Illuminations: National Debt and Wars

National Debt and Wars
Students will collect information about the National debt, plot the data by decade, and determine whether an exponential curve is a good fit for the data. Then student groups will determine and compare common traits and differences in changes in the National debt in three major eras: the Civil War, World War I, and World War II.

Learning Objectives
 
Students will:
  • Determine if an exponential function adequately describes the changes in the National Debt from its inception;
  • Use technology (such as a spreadsheet program) to sketch graphs of functions and bar graphs of percent changes in the National Debt around war times;
  • Choose an appropriate scale and compare percent changes in the Debt around three major wars (the Civil War, World War I, and World War II).

Materials
 
Graphing calculator or spreadsheet program which will calculate curve of best fit
Computers with spreadsheet graphing capability, such as Microsoft Excel®
The National Debt and Wars Activity Sheet
The National Debt and Wars Overhead
Suggested Solutions to the Activity Sheet

Instructional Plan
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 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 do 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

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
 
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.]

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.]

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.]

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

National Debt and Wars Overhead National Debt and Wars Overhead

Assessment Options
 
  1. Observe student written work and graphs. Is the work comprehensive and accurate? How do you know?

  2. Determine whether the presentations identified major points of about the graphs of the wars and exponential growth and percent change.

  3. Listen to student discussions. Are students identifying the major math points? Can they identify when exponential growth functions appear to be good models and when they do not? Are they discussing the major points of comparisons in changes in the national debt during the three wars?

Extensions
 
  1. Have students explore the national debt and its changes by 4-year presidential terms for different time periods—since the 1960s, between the World Wars, etc.

  2. Compare the national debt to Gross Domestic Product (as a ratio) to see how the Debt has changed relative to the GDP standard.

  3. Compare the national debt to Per Capita Income (as a ratio) to see how the debt has changed relative to Per Capita Income.

  4. Have students investigate potential reasons why particular sharp increases or decreases in the national debt have occurred. (For example, there was one time in period when a private citizen paid off the national debt.)

  5. Investigate the apparent connection (or lack of connection) to changes in the national debt related to other US wars (War of 1812, Mexican-American War, Indian Wars or Relocation, Iraq I, Iraq II).

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?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
  2. Approximate and interpret rates of change from graphical and numerical data.
This lesson prepared by Martha Haehl.
  
2 periods   

NCTM Resources

Navigating through Algebra in Grades 9‑12

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