Last year I rattled through some frequently asked questions.

People point to me as a user of interactive notebooks. I would not call my notebooks interactive in the same category as what the internet considers an interactive notebook. If you're looking for foldables, tables of contents, and scripted notebook pages, please go check out Sarah Hagan. She even had a whole day celebrating her.

Anyway, this is what I do to get students to do stuff. Time for bullet points!

Gallery

Here's some pictures. I'll do a little explaining after.

Structure

  • composition books preferred, hold up better over time
  • students can keep them in the room, I use a 30 qt. tub for each class
  • colored duct tape identifies the class period
  • a manilla folder fragment stapled to the back holds old tests
  • students track their SBG scores in the front, it's not a table of contents

Procedures

  • students use it every day, no questions
  • all classwork and notes are contained in the book
  • from time to time work is pre-typed and handed out on 1/3 or 1/2 sheets
  • students do a lot of work copied from the board
  • students don't have homework, they work when they are in class
  • students can use their books on tests, tests are designed to be about thinking, not parroting, we don't do review days
  • class time is intended to provide as much student time as possible, stop talking
  • books are checked every 3 weeks for updated SBG charts and cleanliness
  • students can put things wherever they want
  • assignments are graded a day or so after completion, I walk around with a clipboard while students are working on something newer
  • it takes time to cut things out and glue, I mean, just accept it
  • I never take them home
  • kids forget them, it happens, usually not a chronic issue

Critics of the notebook thing dislike the time required for the cutting, taping, gluing, and preparation. They'd prefer a binder full of daily worksheets or something. Mine don't suffer those issues. I don't find myself spending an inordinate amount of time waiting on kids to do stuff. Of course, I've set up class to be for them. You don't see me talking much. Assignments are short and sweet so they can do all the cutting and still accomplish something. I'm allergic to full page handouts. Over time it's just so much paper. That volume of copying is just unnecessary to me. Same with packets. Yuck. One year I'm going to count how little I copy as proof.

After five years with these things I'd never do binders or folders, ever. Student productivity is at ridiculous levels since I made it a requirement.

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This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students define sin, cos, tan, csc, sec, and cot as the ratio of sides in a right triangle. Students identify certain special ratios for 30º, 45º, and 60º angles and organize them in the unit circle. Students can determine unknown sides and angles in a right triangle. Students should see that the size of an angle has a direct or inverse relationship to the value of certain side ratios.

Progression

Other good resources on the topic come from Sam Shah (with a follow up) and Kate Nowak.

My approach and theirs have some similar messages. Students should understand that trig ratios mean something and that while 30/45/60 ratios are considered special, there's nothing particularly special about them other than reducing to nice-ish numbers.

First, I start with the "special" triangles. Have them draw a set of right triangles (4-5) of arbitrary size for 30/60/90, 60/30/90, and 45/45/90. Establish the definition of the opposite, adjacent, and hypotenuse. Measure the ratio of opposite/hypotenuse and adjacent/hypotenuse for your triangles and record them in a table.

They should see the ratios are the same for an angle regardless of triangle size. We use the three triangles and build a collection we call the Unit Circle where the hypotenuse is 1. I offer them things like root(2)/2 as the exact value of the ~0.71 number they see. We define opposite/hypotenuse as the sine of an angle, adjacent/hypotenuse as the cosine. It's important to take a moment and talk about opposite and adjacent as relative definitions. Some will catch that, others might not. They spend a week finding ratios on the unit circle and defining them as sin/cos/tan/csc/sec/cot.

Now we take a known angle and find the missing side of a right triangle. This is an exercise in vocabulary. Given the location of angle, what is the relative definition of the three sides? What trig ratio is relevant to the given information? This is the first time they compute sin/cos/tan values with a calculator. You could choose not to and save it for later.

Next, using a calculator I have them complete a table of values for sin/cos/tan of the angles 0 to 90 in increments of 5 (this is similar to Sam's large packet of triangles to build a database of ratio values). I give them a set of right triangles of arbitrary size and they use their table of ratios to approximate the angles. I avoid using the inverse trig functions on the calculator until we start talking about trig equations later on. I want them to make a connection between ratio value and angle value. I want them to have an idea of sin and cos trends to help when we graph those functions later.

