To be successful with this lesson, students will need the following prior skills and knowledge:

- How to calculate percent change
- How to use technology to find an exponential or linear curve of best fit and interpret the
*r* value (correlation)
- How to use a spreadsheet to sketch line and bar graphs
- How to calculate values in a spreadsheet using a formula and "copy that formula down"

If students need help with a spreadsheet, help them record the
independent variable values in the left column, then in the next
column, use their function to calculate dependent variable values. In
the next column, starting with the second piece of data, calculate
percent change using the following formula:

Students should then copy the formula down the column.

If some students tend to be the "computer experts" and are doing
all of the work while other students are disengaged, try having the
"experts" become teachers. Have the students who are sitting back be
the ones who work the keyboard and mouse while the experts instruct the
workers in what to do.

### Pre-Activity

Distribute the National Debt and Wars Activity Sheet to the students.

National Debt and Wars Activity Sheet

The pre-activity questions could be assigned as homework the
night before, then discussed as a class at the beginning of the lesson.

### Activity I

Students should recognize that the percent change of an
exponential growth (decay) curve is constant over uniform intervals.
They should also recognize visually the basic shape of an exponential
curve.

All groups should complete Activity I. As groups finish, have them compare
results. Did they use the same or different scales? If so, have them
explain why they chose their particular scale. Was the data all the
same—did some groups pro-rate the data, or did they record the data for
the year regardless of the date given by the US Treasury National debt
website?

Note: When recording the data by
decade, a fairly accurate picture occurs if students do not account for
the differences in reporting dates. However, the following processes
yield a more accurate set of data to analyze and to standardize the
Debt as of January 1 for each year. When the data is given for December
30 or 31, record the data from the preceding year. When the data is
given as of June 30 or July 1, estimate the Debt as of January 1 by
averaging the two adjacent years. When the data is given as of
September 30, estimate the debt as of January 1 by a weighted average
of the two adjacent years (0.25 of preceding year + 0.75 of current
year). The averaging can be done using spreadsheet formula features.
(For 1861, a close look at trends indicate that the Civil War had a
huge impact on the debt. Since the Civil War began early in the year,
in February, an estimate that follows the trends of change in preceding
years is likely a better estimate that the weighted average.)

The independent variable should be the year, and dependent variable the National debt as of January 1 of the corresponding year.

*Note:* The activity sheet allows students to
research the amount of the National Debt over the history of the United
States. If you wish for students to do this, then photocopy the last
page of the activity sheet. If you would rather have the students focus
on working with the information given to them on the Activity Sheet, it
has been provided for them already.

### Activity II

In Activity II, assign different groups to work with data from
different wars. Initially, students will likely have different scales
for the three wars on both the graph of the debt over time and the bar
(or column) graph of the percent changes. For final comparison of
percent change, use the scale of the percent change for the Civil War,
which was the period with the highest percent change.

Students should recognize the following:

- The years preceding, during and after each of the 3 wars produced
similar graphs—with fairly level curves leading up to and following the
war compared to steep growth in the national debt during the war.
- Piecewise linear functions would be a better curve fit for the data than an exponential curve.
- The percent change in the national debt is variable (another
indicator that an exponential function is not a good fit for the data).

After students have completed the Activity II questions on the
Activity Sheet, they can work in groups to prepare a presentation of
the highlights of their findings. The presentation should include
appropriate graphs and tables.

As a class, discuss how wars seemed to be similar or different
in their impact on the national debt. To tweak students' comparisons,
decide on a scale for the percent change bar chart that could represent
the data in all three wars. Rescale those graphs and compare the
results. Which war seemed to have to most significant impact? Students
should justify how you decided and summarize the main points of the
discussion.

### Summary

Suggested Solutions for the National Debt and Wars Activity Sheet

Students should complete the Summary Questions found on the Activity Sheet.

Sample graphs for the Activity Sheet are available on the Suggested Solutions for the National Debt and Wars Activity Sheet.

**Questions for Students**

1. What is the basic shape of an exponential growth curve? An exponential decay curve?

[An exponential growth curve starts gradually but then speeds up as x gets larger. A decay curve starts at a high value but eventually gets closer and closer to some smaller value.]

2. It has been said that any curve can be approximated as linear over short intervals. Discuss what this means and whether you think it is true.

[If a small enough interval is considered, a portion of a curve may appear to be a straight line.]

3. For an exponential growth (or decay curve), describe the graph of percent change in the dependent values over uniform time intervals.

[In equal time intervals, the percent change is constant.]

4. From 1920 to 1930, the debt had declined from $25,952,000,000 to $16,185,000,000. By 1940, the debt had jumped to $42,968,000,000. What was going on in the country economically during the 1920s and 1930s to cause the decline and then increase in the debt?

[During this time period, the country was suffering from the Great Depression.]

(The questions above are also available on The National Debt and Wars Overhead.)

**Teacher Reflection**

- Observe whether all students are engaged. Are different students
contributing different strengths? Are students who are strong in
historical knowledge interjecting that knowledge? Are students who are
proficient in use of technology helping students who need to improve
their skills?
- Are all students engaged in the presentation?
- Did students pull from the lesson a greater understanding of exponential growth?
- Did student make connections between math and historical events or economics?
- Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it?