I’m on a mission to fix all kinds of things about Calculus, especially Calculus AB. Next on deck is Curve Sketching. Previously, while working on integration of Intermediate Value Theorem and Mean Value Theorem, I integrated some curve sketching ideas. Students calculated average rates of change and created rudimentary graphs of a function’s first derivative.

A few weeks later, students now have more familiarity with the first and second derivative and we can talk about how those tables help us analyze a function.

Start with an arbitrary function and interval, and create a table of f’(x) and f’’(x) in Desmos.

At the moment we aren’t concerned with the graphs, those this will be useful later on. Have students recreate the table, but we’re going to declutter the results. Rather than worry about all the values generated, let’s look at whether the first and second derivative were positive or negative at the point.

Having discussed the Intermediate Value Theorem, we have a discussion about where values of zero should appear on our table. For the first derivative, we reestablish a connection we made before, that if the slope of a function is positive, it must be increasing. A zero should mark the transition between increasing/decreasing or decreasing/increasing and these points are important enough to have names.

Next we have a discussion about the second derivative, which is a newcomer to the party. Some days before this activity, we plotted tangent lines, computed second derivatives, and looked at whether the tangent line was an overestimate or underestimate. That opened up the idea of concavity, that the concavity of a function plays a role in how accurate a tangent line will be.

Now it’s time to define concavity a little better. We look for points where the second derivative must be zero and what that could mean. At this point I’m talking with the table and graphs in view, so students can see that something is happening to f(x) at the point where there should be a zero on f’’(x).

Going back to their horizontal table, we now annotate the table with our findings. Based solely on sign value, we can quickly determine where a function is increasing, decreasing, concave up, concave down, and the role of the various critical numbers.

The purpose of all this is to improve a HUGE weakness I’ve seen over the years. For whatever reason, while I could get students sketching f, f’, and f’’ like geniuses, there was a disconnect between how they were making their sketches and what they represented. If a non-sketching question said something about the first derivative being positive, I’d get nothing but blank stares. Very few of them were able to determine that corresponded to increasing behavior.

By building this competency with tables AND graphs, I’m hoping things improve quite a bit. By sticking with equations of tangent lines and tables as recurring themes, I’m hoping free response style questions are more comfortable. It’s way too early to tell, but I’ve really liked how this is going.

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AuthorJonathan Claydon

College Algebra is an interesting course to teach because for the kids involved, the topics aren’t really new, but there are certainly new things they can discover within them, or get better insight hearing about something a second time. This last week we were starting an introduction to transformations. That prompted this bit of lesson planning:

Opening Acts

I used a set of three pre-built Desmos Activities with the group. I intended to use these last year but just never forced myself to do so.

Opener: Transformation Golf

Middle Innings: Translations with Coordinates

Closer: Practice with Symbols

For the ability level of the kids involved, these went really quickly. They are simple, straightforward activities but do present some interesting challenges. The kids really enjoyed transformation golf in particular.

It prompted a lot of good discussion and offered just enough challenge for everyone. We completed that activity in one 50 minute class period (about 40 minutes or actual working time). The other two (coordinates and symbols) were done in a single 50 minute class period. The combination of these three activities were just to job some memories and reacquaint with transformation vocabulary.

Proving Activity

A longer version came later, but using the polygon() tool in Desmos, we did a short proving behavior. We built polygons using a table, and applied some coordinate rules to those polygons. Students had to modify their polygon in 4 ways, writing down what they did. Then they submitted a link to their graph (my subtle way of teaching them how to sign-in with Desmos and save things). This took about 25-30 minutes of real class time:

I really liked the progression. Kids got to take a familiar skill and learn something new about the calculator. A few days later they did a more involved polygon transformation and applied what they learned to transformations of various parent functions (quadratic, absolute value, radical, natural log). The best part? These three days worth of lessons only took 15 minutes to map out thanks to the great resources in the Desmos Activity Library and the incredibly slick polygon command (launched only a few months ago).

Really happy with how all this came together.

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AuthorJonathan Claydon

Part of my Calculus procedure has been taking some benchmark data on my kids throughout the years. Other than improving student attitudes about Calculus, the second big priority is making sure students have an informed opinion about how they might do on the AP Exam. Kids are always free to do what they want, but I want to make sure if they're going to spend the money on that thing that they have a shot. Our results have been creeping upwards, and we are poised for a breakthrough, at least I hope so.

My data collection schemes have been problematic though. I think I've been a little too aggressive, giving questions that students probably aren't ready for in December. With the significant hurricane delay, we weren't even ready for what I've tried in the past, so I needed a new scheme. And with 75 students in AB, I needed something that'd be efficient to process so students could get feedback quickly.