Lastly, they create a set of right triangles, one in each quadrant, of arbitrary size where they demonstrate the values of the six trig functions for one interior angle and use those ratios to approximate the value of the interior angle.

They have an assignment with a set of right triangles where sides and angles are missing and they have to find the missing information using any of the methods we discussed.

Activities

Most of their work through this unit is a series of activities. Here's a picture of the arbitrary special right triangles and the table of ratios:

Here's a picture of the ratio table they complete with the calculator:

Here's a picture of the final assignment, an arbitrary set of right triangles with all ratios computed and angles approximated:

Another option is the classic "how tall is the tall thing?" activity. Take them somewhere where you have some tall objects. Have them measure 1 side and 1 angle. Use a relevant trig ratio to approximate the height of the object and compare results.

Assessment

Triangles cover 5 topics in my SBG system. Student demonstrate knowledge similar to classwork and in a couple cases are asked to draw triangles oriented correctly in a quadrant, label sides based on a defined trig ratio, and determine the rest of the information.

Test 7, Test 8, Test 9

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There are rumors big scale Chromebook deployments could come to the district. There were some pilot programs last year in English IV and Government. I was given a Dell Chromebook 11 to see what possibilities there were for high school math.

Hardware

My primary point of comparison is an iPad. The Chromebook I used sells for $249. You can get refurbished iPad minis for that price, but not with retina screens. Definite price advantage. The keyboard is fine. The case and hinge feel sturdy. Battery lasts for quite a while, with infrequent use would only need a charge once a week. But man, the trackpad is no good. Plugging in a mouse was necessary.

Software

Google Drive and its companion apps are definitely meant for a desktop browser. If given a set of these, I could have students save a lot more of their work. I'm still not sold on handing things in digitally. I like physical products. If anything being able to save things like custom trig graphs or piecewise functions would let students spend time on them outside the classroom if they don't finish. I could add some more elaborate write up portions perhaps?

I couldn't try out Google Classroom. It requires inviting accounts enabled for it through Google Apps for Education. And I have no children at the moment. It doesn't seem to be much more than a central point for students to grab things from you. The add-on Doctopus appears to offer similar management features. I didn't explore add-ons much.

Printing was among my major questions. Google handles this through Cloud Print, a work in progress. In theory you tie a wireless printer or a printer tethered to a desktop you control to your Google account and any subsequent device you sign into can print to that device. I successfully sent stuff to the printer many times, but some OS X weirdness prevented it from working. I don't blame Google here. Everything on their end seemed to work.

Products

I set out to reproduce things I've done with iPads. Regardless of hardware, when it comes to technology the application has got to really trump the quickness of pencil/marker and paper. Remaking things you can do in Desmos on an iPad was simple. My experience with Desmos classroom activities was no different than with an iPad.

Clockwise from top left: Identify a region between curves; determine linear equations for a street map; import data and perform a regression; prepping Desmos output for print

The Chromebook excelled in two scenarios. For a couple years I had Pre-Cal kids wander around and take pictures of random right triangles, dimension them, and then give me a the set of trig ratios associated with an angle in the triangle. The drawing tools in Drive make for some better results. A student submission from that version is on the left, my recreation is on the right.

The desktop implementation of Drive is what wins here. You have do a lot of import/export steps on the iPad to get it sent to the right place, part of the reason I abandoned this idea.

The second improvement came with Geogebra. I don't have much experience with it, but I've known it to be a little fiddly and it works better with a mouse. Were I geometry teacher, I'd be very happy at the thought of student computers capable of using it. Geogebra installs a Chrome app (nothing but a bookmark really) version of itself.

The 3D graphing features seem worthwhile. It could offer a lot of enhancement to the modest discussion of 3D I'm able to do today.

Conclusion

At first pass, a Chromebook would allow me to do the same things I can do with my iPads. As stated before, the best thing I've found for technology is off loading the heavy lifting parts of graphing. That's roughly 10 to 12 class days (once every 4-5 weeks) throughout the course of the year. Putting a desktop experience in a cheap package gives me a chance to do those activities and a little more. They still wouldn't come out everyday.

