"Undoing" the Chain Rule- Intro to Integration by U Substitution

This year I've revised how I'm going to introduce u substitution to my Calculus kiddos. It's an intimidating topic the first year teaching the course, knowing that it's an area in which students can sometimes struggle. Take 2 and I'm much more ready to let my kids do the directing. I'm using the chart below as a discovery activity for students to being to piece together the pieces (u and du) that will be integral to our future study:

After that, we're doing the following activity to practice identifying useful u's and du's. I know I got this activity from somewhere, but can't for the life of me remember where. If you know where it originated, please let me know! 

My kids are taking a quiz next class on basic integration and we'll start u sub as our send off to winter break. Do you have any ways that you love introducing u subs? Any tips or tricks from years of experience I'm in the process of acquiring? 

Happy Monday! 

A Little Bit of Thanks

Y'all....how do you not get the feels from this?

This is a student who went on from my class and is in one of the top engineering programs in the country. They felt the need to email me when they got their Calc II final grade back! 99.999999% of this student's success is from tremendous work ethic and awesomeness, but I am so grateful to hear a thanks for being a small part of that 0.000001% left over.

And also, how awesome is it to know that your kids go on from you and are super competitive and ready to tackle the world?! How awesome are our kids?!?!

Teaching with Anxiety: A (Complicated) Love Story

Anyone who knows me knows I love a good Netflix binge. Last year during a glorious string of hours with as little voluntary muscle movement as possible, I watched the first 2 seasons of the show You're the Worst. It seemed to be at first to be an irreverent look at the transition from the freedom of your 20's to the "settling" of your 30's, centered around 2 self-destructive people who fall in love against their better judgement. As the series went on, it started to take a more serious turn. It's revealed that the main character battles clinical depression and begins to chronicle the impact her battle with her own mind has on her relationships, career, and life. I found it poignant, hilarious, and an interesting break from my usual Bravo nonsense (#loveyouvanderpumprules). 

So what does this all have to do with me and more importantly with teaching?

I have always considered myself a "worrier." Not kidding- I remember being 3 or 4 years old listening to a cassette of "Don't Worry, Be Happy" before bed, scared to death that someone was coming to take my bed if I fell asleep. (Why would they put those lyrics in a song that could possibly be deemed appropriate for my bed time??) It was an endearing personality quirk my family and friends had learned to accept. I had also accepted it about myself and had, as a result, made deliberate choices to challenge my anxieties in hopes of conquering them. I seized opportunities to try new things, study abroad, move across the country, and challenge myself professionally, knowing the whole time that I'd hate the experience until I got comfortable. 

Fast forward to last winter: A series of incredibly unfortunate personal events led to me starting treatment for panic attacks and post-traumatic anxiety. Activities that had been commonplace in my day to day life were now terrifying. I'd spend all my energy to get up and put a smile on for my students and colleagues, only to come home and have no energy left for my family or myself. I started to understand the very real struggle people with anxiety and depression face everyday and I hated it. And I also started to realize that I wasn't alone in this- it was all around me.....especially in my students. With the help of doctors, friends, family, and my amazing husband and pup, I've adjusted to a new normal, one that always will have a little more anxiety. 

It's easy to look longingly at those people who seem to breeze around the school with a huge smile, charisma oozing from their pores. To long for a life where you didn't battle with your own irrational stressors. But the more I've reflected on my experiences with anxiety, the more I see how it has shaped the teacher I've become. And although I could have done without the panic attack before my first observation at a brand new district a few weeks ago (thanks a lot, brain), there are things that I'm learning to love about the way my brain works. Are there cons? Duh. But there are pros, too. 

Downside

Silver Lining

New situations and interactions can be stressful

I've spent the better part of my life making deliberate choices for this exact reason. I started living by the manta "Do one thing every day that scares you" (which isn't hard when you have anxiety) during my freshman year of college and it's taken me on some incredible journeys. You get used to the fact that you're going to feel scared and you're going to hate it, but hopefully it will be fine.  It's the reason I became a TA in college, the reason I studied abroad, the reason I moved across the country to an apartment I'd never seen (twice), and the reason I push myself constantly to do new things in the classroom.

This is how one looks after fear of volcano boarding on one of the most active volcanos 
in Nicaragua is conquered. I would say I volcano-tumbled more than I "boarded," but I still did it. 

Downside

Silver Lining

Living out every possible way a situation can go wrong in your mind

No one can say I'm unprepared.  I had a professor in graduate school who advised us to always overplan; this has never been as issue for me. When we first started teaching, I remember my husband saying he had 10 minutes left at the end of a block....unimaginable. Since I've always got so much planned, I have a plethora of resources and "other options" to use if needed. It's like a choose your own adventure some days- find out what the struggles are, choose the appropriate course of action.   I also tend to tweak and tweak and tweak my lessons, continually thinking how to make it less likely to go wrong. Can it be obnoxious? Sure. But 9 times out of 10 it does help.

Downside

Silver Lining

A constant feeling that things "could've gone better"

I'm my own worst enemy when it comes to criticism. I have had to teach myself when to say "this is as good as it's going to get right now" and be okay with that.  But I know that this sense of perfectionism is what made me a great student in school and what drives me as an educator. I am passionate about improving things around me and passionate about making sure my kids are achieving. I tweak, I read, I talk, I share, I get feedback. It's all in the pursuit of making things just the littlest bit better.

