To The Student I Will Never See Graduate....

Today I watched 522 of my favorite people walk across the stage and on to their next big adventure. I laughed, I cried (no big surprise to anyone who has ever sat through a year of my class or a Subaru/Folgers Coffee commercial with me), I sweat in a giant black garbage bag of a graduation robe....it was beautiful. But the whole time my heart was heavy, and I knew why.

For years prior to this one I had seen you around school, your nearest and dearest never more than a few feet away and all of you with almost constant smiles on your face. Thanks to another teacher, I got to know you through lunch time check ins and quick offers of math help before moving on to ask a kid at the table next to you to stop flipping his water bottle 7 feet in the air for the third time this lunch. And then you became my husband's chemistry "kid" and I knew I loved you before I even really got to know you.

Just a few minutes before lunch ended on the first day of school this year, you and your best friend arrived at my classroom door. You looked a little nervous, a feeling I imagine you might have had formally meeting anyone for the first time. Thinking now, I think of all the amazing ways in which you were different from any other kid I've ever met. But I bet for a 16 year old who longs to live the most normal life they can, there is an uneasiness before you know how someone new will treat you. Luckily for me, you knew and trusted my husband and it gave me an automatic "in" I can imagine would have taken much longer to develop without him (shoutout to him for being the best teacher and person I know).

And then, I had the privilege of being your teacher. I got to watch you and your favorite people giggle together when you came and ate lunch in my room. I saw you surprise yourself with how much math you actually could do when you just had the time to think and try. I laughed as you doodled pictures of your cats when I had a typo on a test that made a question unable to be answered. And I saw how you fought.

Calling you a fighter is an understatement. You fought through days when you didn't feel yourself, coming to school and giving your best effort with a smile. You fought me when I told you that I was exempting you from assignments because I knew you were balancing a lot of other things on your plate. You fought with Khan Academy when it told you that you were wrong after 3 questions in a row correct (even though I'd asked you repeatedly to please stop doing them because you were E-X-E-M-P-T, exempt!). When treatments started to interfere with your confidence and attendance, we talked about if you should drop the class and you fought that too. These were just the surface, the fights I could see. All the while, you were fighting so much harder.

When you started having longer and longer leaves of absence, my relationships with your friends changed drastically as we hoped for the best but braced for the worst. Other teachers and I tried to be a safe space for them, to support them so they could best support you. It was one giant trust fall. And then, on the day before our winter break, neither of your very best friends were in school. And you were gone. I hope I never experience a day like that again in my career. As news spread, waves and waves of students came to find a safe space. We ordered pizza, we played with play doh and colored, we sat in circles and talked and talked. We cried and laughed and cried some more. And then we all left each other for 2 weeks of winter break....each of us experiencing our loss apart from each other.

Once your services were over, we eventually had to come back from our holiday break.  I could barely breath when our class started and I knew I had to say something to address the unbelievable loss our class had just felt. We decided as a class to keep your seat open, because we all knew no one else could fill it. We eased back in to a new sense of normalcy. Some days were easier than others, but somehow time kept passing.

We made it through midterms and prom and finals week and here we are. You would have been celebrating the end of junior year, looking forward to senior year and all it had to offer. And though today wouldn't have been your graduation and though I know next year will be even harder for me, you filled my mind and heart all day today.

So thank you, thank you, thank you. I don't know how Mr G and I were lucky enough to both be your teacher, but you've permanently shaped us as teachers and people. Your infectious laugh, the optimism you displayed in the face of the most daunting challenges, and the joy you brought to every day inspire me to be better than I am. I once marveled to you about your optimism and you said this to me:

This hangs, framed, in our house. I look at it when I'm struggling to see the bright side of things and remember what power your optimism and joy had. When we have children, they will grow up with your words in their childhood home. They may never know you, but they will know your spirit.  Your grit and determination, your refusal to accept defeat, and your tenacity are things to which I aspire.  The depth of your friendships is truly unparalleled and I pray to be the type of friend that surrounded you daily. You've taught me about the way people subconsciously treat others they view as different from themselves and how imperative it is that we bring the same warmth, compassion, and high expectations to every student we teach. To every person we meet. You've made all of us better, just by being in our lives. 

