Precalculus Brainstorm

I was brainstorming for my Precalc class this morning, and I realized that my perception of the Precalc topics has changed since I began teaching IB two years ago! The IB equivalent of this course heavily emphasizes the interconnectedness in applying them, which has in turn changed the way I wish to organize my Precalc class next year. I'd like to organize my Precalc class this year as a sequence of topics that each arises naturally from the previous topic, with the entire course anchored upon modeling as the core skill and purpose for building further functional knowledge.

See flow chart below. (The bold parts are what I think are the more important concepts from the course.) If my new colleagues would support my decision in organizing my course this way, then I hope to start with sequences as a way of re-introducing linear, quadratic, and exponential forms. The kids will see, for example, that if you use summation formulas to capture the sum of second differences, then the resulting n-th element will have a quadratic form in terms of n. At the end of those basic functions re-introduction, my hope is that the kids can do a written project analyzing triangular and stellar numbers, similar to the old IB portfolio task from a few years back. (I've misplaced that prompt now, so I'll have to create one that is similar.)

Then, using their knowledge of these basic functional forms as a basis, we will examine the graphs formed by these basic forms and use that to re-introduce the core concept of transformations. We will learn the other functional types only as necessitated by modeling of different types of data, so that the kids can always remember the importance of contextual analysis and interpretation. Eventually, at the end of the course, each student will do two modeling projects: 

1. a project using GeoGebra or Desmos in order to create a picture with functions and to practice basic functions modeling and specifying domain restrictions. 

2. a real-world modeling project of their choice, in order to practice asymptotic analysis and written communication. In this final project, we can add additional requirements such as analyzing the rates of change, in order to preview some introductory concepts from Calculus.

Thoughts? Do you think this organization would make sense to students?

Differential Calculus Intro - Feedback Please!

I find it difficult to brainstorm without laying down some specifics, so I pulled together a potential Unit 1 for introducing differential Calculus without actually introducing differentiation rules.

Please take a look, Calculus teachers, and let me know what you think! http://bit.ly/differentialCalculusIntro. I'll be trying to meet up with one of my new colleagues soon to discuss some of this material, since she is trying to pull together a projects-based Calculus class (somewhat more slow-paced than the regular Calculus class that I'll be teaching) and she's also working on the explicit incorporation of Habits of Mind this year into all of our curricula. But, in the mean time, I'd love any feedback on whether:  1. you think this is paced too slowly for a "regular" (non-AP) calculus class?   2. Are the worksheets relevant and appropriate for the level of course? Our classes are about 45 minutes each.

Thanks!

After this intro unit, depending on how my colleagues respond to my idea of "teaching backwards", I'll either be planning a similar "integral Calculus intro" unit that focuses on application and skips over the algebra skills, or we'll be doing manual differentiation skills as the next unit...

Addendum 7/31/2013: Thanks to Sam Shah, I will be sharing this link on the Calculus of saying I love you with my students during this first introductory unit to derivatives!

Calculus Project Resources

I am combing the web for project ideas. Here is a cheat sheet for myself, and maybe for you as well:

I like these Calculus writing projects because they're based in realistic context, and the kids have to write a letter to explain their recommendations and to justify them with math explanations.

More of the same type of writing projects, but extra nicely formatted.

Art, technology, and Calculus - what could be better?

Designing a smooth-riding rollercoaster. I know that there are different versions of this project out there, and when the time comes, I'll be doing some shopping around to make sure I'm using the most kid-friendly version. I think my new school already has a version of this project floating around somewhere in the shared files.

Some sciency projects for Calculus.

Some nice activity sheets to spark discussion. (I know, they're not projects, but they're nice, no?)


Also, as a bonus find, I came across an easy-to-read explanation of what makes good math writing , and why writing is necessary in a math class.

Ah, I'm glad it's still just July. Even though I am keeping busy, summer is still cool and bright for another few weeks.

Calculus Brainstorm

I will be teaching Calculus for the first time next year. Yay! So excited about that. In the IB course, I had taught Calculus as a topic, but not as a full year-long course. I am very excited about the prospects of teaching an end-to-end Calculus class and having time to delve into all the nitty gritties. I am spending part of the summer sorting out the overall organization of the course, although I am sure it is to be changed once I start the year and chat with my future colleagues to get feedback.

