# Absolute Minimums and Maximums

Another function relationship project, this time from BC Calculus. Prior to the break we were working with functions defined as the integral of another, and how you could figure out absolute minimums and maximums of a given function. There was a lot of interest in this topic from the AP test last year. A number of items designed to show fluency in function relationships didn’t go well for students at large. The issues involved being very particular with your evidence and how you use notation. At TMC 18 while you were playing games at game night, me and Dave Cesa were sitting in the corner talking about this. We know how to party.

Now, we went through these ideas at the very end of the semester, so to simplify things, we made an assumption that our function would have an initial condition of f(0) = 0 and that all accumulations would happen left to right. First thing for the new semester is revisiting that idea and being a little more flexible.

Here were the student instructions:

Students used Desmos to graph their function and compute integrals at any milestone point, local minimums or maximums of F(x). Students calculated the area of their discrete pieces and kept running totals along the x-axis. In tending to precision, they collected their data into a table of x and F(x) and used a series of integrals to show their computations.

While I’ve covered this topic in years past, I have not been as particular as I need to be when it comes to notation. A real struggle in FRQ for my students is showing their knowledge of notation. Often they will cut corners by not including it, or it will be written incorrectly (integrals without dx, for example).

Like their AB counterparts, students didn’t shy away from tricky situations, including points that would register as “fakes” that they could ignore when rectifying their totals. Desmos allowed us to construct polynomials easily and get a feel for using integral notation with named functions. In parallel, when computing integrals with Desmos I will have students replicate the process on their TI-89 to verify they are proficient with both.

AB Calc will repeat the project in the coming weeks and all of this is a good sign for better efforts with precision and argument structure.

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# Curve Relationships Project

A huge problem in AB Calc has been an understanding of a function, its first derivative, and second derivative in a lot of contexts. Students have be able to infer the behavior of a function from the graph of f’, a table of f’, or just the equation of f’. Being able to construct all three elements when given one is a path to fluency.

Prior to break, students in AB Calculus had to create a polynomial that represented a first derivative. Based on that graph, they had to identify minimums, maximums, and points of inflection for the original function f. In addition, they highlighted the differences in concavity and when f could be expected to be increasing or decreasing. They translated their findings into tables of f’ and f’’ that validated their findings, gave justifications for their findings, and sketched f based on the derivative they created.

We were working on this fluency in class through a variety of prompts. Sometimes we started with a table, other times a picture, and others the equation. Here students created an equation and built out the whole process on their own. The results are a lot more polished than the version I tried last year.

Here’s the full write up students were given:

Most impressive to me is that students didn’t shy away from tricky to analyze functions. A lot of students created derivatives that would generate “fake” maximums, minimums, or points of inflection (where f’ or f’’ has a value of 0 but doesn’t complete the required sign change) and it can be tricky to do sketches based off that. But every student who attempted one was on the right track with their thinking. A few hit a common curve sketching snag of drawing an original function with the right features, but everything was upside down.

Collectively though, this group is doing a great job with the ideas.

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# Ten for Ten: Views

Every so often I stop and realize that I have been at this for a decade. I have never been actively working on such a particular idea for so long. I think it’s safe to say the crisis of career I faced a long time ago has been settled. This is what I’m supposed to do. In recognition of this “holy crap 10 years” and the fact that I like making charts, I present a series of charts about things I’ve been doing for the past decade.

Though I have been writing continuously since 2011, I did a platform migration from Squarespace v5 to Squarespace v6 in late 2012 or so. You wouldn’t have noticed, because barely any one read this back then (people at TMC 13 did, which made it hard not to scream OMG YOU READ WHAT I WRITE the whole time). I did another major migration (that you wouldn’t have noticed aside from a switch from red to blue) to make the Varsity Math store possible. Either way, my currently available view counts only go back to January 2014.

