# Desmos Assessments

In July I spent some time at Desmos HQ driving Eli's Lambo and shooting the breeze with people about how they incorporate Activity Builder and how the Desmos staff see as the role of Activity Builder in the classroom. Two things stuck with me: be thoughtful in AB design, and see how could change the way I assess.

Prior to my visit I had decided to experiment with Activity Builder more. I saw a lot of great work with Pre-Cal kids having to explain their thinking more, and Calculus kids could certainly use the same. The language barriers for making mathematical arguments have been a barrier for my students in the past, and I want to start being more picky about that kind of thing. I also want to push my kids to better understand how the regular Desmos calculator works with regard to restrictions, notation, and such.

In 11 weeks I've done 9 actual, premeditated Desmos activities between Calc AB & BC. There's at least another half dozen instances where they used it to make a project or annotated a pre-made calculator page. Here's a small sample of stuff I tried:

### Calc AB

Being able to let kids sketch is also nice. For an activity on curve sketching, I provided the first derivative and they had to sketch the original function as well as the second derivative:

I really like slides where something has to be added to a graph, makes it easy to see how much a misconception has propagated. Here I can quickly tell that two students misinterpreted the initial picture has f(x) rather than f'(x). Were they working together? Did this idea manifest in separate parts of the room? I can figure it out fast.

### Calc BC

For BC my built activities are part of their regular assessment program. I also incorporate it into their classwork a lot. While doing area between curves and volume, I was able to share a calculator page with them and have them add integrals to it. At regular intervals they complete Activity Builders as closed notes (though collaborative) assessments.

Here they shaded regions of a velocity curve where speed was increasing versus when speed was decreasing:

It's also been easy to adapt free response questions to the format:

Since it's a smaller group we've been able to learn a lot of nitty gritty things about the calculator. How to use folders, make dynamic labels, define variables, etc.

### Takeaway

In all cases I make sure the activities are short and sweet, usually less than 10 screens. I consider what the activities let me do that's not possible with paper (for instance, compute nasty integrals). I also make sure the kids get to see the data that gets collected. In BC for example, we always go over their assessment when it's finished. I'll call up the dashboard and scroll through interesting answers or demonstrate common issues. At no point are kids being called out to mock them for getting something wrong, rather we use it an opportunity to discuss what mistake they might have made and how what we can all learn from it. Kind of like a digital "my favorite no" kind of thing.

There are still some scaling issues I'm working on. For the most part paper assessments are still faster for AB given the size (75), but there's potential there. So far it's working great. Kids can use the system well, I get useful information from it, and the technology gets out of the way.

Posted

# Nine Six Three

As an avid reader of Teacher Internetâ„¢, you may think it's all wild innovation and maker spaces in the math universe. Not so! Every day we all have very uninteresting procedures we have to implement, the real workhorse of any classroom. It's the inescapable part you don't read much about. At some point, you have to put away the VR goggles and 3D printers and give an assignment. In my attempt at College Algebra this year, I do exactly that, a lot.

My audience is ~30 kids split into two classes. That's really small, for me anyway. When there's only 15 people in the room, everyone gets face time for as long as any kid could possibly want.

And some of them take all they can get. Another feature of this audience is they're all extremely capable, but we're all over the place in terms of how long it takes for stuff to click and get assignments done. One kid is over and done with the task at hand in 10 seconds, where another needs 5 minutes. Adding to the challenge is just who is super speedy varies depending on the task. It is a weird environment.

Number one message I send to this group is I don't care at all how long it takes you to do something, I want you to understand what you're doing at the end of the task. If that means we're in 15 different places some times, that's fine. Most of these kids have been fighting the speed thing for years, I think it's time to give them a break.

How do you accommodate such a group? With work time, tons of it. I think I stand and talk in front of this group like 20 minutes a week, tops. The rest of the time they're working on problem sets. It is, by far, incredibly uninteresting. But uninteresting != boring.

Yeah, they work all the time, but the tasks are varied. There's some solving, some listing, some writing, and some graphing depending on what we're doing. And at no point are they working on a task to the point of exhaustion. Enter my go to format this year: 9-6-3.

Take a skill, have them perform it on 9 items, do an extension on 6 of those items, and finish with an analysis task for 3 items. Based on my observations the first few weeks of the year, this format keeps everyone at a similar overall pace despite their varied work speeds. 95% of the kids can complete all of that in the time allotted. Although 5% won't quite finish, they really get the ones they dd because they were given the time to sit and contemplate for a while. There's like 10-15% who finish pretty fast and without any prompting by me will be helping other kids around them.

I think the biggest success in this format is other than the initial "do this for all 9" the remainder of the choices are up to the kid. I'll do similar tasks with "here are 6, pick 4." I'm sure there's something in here about choice theory or something, but the kids put up no protests under this system. The kids take ownership here, managing their time to make sure they can do what's expected. That's the other big component, an inspection always comes and they know it. My inspections aren't a big deal, but it's a little thing that sends a big signal: I want you to try something and I want to see how you tried.

In the end all the kids get in reps, they aren't exhausted, and the ones who need to ask a ton of questions have ample time to ask their questions.

Posted

# Sketching Hunt

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.

### Setup

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.

### Results

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.

### Conclusion

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.

Posted

# Trick or Treat Lock Picking

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.

Posted

# Nixing Calculus Tricks

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.

Posted

# No Magic Tricks

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.

Posted

# Summary Sheet

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.

Posted

# Curve Sketching Crafts

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.

Posted

# Limits and Continuity Crafts

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.

Posted

# Current Assessment Theory

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.

### Conclusion

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.

Posted