Can mathematical representations change the way in which citizens understand economic and social challenges? The news of the day and many public policy decisions involve numbers. Single numbers pepper statements of public policy challenges—the size of the economy, the rate of unemployment, the size of the national debt and budget deficit. Proposals to bail out a bank, eliminate tax cuts, reduce the size of the military budget or cut Medicare are each presented as numbers. In the mathematics section of the Understanding Fiscal Responsibility curriculum, students use functions, modeling, rates of change and data analysis to put those numbers in context and move from single numbers to civic knowledge. They study how such things as different formulations of the income tax; the size and significance of demographic changes in the population; and competing proposals for reducing the amount of money spent on national security would each impact the federal budget and the budget deficit.

This section contains the following lessons:

Demographic Shifts and the Federal Budget


Do mathematical representations solve the problem of sustaining Social Security and Medicare as the U.S. population ages?


The purpose of this lesson is to explore the relationship between population age and Social Security and Medicare expenditures. The lesson is divided into two parts. Part 1 addresses the trend of an aging population, and Part 2 addresses how Social Security and Medicare spending has changed over time.

Mathematical Content: Exponential change, logistical change, functions, trend lines, graphical analysis

Grade Level: 10-12 (Algebra II or Pre-calculus)

Social Studies Links:

Lesson 1.1 (Economics: Social Security and the National Debt)

Lesson 1.2 (Economics: Medicare and the National Debt)

Lesson 2.1 (Civics/Government: Social Security, Governance, and the National Debt)

Lesson 2.2 (Civics/Government: Medicare and the National Debt)

Lesson 3.1 (U.S. History: The History of Social Security)

Lesson 3.2 (U.S. History: The History of Medicare)

Materials Needed: Chart paper and markers, graphing calculators (optional)


Income Tax Models, the Budget Deficit, and the National Debt


Should we raise income taxes to reduce the budget deficit and pay down the national debt?


I think that people at the high end, people like myself, should be paying alot more in taxes. We have it better than we’ve ever had it.

-Warren Buffett (Miller, 2010)

The “Buffett rule,” a formulation for increasing taxes that was discussed seriously by lawmakers in 2012, would have been an adjustment to the tax code that ensures top earners–households with a gross income of $1 million or more–pay at least the tax rate of middle-class workers (30% of their adjusted gross income). It is estimated that the rule would have raised about $500 billion, just 5% of a budget deficit that will top $1 trillion (Pear & Weisman, 2012).

Mathematical Content: Percentages, analyzing data, drawing conclusions from tables, graphs, and charts

Grade Level: 9

Social Studies Links: Lesson 1.4 (Economics: Taxation and the National Debt); Lesson 1.5 (Economics: Balancing the Federal Budget)

Materials Needed: Chart paper and markers, calculators, graph paper, graphing calculators (optional), Excel (optional)


Military Spending: An Analysis


Is the United States spending the right amount of its budget on defense?


Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed. This world in arms in not spending money alone. It is spending the sweat of its laborers, the genius of its scientists, the hopes of its children.

—President Dwight D. Eisenhower, “Chance for Peace” speech, April 16, 1953 (Eisenhower, 2012)

The defense budget is not a pot of gold. We cannot fix the deficit or fund all of our domestic priorities simply with the response of, take it out of defense. Dismantling our military will not solve our domestic problem, but it will destroy our ability to protect our interest and to shape the direction of world events. To cut defense, we have to do it right, [like] a sweeping restructuring of our armed forces that will reduce defense spending while still preserving the military capabilities we need.

—Vice President Richard Cheney, speech at Lawrence Technological University, September 14, 1992 (Cheney, 1992)

Spending on national defense is discretionary, meaning Congress must approve how much is spent on
it each year. This means that each year, Congress, representing the nation, must determine how much
defense is valued relative to other public goods. Note that the money available to fund defense and other
public goods comes from tax revenue. How we spend those revenues should reflect what we value.

The United States spends significantly more on defense than any other nation and it is the largest
discretionary item in the country’s budget. However, no two countries are in the same situation. The purpose of this lesson is to understand that mathematical representations can help students understand a topic more deeply. They should be aware, however, that correlations between variables do not necessarily mean that a change in one variable caused a change in the other and that there may be underlying causes for the changes in both.

Mathematical Content:  Analyzing data, statistics, correlation, drawing conclusions from tables, graphs, and charts

Grade Level:  10-12

Social Studies LinksLessons 1.3 (Economics: The Economics of National Security)Lesson 2.3 (Civics: National Security Goals, the Federal Budget, and the National Debt)

Materials Needed:  Chart paper and markers, calculators, graph paper, graphing calculators (optional), Excel (recommended)


Media, Numeracy, and the National Debt: Using Graphs to Make Decisions


Is it possible to use mathematics to tell the same story in two different ways, both of which are true?


In political debates, it sometimes seems as if someone must be completely wrong, or even lying, when the statistics one party uses to support its arguments are used by the other party to make the opposite point. In May 2012, presidential candidate Mitt Romney claimed that there was a “debt and spending inferno” under the presidency of Barack Obama (Friedman, 2012). In response, the White House cited evidence that such spending “never happened” (Jackson, 2012; Nutting, 2012). Is someone lying? Is it possible that good mathematics can be used to support two different, even opposing, claims?

The purpose of this lesson is to help students explore how mathematical representations are used to support the messages they receive about federal spending. Students begin by examining several graphs before using their own data to develop a graphical representation and message about federal spending. This lesson could complement a lesson on media literacy, and could be supplemented by readings from A Mathematician Reads the Newspaper. Because it requires extensive analysis, this lesson is most appropriate as a beginning of the year review for a course on functions, finite mathematics, or discrete mathematics. It might also be used as an introductory statistics activity to help students begin using mathematical tools for social or political analysis.

Mathematical Content: Percentages, analyzing data, drawing conclusions from tables, graphs, and charts

Grade Level: 9

Social Studies Links: Lesson 2.4 (Civics: Political Beliefs and the Federal Budget)Lesson 2.5 (Civics: Rhetoric)

Materials Needed: Chart paper and markers, calculators, graph paper, graphing calculators (optional), Excel (optional)