For today’s chart, I thought it’d be interesting to compare tweet impressions to pageviews. Twitter analytics were only available starting in Sept 2014, and their scale is reduced by a factor of 10. Otherwise the web page view data is hard to see on the same chart. No y-axis values because this is just about trends. I don’t know that a web page view is the same thing as a twitter impression, but if you compare the raw numbers, 4 years of web views is 8.5% of my twitter impressions in the same time period.

I tweet a lot during TMC, and I tweet A LOT on the last day of TMC. Stats aren’t available, but on a lark at the end of TMC 14 I did some dumb predictions about where TMC 15 would be (having no clue). Enough people liked it I did it again. And now it’s a thing. For fun on the last day of TMC 18 I enabled every twitter notification on my phone just to watch the insanity (I do the tweet storm from a computer). I have amassed a bit of an audience and that’s cool. The goal of my twitter feed is to be very school focused in a very un-serious way. My feed makes a little more sense when you meet me in person. I tweet a lot of gold when I’m grading. Kids are hilarious.

As others have noticed, blogs seem to be less important to teaching, not that mine ever got any huge runway. In fact, the biggest traction I get is from stuff that’s wound up on Pinterest. But yes, I have an audience. And yes, I am very thankful for you. If you’ve ever left a comment or sent me an email, I appreciate them and make a point to try to reply to them all. Other than some spikes here and there, getting traction with a teacher blog is tough. Things just don’t stay in the spotlight very long at all, and it seems like a lot of people have run out of time for reading them. Tweets are easier to parse, but you still have to tweet a lot for people to pay attention (in general a tweet will be “seen” by 10% of your followers if you’re lucky, about 5% will click links). Google Reader dying in 2013 hasn’t helped.

I post less than I used to, but I have less to figure out than I used to. In 2013, 14, and 15 I wrote over 70 entries each year. Now it’s around 50. I was deep in the weeds trying to figure stuff out some years ago, primarily working on improving the quality of student products and exploring technology integration. Those two problems were HUGE and have been kind of solved. I still have things to work on, but I haven’t needed to rely on writing as much to do them.

This site still has a purpose, and it will continue. I still treat myself as the main audience and there are still things I want to see on here. It has been a boon for documenting ideas throughout the school year to discuss at end of year appraisals. And 2018 was a good year for the site. Audience numbers were up and its role as a vehicle for Varsity Math was huge. I have an infrastructure in place to make regular merchandise offerings a reality. I eagerly await the TMC 19 shenanigans.

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# Motivation

In the last year, I had two moments that reinforced my motivations for teaching.

On the plane to TMC18, I had downloaded A Trip to Unicorn Island. It’s a YouTube Premium feature film chronicling Lilly Singh (~14 million subscribers as of today) and how she got from making vlogs in her room to taking a motivational dance show across the world. There was no particular reason I was watching it other than I had heard about it and thought it’d be interesting. The premise of the movie is Lilly decides, arbitrarily and with no experience with anything at this scale, to embark on a world tour to share her happy place with people, something she calls Unicorn Island. There’s skits, dancing, and a serious break in the middle where she talks with the audience about finding a happy place that keeps you motivated. Naturally, there are some low points. With the planning still in infancy, the scale of the project hits her and we have this moment:

“I do a lot of this. I do a lot of sitting by myself in this room, looking at my screen. I feel like, my life is slowly becoming something no one can relate to. And then in moments like that why am I doing this? This sucks.”

And it was one of the most relatable scenes I’ve watched in a movie. Lilly, and highly motivated creators like her, project complete confidence. You know they believe in what they’re doing. At the same time, their struggles can be very individual. Who knows what’s it like to plan a world tour? Who knows what it’s like to plan a world tour while also maintaining a day job uploading well polished YouTube content that several million people have come to expect?