Design

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

I wanted to test three things: Fundamentals (trig values, limits, continuity), Interpretation (curve sketching), and Skills (derivatives rules, Riemann sums). Roughly 12-20 items per section. I wasn't going to belabor any skills, if you can do it once you can do it ten times I figure. I sketched out what I wanted in each section:

To gather all the information, I was going to use Desmos Activity Builder. I didn't want to juggle a lot of papers, and I wanted a better idea of what items were causing problems. With previous benchmarks I had a vague idea of which questions didn't go well, this time I wanted to know for sure.

I included a mix of items: entering answers, typing short answers, multiple choice, plucking data off a graph, sketching on top of a graph, and some screens where problems were presented that students would complete on little cards they'd hand in. I wanted to assess their ability to determine a limit/derivative without making math entry fluency a limiting factor.

Implementation

I initially planned 3 versions with 6 codes, but the reality of sifting through all the dashboards made me reconsider. I settled on 3 versions with 3 codes, randomly distributed among my class periods. There were 44 screens total, and 25 kids on each code.

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The "version numbers" are just arbitrary hexadecimal numbers (go ahead, convert them, see how dumb I am) designed to obscure the number of versions. I was giving this to a lot of kids all day long over multiple days, I knew they were going to discuss it, but I wanted to make it a tiny bit less likely that they could figure out who they were sharing versions with.

Again for data collection simplicity, kids would access the same activity across multiple days. We use Chromebooks with school Google accounts, so linking their accounts to Desmos took 2 seconds and was done earlier in the year. I used pacing to restrict them to the section of the day:

This was one of those features I knew was going to come through, but didn't totally trust until I saw it in action. There wasn't a single technical issue over the three days. Each day the Activity Builder remembered the kid had previously accessed the activity and jumped them right to the section of the day. It was really elegant. Sketch slides with a trackpad still kinda stink, but I was not super critical of the results.

Data Use

At the completion of each day, I did some right/wrong (I was pretty unforgiving here) tallying in a spreadsheet, and determined raw scores for the various sections. I also tallied up incorrect answers to see how questions performed. I would eventually throw out the worst performing questions in each section:

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After three days of data collection, I set out to determine my final product. What were students going to get about their performance on this giant activity?

The intent of the activity was not to assign a grade based on their raw performance, merely to give them a snapshot of where they stood on December 11-13. Yes, this assessment would factor into their course grade somehow, but I wasn't going to cackle in delight as I failed tons of them, that's not what this was here to do.

Being able to click through dashboard screens and tally results was quicker than I thought, maybe 1 hour a day. Generating something meaningful from the data and formatting it nicely took another couple hours.

The other nice thing about this collection method is I could quickly check for version bias. Each of the three versions had questions that were identical, but others that were modified. Codes were distributed at random, and for whatever reason one version registered a higher average raw score. I curved the other two versions up, roughly 1.16x (normal College Board is 1.20x), so that the group average was the same as the highest average. I took the resulting adjusted score, divided it by max points available, which gave each student a percentage and quartile.

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From left to right: class period, fundamentals raw (max 20), curve sketching raw (max 13), skills raw (max 18), raw total (max 48, three questions were deleted), version adjusted total (if required), percentage, quartile. Average was right under 70%. Seven students earned 100%.

I told the students about 10 times, that the percentage was NOT their grade on the assessment, it was merely a tool to see where they landed in the overall population. My message was this is one data point in a series of many and that we would be doing these again. I also wanted to communicate that 1st and 2nd quarter implied you were doing a good job, 3rd quarter meant you needed to study more, and 4th quarter should have been a little wake up call.

After handing out the slips I floated around and had a quick chat with each kid, affirming their work or letting the lower ones know that this number was not a personal judgement, but that something more is required of them.

Conclusion

This went pretty well. The kids took it seriously, the majority of students did well, and I think all of them got useful information out of it. More importantly, this activity was easy to build, easy to manage, and easy to score.

A great experience start to finish.

Posted
AuthorJonathan Claydon

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 a math-based LMS, having access to notation is great. Here I asked AB a series of questions about derivative methods:

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
AuthorJonathan Claydon

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
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.

Posted
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.

Posted
AuthorJonathan Claydon

A series of seven posts on major turning points in my teaching career. A study of where I was, where I am, and where I'm headed.

ALL POSTS IN THIS SERIES

The way I have integrated technology into my practice has to be one of the most radical changes. Rapid progress in the industry and the way my school allocates funds to technology has caused me to reevaluate the tools I use with students on a yearly basis. There is a lot of stuff out there to try, and a lot of wild directions you can head down. Not all of it is an efficient use of your time. The greatest discovery I have made when it comes to classroom technology is to find a simple workflow or two and design projects that use it and use it well. A dozen discrete apps is not the answer, a dozen discrete applications within the same frame of operation produces far better results.

