# Cautious Optimism

One semester of Calculus down, time to see if the rebuilding project is starting to get anywhere. Last year the theme was removing the AP Fear™ from my student's mindset. In previous iterations, students were walking into the AP Calc exam, freaking out, and leaving most of the thing blank. I confronted that problem last year in a number of ways, mock exams being part of it. Being very zen and chanting "calm down" being another part. Oh, and the whole Varsity Math thing. In 18 months taking an AP math class became the coolest thing at my school, go figure.

But can we live up to the hype?

This year I kept the mock exam idea but trotted out a crazy new curriculum. It involves multiple passes through the entire curriculum (we learned about Integrals in October) to build up the familiarity they're going to need in May. I upped the ante on the mock exams. They were going to be longer, focus on multiple choice content, and remove the multiple choices. You can either solve the problem or you can't, no relying on closing your eyes and making a guess.

I discussed this plan at length with people at TMC15. Everyone liked the intention but suggested the kids might need more time with the mock exam material. Last year we just stopped what we were doing and I gave them the exams, for realsies. This year I extended the number of days blocked out for mock AP stuff, and two of them were practice. Kids had the opportunity to discuss the problems with me and each other, and I posted solutions. Then they took a real one, once every six weeks. Before we left for break they took quite a long one.

Here's the interesting part:

Now, there are a LOT of variables involved. Big factors are the way the pacing has changed, and whatever advantages there are in having the 15-16 group a second time (90%+ took Pre-Cal with me). All that aside, I can't help but double take at the results. First of all, this particular 14-15 exam was given in March, the 15-16 equivalent (not a perfect equivalent, but close enough) the kids were ready for in December. That alone is like, whoa.

### Exams as End Goals

It's in the back of your mind just like it's in the back of mine. Why all this focus on exam results? Why let a test dictate what your kids get to learn and not learn about Calculus? A fair argument and one I think about daily.

Standardized exams are helpful in that they are, duh, standardized. The AP Calc exam is administered to a kajillion students all over the world with little variation. My students hit all the metrics you read about when it comes to exam bias. All the data says that because of socioeconomic factors outside of their control, they will do worse than their counterparts in other countries, or heck, on the other side of the freeway in the same school district. Screw that. I watch these kids every day, there's no reason they shouldn't have an equal chance.

If you read through enough iterations of the AP Calc exams, you can tell there is a real emphasis on the part of the exam writers to filter out the kids who memorize steps. You've got to know what you're doing. Vocabulary is huge. Context is huge. Abstraction is huge. Forever years ago as a first time Calculus student I would've killed for better conceptual understanding.

I think the way I've hacked up the curriculum brings that to my students. I want it to make sense for them now, not 15 years in the future, if ever. There is nothing special about Calculus that makes it off limits to individuals who don't "get it" or aren't "math people" or whatever straw man you want to use (driving every kid to take it is a separate issue). My goal is that they can take an intimidating (to them) subject like Caclulus and see how it's built on really simple foundations (our slogan should be: "wait, that's it?"). That kind of critical thinking will serve them well after the exam is over.

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# Neat

Normally I'd have a few items to analyze about the semester. But this was the smoothest semester I can recall. Even with a crazy plan for Calculus, here we are and I don't feel particularly exhausted. Why? If you total the number of hours I've spent actively teaching or working on some aspect of teaching (curriculum and big idea thoughts in the summer, primarily) you get a number that's pretty close to 10,000. That magic number where not only is there confidence in what you're doing, but you know whatever you come up with is going to work out pretty well. A corner has been turned.

A curve like this is a good illustration for students. My Calculus kids are always beside themselves with how difficult the material seems. It just comes down to reps. They're a few thousand hours shy of the turn. I tell them this, it seems to help. Not that I'd ever want to be on their side of the turn again, but still.

The bigger question, is there another steep climb in the future? Am I safe to just coast for the remaining 20 some years? I know there are still parts of my practice I can improve upon, but what's the scale of improvement necessary? Once you solve big questions like assessment, room arrangement, and day-to-day procedures, what are the other challenges?

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Whatever your level of engagement, math teachers on twitter have your back. I saw a retweet about a neat card game a few days ago. In the go go nature of the end of year I forgot how to find it. I sort of remembered the rules but wanted to double check. Naturally the thought didn't occur to me until about 45 minutes before I wanted to play.

