## Tuesday, September 25, 2012

### Life Expectancy, and a lesson that didn't work

Here was something that didn't work with either of my Algebra 2 classes. I'm wondering why it didn't work, and if there's a better draft of this problem to be made.

The lesson was vaguely three-acty.

Act One

How long does the calculator say that a person born in the 90s will live? The 80s? The 1890s? Why the difference?

I asked one class: "When will you have to be born to expect to live to 100?" The other class got "How long can you expect to live if you're born in 2010? 2050?"

Act Two

[Source: CDC]

What do you notice about the table? Take a guess for 1980. My guess is 60 years. You guys like that? Why not?

How good is the following rule: "life expectancy = years * 10"? Is the rule "life expectancy = years * 9" better or worse? How can you find a rule that's better than either of them? What are you changing in the equation?

Can you find an equation that fits it pretty well? How far off would the predictions be?

Act Three

Here's our most recent data. What do your equations predict?

What went wrong?

First, the objective stuff:
• Kids didn't seem into it.
• Kids didn't know where to jump in.
• Kids were confused by the idea that it has to be a rule that gives a line.
• Kids thought it unnatural to make a prediction based only on a few prior data points.
Other issues:
• It wasn't clear to me or them what they were trying to predict. Since we can't check their actual predictions (cuz they're in the future) we have to just limit the data that we make available to them. This seems to be a limitation of the "data analysis of social stuff" type of problem, and an argument for doing regression problems with stuff that we can actually test in the classroom.
• We'd done a similar, and superior, problem last week with data from 100m dash times. I wanted kids to end up with actual equations as models, and I don't think this was different enough to necessitate equations. A lot of kids repeated their tricks from last time: averaging the rate of change, coming up with recursive rules instead of closed-form rules. I didn't feel as if anything, other than my insistence, was pushing them towards closed-form equations.
Improvements?
• Post it as a historical puzzle. Let's say you were in 1960: how far off would your best prediction be for life expectancy in 2010?
• Find a better hook. I needed something like that life expectancy calculator just to make sure kids knew what "life expectancy" means.
Help? Anyone?

1. For the hook: Leave some blanks in the table and have them guess first?

For the extrapolation: Can we use this data (thru 1970) to predict the life expectancy for you guys? Might be far enough out that a closed-form function would make more sense. And, you could verify it on the life expectancy calculator.

I agree though, that it's not obvious the relationship would be linear. There's really no natural reason to assume that, I don't think.

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3. "Kids didn't seem into it."

It's a little grim, in my opinion. Asking them when they would have to be born to have a life expectancy of 100 has the immediate corollary that they and everyone they know were born before this time. In particular, at least some of them must have grandparents on "borrowed time" according to the calculator. Such a realization might be bracing for a mature student, but could be difficult for some to process (especially in front of their peers during math class).

4. I'm thinking it was a lot more abstract than the running thing, and also that even *if* they really understand linear relationships, there's no reason to think this would be an example of one.
I remember being simply (but silently) astounded in physics class that yes, linear relationships really did work out ;)

5. I would probably dump it. The fact that you are predicting what a future prediction will be makes it a little elusive to begin with. Also, it just seems like a lot of speculation about a process that may not be that stable - obesity epidemic, etc. Maybe the life expectancy will actual begin to drop.

6. I guess the data seems a little bit made-up. If you could figure out why they're making those predictions, that could help.

austinmohr says it's grim -- but I think that could be super fun, as well. So many weird jokes to be made.

Also you said this: "Kids thought it unnatural to make a prediction based only on a few prior data points." So cool! What a neat conversation to have. Are they right? How many data points do they want? omg stats.

7. Can you tie in the worlds oldest person? Can we figure out how unlikely a person born in that generation lives that long?

This does seem like a cross curriculum conversation with your health, science, humanities teachers. I say keep tweaking...maybe ask the students how to make it better, they can be creative too.

8. Good points, everybody. I hear those of you who are suggesting that I toss this one out. And why should it be linear?

For what it's worth, I'm going to do this thing next time I do a regression problem: link!