This is new:

Out of nowhere, my epic class sizes have finally retreated. My average is down to 28. My room feels empty like something is wrong. Generally speaking, this is probably a good move long term. As much as I have enjoyed teaching as many kids as can possibly fit in there, the energy level required to watch over 36 kids at once is a little high. Especially when they all feel like making noise. That many kids even just whispering is crazy loud. This year it's like an entire class is missing.

Could this mean we cover more material? Possibly. I feel like Calculus will be a bit more efficient, though you could easily chalk that up to my increased experience with the subject.

It's just weird. I was wandering around the room on the first day and just kept wondering where all the kids went.

AuthorJonathan Claydon
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Time for another one, this time a draft of what I'm looking to do in Pre-Cal.

This is what you'd call a living document, similar to any sort of first draft I do for a course. I'll probably start working with this, mark it up like crazy and produce something a little better next year.

Major design goal was to ditch the Algebra review at the beginning of the year. Diving straight into trig feels like it will be fun. It will be some familiar ideas from geometry, sure, but my history with this course has shown that kids aren't quite grasping the why from geometry. Or at least, it's worn off by the time they get to me.

In all things I want students to understand why something works. Because my math teacher said so is not a sufficient answer.

The Algebra chunk, when it's time, will be a bit more thorough. In the past it's been some review of quadratics at that's it. A couple years with Calculus has shown me that we should go deeper here. There's a lot of topics that only got a surface glance in Algebra II that need the "why?" answered.

Reading this you probably wouldn't say this is the most amazing Pre Cal plan in the world. It's really not. But what I want to do when it's time for the material is really dive in to what makes it tick. For example:

This vector project felt so deep last year. Nearly every vector concept was hidden behind the phrase "draw me a picture." So simple. So not.

How do I bring more of that to the table? Can't wait to figure it out.

AuthorJonathan Claydon

Previously in my Calculus curriculum journey, I hacked together a progression. Lots of positives from the approach. The main idea is that it's valuable to see the entire course up front and slowly iterate on the various ways to apply the two big concepts. In the end, it seemed to work. Room for improvement though, as always.

Presenting the updated version of the plan.

The second page existed in the previous version, but I don't think I shared it. There are subtle modifications throughout, mostly to reflect what actually happened as I worked my way through the original draft. The first semester was pretty jam packed in last year's version, so I altered the goals for some of the grading periods, pushing material down the line. Our pace in the second semester is much more comfortable.

Huge, huge, huge goal for me instructing this year is to focus on the abstract. Applying the chain rule, product rule, and quotient rule when only given "f(x)" and "g(x)" terms for example.

Plus a fair bit of these problems:

Bruce Cohen - TMC16 Morning Session

To the new Calculus student, this feels difficult, but is elegantly simple. Quite honestly it summarizes the AB curriculum quite nicely.

Feeling good about it.

AuthorJonathan Claydon
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The coming school year will be notable for a number of reasons, among them: no more iPads in my classroom.


It sounds like a big deal, but not really.

Use Cases

In 2012 we were given 4 iPads for classroom use by the district. My class sizes were hovering at about 30 and I knew that wouldn't be enough to have much of an impact. Slowly, through donations, personal purchases, and some additional school purchases I peaked at 33 iPads of varying vintages by spring of 2016.

In that four year span, I attempted everything. I look at distributing and collecting things via Dropbox, Google Drive, iCloud photos, you name it. I tried playing around with various apps only to find that the best workflow involved a word processor, Desmos, photos, and a printer. None of those components specifically requires an iPad. It was easy and worked every time, but it could be replicated with other methods if necessary.

Over a year ago I started to realize that there was nothing iPad-y about the way I used them, opening up questions about where my tech use goes from here. In fact, the simple use case I had was beginning to become more tedious as software pushed ever forward but the hardware aged (about 2/3 of the stock had 2011-era internals).


Managing that many iPads yourself is a pain. Mobile Device Management systems require business Apple IDs to use most of the features. Acquiring a business Apple ID requires that you own and operate an actual business, with tax ID numbers and all that. Apple makes an app called Configurator that's allegedly for this purpose, but I'm not linking to it because it never worked. Where does that leave me? Manually updating iOS thirty some odd times, retyping iCloud credentials constantly, retyping the WiFi password when the security certificate expires every week or so, and having to wipe each device at the end of each school year. Plus the cost and time associated with my charging system.

Relatively simple work, but bleh.


As cloud computing became a real, viable thing for student work (Google Docs in 2012 was uh, yeah...and it's still so-so on iOS), and as a 1:1 Chromebook pilot in AP English 4/AP Government proved useful, I started looking at Chromebooks as an alternative. Price wise it's equivalent to a refurbished iPad mini, with none of the maintenance headaches and batteries that last for-e-ver. Also, Summer Camp proved the ideal test market. It was a small group of kids, the librarian had nothing else to do with the Chromebooks for the summer, and in the years since we started tech deployment, every kid has a Google account. In the end? Totally frictionless.

