A couple weeks ago I had a little fun with BC on Halloween. I liked the mechanics of tying a math problem to some bigger puzzle. This is no different than those escape box kits you can buy for way too much money.

For the first time we were looking at a week off for Thanksgiving, so I wanted something to do on Friday that would be a nice closure. Most people go test here, but I was not in the mood to take a big to do list home. In AB we had made some good progress of curve sketching. Students knew how to start at an original and sketch the two "lower" functions as well as start at a second derivative and sketch the "upper" functions. We'd talked about critical points of the first and second derivative as well as what they represent.


In this activity students would work with a partner and be given a random 5th degree polynomial as a starting point. On the back was a word, seemingly random.

I started with 5th degree polynomials because it offered a lot of variety. I mixed up the types of behaviors, repeated x-intercepts, and included some with their reflections to ensure students took the time to analyze the functions as we had done. The students objective was to start with their randomly assigned original, find its first derivative from a bank of choices, and then find the second derivative from another bank of choices. All their cards would have a word on the back.


The hallway set up like this:

Minimums and maximums were marked on the f cards. Minimums, maximums, and x-intercepts were marked on the f' cards. X-intercepts were marked on the f'' cards. As I made a lot of similar functions on purpose, I wanted some anchor to help students determine if they were on the right track. They'd have to find the functions with the appropriately matching critical points as well as behavior in order to complete their set.

My largest AB class has 28 students, so there were 14 sets. SOURCE GRAPHS

In Action

How would they know if they had completed their set?

The words on the back of the cards are seemingly random, but they aren't at all. Some time ago I watched a video on the Remote Associates Test (RAT), a mechanism for studying intuition. When presented with three words, they trigger a related fourth word as a response. Some fourth words are more difficult to determine than others. Sets that comply with the RAT methodology are available right here. Students wishing to validate their set would read me the three words they collected. If they formed a valid RAT item, I proceeded to let them guess the fourth word. If one of the words was incorrect, I would tell them which one.

For example, if a student told me ACTOR, FALLING, DUST they had a valid set. The hurdle to their prize was to correctly guess STAR from the prize wall.


Now in an official RAT test, it's not a multiple choice thing. I wasn't really out to do any kind of behavioral study here, so all the 14 possible final words were available for them to see. To prevent random guessing, students had two chances to guess their fourth word. Failure to do so and they forfeit the prize. If they successfully guessed the fourth word, they could open the locker and retrieve the candy inside.


Students would take pictures of their original and walk between the f' and f'' sets while having discussions. Some would retreat to the classroom and sketch the set and go hunting for something that matched. Some theorized they could try to reproduce the function in Desmos, though most abandoned that idea when they realized there were faster methods. The idea was for them have a discussion about function behavior and how f, f', and f'' are linked by critical points.

All students were able to claim their prize, some in as little as 10 minutes. The longest anyone took was about 20 minutes. Most groups produced a valid set on their first try, those failing to do so usually had the second derivative incorrect. In a handful of cases students made a mistake at the first derivative but found the corresponding (though overall wrong) second derivative, meaning their 2nd and 3rd words, while matching each other, were not valid when grouped with their 1st word. One poor kid had the set right the entire time, but had misremembered the first word, leading me to intervene to figure out what the heck happened.

As a point of comparison, I let BC loose on the task too (mostly because I over bought candy and needed a way to get rid of it). They did curve sketching a solid month ago so I was curious to see what they remembered. Unsurprisingly, all of them cracked it in 5 minutes or less.


This passed with flying colors. Wrapping my head around the decoding scheme took some time, but the use of RAT was really clutch here, a ready made puzzle that provided just enough of a pause point to add a nice challenge at the end. Some students spent many many minutes deliberating what their fourth word was, dancing around it the whole time. The two guess rule really put the pressure on. Designing functions that were similar but not too similar took the most time, as well as all the screenshotting, printing, and cutting. Oh, and a lot of nervous labeling. One misplaced word could derail the whole thing, so would losing the sets I used. I clutched that sucker TIGHT.

Loved it though. I'm not sure I've had a first time task go so smoothly.

AuthorJonathan Claydon

I forget exactly when, but some BC kid suggested we trick or treat on Halloween. I offered the goofy suggestion of hiding candy in the lockers, making them bring candy buckets, and going door to door.

What we actually did was pretty goofy.

I used 8 lockers, and inside was either a trick or a treat.