And lastly, because I wouldn't be talking to you if I didn't say it...

You're the best.

What Do You Meme? (Math Edition!)

I was scrolling through Instagram when I saw this post by iteachalgebra:

Obviously, I had to do this.  

I decided to ask my kids to do a 3 part task: 

1) Your best advice for an AP Calc student

2)  A meme about a particular topic

3) A wildcard! Can be about our class, the ap exam, etc

Did the wildcards get silly? Oh yeah! But I got some winners and we all laughed about what a wild ride this year has been! Many were inside jokes from the class or had pictures of students in them, so I left those out. Here's a smattering of the fun: 

A fun laugh for our time after the AP exam and I am going to print a bunch out to hang up around the room with advice for future AP students. Can totally see this being scaffolded up more and turned into an actual, assessable assignment! 

Explorations in Mathematics Activities (or how to stay sane after the AP exam)

Here is one of the deepest truths about me as an AP teacher:

I can't convince myself to show movies every day. 

Let's be clear: I don't care if YOU are one of those teachers, I really don't! I often walk by your classroom and gaze in, longingly, at the students who are quietly on their phones as you grade papers and get s*** done in the dark corner with 21 playing on your Smart Board. I want to be you....I just can't do it.  

I do a bunch of different things after the exam, including teaching some Calc II topics like Integration by Parts and Partial Fractions.  My biggest successes have come from trying to dive into some upper level mathematics and fun logic puzzles in a fun and non-intimidating way. 

I'm gathering a collection of all the activities I use here, feel free to steal or modify any of them! I will add more as we progress- more to come!

The Josephus Problem

The Zebra Puzzle





The Four Color Theorem


The Collatz Conjecture

The Lagarias Problem

Others I've used in the past but haven't formally written up. Many of these are adapted from Numberphile videos. I'll work on writing up what I've done in the past!

The Secretary Problem & Optimal Stopping

Discrete and Bounded Envy Free Cake Cutting Protocol

Card Tricks & Modular Arithmetic

Optimal Shoe Tying Method & Knot Theory

When to Shoot In a Duel & Game Theory

The Imitation Game & Cryptography

Discovering Average Value in AP Calculus

I saw this tweet ages ago and it's been screenshot and sitting on my desktop for almost a year.

Had a poppin' lesson in calculus today demonstrating the #MeanValueTheoremforIntegrals! #popcornkernels #calculus #rhscalculus #iteachmath #RISDGreatness pic.twitter.com/Cqk3vIEz9T

— Nicole Lalanne (@NHLalanne) January 30, 2018

Before teaching average value this year, I say down to recreate it as best I could.  A few people asked me for it, so I shared my file below, but all credit goes to Nicole Lalanne for this one!

Average Value Discovery Activity

Also, if I could do it again, I'd use split peas or something else that is flat. Even my always responsible and mature AP kids couldn't keep popcorn kernels from rolling all over the classroom at some points (despite their ingenius pencil trick seen below).

Let me know if you make any modifications to it! I'd love to know any and all tweaks we can make! 

Rectilinear Motion & Balloon Rockets in Calculus

We are just wrapping up rectilinear motion in my non-AP Calculus class and I wanted a lab for kids to make the theoretical connections to the motion of real life objects. I also wanted them to get a feel for some rudimentary mathematical modeling. I didn't want them getting to the test feeling like they could ROCKET to an A with their knowledge and then having this kind of result:

I decided to write a performance assessment that would have students model a scenario and analyze the fit of their model based on their actual observations. I'm counting it as 15% of their applications of derivatives unit assessment. 

Rectilinear Motion Performance Assessment 

We work in 3 phases:

• Rocket Construction
• Basic Rocket Motion
• Advanced Rocket Motion

Basic construction is easy and cost me $3 in materials from the dollar store. A few groups struggled to get their rockets to move, based on how they attached their balloons. Most just needed to readjust so the air shot directly back out of the opening instead of up or down. 

