Thanks, y'all.

Nearly 5 years ago, my (then) boyfriend and I graduated from our beloved alma mater with no jobs and nowhere to go but our parents' houses. Unfortunately for us, that meant being hundreds of miles apart as his parents had retired to the Mecca better known as Myrtle Beach. We knew teaching jobs were hard to come by in our home state and decided to begin a new adventure in a place where they needed teachers- North Carolina.

We got a lot of flack when we decided to leave the Northeast....

• "Are you crazy? You don't even know anyone there!"
• "Is everyone going to be stupider?" (Jokes on you all, stupider isn't even a word)
• "Now don't start coming home and saying 'y'all' or anything"
• "You'll love the winters, but aren't the summers the worst?"

But let's face it, when you're 22 and trying to be near someone you love none of that really matters. Not to mention $32K sounds like a TON of money when you've been living off student loan refunds and minimum wage retail jobs. This adventure was going to be great. We knew it. 

We weren't wrong.  We got married, got our first apartment, and adopted the most wonderful dog that's ever lived in our time here. We have made lifelong friends with both people and food groups (pimento cheese is my bffaeaeaeaeae). We've been given the opportunity to grow and lead in supportive environments, surrounded by people who believe in us. 

As I pack up my classroom to begin the next phase of my life-  near family and with the intention of "settling down" a bit- I can't help but reflect on everything my time and the amazing people I've met in North Carolina have taught me. 

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Southern charm is a real thing

There is truly something that is unique about being in the South. No one is kidding when they say that life moves at a different pace. I have honed my North Carolina accent, learned the difference between Eastern and Western BBQ (none of which involve hot dogs or hamburgers....learned that one the hard way), and developed a lifelong relationship with pimento cheese. I've seen the value in small communities of people who grew up together and support each other and I've seen the value of living in a city full of transplants where everyone is looking for a new sort of "family." I now understand what it means to "get your picture made," get a "buggy" at the grocery story, or (my personal favorite) ask a student to take a "toboggan" off their head. It has nothing to do with sleds. I know. I've learned the power of "Yes, Ma'am" or "No, Sir" and have seen students who grew up valuing these ideals. And northerners who haven't experienced this- there are school bus races on a dirt track at Charlotte Motor Speedway- #priceless. 

There is nothing more powerful than the people that surround you

I have been blessed to work with some of the most amazing people in my time here. I took my current job to be near my now husband, but I got so much more than that. I think about my coworkers (who became family) like a cast of characters from a great ensemble sitcom. If anyone wants to make a show about us, seriously....it'd be hilarious. You can't make up the stuff that happens to us. But more than just keeping each other laughing, I met a group of people who challenged me to be better. As a first year middle school teacher, I was all about structure. Inquiry and collaboration were great in the time in my lesson plan that I'd specifically set aside for it. I have learned how to embrace teachable moments, how to have fun with my students, and how to create a culture of collaboration and teamwork in my room. I have seen peers who push themselves constantly to be better, knowing that every small detail of our craft that we hone helps a child. I've had my eyes opened to the connection of my discipline with others and the way which genuine collaboration should work with colleagues. I've been trusted to be part of a team in which everyone knows their role and demonstrates the synergy that comes only from working towards a shared vision. 

Vulnerability makes you a better leader
I'm not someone who loves being wrong. I'm can vividly remember my Grandpa (lovingly and jokingly) saying to me "Why didn't you get a 100%?" when I came home with a 99% on a test. It's just my nature- I strive for perfection. But let's face the facts...no one's perfect. And the only way to innovate and to truly lead is to be able to face this head on and grow from your weaknesses. It has been a lesson that's been learned slowly, but I now see it as key to my teaching. 

If I hadn't come here, I'd never have been able to experiment with STEM education while being supported by a huge federal grant, pilot blended learning in math for an entire district, or design and build my own version of an AP Calculus course to support students that needed it. I felt a huge burden of trust placed upon me and knew that the only way to not let my kids down was to make sure they had a voice. I began to tear down the wall when possible to let students know that I was intentionally designing the things we did in class with a purpose in mind and I was open to their suggestions. I now seek their feedback as often as possible and have used it to better my methods in general and in my specific classes. I've grown so much from doing the things I was scared of doing and I hope to never lose that drive. 

