I have three periods of Calc AB, one of them first thing in the morning. I had the same situation last year and that group was incredibly enthusiastic and ready to start the day. This year...not so much. Lots of tired faces and it takes a while to get them going. Plus, it's the smallest group, only 21 (compared to 25 and 28 in the others).

For the first semester (when it was only 20), they sat like this (the back tables seat 6, the front 7):

Conversations were limited, and performance was poor, scoring about 10% less than the other periods on our first benchmark. In general purpose assessments they were about 5-10% lower than the other classes as well. The table at the top left in particular did a lot of staring their paper, uninterested in finding assistance around them or even asking me.

It was clear things weren't working, so when January rolled around (and we gained 1 kid from another period), I forced them to do this:


It's a little cramped but it has made a world of difference. The low performing table is gone, and there is far more conversation happening when it's time for an assignment. When it came time for benchmark two, this class is now more in line with the others, off by a rounding error.

They always had access to each other, but forcing the issue and making them be more friendly has had the intended effect. Never be afraid to change it up when it's obvious something isn't working.

AuthorJonathan Claydon

As we headed into Spring Break, it was time for a project in AB Calc. I have been trying to stress curve relationships throughout the year. Early on we talked generics, how to indicate a minimum, maximum, and point of inflection. Then came integrals, absolute minimums, and absolute maximums. When we moved into position, velocity, and accleration, I hit those contexts again and again. By far, this is the biggest thing AB students need, how are the interactions between a function and its derivatives the same regardless of context.

Particle motion was a huge problem last year. I found myself having to do a lot of reteach prior to the AP exam over concepts I thought we had gotten (change in direction, particle at rest in particular). In combination with my new thoughts on FRQ, I'm trying to review as we go and combine ideas as much as possible.

Student's objective in this project was to create a polynomial, restrict it, and discuss a variety of properties. Specifically, the difference between a change in position and total distance traveled, where and how a particle is said to change direction, and how to determine what's required for a speed increase/decrease.


The projects were all fairly similar, but here's a particularly outstanding example:

Giving students ownership of the process had really been the best move for my projects. Because there were so many opportunities for variety, students were able to have great conversations about they were or were not meeting the requirements. They also learned a lot of handy Desmos along the way (shading, integrals, absolute value). Project days are my classroom at its very best. The whole room buzzes and I just get to float around assisting as necessary.

Students did outstanding on the free response questions related to this topic. Well, not like everyone got a 9/9 outstanding, but every student got the concepts and had something to contribute when it came to finding the solutions. I'm hoping to roll our review as we go strategy into rate functions next.

AuthorJonathan Claydon

We spent about six weeks on sequences and series in Calc BC. It was an interesting struggle for me personally as I last touched on the topic in 2003 taking Calculus II. The subject appears really intimidating, just search "convergence flowchart" and see what I mean. In the end, it was not nearly as complicated as I thought it would be and the kids and I got through it without any insane flowcharts.

By far the most interesting topic is Taylor/Maclaurin/power series. It's a fascinating bit of math that causes all sorts of problems, because the tiniest variation wrecks the usability of your series completely.

As we headed into Spring Break, I had the kids do a simple exploration of Maclaurin series in particular. Primarily because the series for sin/cos are so easily modified. Their objective: modify a sin/cos function in whatever way they want, iterate a Maclaurin series to four terms, name the iterations appropriately, and take a peak at their error when x = 1.


The coolest part is I don't think the kids (or myself really) was prepared for how wild their constructions could be. A few kids picked functions that worked beautifully at x = 1.

A bigger group got ok results, but had errors over 35%

And then 3 of them picked functions that just went off the rails (the joys of exponents)

450% was hardly the worst. Another was off by 22000+% and the best ever was the student whose error was infinite.

This was a nice, simple project, and I think a nice little eye-opener about how math is sometimes just a shot in the dark.