There are easy ways to modify some of my established practices. Students could do more writing. Students could submit projects to me instead of printing them out. Students have all the benefits of the internet and we could find some research angles. Students could do a lot more self-paced learning (like watching tutorial videos! just seeing who's still with me...) There's tons of unexplored potential with Geogebra and I just teach the wrong subject to exploit it. But, the whole experience doesn't do much to make the pencil and paper needs of math class better or obsolete.

If the testing conglomerate every modernizes I could see using these things all the time in Calculus. A notebook, pencil, and Chromebook/iPad is not a bad way to do math.

So are 1:1 Chromebooks a good idea for math? As part of a larger deployment where ELA/SS have gone paperless, sure. Exclusively for math? Just give me a cart.

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AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students should be comfortable with function notation. Students should see the division of two functions as a ratio. Students should be able to discuss the difference between an asymptote and removable discontinuity. Students should be able to understand why they occur and how they appear on a graph.

Progression

There is a lot here. I talk a lot about algebra, a lot about graphs, and try to cycle between them frequently.

Start where we left off in composite functions. We discussed addition, subtraction, and multiplication but have ignored division. I specify division as the ratio of two independent function and coin the term rational as shorthand. Define some functions f(x) and g(x), both factorable quadratics, and such that the ratio will generate some errors. Have the students create a table of x values from -10 to 10. Have them evaluate f(x) and g(x) for the length of the table. Then evaluate the ratio of f(x) / g(x). They should see a g(x) value of 0 causes a problem. Glenn Waddell has a prepackaged alternative version of this in Desmos.

Graph the ratio of f(x) / g(x) with Desmos (either individually or just you) to provide some explanation for the errors. At this point I'll define asymptote and removable discontinuity.

Now we get into why these errors happen, why some are considered removable, and how you could find their location without a graph. I prepare a number of factorable quadratics as ratios and have them simplify them through factoring to see if any share a binomial in the denominator and numerator. It is tempting to say that if you get something like (x - 3)(x + 4) / (x + 4)(x - 7) that the ratio will fail at -4 and 7 because those are the opposites. It is much better to phrase it as "what value of x will make the binomial zero?" I'm careful to say divide out and not cancel out as well when defining x = -4 as a hole. 

Activities

Graph a Rational without a Calculator: present them with a ratio made of factorable quadratics, have them identify the location of any asymptotes or removable discontinuities. Use these locations as boundaries. Mark the asymptote with a dotted line and notch a spot on the axis where the hole should appear. Use test points on either side of the asymptote and hole to see if the graph is generally positive or negative. Draw curves that follow that behavior.

Identify a ratio from a graph: Provide some graphs of rational functions. Have them work backwards from the location of the asymptotes/holes to write out the possible denominator. Fill in what's possible based on the presence of a hole. Use the idea of a placeholder in the numerator to signify the lack of additional information. Ex: unknown A / (x + 3)(x - 9) for something with two asymptotes but no holes.

Design a Rational: make up two ratios that demonstrate the characteristics we have seen. Demonstrate a function with multiple asymptotes and one of them should have at least one asymptote and a hole. Label the graph with the corresponding ratio and explain why the features appear where they do.

Assessment

Students are tested on mechanics: identifying the location of an asymptote or removable discontinuity from a ratio of factorable quadratics/cubics. Students are tested on context by being given a ratio and possible graph of that ratio and have to develop an argument as to whether or not that graph could represent the ratio. A few students did a fantastic job with those questions. This is tested as two SBG topics.

Test 5, Test 6

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AuthorJonathan Claydon

Lots of people leave the classroom all the time. Some cite unfinished business in their own education, others want to have bigger macro-level impacts, and some write an op-ed for The Atlantic about how poor and untrusted we all are and how they walked out in frustration.

Maybe I'm weird, but I like being exactly where I am. At the moment my only unfinished business relates to making my teaching a tiny bit better each year. I like playing with curriculum, I like playing with procedures, and I really like the day in and day out of running a room. I want the chance to teach 30 years' worth of students at the same school. Call it old fashioned loyalty.