I’ve had these moments as an educator. In particular, earlier in the summer when my year of Calculus efforts got me not a whole lot. On the surface, yeah, I’m a Calculus teacher, lesson planning is challenging. Lesson planning is challenging for everyone. But Calculus, like math, or really education in general, is a reflection on equity problems. I fight these equity problems every day. There are only so many Calculus teachers, and there are only so many Calculus teachers trying to get the work done with one hand while trying to overcome built in inequities with the other. In general, my students have a glass ceiling constructed by economic factors they have no control over, decisions that were made for them, and decisions that were made 10 years ago. And despite all of that, 95% of them are highly motivated to give Calculus a go, and they have a lot of fun doing it. I believe 100% in their ability to register on the scale, there is no reason they can’t. But every year the College Board tells me “good try, maybe next year.”

And I have moments where I’m working on yet another Calculus approach, staring at my screen, thinking why am I doing this? This sucks.

But I keep going, because there are bigger pieces moving. The next group will be better because of the struggles of those that came before. Operating big complicated projects is fun. Varsity Math is a very complicated project. There’s a lot of money, time, and merchandise involved. But I am so confident the results are going to be great that I push through. It is that confidence I admire in Lilly and other creators like her. I know this is going to be great, but you have to trust me. I don’t care how crazy you think I am.

Second, was Won’t You Be My Neighbor? the history of Mr. Roger’s Neighborhood. The whole film spoke to me and reinforced the need to make my classroom a fun and positive atmosphere. Fred Rogers was just the nicest man. He had a respect for children and their opinions that maybe you can only appreciate if you are a teacher. Too many people dismiss children as ill-equipped to speak their minds. I say those people have never taken the time to listen to what the kids have to say.

My specific teaching experience is full of unique situations that no one can understand. Every teacher is the same way. Only you know your kids, know your goals, and know the effort required to get the job done in your room every day. And a LOT of people are going to say you’re doing fine or “that’s neat” with no appreciation for the hundreds of hours you spend grinding to get there. And that’s fine because it’s not about them, it’s about the satisfaction with the work you’re doing. Get things done the way you want to get them done.

Finding the confidence to try something crazy, know it’s going to work, and know it’s going to be great was a huge turning point for me. Letting kids be kids and express themselves was another. When the whole room knows we can do this and that my voice matters, it’s a beautiful thing.

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# What Matters

I have difficult students from time to time. In College Algebra, it’s generally children who are very inefficient with their work time. One kid like to engage his table in conversation. And they’re usually pretty interesting conversations, though it comes at the expense of productivity. This kid is generally pretty polite and will get business taken care of eventually, asking lots of good questions. Sometimes though, when nudged, they throw up a wall. Saying things like “just let me fail, this is my decision” etc. I stay pretty calm through this, citing reasons why letting this kid do nothing isn’t in their best interest. Other kids jump in to help me, though I don’t want it escalating. Kids don’t respond well if ganged up on in full view of a class. Most of the room was lost in their work, for context. This “incident” if you can call it that was isolated to one table.

Eventually I left the kid alone and finished out the class period. Later I took them aside and we had a chit chat. Similar shut down incidents have been rare since, though it still happens. During the initial discussion though, one of the kids at the table asks me “how can you be so patient?”

Four months ago, 7:15am, my birthday. I’m setting up for the day. Phone rings. It was E, a former student, asking “have you heard?”

Spring 2012. I was starting year two of what was to be a 3 year side gig coaching 8th grade soccer at one of our feeder middle schools. I finished up work at the high school and drove down the street in the afternoons. E was a very hard headed young man with a temper. He had a pretty constant string of discipline issues throughout 7th and 8th grade. But he was good at sports and they helped him from being a complete lost cause. The previous year when he was in 7th grade we had a number of incidents with him at games. I was hoping we wouldn’t have a repeat of these incidents.

M was new to the team. He hadn’t been allowed to play in 7th grade, despite the kids saying he was the best in the school. Finally, the athletic director let him tryout in 8th grade. During tryouts we did some shooting drills. M kicked the ball with more certainty than I’d ever seen. He was going to be special.

