Author: samjshah

Function Transformations

I just wanted a quick post to share the documents I created to teach function transformations. All documents are in .doc format. They aren’t flashy, but they really got students thinking about everything. (This is a regular Algebra II class.) They nailed the final assessment, and are now doing amazingly on transformations of exponential functions. In other words, I see my work as a success.

Function Transformations 1 BASIC INTRODUCTION (here): HW (here)
Function Transformations 2 UP! DOWN! LEFT! RIGHT! (here): HW1 (here) , HW2 (here & here)
Function Transformations 3 VERTICAL STRETCHING (here): HW (here)/ Solutions (here)
Function Transformations 3.5 PRACTICING THINGS STEP BY STEP (here)
Function Transformations 4 HORIZONTAL STRETCHING (here): HW (here) / Solution (here)

Also I handed this practice sheet out to all students to practice their 8 base functions (here).

Just so you know, I don’t always teach via handouts. But with all this graphing, I decided it made good sense.
I’m happy if you want to critique them, or make suggestions on how to improve them.

Precious Moment

Today I had one of those great moments which put an impossibly huge smile on my face. Today I had about a zillion student meetings. I had no free periods the entire day! One of the meetings had to take place while I was on “front hall duty” — manning the table where kids sign out to leave the school building for lunch.

While I was helping this student — and if I say so myself, doing an amazing job of explaining the really conceptually hard Fundamental Theorem of Calculus Part II — one of the people who works at the school, the mother of one of my former students, passed us and then doubled back to speak to us. She said “Wow! I just had to say that this image is so great. This is such a great thing. A second semester senior and a teacher working so hard. This is amazing. I wish I had a camera.”

I took stock of the situation, and grinned. I patted my student on the shoulder, made two fists and pumped them in the air, and said “Yeah!”

Teaching seniors is hard. But if you set clear expectations and help them reach them, you too can be as great a teacher as I am. (Just kidding.) But yeah, my faith in my kids’ is on the upswing.

Senioritis

I am cursed[1]. I am surrounded by a sea of transmuted figures, bearing some but not a distinct likeness to their former selves. Slumped and unresponsive, stumbling through the halls of my high school with an arrested gait and in a constant stupor. Hunched over desks as if the desks themselves were magnetized and their brains were oppositely magnetized. The desk seems to call to their heads, inching them closer and closer…

Being spoken to might temporarily break the haze, and elicit a “could you repeat that” or — more tragically comic to me — “yes, definitely” as a response to “What might cause the answer to be 0?”

There are days when these beings don’t come to school. Days, mind you, is plural. There are campus preview weekends, of course. Soon will come AP exams, where mysteriously on the day before the exam, one might be absent. Coupled with the day of the exam, and we have two more days, gone. Mental zombies.

These beings are mere shadows of what they were before.

[1] Okay, okay, I know I’m not cursed. I love my seniors. But fourth quarter is trying for me, because it is trying for them.

Out of Commission

I’m sorry I’ve been out of commission lately. We had comment writing (we write narrative comments for all our students) and I’m moving to a new apartment in two weeks and I’ve been anxious about that.

But before I forget, here are some topics I’ve been musing about posting on in the near future:

1. comment writing (obviously)
2. function transformations
3. senioritis
4. random questions about stuff i don’t fully conceptually get in the last chapter of the multivariable calculus book

I had to write them down here before I forgot about them in the hectic day-to-day that is my life.

Composition of Functions and their Inverses

In Algebra II, we have been talking about inverses, and compositions. We finally got to the point where we are asking:

what is $f^{-1}(f(x))$ and what is $f(f^{-1}(x))$ ?

Last year, to illustrate that both equaled $x$ , I showed them a bunch of examples, and I pretty much said… by the property of it working out for a bunch of different examples… that it was true. However, that sort of hand-waving explanation didn’t sit well with me. Not that there are times when handwaving isn’t appropriate, but this was something that they should get. If they truly understand inverse functions, they really should understand why both compositions above should equal $x$ .

So today in class, we started reviewed what we’ve covered about inverses… I told them it’s a “reversal”… you’re swapping every point of a function $(x,y)$ with $(y,x)$ . That reversal graphically looks like a reflection over the line $y=x$ . Of course, that makes sense, because we’re replacing every $y$ with an $x$ — and that’s the equation that does that. My kids get all this. Which is great. They even get, to some degree, that the domains and ranges of functions and their inverses get swapped because of this.

But then when I say:

“ $f(x)$ means you plug in $x$ and you get out $y$ … but then when you plug that new $y$ into your $f^{-1}(y)$ you’ll be getting $x$ out again”

their eyes glaze over and I sense fear.

So I came up with this really great way to illustrate exactly what inverses are and how the work… on the ground. I put up the following slide and we talked about what actually we were doing when we inputted an $x$ value into both the function and the inverse:

We came up with this:

We then talked about how we noticed that the two sides were “opposites.” Add 1, subtract 1. Multiply by 2, divide by 2. Cube, cube root. And, importantly, that they were in the opposite order.

