Reviewing Inverses

The first few weeks of Pre-Calculus are always a little bit overwhelming for me. The course is designed to fill gaps of knowledge before students get in the ring with Calculus in an AP class or when they're off at college. Great, in theory. However, it means we need to make the gaps glaringly visible so we can address them. Before we tackle any of the really cool concepts, we have to tackle the misconceptions. One that I find the kiddos struggle with the most is inverses.

I've taught inverses so many different ways and always feel like I haven't quite hit the mark. I need a way to efficiently review the concept so we can get into some more skill based practice, but I don't want to do a full blown, hour long exploration. I also don't want to just lecture. I was playing around in Desmos and I decided I'd try a different approach this year. Spend less time trying to remind them of what they've learned in Algebra II myself and give them a little more time to think through it with a partner.  Here are my thoughts right now as to why I'm using Desmos instead of my beloved TI-84:

1) We can examine the concept of an inverse without worrying about the notation or the algebra just yet

The kids always are able to tell me "you just switch x and y," but they often have no idea what an inverse actually is or why switching x and y is important. I like we can build on what they do remember  (switch x and y) to tease out the actual concept. We don't need to get into messy algebra ("YOU EXPECT ME TO USE A LOGARITHM??") to illustrate it....yet. 

2)  The graphs are clear and the commands are user-friendly

We have block scheduling, so I have 2nd semester seniors who haven't seen a graphing calculator since they were first semester sophomores. It's definitely a challenge. Sometimes we spend so much time trying to reintroduce them to the steps ("Okay...2nd....calc....5.....enter....enter....enter") that our intentions get muddled. Yes, I know I need to reteach those, and we will get there. On a conceptual day, I'd rather not. 

We will definitely be examining the fact that the intersections all happen at a point where x=y.....why would that happen? What is demonstrated about the graphs here? This kind of conversations can lead to one to one functions, horizontal line tests, invertibility....all that good stuff. I'm hoping this gives us something concrete to refer back to throughout the year as we repeatedly see inverses in our curriculum. 

This is definitely simple and in the planning stages.  Here's what I have so far. Constructive feedback always welcome! 

A Day in the Life

So week one of the blogging initiative and I'm already behind (see my last post!).

4:30 am  The alarm goes off. I crankily roll over as I fall back to sleep to the sound of my saint of a husband piling on clothes to take the dog on a 25 degree walk. If he wasn't also a teacher I am pretty sure I'd be late to work every day. 

4:40 am Okay, okay....I get it, I have to get up. And shower. And decide what to wear. And think momentarily about putting on makeup before I say "nahhhhhh" and run out the door. It is the first day of classes....don't want to set the bar too high for the rest of the semester without makeup.

5:15 am Out the door dressed like as eskimo, scrape the car, and be thankful I'm not in my hometown buried under snow right now. I'll take 28 degrees at this point. 

5:35 am  Arrive at work and my competitive side is a LITTLE sad I'm not the first car in the lot. God bless the English teacher who, despite having 2 kids, beats my kid-less self to work every day. 

5:38 am  Quality time with the copier

5:51 am Track down a janitor to remind them that I still don't have enough chairs for my first period that starts in just over an hour. Try to hide my minor panic attack that my seating isn't done and ready yet.  

6:15 am Finally set up the 5 new seats that arrive in my classroom. Basically playing Tetris to fit 5 more 16 year olds into the room comfortably....a nice mathematical challenge for the morning.

7:15 am Homeroom starts. I see these kids maybe 5 days per year total. This will be the longest 10 minutes of my day....trying to keep calm 27 sophomore with whom I have no relationship other than I hand out their schedules and try to keep them quiet enough to hear the announcements a few days per quarter. Try to play off the fact that I've forgotten everyone's name since October, the last time we had homeroom. 

7:30 am - 10:35 am Pre-calc time!  I always start the first day with a fun game ( in this case, Math III Taboo - adapted from Pre-Calculus Taboo) to  pre-assess and get the kids talking and then I give them a sheet of skills they're expected to know from previous courses and give them some time to work collaboratively on them. I spend some time calming the panicking kids who haven't had math in over a year because of block scheduling. 

