Man do I hate this table:
If you get the basic principles at play here, the table is needless. And if you don’t get the basic principles at play, you would be much better served studying those principles than studying this table. (How do I go from yearly to daily, for instance? That seems to have been left out of the table.)
And at the College Algebra level, I hate this equation:
That’s because there is no good explanation of e that is justifiable with the tools of College Algebra. You want to describe it as a limiting value of interest rates as number of compounding cycles increases? You’re gonna do a lot of hand-waving at the end there. (Come to think of it, that’s not a bad motivating question for limits in Calculus.)
And any other way you want to introduce e (and mathematicians’ fascination with it) is going to rely even more heavily on Calculus.
But half-lives are a nice, tidy application of an important set of College Algebra topics: exponential and logarithmic relationships.
So I give an extended assignment that deals with half-lives. And I encourage students to think in terms of an annual decay factor, which we can find by means of extracting roots. We know that it takes thirty years before the decay factor gets down to 1/2, so we need to know what number to the 30th power is equal to 1/2.
Basically, I hope to get them to the point where they want to solve this equation:
And in reading their work, I have been blown away by the number of students who have done the whole assignment in terms of half-lives. They are thinking explicitly in terms of half-lives. Rather than blindly plug numbers into , they are figuring out how many half-lives there are and then using that number to calculate. So they’re using the equation, , where x represents the number of half-lives that have passed.
They are putting appropriate scales on their axes, labeling the x-axis half lives. And one even made the observation that, with this setup, the graph describes all half-life situations; the only difference is the length of the half-life.
I don’t hate that.