial to me, but to each his own.