The emphasis of this discussion is on student devices. I have a number of technology bits that help me do my job, but many are unique to my situation and would be hard to apply at scale.

Where It Was

In 2009 we had little to offer students. I had an interactive whiteboard and some student response devices. In 2010 I made a concerted effort to make use of those things as best I could. They aren't bad at getting an idea of what everyone is thinking. Though in practice they had their inefficiencies. Waiting for everyone to submit took time, reliable communication with the devices wasn't guaranteed, and they reinforce an instructor centered model. Kids sit there and work something for a second in their rows and silently interact with me. I kept them around for a few years after before abandoning the whiteboard software. Here they are still kicking (though not in use) in 2012:

In 2012 we got our first deployment of iPads. I tried almost everything you could think of with iPads from 2012 through 2016. In the early days I was trying desperately to avoid the app that was going to put us all out of a job:

I knew that wasn't the future I wanted. We really did try everything. Kids took pictures and drew stuff on top of them. Kids worked problems in interactive PDFs I distributed to them. I tried all sorts of random apps to see if there was anything worthwhile. Eventually I realized I wanted my technology to answer a question. Why is this better than pencil and paper? What better products can this produce?

A year or two into my use of iPads I found the first piece that would answer this question. Technology can improve the speed and quality of our graphs:

We slowly gravitated to more graph-oriented tasks. In 2013 when I taught Algebra II it was the hallmark of the course. We could graph anything and everything quickly and efficiently with our in class devices. I got the iPads working with a printer so we could present our findings.

The iPads shined in this role. We didn't use them every day or even every week. But when it was time to make something nice, it was a workflow they could do well. More importantly, it was a workflow that got out the way. We spent very little time wrestling with technology headaches and most of our time doing something with the technology.

Where It Is

In 2016 I transitioned away from iPads to Chromebooks. In the years since the first iPad deployment my school district adopted Google accounts for all the students and the Google Docs platform had improved tremendously from my first frustrating efforts with it years ago. iPads also had a number of hurdles associated with them, most notably the regular maintenance.

As my stable of iPads grew and grew, the tedium of keeping them on the latest release with appropriately managed permissions just became too tedious. Chromebooks offloaded a lot of that responsibility and were built around the idea of multiple users.

Now when it's time to use technology things look like this:

The main workflow ideas remain. The computers give us the ability to make nice finished products. The computers are the best graphing tools we have available. Let's use them to make the best graphing products we can. At the same time, the computers have allowed a number of other opportunities.

I distribute instructions via Google Docs now, allowing me to be thorough with my expectations. The keyboard makes it easier to have students work through Desmos Activity Builder response questions. And a big feature this year was learning some rudimentary spreadsheet commands to generate information we could graph in Desmos. These computers were a regular feature of class. The workflows were so simple and known that every student understood the expectations when it was time to get a project done. We weren't wrestling with technical hurdles and I wasn't constantly trying to bend some hot new app to my needs. Simple worked and it worked well.

Where It Is Going

Access to computers presents some new opportunities. They have offered new use cases that weren't possible when working with iPads. Maintenance issues are a thing of the past and their batteries last forever. All the same it's the workflow that has shined.

As I continue, the focus is to keep technology use simple. What does it replace well and what does it let us that we couldn't before?

One area I might tackle next is assessment. I experimented a bit with these ideas. Collection and review of student work continues to be a problem. It can be difficult to scan and comment on student work in a meaningful well when collected. Math input on a computer is still nowhere near where it needs to be to replace the cost and efficiency of paper. Discussions are still best done in person than on a message board. One day that may not be true, we shall see.

Takeaway

If you are struggling with how to implement technology, start simple. Find one use case (in my case it was graphing) that you can work on. Come up with a simple procedure (make graph, screenshot, print) that you can implement a few times over the course of a school year. Expand your operation slowly and don't be overwhelmed with the latest new thing. Simple workflows will long outlast any fad app.

Posted
AuthorJonathan Claydon

A couple years ago at TMC15 I got a sneak peek at Desmos Activity Builder. At the time it was fairly limited but there was a lot of promise. At the time of its introduction I wasn't sure I could make use of it and that proved true. I didn't have access to enough devices and the iPads I did have were aging quickly and becoming a pain to manage. I attempted one of the first Marbleslide activities in early 2016 and the hardware just croaked.

Fast forward a bit and now I have access to a fleet of Chromebooks. The number of students who can bring a device from home has increased dramatically. iPad hardware in particularly has accelerated so rapidly in recent years that the struggle I saw before is gone.