But why stay? Like others I'm sure I could be of great service in an education graduate program (having never taken 1 education class, shhh....). I could become an instructional coach. I could try to solve some of the capital B Big Problems with education. I could try to yell louder about how wrong people are about edtech.

All of those prevent me from doing what I like the most, hanging out with five groups of kids in math class every day. My job is not the grind of rattling off the same slides every day, handing out the same worksheet for 10 years, or grading a test guarded under lock and key, all while muttering under my breath about how kids today are doomed. My job is watching kids be goofy, making cool things, and trying to entertain them as best as possible while teaching them a few things. It does wear on me, but usually the first five days of summer fix that problem.

Something else at work here: I already had the career crisis. Lots of people I meet who start teaching as a first job get bigger ambitions because that's what happens at first jobs. You want to make sure you chose correctly before you get too old. Job hopping is what people in their 20s do. I was in a different industry before. After three years it was evident it wasn't fulfilling and I should find something else. If you're moving from teaching to an office job and think paradise awaits or all those things you complain about are behind you, well, there are some things you should know about office jobs...

As far as growth within the education field, I just don't think it's for me. I don't need to reinvent textbooks. I don't want to become an instructional coach, leader, or administrator. I did a bit of research as an undergraduate, enough to tell me I dislike research. Like, I'm the anti-Pershan of research. All of those jobs are motivations for getting a masters. But there's just not enough teaching involved. I'm going to take on the workload and expense of a masters just to do my exact same job for a 2% raise? No thanks. Let's say I departed the classroom to get another degree, what are the odds I could have my exact same job when I return? Given the unique circumstances that got me to my schedule, they are low.

The stuff in the op-ed pieces isn't worth addressing. That angry article with a thousand retweets represents the minority of schools. There are poorly run schools. There are poorly run state systems. I landed in a good situation and it has only proven to get better. I know lots of people who enjoy their situation. Schools like this exist, I want you to find one where you're happy, it is possible.

All this to say I'm staying in the classroom. I will continue to stay in the classroom. If you're a veteran who feels the same way, props to you.

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This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students should become comfortable with function notation. Students should see how functions referenced by their notation can be combined through addition, subtraction, multiplication, and composition. Students relate these actions to perimeter, area, and volume.

Progression

When they reach Pre-Cal, students should have seen f(x) notation for referring to more complicated expressions. I review this notation and what it means to find f(2). Usually all students are ok with this.

First, addition and subtraction. I generate two expressions, say f(x) = 3x + 7 and g(x) = 10x - 2. I pose the question: what do you think f(x) + g(x) would mean? Most are able to arrive at the answer that they need to combine like terms to arrive at f(x) + g(x) = 13x + 5. Subtraction follows a similar progression. They'll practice with a few examples I put on the board.

Next day we will discuss multiplication. I return to f(x) = 3x + 7 and g(x) = 10x - 2 and pose the question: what do you think f(x) * g(x) would mean? There is more struggle here. Some will say 30x - 14 is the answer. A few who thought about it longer will step in and say the answer is more complicated than that. At this point we debate methods. I offer the area model. I offer distribution (arrows flying everywhere). We arrive at 30x^2 + 64x - 14 either way. Students will have a preference. I make no endorsement of either.

I throw a little curveball. Define f(x) = (x+2)^2 - 10 and g(x) = 10x - 2. Any operations would require us to expand the binomial in f(x) first. Often students forget the "-10" there on the back when simplifying f(x), a good talking point.

Students will spend a day practicing addition, subtraction, and multiplication with a set of defined functions. A few of them will require them to expand a binomial first.

Here's an idea:

I don't explicitly mention scaling. Most can easily figure out what I'm after with 2*p(x).

We then work on the mechanics of f(g(x)) and then work another problem set of multiplication and composition. The activity below is designed to help put some context to f(g(x)).

Activities

Ok, that's the mechanics part. Now for something more tangible. Students will find the logic here a bit of a stretch, the trouble of knowing how to multiply f(x) and g(x) without any thought as to where they came from or why we would bother.