Then we calculated $f^{-1}(f(3))$ :

Starting with the inner function: $f(3)$

(1) cube: 27
(2) multiply by 2: 54
(3) add 1: 55

Then we plugged that into the outer function: $f^{-1}(55)$

(1) subtract 1: 54
(2) divide by 2: 27
(3) cube root: 3

This way, the students could actually see how a composition of a function and its inverse actually gives you the original input back. They could see how each step in the function was undone by the inverse function.

I don’t know… maybe this is common to how y’all teach it. But it was such a revelation for me! I loved teaching it this way because the concept became concrete.[1]

[1] I remember reading some blog some months ago that was talking about solving equations, and how each step in an attempt to get $x$ alone was like unwrapping a present. I like that analogy, even though the particular post and blog eludes me. But in those terms, this is like wrapping a present, and then unwrapping it!

Multivariable Calculus Projects

Monday is the start of the 4th — and final — quarter at my school. I desperately want my Multivariable Calculus students to really love the end of the course. In the same spirit as the rest of the course (focusing on basic concepts and applying them to very difficult problems), I have decided to assign half as much homework and have students spend their time really investigating a hard problem, researching an interesting topic, or creating something based on what we’ve learned. Their choice.

I whipped up a draft of my expectations for their end of year project, which I’m embedding below (Go Scribd! I’ve been waiting for this feature to be WordPress.com compatible for eons!). I know this is not an ideal set of expectations. I need to clean up the redundant language, provide examples of good and bad prospectuses, as well as have on hand sample rubrics for students to use as guides when creating their own. (As an aside, if you know of any prospectuses/rubrics for me to show them, I’d very much appreciate you throwing the link down in the comments!) But my thought now is that I’ll see what happens this year, and then tweak it for future years:

View this document on Scribd

I also spent a number of hours coming up with possible Multivariable Calculus projects, posted at my Multivariable Calculus resource site. I hope to add to it as ideas strike me. I’ll copy the projects that I have as of April 11, 2009, below the fold. But click the link above for a list that will hopefully be updated.

(more…)

Take what you don’t know…

In Calculus, I sound like a broken record. Each time we learn something new, I say “take what you don’t know and turn it into what you do know.” I say that at least three times a week. I said it last week when doing integrals like:

$\int \frac{1}{4x^2+1} dx$

We don’t know how to deal with that, but we do know how to deal with

$\int \frac{1}{x^2+1}dx$

So let’s try to turn what we don’t know how to do into something we do know how to do. For those who haven’t taken calculus for a while, the integral above is $\tan^{-1}(x)+C$ . So to do the original problem, we want to somehow get the original integral to look like $\int \frac{1}{(something)^2+1}d(something)$ — the integral of 1 over something squared plus 1. So we rewrite the integral as $\int \frac{1}{(2x)^2+1}dx$ . That’s much closer to what we want to get — it looks more like something we know how to deal with. Next we use $u$ -substitution to finish this beast off ( $u=2x$ ) to get $\frac{1}{2} \int \frac{1}{u^2+1}du$ . Now we have something we know how to deal with, from something we didn’t.

Again today, I showed my students how to solve $\int_0^1 \sqrt{1-x^2}dx$ , and told them to solve: $\int_0^1 5-3\sqrt{1-x^2}dx$ . At first sight, they recoiled, but again, we used the mantra of “take what you don’t know and turn it into what you do know” to solve it. If it looks scary, fine, have a moment of panic, but then ask yourself “what does this look like” and “can I turn it into that with some simple manipulation”?

I was thinking today how this actually could be my refrain in Algebra II also. Example: I could frame quadratics in that way. Students know — or quickly learn — how to solve equations like $(x+1)^2=5$ (hopefully). But what about something like $x^2+6x+1=0$ ? It’s not nearly as easy. But then we can talk about if there is a way to that what we don’t know (that equation) and turn it into something we do know how to solve ( $(x+3)^2=8$ ). It’s not that I don’t do this already, but I am not always explicit about it. It is not my mantra.

But it should be. It’s how we solve math problems. We have something we don’t initially know how to do. And we have to figure out if we can simplify/rewrite/re-envision it to bring it to a place where we know how to do it.It seems stupid and simple and obvious, so much so, that I don’t say all the time. But if I started saying that as my refrain, if students really saw that math is simply this simple process, it might stop seeming like a huge bag of tricks that never fall together. They might see it as the art that it is — where there is creativity in deciding how to get from point A (hard problem) to point B (simple problem they know how to do). And all the specifics that we do in class are giving them the tools which they can use to chisel out a path from A to B. It might finally be us always trying to work out the puzzle: what does this look like that we know how to do, and can we get it to that place?

In other words, we’re now talking processes instead of methods. We’re talking problem solving instead of rote memorization. And whenever a student is stumped on a problem, you can stimulate his/her thought process by saying “we’ve always taken what we don’t know how to do and turned it into something we do know how to do… what similar things does this beast remind you of?”

So yeah, it’s not a huge revelation or anything. But I’m thinking that it might be a really amazing experiment to frame my Algebra II and Calculus classes with this mantra next year. Heck, maybe even in the next few weeks when I’m teaching exponential and logarithmic functions! I mean, yeah $\log(2x+1)+\log(x-1)=2$ may look ugly. But is there a way to turn it into something we do know how to do? Namely something of the form $\log(something)=2$ ? Obvi.

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now