Sometimes the biggest strides come from difficult situations
North Carolina's educational system hasn't always been considered a gleaming beacon of light when compared to it's 49 competitors. In September 2015, Forbes rated this fair state the 2nd worst for teachers in the nation. So did WalletHub. Things aren't great. But I've learned that these trying conditions can lead to truly tremendous results. 

First and foremost, every teacher I've met in my time here has truly been "in it for the kids." They may not spend their whole career teaching and they may not have a ton of advanced degrees, but you don't teach here for the money. These are people who give their time day in and day out making the equivalent of what they could make as a manager at a local gas station. People are in education here because they love kids and that can create a powerful environment for our munchkins to learn.

The other advantage of our educational system being labelled "broken" is that people are trying to do things to fix it. I was amazed during my interview process in other states that I have been at the forefront of technology integration, blended and flipped learning, Professional Learning Community implementation, UBD, and more. Things that I just assumed were commonplace were innovative to some districts with whom I spoke. I have had the experience of facilitating a PLC for years, something many places are just beginning to embrace. I have been a part of EdCamps, I've gotten to learn from and facilitate a huge amount of professional development, and I've taught in a school that encouraged us to try new things. I've been blessed with administrators who believe that if you just teach good math, the test scores should follow.  

I teach kids, not math

I remember one of my graduate professors saying that elementary school teachers love kids, middle school teachers love teaching, and high school teachers love their content. The stereotype of a high school teacher is the slightly professorial expert who is more interested in droning on about their subject than the kids in their room. I love math....I really do. But I've learned that you never get back the moments with kids that you let slip away. I've had to reflect on the things I didn't say to a student who died too young. I've collected supplies for a student who lost everything in house fire. I've sat with a student and helped them fill out their FAFSA when they didn't have the support at home to do so. And as great as some of my math activities may or may not have been, they won't be what I remember most. The hardest part about leaving has been saying goodbye to these kids. My kids. I'm so glad I've learned to connect with them in a way I never expected. 

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Although family (and better pay and more respect for my profession) beckon me back to where I grew up, I am so fortunate to have spent 5 years here learning and growing. I bring so much of what I've learned here back home with me and I'm so thankful for that. 

So consider this my thank you, North Carolina. Thank you for your beaches, your mountains, your quirky neighborhoods, your sunny Saturdays spent with friends, your Panther's games, and your countless new adventures. Thank you for the 60 degree snow days, the pollen all over my car, and the rainy morning that gave us a whole garden to ourselves as my now hubs proposed. Thank you for my puppy, the happy stray that we couldn't have adopted from any other Humane Society. Thanks for teaching me to use the word "y'all" naturally in a sentence (Northerners, get over it; it's so much more efficient). Thank you for your kiddos, who each day brought me joy, laughs, and the occasional headache. Thank you for people who trusted me enough to encourage me to grow and lead. Thank you for giving me friends who I know will stand the test of time. My time here will never be forgotten and you will be missed (especially the first time in 5 years that I have to shovel snow). 

The Final Countdown: AP Week is Here!

Here's the motto I'm trying to instill in my kiddos this week:

During the middle of last year, a colleague and I were discussing the fact that our school only offered BC Calculus and we wished there was an option for students who were interested in pursuing Calculus but might not be ready for the demand of BC. Heck, my bachelor's is in pure mathematics and I did just fine having only been a product of AB Calculus. We approached our administration about this and luckily for both ourselves and our kids, they allowed us to take on the challenge of designing and implementing a full-year AB course. The course would be an "Honors Calculus" course in the fall, leading directly into an AB course in the spring.  It took some selling to get our enrollment up and I am forever grateful to the 20 kids who took the risk of a brand new Calculus course with me. The enrollment is already more than doubled for next year and that definitely wouldn't have been possible without this pilot group. 

I feel a little like the cast of one of those youth sports movies sometimes when we are in class- despite the fact that Emilio Estevez had consistently better hair than me in any number of those movies.  We are an unlikely mix.....17 boys and 3 girls (I promise our demographics are more evenly split for next year!!) with extraordinary variation in past experience with math, intrinsic motivation, and consistency in completing tasks. But they all showed up every day and tried. Cue the Mighty Duck's-esque montage where everyone learns that "Ducks Fly Together" and learn to skate without falling.  I've gotten to watch this unlikely mix of kiddos turn into a team who can function at a really high level (or more concretely, get about 5 to 9 points on the famous 2007 AB 3 problem).