AuthorJonathan Claydon

We've finally covered enough material in AB Calculus to start tackling free response situations. In previous years I haven't done enough of this sort of thing and it's cost us. This year with a fewer days (we're probably ~1 week behind still) I wanted to integrate the language component of AP questions earlier and more often. Last year a big issue was the kinds of things kids wanted to review before the exam were concepts I thought we had nailed already. With fewer days, the less I have to reteach in late April, the better.

A few weeks ago students completed a benchmark in Desmos. We had covered enough material to handle curve sketching and data table FRQ situations. In the days since the benchmark, we have turned to position, velocity, acceleration, and general ideas about rate functions. I ended their previous testing system (short skills based items worth a max of 10 pts) and I'm now using FRQ scenarios as assessment piece.

Last Friday I gave them this:

A small hand out with a couple graphs was included. The idea was to get them used to the language and start getting used to how points are weighted in an FRQ. The class day previous to this, students were given arbitrary v(t) functions and we went through a series of 10 questions to cover the x/v/a relationship.

I gave them 40 minutes (though mentioned in reality they'd have about 30) with no notes, but open discussion. I have to say the results were night and day to previous years. I have had TONS of problems with students just leaving stuff blank, or being stumped by the phrasing. Not so this year. While not every student got 9/9 or anything, the conversations they were having were excellent and it was clear that the majority understand the concepts.

The 40 minutes ended and I collected their papers. Come Monday their papers were returned and they were given a colored pencil. Posted were all to see were some observations I made when flipping through the papers. I have made no marks on the papers. These observations served as feedback to the group at large.

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With a colored pencil, they can use this information to update or amend their work. Then we'll look at the scoring rubric for each question. Finally, I will give them a new question (2009 #1) for them to complete. The plan is to turn this into an FRQ Fridays™ kind of thing where we can minimize how much we talk about these in late April. So far I like the process.

AuthorJonathan Claydon

In December I took a measurement of my Calculus AB group on their knowledge of limits, derivatives, and curve sketching concepts. Students were grouped into quarters and got a score report. Recently, we did this again. Students have to sign up for their AP Exams by the end of next week and one of my stated goals is to make sure they have an informed opinion about what to do.


A public version of the activity can be found here: Calculus Gauntlet 2 Public

I wanted to check a few things. First, could students determine when an integral or derivative was appropriate based on vocabulary clues. Second, could they handle data table, curve sketching, and function behavior free response questions. Third, a raw skills component, looking at their ability to find a series of derivatives and integrals. I also included a couple of multiple choice questions from practice AP exams. Students were given a review with all of this information.

Here's my planning checklist:

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There was a lot of reorganization as I built the activity, and some questions got cut for time. In the end I asked 13 vocab questions (choose integral or derivative for a scenario), a multi-part data table question (1 full FRQ), and released MC questions on integral composition on Day 1 (50 minutes). Day 2 (80 minutes) included multiple curve sketching questions (1 full FRQ), a multi-part questions where an arbitrary curve and areas are given (1 full FRQ), then a series of skills questions about product rule, quotient rule, and a selection of derivative and integral exercises completed on a notecard.


The end result was a 42 (+4 slide homework survey) slide activity. I used Pacing to restrict student access for the various days, and could easily flip it on and off for a few absent students who needed to complete a part they missed. Props to Desmos for including a "randomize choices" option for MC questions. That wasn't available in December.

Having used the structure once before, kids needed no help getting started. With school Google accounts, the student.desmos.com website easily kept track of where they were and got them back in on Day 2 with no issues. To ensure uniformity of results, there was only one version this time. I wanted the data to be as accurate as possible, and didn't want have to fudge on account of versions. Whatever minor details students passed along as they talked about this in between days was a fine trade off. There was zero credible evidence that anyone might have done anything dishonest.


Again, students were ranked and given an overall percentage. This time I improved their data strip to show how they did in each category. I'm hoping to use this as I construct AP Exam review modules, offering students more flexibility in what they choose to review.

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A couple of questions were dropped for technical reasons (written in a way that deviated substantially from how it was taught, causing a high incorrect rate). Students who got the question correct in spite of the error got to keep the point earned.

