# Polynomial Evolution

Another updated lesson from yesteryear enriched by Desmos and years of me tinkering around figuring out what I'm doing. This time, polynomials.

When I taught Algebra II, we spent some time with polynomials: identifying real roots and coming up with possible expressions. It was pretty scripted. I came up with the pictures, distributed a set, had the students pick a couple, copy a paragraph and fill in the blanks for their picture. It looked like this (circa 2012):

Being smarter about this sort of thing now, I removed the script and put the design requirements in the hands of the students. Using Desmos, they were to create a series of 4 polynomials (3rd+ degree with at least one 5th degree) and tell me about them. Two went in their notebook and two went on the wall.

Great products and great discussion along the way. Students also got to play around with scale factors, the secret sauce for making any polynomial graph remotely useful. Prior to this activity we talked about hand sketching the graphs and tried a few. Later on the relevant assessment, I reversed the idea to see if they could work backwards:

Questions like this teach me to trust the students more. Previously I'd fret about asking this sort of thing without explicitly talking about it (the point of the old version of my activity). But if you really want an assessment to do its job, you see what a student can do on the fly with the introductory pieces you have provided.

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# Estimation Wall 2.0

After three years, the mighty Estimation Wall returns.

I use Estimation 180 in my Pre-Calculus and Calculus classes. The kids love (and hate) these things. For every class that screamed "one more, one more, ONE MORE!!!" there was the boy who would see the task and lament "and now to guess and have my dreams crushed."

The kids are pretty familiar with the set up, so it wasn't hard to get them excited about making their own. Each student had to make 2 tasks using four pieces of paper, two for the questions and two for the answers. Pictures had to accompany each question and answer. The answer pictures needed to be presented in a way that no one would argue with the correct answer, or wouldn't be obscure to the point where the viewer would say "well, if you say so."

### The Players

I was going to use one 90 minute block period to crank out the tasks. Prior to the work day, kids were asked to fill out a Google Form telling me their tasks. They also needed to make a plan to take their pictures in advance or bring whatever they needed to school and take the pictures there. They had access to the Chromebooks and my printer.

### The Mess

On work day, we briefly went over expectations, and the room exploded into activity. I wish I had a time lapse to show you the random things going on. All sorts of tasks were being staged around the room, and kids were helping each other modify ideas when something didn't quite work out.

In total there were 114 tasks. I have two classes of Pre-Cal and each class took a bank of lockers for their display area. The work day was right before winter break. On the first day of the second semester, we took about 60 minutes of another 90 minute block day to assemble the walls.

See a question, make a guess, open the locker to check. Simple as can be. So much fun from start to finish. Calculus kids were a little upset they missed out on this one.

My favorite part of the whole thing? I just wandered around enjoying the sites. The kids didn't need much from me other than supplies and the occasional printer troubleshoot. We had spent an entire semester establishing a technology workflow that the kids knew well and worked well for us.

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# Four Phases

Let's say you've been in the business a while. You have routines you like probably a selection of go to activities. At the same time, from time to time you have a new idea and you're not sure if it's going to work out long term. For me, I'm never ready to give up on an idea until I've given it a chance to breathe. I go into understanding that I may not like the first attempt, and that in subsequent years it could make for something great. Forcing yourself to play out an idea over the course of three or four years is a hefty commitment. I thought I'd take a minute and talk through how I develop an activity to decide whether it makes the cut for the long haul.

### Phase 1: Experiment

In 2012 I was teaching Pre-Cal. We were looking at graphs of trigonometric functions. I showed students how to sketch them by hand using what they know about a function's amplitude and frequency. They each sketched two functions from a list I handed out and collected them in a poster by table:

Not bad, a little time consuming. After the activity concluded I decided the students weren't doing quite enough work and had no product demonstrating their thinking. A year later we try again.

### Phase 2: Refine

In 2013 we approached the topic again. I made some adjustments. The posters consumed a lot of space, so they're out. Students should do more than make a bare graph, so we'll make some simple statements about the function properties. The functions being graphed are still chosen from a predetermined list.

Two years of a similar plan showed me that I like this activity, but I don't love this activity. There's still not enough meat. I do all the thinking by predetermining the list of functions. Students miss out on playing with certain types of modifications simply because they didn't pick something off a list. We need a hard reboot.

