Here's a Desmos activity I typed up for my calc kids to find the the derivatives of exponential and logarithmic functions. Sadly...the class set of laptops would not connect to the domain because they hadn't been used all summer, so we did this together as a class, which was not what I wanted, but what can you do? Instead of having the kids click pause on their own screens, I had them yell "STOP" at me...so it was still entertaining.
All this to say, I don't know if this is good or not since I haven't gotten to test it out on students yet, but here it is. Use/modify if you'd like!
In the "notice/wonder" section of f(x)=a^x, one student said he noticed that the derivative was proportional to the given function. This made me a very proud momma and was a perfect segue into finding the derivative when a is different from e. (We explored f(x)=2^x, f'(x), and g(x)=lna(2^x), and found that a=2).
I guess I've been kinda into flow charts this year; I created another one for solving exponential and log equations. The idea is that kids start with the top box, if they can't complete that task, then they go on to the next box. We put examples in each box. I used this for both Algebra II and Pre-Calc this year.
It's not flawless, because mathematics requires more creativity than a flow chart can provide. But it gets the basic ideas across.
You know those lessons/projects you give your students that you look back on and you're like, "Wow. That wasn't half bad"? I had one of those recently in Pre-Calculus.
We've all seen those exercises in textbooks where students are supposed to figure out the time of a person's death using Newton's Law of Cooling and given certain temperatures and times. I always liked those problems, but never really knew what to do with them, more than just present them and say, "See! Math IS applicable to real life."
Then a colleague of mine showed me a literacy activity adapted from Key Curriculum Press. I found a version online that I used (but it was a direct link to the Word document, so I don't know to whom to give credit!). The first page is what I found online (I added Newton's Law of Cooling to the bottom); the second page is the instructions for the kiddos, which includes the rubric:
To start out, we first had to watch a trailer for BCC's Sherlock (LOVE):
I gave the students about a half a day to figure out the math and solve the murder. The next day, I loaned out a laptop cart from the school and the kids finished the story, working in groups of 2-3. I had the students submit their posts on a blog I created for our class via kidblog.org. My principal told me about kidblog, a class-friendly version of Wordpress, and I absolutely love everything about it (except its name). Students don't have to register or sign in with an email account: you just set up usernames and passswords (which can be done in a jiffy) and then they can log in.
On the blog, I posted a sample writing that I found here. (The math is a little off, so be sure to fix it if you use this link--the final t should be negative.) This really eliminated the "I don't get what you want us to do!" comments because the students had an example with which to model their writing. In fact, I didn't get a single such comment (kuddos, kids). However, I also protected this sample with an extra password: students could not get into the post until they had solved the crime, as the password to the post was the time of death (see, kidblog is awesome).
Once all the posts were in, I gave the students a couple days to go back in and comment on their favorite posts. The posts with the most comments received some bonus points.
I was honestly blown away by my students' response to this assignment. Their stories were original, entertaining, and included the required mathematics.
There's obviously room for growth here on my part, but for the first go-around, I was incredibly pleased with this activity. Next time, I may make the crime a bit harder to solve, and I may give different versions. We'll see how motivated I am.
Out of respect to my students, I don't want to post the password to the blog here. But, if you'd like to check out their stories or the blog for instructional purposes, feel free to tweet me (@RebeckaMozdeh) or email me (rebecka dot peterson at gmail dot com).
In Business Calc, we're currently studying exponential growth and decay. I'm rather excited about this since it's something we study in College Algebra, too, and I feel--because it's material I've taught before--that I can expand a bit. I'm learning that it's really, really hard to expand (i.e., go beyond an absolutely dazzling lecture *cough*) when I'm teaching a class for the first time. I sort of feel like I did my first semester as a TA: I just hope I don't screw something up too terribly. But--you gotta start somewhere, right?!
Anyway. Back to exponential growth/decay. In College Algebra, when we study exponential functions, I have my students model the decay of an M&M population. I had planned to do this with my Business Calc class as well. Then Bowman Dickson posted places to find awesome data, which made me want to use data the UN has on the world's populations instead.
The question: How to relate M&M's to population growth or decay?
The answer: I'm not entirely sure. Here's what we did though...
We started out with the M&M project as in College Algebra. Each team found the exponential regression (in the form y=ab^x) and the r^2 value for their data. We talked about the meaning of a and b. Then I asked them to convert their regressions to the form P(t)=P_0e^(kt), which turned out to be very close to the trendlines Excel found (yay!). We talked about what k would mean if it this were a real population and how it's related to the derivative.
Now the challenge: I asked them to do the same types of calculations for an actual population, using data from the UN. They were on their own for this project, which may or may not have been a great idea. Below is what they had to go off of. I focused mainly on finding the exponential regression on a TI as well as understanding growth/decay rates. But there's much, much more to do here (Bowman does a week-long project!).
I keep hearing you use the word exponential in reference to a drastic change from one examined point in time to another. For example, "I slept exponentially better last night than I did the night before." Or, "My coffee tastes exponentially worse at this Starbucks as opposed to the one on Main Street."
I usually just smile and nod. As if I understand what you're saying.
But, I really don't.
How do you know the change is exponential when you're only comparing two points? How do you know it's not linear, perhaps with a steep slope?
