Another function relationship project, this time from BC Calculus. Prior to the break we were working with functions defined as the integral of another, and how you could figure out absolute minimums and maximums of a given function. There was a lot of interest in this topic from the AP test last year. A number of items designed to show fluency in function relationships didn’t go well for students at large. The issues involved being very particular with your evidence and how you use notation. At TMC 18 while you were playing games at game night, me and Dave Cesa were sitting in the corner talking about this. We know how to party.

Now, we went through these ideas at the very end of the semester, so to simplify things, we made an assumption that our function would have an initial condition of f(0) = 0 and that all accumulations would happen left to right. First thing for the new semester is revisiting that idea and being a little more flexible.

Here were the student instructions:

Students used Desmos to graph their function and compute integrals at any milestone point, local minimums or maximums of F(x). Students calculated the area of their discrete pieces and kept running totals along the x-axis. In tending to precision, they collected their data into a table of x and F(x) and used a series of integrals to show their computations.

While I’ve covered this topic in years past, I have not been as particular as I need to be when it comes to notation. A real struggle in FRQ for my students is showing their knowledge of notation. Often they will cut corners by not including it, or it will be written incorrectly (integrals without dx, for example).

Like their AB counterparts, students didn’t shy away from tricky situations, including points that would register as “fakes” that they could ignore when rectifying their totals. Desmos allowed us to construct polynomials easily and get a feel for using integral notation with named functions. In parallel, when computing integrals with Desmos I will have students replicate the process on their TI-89 to verify they are proficient with both.

AB Calc will repeat the project in the coming weeks and all of this is a good sign for better efforts with precision and argument structure.

Posted
AuthorJonathan Claydon

A huge problem in AB Calc has been an understanding of a function, its first derivative, and second derivative in a lot of contexts. Students have be able to infer the behavior of a function from the graph of f’, a table of f’, or just the equation of f’. Being able to construct all three elements when given one is a path to fluency.

Prior to break, students in AB Calculus had to create a polynomial that represented a first derivative. Based on that graph, they had to identify minimums, maximums, and points of inflection for the original function f. In addition, they highlighted the differences in concavity and when f could be expected to be increasing or decreasing. They translated their findings into tables of f’ and f’’ that validated their findings, gave justifications for their findings, and sketched f based on the derivative they created.

We were working on this fluency in class through a variety of prompts. Sometimes we started with a table, other times a picture, and others the equation. Here students created an equation and built out the whole process on their own. The results are a lot more polished than the version I tried last year.

Here’s the full write up students were given:

Most impressive to me is that students didn’t shy away from tricky to analyze functions. A lot of students created derivatives that would generate “fake” maximums, minimums, or points of inflection (where f’ or f’’ has a value of 0 but doesn’t complete the required sign change) and it can be tricky to do sketches based off that. But every student who attempted one was on the right track with their thinking. A few hit a common curve sketching snag of drawing an original function with the right features, but everything was upside down.

Collectively though, this group is doing a great job with the ideas.

Posted
AuthorJonathan Claydon

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.

Posted
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.

Posted
AuthorJonathan Claydon

A few weeks ago I was in a training and during a break I sketched out an idea I have been playing with for a long time. Can you make a single graph that can display all the big ideas of Calculus in such a way that students can work through how and why they relate to one another?

I finally set out to make it real while ticking away the hours before AP scores released (more on that punch in the face later). The result is pretty nice:

The starting function can be whatever and the x-interval is adjustable. On screen is the y-value at each end of the interval, the slope of the graph at each of the interval, a short line segment indicating the slope at each end of the interval, the area accumulated in the interval, and the value of the second derivative at any point in the interval.

The idea is to use this to have a discuss early on in the process about some properties before we put a name to them. That's why other than the y-values the other numbers are unlabeled. This combines a lot of half-hearted attempts at this I've made on the fly in the past.

I have found my students have big problems visualizing how all of this stuff works together. Concavity in particular is a weird one, what with positive/negative readouts at seemingly arbitrary locations.

Play around with it and edit as you please: Calculus in One Picture.

Posted
AuthorJonathan Claydon

Every spring, during a brief period of time when it's very pleasant outside before the ravages of summer, a few hundred students venture outside and make over a half mile of public art. We call it Sidewalk Chalk Day.

Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three
Sidewalk Chalk, the Fourth One
Sidewalk Chalk Five
Sidewalk Chalk Six

Now etched into school tradition, Sidewalk Chalk Day features students from all kinds of math classes displaying graphs of whatever it is they have been learning recently. As one faculty member who was strolling outside as we worked put it, "I always know it's spring when the chalk goes down." The first iteration involved two sections of Pre-Cal. This year 11 classes (AB Calculus, BC Calculus, Pre-Calculus PreAP, Pre-Calculus, and College Algebra) went outside throughout the school day.

Setup

Myself and a colleague pick the day a couple weeks in advance. It's trickier to pick a day than you might think. Because of our bell schedule, only Tuesdays and Fridays work, and, as the subject that started the movement, it has to coincide with Pre-Calculus classes wrapping up Polar Equations. Go to early and the material isn't covered thoroughly enough. Wait too long and suddenly it's testing season and nothing works.

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Day in mind, we incorporate a unit that will culminate with students graphing something. Pre-Calculus students create a pair of polar equations (rose curves/cardioids or something in between); AB Calc creates regions between curves with stated integrals for area and volume using that region; BC Calc creates polar equations with integrals of area for them; and College Algebra showed off logarithms and their corresponding inverses. Each student created their own set of equations/graphs that fit the requirements. They were given two panels (or one double panel) of sidewalk to show off their work. If complete, extra panels could be decorated however they like.

While students worked I flew around a drone and took pictures. Last year there I made a video compilation. The weather didn't cooperate this year, so I had to settle for pictures. Enjoy!

Posted
AuthorJonathan Claydon

As we headed into Spring Break, it was time for a project in AB Calc. I have been trying to stress curve relationships throughout the year. Early on we talked generics, how to indicate a minimum, maximum, and point of inflection. Then came integrals, absolute minimums, and absolute maximums. When we moved into position, velocity, and accleration, I hit those contexts again and again. By far, this is the biggest thing AB students need, how are the interactions between a function and its derivatives the same regardless of context.

Particle motion was a huge problem last year. I found myself having to do a lot of reteach prior to the AP exam over concepts I thought we had gotten (change in direction, particle at rest in particular). In combination with my new thoughts on FRQ, I'm trying to review as we go and combine ideas as much as possible.

Student's objective in this project was to create a polynomial, restrict it, and discuss a variety of properties. Specifically, the difference between a change in position and total distance traveled, where and how a particle is said to change direction, and how to determine what's required for a speed increase/decrease.

FULL WRITE UP

The projects were all fairly similar, but here's a particularly outstanding example:

Giving students ownership of the process had really been the best move for my projects. Because there were so many opportunities for variety, students were able to have great conversations about they were or were not meeting the requirements. They also learned a lot of handy Desmos along the way (shading, integrals, absolute value). Project days are my classroom at its very best. The whole room buzzes and I just get to float around assisting as necessary.

Students did outstanding on the free response questions related to this topic. Well, not like everyone got a 9/9 outstanding, but every student got the concepts and had something to contribute when it came to finding the solutions. I'm hoping to roll our review as we go strategy into rate functions next.

Posted
AuthorJonathan Claydon

We spent about six weeks on sequences and series in Calc BC. It was an interesting struggle for me personally as I last touched on the topic in 2003 taking Calculus II. The subject appears really intimidating, just search "convergence flowchart" and see what I mean. In the end, it was not nearly as complicated as I thought it would be and the kids and I got through it without any insane flowcharts.

By far the most interesting topic is Taylor/Maclaurin/power series. It's a fascinating bit of math that causes all sorts of problems, because the tiniest variation wrecks the usability of your series completely.

As we headed into Spring Break, I had the kids do a simple exploration of Maclaurin series in particular. Primarily because the series for sin/cos are so easily modified. Their objective: modify a sin/cos function in whatever way they want, iterate a Maclaurin series to four terms, name the iterations appropriately, and take a peak at their error when x = 1.

 FULL WRITE UP

The coolest part is I don't think the kids (or myself really) was prepared for how wild their constructions could be. A few kids picked functions that worked beautifully at x = 1.

A bigger group got ok results, but had errors over 35%

And then 3 of them picked functions that just went off the rails (the joys of exponents)

450% was hardly the worst. Another was off by 22000+% and the best ever was the student whose error was infinite.

This was a nice, simple project, and I think a nice little eye-opener about how math is sometimes just a shot in the dark.