Because of the extra time built into our course, I really do feel like I've given the kids all the tools I possibly can at this point in the game. I'm okay with not trying to "cram" any extra information in their brains on the last day....which is a feeling of peace as a teacher that I don't normally feel. With my unlikely group of mathematicians and the mindset that we've prepared with the most gusto we possibly could, I'm taking a very different approach on our last day in class tomorrow. Here's the plan:

1) A sincere "thank you"
Y'all.....I'm so proud of these kids. Because I teach mostly in a STEM program, this is the third straight year I've taught this particular group of 20 and they've been a huge part of my journey as an educator. Beyond all of that, they've been with me through an incredibly difficult year personally. They've accepted me on my bad days and pushed me to be better even on my best days. They've driven me a little crazy, of course. But I am so grateful for the hard work and determination I've seen from each and every one of them. They've made me laugh and learn and grow. I don't know if we ever thank our kids enough. I'm trying to remember to do it more. 

Good Luck Gift Bags (because my inner middle school teacher still hides deep below the surface)

2) A little more shop talk
Because I don't think you can emphasize "DO NOT write 'it' if you mean f'(x)!!!!" enough. I don't mind saying it for the 16,000,001st time if it means they remember. And for Heaven's sake, don't leave off the + C !

3) A real, honest talk about anxiety, nerves, and some brain chemistry
Here's why: I am someone that naturally appears incredibly collected and outgoing. I seem to thrive on being constantly interacting with others. This is a completely intentional choice I make everyday- I am much more of an introvert than anyone would suspect.  It was actually my time as an AP student that started my journey to discovering this. AP US History basically turned me into a robot; it was the highest expectations to which I'd ever been held in a subject area that wasn't my strength. Friends tried to talk some sense into me, but I assumed it was just stress and it would pass. As I got older, I started to become more cognizant of the amount of anxiety I felt on a regular basis (although I'd still fight you if you tried to tell me I was at all abnormal in what I felt). Only this year did my anxiety finally bubble over into something I needed to address and doing so has totally changed my perspective. 

I think a lot of kids feel this kind of anxiety and pressure. Our guidance counselors get crying AP students in their room regularly. We expect to see some mini-breakdowns as exams approach. And let's be real....most 16 and 17 year olds aren't equipped to deal with it. So here's what we're doing:

Activity #1:  Article Reading

I'm going to have the kids read this silently, marking a "?" next to anything they have questions about and a "!" next to anything they want to remember for the test. Then we're just going to talk and see where it goes. 

Activity #2: Journaling Activity

Since my very first standardized testing week as a 1st year teacher, this article has been in my arsenal. I love it. I love the idea that psychologically the symptoms of being "pumped" are the same as being terrified.  I try to get kids to get pumped to show what they know, not fear what they might not know. 

"In a study published last year in the journal Emotion, Dr. Beilock and four co-authors found that with students anxious about math, the more stress hormone they produced, the worse they did on a test; students with low math anxiety did better the more cortisol they produced. “The first group,” she said, “felt the rising anxiety in their bodies and reacted by thinking, ‘I’m really nervous about this test. I’m afraid I’ll fail.’ ” They choked. “The second group told themselves something like, ‘I’m really psyched up for this test! I’m ready to go!’ ” Dr. Beilock recommends consciously adopting positive self-talk. Remind yourself that damp palms and a pounding heart accompany all kinds of enjoyable experiences: riding a roller coaster, winning a sports match, talking to someone you have a crush on."

We are going to end the class by actually doing the journaling activity as an exit ticket. I am encouraging each of them to leave their fears in my classroom by writing them down and putting them into a box. With those left behind, I am going to encourage them to take only their knowledge, skill, and confidence with them into the test. My hope is after the exam we look at some of their fears and see if they were worth "carrying" with us or if in retrospect it was okay to leave them behind. 

I'm incredibly proud of them and am hoping to leave them feeling excited to show all they've learned. Any I will definitely be reflecting with them on how to make the course better. 

