Modeling Prompt

This prompt is very open-ended and a large part of the goal is to provide some national financial literacy. The goal is not to alarm students and after they have worked on their models, it would be worth sharing that as a percentage of gross domestic product, the debt now is comparable to our national debt after World War 2. No one knows what the future holds and the one conclusion that students can safely draw is that if current trends continue we will, within the next decade or two, enter uncharted territory in terms of our national debt. 

Students will need to look up the data for themselves, decide whether to use all of the data (that is, on an annual basis) or just a few data points (for example, every 5 years). Moreover, students need to think carefully about which type of model to use. Choices could include 

• linear
• quadratic (if done later in the course)
• exponential
• piecewise

Students can use statistical tools to produce different models and compare their residuals.

Because there are so many factors involved in the debt (changes to costly programs like Medicare and Social Security, changes to taxation, and natural disasters at home and abroad) there is no way to expect any reliability in a model over the long term. Moreover, growth in the debt needs to be compared to growth in overall wealth (21 trillion dollars would not be a problem if the overall wealth in the country were 150 trillion dollars). So to more fully understand debt issues, students might be prompted to examine issues like inflation, income distribution, or the national budget.

1. Find and graph the United States national debt over the past 30 years. 
2. Choose a function that models that data and justify your choice.
3. What does your model predict for the next few years? For the next two decades?
4. Do you think the predictions will be accurate? Why or why not? What would you suggest the government do to reduce the growing debt?

In this version of the prompt, students are instructed to look at the U.S. debt every other year and two types of models are suggested (linear and exponential). The main work facing students in this version will likely include

• looking up data
• choosing an exponent for an exponential model (especially if they do not use graphing technology to find a good match)
• taking into account that the data is only given every other year when they write an equation for their model

As in the first version, share extra information with students to put the current debt into an appropriate historical perspective as they work on the problem.

1. Find and graph data for the U.S. debt every other year, from 1987 through 2017.
2. Do you think a linear model or an exponential model would be more appropriate for this data? Explain your reasoning and then find the appropriate model.
3. What does your model predict for the future? Do you think it will be accurate? Explain your reasoning.

In this version of the prompt, students are given the data and additionally are told to look at the debt every other year. They are also told that a linear model is not appropriate and asked to explain why. While they are not told that an exponential model can fit the data well, they are asked to find an exponential model that they think does work well. Thus the main lifting for the students in this version is

• finding an appropriate exponential model by hand or with the aid of technology
• dealing with the fact that the data is given in two-year chunks rather than every year if they write the exponential model as a function.

year	debt (trillions)
1987	2.4
1989	2.9
1991	3.7
1993	4.4
1995	5.0
1997	5.4
1999	5.7
2001	5.8
2003	6.8
2005	7.9
2007	9.0
2009	11.9
2011	14.8
2013	16.7
2015	18.1
2017	20.2

1. Plot the U.S. national debt for every other year from 1987 through 2017. 
2. Explain why a linear model does not fit the data well.
3. Find an exponential model that you think models the data well.
4. What does your model predict for the future? Do you think it will be accurate? Explain your reasoning.