A small technical hurdle was solving the Chromebook printing problem, but that wasn't terribly difficult. Some detail on that later, but the easiest way involves a printer with Ethernet (and boy did I purchase a monster).

Having a class set (which the district is providing, for that I am grateful) combined with teaching kids that will be issued one to take home (through other classes) will open up some opportunities. It was already handy for accessing Calculus solutions, and will make some of the deeper parts of Desmos more accessible.

AuthorJonathan Claydon
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On July 16 I presented a My Favorite at Twitter Math Camp. It's a follow up to a presentation I gave the year before. I discuss the ways in which the program expanded at my school. A recording (with the slides embedded) is below:

Main points: Varsity Math is now the blanket term for AP Math classes at my school; membership is up; we did some outreach at our middle schools; laser tag in the cafeteria is the best thing ever; it's a summer program now; YOU'RE INVITED!

If you attended TMC 16, I handed out Varsity Math membership stickers with instructions to 1) come up with a member number 2) write it in the white box 3) take a picture and tweet "I'm on the team! #varsitymath"

But what about you, dear reader who wasn't there and might be upset you missed out on the fun? Well, I have roughly 50 stickers remaining. If you'd like one, get in touch and we can make arrangements. If you're interested in how to start such a branded movement at your school, I'd be happy to help you with that also.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students that shaded regions can form the base of 3D objects with a cross-sectional area of a common geometric shape. Students understand that shaded regions can serve as the profile of objects rotated about the x-axis or y-axis. Students can generate integral expression for the volume of a region for any of these scenarios.

Fear Not the Y-Axis

An early introduction to integrals can make volume expressions particularly smooth. The concept of an integral is not foreign, its representation as an area is well known at this point. The new idea is that much like integrating a velocity functions yields position, integrating an area function yields volume. The integral serves to move us "up" the relationship.

Students grasped the concept fairly quickly, what we spent most of our time on was the variety of scenarios. We discuss 3D objects with square, rectangular, "leg-on" right triangle, "hypotenuse-on" right triangle, equilateral triangle, or semi-circle cross sections set perpendicular to either the x-axis or y-axis. In my first run through Calculus I ignored the y-axis to my detriment. It doesn't have to be a big deal if you don't make it a big deal.

I hand out a set of regions.

For each region, I give them three types of cross sections to consider. The y-axis regions introduce an important concept: what do I do if a part of the region is uniform in size? Seeing that a volume expression can result from two integrals will become useful later.

As students will make all kinds of notes on these pictures while working through them, I have a second set of regions ready later in the week.

This set of regions is for introducing solids of revolution. After all the mixing of shapes before, students find it relatively simple to throw this in the mix. Again I present them with three scenarios to consider for each region, and they determine the appropriate volume expression. Multiple integrals may be required.

Full PDFs are here: Regions 1, Regions 2


To demonstrate understanding of volume expressions, students have be able to apply any scenario to any region. My goal here is to remove any fear of "y-axis" especially since it's not something they've been used to. Revolving a region about the y-axis should be no different than stacking a bunch of square cross sections along the x-axis.

Their (admittedly too lengthy) understanding task looked like this:

Six figures accompanied the task. None of the curves were labeled, I wanted to make a point about innate knowledge of parent functions. None of the given functions were in the appropriate form for y-axis use. I wanted to make a point about knowing how to invert something to suit your needs.

Full PDF: Questions, Regions

Despite the 2016 AP exam not offering a straight forward question on volume, the students exceeded my expectations on this topic. The y-axis was no big deal.

Total time spent was 5 class periods. The assessment took two 2 class periods.

AuthorJonathan Claydon

My intention when making the entry for this post a couple weeks ago is that I'd recount all the great things I learned in the morning sessions and various breakouts I went to during my time at TMC16.

Really, TMC16 had one focus. It's not something I advertised and not something I was going to advertise. It all happened too soon and I was still dealing with it personally, and not prepared to talk about it.

On July 12th, a Calculus student of mine I taught for two years and who had just graduated was admitted to the hospital. The outlook was bad. Within a few minutes of leaving the plane on July 13th in Minneapolis I find out he died. His particular case was a million to one freak occurrence, just absolutely unfair. I briefly considered the emergency flight home, but sitting at home alone was not how I wanted to deal with this, as much as it pained me not to be there.

Seeing old and new faces at TMC helped me work through grieving. To be reminded that there are such positive people out there working in teaching is the renewal I get every year at TMC, to be reminded of why I work so hard for students like him.