I was not a fan of this team. There were constant discipline issues with them. Practices had frequent disruptions. Eventually, a number of them dropped out of school. Some transferred. One went on to rob a car dealership a week before graduation. Every time I’d hear about the new nonsense they’d gotten in to, I wasn’t surprised.

But they’d win games. This particular middle school never won games, in anything. Suddenly we had a shot at being undefeated. Before the game against the primary rival I told M, we get the lead you’re dropping to defense. M nodded aggressively and said absolutely. He knew that’s what we were going to need. Sure enough, we were winning late. M drops back and frustrates the other team for 10 minutes. First win in anything over the rival school in a long time. Eventually, we completed the undefeated season and won middle school district. Some years later the middle school would hang a banner in the gym for this team.

M comes over to the high school and plays for me in the 9th grade. His frustrating teammates and even more frustrating ones from other middle schools follow. I didn’t enjoy a lot about this season. Except for M. He never caused problems. We didn’t have as much success in 9th grade, but he’d put the team on his back when required, you never even had to ask.

M and E are now seniors. Both have made varsity. E just now, M since 10th grade. The frustrating teammates were all gone, washed out because of discipline issues. It was a nicer time. Results wise though, the team wasn’t in great shape. February, M starts missing practice, which was unlike him. We think he has the flu. Later I’d learn his teammates thought he was making excuses because they aren’t doing well.

Late February and M hasn’t been in school for a while. It was leukemia. Treatable, but still. M responds well to the treatment. He is unable to walk at graduation, though he does graduate. He is mentioned in one of the speeches.

Months later, M is strong enough to come to soccer games, and has started working. He says he feels better and can even run a little bit. Late January 2018 I see M again at a soccer game. Again, seems ok, though he doesn’t say much. M wasn’t much of a talker. E was there too.

“No” I say to E. “What happened?”

It’d been 6 months since I’d seen M. Now he was dead. Whether the leukemia was back, or there was a bad response to medicine, or if he’d even been taking his medicine no one knew. After two and a half years, the leukemia won.

I went to the memorial service, but I couldn’t look at him. That’s not a memory I wanted. I’d rather glance at his team pictures on the wall.

Kids lives are complicated. Sometimes it seems the lives of kids I work with are unfairly complicated. The 50 or so minutes we spend with them are just a small window into what they might be dealing with. The kid who tells me “let me fail” got that way for a reason, not because of something I did personally. Those tough seasons with E and M taught me what matters. Winning some argument with a kid choosing to be stubborn does not. There are better ways.

Goodbye, M. ❤️

MAMS 92497 - 82318

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# Ten for Ten: Merch

Every so often I stop and realize that I have been at this for a decade. I have never been actively working on such a particular idea for so long. I think it’s safe to say the crisis of career I faced a long time ago has been settled. This is what I’m supposed to do. In recognition of this “holy crap 10 years” and the fact that I like making charts, I present a series of charts about things I’ve been doing for the past decade.

This time, the saga of how Varsity Math started as a tweet and turned into a massive merchandise brand. A very limited supply was offered to the public this year, but it is but a fraction of what students receive. Here’s units shipped of Varsity Math merchandise for the last 5 years:

Next to each year is a rough approximation of the money making all this possible. Each stack of bills represents \$500 in real money. Students pay for their items. Currently, a base fee of \$20 gets a student a t-shirt, sticker, patch, and sunglasses. No student is ever denied the base package due to an inability to pay. Other items are offered a la carte. If you’re interested in sponsoring a student, options are available.

We started small. Kids like t-shirts, so we got t-shirts. Then Andrew Stadel got me hooked on Sticker Mule and I started ordering stickers. As the whole premise is making a joke about letter jackets, I ordered patches in 2015. These were adhesive backed and the thought was kids could stick/sew them to their actual letter jacket. The concept did not take off. In 2016, I ordered patches with velcro backs and the rest was history. You want to spot a Varsity Math kid at my school? Check the lanyard. This simple addition increased our visibility 10x. Suddenly, kids were hooked on the patches. HOW DO I GET A PATCH? random underclassmen would ask me.