With 71 complete results: 60.58% average (top 10 students averaged 89.29%), 1st quarter 75%+, 2nd 59.82%+, 3rd 46.43%+, and 4th was anything below 46.43%. Students were asked to compare their results to a real AP scale (where a 5 is ~70%+). Other than dropping a couple questions, students percentages reflected their raw performance, no curve was applied.

I have given benchmarks every February for 4 years, here's the historical comparison:

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A ridiculous leap in performance that I think I can explain in a bit.

Generally passing AP scores come from students above the 80% line. I'm hoping we can expand that a bit. Since 2015, I have recommended the AP Exam for any student above the 45% line (2014 is an exception because I had no idea what I was doing). Per College Board policy, all students are free to take or not take the exam regardless of my recommendation.

Finding Promise

Previously, these February benchmarks were composed of questions from released AP Exams. Students struggled mightily with this task. I think time was a factor. Typically I would give a review about a week out, students would work on it in class and on their own. An answer key would be made available and then they'd take a multi-day assessment that was a shorter collection of released AP Exam questions. Results were poor because I think I had the wrong design goals in mind. AP Exams are a massive undertaking for students new to the material. There's generally a good reason most AP classes (even mine!) spend a month on review. A week just isn't enough time. In the end I think I wound up discouraging more kids that probably could've risen to the occasion given appropriate time to prepare.

This year I decided the most important thing was to look for promise. Yes, I would use real AP language and questions, but it would be incredibly focused. I stuck to the scenarios and skills we had covered in class, some of it full blown AP level, some of it not quite. I figure any student who can succeed here can be coached along through exam preparation.

And in this process I saw some great things from the kids. Those in the top worked very hard to stay at the top. Many many kids in the lower ranks redoubled their efforts to show me that December was a fluke. In making recommendations, students who jumped a quarter (or two or even three! some cases) got the benefit of the doubt. If they could improve that much between December and February, what if they were encouraged to keep working until May?

In the end I recommended 55 take the exam without question, 11 to consider it, and 8 to focus their effort on keeping up with classwork. Next week I'll know their final decisions. With a vast majority gearing up for this thing, I think that will provide the "we're all in this together" momentum that could make a difference.

Cautious optimism, as always.

AuthorJonathan Claydon

When I started teaching Calculus, I knew I wanted some regular form of homework. But I wanted to avoid something like "2-40, evens." Primarily, I think the way textbooks are designed don't allow for the kind of focused assignments I want. In 2015 I started rolling my own homework. The goal was something short and to the point that was on older material. I'd give it out on Monday, check it Friday, and provide a solution kids could check at their leisure.

Since implementing that system, my homework completion rates are huge. After 2 or 3, kids were used to the procedure of when it was due. The publicly posted key reduced the number of questions to almost zero, and the familiarity of the material kept kids from getting stuck and giving up.

Here's a sample assignment:

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I skip weeks every so often depending on circumstances. So far I've given 12 assignments like this in AB Calc. This year, I finally decided to gather some data on homework habits. I had some ideas based on solution traffic (it's a bit.ly so I can track usage stats), but I wanted the real numbers.

I have the exact same system for BC, but their homework completion rate is 100% and it counts for a meaningless part of their grade, about 1% per assignment, 4 assignments per grading period. That crowd appreciates the opportunity for extra practice a little differently.

Survey Questions

With 70 AB Calc respondents, here's what the kids had to say:

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Interesting Insight

Strangely, the majority of kids doing homework on the day it's due are in my 1st period. Though perhaps if they stay up past midnight they counted that as Friday?

The even clustering of solution users is fascinating. I thought it'd be more consistent. I really don't think there's a good or bad takeaway from these percentages. I post the solutions intending for the kids to use them, and most of them use them most of the time.

The four kids who said the assignments are too long are the ones who consistently don't do them in the first place, which is kind of funny.

