# 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(π/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(4π) or cos(-π) 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 90º:

I suspect I could extend this more when it comes to evaluating sin 60º. 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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# Baby Shark

Here's a silly classroom thing to brighten your mood.

Last summer I gathered a couple groups of 11th graders together in June for Summer Camp. At some point during the first week, related in no way, I was watching an episode of Casey Neistat's vlogs. In the middle of an episode, for a whopping two seconds or so, he has a moment where his daughter is watching silly kid song videos. Specifically, Baby Shark:

I wish I could remember the episode (and I spent a long time trying to find it again), and the context of the discussion that lead to this. But towards the end of Summer Camp Session 1, I played Baby Shark for the kids. The next day they demanded more. Then on the last day I asked for feedback:

This song is old. It's some traditional summer camp song that I had never heard. But it's infectious, apparently, because by the end of Session 2, kids had learned the dance, were showing their younger siblings, and were bringing me videos of little cousins dancing along to this shark song.

School year starts. Summer camp kids are like, PLAY THE BABY SHARK. The rest are befuddled. But it spreads. Soon the whole cheerleading squad knows the song. Kids in band are singing it to each other. Kids in band who have no idea about the origin are singing it to each other and doing the dance moves. Cheerleaders are CHASING ME DOWN singing the song. Heck, I'm driving the volleyball team back from an event and the older ones sing it FULL VOLUME on the bus ride home. Later on one of my 9th grade players (who would not know the origin) sings it at random in practice.

MISTER THERE'S A HALLOWEEN VERSION

MISTER THERE'S A CLUB VERSION

Now they're asking when the Thanksgiving and Christmas versions come out.

The drum major arranged the song for multiple instruments. He video taped himself and four others playing it on brass instruments. He learned to play it on the guitar. Another student learned it on the piano. The two volunteered to play it in the talent show while a bunch of goof balls dance around in shark costumes (and omg was it not hard to find volunteers for THAT).

Not done.

There was a fall festival at a local elementary school, some of my students were volunteering and MISTER THEY HAD STUFFED SHARKS AND I TOOK SOME FOR YOU.

The kids take turns with them at their table. One kid had them dancing along to Michael Jackson while I was playing music. Another boy, I swear I am not kidding, uses one as a comfort animal because it helps him concentrate (and IT WORKS).

The whole thing has just taken on a life of its own. I just don't even know what to say. Kids are great. I'm not sure what this has to do with teaching math, but it seems to help.

Don't underestimate the selling point of silly things.

Unrelated duct tape shark from long before the madness.

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# Other Ways

Since my Algebra II experiment years ago, I've been obsessed with students to start making connections between graphs and algebra. Fairly consistently I have students who dismiss graphing and something uninteresting because no one seems to making connections about why it's so interesting. That's it's possible the "x= " they're able to find with a calculator represents something. Pre-Cal has some algebra objectives and I've brought what I learned with that experiment along for the ride.

In addition to emphasizing graphing, I've been demanding a lot of explanations from my kids this year. A go to for a while now has been justifying a situation. Recently I gave kids a graph and three possibilities. In class we focused on the quadratic formula, it's super solving powers, and it's ability to show you the x-values of the intersection points for a pair of functions. In the setup for this question, I didn't specifically ask them to use this method, though most went with that approach. Here's a representative sample:

The brute force method. The student solved all three and made a connection to the picture with their work. Nothing wrong with it.

However, some found other ways:

An excellent observation of graph transformations. The functions used were simple enough that spotting some features was a quick way to accomplish the task.

The next student made a similar find:

Not only did they attempt a quadratic formula solution (apparently done on the calculator and not recorded) but they weren't satisfied that the mere presence of an imaginary number would do the trick. The slope of the line was important!

This last one might be my favorite:

Not only do they grind through the quadratic formula here, but they remember that if those intersection points are solutions, they are useful! I think only one other student (out of ~60) tried this out.

I was pleasantly surprised because I only had one method in mind when designing the question, forgetting that kids could wander in so many interesting directions.

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# Presence

The hard part with new initiatives is keeping them going. Sometimes building and maintaining momentum is easy. Five years ago I decided to make notebooks a thing, for example. Am I as fired up about them as I was years ago when I figured everything out? Not exactly. But it's still an idea that needs my support. It continues to be a proven system. It'll last for decades. Take something different, like technology implementation. How do you keep up a strong technology presence for five years? ten years?

Ditching procedures if they no longer motivate you is easy. The thing about school years is your students move on, the knowledge of your procedures with them. My current groups are clueless about how things worked years ago.

What about bigger, organizational goals? When I took over Calculus I had some problems I wanted to solve. By accident it spawned Varsity Math.

New ideas are awesome, but then you realize they come with maintenance. Varsity Math is now a living thing that needs attention or it'll just disappear. How do we make it more permanent? A permanent reminder to myself "HEY KEEP DOING THIS THING."

There was a neglected corner down the hall that never had anything interesting on it. A random gap between unused lockers. For years and years I had visions of putting some kind of installation in place, some legacy to the work I love. Other parts of the school have great art installations from decades past, I wanted something to contribute to the line up. The Estimation Wall was a good first step, but temporary.

Presenting the Hall of Fame.

Painting begins. A day later random passersby are buzzing about the maroon wall.

Piecing together the enormous stencil created with a few hundred 8.5x11 pages.

Arguably the easiest part of stencil construction.

"They painted the wall? What are they doing?" "They" have something big in mind...

Some wall finish technicalities required improvisation to draw the 80" circle.

Hours of letter cutting, taping, and gluing we're ready for the final step.

All done!

This project was completed off and on from October 6 to October 31. It involved the effort of a couple dozen students in my Calculus classes working on random bits at a time. One crew did the maroon base coat. Another did the initial assembly of the letters into something coherent. Others cut and glued the assembly onto stencils boards. Others spent hours hand cutting each letter from the boards. And a final group jumped in for white paint and touch ups. In case you were curious, the capital letters are 2051.94 points tall.

It's awesome, and it's ours.

Class pictures of each Varsity Math crew (including Summer Camp) will go here as well as a notice board of students who pass the exam. I want decades of classes on here.

Coincidentally I saw a former student from the original crew right before this was completed and I showed her pictures of this. She couldn't believe what she help start.

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