# Assessment Upgrade

Last year in Calculus I gave kids an open note assessment on volume with respect to the x and y-axis. There were two handouts. The regions and the questions.

It took.......forever. I had a lot going on here. We had had a jam packed week where we talked about the area between curves, cross-sections stacked on top of the region, AND solids of revolution. Plus there was the x-axis/y-axis context to wrap your head around.

This year I streamlined it a bit. We took our time and focused on area between curves. We did some extensive practice on identifying expressions for various x-axis and y-axis based regions. That was a huge problem last year. My kids weren't comfortable with simply defining the area, so slapping the volume idea on top just added to confusion. Since defining the area is really the hard part of any volume expression, it was important to invest our time here. Afterward we focused solely on built-up solids with known cross sections. We can save rotational solids for later.

To stream line the assessment process I prepared the regions in Desmos and shared the collection with them. I also requested far less information about each region: a single integral expression for each one, either an area or a volume.

For collection I set up an Activity Builder where they can type their integral and its result for submission.

I'm hoping the more methodical approach helps here. Gathering input through Activity Builder is promising as well. Integrating functions with respect to y is probably my favorite Desmos feature these days.

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# What Teachers Seem to Want

After six years, I have put quite a bit of material online. I share tests, problem sets, lesson ideas, and have taken thousands of pictures. Some of that material has taken on a life of its own as teachers continue the great pursuit: what's something I can use tomorrow?

The teachers I watch on Twitter are in great contrast with the teachers I encounter locally. Little focus on the big issues, always looking for the next great idea starter or lesson resource. That seems to bear out in my statistics.

Nothing on this list was written in the last two years. A certain subset of you seem interested in the most recent thing I have to say, which is awesome, I'm glad you like what I have to share. A vast majority are after the archives, the stuff they could potentially use tomorrow. In most cases they're finding their way here from Pinterest, the post in question collected on a board labeled "Teaching Ideas," "Pre-Cal lessons," or something similar.

What does this mean? I don't know, but it seems like there is a constant interest in what other people have to offer. I know I still go looking for it myself:

If you have an archive that's organized in anyway, I suggest making it available, there is certainly an ever present demand. We all have something to learn from each other.

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# In Progress

Here's a little reverse jinx I pulled:

I say reverse jinx because as Februarys go, this one wasn't too bad. Normally I have an increased work load from being soccer season, the push towards AP Exams, and it's all compounded by abysmal weather. With an absolute lack of winter around here, the work load felt a little more tolerable.

While it appears I've been pretty quiet, I have a lot going on:

### Math Department

In October Varsity Math made a splash by taking over a blank wall in the hallway. It now features current class photos and our Hall of Fame. With a load of painting supplies on hand, I turned my attention to an old feature of our math department office.

It dates from the 80s, maybe? I took my painters and we're turning it into a unit circle:

A slow process (we have about 1 hour/week), but soon to be completed.

### Summer Camp

It's almost time to start thinking about things for Varsity Math Summer Camp. Despite the random nature of the topics (from my point of view), all the kids consistently said they enjoyed themselves. One of them even made use of some things we talked about to work on a physics lab this year. I've done the initial advertisements to Pre-Cal students, and several are already convinced this is the thing for them. For \$20 it's a pretty good deal.

Focus is the goal. Fewer topics and more time. I have a better idea of what you can accomplish in a 2.5 hour session now.

### Pre-Calculus

A big change is I have access to Chromebooks this year, making spreadsheets a more realistic tool at my disposal. I learned in Summer Camp that there's a real desire by kids to wrap their heads around spreadsheets. Most having no idea of all the math stuff you can do with them. Both Vectors and Polar Coordinates can make use of these. Recently we've walked through combining vector components and finding magnitudes and directions. A spreadsheet can do this quite nicely:

We jump over to Desmos, and with the help of super slick things like auto-connecting points and labels, we can quickly render our interaction:

It's almost time for the 6th Sidewalk Chalk Day. There's a possibility Calculus will get involved, allowing us to cover a truly massive amount of sidewalk.