Good luck to everyone's AP classes these next 2 weeks! Cheers to all your work to prepare an incredible group of kids for an incredible challenge! 

Learning to Trust the Core: My Weekend with the #CoreAdvocates

I feel very fortunate to have spent the weekend with some amazing math educators from all over my state, digging into the Common Core State Standards. Even though the standards are in their sixth year and we've certainly seen a shift in many classrooms towards a new way of teaching, I think we've all seen that the kinks definitely aren't worked out and there are lots of people who are still reluctant. In all honesty, who could blame them? When you've been doing something for a long time in a particular way (not to mention being a student who learned a particular way), it's difficult to feel like you're being asked to reinvent the wheel. When a huge part of your evaluation is based on a multiple choice test, hard choices sometimes get made. You hope you're doing what's best for kids and try to trust your own professional judgement. 

[Brief aside: I feel so lucky to have taken the path I did to education. My graduate advisor, Dr. Jean Schmittau, was someone who was truly ahead of her time. She advocated for conceptual, student-centered teaching for decades before the words "Common Core" were ever utter together and truly pushed her students to think deeply about mathematics and pedagogy. Let's put it this way....one of our assignments was to make a concept map that showed the concept of multiplication and it's relation to the body of mathematics as a whole. I had to go to Kinko's to get mine printed because it was roughly 5 feet wide. We spent almost a month of 3 hour classes trying to answer the question "What is multiplication?" and often we were given another week to think about it because a room full of people with pure mathematics degrees didn't answer her well enough.  She and the wonderful members of my cohort shaped me into a teacher who is always seeking to improve and challenge myself and my students. I was reminded of her in so many ways this weekend. ]

We opened the weekend with this big question:

Why are we here? 

Pretty existential, right? We were challenged to examine that question from 3 different perspectives and I couldn't think of a better way to share all that's swirling around my brain right now.

Why are we HERE? 
There is something so valuable for a teacher's soul when they are able to congregate with like-minded people in a neutral site. You are able to immerse yourself in a world that centers around what you're studying and have discussions that might be difficult in a place you are too "comfortable." It may seem small, but I think it's a huge part of what makes conferences so powerful.

Why are WE here? 
 A topic that came up over and over again this weekend was the idea that teachers need to "own the standards." It's a picture that I think holds a lot of power in education. If we truly "own" what we are teaching and examine it as practitioners, we will be cognizant of how these standards connect with the conceptual structures our students have created in their schooling. A huge part of this is creating teacher leaders and I was happy to be in a room full of them. I honestly believe that these people are the key to any real change- teacher leaders need to buy in to it and model that these changes can be effective. Ideas that teacher leaders can demonstrate are good for kids (and good for teachers) are the ones that spread like wild fire. 

One aspect that we discussed was the idea of thinking critically about our own instruction and that of our colleagues. My biggest concern about doing this as a department is always that people will feel they are begin "evaluated" if the proper level of trust isn't established beforehand. Teaching is so deeply personal and it takes trust in your peers to allow them into your room to reflect on both your teaching and their own. But what if you do? The trust, collaborative spirit, and sharing of ideas that could happen is almost unfathomable and contrary to the isolationism of teaching in the past. It's using the 21st century skills we ask our students to have every day.

One of the things I'm taking away from the weekend that I felt most passionate about was a really powerful tool to help start these conversations among mathematics coaches, administrators, and peers. 

Achieve the Core Instructional Coaching Tool

This tool provides feedback on what I would consider the 3 most important aspects of any lesson: alignment to standards, appropriate selection of instructional strategies, and student opportunities to engage with the mathematics. Digestible language, jumping off points for conversations, and no wording that rings of "judgement" of the teacher....love. We discussed using it as a goal setting tool in our PLC's and as a tool to start to generate trust among a department. It's something I'm carrying with me as I continue to pursue teacher leadership in mathematics.

WHY are we here? 

I consider myself someone who is pretty well versed in the Common Core, as my master's

centered around conceptual teaching and I completed it right before the Common Core was implemented. We used the standards as our textbook. I've got to be on the savvy-iest end of things, right?
 