The two kids who admitted to 5+ (out of 12) copies are incidentally mediocre in overall classwork. My eyes popped a little when I saw how many had copied at least once.


In general, the system seems to be working as intended. A big majority are doing the assignments and checking their work. A small minority are abusing the system. At roughly 3% per assignment (~4 assignments per grading period), a student does not gain a whole lot from a blind copy.

You could argue I could make changes. Increase the weight or increase the frequency but it's likely to only amplify what is currently a small problem. Increase the frequency and it loses meaning (being seen as constant busy work). Increase the weight and it becomes punishment.

Probably the only clever thing would be to introduce a check digit of sorts into my solutions. I could add one intentional wrong/weird thing that'd signal a blind copy. Though I make natural errors with enough frequency that it could be hard to remember what was supposed to be wrong.

AuthorJonathan Claydon

This school year has been a bit weird. I scaled back my duties yet I still have a lot of work to do outside of school. What gives? Well, each of my preps has presented some interesting challenges.

College Algebra

By far the class I underestimated the most. Not from a material point of view, but how the students would consume it. Turns out, whole group instruction doesn't really work here, I could blab all I want and some percentage is still going to need an individual explanation. In January, I started catering to that need. The challenge has been creating material that offers students a script they can follow and leaves me opportunities to have a conceptual discussion with them. I do not want it to devolve into a "do this, ok great, moving on" kind of thing. I want to retain the discussion aspect and that is requiring a lot of energy, as I have the same conversation 4 times in a class period. Pacing is still weird, many finishing assignments quickly, others taking their time (distracting themselves has started to play a part in this).

But on the plus side, the group in general is understanding new concepts, I'm probably saving time by cutting out the note taking, and it keeps the atmosphere really casual. Everyone still gets tons of time to work and enough face time with me than they can stand.

Calculus AB

Though we are slowly turning a corner, there's a lot of work to do here. I'm using yet another assessment system, and employ Desmos for benchmarks. Until we reach some future performance nirvana, I'm going to keep tweaking my approach in this course. Despite missing all that time at the beginning of the year and then another two days for ice, I think we're in a good place. It's that magical time of year when all the concepts finally start to make sense and you can discuss free response questions without major headaches. In a couple weeks I'll see if I was able to grow my exam participation (last year was 50%).

Calculus BC

We've entered uncharted water. We cruised through the AB material with little problem and now we have to tackle the material that's unique for them. It feels like we have eons of time to get ready. Since the class is small and they're all doing great, there isn't same benchmark mechanic necessary here, they're all taking the exam. As much as it pained me, I decided to bite the bullet and jump into sequences and series now before it starts to feel like we're running out of time. It is my weakest content area, so there's been a lot of studying and restudying and restudying to try and make sure I have it all straight. I knew it was going to be a rough period. But slowly, the light at the end approaches.

Common Theme

What's the problem really then? I have to make TONS of stuff. College Algebra is loosely based on my last attempts at Algebra 2 but the "no talking" structure of the class has really made me rethink a lot of things. Pacing still throws me. Calculus BC is all new territory as I relearn the material myself AND come up with ways to teach it. Calc AB is the one comfort zone, I know what I need to do, but there's still a lot of retooling required.

One day I will take a summer and write some kind of definitive practice book I can use throughout the school year, but that day has yet to come.


AuthorJonathan Claydon

A friendly reminder that we all have our days.

Friday was fairly straightforward, Calc AB and College Algebra had assessments, and we were going to fiddle with Taylor Series in Calc BC. Easy, right?

First I'm five minutes late. I also haven't made copies of anything I need for the day. So, open the room, wave the kids in, dash up to the copier. All this as the first late bell is ringing. On the way up to the copier two kids are trying to make a run for it out an exterior door. Got them back inside and shoo'd about 15 others who were trying to sneak in to avoid being late, a common problem we have.