Posted
AuthorJonathan Claydon

In December I took a measurement of my Calculus AB group on their knowledge of limits, derivatives, and curve sketching concepts. Students were grouped into quarters and got a score report. Recently, we did this again. Students have to sign up for their AP Exams by the end of next week and one of my stated goals is to make sure they have an informed opinion about what to do.

Design

A public version of the activity can be found here: Calculus Gauntlet 2 Public

I wanted to check a few things. First, could students determine when an integral or derivative was appropriate based on vocabulary clues. Second, could they handle data table, curve sketching, and function behavior free response questions. Third, a raw skills component, looking at their ability to find a series of derivatives and integrals. I also included a couple of multiple choice questions from practice AP exams. Students were given a review with all of this information.

Here's my planning checklist:

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There was a lot of reorganization as I built the activity, and some questions got cut for time. In the end I asked 13 vocab questions (choose integral or derivative for a scenario), a multi-part data table question (1 full FRQ), and released MC questions on integral composition on Day 1 (50 minutes). Day 2 (80 minutes) included multiple curve sketching questions (1 full FRQ), a multi-part questions where an arbitrary curve and areas are given (1 full FRQ), then a series of skills questions about product rule, quotient rule, and a selection of derivative and integral exercises completed on a notecard.

Implementation

The end result was a 42 (+4 slide homework survey) slide activity. I used Pacing to restrict student access for the various days, and could easily flip it on and off for a few absent students who needed to complete a part they missed. Props to Desmos for including a "randomize choices" option for MC questions. That wasn't available in December.

Having used the structure once before, kids needed no help getting started. With school Google accounts, the student.desmos.com website easily kept track of where they were and got them back in on Day 2 with no issues. To ensure uniformity of results, there was only one version this time. I wanted the data to be as accurate as possible, and didn't want have to fudge on account of versions. Whatever minor details students passed along as they talked about this in between days was a fine trade off. There was zero credible evidence that anyone might have done anything dishonest.

Data

Again, students were ranked and given an overall percentage. This time I improved their data strip to show how they did in each category. I'm hoping to use this as I construct AP Exam review modules, offering students more flexibility in what they choose to review.

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A couple of questions were dropped for technical reasons (written in a way that deviated substantially from how it was taught, causing a high incorrect rate). Students who got the question correct in spite of the error got to keep the point earned.

With 71 complete results: 60.58% average (top 10 students averaged 89.29%), 1st quarter 75%+, 2nd 59.82%+, 3rd 46.43%+, and 4th was anything below 46.43%. Students were asked to compare their results to a real AP scale (where a 5 is ~70%+). Other than dropping a couple questions, students percentages reflected their raw performance, no curve was applied.

I have given benchmarks every February for 4 years, here's the historical comparison:

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A ridiculous leap in performance that I think I can explain in a bit.

Generally passing AP scores come from students above the 80% line. I'm hoping we can expand that a bit. Since 2015, I have recommended the AP Exam for any student above the 45% line (2014 is an exception because I had no idea what I was doing). Per College Board policy, all students are free to take or not take the exam regardless of my recommendation.

Finding Promise

Previously, these February benchmarks were composed of questions from released AP Exams. Students struggled mightily with this task. I think time was a factor. Typically I would give a review about a week out, students would work on it in class and on their own. An answer key would be made available and then they'd take a multi-day assessment that was a shorter collection of released AP Exam questions. Results were poor because I think I had the wrong design goals in mind. AP Exams are a massive undertaking for students new to the material. There's generally a good reason most AP classes (even mine!) spend a month on review. A week just isn't enough time. In the end I think I wound up discouraging more kids that probably could've risen to the occasion given appropriate time to prepare.

This year I decided the most important thing was to look for promise. Yes, I would use real AP language and questions, but it would be incredibly focused. I stuck to the scenarios and skills we had covered in class, some of it full blown AP level, some of it not quite. I figure any student who can succeed here can be coached along through exam preparation.

And in this process I saw some great things from the kids. Those in the top worked very hard to stay at the top. Many many kids in the lower ranks redoubled their efforts to show me that December was a fluke. In making recommendations, students who jumped a quarter (or two or even three! some cases) got the benefit of the doubt. If they could improve that much between December and February, what if they were encouraged to keep working until May?

In the end I recommended 55 take the exam without question, 11 to consider it, and 8 to focus their effort on keeping up with classwork. Next week I'll know their final decisions. With a vast majority gearing up for this thing, I think that will provide the "we're all in this together" momentum that could make a difference.

Cautious optimism, as always.

Posted
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