US Population 1982-1988

Task

The below table provides some U.S. Population data from 1982 to 1988:

Notice: The change in population from 1982 to 1983 is 2,128,000, which is recorded in thousands in the first row of the 3rd column. The other changes are computed similarly. All population numbers in the table are recorded in thousands.

Source: http://www.census.gov/popest/archives/1990s/popclockest.txt

1. If we were to model the relationship between the U.S. population and the year, would a linear function be appropriate? Explain why or why not.

2. Mike decides to use a linear function to model the relationship. He chooses 2,139, the average of the values in the 3rd column, for the slope. What meaning does this value have in the context of this model?
3. Use Mike's model to predict the U.S. population in 1992.

IM Commentary

This task provides a preliminary investigation of mathematical modeling using linear functions. In particular, students are asked to make predictions using a linear model without ever writing down an equation for a line. As such, the task could be used to motivate or introduce the observation that linear functions are precisely those that change by constant differences over equal intervals. For emphasis of this idea, the task could be used alongside the related tasks F-LE Equal Differences over Equal Intervals I & II.

As with many modeling tasks, there is ample opportunity for a discussion of the plausibility of the model and the methodology that Mike uses to construct it. In particular, while a linear model for population growth appears to fit well over this small time interval, it is unlikely to continue to do so for longer periods. Without constraining factors, population is likely to grow faster than linearly, and so Mike's model would likely overestimate populations before 1982 and underestimate populations after 1988. Indeed, the actual population of the United States in 1992 was 255 million, roughly 2 million above the population that Mike's model predicted. As a further extreme, Mike's model would predict a population of about 295 million for 2012, compared to an actual value of 312 million. Instructors should encourage students to think critically about the plausibility of the model over longer time spans.

Instructors who have already introduced linear functions might expand the modeling content of the task by asking students to plot the data points and use a ruler or software to propose their own linear model.

Solution

1. The table shows that over one year periods, the population increases by approximately the same amount (just a little over 2 million per year). Hence a linear function is appropriate to model the relationship between the population and the year over this short time interval.


2. The slope of the linear function which will model this relationship will measure the change in population per change in time. Its units will be millions-of-people per year in this problem. A value of 2139 thousand = 2,139,000 for the slope would mean that the population is growing by approximately 2,139,000 people per year.


3. The population in 1988 was 244,499,000. Mike's choice for the slope from problem (b) indicates a population growth of about $4\cdot(2,139,000) = 8,556,000$ people between 1988 and 1992. Therefore Mike's model predicts the 1992 population to be approximately 253,055,000 people. Note the actual population of the United States in 1992 was about 255 million.