10:40 am - 12:40 pm AP Calculus Time. I gave back midterms and gave kids time to correct them, identifying and explaining their mistakes. Was going to jump into a new topic today, but first day back from a 5 day weekend and jumping into  a new topic without addressing their mistakes seemed crazy. Take an awkward 25 minute lunch break 30 minutes into class because of our strange 4 lunch schedule. 

12:40- 2:15 pm Planning! I've never had 4th period planning before and I'm not sure how I feel about it yet. It's a good opportunity to reflect on the day and make tweaks, but also to get ready for tomorrow. It also includes all the chores of the day....check mailbox, file papers, walk to office to pick up package with turns out to be the greatest thing a parent could ever drop off...a bulk store's biggest pack of Kleenex! 

Got copies ready for tomorrow, sent lesson plans for Friday and Monday to have them brailled, nursed my Diet Coke to help try to stay focused, set up my new Canvas courses for the semester, sent some parent emails, etc. 

I think it also goes without saying that I refresh the weather report about 35 times during this person too, as some reports are saying we'll get a dusting and some are saying we're going to get 15 inches. 

2:15 pm The last bell rings. New people are starting their afternoon duties and I somehow didn't get parking lot duty. Needless to say, I'm feeling like...

2:40 pm The time I mean to leave school to let me poor puppy out after being home alone since 5:15 am. 

3:10 pm The time I actually manage to sneak out the door. 

3:30 pm Arrive home, long dog walk as an excuse to play in the flurries we are getting right now, settle in for some serious lesson planning to make sure I have time later to relax a little. 

7:00 pm  Realize I've been snacking for 5 hours and actually eat dinner, torture my husband through an episode of Vanderpump Rules, and get ready for bed.

8:45 pm In bed. I realize this is embarrassing, but it's #teacherlife. Alarm is set for 4:30 am...it's coming fast. 

Balancing Act

Saying I've had a trying last 5 months would be putting it nicely...an understatement equivalent to saying that you should probably not put the buffalo chicken dip near me at a party if you want anyone else to get some of it (I'm good at sharing with others until it involves cheese and buffalo-y goodness).   It's been unbelievably challenging personally and the thing that's hit me the hardest is the way it has made me feel as a professional. We go into this profession because we love to help. If you're teaching and you don't love kids and seeing them light up, you need to find another line of work. We work 60....70....80 hour weeks for the first few years of our career to try to prove ourselves. We do all those things you see teachers do in the movies- bring kids things from your own home to make sure they don't go without, commiserate with kids who are struggling with situations that are more than just academic, and get truly invested in a group of munchkins who you will have to say goodbye to in just a few short weeks.

When life happens, it's hard to strike a balance between that "movie" teacher we want to be and the reality of what we can handle. I had to spend a few weeks working to my contract hours only and it's almost unimaginable the amount of things that couldn't get done- the grading, the tutoring, the formative assessment and feedback, the parent emails, the committee meetings, the re-writing of lessons to help them to meet the needs of different kids, the club meetings that had to be cancelled, the bathroom breaks (yes, those and lunch go right out the window first). I don't think in my years of trying to prove myself I realized just how much of my own time I was giving up to be the type of teacher I wanted to be. And I didn't realize the guilt I'd feel when I needed to invest my own time in myself and not in my kids. 

I'm still working on getting back in balance....on being able to take some time out of my own personal life to let myself heal without feeling like I'm letting kids down. On being able to be okay with working only 50 hours a week. If anyone has found a magic pill to cure this, let me know. Until then, I'm taking it one day at a time and knowing that I'm doing the best I can. I'm also saluting everyone who gets up every morning and smiles at kids as they walk into homeroom while putting aside their concerns about their own children, their own family, a chronic illness,  a whole history of personal tragedies and fears. This career is not for the faint of heart and the only way to sustain it is to find a way to be both a teacher and a person. 

