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

Curriculum is a current focus of mine. In recent years, I worked through thought experiments on Algebra II and Calculus curriculum. What happens if you ignore a textbook publisher? What if you start with the standards and find a spiraling pattern that makes sense? Can you find the themes? How will what you teach in September help your students in May?

Viewing your course in broad strokes has proven to be very valuable. In Algebra II I only got one year to play with it, unfortunately there was no follow up. Calculus is very much a continuing experiment. Upon taking over in 2014, I knew it was a five year project minimum. Deploying a curriculum of my own design was going to be a really big piece.

With another round of AP scores to analyze, what needs to change in that vision?

More pressing, what has crafting a vision for Calculus taught me about Pre-Calculus?

Pre-Calculus has some themes: raw algebra, trig, polynomials, and a random bin of parts. Can we tie those together better?

At TMC 16, on Sunday July 17, I'm going to offer an opportunity to participate in this thought exercise. Does the curriculum we work through have to be something we actively teach? Do the standards have to match the one from our state? I don't think so.

I've worked through 8th grades standards and Algebra I standards before, neither are things I've ever taught. If you want to broaden your horizon of how math concepts go together, I think it's well worth the trouble to read about all the things you don't teach.

AuthorJonathan Claydon

I've stumbled upon a lot of random things in the midst of camp, and this is a good one.

If you're of a certain vintage, you probably played a text adventure or two. You remember, the "use lamp" and "exit north" variety as you try not to die. Well these folks Memento Mori make a variety of "text adventures" that can be played out loud with a group of people. The maps/situations are small enough to fit on a single sheet of paper and you can run through the game (with plenty of error) in about 40 minutes.

Best part? There are a ton of scenarios and the PDFs cost a whopping $1.99.

It's a fun introduction to some old school computer programming, as your players have to speak in simple commands such as "go north," "save game" and "examine backpack." As the moderator it's great to stymie them with a "I don't understand that."

In camp this week I have 8 pairs of 2. Each pair gets to say one command then the next pair goes. Watch as the group hopelessly wanders in and out of the same place as the other side of the room screams in frustration!

If you'd like to hear a few adults wander around the jungle aimlessly, listen to The Incomparable Game Show #33.

AuthorJonathan Claydon

Session 1 of my camp ended a couple days ago. Within seconds of the first day ending I knew we were doing it again next year. Such a great week.

The Good

There's always that small amount of dread when you plan a big event that attendance will disappoint. But no, everyone showed and were eager to see what I had prepared for them (no pressure).

I was equally nervous about how much I had planned, and maybe it was the time off, but I forgot that it's actually pretty easy to fill class time, students take (and need) a lot more time with things that you anticipate.

Opener? Solid.

Quadcopters? A little quirkier than I expected. I only had a couple kids really into them, drones in the $40-50 range are tricky to control well. But, we had a good physics discussion about them.

Space? Great discussions here. I had to cut out a lot of my planned material. We focused on a few things. Main idea: space is hard. I had research some of the launch systems that have been used, the new ones under development, and we watched a bunch of videos. We briefly discussed the Curiosity and New Horizons missions with a few spoilers for the The Martian thrown in there. There was just too much here.

Engineering? We had a great discussion about my previous career and we scaled it down a bit with some spaghetti constructions. Materials had a cost associated with them ($1 for thin spaghetti, $2 for thicker stuff, tape was $5/ft), then we compared output and cost of materials. I had some research planned here but had to cut it.

Financial Literacy? Probably the most successful part. We played with Sheets for two days. We played with some basic formulas/functions and I gave them a task designed by my sister (an Excel genius) to work through.

Computer Science? Playing with Snap! was minimal, unfortunately. A longer camp would be required to do it justice. The Sheets stuff was a good, immediately practical substitute.

The Great

The kids had a blast. Many of them wished it was longer, and quite honestly, I wish it was too. Such a great time with these inaugural campers. We had a lengthy discussion about how to give the non-attendees #summerjealousy when they return for Calculus in August.

Next year will be huge (and probably backed by a grant).

Extra exciting? There's another session, it starts Monday, OMG!