### Phase 3: Redeploy

In 2014 I had a stronger plan. Students were going to do more of the mental lifting. This is the make or break year for the activity. Either this is going to have what I'm after or it's time to just forget and think of something else entirely. Students came up with six functions, all of which demonstrated increased/decreased amplitude and increased/decreased frequencies. They would make simple descriptions about which one was faster or slower. Bonus, we had iPads and a printer now, so the product could look a lot nicer.

Having the students come up with the functions was the winning piece here. There were good conversations and a lot of good clarifying questions about how to create what we were looking for. No more repetitions of the same functions over and over, each person had something unique. With that achieved, it's time to add features that will make this an activity I really look forward to.

### Phase 4: Keep or Scrap

In 2015 and 2016 this activity came back again but now it's really something great. Students design their own functions, have to write detailed descriptions, and now have more features to include in their pictures such as translations and centerlines.

2015

2016

The products this year are really something. My students from 2012 would scream at the demands of today's kids. You might be saying I iterated too slow and you might be right. Each year I picked a small aspect to change. I focused. If you're inexperienced writing an elaborate activity you've never watched play out before is highly likely to cause problems. Playing my trigonometric graphing activity out over the course of five school years has taught me what questions to anticipate, what concepts to stress more in our initial discussions, and what kind of structure kids need to safely string themselves out on a ledge and come back with something great.

The biggest advantage to slow, methodical iteration early on is that my subsequent activities have benefited tremendously. A new idea I have to today is far more likely to have the heavy lifting features I want by version two or even version one.

Start simple. Focus. Give yourself a big pat on the back when the kids of today help make the kids of three years from now even better.

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# Essential Pre-Calculus

Some more "why you Twitter" evidence here. Our gracious host for TMC14 is teaching Pre-Cal and every so often wants to know connections to Calculus. Maybe you do to?

I've taught Pre-Cal for a while now and Pre-Cal/Calculus concurrently for several years. I have learned so much about how to teach Pre-Cal by observing what really happens the next year (assuming AP Calc specifically). Sidenote: throw your on campus Stats teacher a bone and carve out some time for a Stats primer too, yeah?

In no particular order, here's what I have found to actually matter in Pre-Calculus:

### Rational Functions

The concept of discontinuity comes up here. The key points are horizontal/vertical asymptotes and removable discontinuities (holes). You can talk about continuity in general here. I describe holes as glitches that can be compensated for (there's a problem at x=1, but is that true at x=0.99999?).

### Piecewise Functions

Strictly for the notation and the idea that functions can have multiple behaviors. Piecewise functions aren't particularly practical things on their own other than giving you an opportunity to talk about increasing, decreasing, constant, and undefined behaviors over intervals. Piecewise functions pop up when determining if a function is continuous/differentiable at a point. It's also quite common to use piecewise functions to demonstrate the additive properties of integrals.

Trig is not the focus of Calculus, but it is an assumption that a student has a working knowledge of trig functions. They should know that sin/cos/tan/sec are things with a unique class of behavior. They should be able to spot the graphs of those four easily. They should also be aware that sin(Ï€/4) and other trig values of special angles can be simplified. This is about all the unit circle you need to know:

While students are in Pre-Cal with me they have free use of a full unit circle diagram. In Calc that's not the case and I have them learn this particular subset. It covers 95% of the random trig values you need to apply along the way. Knowing how to find sin(4Ï€) or cos(-Ï€) is a smart move too.

### Exponential/Logarithmic Functions

Exponentials and natural logs are real late-game Calculus, the curriculum has an affinity for them when it comes to separable differential equations. Also a fan favorite with volume problems where it may be necessary to solve something like y = e^(2x) for x. The general use case is a need to know how to integrate/derive functions that make use of them. I use these functions as a part of some algebra review, really just as a "hey, remember these things?" Basic knowledge of their graphs is useful.

### Polynomial Sketching

Function behavior and the connections between f, f', and f'' are the bulk of Calc AB. While it is not necessary to try and introduce the idea of a derivative, it is helpful if a student can roughly sketch f(x) = x (x - 3) (x + 5)^2, note the type of zeros present, and see how the function being even or odd helps them with the sketch. Solving some kooky polynomial for all its real/imaginary roots is not a thing that will happen.