His funeral was on Saturday the 16th at 1 pm. Part of the reason I stayed was because I was slotted for a My Favorite at the exact same time. The topic was Varsity Math, an initiative that brings me and the kids so much joy. If you were there, he was in some of the photos you saw. It felt right to include him. Here's his class photo, no red boxes this time. I'm not going to point him out, but here's there, happy as could be.

He is one of the happiest kids I ever taught. The kids he sat with loved him.

Thank you to everyone I interacted with at TMC this year. You made more of an impact than you know. It's the community, stupid.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students understand the relationship between an original function, its derivative, and its second derivative. Students understand that a value of zero on one graph implies meaning about the equivalent location at steps "above" and "below" that graph.


I found great success with curve sketching, a subject I found difficult as a student, through the early introduction of integrals. I establish an "up" and "down" narrative about the role of integrals and derivatives. I find it helpful to introduce this with a bit of physics and introduce the connections between position, velocity, and acceleration.

Eventually, the "up" and "down" visual looks like this:

I do a little physics demonstration with a tennis ball, discussing its velocity and direction and various moving away or towards a table. It's a remarkable "OHHHHHHHHHHH" kind of moment that I wish your average physics class included.

Units are a critical part of the discussion. Later you need students to understand which operator is necessary based on interpreting a lot of language. Unit awareness helps with some of this. Being given information noted in meters/second with an answer requested in meters should be a huge flag that going "up" or integrating is appropriate. See: What Helps Me?


This takes a lot of repetition and discussion to get right. I find this Desmos graph helpful. I place an emphasis on values vs. behaviors, a tricky idea conceptual. For example: a piece of a function can have positive values yet negative behavior. Say that piece belonged to a graph labeled f'(x). What graph can we construct from those values? What graph depends on the behaviors? Catching that difference takes a minute.

Primarily we spend a lot of time drawing families. Students are given a starting graph at any point in the family.

Eventually we arrive at the idea that my pattern of behaviors at one level, say f'(x), should be the values of my graph at the f''(x) level. If we moved "up" the family, the relationship is inverted.


Curve sketching is great and all, but it means nothing if you can't interpret the drawings. Sequence wise, I go Riemann Sums > Curve Sketching > Integration > Curve Interpretation. At this point we do a lot of heavy vocabulary lifting: minimums, maximums, points of inflection. concavity, and justifications for the appearance of all these things. Justifications in Calculus are relatively simple, but teaching students to make simple arguments is strangely difficult.

At this point we've reached December and students are ready for a concluding task:

Going Further

Soon enough, I've introduced the concept of algebraic integration and the +C terms that follow. Students will start asking questions throughout curve sketching. When going "up" a level, many will ask how they know where to put the graph along the y-axis. I play a little dumb and say the answer is coming soon.

Once you've established +C as a class norm and they understand the idea of what that constant represents, you get into the numerous possibilities with what the graph one level "up" could look like, and bam, you're ready for slope fields and differential equations.

Early access to integrals saves so many headaches later. The idea has had time to marinate in their minds. It's hard to talk about slope fields if kids aren't square on the base concept on why there could be so many options in the first place.

AuthorJonathan Claydon

AP scores are in, it's time to push my five year Calculus plan into Year Three.

Only recently have I thought about what the full five years looks like:

Year 1, 2014-15
Figure out what I'm doing, address some of the biggest problems: the AP test is scary, students quit when things are scary, running for a schedule change when things get tough. Solved mostly with the happy accident that was Varsity Math. I had 50% (25 of 50) students take the test, exam results had a faint, but present, pulse.

Year 2, 2015-16
Hammer out a curriculum that makes sense. Address the literacy demands of the course. Grow everything: overall numbers, exam takers, results, etc. Varsity Math was straight fire in Year 2. It's an established bit of school culture. Kids weren't scared anymore and were just amazing. I had 72% (49 of 68) students take the test, exams results below (such a tease).

Year 3, 2016-17
Push harder when it comes to literacy demands. My AP benchmarking system was effective, but has room to improve. Some items of the curriculum need a little polish: sketching polynomials, and a touch more algebra. We've got a BC student this year! That's going to be a learning experience for us both. I want 80% of students to take the exam and 30% of them to pass.

Year 4, 2017-18
We really grow the program now. I hope to have identified 2-3 students who could take BC straight up, without AB initially. This will involve some diligence on the part of me and the other Pre-Cal teachers. For BC to be a standard part of the program by Year 10, the first few years have to have the right students involved. AB goals remain the same, keep test takers about 80%, push on the passing rate. Could we nudge up to 35% here? More? We diversify options for other 12th graders and start offering a section of academic Calculus. We have a healthy set of students who would enjoy the Calculus curriculum, but at a more relaxed pace.

Year 5, 2018-19
Firing on all cylinders. Another set of 2-3 BC students, fully stocked AB sections, and an academic offering. Could passing the AB exam be a normal expectation of the majority at this point?