I constructed a little puzzle. I borrowed some locks from athletics which have serial numbers. Bulk combination locks have the serial number and a key slot on the back so you can recover the combination or manually open the lock if a kid forgets the way in (or if a random lock appears). The task would have two parts: decode the combination, and find the lock it corresponded to.

For the clues, I had this idea that the lock would be "worth its weight in gold." The serial number would represent a weight, in grams, of gold. If you translate that into moles you get a cryptic looking number. I used that molar count of gold as the envelope label. Inside were three problems, either two derivatives and an integral or two integrals and a derivative. Inside was also a clue mentioning that "two like expressions wouldn't be caught dead next to one another." The intent was kids would realize the combination went D I D or I D I. Evaluating the three problems would yield three numbers that opened the lock in some order. Multiplying the number on the envelope by the g/mol of gold would convert it to the serial number.

I sent out some cryptic hints on our class Remind.

It worked pretty well. They were really confused/interested as to what the heck might be happening. On Halloween, I read some brief instructions, turned off the lights, and they went scrambling. They picked a clue at random and a partner. Five lockers had candy, three did not, but they'd have no idea until cracked the puzzle.

Admittedly I was a little too clever. They figured out the numbers were the combination. Translating the number on the envelope into a serial number had them chasing a lot of random ideas and eventually I had to offer some hints. Most chose to brute force the combination (trying the numbers in various orders until it worked) rather than think about the clue about like expressions. One kid wanted to try all possible versions of his combination on all the locks, a process he didn't realize would take forever.

Overall it worked. The idea just needs a little more refinement. The lock opening phase was particularly intense once they realized it might have all been for nothing.

I think there's a version here that scales for AB.

AuthorJonathan Claydon

A follow up from last week. You may have heard of Nix The Tricks, a book that started as a community effort to identify common pitfalls in math concepts.

Tina, compiler and grand Trick Nixer, took notice:

And so we'll see if that idea has some legs. I know the Calculus community is vast, and I have no idea what concepts/progressions need some love. I know what I think is weird, but you, the more experienced Calculus teacher probably have some input.

Help us won't you? Contribute whatever you want to NIX THE TRICKS CALCULUS.

AuthorJonathan Claydon

A Calculus goal was to be more upfront about where some of our patterns come from this year. I really disliked Named Theorems. For example, I have found no purpose if having kids learn the name "Fundamental Theorem of Calculus." It becomes a parrot moment. You say "what's the Fundamental Theorem of Calculus?" and I'm sure they can rattle off the pattern. But is there any connection to what that actually does? Does it improve their ability to calculate a definite integral with that name? I think it's debatable. Talking about final minus initial position has served me better.

Shoutout to my time at the Desmos Fellowship for the following revelations. Sam and Sarah blew my mind with the area model for a derivative of two products. The proof is so beautiful and perfectly suited for high school kids. WAY better than the awful derivation from the limit definition that flew over my head. I used it during Hurricane School with great effect. The question was prompted, well what about the derivative of a quotient? Does that have a proof? I didn't have an immediate answer, but YouTube to the rescue. The relationship made at the beginning was brilliant.

Here are both proofs written out:

Just so simple. And easily discussed in a few minutes. AB got to the product and quotient cases some time later, and I made a point to show them where the patterns come from. I wasn't super interested in them regurgitating the proof, I just wanted them to follow the logic. We were able to have some great discussions about the communicative nature of the two setups. The terms in a product derivative can go in whatever order we choose. A quotient requires us to be more careful. Later we saw that the product can scale to more than two functions.

I decided to cover all our derivative use cases now, rather than coming back to them. Next was exponentials and logs. Twitter to the rescue:

Later Dave Cesa dropped this bomb:

Once you establish e^x is its own derivative, you can show the logic for a natural log:

Again, so great and simple. Props to me for having the foresight to cover implicit derivatives (and isolating dy/dx terms) prior to this, so the dy/dx logic wasn't crazy bananas to them.

In the end, yes, most kids will just learn the patterns as a matter of faith. They will forget these proofs and will probably never reproduce them unless they grow up to be math teachers. But it was important to me to show them that math isn't magic, that we can reason our way to new ideas. Don't take the product rule on faith because I said so.

I can tell you as a student seeing something like this would've been so helpful. Calculus, despite having a great teacher, was a series of rules I felt I had to learn for the sake of learning them. It was many years before I realized how it all fit together.

AuthorJonathan Claydon
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My College Algebra students can also benefit from my project structures. I have more flexible goals with this group of students, and a main feature is letting them have as much class time as possible to get work done. I set up very brief lessons and let them spend time working. Most of the material is not new and students in here could use more reps.