### Limits

If you want to save your local Calc teacher some time, covering Limits is a great way to spend a week and a half at the end of the year. It's a good time to bring back continuity from Rational Functions and discuss how Limits are concerned with the "neighborhood" around a function. Removable discontinuities as merely a "glitch" gets justified here since we can now talk about how right-side and left-side behavior agree, letting us "remove" the offending point from the discussion. There is not need for an exhaustive study. Finding limits graphically, algebraically, one-sided, and at infinity (end behavior! horizontal asymptotes! same thing!) does the job.

### Stuff That Only Matters to Calc BC

If you cover all of the previous topics, you will still have a lot of time during the year. And you might be wondering "what about [blank] which so and so SWEARS we have to teach?" Well, it's probably in the list of things that only matter to students taking Calc BC. Roughly:

Vectors, polar coordinates, parametric functions, summations, converging/diverging series, partial fractions, and inverse trig functions.

Before you freak out, I am not saying you are free to ignore those topics completely. Among them are some of my favorite things to teach in Pre-Cal. Depending on the size of your potential Calc BC population, you aren't doing any harm in condensing or focusing these topics into something you like to teach. If partial fractions leave a bad taste in your mouth and someone's trying to tell you students will suffer if we don't spend three weeks on it, kindly tell them to chill out.

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# Room to Improve

I set a goal to start eliminating questions from my Pre-Cal assessments that could be defeated with modern tools. Sketching variations on sin x and cos x were the first target. Nowadays we write about them:

Next on the list is special angle trig, you know what I mean:

I do like these questions if you spend time emphasizing the role of right triangles in the unit circle (the aforementioned trig diagram in the instructions). Students will spend some time thinking about what the ratio is supposed to be and simplify it. However, this year I placed an emphasis on recognizing the decimal equivalents for these ratios (sqrt(3)/2 being approximately 0.866 for example). In theory these questions are defeated pretty quickly by a calculator.

I did like what I got from a discussion question about how right triangles played a role at 90Âº:

I suspect I could extend this more when it comes to evaluating sin 60Âº. How are you arriving at that answer? How are you interpreting the diagram to make that conclusion?

Pre-Cal is such a delicate balance of explanation and drill. Students have seen enough big kid math to start putting some previously distant ideas together, yet you want them to be good at the bare mechanics of it all too.

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# Electronic Finals

I know this is not a new concept, but as an experiment, I gave Calculus an electronic "final exam." I use quotes because with our exemption system it was not a real graded thing, but an activity they completed during the traditional final exam period. Several people I know out there have used Desmos Activity Builder to construct a proper final. With the additional of multiple choice and other screen types, it's a natural fit. You can also do similar stuff with a Google Form. I combined both ideas just to see what there was to see.

### Part 1: Form

Students were directed to a quick Google Form with 13 questions, 8 of them were specific to Calculus, the others were things like input on seating changes, feedback on their in class support, etc.

It's always interesting to ask kids about terms you use to refer to things. Although I swear I used the phrase "stack" when referring to f, f', and f'' enough that they should know what I'm talking about, several needed hints. The intent was to try the delivery method, not grade it, so providing hints or letting them talk about the questions was fine with me.

Exporting the answers to a spreadsheet when everyone had finished is great. I don't know how efficient it would be to grade a series of short answers in a spreadsheet, but probably no worse than combing through a stack of papers.

And of course, I got a proper answer to the most burning question:

I'm not too surprised.

### Part 2: Activity Building

Second, they were required to work through an Desmos Activity. The intent here was to check out how sketching slides worked with Chromebooks. The last big topic of the semester was curve sketching, so it was a no brainer to have them attempt this:

This 80 response overlay is impressive to look at, and gives you an idea of how many students went the wrong direction (graphing f''(x) instead of f(x)). It also pointed out the wobbly nature of drawing with a trackpad. I suppose buying a million regular mice would fix the problem, or touchscreens.

The unexpected "oooooooh" came from a response screen where students had to enter a math expression. I had no idea their responses could be grouped like this:

Interesting to see that there were several types of correct notation in use, and really nice to see how frequently certain kinds of errors are made (lack of dx for example). The actual grouping I got goes on for a little longer, and the errors start becoming more egregious, but you get the idea. Also unexpected and awesome that pressing the Anonymize button works here.

### Conclusion

Both aspects to this "final" went smoothly. No one had any trouble and the kids really wanted to stick with it when it came to sketching. You can probably chalk this up to classroom culture, but props to the kids for working on this despite knowing there would be no grade. If you can reach "do this because I would like you to do this, I'm interested in what happens" status with kids you've really got something. When we return in January I think I will show them the teacher overlays and see if they can figure out why people chose to sketch what they did.