In my mind, I picture two contrasting ways of structuring a Calculus course (and a hybrid way of doing something in between):

1. The traditional way: You introduce limits/infinity, followed by derivatives and their applications, followed by integrals and their applications, followed by mixed Calculus applications practice. Maybe you'd have a research component in the start of the year, in which kids do some research on all the modern applications of Calculus. From the various areas of applications, each student would choose one area to complete a capstone project on. This gives you chance to give them written feedback on their capstone project throughout the year, and their final paper to be submitted at the end of the year would have to contain an explanation of the relevant Calculus concepts, show example calculations that are applied within their area of research, incorporate technology as a way of enhancing/verifying their results, and they would present their projects to their peers as a way of solidifying the entire class's understanding of the applicability of Calculus.

2. Working backwards: You first spend some weeks taking the kids through the many ways of analyzing real-world problems using basic differential and integral concepts. They do everything on their calculator only, and they don't learn any of the algebra skills until after they have mastered the interpretation/application of results in the setup and analysis of a problem. Then, you peel away one layer of the magic and you work on all the core algebra skills of differentiation and integration without calculator, while consistently referring back to the applications that they have already seen, so that they don't forget the point of these algebra skills. Then, after they master doing all the same contextual applications manually and verifying their differentiated/integrated results via graphing calculators, you go on to more advanced concepts such as related rates problems, which cannot be explored solely on a calculator. Eventually, once the students have learned all the ins and outs of the core Calculus skills, you finally peel away the last layer of the magic and tie those "algebra shortcut" skills to the ideas of limits and infinity. The advantage of leaving limits to last is that the kids would already know what the derivative and integral formulas should look like by this point, and in taking the limits, they can self-monitor correctness. This "backwards" structure provides them with the full picture by the end of the course, of where everything comes from, but hopefully you have provided so much reinforcement of the applications that no kid will walk away from your class with only vague ideas of where Calculus is useful or how all the pieces are interconnected.

I talked briefly to one of my department chairs, and he is very supportive of the latter approach as an experiment. That makes me pretty excited! I will wait until I have chatted with the other Calculus teachers to start specific planning, since I have never taught Calculus as a full course before, but I think that our school culture is very supportive of teachers trying new things. So, I am going to try to wiggle my way towards a somewhat backwards organization of the course, because that makes the most sense to me.

I did some quick research, and these are the Calculus applications that I would like to introduce, either at the beginning of the year or at the start of each unit prior to developing those specific skills. Are there other ones that I am leaving out, but that are very common/useful for students to know?

Calculus applications

1. Optimization
    * Maximum enclosed area
    * Build a box (hands-on activity)
    * Max/min rotating distances from a fixed point (incorporate technology)

2. Instantaneous rates / steepness
    * Falling ladder analysis (exploratory starting with diagrams/tables)
    * Learning curve, length of list vs. memorization time (lead-in with experiment)
    * Marginal cost / marginal revenue / marginal profit
    * Incremental effect of education/experience on income
    * Rates of change within different savings account setups after fixed time

3. Summation of variable quantity
    * Physics applications (distance, velocity, acceleration)
    * Total work required/done in stretching a spring
    * Total interests paid on a loan over time
    * Supply and demand analysis – total opportunity cost for producer
    * Supply and demand analysis – total consumer surplus, total producer surplus
    * Architectural integration for accurate surface area
    * Center of mass (difficult)

4. Average of a function
    * Average velocity 
    * Average day length over a season
    * Average power of an AC circuit
    * Moving average of stocks (mini-project?)

5. Related rates (we won't work on these until later in the year)

About Optimism

Optimism is something that we have to work hard at. If you are as fortunate as I to live with a deeply optimistic person, you will learn that this person always thinks carefully before they speak, in order to filter out negative thoughts internally. It's not simply a random personal trait, but an accumulation of ongoing internal choices. There are volumes written on how to maintain your own optimism in the most trying of times, and the people who are the most inspiring optimists -- whether they are religious or not -- are often simply the people who work the hardest at maintaining that pure and literal semper fi (not referring to its use in the military context), or constant faith. Optimism, curiously, is not just a fluffy and unimportant personal trait either. It's connected to our health, our ability to bounce back from setbacks, and, in turn, our ability to grow over time and to lead other people. In a way, I see optimism as being closely tied to your personal spirituality. It speaks to your willingness to believe -- in what, I don't think is really that important -- something ranging from your religious belief to your belief in human ingenuity or humanity, but something that will bring you solace and a positive outlook to keep going productively despite the challenges that may come.