The game worked great too. Definitely something I want to keep in my back pocket for a rainy day. Thanks to Sara for sharing! And of course to Kate the Great for saving the day.

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# Interesting Problem

Graphs of trig functions always hit at the end of the first semester, and I have a pretty good progression in place right now. However, my push for modernism bit me a little bit recently. Early on the semester I gave the kids the gift of Desmos. Many of them proactively would pull out their phone to verify their algebra with a quadratic system or what have you. Some of them were so clever in their choice of Desmos for the given problem that I didn't want to discourage them.

Flash forward a few weeks and I had questions like this on the last test of the semester:

And within a few minutes of reaching this section, I start getting THE question: "can I use Desmos?" to which I wasn't sure how to reply. Here they are, instantly realizing they know the best way to graph these things and would like to demonstrate that to me. If you write your own tests (I've written 92 Pre-Cal tests with multiple versions up until now), there's always the nagging question, what am I trying to test here?

On the surface, I'd say the important things I was looking for in these questions is do they understand what the parameters affect and how thing are modified by vertical translations. The statement part takes care of that. Other than that, in an era with such nice graphing tools, I'm not sure what else I was going for here.

I can tell you that it was obvious who really understood how to annotate the graphs and who didn't. The kids who just slapped the thing in Desmos and copied the picture were pretty obvious in their inability to show what the frequency of a function really meant. Many just randomly plotting points on the x-axis (one student graphed a function with a period of pi/2 and the only point labeled on their x-axis was a tick at 10pi, because...reasons). A different set did base their graph off Desmos but were able to note what should line up where on the x-axis and how to properly illustrate a vertical shift. And then a third set did the whole thing by hand, like the 90s would've wanted.

The lesson, I think, is that in the future I need to change the emphasis of this assessment section. The time for assessing graphing questions like this may have ended. We've reached the point where I can assume they'll know how to generate a graph given any sort of tool and it'll look the part. Now I need a higher concept wrapper around the subject. Things like "what equation would cancel the functions?" or "what equation would amplify the functions?"

I've always wanted to restructure Pre-Cal around bigger, higher concept ideas (ie instead of Trig Graphs we study signals, instead of vectors we study static mechanics, etc), and I think this moment is going to push me to finally do it.

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Last week, my power company wanted to cut me a deal.

What would I need to verify this is a good idea? A couple clicks to bring up my billing history solved the problem.

I spend about \$145 MORE over the course of the year with the savings plan. That's a real world math problem that I was interested in solving. The offer is tempting. No more power bill anxiety! Click!

Would you consider this financial literacy? I'd say so. No different than understanding a credit card cash back system in my opinion.

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# So Crazy It Might Work

This year is highlighted by a bold Calculus curriculum. I chopped everything up into random bits loosely based around a set of themes. I ditched the textbook and write my own homework. I'm grading on an A/B/NY scale. It's crazy! And I think, just maybe, we are making significant progress.

Reality has drifted away from expectations. Here's the curriculum progression as we close the first semester.

I didn't quite hit my marks. And a couple of things are getting pushed to later. Primarily because the first semester is so dense with material, the 2nd six weeks in particular. But, if you look really hard at what's coming up in semester two (4th and 5th six weeks), you can imagine that there will be lots of room to play with there.

The most positive aspect has been showing them what an integral is in October. It's made a world of difference as far as my ability to dive into the relationships between f, f prime, and f double prime. Also great to further connect position/velocity/acceleration when you can show them an abstract area under the curve that will produce the units of the function "above" it.

How does this compare to last year? I think I'm easily five weeks ahead. Our emphasis on the conceptual ideas has been huge. Mock AP performance has been through the roof too in comparison.

Fingers continue to be crossed.

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# Power Bills

My power company has some new offers for me. Among the variety of choices, they want to reduce the uncertainty that comes from seasonal power needs.

Seems like a good deal. My power bill is \$110.00 no matter how low I crank the AC in the summer or how high I crank the heater for the 1 week a year that you need those around here.

Should I take them up on the offer? Here's my last year of usage and the highest bill I paid highlighted.

What extra information would you want? What kind of discussion can you have around this? What's up with January?

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# Extra Time

A more clickbait headline would be Pinterest Greatest Hits. For whatever reason, these activities pop up time and time again in the list of things people pin from me.

