# Sidewalk Chalk Seven

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.

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.

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!

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# Guesspionage

I have always liked introducing games into the classroom. Often I take trips to Target just to see what kind of new games they have that might work with kids. Some are relevant to math, and some are just fun to play in big groups. With the advent of internet connected devices everywhere, a new kind of genre has opened up, trivia games played on a console or computer that lets students enter a room code and participate in front of everyone for fame and fabulous prizes (well, not so much the prizes). You know, like Kahoot, but better. WAY better.

I have gotten endless value from the Jackbox Party Pack, specifically Party Pack 3. For \$25 you get 5 games, and two of them make FANTASTIC classroom games. Right before Spring Break, with a confluence of a blood drive, field trips, and general maybe-lets-not-introduce-something-new-right-before-a-giant-holiday, I played Guesspionage with all my classes. Participants take turns approximating answers to survey questions, the other players get an opportunity to decide whether the guess is much higher, higher, lower, or much lower. Points are awarded based on the accuracy of the guesser, and who correctly said higher or lower, with a bonus for much higher and much lower. In the final round, everyone is given the same question and has to choose the three most popular answers from a set of 9 (for example: which decade would people like to live in the most?).

Thanks to The Array™, I brought in my Nintendo Switch and we were off.

The questions are great. Every kid has an opinion and some of the answers will surprise you. There's also the collective FREAK OUT if a participant manages to get the question exactly right. Guesspionage allows 8 players to compete in the main game, with room for an audience who are also able to answer the questions. You can also set a family filter if you have a younger audience.

Here's a gameplay video:

Once kids understand the rules, the game runs itself. I just sit back and enjoy the arguing. When prompted to play a second round, every class was a unanimous "YES MORE GUESS NOW."

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# The Array

My most elaborate creation has got to be what I affectionately call The Array™. It takes a Mac mini, my dedicated teaching computer, and duplicates its screen all around the room to six little pods. It was constructed one piece at a time starting in 2011 and might be nearing its final form, well until fancier TVs get cheaper or school becomes a VR head set experience anyway.

Here's a schematic

Each TV serves as a focus point for discussion in each little pod. I can have something on display and reference it anywhere in the room. My ability to turn College Algebra into a no lecture zone works great thanks to The Array™. While working on computers, kids have access to the room's printer and can reference the screens for examples or instructions or whatever. I can't count the number of times I've caught kids teaching each other using their local screen. It makes my big classes tiny.

Key modification this year:

These screens used to sit on their own tables, limiting my mobility at the back of the room. I found wall mounts that use picture nails (a LOT of picture nails) to do the job of cumbersome bolts, allowing these TVs to hang in place without having to damage studs or do any significant hacking away at the walls. Though it took forever, routing the cables along the wall was the smartest thing I think I've ever done.

I'm not saying go out and build your own array, but finding ways to decentralize your classroom opens you up to all kinds of cool possibilities. Some of which I will elaborate on soon...

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# Time Off

Last school year was very demanding. Through a lot of circumstances I had a full teaching load plus coaching duties for two very time consuming sports. Prior to that school year, I had already made the decision to step back from my school responsibilities and call the 16-17 my last year coaching. Overall, it has proven to be a very good decision. Not only have I been able to focus on my teaching more, but I have been able to incorporate a lot more time off into my schedule. This is something that gets talked about a lot and remains the most important thing, you need to have set times where you say "no" and do something, anything, else other than work.

To co-op a discussion from the fall, here's my work load #yearinmath measured in Stress Units™

While teaching still remains demanding, mostly because I have to make all kinds of stuff, it has been incredibly more manageable this year. Two nights a week (Wednesday and Friday) I don't do anything school related after hours. Saturdays are generally off limits too. As much as I like to think about school constantly (and I'm usually thinking about school constantly), I don't act on any of it when it's a night off. Secondary to this, I have become comfortable in my ability to do the job. As I told a panel of aspiring student teachers a little while ago, eventually you can plan 3 completely different 90 minutes blocks in about 20 minutes.

Regardless of experience level, make sure you're forcing yourself to take time off. Whatever it is can wait, and whatever it is can be done faster than you think.

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# Rearrangement

I have three periods of Calc AB, one of them first thing in the morning. I had the same situation last year and that group was incredibly enthusiastic and ready to start the day. This year...not so much. Lots of tired faces and it takes a while to get them going. Plus, it's the smallest group, only 21 (compared to 25 and 28 in the others).

For the first semester (when it was only 20), they sat like this (the back tables seat 6, the front 7):

Conversations were limited, and performance was poor, scoring about 10% less than the other periods on our first benchmark. In general purpose assessments they were about 5-10% lower than the other classes as well. The table at the top left in particular did a lot of staring their paper, uninterested in finding assistance around them or even asking me.