We spent the first month of the school year talking about linear systems, quadratic systems, and radical equations. In all cases my goal was to show them the importance of a graph and how it related to work they do by hand. Students were to create 3 problems for a set of 4 possibilities: a linear system, a quadratic system with real solutions, a quadratic system with non-real solutions, and a radical equation. In all cases they stated the problem, did the work by hand, and graphed the equation to prove their work. Then they explained their process.

Students were able to use previous classwork as a starting point if they weren't sure how to make up a problem. For most of them they had rarely, if ever, been asked to do something like this. As we have progressed, students have been persevering through their work because checking themselves is so accessible. They don't need me to share an answer key, they have the ability to do it on their own.

As with Pre-Cal and Calculus, I got a lot of variety in the kind of work students turned in and all of them had great conversations along the way thinking about how to represent the situations they chose.

AuthorJonathan Claydon

Off in the BC side of things, some weeks ago we were talking about curve sketching. In my local parlance, I refer to f, f', and f'' as a stack. We need to learn where we are in the stack, and what information can be used to take us "up" or "down" that stack.

We started with sketching a polynomial and identifying regions where the behavior was increasing or decreasing. We translate that into positive or negative values for the function "down" one level in the stack. Similarly, though they didn't know algebraic integration at the time, we talked about how to make connections "up" between the values of a graph and the behavior those values were connected to.

Normally it's a long process in AB, but with the size of my BC class we knocked it out in a couple days. Throughout the sketching we added the ability to justify minimums, maximums, and points of inflection as well as identify regions that were concave up or down. For their design project, they used Desmos to create a polynomial, use prime notation to plot its first and second derivative, and then split it by its critical points. At the end they had to justify all the important features of their original, or "top" curve.

I've been using this class to integrate use of notation better. Desmos support for derivatives works great here, because although they could expand their initial polynomial and manually determine the first and second derivative, that was not the point of the exercise. I wanted them to work on how to make a mathematical justification for various points of interest on a function.

It's likely the AB students will follow up with a version of this project. It may not be as dense, but the "create your own stack of functions" aspects will play a key part.

AuthorJonathan Claydon

For a couple years in Pre-Cal, I had a lot of success with design projects. Students would take an aspect of the curriculum we had done recently, and complete an assignment that showed me a lot of aspects of that topic (Vectors, Polynomials, Polar Conics). What they used to accomplish it was up to them. It was a way to defeat identical project syndrome.

Adapting that idea to Calculus has taken some time as I tried to come up with ideas that suited the format. A few weeks ago I had students complete the first one. We had spent time discussing continuity and limits, so I had students design a piecewise function that demonstrated a lot of aspects of the topic. The function had to have five continuity problems (left/right disagreement, removable discontinuity, and unbounded), they had to demonstrate they could find the limit at various points on their function, and they had to explain the situation.

The lower one has a lot more detail to offer since I can see the functions used and all the boundary points are labeled. Students had a good opportunity to play with function restrictions in Desmos and could use a wide variety of function types to accomplish the goal.

Every time I do one of these, the students spend of a time thinking about how to make the requirements happen. Common problems throughout this project were multiple functions in the same domain, figuring out how to translate unbounded functions, and getting a handle on restrictions. As I keep my students in groups, usually one student is able to crack the issue and spread the knowledge around.

AuthorJonathan Claydon

I find assessment to be the most fascinating part of the job. For what I thought was a very rigid, established, system, there is really a lot of room to very creative. Figuring out how students turn your word salad into their own knowledge is magical. I mused on what I look for in assessment a year ago. I still agree with the general idea: the grade part is irrelevant, I'm just curious to see what you think. I still have to assign grades in some thoughtful way, incorporating that continues to evolve.

Most of my assessment methods were driven by necessity. When I first implemented SBG a long time ago, it was driven by being more efficient with my time due to athletics. I needed assessments that were short and simple to grade. I also needed to reduce the amount of things I graded. Two years ago I dabbled with A/B/Not Yet for Calculus. Through the year I stopped reviewing student papers and had them self determine their level. Most of the self-ratings were pretty honest, in general I've found students are harder on themselves than you'd think. Last year, also due to athletic constraints, I put all of the determination on my Calculus students. I still had piles of Pre-Cal stuff to deal with an two sports consuming all my extra time, so that's just how it needed to be.