AuthorJonathan Claydon
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Varsity Math Summer Camp is here, finally. By the end I hope to have an interesting answer to the question "what happens when you have no official curriculum?" In addition, I get to see what's it like to teach small groups for the first time in a while. Around 35 kids/class has become the norm. Long ago when I was less adept I maxed out at 20. What could I do with small numbers now? Session 1 has 20 kids, Session 2 not but 16. My room is going to feel massive.


Big Ideas: Space Travel, Computer Science
Small Projects: Quadcopter mechanics, engineering scenarios, financial literacy

I have 2.5 hours for four days with each crew. In my planning, the first thing on my list bolded and underlined was "OMG you better not lecture for 45 minutes on any of this." A lot of what we're studying the students have limited prior knowledge, so it would be easy to sit there and wax on about the nature of the Apollo launch system or whatever, but it'd be far more productive to let them find out how it worked on their own.

Second, I like the results when students are given multiple passes on a topic. Each 2.5 hour session is broken into small pieces to facilitate one talking point. Rather than spend a marathon on space, we'll do a little space over the course of the camp.


I thought we'd have a little fun at the beginning of each day. I subdivided the room into 4 groups. Each team competing for a varying number of Starbursts. Through happenstance, I found out about the Remote Associates Test, a old instrument for measuring creative thinking. Given three words, what's a fourth word that ties them all together?

In addition to that, a BrainQuest pack of 7th grade trivia. Should be fun with a group that just finished 11th grade.


You can find a billion of these of Amazon. I bought four of a model that was highly rated and not super expensive. Handy bonus, the radio is 4-channel allowed all four to operate simultaneously. The batteries last 5-7 minutes, but extras are easy to come by.

For roughly 30 minutes for 3 days, I will hand each group a radio, copter, and manual. They figure out how to fly it, why the propellers are oriented the way they are (two spin clockwise, two counterclockwise), what "six-axis" motion means, and how the copter might be changing its orientation in any axis. Then we race them. And I show them the big one.


This batch of students was in 7th grade when the Shuttle retired. They don't know much about it, or the litany of vehicles that get things into space, or even where we've been in space.

For two days, the students will do some research on the methods NASA and others have used to fling things into space. Plus we'll throw in some Flappy Space Program. Day 1, the variety of launch platforms used since the 60s. What were the payloads? Which ones had manned missions? Who are the new players? Can anyone find the sweet video of the SpaceX auto landing? Day 2, we get more specific and discuss the missions that went to Mars. Which ones failed? Which ones are currently ongoing? How did they get the vehicles to the surface? How insane is that Curiosity landing rig anyway?h

Each group will have a chunk of research to do and then they'll present to rest of their group. Initially I wanted to make some display pieces, but this is summer and anything we put in the halls will eventually be destroyed in the back to school cleaning. Though maybe we do something here.

Lastly, we spend some time on the STS specifically and a look at the technical problems that lead to its retirement.

I might modify this section a bit for Session 2, perhaps with a focus on the International Space Station, for the sake of variety.


A brief paper airplane contest, then a spaghetti construction project. Teams will be given an objective. Supplies will have costs associated with them. Structures of different heights will have different profit payouts. It's like Hotel Snap but with pasta.

Financial Literacy

Over two days we'll run through the basics of a spreadsheet, learning how to manipulate cells with formulas, sorting things, etc. In the end we'll look at how to build a budget, how recurring expenses affect a budget, and how you can determine the amount of money you make each day.

Second, we'll do a overview the housing market. Why are they so expensive? Am I really a sucker for renting?

Computer Science

The big one. We explore the Snap programming language. Everyone is a novice with programming, so there's going to be a lot of start up exercises. Eventually I hope to get them to the point where they can develop a maze. Or at a minimum draw some geometrically styled art.


I really want to value students time here. The kind of learning you do of your own free will should be engaging, interesting, and active. Bringing kids to school in the summer just to do all the talking would be a waste of this opportunity. I'm hoping this experience trickles its way into the regular school experience. The quadcopters already have me thinking about new ways to introduce 3D Vectors. What else could benefit?

AuthorJonathan Claydon
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