I wanted to share with you something that I found on the web today, that gives me a bit of pick-me-up in a trying time. I cannot say why this quote picks me up, except that it speaks to doing our best and then simply believing.

“Promise Yourself

To be so strong that nothing
can disturb your peace of mind.
To talk health, happiness, and prosperity
to every person you meet.

To make all your friends feel
that there is something in them
To look at the sunny side of everything
and make your optimism come true.

To think only the best, to work only for the best,
and to expect only the best.
To be just as enthusiastic about the success of others
as you are about your own.

To forget the mistakes of the past
and press on to the greater achievements of the future.
To wear a cheerful countenance at all times
and give every living creature you meet a smile.

To give so much time to the improvement of yourself
that you have no time to criticize others.
To be too large for worry, too noble for anger, too strong for fear,
and too happy to permit the presence of trouble.

To think well of yourself and to proclaim this fact to the world,
not in loud words but great deeds.
To live in faith that the whole world is on your side
so long as you are true to the best that is in you.”
― Christian D. Larson, Your Forces and How to Use Them

Goodbye, Expat Life; Hello, Repat Life?

As I type, I am sitting in Geoff's hometown in NJ. We're en route from Berlin to Seattle, stopping over just long enough on the East Coast to attend a wedding. It has been my first real break since January. The first time that I have had real time off, not thinking about wedding or work or looking for work or moving logistics. And the sun is beaming beautifully outside; even the sweltering tri-state humidity cannot begin to bother me when I am sleeping 12 hours a day. I am spending most of my time just hanging out with the in-laws in Jersey, but am also idly looking up friends in NYC during the week. During the coming weekend, Geoff and I will be visiting the Museum of Math, as well as catching the "new" musical Once. Although I love New York, I cannot help but feel relieved that we don't live there anymore. The city is a total wallet-zapper!

On a more personal note, things have been pretty rough at home, since my mom has been in the hospital for a couple of weeks now, and Geoff's parents have had health scares of their own recently. I think this is the start of an era -- the years when we feel lucky whenever our parents get over a scary episode of something without it becoming life-threatening; it's no longer the norm that our parents get sick and they would get better. It's scary, and we're still waiting for biopsy results to see whether my mom will get lucky this time. I really hope so, but as my mom has already said, if it's not this and not this time (that becomes life-threatening), then it will be something else at another time. That realization has hit me very hard lately, and I don't know how to navigate through my web of feelings about it. I think grief is a very selfish thing, because as soon as you start to dwell on your own grief, you're already prioritizing your own fears and needs over the tremendous needs of the person who is actually ill. So I have tried to keep everything latent, because I don't know when my mom will need that extra boost of positivity from me. The wait for diagnosis has been agonizing, but I keep reminding myself that it is 10 times more difficult for my mom than it is for the rest of us. It helps to keep other things in perspective.

So, it was with a heavy heart that I had said goodbye to Berlin. Amidst all the furniture-selling, packing, cleaning, painting, paperwork logistics, and long-distance phone calls to Shanghai, the time simply flew. There are many things that I could say about Berlin, but most of all, I will remember the wonderful friends that we've made there. Although it was quite random that Geoff and I ended up there, we were fortunate to experience the city on its way to becoming -- truly -- one of the greatest cities in the world. When I think back about my two years there, I will always remember how charming the neighborhoods were, with their parks, biergartens, craft and flea markets, and slowly savored Sunday brunches. The city can throw a helluva open-air party, or two or three. And I've never been in any other city where the train is habitually still packed to standing-room at 5am, with a diversity of languages to rival that of NYC.

So long, Berlin! Thanks for all the wonderful memories. I wish that there was more time, and that the goodbye wasn't quite so hasty and so distracted, but I am sure that we will visit again soon.

PS. Next year, I will be teaching Algebra 2, Precalculus, and Calculus. To keep myself from being overly lazy all summer, I plan to do some Calculus planning starting in late July or early August. Any resources that you can point me at would be awesome!

PPS. On a different note, if you visit Germany at any point as a tourist, I highly recommend talking to the locals and going to a traditional German sauna to try their Aufguss experience. It's not to be missed! The experience is so exclusively German that you can hardly find any English info about it on the web, and the descriptions that you do find do not adequately describe it...