In retrospect, it was a swing too far in the "grade how you feel" direction. I have since retired from my major athletics duties, and now have the time to give students a greater amount of attention. You might frown at slacking in this area, and believe me I wasn't happy about it, but to you I gently say ask a coach what the grind is like.

Here's how I handle assessment in my three subjects:

College Algebra

I use a stock SBG (0-4 scale with two required attempts) system here. It's something I know, and the multiple attempts play great for this audience. Kids are encouraged to keep their resources (notebook) organized by using them on the assessment. Kids can also work on them together. Multiple versions are scattered around. The sections are short and sweet: demonstrate a skill and explain it to me. All of the kids in this class are very capable, some require more time than others. To discourage "speed = smart" I keep the problem load low and make them do a lot of explanations.

My intent with this class is to dedicate their time to classwork as much as possible. The classes are small enough (18 and 15) that some intense differentiation is feasible and probably best. I don't have a plan for that yet. Assessments are just a piece of classwork I happen to look at and shouldn't be feared. A post-secondary goal for these kids is increasing the number that can qualify for college level math courses. Though this class is College Algebra in name, it does not award any credit (since I lack a master's degree I'm obviously unqualified to grant credit, apparently).

Calculus AB

I am keeping some of the ideas from my A/B/Not Yet experiment, namely the self-determination aspects. Assessments are short (half piece of paper) and sweet, with a combination of skills and explanations. There's a max of 10 questions and each assessment is rated on a 0-10 scale. Though 1 question does not necessarily equal 1 point. Students have about 35 minutes to complete the assessment. When I call time they grab a marker and I show the answer keys. Students give themselves as much feedback as they desire. Though unlike my previous system, I collect the papers and assign the rating. I take a holistic view of the paper when determining the score. I put a sticker on there if they get an 8, 9, or 10. The kids making a pre-check helps the rating process go faster.

Right now things are still pretty introductory, we don't know enough to tackle legit AP stuff. Eventually we will transition to AP-style problems on these. They will still take benchmarks to help them make an informed decision about the AP Exam. I expect about 60-70% (~50 out of 75) of these kids to opt-in to the exam.

Calculus BC

It's a small class (16) and a group of kids who put enough pressure on themselves without me being hardcore about tests. Their assessment system is divided into open book (35% of the grade) and closed book (45% of the grade) activities. In all cases they may collaborate on the task at hand. Open book assignments are chopped into a few sections worth a varying amount of points a piece (anywhere from 2 to 9) based on length or complexity. Each section is its own gradebook entry. Because of the speed required, we have already dabbled with legit AP material, so often I use the open book questions to give them a shot at FRQ style situations. The goal is to be diligent about how to make a math argument.

Closed book assessment has been done via Desmos Activity Builder. Topics vary and are usually conceptually in nature. Though the College Board is stuck in the stone ages with calculator technology, I use these as an opportunity to get them better at typing math notation, among other things. Use of Activity Builder spawns from a comment last year that students were fine with all the writing, but typing might be more preferable. These activities are about 15 slides long and I use the dashboard to assess how they rate on a 0-15 scale.

As this group transitions towards even more AP material, closed book will become a little more intense. These students will also take benchmarks to give them an idea about how they'll do on the AP Exam. I would be wildly surprised if the opt-in rate for the Exam was under 100% here.


Assessment can be whatever you want. Find a system you like. Experiment with one you're not sure about. You can always make changes. I don't love points systems, but if you don't want to work within 0-100, don't. There are many many ways to see what kids know and don't know.

AuthorJonathan Claydon
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Calculus BC is presenting an interesting challenge. Pacing is really hard to nail down. With a such a small, equally capable group, stuff that was normally a week in AB is taking us a day, tops. It reminds me of when I had a class of two (yes, two, though it eventually became ten) students one time. It's really easy to go "you got it?" and get 100% agreement.

Compounding the matter, we are wandering in all kinds of weird directions because, hurricane. My initial motives behind BC were to write a narrative around coordinate systems: rectangular, parametric, vector, and polar. When we missed 10 days, I rewrote the narrative entirely to get something productive out of Hurricane School. With a super fast grading period (an entire post of its own), I've been winging it for a month.

Other than logs/exponentials, we've knocked out all the derivative rules. We covered nearly the entirety of curve sketching (concavity, extrema identification, etc) in like 2 days, and we even introduced integrals via Riemann and trapezoidal sums. Next on the list is algebraic integration, but first a dabble into the abstract idea of integration, computing areas geometrically. But there are a lot of concepts at play here. For this, I started with six graphs:

I covered several situations, all based on some reflections I made a couple years ago. If you really want to demonstrate you know Calculus, you have to do it symbolically or all the algebra is just irrelevant.