For both phases- the basic and the ridiculous- students used Logger Pro to perform video analysis of the motion. They did a great job and the directions seemed to be clear to them. These were modified from directions to a Related Rates project I stole from Sam Shah, which he stole from someone else. Sam's version of the project is here and the original Logger Pro directions for that project are here. (Also I've used that project in my class before and it's fun if you're looking for a related rates adventure in modeling). 

What I love about Logger Pro is how easily students can look at best fit models for various functions. They were able to easily analyze the correlation and make decisions on which model was most accurate. 

I limited their options, as seen in the table below, based on which they could actually differentiate based on their limited algebra and calculus knowledge. 

Students then used their models and their knowledge of calculus to answer these....

Simple Motion:

1. At t=2, what direction is your particle moving? Use calculus to support your answer.

2. Use calculus to prove whether the particle is speeding up or slowing down at t=4 seconds. Explain how you know.

3. Based on your model, after how many seconds should the particle be at rest? (Use your calculator to solve, if necessary)

4. What should the position of the particle be when it comes to a stop?

5. How closely did the mathematical model represent your rocket’s path? Give evidence to support your argument.

6. How could we make another trial run more accurate?

Advanced Motion: 

1. When should your rocket be at rest?

2. On what intervals should your rocket be moving right? Left? Use a sign chart to make your conclusions and explain how you know.

3. On what intervals is your particle accelerating in a positive direction? Negative direction? Use a sign chart to make your conclusions and explain how you know.

4. On what intervals should your rocket be speeding up? Slowing down? How do you know?

5. How closely did the mathematical model represent your rocket’s path? Give evidence to support your argument.

So far, so good! If you try using this with your students, please send me any tweaks you make! I'm presenting this to some of my colleagues this week and can tell you there's lot so applications you can make of this lab to quadratics (at any level), parametric equations, and more! It's a fun way to bring math to life! 

Teaching Derivatives of Inverse Functions - The Easy Way!

I told someone last week that I'd send them a summary of how I teach derivatives of inverse functions, so I figured I would post it here too in case it could also help others.  My first year teaching AB this felt like a small formula that kids would need to have memorized for the potential of 1(or maybe 2) questions on the AP exam- ugh. I had multiple teachers my first year tell me just to skip it....it could be left out and no one would be the wiser. I really struggled with that idea, so I stumbled my way through it.

I was lucky enough to have another teacher share their strategy for teaching derivatives of inverses with me and I've never looked back. My kids rock at it now and it makes sense to them- not another formula to memorize. We are building on the idea of inverses from previous courses....that if (a,b) is a point on f we know (b, a) must be a point on the inverse of f.

I've also linked to a discovery activity where students see that derivatives of inverses are reciprocals by looking at linear functions. It's simple, but lets them do a bit more of the heavy lifting.

Derivatives of Inverses Discovery Activity & Directions with Table

Happy deriving! 

Polynomial Symmetry vs. Polynomial End Behavior

Let me dramatically reenact any given pre-calculus, algebra, or calculus class to you from my past almost decade of teaching....

Scenario #1:

Student: "This function is even."

Me: (Secretly lighting up inside because they understood the lesson!) "Why do you say that?"

Student: "Because both ends go up!"

**Face Palm**

Scenario #2: 

Student: "The function has an odd degree."

Me: (Secretly lighting up inside because they understood the lesson!) "Why do you say that?"

Student: "Because it's symmetric about the origin!"

**Face Palm"

Does this sound familiar to anyone else? 

I am fighting the good fight this year and preemptively planning an activity for my A2 students to work on to get them thinking about the differences between symmetry and end behavior. 

Symmetry vs. End Behavior Activity

It starts with students identifying both the symmetry and degree of given polynomials:

Then, asks them to agree or disagree with some statements about symmetry and degree:

Then, finally, asks them to draw their own pictures to fit constraints. 

Will report back on how this goes. If you have any tricks to help students keep these straight, please share!