Sidewalk Chalk means it's time for polar coordinates, a unit I started teaching after Vectors because so many of the concepts and math are identical. Traditionally I start polar coordinates with some hand calculation and plotting of points:

But computers are so much better at these things. Can we teach a computer to calculate polar coordinates? Let's used what we learned from vectors to speed up the process:

And thanks to the super bananas awesome data table pasting, we can get something far more sophisticated than our markers could accomplish:

Very excited to see how this goes.

### Calculus

Ugh, I don't know where to start here. It's time to register for the AP Exam, and as part of my five year plan, I said I wanted 80% (54 students) to register for the exam and have 30% pass. Well, determined to avoid the absolute fake out that happened to me last year (I had data to suggest that many many students would do well, it was wrong), I have tweaked the process. And well, ugh. But at the same time there are positives.

First, I learned I have a solid 20 kids who don't seem to have learned anything. It's late February. How did this happen? How much of that is my responsibility? At the same time, I have 25 who seemed to have learned everything. I'll spare you the details of my benchmarking calculations (the older a benchmark, the less it's weighted in a student's rating), but the data identified 8 highly proficient students last year. Using more difficult assignments, that same method has identified 16 individuals this year, with a higher average than the previous 8.

While there was a lot wrong about my methods last year, those top 8 all registered a 2+ on the exam. To have doubled that group is a positive.

The real disheartening thing is at the other end of the spectrum. 16 nailed it, another 9 did alright, and the remaining 55 are just wandering in the wilderness.

I have to make some hard decisions about what happens next and what is best for each student. My first year of teaching Calculus taught me that allowing kids to leave knowing nothing is a disservice. But slowing everyone down is a similar disservice.

### Calculus BC

We identified 16 individuals willing to start the first full Calculus BC course in the history of my school. They're excited. I'm excited.

### Classroom

Lastly, in 3-4 weeks I'm getting all new furniture. It will be a bit more flexible than what I have now but will still let me establishing the grouping methods I have come to like. More on that when it arrives.

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# Subtle Tweaks

Last year I had a pretty well thought out AP Review program. The intent was to give students exposure to the most common material. In the end, it proved somewhat effective, but time was the enemy. A big lesson I've learned in Calculus is knowing when to stop and let the students rehash through a bunch of material. Last year I gave them a huge stack of free response and integral/derivative mechanical work a few weeks before the exam hit. The eternal truth is that students take forever to do anything. As we progressed through the material students were waiting for my review sessions to get the answers straight from the source. Or they were working problems alongside the answer key, nodding along on the assumption they were doing all sorts of learning.

That learning didn't happen. I answered way too many questions. It was too late in the game to be answering so many rudimentary questions.

New strategy for this year, recognize that the review material I came up with is still good, but introduce it earlier. Force the kids to work through it before giving up the goods. Minimize questions. And more importantly, expose them to free response material in smaller batches, allowing them time to internalize some things.

Currently my kids are working through my big set of integral and derivative mechanics and two FRQs that are similar in scope. I allotted 3 class days plus a weekend for this. My observations so far is that kids are able to progress better on their own because a) the material is more current b) I haven't overloaded them and c) it's February and time is not yet the enemy.

If this goes well, I free up more precious time in April to let them work through a (now) less dense set of review material.

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# Agenda Days

You ever have a run where you feel like there are lots of loose ends? Assignments you haven't checked? Assignments kids haven't finished? Kids who have been in and out because of field trips or something? Or maybe all of the above and your personal to-do list has piled up?

From time to time I resolve this matter with an Agenda Day. For whatever reason Calculus benefits from the tactic to the most. The premise is pretty simple: no new material, an explicit agenda on the board with time frames.

For example:

At my school we stand at the door as kids enter. Having a prescribed agenda ready to go helps them get started without my usual "welcome to today's episode of math class" preamble. I constantly fight a feeling of being rushed, and taking a day to collect ourselves helps the problem from getting out of control.

I don't use this technique very often. It's usually at the end of a grading period, or after we had a bunch of assignments started or whatever. My only recommendation is offering gentle reminders "Hey yo, [x] [y] [z] you haven't talked to me yet!" throughout the process.

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# 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.

### The Task

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.

At the conclusion, I had collected a small mountain of tasks all ready to for display.

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.

### Radian-based Trig

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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