TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

We have used this assignment in a number of calculus and above classes, e.g., mathematical modelling class in which this effort represents an empirical model of data fitting.

In all cases students usually come to the model P(r) = K/r^a
where P = P(r) is population, r is rank, K is often a parameter to be determined or set to the rank of the largest city (P(1) = K/1^a = K).

One year our calculus students did go after the census data for the United States and it was in that class that we discovered the potential for studying changing a values. The students did get excited about this for their analysis worked decade after decade and they then plotted the changing values of a versus time and made conjectures. Thus there was a meta data analysis problem as well.

Often time students could eyeball the parameters, at least K, and to some extent a. Some students will linearize the data to obtain

log(P(r)) = log(K) - a log(r)

followed by linear regression or straight line fitting. It is in this linearization that one can get different K values other than K = P(1).

Non-calculus students can do a good bit of eyeballing of non-linear and linearized data and get good success here.

Calculus students who can apply optimization techniques may choose to go directly after the least squares function formulation and use software to find the parameter values K and a which minimize the least squares, be it from linearized data or non-linearized data. The K and a values you get from these two approaches, i.e., linearize first then do least squares or non-linear least squares, may differ and cause some discussion. They should not be expected to be the same as in each case we are minimizing the vertical distance squared between different types of functions and data.

Prerequisites

Students should be able to use the logarithm function to "linearize" data . They should be able to plot data, perhaps logged data and estimate slopes.

Time allotment - time management

Expectations

Future payoffs

Extensions

Study the parameter a over time and get a plot of a vs. time. Then conjecture the significance of change, e.g. Industrail Revolution, Depression, Boom, etc.

References and Sources

Population of United States Cities. Source: 1988 World Almanac and Book of Facts. New York: World Almanac. p. 538.