The big question is finding the inflection point. With enough effort, the program will turn the corner, but how soon? I thought it might be this year. We aren't quite ready for prime time yet. It's primed though.

Score Time

Standard disclaimer: I know I write about it a lot, but I am not all consumed with AP results. It's interesting third party data. That said, there's absolutely no reason my students can't enjoy the kind of AP success seen at other high schools in my district. And, that success can be had without your boring old AP approach.

Last four years (2015 is my first group):

Those 44 1s were hard to swallow, especially given the kids who got them. I know those kids, tons of them were dynamite and were humming along with the benchmarks. What happened? Language? Nerves? An interesting mystery, since I'll never know what kind of 1s they got.

But look! A 5! Believe you me, that kid was a rock star and totally earned it. And and and, two other kids passed! When I tell you the AP situation was pretty bad here, I mean it. Three kids passing is crazy pants.

A positive for the benchmarking system I used (questions sourced from secure CollegeBoard releases), those 5 non-1 scores all sat at the top of the charts (1st, 2nd, 3rd, 4th, and 8th overall). Clearly something went right there.

A deeper comparison of 2015 vs 2016:

Aaaaaand here's the source of that big 1 pool. Multiple Choice still appears to be the Doombringerâ„¢. But but but....look at those Highest 4th numbers! Even the Third 4th has more signs of life. Shoutout the growth on integral-based questions.

Biggest win in Free Response was all those 0.00% in the Lowest 4th for half the questions. Crazy stuff. Question 1, 4, and 5 in 2015 covered similar content to Question 1, 4, and 3 in 2016. Small signs of growth in every quarter.

What Now?

Curriculum tweaks and literacy demands are priority one. What of the benchmark system? It definitely has merit, but how can it be adjusted to be a better predictor? I was feeling really good, but got faked out in the end.

I love chewing on this problem. There are so many variables to consider and I feel pushed to get better and better. The kids are SO awesome, how can we prove it and quiet the haters?


The purpose here is to remove some stigma. Even teachers on the internet struggle. It's the nature of the profession. I'm choosing to struggle out loud.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students have a fundamental understanding of integration in all its forms: approximate area through Riemann sums, abstract area through geometric breakdowns, and algebraically through antiderivatives.


My approach is unique, as far as I can tell. In all the Calc AB material I've seen, integration enters the picture really late, after the idea of curve relations (f, f prime, and f double prime). In my opinion this does students a disservice.


Discuss the Riemann sum methods: left, right, midpoint, and trapezoid.

Place emphasis on an integral as the approximation of area under the curve. At this point in time I will use functions defined only with a set of data. I show them definite integral notation and how our Riemann sum values are one option.

The biggest lesson of all needs to happen right here. WHY would a data table defined with x-units of seconds and y-units of meters/second have an integral value measured in meters? Don't walk away from the concept until they see that the dimensions used in your sum approximations have units attached. Multiplying those dimensions to create an area have consequences for the resulting units. This is how your April fortunes are won and lost.

Before moving on, take the opportunity to discuss antiderivatives. Keep the examples simple. Reverse power rules and definition based trig functions without limits of integration. The word "antiderivative" is the next key.


Return to the concept of a definite integral. Review the idea of a Riemann sum. With arbitrary data, we can make approximations of an integral's value. What if we had a function defined solely with a picture?

Knowing your final learning target matters here. Can students recognize when to interpret a function value as an area? In the graph shown here, what, exactly, is that integral symbol implying about the relationship between g(x) and f(t)?

This type of integration scenario immediately follows f, f prime, f double prime relations and the curve sketching exercises associated with the topic. If you're students know about antiderivatives in the midst of curve sketching, you'll be amazed at how quickly they can draw a curve family AND discuss the transition between curves using the concepts of derivation and integration.


Right before you break for winter, briefly return to the antiderivative idea mentioned in October. Using the same simple scenarios (reverse power rule, base trig functions), add limits of integration. Where have we seen those before? What do those imply? Are we still talking about area even though I have no picture? Could I make a picture? What does my algebra demonstrate on that picture?

If you find students struggle with implementing the fundamental theorem, see what happens if they know what an integral is months before you introduce it.


Pat yourself on the back because the heavy conceptual lifting is over. Now you can move to integrals that require chain rule concepts (or u-substitution if that's your thing, it's not mine). A huge problem for Calculus is battling student understanding of skills while at the same time realizing they have no conceptual idea of what's going on. April is the wrong time to realize that students might be able to parrot all the appropriate methods associated with integrals, but can't think their way around the concept at all.


Students need time with this stuff. Cramming the many many facets of integration (we didn't even talk about volume!) into 4 weeks is incredibly strenuous, and I think it breeds a student who can do everything you say but has no idea what they're doing.

Don't make your students become math teachers before they really get this stuff.

AuthorJonathan Claydon