With the end of the semester approaching, it's likely you might have a random dead day here and there. You might not think these activities are something high school kids would enjoy, but believe me, they eat these up.

### Hotel Snap

Something I do as an actual lesson is Hotel Snap. But it's good any time. Fawn provides a set of rules, but you can modify them to suit your needs. It's an interesting discussion in optimization.

### Straw Constructions

An old classic. I keep a few thousand coffee straws in my cabinet. It came in handy this year when a nasty thunderstorm rolled in a 7am and a bunch of kids couldn't get to school.

The number of straws is up to you, typically 20-30. Usually I have them support a tennis ball and the structure can't be attached to the table. I generally set a height requirement of 18 inches off the table surface. I give them anywhere between 25-30 minutes to do this. You can even try bridges or whatever really. It's interesting to see the variety of designs that will work.

### Forest Fires

Something I stole from my own 6th grade teacher. Hand out a forest (a blank grid really) and give each forest a key related to the roll of a die (1 = tree, 2 = blank, etc) but create a few different sets. I have four cards, ranging from 33 - 66% chance that a tree will be planted on the roll. The students roll the dice, and if it comes up tree, they color in the square. I have them complete two forests. It takes a while and is very LOUD.

After the forest is planted, have them light the left-most column on fire. Fire spreads left, right, up, and down but not diagonally. I have them spread the fire to its natural conclusion and count up the percentage of dead trees. The cards with the highly probability of planting a tree are more likely to burn everything to the ground. Imagine that. Tree survival rates are all over the place.

### 24 Game

Simple premise, with all levels of difficulty. Given four numbers and a set of operations (for the set I use, addition, multiplication, subtraction, and division), use each number once to make a total of 24.

I hand out about half a deck to each table, set a timer for 20 minutes and see how many they can solve as a group. The cards vary in difficulty. They get 1 point for the 1 dot, 2 pts for 2 dots, and 3 pts for 3 dots. Someone at each table keeps score and I put them on the honor system as far as the solutions go. You could require them to write them down if you wanted to.

I had a pair of 2 kids manage 17 points on their own and a set of 4 manage 50. For some kids, this is their jam.

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# Approximating Angles

Amidst the many things I do with Right Triangles, I added a bit last year to try an add an idea of what trends in trig ratios mean for angle values. Eventually we use the inverse buttons on the calculator, but it helps to add some process to the magic. Yes, the calculator can tell you, but how is the calculator coming to its conclusion?

First, have them crank out a trig table, 0-90, increments of 5:

A few will mention the patterns. I have introduced the Unit Circle at this point, so I've been hinting that trig is a pattern lovers dream come true. All the patterns! A lot find it interesting that sin and cos have the same values but in a different order.

Next I demonstrate how you'd use the table given a few dimensions of a right triangle. A good moment to discuss the relativity of the words "opposite" and "adjacent."

Random new addition, I gave them a challenge. Using a person (160-190cm) as the leg of a triangle, and with only a tape measure, can you plot out a triangle with a 20º, 50º, and 65º angle with the floor? I set them loose in the hallway to experiment and then had them determine how well they did.

I have them in groups of 5-6 all the time, so to ensure there was enough to do, they had to split their table into two teams.

Some interesting results. A lot of groups were able to get into the neighborhood pretty well. Some were consistently off (see: the Y2 45 40 43 and R2 51 57 48 squads up there). I overhead a lot of interesting strategies. Many correctly guessing that creating a 20º angle would require quite the triangle.

Now were they mindful of the table while doing this? Not really. Most used their own ideas of what 20º, 50º, and 65º looked like and used the table to verify the experiment. One intrepid group immediately sat on the floor and worked backwards from the table to determine everything in advance (W1 on the purple card). Another was similarly calculating, although didn't think about the work backwards part (G1 on the purple card).

For some added practice they tackled this the next day (PDF link):

A few weeks ago we used the tape measures for a linear speed challenge. Fun to see the "today's going to be GOOD" reactions when they saw them set out again.

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# Birthday Crowns

This started, like, 3 years ago? Something like that. At one point I found myself with access to colored paper and a birthday boy/girl in class. I made them a crown. Then it became a thing. Having kids for multiple years doesn't help this. Calculus expects them. I'm getting pretty good at crafting them.

Depending on the timing, it may or may not include an embarrassing rendition of the Happy Birthday song.

There's a giant duct tape shark on my floor. Don't act surprised by this.

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