Mathematically Correct vs. Mathematically Helpful: Polar & Rectangular Forms

There are some topics, especially in Pre-Calculus, where I feel like I need to help kids realize the difference between making mathematically correct moves and making mathematically helpful moves. Trigonometric identities is one of these. Converting between polar and rectangular equations is another. If you present it the right way, kids appreciate the elegance of a mathematical "trick"- something one of my beloved mathematics professors would call a "cute" move. These moves the ones that are more than just correct- they are actually helpful in getting you towards your final goal in a problem. 

I have always found that in these kinds of situations learners respond in one of two ways:
either with excitement and curiosity or resentment and bitterness. Students who have come to my class from a previous course that emphasized rote memorization and procedures sometimes feel like they're "doing everything right" but still can't come up with the right answer- a huge source of frustration. Students who have been asked to reason on their own, look for patterns, and solve puzzles in previous courses often find these types of problems invigorating and exciting; they live for the "thrill" of discovering the right move to make.

One way I've tried to combat this and get everyone more on the "I can solve it!" page is to try to make these problems into puzzles whenever possible. I love the way this Time Magazine article discussed the human desire to problem solve:

Human brains have an extreme form of consciousness: they’re ravenous for new innovative solutions to problems in the world, ravenous for optimizing our lives, for building pyramids of knowledge. 

I have found that the more I try to celebrate the "cute" and elegant mathematical moves and help kids see the repeated reasoning that goes into these problems, the less resentful they become. 

Here is a puzzle activity for converting polar and rectangular equations. Directions I give the students: 

I have partners sort them into groups and try to put the "steps" in order. They then need to create a "booklet"  (fold a piece of paper in half so there are 4 sections- 2 on the front, 2 on the back) with one problem on each "page", glue the steps down in order, and explain the reasoning between each step in their own words. I love the conversations I hear between partners, especially on a problem where they "just know" what the next line should be because it seems correct, but they need to think about how to justify it. This really helped my kids dig into the "why" of these sorts of problems. 

Polar and Rectangular Conversion Sort

Coherence & Canoe Trips: Common Language for Understanding Vectors & Polar Coordinates

First, let me begin with this:

Okay, now that we've gotten that out of the way...

One challenge that I find with teaching Pre-Calc is that it's the culmination of all the content the students have learned up to this point. Very few of the topics are completely new to the students, so often there is a "we already know this" attitude when we intro a topic. However, the course requires more than just the knowledge of each specific skill; it demands an integrated understanding of the concepts. Disconnected thinking just isn't going to cut it here....coherence among pedagogical approaches needs to be a priority if we want kids to have an integrated conceptual framework in their minds. Kate Nowak mentioned it in her talk at NCTM Nashville- you don't want to piece together a string of "short stories" in your class; you want to reveal to students the "whole novel." 

My goal years ago was to develop some sort of common experience to use with my kids to talk about the concepts of vectors and polar coordinates since they are so closely related but often taught separately. I wrote and have been tweaking these explorations for the past few semesters.

Canoe Trip Part I- Finding Vector Components


Canoe Trip Part II- Converting Between Rectangular and Polar Coordinates

Photo from http://smsta.msta.org/wp-content/uploads/2014/09/460507657.jpg

Both of these are introductory level activities and I've found that some kids latch on to the "rule" that they develop quite easily, while others cling to the "drawing a triangle will always work" approach. Both methods are 100% valid to me, so I don't make much a distinction at this point in the concept (although the "rule" method is much faster!). When I see those right triangles being drawn out, I get the "teacher stock photo" feeling. My heart feels like this picture- a student choosing to use the concept instead of just blindly memorize the formula! I have a "moment", for sure. 

I like the way these work right now, but I'd like to make them more open ended in the future. Maybe give different groups different scenarios and have them critique each other. Maybe ask them to create their own scenario to model a situation. Unfortunately, these topics always wind up getting smushed at the end of the course and we don't get to have the "play time" I'd like. Anyone have any tweaks or any ways they love to teach these topics? I've love anything to improve or enhance these! Comment away! 

