Student worksheet – Teacher version
Piecewise modeling with multiple functions: Federal Debt

Loose change (toonies, loonies, quarters, ...) on Canadian flagName: Teacher 

Time period: 1989 to 1997


  1. What function type do you think best approximates your data?

  2. Why did you select this function type? Be sure to include difference tables and r2 values in your explanation.

    Question 2.  Federal debt
    Year Debt First difference
      $ millions
    1989 379,993 26,613
    1990 406,606 37,951
    1991 444,557 31,547
    1992 476,104 38,253
    1993 514,357 43,247
    1994 557,604 38,273
    1995 595,877 39,062
    1996 634,939 16,185
    1997 651,124  

    The r2 value of the least-squares line (line of best fit) is 1.00, which means the data are approximated by a linear model very well. Although the first differences are not exactly equal, this is to be expected because we are using real data. The first differences are close though, and the data are best approximated by a linear model.
  3. What is the basic form of the equation (e.g., y = mx + b) for the type of function you selected?

    y = mx + b
  4. What does each variable represent in the equation? (e.g., What do m and b represent in a linear equation?)

    m represents the slope of the line.
    b represents the y-intercept of the line.
  5. For this dataset, what does each variable represent in the real world?

    For this dataset, m represents the rate at which the federal debt is increasing or decreasing annually whereas b represents the value of the federal debt in 1989, the first year of the period.
  6. What are your best values for each variable?

    My best value for m is 35,900 and my best value for b is 374,000.
  7. What do the values of these variables mean in the real world?

    The m value of 35,900 means that the federal debt is increasing at a rate of $35,900,000 per year, whereas the b value of 374,000 means that the value of the federal debt in the first year is $374,000,000. (Note: Federal debt values are provided in thousands of dollars.)
  8. Write your equation here, with your best values inserted.

    y = 35,900x + 374,000
  9. Consult with the students who analysed the data for the time periods immediately before and after your assigned time period. Compare their data and functions with yours. What are the similarities and differences?

    Previous time period: 1975 to 1988
    In this time period, the debt was best modelled by a quadratic or an exponential function, which means it changed more rapidly each year than during my time period. However, in both their and my time period, the debt is increasing.

    Next time period: 1998 to 2008
    In this time period, debt decreased quadratically, whereas in my period, debt increased linearly. In the next time period, the data points do not follow the function model as well as the data in my time period.
  10. What historical events may have contributed to the shape of your graph, and how?

    Answers vary.
  11. Print a copy of your graph (with the function model and best values for your variables showing) and hand it in with your answers to the above questions.

    Fathom graph
    (Fathom is licensed by the Ministry of Education and used by schools in some provinces. Providing the Fathom format is in no way an endorsement or recommendation of the Fathom software by Statistics Canada.)

    Fathom graph: Federal debt scatter plot
    Note: this is the least squares line calculated by Fathom.

    Excel graph
    To generate the graph below, select XY (Scatter) and Add Trendline (linear).
    (Providing the Excel format is in no way an endorsement or recommendation of the Excel software by Statistics Canada.)

    Fedral debt, 1989 to 1997: Least squares line from Fathom