WRONG! But let's be honest, I didn't know what I didn't know. 

http://www.360results.com.au/wp-content/
uploads/2013/06/tumblr_mh6eljSrzR1rx06nvo1_500.jpg

The picture here rings so true to me....we examined the standards as a whole, not just our own grade level. And more importantly, we examined them from a perspective of 2 of the 3 major shifts- Focus and Coherence.  So often as teachers we are focused on the "checklist" of topics we need to teach, the day by day "calendar" we've created, and not looking at what's happening outside our own course. For this weekend, I hit "pause" on my high school teacher mentality and tried to examine the standards from a mathematics educator perspective. 

Here's my biggest takeaway from this weekend:

We need to learn to trust the standards. 

The key word I've started to imagine with the regards to the standards is this: design.  These were written, revised, and critiqued by people who have spent their careers immersed in mathematics education. These were designed intentionally to create a cohesive understanding of mathematics that balances conceptual understanding, procedural fluency, and applications.  They are saying this: "Trust me. There is a method to my madness." These were my "ah-ha" moments: 

•  Situation: A 3rd grade teacher who is introducing multiplication teaches how to "carry" because it makes it so much easier for students to multiply big numbers!
• Explanation: This teacher is trying to do right by their students. They are trying to give them a strategy that will always work. However, in doing so, they're neglecting the coherence that is built in the standards to really develop students' conceptual understanding of multiplication so they can apply it later. Fluency with the standard algorithm for multiplication isn't expected for 2 more years (5.NBT) and if we introduce a "trick" like this too early it can actually be a detriment to the understanding our students are building. 
• Situation: A teacher is so used to teaching surface area in 7th grade that they continue to teach it. It's in the textbook, right?
• Explanation: The core of the major work in elementary school is focused on developing students' understanding of number and operation. Geometry can be a tool to do that, but it is regularly listed as a supporting standard. This doesn't mean it should be ignored; it just means it can be used as a lens through which you can refer back to the major topics. It isn't until 8th grade that a geometric topic is considered part of the "major work" and with good reason- the lower grades are trying to go deeper into content. As we advance in the grades, we can have a wider focus. Surface area formulas are not specifically memorized in any grade because it lacks the conceptual basis that the Common Core demands. Just because it's in our textbook doesn't mean it needs to be our focus.
• Situation: High school teachers have always heard that there is supposed to be "less content" in the Common Core, but our courses seem to be even more jam packed.
•  Explanation: The high school standards were not written to decrease the amount students learn each year. This was never their intention. By focusing on fluency in the younger grades, high school is able to become a time where concepts are connected. Mathematics is a story and high school is the time when we weave together all of the plot points.

These standards were not written arbitrarily. They have a plan and every teacher has a role to fill to make sure students are getting the most from the instructional shifts. We need to see ourselves as part a whole; we are a K-12 team that is created to support the success of each individual student. Synergy, synergy, synergy.

This is by no means comprehensive of all the things I'm leaving this weekend having learned, but it's certainly where my mind is focused right now. I feel so inspired to keep learning and leading and I'm grateful I was able to be explore so much this weekend!

Marbleslides 4 Lyfe

Today we played with Desmos Marbleslide for the first time and I've fallen in love. Truly, madly, deeply. I know the MTBoS has been singing its praises for a while and I'm so glad we took the leap today. Here's why I'm lovin' it right now:

1. It encourages experimentation & a growth mindset

When I asked my kids what their favorite part of the activity was in our feedback session, I was relatively sure it would be answers like this one (arguably the most honest of the day): 

Q: What did you like the most about this activity?

A: Watching the marbles

I was shocked that so many of my kids keyed in on the ability to learn from your mistakes and get visual feedback to make improvements. They explained that they loved being able to try again....it just made my heart sing. If a kid says their favorite part is being able to put forth effort to improve themselves, I am one happy teacher.  

2. It made the students think....hard

Q: What did you like the least about this activity?

A: "I had to do a lot of thinking and that's not something I'm fond of at 9:30 in the morning."

When I was reading my students' negative feedback I felt the familiar feeling of being asked your biggest weakness in an interview. "Oh, I just work so hard that sometimes I struggle to balance my life outside of work"...."I care too much".....you know the answers. Everything that my kids had listed as a challenge was something I was extremely proud to have asked them to do. Here's just a few of the things they liked least:

• "I had to do a lot of thinking and that's not something I'm fond of at 9:30 in the morning."
• "The open ended questions"
• "How frustrating it is... but i actually do like it.. so I really don't have one. I guess the worst part is when we had to get off :("
• "I got really frustrated, but I'm just really competitive"

Also surprising, the number of students who said "It made me think" as a positive was triple the number of students who said it as a negative. 

3. It required critical analysis of mathematicals- a CCSS gold mine

This activity is all over the 8 Standards for Mathematical Practice. I watched my students makes sense of a situation (both physically and mathematically) and persevere through LOTS of frustration. I watched them critique the reasoning of people around them as they competitively worked to solve a puzzle fastest. They looked for a made use of structure, recognizing the way that transformations have affected functions they've previously mastered and applying it to the functions we were currently practicing (periodic ones!). They took all these practices and started to express regularity in repeated reasoning, weaving together their knowledge from class with the patterns and peculiarities they noticed. Here are few of my favorite kid comments: 

• "[I liked] Being able to see how the graphs changed as we went"
• " I liked that it challenged me. It really made me think about the separate parts of the equation and what each of them do."

If you haven't already tried it, please do! Desmos has pre-made ones for lines, periodics, parabolas, exponentials, and rationals. It's a Pre-Calculus or Algebra II dream! 

Volumes Performance Task: A Love Story

So I just need to brag about my kiddos for a second. As I've mentioned before, I teach a Calculus class that is totally new to our school this year. The idea is to give kids 90 minutes per day for an entire year to master AB Calculus. We spend lots more time on practice and on exploring concepts than a traditional class would because we have double the time. Test (and frankly math) anxiety is still a very real thing for some. We just took our volumes test yesterday and I wanted to do something fun on our half day before spring break. I loved the idea of a performance task for these kids so they could jump in and show me what they know in another way, while engaging with each other and using some creativity. 

I gave each group a choice between the clear vases or the colored vases pictured here.

The clear were worth up to 10 points extra credit and the colored were worth up to 14 since they were more intricate. For every 10 mL that the group was off from the actual volume of the vase, I would take away one of those extra credit points. I gave them 35 minutes to plan and calculate, requiring that each group sketch a graph and show their calculus work. 

Then we tested. And it worked. 

I'll be honest...I expected this to go way differently. I did this activity with a bunch of math teachers (or more accurately watched as the veterans took the lead) when I was just getting back into my calculus groove and planning for my course and we overflowed the crap out of that vase. By my standards, I don't think we would've gotten any points. But my kids were amazing!!! They used all different approaches- some did Riemann sums to find the volume, some used regression to figure out a function that matched closely and then integrate that, others broke it into piece-wise functions and integrated each. They used calipers to measure exact diameters and used a light table to get an exact trace on a graph.  I left it totally open ended and they all took a different approach- but every group nailed it! Each group was within 50 mL of the actual volume- and were so excited to see who would be the closest. I never told them I knew the actual answers, we just tested by pouring the water into the vase.  We talked about over and underestimates. We talked about washers vs. disks and trying to account for the width of the glass. 

Such a rich 40 minutes to send us off for a week and a half. Happy spring break, everyone! 

What's a Parabola, Anyway?

One of my favorite conversations with my students is always when we discuss the geometric definitions of conics. It stems from my middle school teaching years, where I would put a bucket of candy in the middle of room and say "first one to get to it wins!" Inevitably, there was always a riot thrown by students who were sitting in the back and felt they were being treated unfairly.

My challenge to them was always this: "Okay...make it fair."

I would watch them as a class try to negotiate how they should stand. Almost always they would try to stand in a straight line in front of it, until the people at the end complained that they were too far away. Someone would suggest making an oval and eventually they'd maneuver themselves into an almost perfect circle, being sure to chastise anyone who was trying to inch closer than the perfect radius they'd formed. So what's a circle? The set of all points that are an equal distance from a center....or a really valuable bucket of candy. 

This idea of identifying the locus of points defined by a certain characteristic carries over in such a beautiful way into conics. I tried something new this year with parabolas (that admittedly I thought of in front of my first period today mid-class) and it seemed to work well and made finding focus and directrix afterwards much easier. 

I started by putting down painter's tape in the hallway to form a line and a point (and build some buy in with the kids because who doesn't love a field trip to the hallway?)

From there I picked a student and asked them to put a cup down at a point that was exactly equidistant from the line and the point. Inevitably, they choose the vertex (which we don't name that....we call it "Jimmy's Point" if Jimmy put the cup down there). Perfect start. 






Next, I challenge another student to do the same, without stacking the cup on top of the other. First try is always to put it right next to it. You better believe this stirs an argument in the group. Someone always runs for the meter stick since we can't just count tiles now. We discuss which it's actually closer to and the class gives suggestions to that student of where to re-position it. I continue giving each student a cup and letting the class hash it out until we've started to notice a pattern. 

It's a parabola! 

We discussed the idea of the focus and directrix at this point and talk about why Jimmy's cup was so important, finally assigning its proper mathematical name. It gave us a great, hands-on experience to tie our knowledge to the rest of class and I saw a big difference in how comfortable my kids felt finding focus and directrix, even from standard form.  

Thinking of turning this into a small group exploration next year where kids can then use the cups to actually do some measuring and explore some traits like focal width or width of the parabola! If anyone has done anything like that before, please feel free to share! 

Playing Around with Volumes of Revolution

It's finally here...our final topic in AP Calculus!! With the time before spring break winding down, I am wondering where this whole year went and am amazed at how hard my students have pushed themselves. Calculus isn't easy and the course I'm teaching this year is an AP/Honors combo class to help give access to Calculus to students who might not otherwise have felt prepared to take it.  Just wanted to share a few things we're doing this unit in class to practice and cement understanding for the kiddos:

Get the Picture! 
Visualizing 3D volumes is tough. I started the conversation with my kids by looking at some party decorations I'd bought at the dollar store. We looked at the folded shape and the kids predicted what shape decorations I had purchased. This was super intuitive for them and made them start to connect the idea of volumes caused by rotation. 


We also used these to discuss the cross sections that would be formed. Specifically, we talked about a pineapple decoration.  When we slice a pineapple, we know we get circular slices. Duh, they come in a can at the grocery store! That gave us a great frame of reference for discussing why all the cross sections are circular. I'm looking forward to using pineapple slices to talk about washers too- where you can find the whole area and then subtract the area of the core. And anytime I can talk about food, it's a good pedagogical move in my book.


My wheels are turning for a project after the AP exam:

• Use Desmos to design a 2D shape and determine an axis of revolution
• Calculate solid of revolution's volume
• Actually build 3D solid honeycomb decoration (This video is making me think...)

More to come on this later! 

Washer or Disk?
As I mentioned before, my calculus class is an honors/AP full year combo. I have double the time to teach the standard AB Calculus curriculum, which gives me lots of time for practice activities and conceptual explorations to make sure students are understanding. I am always surprised by the concepts with which my kiddos struggle....the topics in which I anticipate difficulty are sometimes easy for them and it's often the background skills that shouldn't be threatening are hard. I am anticipating some amount of practice needed on distinguishing between when to use washer vs. disk method. I think they'll really need to think through and visualize the areas of a few to see the differences clearly. I designed this activity so the kids could sketch the regions for each graph, sort them by the methods used, and then actually calculate the volumes after I'm sure they understand why they would use each method.

Washer vs. Disks Sort

Volumes Performance Task 

I've mentioned this task before and I have to tell you that I LOVE it! Our last day before spring break is a half day and what better way to fill it than throwing in some competition and a real-life performance task? 

The basic idea is that students are given a vase and need to calculate the volume of it using calculus. I bought a variety of vases from the dollar store that students will need to analyze. 

What's the Volume Performance Task

The link above is the original performance task that I first tried at NCTM Regionals this year. I will be modifying it slightly and plan to write about my modifications and how it went in my room after! 

I'm excited to get the kids thinkings. They can do this with vertical or horizontal slicing.  they can think in terms of Riemann Sums. They can treat it as a disk or as a washer (if they're analyzing the thickness of the glass). And, at the end of the day, we get to actually test their results and see how accurate they were. It'll a fun way to send them off to spring break and a fun last activity before I hand them their enormous AP review binders. 

Should be a fun last week of content!