Thanksgiving

We have an amazing art teacher at my school who dedicates the day before Thanksgiving Break to gratitude each year. She has each student write notes, some anonymous and some signed, thanking someone who has influenced them at school. This year, one of her students chose to write this to ALL teachers instead of just picking one she sent it out to all of us. It was my email I received all day:

I'll be the first to admit that I love a good pity party. I think about not having to choose between paying off my student loans and using my 20's to travel the world. I think about having evenings and weekends to relax....maybe even go out! I think about being evaluated on my own merit alone, not on the amount that I'm able to overcome the host of outside influences on a  group of teenagers. It all sounds pretty appealing, especially when you're in a job that even said teenagers can see is under-appreciated.

But then I take a second and I think about those teenagers- the fun that I have with them on 8 out of 10 days, getting to witness them understand something for the first time, earning their trust enough to have tough conversations about what's going on at home or with their friends, the list goes on and on. I can't imagine a career where I don't get to share dynamic moments like that with a group of people who are excited about all that the future holds. We talk about "kids these days," but I have been lucky enough to know great kids who are going to make the world better.

So here's my yearly list of things for which I am grateful, work edition:

• The opportunity to teach in a program that supports innovation, most notably in it's allowance for failure in the interest of growth
• That I have spent my entire career surrounded by the most incredible colleagues
• The people who have shaped my teaching, whether it was in person or through online collaboration (I'm looking at you, MTBOS). The whole "standing on the shoulders of giants" thing is so true- I am truly a better teacher for having worked with each and every one of them. 
• The trust of those around me which has opened new doors to me as a teacher
• That I married an educator who not only understands my insane work schedule, but pushes me to think differently about my own practice and my kiddos
• Caffeine....in nearly any form.
• The truly amazing kids who challenge me every single day to bring my best and forgive me when I don't

Time to put on my stretchy pants and prep for my 3 favorite things: family, stuffing, and football. Happy Thanksgiving! 

Prunes and Cowboy Boots: Tales from NCTM Nashville

I've been home from Nashville for 2 days and trying to somehow simultaneously process all the awesomeness I just experienced and entertain family that is in town for the holidays....not an easy task so far. After packing up the in-law's van this morning and eating about 15 candy canes (it's the holidays, isn't it?), I'm hoping a little writing will do the trick. 

Nashville or Bust! 
This year has been my toughest yet teaching, without a doubt. I have been facing that typical year 4/5 burnout- taking on a huge number of responsibilities while trying to juggle being a newlywed and have a personal life. The balance between doing what's good for my students every minute of every day and doing what's good for my own life and sanity....it's been a tough one to try to strike. In addition to all of that, I've had a heck of a fall personally. I sliced the top part of my finger off about 4 days before the school year started trying to be an iron chef and not use a hand guard on my mandolin slicer. We've been battling puppy cancer with our (way too spoiled) rescue mutt since September- and seem to be on the other side of it right now, thankfully.  We've had other personal trials we never saw coming and all of this has such a huge impact on life in the classroom. You don't get an "off" day in teaching and when you have one, it just makes you feel guilty that you're letting the kids down. I needed some time away from the classroom to get some perspective back, to reflect, and to immersed in a group of people who understand all too well the challenges I'm facing. I'd been looking forward to NCTM Regionals as that opportunity and it was that, for sure. 

Side Note: If you've felt this way, I highly recommend this blog post by Vicki Davis. We definitely all walk wounded in teaching....gotta stay in the game. 

Prunes? Why Would You Put That in a Blog Title?
Andrew Stadel (Divisible by 3) told a wonderful story in the opening session about how advice is taken best when it is timely. Or more precisely, how another exhausted parent at a playground solved his toddler's constipation issues by telling him how they'd gone through the same issue and the answer was PRUNES! If someone had told them this advice at their baby shower or months before, it wouldn't have carried or the same weight or perhaps might have been forgotten completely.  I think about it like this: In high school, I read the book The Unbearable Lightness of Being. Or should I say I skimmed "it"....."it" being the Sparknotes. It didn't resonate with me; I didn't have the life experience to be able to connect with it. In my senior year of college, I was assigned it again and was dreading it. I was also right in the middle of a breakup of the cliche big college relationship....cue the timely part. I didn't even know if I think the book was all that "good", but it was the right book at the right time for me and it helped shape my experience at that point in my life. It was my "prunes" when I was holding the constipated toddler in the middle of the playground. 