But oh….we weren’t done. Summer 2016 we added Summer Camp and a merchandise line to go with it. These shirts and stickers are among the most exclusive because you had to be there. Before you know it, we’re in the sunglasses business, the sweatshirt business, the sock business, and the shoe business.

Eleven kids signed up to make custom Vans with a Varsity Math logo. These shoes weren’t cheap and took 4 weeks to arrive. For this year’s round, I started collecting merchandise money in August 2018, and by Dec 1 over 800 items had been delivered. Simply put, the kids are merch crazy.

I design all the merchandise (force teaching myself Illustrator has worked wonders) and developed a whole slate of contacts who can make me just about anything. It has been a fascinating study of design, manufacturing, and logistics. By no means do I suggest it if you’re looking for a casual hobby, but if you’ve ever seen the tweets and were curious about what kind of scales I operate at, well here you go.

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# Ten for Ten: Stress

Every so often I stop and realize that I have been at this for a decade. I have never been actively working on such a particular idea for so long. I think it’s safe to say the crisis of career I faced a long time ago has been settled. This is what I’m supposed to do. In recognition of this “holy crap 10 years” and the fact that I like making charts, I made a little stress diagram of those 10 years.

The taller the bar, the more work I felt like I was doing. Underneath is the variety and quantity of preps I had. Soccer balls and volleyballs represent coaching years. Trophies are major awards. Years where I took on a new prep or big responsibility spiked the work load, as I figured out how to do something for the first time. I put a LOT of effort into things when I’m doing them for the first time. REALLY quickly I figured out I like being thoughtful about my assignments and not just taking them from a binder, and that took a lot of time. Now I reap the rewards of that investment constantly. In the case of a new prep, there’s curriculum to map, assignments to make, and unknowns to solve. With a new responsibility, the time management needs a rebalance.

Along the way my confidence grew. You start to see that kids are buying what you’re selling, and that you can sell it really well. You get comfortable in the space, adapting good ideas to any old prep. College Algebra (Algebra 3 in local parlance) is this self-paced little wonderland because of all the grind that came before.

The first five years I felt I had something to prove. I was an outsider to education, a random guy with an alternative certification who did not know what he was doing. My first group of kids were very kind and said I did a good job, but I really did not know what I was doing. I wanted to show my school that I belonged, and that I could be trusted. Earning trust in the workplace is the hardest thing to do, and is so valuable once you have it.

Now though? Man, this is just the best. Yes I have three preps. Yes I coach a sport. Yes I’m co-running our National Honor Society. But it’s just so smooth. I’m not really sure what the shift was, but it’s an enjoyable place.

I want to deep dive some more into some other trends, so prepare your self for 9 more emoji-laden charts.

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# Assessment Review

Some of my biggest experiments have been with assessment. It started with an SBG adoption in Pre-Cal and Algebra 2 seven years ago and it really changed the way I view assessing students. Scores out of 100 are silly and arbitrary, so I don’t bother. These days Calculus takes assessments that are segmented by a particular topic, usually integrating a variety of skills into a short set of questions. Normally when you have an assessment, tradition says you should review.

Long ago before SBG I would write up reviews that students would complete the day before an assessment, you’ve probably done the same. In practice creating a review is almost as much work as writing the assessment. I quit doing stand alone reviews years ago because I think they send a signal that classwork isn’t as important, this review is all you should care about. I want students to be diligent about completing classwork and seeing its purpose, so I’ve designed my “review” around that idea.

Both flavors of Calculus are taking an assessment today, here was their “review:”

I put the list on the screen and kids can take a picture. That’s it. We spend zero class time on this because everything on the list is represented in some piece of classwork we did in the days before. Students who were diligent about organizing their classwork should be able to find anything. The only thing that takes time is if a student has a question about what I mean by a topic. For example, they may want to clarify what I mean by “factor a polynomial into a sketch-able equation.”