Downside	Silver Lining
You have to experience how crippling anxiety feels.	I have a much more profound sense of empathy for those around me than I ever did before my anxiety peaked. I can recognize the day to day struggles and identify small triggers that I never saw before. Working with teenagers? This gives you a whole new perspective. It gave me a bigger heart and bigger ability to teach kids, not just math.

When I'm feeling really annoyed by my own anxiety, I like this article too. 

I'm not 100% sure why I felt the need to write this, but it's been sitting unpublished for weeks in my drafts. I feel like there's more of us out there in teaching than we'd often like to admit. I've had to learn to give myself a higher level of self-care as I become more aware of my own anxiety and I hope others who need it can do the same. We are lucky to be in a profession where our weakness can often make us stronger, helping us connect with our kiddos in a new and more profound way. 

Applications of Derivatives Slap Jack Review

I've written before about my Slap Jack review game. It's one of my favorites for getting kids engaged and working together! 

The basic rules:

• Students play in 2 vs. 2 games (so 4 students per group)

• Students look at the 1st card and work with their partner to solve the problem completely on their whiteboard or in their notes

• First group to have a complete solution written can "slap" the deck to indicate they're done. They then have to defend their answer. If they're right, they get to keep the card. If not, the other group gets a chance to "steal" by getting the question right. 

• If no one gets it right, the whole group has to work to figure the problem out. No one gets the card, but they're still responsible for the topic. 

• Group with the most cards at the end wins!!

Here's one I wrote recently for my AP Calculus Class to get them ready for their big applications of derivatives test. It's a mix of released AP questions, questions from review books, and questions from worksheets. Lots of mixed practice on everything from curve sketching to related rates to tangent line approximations.

Applications of Derivatives Slap Jack

Triangle Congruence Theorems- Why Not ASS?

First of all, kids, yes. We are going to write the word ASS. We are going to write it repeatedly. I can insist you write it as SSA, but we all know that isn't going to happen OR help you remember it nearly as well. Get your giggling out now. 

We've been digging into triangle congruence proofs in Geometry and I wanted to make sure were digging into WHY, not just HOW. 

We started examining why the Side-Side-Side Criteria works, inspired by this post from Math Giraffe. I displayed each step one at a time so groups couldn't rush through and to make it a bit of a game. I offered candy to any group that could create a triangle that was different from the first on they created. Any group who felt like they had done it presented their ideas to the class and we examined each one. Something crazy happened....we could ROTATE, REFLECT, or TRANSLATE them all to be on top of the others. This was handy, as we'd defined congruence using the existence of a rigid motion between the 2 shapes that would match one onto the other. 

I knew this trick wouldn't work again (you can't fool kids out of candy twice). So for round 2, we started comparing SAS with ASS. This time, I went pasta! I literally woke up in the middle of the night thinking about this and wrote up my directions when I got to school on the board, so no file to share there. Here's the gist:

1) Each group gets a piece of poster paper & folds it into 4 sections 

2) Have kids break pasta into 4 pieces of 2 lengths (you can specify what lengths with your kids or just let them choose.....I think I did 4 4" pieces and 4 5" pieces). 

3) In each section on the paper, have kids draw an given of the same size. I did 30 degrees and had kids extend it out to the edge of that section. 

4) Challenge the students to use one of each length of the pasta to create 2 different triangles if the angle given is included between the 2 sides. (They can't. SAS for the win here.) 

5) Challenge the students to use the remaining pasta to create 2 different triangles if the angle given is NOT included. Don't let them glue until they have tried a bunch of different options- some will try to say this is impossible too. 

6) Have students summarize the difference between the 2 scenarios and name them as SAS and ASS on their poster. 

7) DECORATE! You have awesome new posters! 

Here's some of the resources I've used throughout the unit. I'll continue to add if I see anything else unique: 

• Triangle Congruence Start Up Worksheet from the wonderful Math Teacher Mambo
• Triangle Congruence Warm Up that I made to help differentiate between possible ways to prove triangles congruent. 
• Triangle Proof Planning Guide that I created this as a triage for some kids who were struggling with the basics, but wound up being really helpful as a conversation piece for lots of my students.

I haven't taught triangle congruence in 5 years, so I'm slowly starting to get back into the groove. So many things I'll change for next year! 

Tangent Line Approximation Discovery

You guys, what is it with me and blogging this year? What are your best tips when you feel like you just can't get into a groove with your blog?! I miss it! 

We just wrapped up our applications of derivatives unit in AP Calc and I wrote this little activity to introduce local linearization and tangent line approximations. It's one of the simplest but most important ideas in all of AP Calc- that if we look close enough, a tangent line at a particular point will be almost the same as the function since functions are locally linear. 

I knew my kids were more than capable of discovering this phenomenon and explaining it to each other without much help from me, so I wrote up this very simple activity for them:

We wrapped up with a discussion about what students had discovered, focusing heavily on part (f). I love seeing kids have these "duhhhh" moments- when something seems so simple that it's barely worth mentioning. It reminds me of those moments in college when a professor calls something trivial that was anything but trivial to you. When my students can start to view concepts in Calculus as so obvious they barely deserve explanation, I'm a happy teacher. 

Here's a link to the document! 

Tangent Line Approximation Discovery