To illustrate my point, let's say on Day 1 you rate your sleep as a 2. On day 2, you rate your sleep as a 4. (I have no idea what these numbers indicate, but you must be able to rate your sleep on some kind of scale if you can make a statement like the aforementioned one.) Well, then, maybe your sleep pattern, indeed, is following an exponential trend line:
An exponential function: y=2^x
Or...you could have trend lines such as:
A linear function: y=2x
A quadratic function: y=(2/3)x^2+(4/3)
A logarithmic function: y=2+2.88539ln(x)
And these only represent a few of the possibilities.
So, residents of the great Midwest: I urge you--be more creative in your comparisons. Don't assume you need to use the phrase "exponentially better." Nay. Why not try something like, "quadratically better," or "logarithmically better"?
I've written before about how review days are a continual source of stress for me. To review or not to review? that is the question. And if to review, how do you make it interesting and beneficial for the upcoming test? I have no profound answers yet. But, I am putting a lot more time and effort into my review days now (mostly because I've taught College Algebra enough times so that I have the extra time to do that).
That said, I've been very much looking forward to this review day for quite some time. The unit has been on exponential and logarithmic functions. When I talk to my husband about this unit (who patiently listens to all my teacher talk--I found a really good man, let me tell you) he always reminds me, "It would be much less scary if it weren't called a LOGARITHM."
And he's completely correct.
So, I've been hyping up this review day nearly all unit: "We'll talk about who invented logarithms, why in the world he did so (just to make your lives miserable?), and why they're called what they're called."
Here are the slides that took us through a short history of logs. The SMART notes didn't convert perfectly to PowerPoint, but email me if you want the SMART notes as well.
I gave the students a sheet that corresponded with the notes so they could follow along. After we went through John Napier's method of multiplying two numbers we worked the same multiplication on slide rules! A few of my colleagues were kind enough (and...ahem...old enough) to loan me enough slide rules so that nearly every student could have his/her own for the day. Next, as review for the test and as further proof of how quickly exponential functions grow, I had the class break into four groups and choose one of the following problems below. I got the first problem from Ethan Siegel's blog. Unit 4 Review Problems
I think this idea originally came from Virginia Tech, but I may be wrong. In any case, here's how we did exponential regressions in College Algebra this semester.
Each
student gets a "Fun Size" bag of M&Ms. Students divide into teams of 3-4. Each team gets a napkin that they're asked to unfold
completely. The teams spill out their M&Ms on their napkins, making
sure all candy pieces are lying flat on the napkin. Now for the
math...
Count the total number of M&Ms on the napkin (this will correspond to x=0, where x is the number of "shakes").
Fold the napkin over the M&Ms and shake, shake, shake so that
the candies get mixed up well. When done, make sure all pieces are
lying flat. Take away any M&Ms that don't have the M facing up.
Eat them. Now count how many M&Ms are left (this will correspond to
x=1).
Fold up the napkin, shake, remove M&Ms that don't have the M facing up, eat them, and count the leftovers (x=2).
Repeat Step 3 until M&Ms are gone.
After each turn, I asked all teams to tell me how many M&Ms
were left, and I inputted their results into an Excel spreadsheet, which
was being projected on the board. I created a template so that with
each number I inputted, a scatterplot began to form. [Download the template.] After five turns,
one group's data (whose M&Ms cooperated quite nicely) looked like this:
After the M&Ms were gone, I asked each team to find an
exponential regression using their graphing calculators that fit their
particular data (they could look up at the Excel spreadsheet, where the data
had been recorded for them). In a perfect world, their regressions
would look something like y=a(0.5)^x, where a is the number of M&Ms
they started with. Of course, the number of M&Ms doesn't diminish
perfectly to half its previous size every time, so we got results that
looked more like this (again, this was a rather good trial):
But the imperfection is good. For one, that's life. For two, it
makes it a little less obvious as to what's going on and creates a nice starting point for some discussion.
After the students gave me the regression equations, I plotted
the regressions on Excel (which you can do easily in just a couple
clicks). I asked which team looked like they had the best regression
and then we compared r^2 values to see if they matched the students'
intuition.
I really wanted to use Skittles for this project so I could call
it "Skittles: Taste the Exponential Regression." Alas, M&Ms were
half the price of Skittles and my frugality won over. Maybe next semester.
I've never been all that happy with my presentation of exponential functions. I usually slap a couple of functions on the board, plot some points with the class, and then write up a list of these functions' characteristics. Boring.
Thankfully, I recently came across Sam Shah's method of exploring exponential functions using paper folding. I like this idea so much more than plain old point-plotting (though I ask the students to do that as well). Here's the worksheet I had my classes fill out before introducing the formal definition of an exponential function:
We spent 15-20 minutes on this worksheet in class before diving into the lecture. That's a lot of time for a college class, and I felt rushed the rest of period, which is never good. So, next semester I may assign it for homework the day before instead to save some time. I would have to give some kind of hint as to how to find f(x) and g(x) though, because the students were a bit unsure of the pattern going on. I don't think anyone got 2^x, but most did see that we were multiplying (or dividing) by two each time.
Once we got into the lecture part, I gave the students a great cheesy word problem:
One of my students noticed that it didn't take much time for the spending to increase significantly, which I was quite happy about. "So the moral of the story," I began to say...
"Is not to be exponential when you're shopping!" she finished for me.