I’ve found the practice effective. Previously students have said I’m too vague about what might be on an assessment, so they can’t focus their time. In fact, I’ve been too vague because usually the assessment isn’t written until later. I think these simple lists give them the focus they want without spending a ton of time on a purpose-built review. More importantly, it helps me focus when writing the assessment, to make sure I stick with whatever ideas I had when writing this list. I am incredibly bad about changing my mind constantly about approaching things. This has brought some much needed focus to my work as well.

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# Casual Friday

Every so often the stars align and we have a real casual experience all day long in my room. Yesterday was the end of a marking period and every kid was in “finish this stuff” mode.

In Calculus we were wrapping up a function analysis project:

BC Calculus was up to something similar, and we talked about a benchmark they took recently.

Over in College Algebra kids had two tasks to accomplish, wrap up a project on polynomials:

…and take a test. After submitting their project, they grabbed their weekly (ish) assessment and completed it on their own time. College Algebra is a very casual environment. Whole group instruction isn’t really a thing in there. In any given week kids have a to do list (new lesson, assignment, assessment) and have to complete it by the end of the week. There are 23 kids in this class and they work at 23 different paces. Often that means a handful are done early. Most of the time they’ll get a bonus activity (via Desmos or something else), or sometimes I’ll bust out the puzzles:

So, in yesterday’s 50 minute period, lots of kids were wrapping up their assessment, a few handed in their projects and started/finished the assessment, and others were finished, all with some Weezer in the background. 23 kids, all hanging out, taking care of whatever they needed to get done.

And it played out that way all day long. Hanging out, doing some math, listening to music. It is my favorite learning environment. To spend the whole day in it is simply sublime.

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# Function Analysis with Tables

I’m on a mission to fix all kinds of things about Calculus, especially Calculus AB. Next on deck is Curve Sketching. Previously, while working on integration of Intermediate Value Theorem and Mean Value Theorem, I integrated some curve sketching ideas. Students calculated average rates of change and created rudimentary graphs of a function’s first derivative.

A few weeks later, students now have more familiarity with the first and second derivative and we can talk about how those tables help us analyze a function.

Start with an arbitrary function and interval, and create a table of f’(x) and f’’(x) in Desmos.

At the moment we aren’t concerned with the graphs, those this will be useful later on. Have students recreate the table, but we’re going to declutter the results. Rather than worry about all the values generated, let’s look at whether the first and second derivative were positive or negative at the point.

Having discussed the Intermediate Value Theorem, we have a discussion about where values of zero should appear on our table. For the first derivative, we reestablish a connection we made before, that if the slope of a function is positive, it must be increasing. A zero should mark the transition between increasing/decreasing or decreasing/increasing and these points are important enough to have names.

Next we have a discussion about the second derivative, which is a newcomer to the party. Some days before this activity, we plotted tangent lines, computed second derivatives, and looked at whether the tangent line was an overestimate or underestimate. That opened up the idea of concavity, that the concavity of a function plays a role in how accurate a tangent line will be.

Now it’s time to define concavity a little better. We look for points where the second derivative must be zero and what that could mean. At this point I’m talking with the table and graphs in view, so students can see that something is happening to f(x) at the point where there should be a zero on f’’(x).

Going back to their horizontal table, we now annotate the table with our findings. Based solely on sign value, we can quickly determine where a function is increasing, decreasing, concave up, concave down, and the role of the various critical numbers.

The purpose of all this is to improve a HUGE weakness I’ve seen over the years. For whatever reason, while I could get students sketching f, f’, and f’’ like geniuses, there was a disconnect between how they were making their sketches and what they represented. If a non-sketching question said something about the first derivative being positive, I’d get nothing but blank stares. Very few of them were able to determine that corresponded to increasing behavior.

By building this competency with tables AND graphs, I’m hoping things improve quite a bit. By sticking with equations of tangent lines and tables as recurring themes, I’m hoping free response style questions are more comfortable. It’s way too early to tell, but I’ve really liked how this is going.

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