We spent a couple days with these graphs. Prior to playing with them, I showed them something I rigged up in Desmos:

Screen Shot 2017-10-06 at 10.57.31 PM.png

I move a slider and I have them keep an eye on the black number (reading 9.28 in this shot). I stop the slider at various benchmark points along the way and we note the value of the black number. At no point do I tell them what the black number means. Eventually they reason out that the black number represents the area at any given time from x = -10 to x = 17. Further, they're able to determine how areas increase or decrease the total depending on position above/below the x-axis.

Concept 1: Integral as Area

Students take a'(t) and f'(t) and determine the area from left to right. We talk about how the function could be subdivided to accomplish this (rectangles, trapezoids, triangles, circles, etc). I redefine their answers as the integrals of a'(t) and f'(t) from far left to far right.

Concept 2: Area is Relative

Students use b'(t) and I define an arbitrary starting point, in this case I chose t = 0. This axis is measured in steps of 3, so I have them determine an integral from 0 to 30 and from 0 to -27. One student wondered out loud that 0 to -27 seemed off, as if the limits of the integral were backwards. I shrugged and played dumb.

I went back to the Desmos graph and we talked about "reversing" contributions. If I slid "backwards" in time, any area that was previously added is taken away from the total, and any area taken away is now added back. Thus, if our integrals moves left along the x-axis, the notion of what increases or decreases the total is reversed. We then mark the value of b'(t) from 0 to -27 as negative overall. We have gone "backwards" in time 27 units.

Concept 3: End Points are Flexible

With g'(t), I define a new function k(x). I define k(x) as the integral of g'(t) from an arbitrary starting location (in this case I chose -3), and an arbitrary ending location "x." I ask, given that definition, how would we determine k(3)? k(-8)? They think about it for a bit and see that "x" gives them flexibility to define the endpoint of the integral, and k(3) and k(-8) represent choices on how much area to find. In my scenario where the reference point for the integral is t = 3, k(-8) will follow our "backwards" in time model.

Concept 4: Absolute Extrema

Lastly, with c'(t) and h'(t), I bring them back to curve sketching. Based on these pictures, where would c(t) and h(t) have extrema? Currently we know how to identify minimums, maximums, and points of inflection. I introduce the concept of a relative min/max with these pictures. And then branch out into determine absolute minimums and maximums. For c'(t), I offered an initial condition at the far left, and we calculated area at four places: the far left, local max, local min, and the far right. Those numbers determined, we could see how end points come into consideration when making a conclusion about what's "absolute" and what's not.

Effectively, three days and six pictures give kids a symbolic look at almost all the major concepts in the AB course.

AuthorJonathan Claydon

A small, but subtle change I'm forcing myself to make this year.

I don't know specifically when, but some years ago I started making work time a priority. Kids should do stuff while I walked around. The urge to intervene when you watch kids work is very strong and it has taken me a long time to find the patience to leave them be.

The truest mantra I have learned in education is that it takes kids forever to do stuff. No matter how much time I seem to budget, kids need more. Now, I've gotten better at predicting just how much more. I've also gotten better at setting up their work time so that they can be more efficient.

What I have been bad about for a long time is what happens when a kid asks me to intervene. Most of the time it's simple question and answer. That student or several students need something re-explained. My little group units make answering a question for several kids pretty easy to do. If a kid requires more assistance, however, for years and years my response was "can I see your pencil?"

That question had good intentions. There is a clear misconception, the student would like help, I can show them or a make a correction quickly if I do the writing. But I was robbing the kid of the opportunity to make the correction on their own, simply because I didn't want it to take forever.

Now I've been forcing a different response, "do this for me..." and I'll do a little dictation depending on the issue. Sometimes the kid knows exactly what to do, they just need something set up. Other times they need to go through a much longer process. Regardless of need, I'm making sure they do all the writing.

Many, many of you probably read that and went "uh, duh, of course that's what you should do." I know this isn't a giant revelation, but it's a subtle difference I've known I needed to make for a while. I have no empirical evidence that changing who does the writing makes a huge difference, but it just feels like the right move. More of the thinking burden transfers to the student. And maybe, just maybe, they get a more robust answer to their question than just nodding along as I write stuff out.

Is this related to the phenomenon of "I get it when you explain with me, but then you walk away and I'm confused" ? It could be. That statement is something I've been trying to put a damper on too.

AuthorJonathan Claydon
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