Half lives

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:

f(x)=e^{-kx}.

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 (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:

\frac {1}{2}=r^{30}

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 f(x)=0.5^{\frac{x}{30}}, they are figuring out how many half-lives there are (\frac{x}{30}) and then using that number to calculate. So they’re using the equation, f(x)=0.5^x, where 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.

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6 responses to “Half lives

  1. …there is no good explanation of e that is justifiable with the tools of College Algebra.

    THANK YOU. The worst part of teaching exponential concepts (which are otherwise my favorite part of College Algebra) is being forced to introduce e. And the awful textbook we use actually states that if a quantity A starts (for example) at 1000 and decays at 2.8% per year, then the function to model this is A(t) = 1000 e^(-0.028 t). Utter nonsense.

    The topic of continuously compounded interest is so painfully dull that I generally just skip it and introduce the base e by saying that it has some very nice properties that are important in calculus. But I’m not sure I agree with your comment on hand-waving; if the topic of continuously compounded interest weren’t so unpalatable, that origin of the number e could easily be explained to students pretty thoroughly, I think.

  2. Roy:

    But I’m not sure I agree with your comment on hand-waving; if the topic of continuously compounded interest weren’t so unpalatable, that origin of the number e could easily be explained to students pretty thoroughly, I think.

    So I’m curious about this. For me, the whole compound interest thing is distasteful precisely because of the hand-waving. The argument depends on students having decent intuition about limits at infinity. I’m quite sure these intuitions are weak to non-existent. So if the first time they are really exposed to limits at infinity is by having that limit be transcendental and non-intuitive? That’s where the hand-waving comes in.

    Now this gets me thinking about other limits at infinity in College Algebra, and rational functions are the other place they appear. So maybe this is a reasonable use for rational functions (which, in my view, are way overemphasized in many College Algebra texts)-helping students form some intuitions about limits at infinity. Then we draw on that when we do compound interest to introduce e. I’ll have to play around with that in my mind.

    So help me out-what do you hate about compound interest as the context for introducing e?

    • As a context for introducing e in College Algebra, continuously compounded interest is possibly just about the best relatively simple option available. My problem with continuously compounded interest is just that it’s an incredibly boring topic with no grounding in almost anyone’s personal experience. That it may be the best context for introducing e to College Algebra students is a pretty good indicator that e has no rightful business being mentioned in College Algebra.

  3. I leave out most of the e stuff myself.

    Is that wretched table in your textbook?

    >I give an extended assignment that deals with half-lives.

    I’d like to see it. I give an extended assignment that deals with finding the time of death based on the idea (not really accurate) that “a dead body cools off just like a hot cup of joe”. It’s on my blog, here.

  4. Roy:

    My problem with continuously compounded interest is just that it’s an incredibly boring topic with no grounding in almost anyone’s personal experience.

    I’m not convinced the topic has to be boring. Nor am I convinced that if we solve the dullness problem we’ve solved the teaching problem here.

    I have actually made a bit of headway with the compound interest in College Algebra. I wrote about it last spring. The entry point is pointing out to students that there are two rates posted at their local banks: APY and APR. When I ask about the difference between these, I get blank looks all around. Nobody knows-even those who can recite the interest formula.

    So we approach the interest formula as a way of getting at the questions, How should banks figure interest multiple times per year? and How do banks figure interest multiple times per year? That turns out to be pretty interesting for my students. They have a lot to say about how they think it should be done, and the math helps us to decide whether it’s fair, etc.

    But then there are two leaps of faith on the way to e that haven’t figured out how to make reasonable. (1) continuous compounding-this is where limits at infinity come in, and (2) getting from there to e itself, since continuous compounding on a reasonable interest rate doesn’t reveal the value 2.71828…

    Sue:

    Is that wretched table in your textbook?

    Yes. I have made my peace with this particular text. It seems to support most of what I do rather than impede my goals (in contrast to the college-wide adopted text). But it’s far from being the text of my dreams.

    I’d like to see [your extended assignment]. I give an extended assignment that deals with finding the time of death based on the idea (not really accurate) that “a dead body cools off just like a hot cup of joe”.

    I’ll take a look at yours soon. I’ll send you a pdf of the current version of the half-life assignment.

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