It was clear things weren't working, so when January rolled around (and we gained 1 kid from another period), I forced them to do this:

It's a little cramped but it has made a world of difference. The low performing table is gone, and there is far more conversation happening when it's time for an assignment. When it came time for benchmark two, this class is now more in line with the others, off by a rounding error.

They always had access to each other, but forcing the issue and making them be more friendly has had the intended effect. Never be afraid to change it up when it's obvious something isn't working.

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# Speed

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.

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# Extreme Error

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.

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.

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# New FRQ Thoughts

We've finally covered enough material in AB Calculus to start tackling free response situations. In previous years I haven't done enough of this sort of thing and it's cost us. This year with a fewer days (we're probably ~1 week behind still) I wanted to integrate the language component of AP questions earlier and more often. Last year a big issue was the kinds of things kids wanted to review before the exam were concepts I thought we had nailed already. With fewer days, the less I have to reteach in late April, the better.

A few weeks ago students completed a benchmark in Desmos. We had covered enough material to handle curve sketching and data table FRQ situations. In the days since the benchmark, we have turned to position, velocity, acceleration, and general ideas about rate functions. I ended their previous testing system (short skills based items worth a max of 10 pts) and I'm now using FRQ scenarios as assessment piece.

Last Friday I gave them this:

A small hand out with a couple graphs was included. The idea was to get them used to the language and start getting used to how points are weighted in an FRQ. The class day previous to this, students were given arbitrary v(t) functions and we went through a series of 10 questions to cover the x/v/a relationship.

I gave them 40 minutes (though mentioned in reality they'd have about 30) with no notes, but open discussion. I have to say the results were night and day to previous years. I have had TONS of problems with students just leaving stuff blank, or being stumped by the phrasing. Not so this year. While not every student got 9/9 or anything, the conversations they were having were excellent and it was clear that the majority understand the concepts.

The 40 minutes ended and I collected their papers. Come Monday their papers were returned and they were given a colored pencil. Posted were all to see were some observations I made when flipping through the papers. I have made no marks on the papers. These observations served as feedback to the group at large.

With a colored pencil, they can use this information to update or amend their work. Then we'll look at the scoring rubric for each question. Finally, I will give them a new question (2009 #1) for them to complete. The plan is to turn this into an FRQ Fridays™ kind of thing where we can minimize how much we talk about these in late April. So far I like the process.

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# Second Mega Giant Desmos Assessment

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:

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.

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:

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.

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# Homework Survey

When I started teaching Calculus, I knew I wanted some regular form of homework. But I wanted to avoid something like "2-40, evens." Primarily, I think the way textbooks are designed don't allow for the kind of focused assignments I want. In 2015 I started rolling my own homework. The goal was something short and to the point that was on older material. I'd give it out on Monday, check it Friday, and provide a solution kids could check at their leisure.

Since implementing that system, my homework completion rates are huge. After 2 or 3, kids were used to the procedure of when it was due. The publicly posted key reduced the number of questions to almost zero, and the familiarity of the material kept kids from getting stuck and giving up.

Here's a sample assignment:

I skip weeks every so often depending on circumstances. So far I've given 12 assignments like this in AB Calc. This year, I finally decided to gather some data on homework habits. I had some ideas based on solution traffic (it's a bit.ly so I can track usage stats), but I wanted the real numbers.

I have the exact same system for BC, but their homework completion rate is 100% and it counts for a meaningless part of their grade, about 1% per assignment, 4 assignments per grading period. That crowd appreciates the opportunity for extra practice a little differently.

### Survey Questions

With 70 AB Calc respondents, here's what the kids had to say:

### Interesting Insight

Strangely, the majority of kids doing homework on the day it's due are in my 1st period. Though perhaps if they stay up past midnight they counted that as Friday?

The even clustering of solution users is fascinating. I thought it'd be more consistent. I really don't think there's a good or bad takeaway from these percentages. I post the solutions intending for the kids to use them, and most of them use them most of the time.

The four kids who said the assignments are too long are the ones who consistently don't do them in the first place, which is kind of funny.

The two kids who admitted to 5+ (out of 12) copies are incidentally mediocre in overall classwork. My eyes popped a little when I saw how many had copied at least once.

### Takeaways

In general, the system seems to be working as intended. A big majority are doing the assignments and checking their work. A small minority are abusing the system. At roughly 3% per assignment (~4 assignments per grading period), a student does not gain a whole lot from a blind copy.

You could argue I could make changes. Increase the weight or increase the frequency but it's likely to only amplify what is currently a small problem. Increase the frequency and it loses meaning (being seen as constant busy work). Increase the weight and it becomes punishment.

Probably the only clever thing would be to introduce a check digit of sorts into my solutions. I could add one intentional wrong/weird thing that'd signal a blind copy. Though I make natural errors with enough frequency that it could be hard to remember what was supposed to be wrong.

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