I felt that familiar feeling of resonance in Nashville....a recognition that this was the right advice at the right time for me. I am feeling re-energized not only by the presentations I heard (most of which were great), but by the people I met. I was ready to hear these things and meet these people and I'm so thankful that I had the opportunity. 

What Did I Walk Away from Nashville With?

• Formative Assessment I got to attend the pre-conference workshop on formative assessment and left with tons of research-based ideas on how to involve my kids more in peer and self assessment and also with a new lens on the amount of grading I'm doing. I'm going to start incorporating way more self-graded or ungraded assignments which I can leave feedback on without muddling it with a grade at the top. I teach honors kids. If it's an A, they file it away. If it's not an A, they hide it in hopes no one else sees it. Either way, I spend hours grading and wind up finding papers on the floor or in the recycling bin.....frustrating to say the least. Get rid of the grade and they need to fish for the comments and the feedback more. So simple, but filled with so much power.
• Modeling I love modeling tasks...it's my bread and butter as a STEM teacher. While It might not be a new idea to me, I definitely left with a host of new ideas for every course (including Calculus! Yay!). A huge thank you to Brian Shay (@mrbrianshay), Brett Doudican (Coordinated Achievement), and Robert Kaplinsky (RobertKaplinsky.com) for inspiring modeling sessions. 
• Blended & Flipped Learning I've been doing blended learning for two years in my reach-model classroom, piloting it for our district. I've proven that the reach model works for the right kiddo- one who is self-motivated and takes the time to do their work outside of class. I learned that I really am towards the forefront of thinking on much of this stuff (which is nice validation), but I had the chance to really push my thinking further with great colleagues. One session in particular that I attended examined how people are using flipped models, examining the difference between videos that are purely instructive vs. videos that still drive student thinking using the ideas from Principles to Action. Yes, the videos replace the lecture. But in particular, we discussed using the video as a driving factor the conversation the next day. The video does not need to answer questions- it needs to generate them. So small, but so powerful. Thank you to Jeremy Strayer from Middle Tennessee State University for having such a productive conversation in his session. I also attended the packed session on flipping AP Calculus with Joel Evans and left feeling like my dive into flipped AP Calc (which I promised my class we'd start after Thanksgiving) won't be as scary as I thought. 
• Networking I finally got to meet a lot of the people whom I've been following for a long time and who have inspired a lot of my lessons....got to have my little math teacher celebrity moments. Getting to chat with so many of MTBoS colleagues who have supported me without ever meeting me was amazing.  And my final math teacher celebrity moment- finding out Kate Nowak (Function of Time) and I had the same graduate advisor. My graduate experience shaped so much of who I am as a teacher....no wonder I find inspiration from someone who worked closely with the same people I did. 
• FREE STUFF!! I won a copy of Principles to Action and have already highlighted the crap out of it. I ate about 37 tootsie rolls and 113 mints from vendor booths. I signed up for copious amounts of free resources from NASA. I got a brain stress ball from Gizmos when I told them I was part of a grant. I got a student sized whiteboard (because I don't already have 30 of these in my classroom....). And I heard lots of great country music in our time outside the conference. 

My Favorite "Overheard at NCTM Nashville" Moments

Pretty sure this was Robert Kaplinsky right before his session started to a participant talking about fear of things not working

"If everything always works in your classroom, you're not innovating enough" 

An awesome lens for viewing flipped instruction, when people were asking questions too specific about "videos"

"It's not about the videos, it's about the pedagogy." 

God Bless Puckett's Nashville

"Mac and cheese is the vegetable of the day"

A participant asked Kate Nowak how to do a particular problem she'd offered to us as a challenge. Her response was the perfect teacher answer. 

"It's pretty satisfying and magical when you figure it out, so I don't want to steal that from you by telling you."

So to NCTM, all the conference attendees, my coworkers who tolerated my crankiness on the traffic-filled drive home, and the city of Nashville, I say: Thank you, thank you, thank you!!!