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June 14, 2013

Are colleges affordable? – Lesson Plan

By Mary F. Klein a mathematics teacher from Urbana, Illinois


Second-year algebra courses and beyond, high school mathematics


One to two class periods


A mathematical model is a method of understanding a complex real world process involving multiple variables. To design a model, important features are identified and quantified symbolically using a number of mathematical tools. Once designed, a mathematical model must be evaluated and critiqued for accuracy, advantages, and limitations.

In this lesson, the students investigate a mathematical model that compares the cost of education to potential earnings in order to decide if the investment is a good one. The students compute the amount to be repaid each year for the life of the loan and then they compare the answer to some probable starting salaries. Finally, the students critique the model discussing which other factors (demand for the job, other costs, etc.) should be taken into consideration.


This lesson and activity are designed to help students understand the idea of a mathematical model and, at the same time, provide a tool that may be useful to them when they choose a college and evaluate the potential debt that may result.


  1. Distribute the Online NewsHour story, ‘Paying More For College, Getting Less’ for students to read. Briefly discuss the story. Bring up the idea that one goes into debt to finance a house. Does it pay to borrow to finance future earnings? Point out that borrowing for a house or an education can be considered an investment, while borrowing for a car is not an investment since the car depreciates in value. Tell the class that today’s goal is to use mathematics to investigate the relationship between how much we borrow and how much we can afford to borrow.
  2. Tell students that the power of mathematics is that it can be used to help us understand our world and that an important tool is the mathematical model. Define ‘mathematical model’. Distribute the handout, ‘The Mathematical Model,’ and the worksheet.
  3. Go over the parts of the model and then work out the details of the first problem on the worksheet.
  4. Point out that since we know the amount of the yearly payment, we can consider the yearly payment to be a constant. Equation D can be simplified to y = (205,800) / x. What sort of graph can we expect? A hyperbola! Graph equation D. (Let 0 < x < $60,000 using increments of $10,000 and let 0 < y < 100% using increments of 10%.)
  5. Discuss how to interpret the graph. (As earnings increase the percent of income used to repay the debt decreases). The financial industry standard is that total debt (including housing) should be less that 36% of gross earnings. Is the value we found for A in fact a reasonable debt ratio for a first year teacher earning $28,132 (the median income for a teacher with a BA in Illinois)? The minimum starting salary for such a teacher is $20,299. Discuss how low that first year starting salary could be and still support a reasonable debt ratio. (Use a graphing utility to plot y = 36. Find the intersection of the graphs.) Would it matter if there were a glut or a shortage of teachers in a particular field?
  6. Assign students to groups and have them present their findings. (To save time, you might assign just 1 or 2 problems to each group.)
  7. Check answers.
  8. Discuss the model itself. Is it a good model? What hidden assumptions does it make? What are its limitations? What would the students change?

Extension Activity

  1. Investigate the effect on the debt ratio if a student charges daily expenses to a credit card and makes only the minimum payment each month.
  2. Borrow a copy of Peterson’s Complete Guide to Colleges from a high school counselor. Choose a number of colleges or universities and look up ‘the average debt incurred by students the previous year’ for each school. Compare the debt ratios for the different schools.

Robert Andersen, Associate Director of Financial Aid, University of Illinois at Urbana-Champaign

Chris Hopkins, Counselor, Urbana High School

Author Mary F. Klein is a mathematics teacher with experience at the high school, middle school, and junior high school levels. She is the author of several Web-based lessons on mathematics and basketball. She lives with her husband and son in Urbana, Illinois.

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    Relevant National Standards:
      National Council of Teachers of
      Mathematics Standards
    • Students should use mathematical models to represent and understand quantitative relationships
    • Students should analyze and evaluate the mathematical thinking and strategies of others
    • Students should represent and analyze mathematical situations and structures using algebraic symbols

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