Copy room! No line! Yes! Except the first machine I try throws a bunch of errors. Don't have time, move on to another. I get things copied and cut, and at this point it's almost 8:00, nearly 15 minutes into class. My first class needs to take the assessment that is currently in my hands. Fortunately it's short (Calc AB takes ~30 minute assessments that they check at the end). I manage to check homework and get them started such that they have plenty of time to work.

I didn't make the answer key in advance (each AB class has their own version of the test for which there are 2 sub versions, so 1 AB test = 6 keys), so I work on that while they work. Time's up, we check, no errors on my part.

Second Calc AB group comes in. Things are calmer now that I'm ready. Check homework, pass stuff out, work on their answer key. Time's up, we check, aaaaand I screwed up a couple of them. A small riot breaks out because kids are very adamant that there's no way what I have is the answer. And good for them being confident in their work enough to call me out. This will be important later.

Fortunately there's enough time before they leave to correct the mistake. They calm down and are able to check their stuff accurately. The rest of the day passes with minimal incident. College Algebra and Calc BC proceed with no problems.

Last class of the day! I even made the answer key while College Algebra was working so I didn't have to scramble. I chill out while they work. Time's up, let's check!

"um, what...?"
"I'm going to cry..."

Actually, I don't remember anyone boo'ing, but there was, simply put, a scene. I don't know what funky logic I was using while completing their key, but MAN, there were some problems. Unfortunately there was not enough time to figure out everything that was wrong, so they were only able to accurately gauge a couple of them.

It all worked out in the end, I found all the errors in the various keys, kids got graded accurately and there was a greater calm on Monday when I passed them back (kids do an initial check, I then verify and assign a score).

You never know when these moments of crisis will arise, the important thing is to trust the procedures you have in place. Kids were confident enough in their work to call me out. I was confident enough in them to take a second look, and not so confident that I can't admit fault in front of them. It's fine for kids to know that yes, you bleed.

AuthorJonathan Claydon

Though I don't write about it much, I've been teaching College Algebra this year. The kids aren't earning college credit for it, it's more of an alternative to Pre Cal we're offering for students that need another year of algebra. The Pre Cal we teach isn't like some places where you're doing Algebra 2 all over again, so our version of College Algebra makes a nice option.

Interestingly enough, I have two very small sections, 18 and 16 kids in each. It's been a long time since I've had classes this small. We had a bit of a population boom since I started and some quirks happened to bring class sizes down this year despite a very high overall population. The biggest lesson I learned from those early years with small classes is that they were totally wasted on early teaching me. I always wanted a chance to do a better job with that kind of environment.

The first challenge was their work pace. These kids are great, but they're all over the place. I developed a technique for dealing with that last semester.

The new challenge became holding their attention. For whatever reason, they aren't interesting in listening to me speak in front of the group in a traditional sense. I noticed this back in 2013 when I reframed Algebra II around lots and lots and lots of classwork. The kids got so used to knowing math class would require them to do something, that listening became problematic. They just wanted the classwork and it was crazy. Same thing here, this College Algebra crew eats up their classwork. But lecturing? Forget it.


How did I fix it? Let's look at a multi-part assignment I gave for inverses:

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On the surface, this looks like bad textbook worksheet moves. Steps are labeled. Kids follow the steps. Repeat.

The Mini Lesson

As mentioned, these groups work at varying paces. As the semester has gone on, they've gravitated to sitting with kids who move at their same pace. Generally, I know which table is going to finish first and which takes some time. This frees me to wander, stop, and initiate a discussion.

This multi-part assignment came with three discussion phases. As a group finished a part, we discussed what they learned. In Part 1, we made some observations about the data points from the table. We ironed out the relationship I asked them to observe, many noticing right away that it looked like some kind of reflection. I floated around and had the Part 1 discussion several times, with audiences of 3-6 kids each time.

For Part 2, we expanded the idea. Can entire equations be reflections of one another? Do the observations about points still hold true? Are the points of one equation just reflections of another? Could we come up with a counter example?

Part 3 is a culmination of the first two discussions. Now that we seem to have a definition of inverses in general, can you determine them on your own and check your work with a graph.