Show the Dan Meyer tank filling video:

I simplify the tank as a cylinder, but you don't have to. Area of an octagon is not as readily retrievable. We work towards a definition of V(t), the volume of the water in the tank. We discuss the volume of a cylinder and define A(r), the area of the top, and h(t), the height of the water. V(t) is the result of A(r) * h(t).

The idea stretches further if you talk about V(h(t)) since the area of the top is a constant. They get weirded out by this idea because it looks overly complex (and kind of is). I have screwed up this part.

As a follow up, I will relate addition to finding the perimeter of a figure and relate multiplication to finding the area.

The idea is not to convince them that this stuff is magically more real, just to show them how more sophisticated notation can be used for a problem they recognize.

Assessment

Students are assessed on the mechanics and context of these problems. For context, students are given some functions defined as the height and width of an unknown figure and they have to illustrate the scenario. It is tested across three SBG topics which takes three tests to complete.

Test 3, Test 4, Test 5

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Students always want to know if they're you're favorite. Every class wants to be your favorite. The standard (and true) answer is that I like them all for different reasons. There's only one class I didn't really care for, but that was forever ago. Plenty of groups have been a hot mess, but it's an endearing kind of hot mess.

Secretly though, I do have an actual favorite class. And they stumbled into that status for lots of reasons. It was a random assortment of 1 freshman, 2 sophomores, 14 juniors, and 9 seniors in on-level Algebra II. A mix of abilities all up and down the scale. The needs of the seniors were far different from the (lack of) needs for the freshman.

What'd they do that was so great?

Pioneers

Through no action of their own, they stumbled into a grand experiment. For years I taught multiple sections of Algebra II and had decided it was time to really shake things up. Having a couple classes to play with would make it interesting. Funny thing though, when schedules posted, I had but one section of Algebra II. It strengthened my excitement for the project. No worries about pacing. We'd move as fast as we wanted to. If I ever get the opportunity to teach a single section of anything I'd do it immediately. It's so fun.

Makers

This group was the first to make me realize how valuable it is to set an expectation of work. We were going to be exploring a lot of new stuff in rapid succession and they'd need to practice it a lot. By October they figured out math time was work time. It was difficult to get them to sit still for the small bits of instruction I needed to add. A lot of days were no different than an experience Tina wrote about. I don't know if you've ever taught seniors in Algebra II, but to watch that group sit there at try at something consistently day in day out was so great. A couple of them were so proud of themselves when it was all over.

Their special status gave special opportunities. Not only were we going after a new curriculum, but we were going to play around with Estimation 180 all year too. And they built their own.

Not content with that, during the second semester they showed off the whole family of Algebra they've come to know (and not fear).

Their class time was a flurry of activity. They made the biggest messes.

Fearless

The most important thing is the confidence they had when it was all over. I'd like to say the story ended with all of them getting an A for the semester and being crazy algebra ninjas. While a lot of them did enjoy that status, a few did struggle and were in over their head for lack of understanding or lack of trying. But that was a minority opinion. The ones that bought in were not scared of whatever I threw at them. Bring on the decimals, bring on the piles of parenthesis, bring on the quadratic wrapped in a log, it doesn't matter. They didn't give up.

Many of them said it was their favorite part of the day (math class, imagine) because of the atmosphere. It was my favorite part of the day. It made all the effort I was putting into it worth it. They were so hard to plan for and I made a lot of mistakes, but they did a great job with everything we did.

I hope everyone gets to have that one great group at some point in their career.

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AuthorJonathan Claydon

Summer time, or 12 weeks of unpaid leave, however you want to look at it. As always, I have projects.

The carpet in there isn't aging well. I have less official things on the agenda this year, so there's a little more time for planning. Pre-Cal is overdue for some examination and there is lots to do in Calculus.