I experimented with a few use cases this year, just to see what there was to see.

Match Me

Started simple. I took an activity I had done previously where students had various graphs on paper and had to recreate the pictures in Desmos. It looked like this:

Not bad, worked pretty well for a couple years. With Activity Builder I could work through the same idea but get students to add more detail and learn a bit more about the functions of the calculator.

Students could more in the matching realm, in this case finding a sin and cos function that matched the black line. Eventually they could create projects that included center lines, amplitude markers, etc.

Pretty good. Being able to build more complex prompts let more students know more fiddly details about the calculator. I liked that a lot.

Assessment

I used Google Forms quite a bit this year, and realized (well ok, Dan nudged me) that Activity Builder can be used to gather the same kind of information, though one screen at a time. Bonus, it understand math notation natively. I experimented on Calculus and used sketching screens for part of their final exam.

Sketching with the sub-par Chromebook trackpads isn't the best, but that can be remedied with some cheap wired mice. Pretty cool to see 89 sketches on top of each other.

I also used it for a two-fold assessment. Students were given access to a saved calculator with a bunch of regions on it, they had to determine expressions for the area or volume of that region, and then enter it in a separate Activity Builder.

Also cool. I really appreciate the detail that has gone into the teacher dashboard screen. Though there is room for improvement. Examining student responses screen by screen wound up being a little tedious here. Though I'm not sure a spreadsheet generated by a Google Form would've been any more efficient.

Going Further

I really liked what I learned using Activity Builder this year. Though Dylan Kane and others dropped some quality thought bombs on the subject. There can be a lot of silence while students work through these. Instant gratification may not lead to the most genuine student guesses. A subset of students may just hammer away at parameters until it works. I tried to counter the idea by requiring explanations after students had a chance to experiment. I think it helped a bit. Though Dylan's Conics activity is really something. You get no idea what your submission looks like until hitting a button.

There's a lot to think about here. The challenge of drawing a circle around a subset of dots is just brilliant. I need to bring more of this to the way I design activities.

Conclusion

Really excited to see how this evolves over the next year. There has been a lot of effort put into the feature and it's impressive how far it has come since I first saw it. Desmos curated activities are top notch. I think Activity of the Year should go to Jennifer Vadnais and her mini-golf game:

I know the intended audience is a younger crowd, but I had plenty of juniors and seniors cursing this thing. Bravo.

Posted
AuthorJonathan Claydon

Post AP test, I try to give the older ones some access to the kind of adult information they all hear whispers about but never learn until it's too late.

For the last few days we've spent some time on financial literacy. Specifically, building a spreadsheet that would make it easy to compare the real costs of owning a home, with some generous assumptions built in.

Using the (apparently secret) mortgage formula I dug up a few years ago, we built a tool. Each kid made a version of this. The input variables are the sale price and the appraisal value (sourced from databases local to the area). We plugged in some assumptions about property tax calculation and insurance. The point is to have them think about a house as more than the base line mortgage payment. And to see how ungodly high interest can pile up on a 15 or 30 year loan. Also, property tax? Say what?

This spreadsheet has other modules in it, but this first one served our purposes. It can also be extended to part two which involves buying a car and then ballparking what kind of monthly income it would take to afford the car and the house.

Task

They were given a link to the following instructions. The price range is mid-tier for this part of the city. Another assumption in the scenario is that we're about 15 years in the future and have acquired the cash to make a 20% down payment on something.

Note the mandatory fidget spinner.

Over in the corner I set up the bank. I had them make a few blind choices to acquire a random set of loan terms. Then they chose between a 15 year and 30 year option.

I had fun with it. They were required to begin the dialog with "Excuse me Mr. Generous Banker, may I have a loan please?" The bank was only open for 40 minutes (we had about 80). I'd get up and go on break at random, even in the middle of a transaction. Kids would play along and complain when there was a line. "There's always a line!" One regret is not having a bowl of lollipops.

I had a couple kids walk out and then yell that they didn't like the terms of their loan. I directed them to the complaint department.

Super fun task. A more complicated version might subdivide the loans based on the amount they're trying to borrow. I think that's how part 2 with car loans will go. Terms dependent on what kind of cash they're trying to drop. Kids are doing some dynamite stuff with their collection of materials. The hilarious part was listening to others watch someone get their loan terms at the bank and react with "ooh, nice one! or, oooh you got the bad one!" without any real idea what they were talking about. Listening to them chit chat about their interest rate was hilarious. "You know this is what adults do, right? This is what we talk about?" "Ew, gross."

Gross, indeed.

Posted
AuthorJonathan Claydon