In both cases my job was made easier. I was able to collect information fast and get it organized unlike the scraps of index cards I've used previously.

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# Day 100

Earlier this week was my 100th day on duty. I'm at school quite a bit. While walking through a common area the other day, a kid asked me if I was "low key homeless" because I can be seen at school all hours of the day it seems. Other than being a little tired, it's been fine. In fact, the main portion of the school day in my classroom is great. It's all the work that happens after hours to make that great that can be a bit of grind (looking at you endless cycle of AP benchmark typo correction).

Here are some things that I've been pushing in my classroom:

### Literacy

Pre-Cal assessment requires students to give a lot of explanations. As someone said on Twitter the other day, Pre-Cal is one of the first opportunities students have to see the skills they get from Algebra and Geometry put into practice. I have been really impressed with what some of my students have been bringing to the party here.

### Technology

We moved on from iPads and have a class set of Chromebooks now. I bought a massive enterprise level network printer and make it available to my students via Google CloudPrint. Through some magic on my classroom computer they can print from their phones as well. Pre-Cal students have an assigned Chromebook. They've done a number of activities in Desmos that are making them more fluent in the calculator than any of my previous groups. It is the second language of my classroom. Kids bust it out on their phone when they want to prove something. It's their preferred method of graphing. If you ever wondered what that "students consistently use technology in meaningful ways" objective on your technology goal rubric looks like, give us a visit.

### Presence

Varsity Math is here to say hello. I spot a kid in one of the t-shirts almost every day. They've got patches on their lanyards. I have random younger students asking me how to join. A small batch of students have just been invited to be our first ever straight BC class next year. As one kid put it in their end of semester survey "this is the only AP course I feel I have a chance to brag about."

Pre-Cal has something to say too. Several years ago I had some Algebra 2 students build their own estimation tasks. This year, during our final block period before the break, students brought in ideas for a brand new round of tasks (over 100). We made a giant mess. Kids came up with some awesome ideas. Early next semester we will install mega Estimation Wall 2.0.

### Acknowledgments

Shout out to the students for being awesome as always. Shout out to those of you in Twitter land who give me new ideas. Special shout out to Team Desmos for having a really polished product that just works.

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# Stacks

I teach integrals early in Calculus, super early really. Roughly two months ago I introduced the concept. It sounds a little crazy, but it's very handy when it comes time to do curve sketching. Somewhere along this way I coined the term "the stack" for the relationship between f, f', and f'' with derivatives and integrals serving at the means of transport between layers in the stack.

Previously, I've had students graph f, f', and f'' on separate axes, this year I changed it up. I don't really know if it's different or better, but it is more space efficient. Super bonus fact, I introduced curve sketching with a review of sketching polynomials (of the factored variety).

Here was the arts and crafts project of the week, given a graph of varying points in the stack, finish the stack:

Introducing integrals prior to this is helpful when having students determine if their stack is reasonable: if I start with a linear function of f'', I know integrating that function should yield a quadratic, and integrating again should give me a cubic. The followup activity involves interpreting the sketches: validating minimums, maximums, and points of inflection. A common narrative in Calculus is starting at an arbitrary point in the stack and asking students to interpret information about a different function entirely (given f', what's up with f?).

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# Struggle

One of those "teaching is such a grind" moments. I don't know how common this is, but I'm constantly thinking about my courses in the long term and super immediate short term. The march toward May constantly at odds with what we're able to accomplish in any given 50 minutes. Generally speaking I think I know how to help students learn something, but what gets internalized and how students make personal breakthroughs is just such a mystery.

Compounding the problem is an issue I think teachers fail to recognize sometimes, the ever growing gap between my fluency in a subject, and where students are when they get to me:

The longer you teach a subject, the greater your comfort level. At a certain point, you can recite the entire curriculum without reference, and walk someone through the connections to past and future courses. You can Calculus in your sleep.

Students don't have the benefit of running through Calculus over and over and over again. They're seeing it for the first time, expecting you to be their faithful guide. Their base fluency in math will be about the same year after year. And that's where it can become frustrating. You forget what it's like to learn derivatives for the first time, and what do you mean I have to explain integration again, don't you get it yet!?