A summary of upcoming events:

  • There are rumors of large scale Chromebook deployments in a few years, and there are already pilot programs. I was asked to explore and was given a Chromebook. Then I made some appointments with Vaudry, Stevens, and the Butler to see what might work.
  • I want to take Pre-Cal down to foundations and see if there are ways to do better, something I've been calling the Pillars of Pre-Cal in my head. I drew inspiration from a final review I had lying around where the second semester fit into 4 nice little areas. I really liked how much Calculus we were able to do.
  • The Calculus remodel involves taking a page from the Pivot Algebra Two project. I need to pass through the material a lot more times. Student feedback indicated that AP level stuff would've been nice a little sooner, and the most challenging thing was when we started adding context (what does an integral mean, etc). The sooner I can start throwing that stuff around the better. I see them 5 days a week next year. Throwback Thursday is sticking around. We're getting new textbooks too with more online access for the kids. Will homework stick around? This year's group said it should. But what role will the textbook have, if at all?
  • I hoping to be able to celebrate some progress when AP scores release on July 6.
  • This site has always been geared for me as the primary audience, but over the last year a lot of people have started using it as a resource. The Top 10 items every month involve my various content areas. Something I've also noticed is that I don't have the best explanations for how I go through a topic. Glenn has started explaining his Algebra 2 methods in better detail, I'm going to sketch out a series for Pre-Cal. Hopefully that helps the "how do you...." crowd find some better answers.
  • No major purchases! Classroom build out is pretty much done. Other than reload some supplies, it's in good shape.
  • I might have found a way to manage the stuff on all the iPads I have lying around. Maybe, almost.
  • Staff development slows down a little bit too. TMC15 in Claremont is going to be a good time. I'm not presenting this time so maybe I'll be less exhausted when it's over. I'm instructing a few courses locally in August over the standard stuff: notebooks, SBG, and Desmos.

I'm looking forward to what these projects might yield. Will any of it surpass eating cheeseburgers with Stadel? To be determined.

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AuthorJonathan Claydon

One more time, with feeling.

I keep a photo collection. I'm the primary visitor. None of you view it, I have the stats to prove it. But it's ok. It's my favorite part of the site.

Even if you aren't going to through the trouble of setting up a site and show all this stuff to the world, you should still take pictures.

Primarily, it improves my memory tremendously when it comes to next year. I remember the lesson and I remember the pictures I took during the lesson and what the intended output was.

If you get to be the kind of person who leads staff development, having a library of classroom photos is immensely helpful. I think at least on four occasions during meetings I've been able to reference one of my kooky ideas by showing the photo that went along with it.

You'll thank yourself at the end of the year when it's time to submit evidence for an appraisal.

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A popular question on the official AP forums is "what in the world do you do after the test?" and there are a variety of answers. Some of the projects are intense, others not so much. Those of ending school deep in June over a month after this thing have my sympathies. My kids desperately wanted to make a movie, but there was no way we had the time. I also had just finished the batch for this year and was not in the mood for more.

Lots of stuff worked against me after the AP test. For one, they had other tests to take, so random handfuls were out depending on the day. Then, college placement exam testing cropped up taking more random handfuls away. In the two weeks since the exam I've had them all together twice. It's better now, but they're out of here in 6 schools days.

Random Activities List

Financials - discussed doing your taxes (primarily that high tax refunds aren't some kind of magic bonus) and the basic of credit cards/borrowing money.

Engineering - support tennis balls with coffee straws, the ball had to be 12" off the ground and you couldn't tape your creation to the table for added stability

Field Day - just take them outside and let them hang out

Exit Interviews - a few questions just to see what they thought about the class, and to help me make some improvements for next year. I never learn a ton for these, but it's more about one last little relationship building item before they leave forever.

Sidewalk Chalk - why not? A lot of them did it when they were in Pre Cal. I generated some regions between curves and had them pick sets at random. They had to sketch the graph and compute the area of the region. With calculators and desmos the work part took about 15 minutes if you focused the whole time (hard for a senior in May). Later in the week they'd reproduce their findings on the sidewalk. Two regions per kid, about 4 blocks per kid to show the work properly and everything.

The only shame here is that the second group finished a little after noon. At 2:50 a storm rolled in and the whole installation vanished before anyone really noticed we did it.

Conclusion

Not a bad set of things. I will add to this list next year because I'll see them a little more and I know how to plan around the random absences a little better.

Goodbye Varsity Math. It's been real.

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AuthorJonathan Claydon