My classroom is uniquely designed for this kind of set up. Near each table is a screen replicating my teaching computer. I can carry a keyboard and trackpad with me to manipulate stuff from wherever. I can also make use of their work as we talk since it's right there with us at their table.

I think it went really well. It was easy to get the kids to focus because there's no hiding when it's a group of three, or even six. The kids were eager to share their observations and made some good ones, all catching on quickly. The pacing allowed kids to work independently and have a piece of my attention. In this classwork heavy setting, I have been very free to offer more face time than these kids could ever want.

From an outside perspective it probably looks chaotic. From a breakthrough perspective it's not that special. It's just some scripted tasks leading kids towards an observation. Kids do some work, practice some things, get tested over those things. The key part I think is that the teaching element hasn't gone away. I'm still their main source of information, rather than videos or whatever. I'm still facilitating discussion and offering new ideas even if the tasks aren't particularly exciting, such is the nitty gritty algebra sometimes. And the kids are more than willing to give everything a try, mistakes are no big deal when it's just a few people sitting around a table.

I like where this is going.

AuthorJonathan Claydon

Have you resolved to document more of your classroom experience this year? Or do you want to start making some more organized reflections? Let me offer some unspoken policies about how I operate around here. Maybe these will help you find your own voice. This is not some set in stone set of rules and feel free to ignore one or any of them, but it's the pattern I've settled into.


Write about the good things. A lesson that went well, a positive moment in your career, or a positive development you've made in planning something long term. If you're going to be negative, make it about something that was in your control. A procedure that could use work, an end product that's missing an extra something, or a better seating arrangement. Being negative has its place, and I certainly write about lessons that don't go well, but the intimate ins and outs of events on your campus may not be best suited for the internet at large. Dealing with local issues is usually best done locally. I have work place ups and downs, but this isn't the place for them.


I write up lessons and projects that I do throughout the year. I include some information about what happened before, what the goal of the project was, and how it might be an iteration of a version that came before. I document lessons so I can track their evolution over the year. Nothing pushes you to better ideas than looking back on the ones from last year and seeing what can be done differently. These are the bulk of my posts and done for my benefit primarily. They help me remember things for school years in the future and help me compile evidence for end of the year appraisals. If someone wants to see how I'm making use of Chromebooks and they didn't happen to see it during an in person observation, I can have 6 links ready to go in 5 minutes because I spent the year keeping track. Since starting a site, submitting stuff for appraisals has been an absolute breeze.


I keep the opinions to things I have control over. I have opinions on lessons I gave, what was done well and what was not. I have opinions on content, what's important and what could be done better. I have opinions on curriculum, and how it can be worked through better. I also have opinions about local on campus issues, but again, local problems are best dealt with locally. The internet is not the place for me to go on a rant about class size, or cafeteria menus, or whatever. Unless it's "I tried the school lunch fried chicken and let me tell you, WOW!"


I don't write up stuff in exchange for money. I don't write up stuff because an enterprising ed-tech person e-mailed me about it. If I discover or am directed to a product and I believe in it, I'll write about my experiences. I also don't have intentions of offering stuff for sale or taking donations. I'm in this for the betterment of my practice and the practice of others.


Lessons and projects are great, but I also play around with themes. Maybe I do a series on Pre Cal lessons that have worked well. Maybe I talk out loud about a new way to organize a class. Maybe I make a post about what's inside my classroom closet (which has totally happened and probably needs a version 2). There is no set requirement about what makes something worth writing about. If I'm in the mood to write it up, it gets written up.


I am the primary consumer of my site. You will be the same. Write what you want to write. Write what's interesting to you at the moment. Write what will help you the most. If you need to write up a million lessons to work through some things, do it. If you want to write about classroom furniture, do it. If you just want to post a daily picture with a caption to help you remember sequencing or something, do it. Your platform is yours and it should please you the most. If other people start to enjoy it, that's a bonus.

AuthorJonathan Claydon