I've never ranted out loud, but we've all been there, explaining something that just feels so simple for the 5th time to someone who isn't there yet. Constantly reminding myself of the fluency gap keeps me sane. You can't teach 15 years of math grind in 8 months.

And yet, the struggle remains real. Calculus has moments where we make immense progress, and then I set them loose on an assignment and I seem to be answering very basic questions over and over again. I have options. I can push forward because darn it we have a curriculum to get through, get mad and rant about their dedication, or find ways to give them time. In most cases the students are asking the questions from a genuine place. They really want to make the connection, they know the explanation was good but gosh darn it Mister why isn't this making sense to me?

I've run through the AP ringer twice now. I know the simple things are the enemy. It's not worth bothering with the complex scenarios and phrasing if they can't do the simple things. What good is recognizing an integral is necessary if you have no idea how to integrate?

So we stop. I give them a small mountain of derivatives and integrals to chew on for a week. They get better, and we slowly start moving forward again. Despite having to take a couple of pauses, we're still on track pretty well. But there's a balance here. You can't do this all the time. Kids need a push. It's very easy for them to say "we're good" and bring the pace to a crawl.

The data seems to indicate this is mutually beneficial. We've done two benchmarks in Calculus so far, and the students are doing about 15% better year over year. But it's easy to get excited about that data. Sure, averages are up, but what does that mean? Last year I felt super confident with my data, and got burned. I now read the numbers with a more skeptical eye. If only 10% of my students scraped together a decent performance last year, what does 15% better mean? At the same time, I know I've pushed this group harder and placed more emphasis on vocabulary. So what does 15% better on harder material mean? I have no idea.

The point is to say that it's very easy to obsess over the end goal, whether that's a district-level or campus-level goal, state testing or national testing watermark. Those end goals are good external motivators. But the struggle will remain. You've still got inexperienced learners who are going to take exactly the amount of time they need before they start improving their fluency. Kids just take forever sometimes, it's what they do. There are tools to speed this up, there are more efficient ways to teach a subject the next year, but if they just need a few days to work things out, are you willing to give it to them?

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# Group Assessment

Wendy was looking for help on assessment the other day:

My thoughts on assessing in Calculus are ever changing. I attempted to adapt it to two-attempt SBG with a colleague, it didn't work super great. Then we tried an SBG-ish hybrid system. Then I went A/B/Not Yet, and now their assessments are graded on an A/B/Not Yet scale but I don't do a lot of the grading.

For the purposes of reporting, I keep track of grades. The assessments are about once a week, sometimes longer. It takes a while to cover enough unique material in Calculus for it to be worth assessing, part of the reason double SBG clunked. Each one has two or three sections. These assessments are purely for mechanical stuff. I cover all the phrasing and conceptual stuff through AP style benchmarks. These sections are recorded individually and students can earn an A (95), B (85), or Not Yet (0 or 50). All the sections added together are worth 50% of the grade. I shoot for 5 or 6 in a grading period (six weeks).

### Practice

Early on I realized that little grades like this are a big pile of whatever. Since there are two and a half topics in Calculus we are constantly addressing the same things over and over, just refining our applications of them. We don't really have the full picture until April. I use these graded assessments as little checks along the way. What are we doing well? What could we do differently? What topics can students comfortably explain to one another?

The explanation part is what I want to get at. I incorporate a lot more discussion into assessment this year. We've done 5 so far and in each cases the students had a period of time where they could talk to each other about the task at hand. Sometimes the entire time, sometimes for only a few minutes. Then we'd discuss. Then I'd force them to go back and look at everything until they had a decent idea. No leaving questions blank or giving up. You aren't allowed to declare intellectual bankruptcy.

Grading is done by the students. I give them access to an answer key and they spent part of their time sifting through it. I ask for honesty in their ratings and I think for the most part I get it. Some students will ask for my opinion of their work before committing to a rating.

This is a time consuming process, a reason I minimize these assessments and stop doing them altogether at the end of January. Planning an assessment that is comprehensive, challenging, and completable in the time allotted is hard enough. Accounting for 10-15 minutes of discussion and 10-15 minutes of grading is equally difficult.

If you have a goal of assessing once a week or moving through a very long list of topics (like the way I do Pre-Cal), you will find this method rough. If you have a class that is pretty focused in scope and you have some flexibility in your time table, give it a shot.

The discussions are fascinating to listen to.

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