I have a question for you. In a day or two I’m going to be teaching exponential functions, followed by logarithmic functions. These are historically very difficult for our students (and I’m assuming your students too). One idea I had to making things easier is to separate “e” from the lessons totally. I’m thinking that teaching a difficult topic, and then integrating another foreign concept at the same time, has been part of the problem.
(Just like last year I taught quadratics, and when we saw the square root of negative numbers, I taught complex numbers. It arose naturally, but it was too much for my students. This year I taught complex numbers as a short mini unit before embarking on quadratics, and things went amazingly!)
I was going to teach exponential equations and logarithmic equations without “e.” Then when students had done everything, I was going to spend a few days introducing e and applying what we had learned to it.
What do you think? Do you do something similar?
Also, if you have any good ideas on teaching these topics, or good activities, or good resources, or even good questions/problems, please leave them in the comments. I want my students to really get and appreciate the concepts this year.
I’d start with exponential tables.
You have $1,000 that earns 4% interest every year.
Have them fill in the table, then look for patterns in the table. (I had one student get to the 4th level difference before concluding it would never be constant, but that there is a constant ratio in successive outputs).
Then give them a table of outputs. Then one that starts at t=4 and have them work backwards for the initial amount.
They should be able to come up with the general rule of a*b^x.
Then give them a few side by side tables, one of a*b^x, a*b^x + 1, a*b^x +3 (same a’s and b’s in each).
Ask them to plot each table too.
Seems to have worked for my cherubs this year. (I’ve actually got nothing typed up, my textbook does it all very nicely).
What level is this? In calculus, I like to do the real way of discovering “e” via the area under 1/x. So, logs come first, then exponentials as their inverse.
For algebra, I like the idea of “real world” problems bringing them in. Population modeling, interest rates (for making money or debt increase), etc.
My only “fun” contribution to “e” is memorizing some digits from it. Especially from TN, I like this one:
2 – Just gotta remember this one
7 – 7th president from our very own state?
1828,1828 – Elected for TWO terms, the second term started on what year?
45-90-45 : your favorite right triangle
I don’t bother with e either. All the exponential growth and decay models can be done with more natural bases. Nuclear decay: 1/2, bacterial growth: 2, a population that increases by 4% each year: 1.04
At the end of all this I do mention ‘e’ a bit. But you can do loads without it.
Also check out the International Data Base (http://www.census.gov/ipc/www/idb/tables.html) for population records for the past 60 years for most of the world. Great stuff.
I wouldn’t touch e in precalc or Algebra II (except perhaps a parenthetical mention). I’d also separate logs from exponentials.
I find that kids can learn how to transform the graph of f(x) = ab^(x) + c pretty easily, and then can reverse engineer. One nice task, eventually, is to take three points and decide if they’re most easily fit to an exponential, a parabola, or a line, and writing the equation.
Lots of practice with manipulating exponents, which then sets the stage for logs.
Anyway, that’s more my style.
Thanks all for your comments.
Based on my own concerns, and from what you said, I spoke with my department head about not teaching e and ln until precalculus, and we’ve decided to do that. Which is great, because we get to focus a lot on the basics. Plus I get to spend time doing my fun lessons on graphing on a logarithmic scale!
@Jackie: Me likey. I’ve done something similar in terms of “you sock away $X in savings for retirement with an interest rate of …”
@David P: I’ve never learned that mnemonic to memorize the digits of e. It would be a great extra credit question for a test… 1/4 pt for every digit. And I don’t even believe in extra credit and rarely offer it! So that’s how cool I think this is.
@kevin: Thanks for the link to the populations. It’s an EXCELLENT site. I just bookmarked it to my delicious acct.
@jonathan: I was definitely planning on separating exponential functions and logs.
Now, are you going to teach them what would happen if they put a log scale ruler next to a log scale ruler, and perhaps advanced one by the log of 2?
(apparently they sell these, as ready-made kits…)
@Jonathan: what would happen? their heads explode!
I respectfully disagree with what I’m hearing here. I teach honors Precalc and am just about to introduce exponentials with logs next week. While it seems to be common, I do not think it really makes sense to separate the two, as long as kids are comfortable with the idea of exponentials (which they should be from earlier courses as this repeated multiplication or division… depending on your direction).
Then, when thinking about how we can certainly reverse a constant nth power by taking the nth root (with ‘n’ being some constant), how can we take an xth root if we are trying to reverse a constant being raised to a variable power? This way, we relate back to inverse functions as being what you do to reverse some set of operations to get back to where you started.
Now, if we want to reverse a number being raised to some variable power, we can introduce this operation that mathematicians introduced called a “logarithm.” The name is not important where it came from, but it essentially just answers the question, “What power must we raise some number that we know to get some other number that we also know?”
From that, we see that exponential and logs are inverses of each other, which has other implications (algebraic and graphical), with which students should be comfortable already.
I agree with emre007 – they cannot be separated. They are inverses and are used to undo each other when solving equations. Logs arise naturally when you try to solve exponential equations and that is how I introduced them this year (last week) to my Algebra II classes. It was all very natural and not contrived at all. Then there was an earthquake in Alaska this weekend, so there you go – logarithms on a platter.
And I don’t know if Meghan Trainor knows anything about logs, but she sure helped my kids remember when to use a logarithm last year!
I’ve been reading this website for a few weeks and have found their explanation of logarithms and exponentials to be interesting: